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of End Chapter Solutions
Essentials of Corporate Finance 6th edition
Ross, Westerfield, and Jordan
Updated 08-01-2007
CHAPTER 1
INTRODUCTION TO CORPORATE
FINANCE
Answers to Concepts Review and Critical Thinking Questions
1.
Capital budgeting (deciding on whether to expand a manufacturing plant), capital structure (deciding
whether to issue new equity and use the proceeds to retire outstanding debt), and working capital
management (modifying the firms credit collection policy with its customers).
2.
Disadvantages: unlimited liability, limited life, difficulty in transferring ownership, hard to raise
capital funds. Some advantages: simpler, less regulation, the owners are also the managers,
sometimes personal tax rates are better than corporate tax rates.
3.
The primary disadvantage of the corporate form is the double taxation to shareholders of distributed
earnings and dividends. Some advantages include: limited liability, ease of transferability, ability to
raise capital, and unlimited life.
4.
The treasurers office and the controllers office are the two primary organizational groups that
report directly to the chief financial officer. The controllers office handles cost and financial
accounting, tax management, and management information systems. The treasurers office is
responsible for cash and credit management, capital budgeting, and financial planning. Therefore,
the study of corporate finance is concentrated within the functions of the treasurers office.
5.
To maximize the current market value (share price) of the equity of the firm (whether its publicly
traded or not).
6.
In the corporate form of ownership, the shareholders are the owners of the firm. The shareholders
elect the directors of the corporation, who in turn appoint the firms management. This separation of
ownership from control in the corporate form of organization is what causes agency problems to
exist. Management may act in its own or someone elses best interests, rather than those of the
shareholders. If such events occur, they may contradict the goal of maximizing the share price of the
equity of the firm.
7.
A primary market transaction.
8.
In auction markets like the NYSE, brokers and agents meet at a physical location (the exchange) to
buy and sell their assets. Dealer markets like Nasdaq represent dealers operating in
CHAPTER 2 B-3
dispersed locales who buy and sell assets themselves, usually communicating with other dealers
electronically or literally over the counter.
9.
Since such organizations frequently pursue social or political missions, many different goals are
conceivable. One goal that is often cited is revenue minimization; i.e., providing their goods and
services to society at the lowest possible cost. Another approach might be to observe that even a notfor-profit business has equity. Thus, an appropriate goal would be to maximize the value of the
equity.
10. An argument can be made either way. At one extreme, we could argue that in a market economy, all
of these things are priced. This implies an optimal level of ethical and/or illegal behavior and the
framework of stock valuation explicitly includes these. At the other extreme, we could argue that
these are non-economic phenomena and are best handled through the political process. The
following is a classic (and highly relevant) thought question that illustrates this debate: A firm has
estimated that the cost of improving the safety of one of its products is $30 million. However, the
firm believes that improving the safety of the product will only save $20 million in product liability
claims. What should the firm do?
11. The goal will be the same, but the best course of action toward that goal may require adjustments
due different social, political, and economic climates.
12. The goal of management should be to maximize the share price for the current shareholders. If
management believes that it can improve the profitability of the firm so that the share price will
exceed $35, then they should fight the offer from the outside company. If management believes that
this bidder or other unidentified bidders will actually pay more than $35 per share to acquire the
company, then they should still fight the offer. However, if the current management cannot increase
the value of the firm beyond the bid price, and no other higher bids come in, then management is not
acting in the interests of the shareholders by fighting the offer. Since current managers often lose
their jobs when the corporation is acquired, poorly monitored managers have an incentive to fight
corporate takeovers in situations such as this.
13. We would expect agency problems to be less severe in other countries, primarily due to the relatively
small percentage of individual ownership. Fewer individual owners should reduce the number of
diverse opinions concerning corporate goals. The high percentage of institutional ownership might
lead to a higher degree of agreement between owners and managers on decisions concerning risky
projects. In addition, institutions may be able to implement more effective monitoring mechanisms
than can individual owners, given an institutions deeper resources and experiences with their own
management. The increase in institutional ownership of stock in the United States and the growing
activism of these large shareholder groups may lead to a reduction in agency problems for U.S.
corporations and a more efficient market for corporate control.
CHAPTER 2 B-4
14. How much is too much? Who is worth more, Steve Jobs or Tiger Woods? The simplest answer is
that there is a market for executives just as there is for all types of labor. Executive compensation is
the price that clears the market. The same is true for athletes and performers. Having said that, one
aspect of executive compensation deserves comment. A primary reason executive compensation has
grown so dramatically is that companies have increasingly moved to stock-based compensation.
Such movement is obviously consistent with the attempt to better align stockholder and management
interests. In recent years, stock prices have soared, so management has cleaned up. It is sometimes
argued that much of this reward is simply due to rising stock prices in general, not managerial
performance. Perhaps in the future, executive compensation will be designed to reward only
differential performance, i.e., stock price increases in excess of general market increases.
15. The biggest reason that a company would go dark is because of the increased audit costs
associated with Sarbanes-Oxley compliance. A company should always do a cost-benefit analysis,
and it may be the case that the costs of complying with Sarbox outweigh the benefits. Of course, the
company could always be trying to hide financial issues of the company! This is also one of the costs
of going dark: Investors surely believe that some companies are going dark to avoid the increased
scrutiny from SarbOx. This taints other companies that go dark just to avoid compliance costs. This
is similar to the lemon problem with used automobiles: Buyers tend to underpay because they know
a certain percentage of used cars are lemons. So, investors will tend to pay less for the company
stock than they otherwise would. It is important to note that even if the company delists, its stock is
still likely traded, but on the over-the-counter market pink sheets rather than on an organized
exchange. This adds another cost since the stock is likely to be less liquid now. All else the same,
investors pay less for an asset with less liquidity. Overall, the cost to the company is likely a reduced
market value. Whether delisting is good or bad for investors depends on the individual
circumstances of the company. It is also important to remember that there are already many small
companies that file only limited financial information already.
CHAPTER 2
WORKING WITH FINANCIAL
STATEMENTS
Answers to Concepts Review and Critical Thinking Questions
1.
Liquidity measures how quickly and easily an asset can be converted to cash without significant loss
in value. Its desirable for firms to have high liquidity so that they can more safely meet short-term
creditor demands. However, liquidity also has an opportunity cost. Firms generally reap higher
returns by investing in illiquid, productive assets. Its up to the firms financial management staff to
find a reasonable compromise between these opposing needs.
2.
The recognition and matching principles in financial accounting call for revenues, and the costs
associated with producing those revenues, to be booked when the revenue process is essentially
complete, not necessarily when the cash is collected or bills are paid. Note that this way is not
necessarily correct; its the way accountants have chosen to do it.
3.
Historical costs can be objectively and precisely measured, whereas market values can be difficult to
estimate, and different analysts would come up with different numbers. Thus, there is a tradeoff
between relevance (market values) and objectivity (book values).
4.
Depreciation is a non-cash deduction that reflects adjustments made in asset book values in
accordance with the matching principle in financial accounting. Interest expense is a cash outlay, but
its a financing cost, not an operating cost.
5.
Market values can never be negative. Imagine a share of stock selling for $20. This would mean
that if you placed an order for 100 shares, you would get the stock along with a check for $2,000.
How many shares do you want to buy? More generally, because of corporate and individual
bankruptcy laws, net worth for a person or a corporation cannot be negative, implying that liabilities
cannot exceed assets in market value.
6.
For a successful company that is rapidly expanding, capital outlays would typically be large,
possibly leading to negative cash flow from assets. In general, what matters is whether the money is
spent wisely, not whether cash flow from assets is positive or negative.
7.
Its probably not a good sign for an established company, but it would be fairly ordinary for a startup, so it depends.
8.
For example, if a company were to become more efficient in inventory management, the amount of
inventory needed would decline. The same might be true if it becomes better at collecting its
receivables. In general, anything that leads to a decline in ending NWC relative to beginning NWC
would have this effect. Negative net capital spending would mean more long-lived assets were
liquidated than purchased.
CHAPTER 2 B-6
9.
If a company raises more money from selling stock than it pays in dividends in a particular period,
its cash flow to stockholders will be negative. If a company borrows more than it pays in interest, its
cash flow to creditors will be negative.
10. The adjustments discussed were purely accounting changes; they had no cash flow or market value
consequences unless the new accounting information caused stockholders to revalue the company.
11. The legal system thought it was fraud. Mr. Sullivan disregarded GAAP procedures, which is
fraudulent. That fraudulent activity is unethical goes without saying.
12. By reclassifying costs as assets, it lowered costs when the lines were leased. This increased the net
income for the company. It probably increased most future net income amounts, although not as
much as you might think. Since the telephone lines were fixed assets, they would have been
depreciated in the future. This depreciation would reduce the effect of expensing the telephone lines.
The cash flows of the firm would basically be unaffected no matter what the accounting treatment of
the telephone lines.
Solutions to Questions and Problems
NOTE: All end-of-chapter problems were solved using a spreadsheet. Many problems require multiple
steps. Due to space and readability constraints, when these intermediate steps are included in this
solutions manual, rounding may appear to have occurred. However, the final answer for each problem is
found without rounding during any step in the problem.
Basic
1.
The balance sheet for the company will look like this:
Current assets
Net fixed assets
Total assets
Balance sheet
$1,850
Current liabilities
8,600
Long-term debt
Owner's equity
$10,450
Total liabilities & Equity
$1,600
6,100
2,750
$10,450
The owners equity is a plug variable. We know that total assets must equal total liabilities &
owners equity. Total liabilities and equity is the sum of all debt and equity, so if we subtract debt
from total liabilities and owners equity, the remainder must be the equity balance, so:
CHAPTER 2 B-7
Owners equity = Total liabilities & equity Current liabilities Long-term debt
Owners equity = $10,450 1,600 6,100
Owners equity = $2,750
Net working capital is current assets minus current liabilities, so:
NWC = Current assets Current liabilities
NWC = $1,850 1,600
NWC = $250
2.
The income statement starts with revenues and subtracts costs to arrive at EBIT. We then subtract
out interest to get taxable income, and then subtract taxes to arrive at net income. Doing so, we get:
Income Statement
Sales
Costs
Depreciation
EBIT
Interest
Taxable income
Taxes
Net income
3.
$625,000
260,000
79,000
$286,000
43,000
$243,000
85,050
$157,950
The dividends paid plus addition to retained earnings must equal net income, so:
Net income = Dividends + Addition to retained earnings
Addition to retained earnings = $157,950 60,000
Addition to retained earnings = $97,950
4.
Earnings per share is the net income divided by the shares outstanding, so:
EPS = Net income / Shares outstanding
EPS = $157,950 / 40,000
EPS = $3.95 per share
And dividends per share are the total dividends paid divided by the shares outstanding, so:
DPS = Dividends / Shares outstanding
DPS = $60,000 / 40,000
DPS = $1.50 per share
5.
To find the book value of assets, we first need to find the book value of current assets. We are given
the NWC. NWC is the difference between current assets and current liabilities, so we can use this
relationship to find the book value of current assets. Doing so, we find:
NWC = Current assets Current liabilities
Current assets = $100,000 + 780,000 = $880,000
CHAPTER 2 B-8
Now we can construct the book value of assets. Doing so, we get:
Book value of assets
Current assets
$ 880,000
Fixed assets
4,800,000
Total assets
$5,680,000
All of the information necessary to calculate the market value of assets is given, so:
Market value of assets
Current assets
$ 805,000
Fixed assets
5,600,000
Total assets
$6,405,000
6.
Using Table 2.3, we can see the marginal tax schedule. The first $25,000 of income is taxed at 15
percent, the next $50,000 is taxed at 25 percent, the next $25,000 is taxed at 34 percent, and the next
$215,000 is taxed at 39 percent. So, the total taxes for the company will be:
Taxes = 0.15($50,000) + 0.25($25,000) + 0.34($25,000) + 0.39($315,000 100,000)
Taxes = $106,100
7.
The average tax rate is the total taxes paid divided by net income, so:
Average tax rate = Total tax / Net income
Average tax rate = $106,100 / $315,000
Average tax rate = .3368 or 33.68%
The marginal tax rate is the tax rate on the next dollar of income. The company has net income of
$315,000 and the 39 percent tax bracket is applicable to a net income of $335,000, so the marginal
tax rate is 39 percent.
8.
To calculate the OCF, we first need to construct an income statement. The income statement starts
with revenues and subtracts costs to arrive at EBIT. We then subtract out interest to get taxable
income, and then subtract taxes to arrive at net income. Doing so, we get:
Income Statement
Sales
Costs
Depreciation
EBIT
Interest
Taxable income
Taxes (35%)
Net income
$16,550
5,930
1,940
$8,680
1,460
$7,220
2,527
$4,693
CHAPTER 2 B-9
Now we can calculate the OCF, which is:
OCF = EBIT + Depreciation Taxes
OCF = $8,680 + 1,940 2,527
OCF = $8,093
9.
Net capital spending is the increase in fixed assets, plus depreciation. Using this relationship, we
find:
Net capital spending = NFAend NFAbeg + Depreciation
Net capital spending = $2,120,000 1,875,000 + 220,000
Net capital spending = $465,000
10. The change in net working capital is the end of period net working capital minus the beginning of
period net working capital, so:
Change in NWC = NWCend NWCbeg
Change in NWC = (CAend CLend) (CAbeg CLbeg)
Change in NWC = ($910 335) (840 320)
Change in NWC = $55
11. The cash flow to creditors is the interest paid, minus any new borrowing, so:
Cash flow to creditors = Interest paid Net new borrowing
Cash flow to creditors = Interest paid (LTDend LTDbeg)
Cash flow to creditors = $49,000 ($1,800,000 1,650,000)
Cash flow to creditors = $101,000
12. The cash flow to stockholders is the dividends paid minus any new equity raised. So, the cash flow
to stockholders is: (Note that APIS is the additional paid-in surplus.)
Cash flow to stockholders = Dividends paid Net new equity
Cash flow to stockholders = Dividends paid (Commonend + APISend) (Commonbeg + APISbeg)
Cash flow to stockholders = $70,000 [($160,000 + 3,200,000) ($150,000 + 2,900,000)]
Cash flow to stockholders = $240,000
13. We know that cash flow from assets is equal to cash flow to creditors plus cash flow to stockholders.
So, cash flow from assets is:
Cash flow from assets = Cash flow to creditors + Cash flow to stockholders
Cash flow from assets = $101,000 240,000
Cash flow from assets = $341,000
CHAPTER 2 B-10
We also know that cash flow from assets is equal to the operating cash flow minus the change in net
working capital and the net capital spending. We can use this relationship to find the operating cash
flow. Doing so, we find:
Cash flow from assets = OCF Change in NWC Net capital spending
$341,000 = OCF ($135,000) (760,000)
OCF = $341,000 135,000 + 760,000
OCF = $284,000
Intermediate
14. a. To calculate the OCF, we first need to construct an income statement. The income statement
starts with revenues and subtracts costs to arrive at EBIT. We then subtract out interest to get
taxable income, and then subtract taxes to arrive at net income. Doing so, we get:
Income Statement
Sales
$138,000
Costs
71,500
Other Expenses
4,100
Depreciation
10,100
EBIT
$52,300
Interest
7,900
Taxable income
$44,400
Taxes
17,760
Net income
$26,640
Dividends
Addition to retained earnings
$5,400
21,240
D ividends paid plus addition to retained earnings must equal net income, so:
Net income = Dividends + Addition to retained earnings
Addition to retained earnings = $26,640 5,400
Addition to retained earnings = $21,240
So, the operating cash flow is:
OCF = EBIT + Depreciation Taxes
OCF = $52,300 + 10,100 17,760
OCF = $44,640
b. The cash flow to creditors is the interest paid, minus any new borrowing. Since the company
redeemed long-term debt, the new borrowing is negative. So, the cash flow to creditors is:
Cash flow to creditors = Interest paid Net new borrowing
Cash flow to creditors = $7,900 ($3,800)
Cash flow to creditors = $11,700
CHAPTER 2 B-11
c. The cash flow to stockholders is the dividends paid minus any new equity. So, the cash flow to
stockholders is:
Cash flow to stockholders = Dividends paid Net new equity
Cash flow to stockholders = $5,400 2,500
Cash flow to stockholders = $2,900
d. In this case, to find the addition to NWC, we need to find the cash flow from assets. We can then
use the cash flow from assets equation to find the change in NWC. We know that cash flow from
assets is equal to cash flow to creditors plus cash flow to stockholders. So, cash flow from assets
is:
Cash flow from assets = Cash flow to creditors + Cash flow to stockholders
Cash flow from assets = $11,700 + 2,900
Cash flow from assets = $14,600
Net capital spending is equal to depreciation plus the increase in fixed assets, so:
Net capital spending = Depreciation + Increase in fixed assets
Net capital spending = $10,100 + 17,400
Net capital spending = $27,500
Now we can use the cash flow from assets equation to find the change in NWC. Doing so, we
find:
Cash flow from assets = OCF Change in NWC Net capital spending
$14,600 = $44,640 Change in NWC $27,500
Change in NWC = $2,540
15. Here we need to work the income statement backward. Starting with net income, we know that net
income is:
Net income = Dividends + Addition to retained earnings
Net income = $915 + 2,100
Net income = $3,015
Net income is also the taxable income, minus the taxable income times the tax rate, or:
Net income = Taxable income (Taxable income)(Tax rate)
Net income = Taxable income(1 Tax rate)
We can rearrange this equation and solve for the taxable income as:
Taxable income = Net income / (1 Tax rate)
Taxable income = $3,015 / (1 .40)
Taxable income = $5,025
CHAPTER 2 B-12
EBIT minus interest equals taxable income, so rearranging this relationship, we find:
EBIT = Taxable income + Interest
EBIT = $5,025 + 1,360
EBIT = $6,385
Now that we have the EBIT, we know that sales minus costs minus depreciation equals EBIT.
Solving this equation for EBIT, we find:
EBIT = Sales Costs Depreciation
$6,385 = $42,000 28,000 Depreciation
Depreciation = $7,615
16. We can fill in the balance sheet with the numbers we are given. The balance sheet will be:
Balance Sheet
Cash
Accounts receivable
Inventory
Current assets
$167,000
241,000
498,000
$906,000
Tangible net fixed assets
Intangible net fixed assets
$4,700,000
818,000
Total assets
$6,424,000
Accounts payable
Notes payable
Current liabilities
Long-term debt
Total liabilities
$236,000
176,000
$412,000
913,000
$1,325,000
Common stock
Accumulated retained earnings
Total liabilities & owners equity
??
4,230,000
$6,424,000
Owners equity has to be total liabilities & equity minus accumulated retained earnings and total
liabilities, so:
Owners equity = Total liabilities & equity Accumulated retained earnings Total liabilities
Owners equity = $6,424,000 4,230,000 1,325,000
Owners equity = $869,000
17. Owners equity is the maximum of total assets minus total liabilities, or zero. Although the book
value of owners equity can be negative, the market value of owners equity cannot be negative, so:
Owners equity = Max [(TA TL), 0]
a. If total assets are $8,700, the owners equity is:
Owners equity = Max[($8,700 7,500),0]
Owners equity = $1,200
b. If total assets are $6,900, the owners equity is:
Owners equity = Max[($6,900 7,500),0]
Owners equity = $0
CHAPTER 2 B-13
18. a. Using Table 2.3, we can see the marginal tax schedule. For Corporation Growth, the first $50,000
of income is taxed at 15 percent, the next $25,000 is taxed at 25 percent, and the next $25,000 is
taxed at 34 percent. So, the total taxes for the company will be:
TaxesGrowth = 0.15($50,000) + 0.25($25,000) + 0.34($8,000)
TaxesGrowth = $16,470
For Corporation Income, the first $50,000 of income is taxed at 15 percent, the next $25,000 is
taxed at 25 percent, the next $25,000 is taxed at 34 percent, the next $235,000 is taxed at 39
percent, and the next $7,965,000 is taxed at 34 percent. So, the total taxes for the company will
be:
TaxesIncome = 0.15($50,000) + 0.25($25,000) + 0.34($25,000) + 0.39($235,000)
+ 0.34($7,965,000)
TaxesIncome = $2,822,000
b. The marginal tax rate is the tax rate on the next $1 of earnings. Each firm has a marginal tax rate
of 34% on the next $10,000 of taxable income, despite their different average tax rates, so both
firms will pay an additional $3,400 in taxes.
19. a. The income statement starts with revenues and subtracts costs to arrive at EBIT. We then subtract
interest to get taxable income, and then subtract taxes to arrive at net income. Doing so, we get:
Income Statement
Sales
$2,700,000
Cost of goods sold
1,690,000
Other expenses
465,000
Depreciation
530,000
EBIT
$ 15,000
Interest
210,000
Taxable income
$195,000
Taxes (35%)
0
Net income
$195,000
The taxes are zero since we are ignoring any carryback or carryforward provisions.
b. The operating cash flow for the year was:
OCF = EBIT + Depreciation Taxes
OCF = $15,000 + 530,000 0
OCF = $545,000
c. Net income was negative because of the tax deductibility of depreciation and interest expense.
However, the actual cash flow from operations was positive because depreciation is a non-cash
expense and interest is a financing, not an operating, expense.
CHAPTER 2 B-14
20. A firm can still pay out dividends if net income is negative; it just has to be sure there is sufficient
cash flow to make the dividend payments. The assumptions made in the question are:
Change in NWC = Net capital spending = Net new equity = 0
To find the new long-term debt, we first need to find the cash flow from assets. The cash flow from
assets is:
Cash flow from assets = OCF Change in NWC Net capital spending
Cash flow from assets = $545,000 0 0
Cash flow from assets = $545,000
We can also find the cash flow to stockholders, which is:
Cash flow to stockholders = Dividends Net new equity
Cash flow to stockholders = $500,000 0
Cash flow to stockholders = $500,000
Now we can use the cash flow from assets equation to find the cash flow to creditors. Doing so, we
get:
Cash flow from assets = Cash flow to creditors + Cash flow to stockholders
$545,000 = Cash flow to creditors + $500,000
Cash flow to creditors = $45,000
Now we can use the cash flow to creditors equation to find:
Cash flow to creditors = Interest Net new long-term debt
$45,000 = $210,000 Net new long-term debt
Net new long-term debt = $165,000
21. a. To calculate the OCF, we first need to construct an income statement. The income statement starts
with revenues and subtracts costs to arrive at EBIT. We then subtract out interest to get taxable
income, and then subtract taxes to arrive at net income. Doing so, we get:
Income Statement
Sales
Cost of goods sold
Depreciation
EBIT
Interest
Taxable income
Taxes (35%)
Net income
$18,450
13,610
2,420
$ 2,420
260
$ 2,160
756
$ 1,404
CHAPTER 2 B-15
b. The operating cash flow for the year was:
OCF = EBIT + Depreciation Taxes
OCF = $2,420 + 2,420 756 = $4,084
c. To calculate the cash flow from assets, we also need the change in net working capital and net
capital spending. The change in net working capital was:
Change in NWC = NWCend NWCbeg
Change in NWC = (CAend CLend) (CAbeg CLbeg)
Change in NWC = ($4,690 2,720) ($3,020 2,260)
Change in NWC = $1,210
And the net capital spending was:
Net capital spending = NFAend NFAbeg + Depreciation
Net capital spending = $12,700 12,100 + 2,420
Net capital spending = $3,020
So, the cash flow from assets was:
Cash flow from assets = OCF Change in NWC Net capital spending
Cash flow from assets = $4,084 1,210 3,020
Cash flow from assets = $146
The cash flow from assets can be positive or negative, since it represents whether the firm raised
funds or distributed funds on a net basis. In this problem, even though net income and OCF are
positive, the firm invested heavily in both fixed assets and net working capital; it had to raise a
net $146 in funds from its stockholders and creditors to make these investments.
d. The cash flow from creditors was:
Cash flow to creditors = Interest Net new LTD
Cash flow to creditors = $260 0
Cash flow to creditors = $260
Rearranging the cash flow from assets equation, we can calculate the cash flow to stockholders
as:
Cash flow from assets = Cash flow to stockholders + Cash flow to creditors
$146 = Cash flow to stockholders + $260
Cash flow to stockholders = $406
Now we can use the cash flow to stockholders equation to find the net new equity as:
Cash flow to stockholders = Dividends Net new equity
$406 = $450 Net new equity
Net new equity = $856
CHAPTER 2 B-16
The firm had positive earnings in an accounting sense (NI > 0) and had positive cash flow from
operations. The firm invested $1,210 in new net working capital and $3,020 in new fixed assets.
The firm had to raise $146 from its stakeholders to support this new investment. It accomplished
this by raising $856 in the form of new equity. After paying out $450 in the form of dividends to
shareholders and $260 in the form of interest to creditors, $146 was left to just meet the firms
cash flow needs for investment.
22. a. To calculate owners equity, we first need total liabilities and owners equity. From the balance
sheet relationship we know that this is equal to total assets. We are given the necessary
information to calculate total assets. Total assets are current assets plus fixed assets, so:
Total assets = Current assets + Fixed assets = Total liabilities and owners equity
For 2007, we get:
Total assets = $2,050 + 9,504
Total assets = $11,554
Now, we can solve for owners equity as:
Total liabilities and owners equity = Current liabilities + Long-term debt + Owners equity
$11,554 = $885 + 5,184 + Owners equity
Owners equity = $5,485
For 2008, we get:
Total assets = $2,172 + 9,936
Total assets = $12,108
Now we can solve for owners equity as:
Total liabilities and owners equity = Current liabilities + Long-term debt + Owners equity
$12,108 = $1,301 + 6,048 + Owners equity
Owners equity = $4,759
b. The change in net working capital was:
Change in NWC = NWCend NWCbeg
Change in NWC = (CAend CLend) (CAbeg CLbeg)
Change in NWC = ($2,172 1,301) ($2,050 885)
Change in NWC = $294
c. To find the amount of fixed assets the company sold, we need to find the net capital spending,
The net capital spending was:
Net capital spending = NFAend NFAbeg + Depreciation
Net capital spending = $9,936 9,504 + 2,590
Net capital spending = $3,022
CHAPTER 2 B-17
To find the fixed assets sold, we can also calculate net capital spending as:
Net capital spending = Fixed assets bought Fixed assets sold
$3,022 = $4,320 Fixed assets sold
Fixed assets sold = $1,298
To calculate the cash flow from assets, we first need to calculate the operating cash flow. For the
operating cash flow, we need the income statement. So, the income statement for the year is:
Income Statement
Sales
Costs
Depreciation
EBIT
Interest
Taxable income
Taxes (35%)
Net income
$30,670
15,380
2,590
$12,700
480
$12,220
4,277
$ 7,943
Now we can calculate the operating cash flow which is:
OCF = EBIT + Depreciation Taxes
OCF = $12,700 + 2,590 4,277 = $11,013
And the cash flow from assets is:
Cash flow from assets = OCF Change in NWC Net capital spending.
Cash flow from assets = $11,013 ($294) 3,022
Cash flow from assets = $8,285
d. To find the cash flow to creditors, we first need to find the net new borrowing. The net new
borrowing is the difference between the ending long-term debt and the beginning long-term debt,
so:
Net new borrowing = LTDEnding LTDBeginnning
Net new borrowing = $6,048 5,184
Net new borrowing = $864
So, the cash flow to creditors is:
Cash flow to creditors = Interest Net new borrowing
Cash flow to creditors = $480 864 = $384
CHAPTER 2 B-18
The net new borrowing is also the difference between the debt issued and the debt retired. We
know the amount the company issued during the year, so we can find the amount the company
retired. The amount of debt retired was:
Net new borrowing = Debt issued Debt retired
$864 = $1,300 Debt retired
Debt retired = $436
23. To construct the cash flow identity, we will begin cash flow from assets. Cash flow from assets is:
Cash flow from assets = OCF Change in NWC Net capital spending
So, the operating cash flow is:
OCF = EBIT + Depreciation Taxes
OCF = $139,833 + 68,220 40,499
OCF = $167,554
Next, we will calculate the change in net working capital which is:
Change in NWC = NWCend NWCbeg
Change in NWC = (CAend CLend) (CAbeg CLbeg)
Change in NWC = ($72,700 33,723) ($57,634 30,015)
Change in NWC = $11,358
Now, we can calculate the capital spending. The capital spending is:
Net capital spending = NFAend NFAbeg + Depreciation
Net capital spending = $507,888 430,533 + 68,220
Net capital spending = $145,575
Now, we have the cash flow from assets, which is:
Cash flow from assets = OCF Change in NWC Net capital spending
Cash flow from assets = $167,554 11,358 145,575
Cash flow from assets = $10,621
The company generated $10,621 in cash from its assets. The cash flow from operations was
$167,554, and the company spent $11,358 on net working capital and $145,575 in fixed assets.
The cash flow to creditors is:
Cash flow to creditors = Interest paid New long-term debt
Cash flow to creditors = Interest paid (Long-term debtend Long-term debtbeg)
Cash flow to creditors = $24,120 ($190,000 171,000)
Cash flow to creditors = $5,120
CHAPTER 2 B-19
The cash flow to stockholders is a little trickier in this problem. First, we need to calculate the new
equity sold. The equity balance increased during the year. The only way to increase the equity
balance is to add addition to retained earnings or sell equity. To calculate the new equity sold, we
can use the following equation:
New equity = Ending equity Beginning equity Addition to retained earnings
New equity = $356,865 287,152 63,214
New equity = $6,499
What happened was the equity account increased by $69,713. $63,214 of this came from addition to
retained earnings, so the remainder must have been the sale of new equity. Now we can calculate the
cash flow to stockholders as:
Cash flow to stockholders = Dividends paid Net new equity
Cash flow to stockholders = $12,000 6,499
Cash flow to stockholders = $5,501
The company paid $5,120 to creditors and $5,500 to stockholders.
Finally, the cash flow identity is:
Cash flow from assets = Cash flow to creditors + Cash flow to stockholders
$10,621
=
$5,120
+
$5,501
The cash flow identity balances, which is what we expect.
Challenge
24. Net capital spending
= NFAend NFAbeg + Depreciation
= (NFAend NFAbeg) + (Depreciation + ADbeg) ADbeg
= (NFAend NFAbeg)+ ADend ADbeg
= (NFAend + ADend) (NFAbeg + ADbeg)
= FAend FAbeg
25. a. The tax bubble causes average tax rates to catch up to marginal tax rates, thus eliminating the tax
advantage of low marginal rates for high income corporations.
b. Taxes = 0.15($50K) + 0.25($25K) + 0.34($25K) + 0.39($235K) = $113.9K
Average tax rate = $113.9K / $335K = 34%
The marginal tax rate on the next dollar of income is 34 percent.
For corporate taxable income levels of $335K to $10M, average tax rates are equal to marginal
tax rates.
Taxes = 0.34($10M) + 0.35($5M) + 0.38($3.333M) = $6,416,667
CHAPTER 2 B-20
Average tax rate = $6,416,667 / $18,333,334 = 35%
CHAPTER 2 B-21
The marginal tax rate on the next dollar of income is 35 percent. For corporate taxable income
levels over $18,333,334, average tax rates are again equal to marginal tax rates.
c. At the end of the tax bubble, the marginal tax rate on the next dollar should equal the average
tax rate on all preceding dollars. Since the upper threshold of the bubble bracket is now $200,000,
the marginal tax rate on dollar $200,001 should be 34 percent, and the total tax paid on the first
$200,000 should be $200,000(.34). So, we get:
Taxes
X($100K)
X
X
= 0.34($200K) = $68K = 0.15($50K) + 0.25($25K) + 0.34($25K) + X($100K)
= $68K 22.25K = $45.75K
= $45.75K / $100K
= 45.75%
CHAPTER 3
WORKING WITH FINANCIAL
STATEMENTS
Answers to Concepts Review and Critical Thinking Questions
1.
a. If inventory is purchased with cash, then there is no change in the current ratio. If inventory is
purchased on credit, then there is a decrease in the current ratio if it was initially greater than 1.0.
b. Reducing accounts payable with cash increases the current ratio if it was initially greater than 1.0.
c. Reducing short-term debt with cash increases the current ratio if it was initially greater than 1.0.
d. As long-term debt approaches maturity, the principal repayment and the remaining interest
expense become current liabilities. Thus, if debt is paid off with cash, the current ratio increases
if it was initially greater than 1.0. If the debt has not yet become a current liability, then paying it
off will reduce the current ratio since current liabilities are not affected.
e. Reduction of accounts receivables and an increase in cash leaves the current ratio unchanged.
f.
Inventory sold at cost reduces inventory and raises cash, so the current ratio is unchanged.
g. Inventory sold for a profit raises cash in excess of the inventory recorded at cost, so the current
ratio increases.
2.
The firm has increased inventory relative to other current assets; therefore, assuming current liability
levels remain mostly unchanged, liquidity has potentially decreased.
3.
A current ratio of 0.50 means that the firm has twice as much in current liabilities as it does in
current assets; the firm potentially has poor liquidity. If pressed by its short-term creditors and
suppliers for immediate payment, the firm might have a difficult time meeting its obligations. A
current ratio of 1.50 means the firm has 50% more current assets than it does current liabilities. This
probably represents an improvement in liquidity; short-term obligations can generally be met completely with a safety factor built in. A current ratio of 15.0, however, might be excessive. Any excess
funds sitting in current assets generally earn little or no return. These excess funds might be put to
better use by investing in productive long-term assets or distributing the funds to shareholders.
4.
a. Quick ratio provides a measure of the short-term liquidity of the firm, after removing the effects
of inventory, generally the least liquid of the firms current assets.
b. Cash ratio represents the ability of the firm to completely pay off its current liabilities balance
with its most liquid asset (cash).
CHAPTER 3 B-23
c. The capital intensity ratio tells us the dollar amount investment in assets needed to generate one
dollar in sales.
d. Total asset turnover measures how much in sales is generated by each dollar of firm assets.
e. Equity multiplier represents the degree of leverage for an equity investor of the firm; it measures
the dollar worth of firm assets each equity dollar has a claim to.
f.
Long-term debt ratio measures the percentage of total firm capitalization funded by long-term
debt.
g. Times interest earned ratio provides a relative measure of how well the firms operating earnings
can cover current interest obligations.
h. Profit margin is the accounting measure of bottom-line profit per dollar of sales.
i.
Return on assets is a measure of bottom-line profit per dollar of total assets.
j.
Return on equity is a measure of bottom-line profit per dollar of equity.
k. Price-earnings ratio reflects how much value per share the market places on a dollar of
accounting earnings for a firm.
5.
Common size financial statements express all balance sheet accounts as a percentage of total assets
and all income statement accounts as a percentage of total sales. Using these percentage values
rather than nominal dollar values facilitates comparisons between firms of different size or business
type.
6.
Peer group analysis involves comparing the financial ratios and operating performance of a
particular firm to a set of peer group firms in the same industry or line of business. Comparing a firm
to its peers allows the financial manager to evaluate whether some aspects of the firms operations,
finances, or investment activities are out of line with the norm, thereby providing some guidance on
appropriate actions to take to adjust these ratios, if appropriate. An aspirant group would be a set of
firms whose performance the company in question would like to emulate. The financial manager
often uses the financial ratios of aspirant groups as the target ratios for his or her firm; some
managers are evaluated by how well they match the performance of an identified aspirant group.
7.
Return on equity is probably the most important accounting ratio that measures the bottom-line
performance of the firm with respect to the equity shareholders. The Du Pont identity emphasizes the
role of a firms profitability, asset utilization efficiency, and financial leverage in achieving a ROE
figure. For example, a firm with ROE of 20% would seem to be doing well, but this figure may be
misleading if it were a marginally profitable (low profit margin) and highly levered (high equity
multiplier). If the firms margins were to erode slightly, the ROE would be heavily impacted.
8.
The book-to-bill ratio is intended to measure whether demand is growing or falling. It is closely
followed because it is a barometer for the entire high-tech industry where levels of revenues and
earnings have been relatively volatile.
CHAPTER 3 B-24
9.
If a company is growing by opening new stores, then presumably total revenues would be rising.
Comparing total sales at two different points in time might be misleading. Same-store sales control
for this by only looking at revenues of stores open within a specific period.
10.
a. For an electric utility such as Con Ed, expressing costs on a per kilowatt hour basis would be a
way comparing costs with other utilities of different sizes.
b. For a retailer such as JC Penney, expressing sales on a per square foot basis would be useful in
comparing revenue production against other retailers.
c. For an airline such as Delta, expressing costs on a per passenger mile basis allows for
comparisons with other airlines by examining how much it costs to fly one passenger one
mile.
d. For an on-line service such as AOL, using a per call basis for costs would allow for
comparisons with smaller services. A per subscriber basis would also make sense.
e. For a hospital such as Holy Cross, revenues and costs expressed on a per bed basis would be
useful.
f. For a college textbook publisher such as McGraw-Hill/Irwin, the leading publisher of finance
textbooks for the college market, the obvious standardization would be per book sold.
11.
As with any ratio analysis, the ratios themselves do not necessarily indicate a problem, but simply
indicate that something is different and it is up to us to determine if a problem exists. If the cost of
goods sold as a percentage of sales is increasing, we would expect that EBIT as a percentage of sales
would decrease, all else constant. An increase in the cost of goods sold as a percentage of sales
occurs because the cost of raw materials or other inventory is increasing at a faster rate than the sales
price.
This is may be a bad sign since the contribution of each sales dollar to net income and cash flow is
lower. However, when a new product, for example, the HDTV, enters the market, the price of one
unit will often be high relative to the cost of goods sold per unit, and demand, therefore sales,
initially small. As the product market becomes more developed, price of the product generally drops,
and sales increase as more competition enters the market. In this case, the increase in cost of goods
sold as a percentage of sales is to be expected. The maker or seller expects to boost sales at a faster
rate than its cost of goods sold increases. In this case, a good practice would be to examine the
common-size income statements to see if this is an industry-wide occurrence.
12.
If we assume that the cause is negative, the two reasons for the trend of increasing cost of goods sold
as a percentage of sales are that costs are becoming too high or the sales price is not increasing fast
enough. If the cause is an increase in the cost of goods sold, the manager should look at possible
actions to control costs. If costs can be lowered by seeking lower cost suppliers of similar or higher
quality, the cost of goods sold as a percentage of sales should decrease. Another alternative is to
increase the sales price to cover the increase in the cost of goods sold. Depending on the industry,
this may be difficult or impossible. For example, if the company sells most of its products under a
long-term contract that has a fixed price, it may not be able to increase the sales price and will be
CHAPTER 3 B-25
forced to look for other cost-cutting possibilities. Additionally, if the market is competitive, the
company might also be unable to increase the sales price.
CHAPTER 3 B-26
Solutions to Questions and Problems
NOTE: All end-of-chapter problems were solved using a spreadsheet. Many problems require multiple
steps. Due to space and readability constraints, when these intermediate steps are included in this
solutions manual, rounding may appear to have occurred. However, the final answer for each problem is
found without rounding during any step in the problem.
Basic
1.
To find the current assets, we must use the net working capital equation. Doing so, we find:
NWC = Current assets Current liabilities
$1,350 = Current assets $4,290
Current assets = $5,640
Now, use this number to calculate the current ratio and the quick ratio. The current ratio is:
Current ratio = Current assets / Current liabilities
Current ratio = $5,640 / $4,290
Current ratio = 1.31 times
And the quick ratio is:
Quick ratio = (Current assets Inventory) / Current liabilities
Quick ratio = ($5,640 1,820) / $4,290
Quick ratio = 0.89 times
2.
To find the return on assets and return on equity, we need net income. We can calculate the net
income using the profit margin. Doing so, we find the net income is:
Profit margin = Net income / Sales
.08 = Net income / $27,000,000
Net income = $2,160,000
Now we can calculate the return on assets as:
ROA = Net income / Total assets
ROA = $2,160,000 / $99,000,000
ROA = 0.1137 or 11.37%
We do not have the equity for the company, but we know that equity must be equal to total assets
minus total debt, so the ROE is:
ROE = Net income / (Total assets Total debt)
ROE = $2,160,000 / ($19,000,000 6,400,000)
ROE = 0.1717 or 17.14%
CHAPTER 3 B-27
3.
The receivables turnover for the company was:
Receivables turnover = Credit sales / Receivables
Receivables turnover = $5,871,650 / $645,382
Receivables turnover = 9.10 times
Using the receivables turnover, we can calculate the days sales in receivables as:
Days sales in receivables = 365 days / Receivables turnover
Days sales in receivables = 365 days / 9.10
Days sales in receivables = 40.12 days
The average collection period, which is the same as the days sales in receivables, was 40.12 days.
4.
The inventory turnover for the company was:
Inventory turnover = COGS / Inventory
Inventory turnover = $8,493,825 / $743,186
Inventory turnover = 11.43 times
Using the inventory turnover, we can calculate the days sales in inventory as:
Days sales in inventory = 365 days / Inventory turnover
Days sales in inventory = 365 days / 11.43
Days sales in inventory = 31.94 days
On average, a unit of inventory sat on the shelf 31.94 days before it was sold.
5.
To find the debt-equity ratio using the total debt ratio, we need to rearrange the total debt ratio
equation. We must realize that the total assets are equal to total debt plus total equity. Doing so, we
find:
Total debt ratio = Total debt / Total assets
0.70 = Total debt / (Total debt + Total equity)
0.30(Total debt) = 0.70(Total equity)
Total debt / Total equity = 0.70 / 0.30
Debt-equity ratio = 2.33
And the equity multiplier is one plus the debt-equity ratio, so:
Equity multiplier = 1 + D/E
Equity multiplier = 1 + 2.33
Equity multiplier = 3.33
CHAPTER 3 B-28
6.
We need to calculate the net income before we calculate the earnings per share. The sum of
dividends and addition to retained earnings must equal net income, so net income must have been:
Net income = Addition to retained earnings + Dividends
Net income = $530,000 + 190,000
Net income = $720,000
So, the earnings per share were:
EPS = Net income / Shares outstanding
EPS = $720,000 / 570,000
EPS = $1.26 per share
The dividends per share were:
Dividends per share = Total dividends / Shares outstanding
Dividends per share = $190,000 / 570,000
Dividends per share = $0.33 per share
The book value per share was:
Book value per share = Total equity / Shares outstanding
Book value per share = $6,800,000 / 570,000
Book value per share = $11.93 per share
The market-to-book ratio is:
Market-to-book ratio = Share price / Book value per share
Market-to-book ratio = $39 / $11.93
Market-to-book ratio = 3.27 times
The P/E ratio is:
P/E ratio = Share price / EPS
P/E ratio = $39 / $1.26
P/E ratio = 30.88 times
Sales per share are:
Sales per share = Total sales / Shares outstanding
Sales per share = $16,000,000 / 570,000
Sales per share = $28.07
The P/S ratio is:
P/S ratio = Share price / Sales per share
P/S ratio = $39 / $28.07
P/S ratio = 1.39 times
CHAPTER 3 B-29
7.
With the information given, we must use the Du Pont identity to calculate return on equity. Doing
so, we find:
ROE = (Profit margin)(Total asset turnover)(Equity multiplier)
ROE = (.08)(1.32)(1.60)
ROE = 0.1690 or 16.90%
8.
We can use the Du Pont identity and solve for the equity multiplier. With the equity multiplier we
can find the debt-equity ratio. Doing so we find:
ROE = (Profit margin)(Total asset turnover)(Equity multiplier)
0.1570 = (0.10)(1.35)(Equity multiplier)
Equity multiplier = 1.16
Now, using the equation for the equity multiplier, we get:
Equity multiplier = 1 + Debt-equity ratio
1.16 = 1 + Debt-equity ratio
Debt-equity ratio = 0.16
9.
To find the days sales in payables, we first need to find the payables turnover. The payables
turnover was:
Payables turnover = Cost of goods sold / Payables balance
Payables turnover = $48,813 / $11,816
Payables turnover = 4.13 times
Now, we can use the payables turnover to find the days sales in payables as:
Days sales in payables = 365 days / Payables turnover
Days sales in payables = 365 days / 4.13
Days sales in payables = 88.35 days
The company left its bills to suppliers outstanding for 88.35 days on average. A large value for this
ratio could imply that either (1) the company is having liquidity problems, making it difficult to pay
off its short-term obligations, or (2) that the company has successfully negotiated lenient credit terms
from its suppliers.
10. With the information provided, we need to calculate the return on equity using an extended return on
equity equation. We first need to find the equity multiplier which is:
Equity multiplier = 1 + Debt-equity ratio
Equity multiplier = 1 + 0.80
Equity multiplier = 1.80
CHAPTER 3 B-30
Now we can calculate the return on equity as:
ROE = (ROA)(Equity multiplier)
ROE = 0.089(1.80)
ROE = 0.1602 or 16.02%
The return on equity equation we used was an abbreviated version of the Du Pont identity. If we
multiply the profit margin and total asset turnover ratios from the Du Pont identity, we get:
(Net income / Sales)(Sales / Total assets) = Net income / Total assets = ROA
With the return on equity, we can calculate the net income as:
ROE = Net income / Total equity
0.1602 = Net income / $590,000
Net income = $94,518
11. To find the internal growth rate, we need the plowback, or retention, ratio. The plowback ratio is:
b = 1 0.20
b = 0.80
Now, we can use the internal growth rate equation to find:
Internal growth rate = [(ROA)(b)] / [1 (ROA)(b)]
Internal growth rate = [0.11(0.80)] / [1 0.11(0.80)]
Internal growth rate = 0.0965 or 9.65%
12. To find the internal growth rate we need the plowback, or retention, ratio. The plowback ratio is:
b = 1 0.25
b = 0.75
Now, we can use the sustainable growth rate equation to find:
Sustainable growth rate = [(ROE)(b)] / [1 (ROE)(b)]
Sustainable growth rate = [0.142(0.75)] / [1 0.142(0.75)]
Sustainable growth rate = 0.1192 or 11.92%
13. We need the return on equity to calculate the sustainable growth rate. To calculate return on equity,
we need to realize that the total asset turnover is the inverse of the capital intensity ratio and the
equity multiplier is one plus the debt-equity ratio. So, the return on equity is:
ROE = (Profit margin)(Total asset turnover)(Equity multiplier)
ROE = (0.74)(1/0.55)(1 + 0.30)
ROE = 0.1749 or 17.49%
CHAPTER 3 B-31
Next we need the plowback ratio. The payout ratio is one minus the payout ratio. We can calculate
the payout ratio as the dividends divided by net income, so the plowback ratio is:
b = 1 ($25,000 / $70,000)
b = 0.64
Now we can use the sustainable growth rate equation to find:
Sustainable growth rate = [(ROE)(b)] / [1 (ROE)(b)]
Sustainable growth rate = [0.1749(0.64)] / [1 0.1749(0.64)]
Sustainable growth rate = 0.1267 or 12.67%
14. We need the return on equity to calculate the sustainable growth rate. Using the Du Pont identity, the
return on equity is:
ROE = (Profit margin)(Total asset turnover)(Equity multiplier)
ROE = (.083)(1.75)(1.85)
ROE = .2687 or 26.87%
To find the sustainable growth rate, we need the plowback, or retention, ratio. The plowback ratio is:
b = 1 .40
b = .60
Now, we can use the sustainable growth rate equation to find:
Sustainable growth rate = [(ROE)(b)] / [1 (ROE)(b)]
Sustainable growth rate = [.2687(.60)] / [1 .2687(.60)]
Sustainable growth rate = .1922 or 19.22%
15. To calculate the common-size balance sheet, we divide each asset account by total assets, and each
liability and equity account by total liabilities and equity. For example, the common-size cash
percentage for 2007 is:
Cash percentage = Cash / Total assets
Cash percentage = $18,288 / $748,879
Cash percentage = 0.0244 or 2.44%
CHAPTER 3 B-32
Repeating this procedure for each account, we get:
2007
Assets
Current assets
Cash
Accounts receivable
Inventory
Total
Fixed assets
Net plant and equipment
Total assets
$18,288
44,062
104,339
$166,689
2.44%
5.88%
13.93%
22.26%
$22,455
55,457
144,696
$222,608
2.86%
7.07%
18.44%
28.37%
582,190
$748,879
77.74%
100%
561,988
$784,596
71.63%
100%
20.02%
9.25%
29.27%
25.37%
$144,722
101,134
$245,856
131,250
18.45%
12.89%
31.34%
16.73%
21.37%
23.99%
45.36%
100%
$160,000
247,490
$407,490
$784,596
20.39%
31.54%
51.94%
100%
Liabilities and owners' equity
Current liabilities
Accounts payable
$149,940
Notes payable
69,246
Total
$219,186
Long-term debt
190,000
Owners' equity
Common stock and paid-in
surplus
$160,000
Accumulated retained earnings
179,693
Total
$339,693
Total liabilities and owners' equity
$748,879
16. a.
2008
The current ratio is calculated as:
Current ratio = Current assets / Current liabilities
Current ratio2007 = $166,869 / $219,186
Current ratio2007 = 0.76 times
Current ratio2008 = $222,608 / $245,856
Current ratio2008 = 0.91 times
b.
The quick ratio is calculated as:
Quick ratio = (Current assets Inventory) / Current liabilities
Quick ratio2007 = ($166,689 104,339) / $219,186
Quick ratio2007 = 0.28 times
CHAPTER 3 B-33
Quick ratio2008 = ($222,608 144,696) / $245,856
Quick ratio2008 = 0.32 times
c.
The cash ratio is calculated as:
Cash ratio = Cash / Current liabilities
Cash ratio2007 = $18,288 / $219,186
Cash ratio2007 = 0.08 times
Cash ratio2008 = $22,455 / $245,856
Cash ratio2008 = 0.09 times
d.
The debt-equity ratio is calculated as:
Debt-equity ratio = Total debt / Total equity
Debt-equity ratio = (Current liabilities + Long-term debt) / Total equity
Debt-equity ratio2007 = ($219,186 + 190,000) / $339,693
Debt-equity ratio2007 = 1.20
Debt-equity ratio2008 = ($245,856 + 131,250) / $407,490
Debt-equity ratio2008 = 0.93
And the equity multiplier is:
Equity multiplier = 1 + Debt-equity ratio
Equity multiplier2007 = 1 + 1.20
Equity multiplier2007 = 2.20
Equity multiplier2008 = 1 + 0.93
Equity multiplier2008 = 1.93
e.
The total debt ratio is calculated as:
Total debt ratio = Total debt / Total assets
Total debt ratio = (Current liabilities + Long-term debt) / Total assets
Total debt ratio2007 = ($219,186 + 190,000) / $748,879
Total debt ratio2007 = 0.55
Total debt ratio2007 = ($245,856 + 131,250) / $784,596
Total debt ratio2007 = 0.48
CHAPTER 3 B-34
17. Using the Du Pont identity to calculate ROE, we get:
ROE = (Profit margin)(Total asset turnover)(Equity multiplier)
ROE = (Net income / Sales)(Sales / Total assets)(Total asset / Total equity)
ROE = ($132,186 / $2,678,461)($2,678,461 / $784,596)($784,596 / $407,490)
ROE = 0.3244 or 32.44%
18. One equation to calculate ROA is:
ROA = (Profit margin)(Total asset turnover)
We can solve this equation to find total asset turnover as:
0.12 = 0.07(Total asset turnover)
Total asset turnover = 1.71 times
Now, solve the ROE equation to find the equity multiplier which is:
ROE = (ROA)(Equity multiplier)
0.17 = 0.12(Equity multiplier)
Equity multiplier = 1.42 times
19. To calculate the ROA, we first need to find the net income. Using the profit margin equation, we
find:
Profit margin = Net income / Sales
0.075 = Net income / $26,000,000
Net income = $1,950,000
Now we can calculate ROA as:
ROA = Net income / Total assets
ROA = $1,950,000 / $19,000,000
ROA = 0.1026 or 10.26%
20. To calculate the internal growth rate, we need to find the ROA and the plowback ratio. The ROA for
the company is:
ROA = Net income / Total assets
ROA = $11,687 / $86,900
ROA = 0.1345 or 13.45%
And the plowback ratio is:
b = 1 .40
b = .60
CHAPTER 3 B-35
Now, we can use the internal growth rate equation to find:
Internal growth rate = [(ROA)(b)] / [1 (ROA)(b)]
Internal growth rate = [0.1345(0.60)] / [1 0.1345(.60)]
Internal growth rate = 0.0878 or 8.78%
21. To calculate the sustainable growth rate, we need to find the ROE and the plowback ratio. The ROE
for the company is:
ROE = Net income / Equity
ROE = $11,687 / $58,300
ROE = 0.2005 or 20.05%
Using the sustainable growth rate, we calculated in the precious problem, we find the sustainable
growth rate is:
Sustainable growth rate = [(ROE)(b)] / [1 (ROE)(b)]
Sustainable growth rate = [(0.2005)(0.60)] / [1 (0.2005)(0.60)]
Sustainable growth rate = 0.1367 or 13.67%
22. The total asset turnover is:
Total asset turnover = Sales / Total assets
Total asset turnover = $17,000,000 / $7,000,000 = 2.43 times
If the new total asset turnover is 2.75, we can use the total asset turnover equation to solve for the
necessary sales level. The new sales level will be:
Total asset turnover = Sales / Total assets
2.75 = Sales / $7,000,000
Sales = $19,250,000
23. To find the ROE, we need the equity balance. Since we have the total debt, if we can find the total
assets we can calculate the equity. Using the total debt ratio, we find total assets as:
Debt ratio = Total debt / Total assets
0.70 = $265,000 / Total assets
Total assets = $378,571
Total liabilities and equity is equal to total assets. Using this relationship, we find:
Total liabilities and equity = Total debt + Total equity
$378,571 = $265,000 + Total equity
Total equity = $113,571
CHAPTER 3 B-36
Now, we can calculate the ROE as:
ROE = Net income / Total equity
ROE = $24,850 / $113,571
ROE = 0.2188 or 21.88%
24. The earnings per share are:
EPS = Net income / Shares
EPS = $5,150,000 / 4,100,000
EPS = $1.26
The price-earnings ratio is:
P/E = Price / EPS
P/E = $41 / $1.26
P/E = 32.64
The sales per share are:
Sales per share = Sales / Shares
Sales per share = $39,000,000 / 4,100,000
Sales per share = $9.51
The price-sales ratio is:
P/S = Price / Sales per share
P/S = $41 / $9.51
P/S = 4.31
The book value per share is:
Book value per share = Book value of equity / Shares
Book value per share = $21,580,000 / 4,100,000
Book value per share = $5.26 per share
And the market-to-book ratio is:
Market-to-book = Market value per share / Book value per share
Market-to-book = $41 / $5.26
Market-to-book = 7.79
CHAPTER 3 B-37
25. To find the profit margin, we need the net income and sales. We can use the total asset turnover to
find the sales and the return on assets to find the net income. Beginning with the total asset turnover,
we find sales are:
Total asset turnover = Sales / Total assets
2.10 = Sales / $10,500,000
Sales = $22,050,000
And the net income is:
ROA = Net income / Total assets
0.13 = Net income / $10,500,000
Net income = $1,365,000
Now we can find the profit margin which is:
Profit margin = Net income / Sales
Profit margin = $1,365,000 / $22,050,000
Profit margin = 0.0619 or 6.19%
Intermediate
26. We can rearrange the Du Pont identity to calculate the profit margin. So, we need the equity
multiplier and the total asset turnover. The equity multiplier is:
Equity multiplier = 1 + Debt-equity ratio
Equity multiplier = 1 + .25
Equity multiplier = 1.25
And the total asset turnover is:
Total asset turnover = Sales / Total assets
Total asset turnover = $9,980 / $3,140
Total asset turnover = 3.18 times
Now, we can use the Du Pont identity to find total sales as:
ROE = (Profit margin)(Total asset turnover)(Equity multiplier)
0.16 = (PM)(3.18)(1.25)
Profit margin = 0.403 or 4.03%
Rearranging the profit margin ratio, we can find the net income which is:
Profit margin = Net income / Sales
0.0403 = Net income / $9,980
Net income = $401.92
CHAPTER 3 B-38
27. This is a multi-step problem in which we need to calculate several ratios to find the fixed assets. If
we know total assets and current assets, we can calculate the fixed assets. Using the current ratio to
find the current assets, we get:
Current ratio = Current assets / Current liabilities
1.30 = Current assets / $900
Current assets = $1,170.00
Now, we are going to use the profit margin to find the net income and use the net income to find the
equity. Doing so, we get:
Profit margin = Net income / Sales
0.09 = Net income / $6,590
Net income = $593.10
And using this net income figure in the return on equity equation to find the equity, we get:
ROE = Net income / Total equity
0.16 = $593.10 / Total equity
Total equity = $3,706.88
Now, we can use the long-term debt ratio to find the total long-term debt. The equation is:
Long-term debt ratio = Long-term debt / (Long-term debt + Total equity)
Inverting both sides we get:
1 / Long-term debt = 1 + (Total equity / Long-term debt)
1 / 0.60 = 1 + (Total equity / Long-term debt)
Total equity / Long-term debt = 0.667
$1,907.03 / Long-term debt = 0.667
Long-term debt = $5,560.31
Now, we can calculate the total debt as:
Total debt = Current liabilities + Long-term debt
Total debt = $900 + 5,560.31
Total debt = $6,460.31
This allows us to calculate the total assets as:
Total assets = Total debt + Total equity
Total assets = $6,460.31 + 3,706.81
Total assets = $10,167.19
CHAPTER 3 B-39
Finally, we can calculate the net fixed assets as:
Net fixed assets = Total assets Current assets
Net fixed assets = $10,167.19 1,170
Net fixed assets = $8,997.19
28. The childs profit margin is:
Profit margin = Net income / Sales
Profit margin = $1 / $25
Profit margin = 0.04 or 4%
And the stores profit margin is:
Profit margin = Net income / Sales
Profit margin = $13,200,000 / $660,000,000
Profit margin = 0.02 or 2%
The advertisement is referring to the stores profit margin, but a more appropriate earnings measure
for the firms owners is the return on equity. The stores return on equity is:
ROE = Net income / Total equity
ROE = Net income / (Total assets Total debt)
ROE = $13,200,000 / ($280,000,000 151,500,000)
ROE = 0.1027 or 10.27%
29. To calculate the profit margin, we first need to calculate the sales. Using the days sales in
receivables, we find the receivables turnover is:
Days sales in receivables = 365 days / Receivables turnover
29.70 days = 365 days / Receivables turnover
Receivables turnover = 12.29 times
Now, we can use the receivables turnover to calculate the sales as:
Receivables turnover = Sales / Receivables
12.29 = Sales / $138,600
Sales = $1,703,333
So, the profit margin is:
Profit margin = Net income / Sales
Profit margin = $132,500 / $1,703,333
Profit margin = 0.0778 or 7.78%
CHAPTER 3 B-40
The total asset turnover is:
Total asset turnover = Sales / Total assets
Total asset turnover = $1,703,333 / $820,000
Total asset turnover = 2.08 times
We need to use the Du Pont identity to calculate the return on equity. Using this relationship, we get:
ROE = (Profit margin)(Total asset turnover)(1 + Debt-equity ratio)
ROE = (0.778)(2.08)(1 + 0.60)
ROE = 0.2585 or 25.85%
30. Here, we need to work the income statement backward to find the EBIT. Starting at the bottom of
the income statement, we know that the taxes are the taxable income times the tax rate. The net
income is the taxable income minus taxes. Rearranging this equation, we get:
Net income = Taxable income (tC)(Taxable income)
Net income = (1 tC)(Taxable income)
Using this relationship we find the taxable income is:
Net income = (1 tC)(Taxable income)
$10,508 = (1 .34)(Taxable income)
Taxable income = $15,921.21
Now, we can calculate the EBIT as:
Taxable income = EBIT Interest
$15,921.21 = EBIT $3,685
EBIT = $19,606.21
So, the cash coverage ratio is:
Cash coverage ratio = (EBIT + Depreciation expense) / Interest
Cash coverage ratio = ($19,606.21 + 4,382) / $3,685
Cash coverage ratio = 6.51 times
31. To find the times interest earned, we need the EBIT and interest expense. EBIT is sales minus costs
minus depreciation, so:
EBIT = Sales Costs Depreciation
EBIT = $378,000 95,400 47,000
EBIT = $235,600
CHAPTER 3 B-41
Now, we need the interest expense. We know the EBIT, so if we find the taxable income (EBT), the
difference between these two is the interest expense. To find EBT, we must work backward through
the income statement. We need total dividends paid. We can use the dividends per share equation to
find the total dividends. Dong so, we find:
DPS = Dividends / Shares
$1.40 = Dividends / 20,000
Dividends = $28,000
Net income is the sum of dividends and addition to retained earnings, so:
Net income = Dividends + Addition to retained earnings
Net income = $28,000 + 48,750
Net income = $76,750
We know that the taxes are the taxable income times the tax rate. The net income is the taxable
income minus taxes. Rearranging this equation, we get:
Net income = Taxable income (tC)(EBT)
Net income = (1 tC)(EBT)
$76,750 = (1 .34)(EBT)
EBT = $116,288
Now, we can use the income statement relationship:
EBT = EBIT Interest
$116,288 = $235,600 Interest
Interest = $119,312
So, the times interest earned ratio is:
Times interest earned = EBIT / Interest
Times interest earned = $235,600 / $119,312
Times interest earned = 1.97 times
32. To find the return on equity, we need the net income and total equity. We can use the total debt ratio
to find the total assets as:
Total debt ratio = Total debt / Total assets
0.30 = $648,000 / Total assets
Total assets = $2,160,000
Using the balance sheet relationship that total assets is equal to total liabilities and equity, we find
the total equity is:
Total assets = Total debt + Equity
$2,160,000 = $648,000 + Equity
Equity = $1,512,000
CHAPTER 3 B-42
We have the return on equity and the equity. We can use the return on equity equation to find net
income is:
ROE = Net income / Equity
0.1650 = Net income / $1,512,000
Net income = $249,480
We have all the information necessary to calculate the ROA, Doing so, we find the ROA is:
ROA = Net income / Total assets
ROA = $249,480 / $2,160,000
ROA = 0.1155 or 11.55%
33. The currency is generally irrelevant in calculating any financial ratio. The companys profit margin
is:
Profit margin = Net income / Sales
Profit margin = 16,182 / 238,165
Profit margin = 0.0679 or 6.79%
As long as both net income and sales are measured in the same currency, there is no problem; in fact,
except for some market value ratios like EPS and BVPS, none of the financial ratios discussed in the
text are measured in terms of currency. This is one reason why financial ratio analysis is widely used
in international finance to compare the business operations of firms and/or divisions across national
economic borders.
We can use the profit margin we previously calculated and the dollar sales to calculate the net
income. Doing so, we get:
Profit margin = Net income / Sales
0.0679 = Net income / $454,058
Net income = $30,850.74
34. Here, we need to calculate several ratios given the financial statements. The ratios are:
Short-term solvency ratios:
Current ratio = Current assets / Current liabilities
Current ratio2007 = $14,626 / $3,375
Current ratio2007 = 4.33 times
Current ratio2008 = $16,536 / $3,714
Current ratio2008 = 4.45 times
Quick ratio = (Current assets Inventory) / Current liabilities
Quick ratio2007 = ($14,626 8,856) / $3,375
Quick ratio2007 = 1.71 times
CHAPTER 3 B-43
Quick ratio2008 = ($16,536 9,873) / $3,714
Quick ratio2008 = 1.79 times
Cash ratio = Cash / Current liabilities
Cash ratio2007 = $2,351 / $3,375
Cash ratio2007 = 0.70 times
Cash ratio2008 = $2,505 / $3,714
Cash ratio2008 = 0.67 times
Asset utilization ratios:
Total asset turnover = Sales / Total assets
Total asset turnover = $124,380 / $68,657
Total asset turnover = 1.81 times
Inventory turnover = COGS / Inventory
Inventory turnover = $64,805 / $9,873
Inventory turnover = 6.56 times
Receivables turnover = Sales / Receivables
Receivables turnover = $124,380 / $4,158
Receivables turnover = 29.91 times
Long-term solvency ratios:
Total debt ratio = (Current liabilities + Long-term debt) / Total assets
Total debt ratio2007 = ($3,375 + 11,500) / $49,560
Total debt ratio2007 = 0.30
Total debt ratio2008 = ($3,714 + 12,500) / $68,657
Total debt ratio2008 = 0.24
Debt-equity ratio = (Current liabilities + Long-term debt) / Total equity
Debt-equity ratio2007 = ($3,375 + 11,500) / $34,685
Debt-equity ratio2007 = 0.43
Debt-equity ratio2008 = ($3,714 + 12,500) / $52,443
Debt-equity ratio2008 = 0.31
Equity multiplier = 1 + D/E ratio
Equity multiplier2007 = 1 + 0.43
Equity multiplier2007 = 1.43
CHAPTER 3 B-44
Equity multiplier2008 = 1 + 0.31
Equity multiplier2008 = 1.31
Times interest earned = EBIT / Interest
Times interest earned = $55,993 / $980
Times interest earned = 57.14 times
Cash coverage ratio = (EBIT + Depreciation) / Interest
Cash coverage ratio = ($55,993 + 3,582) / $980
Cash coverage ratio = 60.79 times
Profitability ratios:
Profit margin = Net income / Sales
Profit margin = $35,758 / $124,380
Profit margin = 0.2875 or 28.75%
Return on assets = Net income / Total assets
Return on assets = $35,758 / $68,657
Return on assets = 0.5208 or 52.08%
Return on equity = Net income / Total equity
Return on equity = $35,758 / $52,443
Return on equity = 0.6818 or 68.18%
35. The Du Pont identity is:
ROE = (PM)(Total asset turnover)(Equity multiplier)
ROE = (Net income / Sales)(Sales / Total assets)(Total assets / Total equity)
ROE = ($35,758 / $124,380)($124,380 / $68,657)($68,657 / $52,443)
ROE = 0.6818 or 68.18%
36. To find the price-earnings ratio we first need the earnings per share. The earnings per share are:
EPS = Net income / Shares outstanding
EPS = $35,758 / 10,000
EPS = $3.58
So, the price-earnings ratio is:
P/E ratio = Share price / EPS
P/E ratio = $73 / $3.58
P/E ratio = 20.42
CHAPTER 3 B-45
The sales per share are:
Sales per share = Sales / Shares outstanding
Sales per share = $124,380 / 10,000
Sales per share = $12.44
So, the price-sales ratio is:
P/S ratio = Share price / Sales per share
P/S ratio = $73 / $12.44
P/S ratio = 5.87
The dividends per share are:
Dividends per share = Total dividends /Shares outstanding
Dividends per share = $18,000 / 10,000 shares
Dividends per share = $1.80 per share
To find the market-to-book ratio, we first need the book value per share. The book value per share is:
Book value per share = Total equity / Shares outstanding
Book value per share = $52,443 / 10,000 shares
Book value per share = $5.24 per share
So, the market-to-book ratio is:
Market-to-book ratio = Share price / Book value per share
Market-to-book ratio = $73 / $5.24
Market-to-book ratio = 13.92 times
37. The current ratio appears to be relatively high when compared to the median; however, it is below
the upper quartile, meaning that at least 25 percent of firms in the industry have a higher current
ratio. Overall, it does not appear that the current ratio is out of line with the industry. The total asset
turnover is low when compared to the industry. In fact, the total asset turnover is in the lower
quartile. This means that the company does not use assets as efficiently overall or that the company
has newer assets than the industry. This would mean that the assets have not been depreciated, which
would mean a higher book value and a lower total asset turnover. The debt-equity ratio is in line with
the industry, between the mean and the lower quartile. The profit margin is in the upper quartile. The
company may be better at controlling costs, or has a better product which enables it to charge a
premium price. It could also be negative in that the company may have too high of a margin on its
sales, which could reduce overall net income.
38. To find the profit margin, we can solve the Du Pont identity. First, we need to find the retention
ratio. The retention ratio for the company is:
b = 1 0.25
b = 0.75
CHAPTER 3 B-46
Now, we can use the sustainable growth rate equation to find the ROE. Doing so, we find:
Sustainable growth rate = [(ROE)(b)] / [1 (ROE)(b)]
0.08 = [ROE(0.75)] / [1 ROE(0.75)]
ROE = 0.0988 or 9.88%
Now, we can use the Du Pont identity. We are given the total asset to sales ratio, which is the inverse
of the total asset turnover, and the equity multiplier is one plus the debt-equity ratio. Solving the Du
Pont identity for the profit margin, we find:
ROE = (Profit margin)(Total asset turnover)(Equity multiplier)
0.0988 = (Profit margin)(1 / 1.40)(1 + 0.45)
Profit margin = 0.0954 or 9.54%
39. The earnings per share is the net income divided by the shares outstanding. Since all numbers are in
millions, the earnings per share for Abercrombie & Fitch was:
EPS = $422.19 / 87.69
EPS = $4.81
And the earnings per share for Ann Taylor were;
EPS = $142.98 / 69.37
EPS = $2.06
The market-to-book ratio is the stock price divided by the book value per share. To find the book
value per share, we divide the total equity by the shares outstanding. The book value per share and
market-to-book ratio for Abercrombie & Fitch was:
Book value per share = $1,405.30 / 87.69
Book value per share = $16.03
Market-to-book = $80.77 / $16.03
Market-to-book = 5.04
And the market-to-book ratio for Ann Taylor was:
Book value per share = $1,049.91 / 69.37
Book value per share = $15.13
Market-to-book = $35.33 / $15.13
Market-to-book = 2.33
And the price-earnings ratio for Abercrombie & Fitch was:
P/E = $80.77 / $4.81
P/E = 16.78
CHAPTER 3 B-47
And for Ann Taylor, the P/E was:
P/E = $35.33 / $2.06
P/E = 17.14
40. To find the total asset turnover, we can solve the ROA equation. First, we need to find the retention
ratio. The retention ratio for the company is:
b = 1 0.35
b = 0.65
Now, we can use the internal growth rate equation to find the ROA. Doing so, we find:
Internal growth rate = [(ROA)(b)] / [1 (ROA)(b)]
0.07 = [ROA(0.65)] / [1 ROA(0.65)]
ROA = 0.1006 or 10.06%
Now, we can use the ROA equation to find the total asset turnover is:
ROA = (PM)(TAT)
0.1006 = (0.08)TAT
Total asset turnover = 1.26 times
41. To calculate the sustainable growth rate, we need to calculate the return on equity. We can use the
Du Pont identity to calculate the return on equity if we can find the equity multiplier. Using the total
debt ratio, we can find the debt-equity ratio is:
Total debt ratio = Total debt / Total assets
0.20 = Total debt / Total assets
1 / 0.20 = Total assets / Total debt
1 / 0.20 = (Total debt + Total equity) / Total debt
1 / 0.20 = 1 + Total equity / Total debt
Total equity / Total debt = (1 / 0.20) 1
Total debt / Total equity = 1 / [(1 /0.20) 1]
Total debt / Total equity = 0.25
Debt-equity ratio = 0.25
So, the equity multiplier is:
Equity multiplier = 1 + Debt-equity ratio
Equity multiplier = 1 + 0.25
Equity multiplier = 1.25
Using the Du Pont identity, the ROE is:
ROE = (Profit margin)(Total asset turnover)(Equity multiplier)
ROE = (.1050)(1.40)(1.25)
ROE = 0.1838 or 18.38%
CHAPTER 3 B-48
To calculate the sustainable growth rate, we also need the retention ratio. The retention ratio is:
b = 1 0.15
b = 0.85
Now we can calculate the sustainable growth rate as:
Sustainable growth rate = [(ROE)(b)] / [1 (ROE)(b)]
Sustainable growth rate = [0.1838(0.85)] / [1 0.1838(0.85)]
Sustainable growth rate = 0.1851 or 18.51%
And the return on assets is:
ROA = (Profit margin)(Total asset turnover)
ROA = (0.1050)(1.40)
ROA = 0.1470 or 14.70%
42. To find the sustainable growth rate, we need the retention ratio and the return on equity. The payout
ratio is the dividend payment divided by net income, so:
b = 1 ($4,500 / $19,000)
b = 0.7632
And the return on equity is:
ROE = Net income / Total equity
ROE = $19,000 / $72,000
ROE = 0.2639 or 26.39%
So, the sustainable growth rate is:
Sustainable growth rate = [(ROE)(b)] / [1 (ROE)(b)]
Sustainable growth rate = [0.2639(0.7632)] / [1 0.2639(0.7632)]
Sustainable growth rate = 0.2522 or 25.22%
The total assets of the company are equal to the total debt plus the total equity. The total assets will
increase at the sustainable growth rate, so the total assets next year will be:
New total assets = (1 + Sustainable growth rate)(Fixed assets)
New total assets = (1 + 0.2522)($72,000 + 49,000)
New total assets = $151,513.04
We can find the new total debt amount by multiplying the new total assets by the debt-equity ratio.
Doing so, we find the new total debt is:
New total debt = [Total debt / (Total debt + Total equity)](New total assets)
New total debt = [$49,000 / ($49,000 + 72,000)]($151,513.04)
New total debt = $61,356.52
CHAPTER 3 B-49
The additional borrowing is the difference between the new total debt and the current total debt, so:
Additional borrowing = New total debt Current total debt
Additional borrowing = $61,356.52 49,000
Additional borrowing = $12,356.52
The growth rate that could be achieved with no outside financing at all is the internal growth rate. To
find the internal growth rate we first need the return on assets, which is:
ROA = Net income / Total assets
ROA = $19,000 / ($49,000 + 72,000)
ROA = 0.1570 or 15.70%
So, the internal growth rate is:
Internal growth rate = [(ROA)(b)] / [1 (ROA)(b)]
Internal growth rate = [(0.1570)(0.7632)] / [1 (0.1570)(0.7632)]
Internal growth rate = .1362 or 13.62%
43. We can find the payout ratio from the sustainable growth rate formula. First, we need the return on
equity. Using the Du Pont identity, we find the return on equity is:
ROE = (Profit margin)(Total asset turnover)(Equity multiplier)
ROE = (0.06)(1.05)(1 + 0.40)
ROE = 0.0882 or 8.82%
Now we can use the sustainable growth rate equation to find the retention ratio, which is:
Sustainable growth rate = [(ROE)(b)] / [1 (ROE)(b)]
0.13 = [0.0882(b)] / [1 0.0882(b)]
b = 1.30
So, the payout ratio is:
Payout ratio = 1 b
Payout ratio = 1 1.30
Payout ratio = 0.30 or 30%
This is a negative dividend payout ratio of 130%, which is impossible; the growth rate is not
consistent with the other constraints. The lowest possible payout rate is zero, which corresponds to
retention ratio of one, or total earnings retention. The maximum sustainable growth rate for this
company is:
Sustainable growth rate = [(ROE)(b)] / [1 (ROE)(b)]
Sustainable growth rate = [0.0882(1)] / [1 0.0882(1)]
Sustainable growth rate = 0.0967 or 9.67%
CHAPTER 3 B-50
44. Using the beginning of period total assets, the ROA is:
ROABegin = $1,140 / $10,294
ROABegin = .1107 or 11.07%
Using the end of period total assets, the ROA is:
ROAEnd = $1,140 / $11,864
ROAEnd = .0961 or 9.61%
The ROE using beginning of period equity is:
ROEBegin = $1,140 / $4,449
ROEBegin = .2562 or 25.62%
The ROE using the end of period equity is:
ROEEnd = $1,140 / $5,257
ROEEnd = .2169 or 21.69%
The retention ratio, which is one minus the dividend payout ratio, is:
b = 1 Dividends/Net income
b = 1 $151 / $1,140
b = .8675 or 86.75%
With the growth rate equations, we need to use the ROA and ROE based on the end of period assets
or equity, so the internal growth rate is:
Internal growth rate = [(ROA)(b)] / [1 (ROA)(b)]
Internal growth rate = [(.0961)(.8675)] / [1 (.0961)(.8675)]
Internal growth rate = .0909 or 9.09%
And the sustainable growth rate is:
Sustainable growth rate = [(ROE)(b)] / [1 (ROE)(b)]
Sustainable growth rate = [(.2169)(.8675)] / [1 (.2169)(.8675)]
Sustainable growth rate = .2317 or 23.17%
Using ROA b and end of period assets to find the internal growth rate, we find:
Internal growth rate = ROAEnd b
Internal growth rate = .0961 .8675
Internal growth rate = .0834 or 8.34%
CHAPTER 3 B-51
And, using ROE b and the end of period equity to find the sustainable growth rate, we find:
Sustainable growth rate = ROEEnd b
Sustainable growth rate = .2169 .8675
Sustainable growth rate = .1881 or 18.81%
Using ROA b and beginning of period assets to find the internal growth rate, we find:
Internal growth rate = ROABegin b
Internal growth rate = .1107 .8675
Internal growth rate = .0961 or 9.61%
And, using ROE b and the beginning of period equity to find the sustainable growth rate, we find:
Sustainable growth rate = ROEBegin b
Sustainable growth rate = .2562 .8675
Sustainable growth rate = .2223 or 22.23%
45. The expanded Du Pont table is shown on the next page. The ROE is 81.80%.
Return on
equity
81.80%
Return on
assets
13.45%
Profit margin
11.31%
Net income
$559.061
divided by
multiplied
by
Equity
multiplier
6.083
multiplied by
Total asset turnover
1.189
Sales
$4,944.230
Sales
$4,944.230
divided by
Total assets
$4,157.565
Sales
$4,944.230
Fixed assets
$2,739.753
plus
Current assets
$1,417.812
subtracted from
Total costs
$4,385.169
Cost of goods sold
$3,076.718
Other expenses
$692.234
Cash
$97.141
Depreciation
$181.038
Interest
$117.738
Taxes
$317.441
Accounts rec.
$584.033
Inventory
$736.638
CHAPTER 4
INTRODUCTION TO VALUATION: THE
TIME VALUE OF MONEY
Answers to Concepts Review and Critical Thinking Questions
1.
Compounding refers to the growth of a dollar amount through time via reinvestment of interest
earned. It is also the process of determining the future value of an investment. Discounting is the
process of determining the value today of an amount to be received in the future.
2.
Future values grow (assuming a positive rate of return); present values shrink.
3.
The future value rises (assuming a positive rate of return); the present value falls.
4.
It depends. The large deposit will have a larger future value for some period, but after time, the
smaller deposit with the larger interest rate will eventually become larger. The length of time for the
smaller deposit to overtake the larger deposit depends on the amount deposited in each account and
the interest rates.
5.
It would appear to be both deceptive and unethical to run such an ad without a disclaimer or
explanation.
6.
Its a reflection of the time value of money. TMCC gets to use the $1,163. If TMCC uses it wisely, it
will be worth more than $10,000 in thirty years.
7.
This will probably make the security less desirable. TMCC will only repurchase the security prior to
maturity if it to its advantage, i.e. interest rates decline. Given the drop in interest rates needed to
make this viable for TMCC, it is unlikely the company will repurchase the security. This is an
example of a call feature. Such features are discussed at length in a later chapter.
8.
The key considerations would be: (1) Is the rate of return implicit in the offer attractive relative to
other, similar risk investments? and (2) How risky is the investment; i.e., how certain are we that we
will actually get the $10,000? Thus, our answer does depend on who is making the promise to repay.
9.
The Treasury security would have a somewhat higher price because the Treasury is the strongest of
all borrowers.
10. The price would be higher because, as time passes, the price of the security will tend to rise toward
$10,000. This rise is just a reflection of the time value of money. As time passes, the time until
receipt of the $10,000 grows shorter, and the present value rises. In 2015, the price will probably be
higher for the same reason. We cannot be sure, however, because interest rates could be much
higher, or TMCCs financial position could deteriorate. Either event would tend to depress the
securitys price.
CHAPTER 4 B-54
Solutions to Questions and Problems
NOTE: All end-of-chapter problems were solved using a spreadsheet. Many problems require multiple
steps. Due to space and readability constraints, when these intermediate steps are included in this
solutions manual, rounding may appear to have occurred. However, the final answer for each problem is
found without rounding during any step in the problem.
Basic
1.
The simple interest per year is:
$6,000 0.08 = $480
So, after 10 years, you will have:
$480 10 = $4,800 in interest.
The total balance will be $6,000 + 4,800 = $10,800
With compound interest, we use the future value formula:
FV = PV(1 +r)t
FV = $6,000(1.08)10 = $12,953.55
The difference is:
$12,953.55 10,800 = $2,153.55
2.
To find the FV of a lump sum, we use:
FV = PV(1 + r)t
FV = $3,150(1.18)5
FV = $8,453(1.06)10
FV = $89,305(1.11)17
FV = $227,382(1.05)20
3.
= $ 7,206.44
= $ 15,138.04
= $526,461.25
= $603,312.14
To find the PV of a lump sum, we use:
PV = FV / (1 + r)t
PV = $15,451 / (1.04)12
PV = $51,557 / (1.09)4
PV = $886,073 / (1.17)16
PV = $901,450 / (1.20)21
= $ 9,650.65
= $36,524.28
= $71,861.41
= $19,594.56
CHAPTER 4 B-55
4.
To answer this question, we can use either the FV or the PV formula. Both will give the same answer
since they are the inverse of each other. We will use the FV formula, that is:
FV = PV(1 + r)t
Solving for r, we get:
r = (FV / PV)1 / t 1
FV = $307 = $221(1 + r)6
r = ($307 / $221)1/6 1
r = 0.0563 or 5.63%
FV = $905 = $425(1 + r)7
r = ($905 / $425)1/7 1
r = 0.1140 or 11.40%
FV = $143,625 = $25,000(1 + r)18
r = ($143,625 / $25,000)1/18 1
r = 0.1020 or 10.20%
FV = $255,810 = $40,200(1 + r)21
r = ($255,810 / $40,200)1/21 1
r = 0.0921 or 9.21%
5.
To answer this question, we can use either the FV or the PV formula. Both will give the same answer
since they are the inverse of each other. We will use the FV formula, that is:
FV = PV(1 + r)t
Solving for t, we get:
t = ln(FV / PV) / ln(1 + r)
FV = $1,105 = $250 (1.08)t
t = ln($1,105 / $250) / ln 1.08
t = 19.31 years
FV = $3,700 = $1,941(1.05)t
t = ln($3,700 / $1,941) / ln 1.05
t = 13.22 years
FV = $387,120 = $32,805(1.14)t
t = ln($387,120 / $32,805) / ln 1.14
t = 18.84 years
CHAPTER 4 B-56
FV = $198,212 = $32,500(1.24)t
t = ln($198,212 / $32,500) / ln 1.24
t = 8.41 years
6.
To answer this question, we can use either the FV or the PV formula. Both will give the same answer
since they are the inverse of each other. We will use the FV formula, that is:
FV = PV(1 + r)t
Solving for r, we get:
r = (FV / PV)1 / t 1
r = ($290,000 / $35,000)1/18 1
r = 0.1247 or 12.47%
7.
To find the length of time for money to double, triple, etc., the present value and future value are
irrelevant as long as the future value is twice the present value for doubling, three times as large for
tripling, etc. To answer this question, we can use either the FV or the PV formula. Both will give the
same answer since they are the inverse of each other. We will use the FV formula, that is:
FV = PV(1 + r)t
Solving for t, we get:
t = ln(FV / PV) / ln(1 + r)
The length of time to double your money is:
FV = $2 = $1(1.09)t
t = ln 2 / ln 1.09
t = 8.04 years
The length of time to quadruple your money is:
FV = $4 = $1(1.09)t
t = ln 4 / ln 1.09
t = 16.09 years
Notice that the length of time to quadruple your money is twice as long as the time needed to double
your money (the slight difference in these answers is due to rounding). This is an important concept
of time value of money.
CHAPTER 4 B-57
8.
To answer this question, we can use either the FV or the PV formula. Both will give the same answer
since they are the inverse of each other. We will use the FV formula, that is:
FV = PV(1 + r)t
Solving for r, we get:
r = (FV / PV)1 / t 1
r = ($1,300 / $0.50)1/102 1
r = 0.0801 or 8.01%
9.
To answer this question, we can use either the FV or the PV formula. Both will give the same answer
since they are the inverse of each other. We will use the FV formula, that is:
FV = PV(1 + r)t
Solving for t, we get:
t = ln(FV / PV) / ln(1 + r)
FV = $160,000 = $30,000(1.047)t
t = ln($160,000 / $30,000) / ln 1.047
t = 36.45 years
10. To find the PV of a lump sum, we use:
PV = FV / (1 + r)t
PV = $800,000,000 / (1.08)20
PV = $171,638,566
11. To find the PV of a lump sum, we use:
PV = FV / (1 + r)t
PV = $2,000,000 / (1.11)80
PV = $473.36
12. To find the FV of a lump sum, we use:
FV = PV(1 + r)t
FV = $50(1.057)108
FV = $19,909.88
CHAPTER 4 B-58
13. To answer this question, we can use either the FV or the PV formula. Both will give the same answer
since they are the inverse of each other. We will use the FV formula, that is:
FV = PV(1 + r)t
Solving for r, we get:
r = (FV / PV)1 / t 1
r = ($1,225,000 / $150)1/111 1
r = 0.0845 or 8.45%
To find what the check will be in 2040, we use the FV of a lump sum, so:
FV = PV(1 + r)t
FV = $1,225,000(1.0845)34
FV = $19,338,380.03
14. To answer this question, we can use either the FV or the PV formula. Both will give the same answer
since they are the inverse of each other. We will use the FV formula, that is:
FV = PV(1 + r)t
Solving for r, we get:
r = (FV / PV)1 / t 1
r ($9,000 = / $3)1/127 1
r = .0651 or 6.51%
15. To answer this question, we can use either the FV or the PV formula. Both will give the same answer
since they are the inverse of each other. We will use the FV formula, that is:
FV = PV(1 + r)t
Solving for r, we get:
r = (FV / PV)1 / t 1
r = ($10,311,500 / $12,377,500)1/4 1
r = .0446 or 4.46%
CHAPTER 4 B-59
Intermediate
16. a. To answer this question, we can use either the FV or the PV formula. Both will give the same
answer since they are the inverse of each other. We will use the FV formula, that is:
FV = PV(1 + r)t
Solving for r, we get:
r = (FV / PV)1 / t 1
r = ($10,000 / $1,163)1 /30 1
r = 0.0744 or 7.44%
b. Using the FV formula and solving for the interest rate, we get:
r = (FV / PV)1 / t 1
r = ($2,500 / $1,163)1 /9 1
r = 0.0888 or 8.88%
c. Using the FV formula and solving for the interest rate, we get:
r = (FV / PV)1 / t 1
r = ($10,000 / $2,500)1 /21 1
r = 0.0682 or 6.82%
17. To find the PV of a lump sum, we use:
PV = FV / (1 + r)t
PV = $160,000 / (1.1075)10
PV = $57,634.51
18. To find the FV of a lump sum, we use:
FV = PV(1 + r)t
FV = $5,000(1.11)45
FV = $547,651.21
If you wait 10 years, the value of your deposit at your retirement will be:
FV = $5,000(1.11)35
FV = $192,874.26
Better start early!
CHAPTER 4 B-60
19. Even though we need to calculate the value in eight years, we will only have the money for six years,
so we need to use six years as the number of periods. To find the FV of a lump sum, we use:
FV = PV(1 + r)t
FV = $15,000(1.08)6
FV = $23,803.11
20. To answer this question, we can use either the FV or the PV formula. Both will give the same answer
since they are the inverse of each other. We will use the FV formula, that is:
FV = PV(1 + r)t
$160,000 = $30,000(1.09)t
t = ln($160,000 / $30,000) / ln 1.09
t = 19.42 years
From now, youll wait 2 + 19.42 = 21.42 years
21. To find the FV of a lump sum, we use:
FV = PV(1 + r)t
In Regency Bank, you will have:
FV = $7,000(1.01)240
FV = $76,247.88
And in King Bank, you will have:
FV = $7,000(1.12)20
FV = $67,524.05
22. To find the length of time for money to double, triple, etc., the present value and future value are
irrelevant as long as the future value is twice the present value for doubling, three times as large for
tripling, etc. We also need to be careful about the number of periods. Since the length of the
compounding is three months and we have 24 months, there are eight compounding periods. To
answer this question, we can use either the FV or the PV formula. Both will give the same answer
since they are the inverse of each other. We will use the FV formula, that is:
FV = PV(1 + r)t
Solving for r, we get:
r = (FV / PV)1 / t 1
r = ($3 / $1)1/8 1
r = 0.1472 or 14.72%
CHAPTER 4 B-61
23. To answer this question, we can use either the FV or the PV formula. Both will give the same answer
since they are the inverse of each other. We will use the FV formula, that is:
FV = PV(1 + r)t
$3,600 = $1,500(1.0045)t
t = ln($3,600 / $1,500) / ln 1.0045
t = 194.99 months
24. To find the PV of a lump sum, we use:
PV = FV / (1 + r)t
PV = $75,000 / (1.0055)120
PV = $38,834.01
25. To find the PV of a lump sum, we use:
PV = FV / (1 + r)t
So, if you can earn 11 percent, you will need to invest:
PV = $1,000,000 / (1.12)45
PV = $6,098.02
And if you can earn 5 percent, you will need to invest:
PV = $1,000,000 / (1.06)45
PV = $72,650.07
Challenge
26. In this case, we have an investment that earns two different interest rates. We will calculate the value
of the investment at the end of the first 20 years then use this value with the second interest rate to
find the final value at the end of 40 years. Using the future value equation, at the end of the first 20
years, the account will be worth:
Value in 20 years = PV(1 + r)t
Value in 20 years = $10,000(1.08)20
Value in 20 years = $46,609.57
Now we can find out how much this will be worth 20 years later at the end of the investment. Using
the future value equation, we find:
Value in 40 years = PV(1 + r)t
Value in 40 years = $46,609.57(1.12)20
Value in 40 years = $449,609.59
CHAPTER 4 B-62
It is irrelevant which interest rate is offered when as long as each interest rate is offered for 20 years.
We can find the value of the initial investment in 40 years with the following:
FV = PV(1 + r1)t (1 + r2)t
FV = $10,000(1.08)20(1.12)20
FV = $449,609.59
With the commutative property of multiplication, it does not matter which order the interest rates
occur, the final value will always be the same.
Calculator Solutions
1.
Enter
8%
10
N
$6,000
I/Y
PV
PM
FV
T
Solvefor
$12,953.55
$6,953.55 10($480) = $2,153.55
2.
Enter
5
18%
N
I/Y
$3,150
PV
PM
FV
T
Solvefor
Enter
$7,206.44
6%
10
N
$8,453
I/Y
PV
PM
FV
T
Solvefor
Enter
$15,138.04
17
11%
N
I/Y
$89,305
PV
PM
FV
T
Solvefor
Enter
$526,461.25
20
5%
N
I/Y
$227,382
PV
PM
FV
T
CHAPTER 4 B-63
Solvefor
3.
Enter
$603,312.14
12
4%
N
I/Y
$15,451
PV
PM
FV
T
Solvefor
$9,650.65
CHAPTER 4 B-64
Enter
4
9%
N
I/Y
$51,557
PV
PM
FV
T
Solvefor
Enter
$36,524.28
16
17%
N
I/Y
$886,073
PV
PM
FV
T
Solvefor
Enter
$71,861.41
21
20%
N
I/Y
$901,450
PV
PM
FV
T
Solvefor
4.
Enter
$19,594.56
$221
6
N
I/Y
PV
$307
PM
FV
T
Solvefor
Enter
5.63%
$425
7
N
I/Y
PV
$905
PM
FV
T
Solvefor
Enter
11.40%
$25,000
18
N
I/Y
PV
$143,625
PM
FV
T
Solvefor
Enter
10.20%
$40,200
21
N
I/Y
PV
$255,810
PM
FV
CHAPTER 4 B-65
T
Solvefor
9.21%
5.
Enter
8%
N
$250
I/Y
PV
$1,105
PM
FV
T
Solvefor
19.31
Enter
5%
N
$1,941
I/Y
PV
$3,700
PM
FV
T
Solvefor
13.22
CHAPTER 4 B-66
Enter
14%
N
I/Y
$32,805
PV
$387,120
PM
FV
T
Solvefor
18.84
Enter
24%
N
I/Y
$32,500
PV
$198,212
PM
FV
T
Solvefor
6.
Enter
8.81
$35,000
18
N
I/Y
PV
$290,000
PM
FV
T
Solvefor
12.47%
7.
Enter
9%
N
I/Y
$1
PV
$2
PM
FV
T
Solvefor
8.04
Enter
9%
N
I/Y
$1
PV
$4
PM
FV
T
Solvefor
8.
Enter
16.09
$0.50
102
N
I/Y
PV
$1,300
PM
FV
T
Solvefor
8.01%
9.
Enter
4.7%
N
I/Y
$30,000
PV
$160,000
PM
FV
CHAPTER 4 B-67
T
Solvefor
10.
Enter
36.45
20
8%
N
I/Y
$800,000,000
PV
PM
FV
T
Solvefor
11.
Enter
$171,638,566
80
11%
N
I/Y
$2,000,000
PV
PM
FV
T
Solvefor
$473.36
CHAPTER 4 B-68
12.
Enter
108
N
5.70%
I/Y
$50
PV
PM
FV
T
Solvefor
13.
Enter
$19,909.88
$150
111
N
I/Y
PV
$1,225,000
PM
FV
T
Solvefor
Enter
8.45%
34
8.45%
N
I/Y
$1,225,000
PV
PM
FV
T
Solvefor
14.
Enter
$19,338,380
$3
127
N
I/Y
PV
$9,000
PM
FV
T
Solvefor
15.
Enter
6.51%
$12,377,500
4
N
I/Y
PV
$10,311,500
PM
FV
T
Solvefor
16. a.
Enter
4.46%
$1,163
30
N
I/Y
PV
$10,000
PM
FV
T
Solvefor
b.
Enter
7.44%
$1,163
9
N
I/Y
PV
$2,500
PM
FV
CHAPTER 4 B-69
T
Solvefor
c.
Enter
8.88%
$2,500
21
N
I/Y
PV
$10,000
PM
FV
T
Solvefor
17.
Enter
6.82%
10
10.75%
N
I/Y
$160,000
PV
PM
FV
T
Solvefor
$57,634.51
CHAPTER 4 B-70
18.
Enter
45
11%
N
I/Y
$5,000
PV
PM
FV
T
Solvefor
Enter
$547,651.21
35
11%
N
I/Y
$5,000
PV
PM
FV
T
Solvefor
19.
Enter
$192,874.26
6
8%
N
I/Y
$15,000
PV
PM
FV
T
Solvefor
$23,803.11
20.
Enter
9%
N
I/Y
$30,000
PV
$160,000
PM
FV
T
Solvefor
19.42
You must wait 2 + 19.42 = 21.42 years.
21.
Enter
240
N
1%
$7,000
I/Y
PV
PM
FV
T
Solvefor
Enter
$76,247.88
20
12%
N
I/Y
$7,000
PV
PM
FV
T
Solvefor
22.
Enter
$67,524.05
8
$1
$3
CHAPTER 4 B-71
N
I/Y
PV
PM
FV
T
Solvefor
14.72%
23.
Enter
0.45%
N
I/Y
$1,500
PV
$3,600
PM
FV
T
Solvefor
24.
Enter
194.99
120
N
0.55%
I/Y
$75,000
PV
PM
FV
T
Solvefor
$38,834.01
CHAPTER 4 B-72
25.
Enter
45
12%
N
I/Y
$1,000,000
PV
PM
FV
T
Solvefor
Enter
$6,098.02
45
6%
N
I/Y
$1,000,000
PV
PM
FV
T
Solvefor
26.
Enter
$72,650.07
20
8%
N
I/Y
$10,000
PV
PM
FV
T
Solvefor
Enter
$46,609.57
20
12%
N
I/Y
$46,609.57
PV
PM
FV
T
Solvefor
Enter
$449,609.59
1
N
I/Y
$10,440
PV
$12,000
FV
PM
T
CHAPTER 5
DISCOUNTED CASH FLOW VALUATION
Answers to Concepts Review and Critical Thinking Questions
1.
Assuming positive cash flows and a positive interest rate, both the present and the future value will
rise.
2.
Assuming positive cash flows and a positive interest rate, the present value will fall, and the future
value will rise.
3.
Its deceptive, but very common. The deception is particularly irritating given that such lotteries are
usually government sponsored!
4.
The most important consideration is the interest rate the lottery uses to calculate the lump sum
option. If you can earn an interest rate that is higher than you are being offered, you can create larger
annuity payments. Of course, taxes are also a consideration, as well as how badly you really need $5
million today.
5.
If the total money is fixed, you want as much as possible as soon as possible. The team (or, more
accurately, the team owner) wants just the opposite.
6.
The better deal is the one with equal installments.
7.
Yes, they should. APRs generally dont provide the relevant rate. The only advantage is that they are
easier to compute, but, with modern computing equipment, that advantage is not very important.
8.
A freshman does. The reason is that the freshman gets to use the money for much longer before
interest starts to accrue.
9.
The subsidy is the present value (on the day the loan is made) of the interest that would have accrued
up until the time it actually begins to accrue.
10. The problem is that the subsidy makes it easier to repay the loan, not obtain it. However, the ability
to repay the loan depends on future employment, not current need. For example, consider a student
who is currently needy, but is preparing for a career in a high-paying area (such as corporate
finance!). Should this student receive the subsidy? How about a student who is currently not needy,
but is preparing for a relatively low-paying job (such as becoming a college professor)?
CHAPTER 5 B-74
Solutions to Questions and Problems
NOTE: All end-of-chapter problems were solved using a spreadsheet. Many problems require multiple
steps. Due to space and readability constraints, when these intermediate steps are included in this
solutions manual, rounding may appear to have occurred. However, the final answer for each problem is
found without rounding during any step in the problem.
Basic
1.
To solve this problem, we must find the PV of each cash flow and add them. To find the PV of a
lump sum, we use:
PV = FV / (1 + r)t
PV@10% = $800 / 1.10 + $500 / 1.102 + $1,300 / 1.103 + $1,480 / 1.104 = $3,128.07
PV@18% = $800 / 1.18 + $500 / 1.182 + $1,300 / 1.183 + $1,480 / 1.184 = $2,591.65
PV@24% = $800 / 1.24 + $500 / 1.242 + $1,300 / 1.243 + $1,480 / 1.244 = $2,278.18
2.
To find the PVA, we use the equation:
PVA = C({1 [1/(1 + r)]t } / r )
At a 6 percent interest rate:
X@5%: PVA = $5,500{[1 (1/1.06)9 ] / .06 } = $37,409.31
Y@5%: PVA = $8,000{[1 (1/1.06)5 ] / .06 } = $33,698.91
And at a 22 percent interest rate:
X@22%: PVA = $5,500{[1 (1/1.22)9 ] / .22 } = $20,824.57
Y@22%: PVA = $8,000{[1 (1/1.22)5 ] / .22 } = $22,909.12
Notice that the PV of Investment X has a greater PV at a 6 percent interest rate, but a lower PV at a
22 percent interest rate. The reason is that X has greater total cash flows. At a lower interest rate, the
total cash flow is more important since the cost of waiting (the interest rate) is not as great. At a
higher interest rate, Y is more valuable since it has larger annual payments. At a higher interest rate,
getting these payments early are more important since the cost of waiting (the interest rate) is so
much greater.
CHAPTER 5 B-75
3.
To solve this problem, we must find the FV of each cash flow and sum. To find the FV of a lump
sum, we use:
FV = PV(1 + r)t
FV@8% = $700(1.08)3 + $900(1.08)2 + $1,400(1.08) + $2,000 = $5,443.56
FV@11% = $700(1.11)3 + $900(1.11)2 + $1,400(1.11) + $2,000 = $5,620.23
FV@24% = $700(1.24)3 + $900(1.24)2 + $1,400(1.24) + $2,000 = $6,454.48
Notice, since we are finding the value at Year 4, the cash flow at Year 4 is simply added to the FV of
the other cash flows. In other words, we do not need to compound this cash flow.
4.
To find the PVA, we use the equation:
PVA = C({1 [1/(1 + r)]t } / r )
PVA@15 yrs:
PVA = $7,000{[1 (1/1.09)15 ] / .09} = $56,424.82
PVA@40 yrs:
PVA = $7,000{[1 (1/1.09)40 ] / .09} = $75,301.52
PVA@75 yrs:
PVA = $7,000{[1 (1/1.09)75 ] / .09} = $77,656.48
To find the PV of a perpetuity, we use the equation:
PV = C / r
PV = $7,000 / .09
PV = $77,777.78
Notice that as the length of the annuity payments increases, the present value of the annuity
approaches the present value of the perpetuity. The present value of the 75-year annuity and the
present value of the perpetuity imply that the value today of all perpetuity payments beyond 75 years
is only $121.30.
5.
Here we have the PVA, the length of the annuity, and the interest rate. We want to calculate the
annuity payment. Using the PVA equation:
PVA = C({1 [1/(1 + r)]t } / r )
PVA = $20,000 = $C{[1 (1/1.085)12 ] / .085}
We can now solve this equation for the annuity payment. Doing so, we get:
C = $20,000 / 9.46334
C = $2,723.06
CHAPTER 5 B-76
6.
To find the PVA, we use the equation:
PVA = C({1 [1/(1 + r)]t } / r )
PVA = $50,000{[1 (1/1.0825)9 ] / .0825}
PVA = $309,123.20
The present value of the revenue is greater than the cost, so your company can afford the equipment.
7.
Here we need to find the FVA. The equation to find the FVA is:
FVA = C{[(1 + r)t 1] / r}
FVA for 20 years = $3,000[(1.09520 1) / .095]
FVA for 20 years = $162,366.70
FVA for 40 years = $3,000[(1.09540 1) / .095]
FVA for 40 years = $1,159,559.98
Notice that doubling the number of periods more than doubles the FVA.
8.
Here we have the FVA, the length of the annuity, and the interest rate. We want to calculate the
annuity payment. Using the FVA equation:
FVA = C{[(1 + r)t 1] / r}
$30,000 = $C[(1.05258 1) / .0525]
We can now solve this equation for the annuity payment. Doing so, we get:
C = $30,000 / 9.63492
C = $3,113.68
9.
Here we have the PVA, the length of the annuity, and the interest rate. We want to calculate the
annuity payment. Using the PVA equation:
PVA = C({1 [1/(1 + r)]t } / r)
$50,000 = C{[1 (1/1.08)7 ] / .08}
We can now solve this equation for the annuity payment. Doing so, we get:
C = $50,000 / 5.20637
C = $9,603.62
CHAPTER 5 B-77
10. This cash flow is a perpetuity. To find the PV of a perpetuity, we use the equation:
PV = C / r
PV = $25,000 / .07 = $357,142.86
11. Here we need to find the interest rate that equates the perpetuity cash flows with the PV of the cash
flows. Using the PV of a perpetuity equation:
PV = C / r
$400,000 = $25,000 / r
We can now solve for the interest rate as follows:
r = $25,000 / $400,000
r = .0625 or 6.25%
12. For discrete compounding, to find the EAR, we use the equation:
EAR = [1 + (APR / m)]m 1
EAR = [1 + (.09 / 4)]4 1
= .0931 or 9.31%
EAR = [1 + (.13 / 12)]12 1
= .1380 or 13.80%
EAR = [1 + (.16 / 365)]365 1 = .1735 or 17.35%
EAR = [1 + (.19 / 2)]2 1
= .1990 or 19.90%
13. Here we are given the EAR and need to find the APR. Using the equation for discrete compounding:
EAR = [1 + (APR / m)]m 1
We can now solve for the APR. Doing so, we get:
APR = m[(1 + EAR)1/m 1]
EAR = .10 = [1 + (APR / 2)]2 1
APR = 2[(1.10)1/2 1]
= 9.76%
EAR = .14 = [1 + (APR / 12)]12 1
APR = 12[(1.14)1/12 1]
= 13.17%
EAR = .09 = [1 + (APR / 52)]52 1
APR = 52[(1.09)1/52 1]
= 8.62%
EAR = .16 = [1 + (APR / 365)]365 1
APR = 365[(1.16)1/365 1]
= 14.85%
CHAPTER 5 B-78
14. For discrete compounding, to find the EAR, we use the equation:
EAR = [1 + (APR / m)]m 1
So, for each bank, the EAR is:
First National: EAR = [1 + (.124 / 12)]12 1 = .1313 or 13.13%
First United:
EAR = [1 + (.127 / 2)]2 1 = .1310 or 13.10%
For a borrower, First United would be preferred since the EAR of the loan is lower. Notice that the
higher APR does not necessarily mean the higher EAR. The number of compounding periods within
a year will also affect the EAR.
15. The reported rate is the APR, so we need to convert the EAR to an APR as follows:
EAR = [1 + (APR / m)]m 1
APR = m[(1 + EAR)1/m 1]
APR = 365[(1.18)1/365 1] = .1656 or 16.56%
This is deceptive because the borrower is actually paying annualized interest of 18% per year, not
the 16.56% reported on the loan contract.
16. For this problem, we simply need to find the FV of a lump sum using the equation:
FV = PV(1 + r)t
It is important to note that compounding occurs semiannually. To account for this, we will divide the
interest rate by two (the number of compounding periods in a year), and multiply the number of
periods by two. Doing so, we get:
FV = $1,280[1 + (.11/2)]26
FV = $5,149.61
17. For this problem, we simply need to find the FV of a lump sum using the equation:
FV = PV(1 + r)t
It is important to note that compounding occurs daily. To account for this, we will divide the interest
rate by 365 (the number of days in a year, ignoring leap year), and multiply the number of periods by
365. Doing so, we get:
FV in 5 years = $5,000[1 + (.045/365)]5(365) = $6,261.53
FV in 10 years = $5,000[1 + (.045/365)]10(365) = $7,841.34
FV in 20 years = $5,000[1 + (.045/365)]20(365) = $12,297.33
CHAPTER 5 B-79
18. For this problem, we simply need to find the PV of a lump sum using the equation:
PV = FV / (1 + r)t
It is important to note that compounding occurs on a daily basis. To account for this, we will divide
the interest rate by 365 (the number of days in a year, ignoring leap year), and multiply the number
of periods by 365. Doing so, we get:
PV = $75,000 / [(1 + .12/365)6(365)]
PV = $36,510.74
19. The APR is simply the interest rate per period times the number of periods in a year. In this case, the
interest rate is 20 percent per month, and there are 12 months in a year, so we get:
APR = 12(20%)
APR = 240%
To find the EAR, we use the EAR formula:
EAR = [1 + (APR / m)]m 1
EAR = (1 + .20)12 1
EAR = 7.9161 or 791.61%
Notice that we didnt need to divide the APR by the number of compounding periods per year. We
do this division to get the interest rate per period, but in this problem we are already given the
interest rate per period.
20. We first need to find the annuity payment. We have the PVA, the length of the annuity, and the
interest rate. Using the PVA equation:
PVA = C({1 [1/(1 + r)]t } / r)
$69,500 = $C[1 {1 / [1 + (.086/12)]60} / (.086/12)]
Solving for the payment, we get:
C = $69,500 / 48.62687
C = $1,429.25
To find the EAR, we use the EAR equation:
EAR = [1 + (APR / m)]m 1
EAR = [1 + (.086 / 12)]12 1
EAR = .0895 or 8.95%
CHAPTER 5 B-80
21. Here we need to find the length of an annuity. We know the interest rate, the PV, and the payments.
Using the PVA equation:
PVA = C({1 [1/(1 + r)]t } / r)
$13,850 = $500{ [1 (1/1.011)t ] / .011}
Now we solve for t:
1/1.011t = 1 [($13,850)(.011) / ($500)]
1.011t = 1/(0.6953) = 1.4382
t = ln 1.4382 / ln 1.011
t = 33.22 months
22. Here we are trying to find the interest rate when we know the PV and FV. Using the FV equation:
FV = PV(1 + r)
$5 = $4(1 + r)
r = $5/$4 1
r = .2500 or 25.00% per week
The interest rate is 25.00% per week. To find the APR, we multiply this rate by the number of weeks
in a year, so:
APR = (52)25.00% = 1,300.00%
And using the equation to find the EAR, we find:
EAR = [1 + (APR / m)]m 1
EAR = [1 + .2500]52 1
EAR = 109,475.4425 or 10,947,544.25%
23. Here we need to find the interest rate that equates the perpetuity cash flows with the PV of the cash
flows. Using the PV of a perpetuity equation:
PV = C / r
$160,000 = $2,500 / r
We can now solve for the interest rate as follows:
r = $2,500 / $160,000
r = .0156 or 1.56% per month
The interest rate is 1.56% per month. To find the APR, we multiply this rate by the number of
months in a year, so:
APR = (12)1.56%
APR = 18.75%
CHAPTER 5 B-81
And using the equation to find the EAR, we find:
EAR = [1 + (APR / m)]m 1
EAR = [1 + .0156]12 1
EAR = .2045 or 20.45%
24. This problem requires us to find the FVA. The equation to find the FVA is:
FVA = C{[(1 + r)t 1] / r}
FVA = $300[{[1 + (.12/12) ]360 1} / (.12/12)]
FVA = $1,048,489.24
25. In the previous problem, the cash flows are monthly and the compounding period is monthly. This
assumption still holds. Since the cash flows are annual, we need to use the EAR to calculate the
future value of annual cash flows. It is important to remember that you have to make sure the
compounding periods of the interest rate times with the cash flows. In this case, we have annual cash
flows, so we need the EAR since it is the true annual interest rate you will earn. So, finding the EAR:
EAR = [1 + (APR / m)]m 1
EAR = [1 + (.12/12)]12 1
EAR = .1268 or 12.68%
Using the FVA equation, we get:
FVA = C{[(1 + r)t 1] / r}
FVA = $3,600[(1.126830 1) / .1157]
FVA = $992,065.28
26. The cash flows are simply an annuity with four payments per year for four years, or 16 payments.
We can use the PVA equation:
PVA = C({1 [1/(1 + r)]t } / r)
PVA = $2,200{[1 (1/1.009)16] / .009}
PVA = $32,646.61
27. To solve this problem, we must find the PV of each cash flow and add them. To find the PV of a
lump sum, we use:
PV = FV / (1 + r)t
PV = $300 / 1.095 + $900 / 1.0952 + $700 / 1.0953 + $600 / 1.0954
PV = $1,975.08
CHAPTER 5 B-82
28. To solve this problem, we must find the PV of each cash flow and add them. To find the PV of a
lump sum, we use:
PV = FV / (1 + r)t
PV = $2,400 / 1.0816 + $3,200 / 1.08162 + $6,800 / 1.08163 + $8,100 / 1.08164
PV = $16,247.04
Intermediate
29. The total interest paid by First Simple Bank is the interest rate per period times the number of
periods. In other words, the interest by First Simple Bank paid over 10 years will be:
.07(10) = .7
First Complex Bank pays compound interest, so the interest paid by this bank will be the FV factor
of $1, or:
(1 + r)10
Setting the two equal, we get:
(.07)(10) = (1 + r)10 1
r = 1.71/10 1
r = .0545 or 5.45%
30. We need to use the PVA due equation, which is:
PVAdue = (1 + r) PVA
Using this equation:
PVAdue = $48,000 = [1 + (.0745/12)] C[{1 1 / [1 + (.0745/12)]60} / (.0745/12)
$47,703.84 = $C{1 [1 / (1 + .0745/12)60]} / (.0745/12)
C = $954.75
Notice, when we find the payment for the PVA due, we simply discount the PV of the annuity due
back one period. We then use this value as the PV of an ordinary annuity.
31. Here we need to find the FV of a lump sum, with a changing interest rate. We must do this problem
in two parts. After the first six months, the balance will be:
FV = $10,000 [1 + (.015/12)]6
FV = $10,075.23
CHAPTER 5 B-83
This is the balance in six months. The FV in another six months will be:
FV = $10,075.23 [1 + (.18/12)]6
FV = $11,016.70
The problem asks for the interest accrued, so, to find the interest, we subtract the beginning balance
from the principal. The interest accrued is:
Interest = $11,016.70 10,000.00
Interest = $1,016.70
32. We will calculate the time we must wait if we deposit in the bank that pays simple interest. The
interest amount we will receive each year in this bank will be:
Interest = $89,000(.05)
Interest = $4,450 per year
The deposit will have to increase by the difference between the amount we need by the amount we
originally deposit with divided by the interest earned per year, so the number of years it will take in
the bank that pays simple interest is:
Years to wait = ($175,000 89,000) / $4,450
Years to wait = 19.33 years
To find the number of years it will take in the bank that pays compound interest, we can use the
future value equation for a lump sum and solve for the periods. Doing so, we find:
FV = PV(1 + r)t
$175,000 = $89,000 [1 + (.05/12)]t
t = 162.61 months or 13.55 years
33. Here we need to find the future value of a lump sum. We need to make sure to use the correct
number of periods. So, the future value after one year will be:
FV = PV(1 + r)t
FV = $1(1.0105)12
FV = $1.13
And the future value after two years will be:
FV = PV(1 + r)t
FV = $1(1.0105)24
FV = $1.28
CHAPTER 5 B-84
34. Here we are given the PVA, number of periods, and the amount of the annuity. We need to solve for
the interest rate. Even though the currency is pounds and not dollars, we can still use the same time
value equations. Using the PVA equation:
PVA = C({1 [1/(1 + r)]t } / r)
440 = 60[{1 [1 / (1 + r)]31}/ r]
To find the interest rate, we need to solve this equation on a financial calculator, using a spreadsheet,
or by trial and error. If you use trial and error, remember that increasing the interest rate decreases
the PVA, and decreasing the interest rate increases the PVA. Using a spreadsheet, we find:
r = 13.36%
Not bad for an English Literature major!
35. Here we need to compare two cash flows. The only way to compare cash flows is to find the value of
the cash flows at a common time, so we will find the present value of each cash flow stream. Since
the cash flows are monthly, we need to use the monthly interest rate, which is:
Monthly rate = .07 / 12
Monthly rate = .0058 or .58%
The value today of the $6,200 monthly salary is:
PVA = C({1 [1/(1 + r)]t } / r)
PVA = $7,500{[1 (1 / 1.0058)24 ] / .0058}
PVA = $167,513.24
To find the value of the second option, we find the present value of the monthly payments and add
the bonus. We can add the bonus since it is paid today. So:
PVA = C({1 [1/(1 + r)]t } / r)
PVA = $6,000{[1 (1/1.0058)24] / .0058}
PVA = $134,010.60
So, the total value of the second option is:
Value of second option = $134,010.60 + 30,000
Value of second option = $164,010.60
The difference in the value of the two options today is:
Difference in value today = $167,513.24 164,010.60
Difference in value today = $3,502.65
CHAPTER 5 B-85
What if we found the future value of the two cash flows? For the annual salary, the future value will
be:
FVA = C{[(1 + r)t 1] / r}
FVA = $7,500{[(1 + .0058)24 1] / .0058}
FVA = $192,607.74
To find the future value of the second option we also need to find the future value of the bonus as
well. So, the future value of this option is:
FV = C{[(1 + r)t 1] / r} + PV(1 + r)t
FV = $6,000{[(1 + .0058)24 1] / .0058} + $30,000(1 + .0058)24
FV = $188,580.37
So, the first option is still the better choice. The difference between the two options now is:
Difference in future value = $192,607.74 188,580.37
Difference in future value = $4,027.37
No matter when you compare two cash flows, the cash flow with the greatest value on one period
will always have the greatest value in any other period. Heres a question for you: What is the future
value of $3,502.65 (the difference in the cash flows at time zero) in 24 months at an interest rate
of .58 percent per month? With no calculations, you know the future value must be $4,027.37, the
difference in the cash flows at the same time!
36. The cash flows are an annuity, so we can use the present value of an annuity equation. Doing so, we
find:
PVA = C({1 [1/(1 + r)]t } / r)
PVA = $15,000[1 (1/1.13)20 / .13]
PVA = $105,371.27
37. The investment we should choose is the investment with the higher rate of return. We will use the
future value equation to find the interest rate for each option. Doing so, we find the return for
Investment G is:
FV = PV(1 + r)t
$120,000 = $75,000(1 + r)6
r = ($120,000/$75,000)1/6 1
r = .0815 or 8.15%
And, the return for Investment H is:
FV = PV(1 + r)t
$220,000 = $75,000(1 + r)13
r = ($220,000/$75,000)1/13 1
r = .0863 or 8.63%
So, we should choose Investment H.
CHAPTER 5 B-86
38. The present value of an annuity falls as r increases, and the present value of an annuity rises as r
decreases. The future value of an annuity rises as r increases, and the future value of an annuity falls
as r decreases.
Here we need to calculate the present value of an annuity for different interest rates. Using the
present value of an annuity equation and an interest rate of 10 percent, we get:
PVA = C({1 [1/(1 + r)]t } / r)
PVA = $10,000{[1 (1/1.10)10] / .10 }
PVA = $61,445.67
At an interest rate of 5 percent, the present value of the annuity is:
PVA = C({1 [1/(1 + r)]t } / r)
PVA = $10,000{[1 (1/1.05)10] / .05 }
PVA = $77,217.35
And, at an interest rate of 15 percent, the present value of the annuity is:
PVA = C({1 [1/(1 + r)]t } / r)
PVA = $10,000{[1 (1/1.15)10] / .15 }
PVA = $50,187.69
39. Here we are given the future value of an annuity, the interest rate, and the number of payments. We
need to find the number of periods of the annuity payments. So, we can solve the future value of an
annuity equation for the number of periods as follows:
FVA = C{[(1 + r)t 1] / r}
$50,000 = $250[{[1 + (.10/12)]t 1 } / (.10/12) ]
200 = {[1 + (.10/12)]t 1 } / (.10/12)
1.667 = (1 + .0083)t 1
2.667 = (1.0083)t
ln 2.667 = t ln 1.0083
t = ln 2.667 / ln 1.0083
t = 118.19 payments
40. Here we are given the PVA, number of periods, and the amount of the annuity. We need to solve for
the interest rate. Using the PVA equation:
PVA = C({1 [1/(1 + r)]t } / r)
$75,000 = $1,600[{1 [1 / (1 + r)]60}/ r]
CHAPTER 5 B-87
To find the interest rate, we need to solve this equation on a financial calculator, using a spreadsheet,
or by trial and error. If you use trial and error, remember that increasing the interest rate decreases
the PVA, and decreasing the interest rate increases the PVA. Using a spreadsheet, we find:
r = .00848 or .848%
This is the monthly interest rate. To find the APR with a monthly interest rate, we simply multiply
the monthly rate by 12, so the APR is:
APR = .00848 12
APR = .1018 or 10.18%
41. To solve this problem, we must find the PV of each cash flow and add them. To find the PV of a
lump sum, we use:
PV = FV / (1 + r)t
PV = $24,000,000 + $24,000,000/1.11 + $24,000,000/1.112 + $24,000,000/1.113 + $27,000,000/1.114
+ $25,000,000/1.115 + $26,000,000/1.116 + $26,000,000/1.117 + $26,000,000/1.118
+ $26,000,000/1.119
PV = $163,141,086.06
42. To solve this problem, we must find the PV of each cash flow and add them. To find the PV of a
lump sum, we use:
PV = FV / (1 + r)t
PV = $16,000,000 + $18,500,000/1.11 + $21,000,000/1.112 + $23,500,000/1.113 + $20,000,000/1.114
+ $20,000,000/1.115 + $18,000,000/1.116 + $20,000,000/1.117
PV = $113,170,277.46
43. Here we are given the PVA, number of periods, and the amount of the annuity. We need to solve for
the interest rate. First, we need to find the amount borrowed since it is only 80 percent of the
building value. So, the amount borrowed is:
Amount borrowed = .80($2,500,000)
Amount borrowed = $2,000,000
Now we can use the PVA equation:
PVA = C({1 [1/(1 + r)]t } / r)
$2,000,000 = $13,400[{1 [1 / (1 + r)]360}/ r]
CHAPTER 5 B-88
To find the interest rate, we need to solve this equation on a financial calculator, using a spreadsheet,
or by trial and error. If you use trial and error, remember that increasing the interest rate decreases
the PVA, and decreasing the interest rate increases the PVA. Using a spreadsheet, we find:
r = .00589 or .589%
This is the monthly interest rate. To find the APR with a monthly interest rate, we simply multiply
the monthly rate by 12, so the APR is:
APR = .00589 12
APR = .0707 or 7.07%
And the EAR is:
EAR = [1 + (APR / m)]m 1
EAR = [1 + .00589]12 1
EAR = .0730 or 7.30%
44. Here, we have two cash flow streams that will be combined in the future. To find the withdrawal
amount, we need to know the present value, as well as the interest rate and periods, which are given.
The present value of the retirement account is the future value of the stock and bond account. We
need to find the future value of each account and add the future values together. For the bond
account the future value is the value of the current savings plus the value of the annual deposits. So,
the future value of the bond account will be:
FV = C{[(1 + r)t 1] / r} + PV(1 + r)t
FV = $9,000{[(1 + .075)10 1] / .075} + $150,000(1 + .075)10
FV = $436,478.52
The total value of the stock account at retirement will be the future value of a lump sum, so:
FV = PV(1 + r)t
FV = $450,000(1 + .115)10
FV = $1,336,476.07
The total value of the account at retirement will be:
Total value at retirement = $436,478.52 + 1,336,476.07
Total value at retirement = $1,772,954.59
This amount is the present value of the annual withdrawals. Now we can use the present value of an
annuity equation to find the annuity amount. Doing so, we find the annual withdrawal will be:
PVA = C({1 [1/(1 + r)]t } / r)
$1,772,954.59 = C[{1 [1 / (1 + .0675)]25}/ .0675]
C = $148,727.69
CHAPTER 5 B-89
45. We need to use the PVA due equation, that is:
PVAdue = (1 + r) PVA
Using this equation:
PVAdue = $61,000 = [1 + (.0815/12)] C[{1 1 / [1 + (.0815/12)]60} / (.0815/12)
$60,588.50 = $C{1 [1 / (1 + .0815/12)60]} / (.0815/12)
C = $1,232.87
Notice, when we find the payment for the PVA due, we simply discount the PV of the annuity due
back one period. We then use this value as the PV of an ordinary annuity.
46. a.
If the payments are in the form of an ordinary annuity, the present value will be:
PVA = C({1 [1/(1 + r)]t } / r)
PVA = $10,000[{1 [1 / (1 + .11)]5}/ .11]
PVA = $36,958.97
If the payments are an annuity due, the present value will be:
PVAdue = (1 + r) PVA
PVAdue = (1 + .11)$36,958.97
PVAdue = $41,024.46
b.
We can find the future value of the ordinary annuity as:
FVA = C{[(1 + r)t 1] / r}
FVA = $10,000{[(1 + .11)5 1] / .11}
FVA = $62,278.01
If the payments are an annuity due, the future value will be:
FVAdue = (1 + r) FVA
FVAdue = (1 + .11)$62,278.01
FVAdue = $69,128.60
c.
Assuming a positive interest rate, the present value of an annuity due will always be larger than
the present value of an ordinary annuity. Each cash flow in an annuity due is received one
period earlier, which means there is one period less to discount each cash flow. Assuming a
positive interest rate, the future value of an ordinary annuity will always higher than the future
value of an ordinary annuity. Since each cash flow is made one period sooner, each cash flow
receives one extra period of compounding.
CHAPTER 5 B-90
47. Here, we need to find the difference between the present value of an annuity and the present value of
a perpetuity. The present value of the annuity is:
PVA = C({1 [1/(1 + r)]t } / r)
PVA = $8,000{[1 (1/1.09)30] / .09}
PVA = $82,189.23
And the present value of the perpetuity is:
PVP = C / r
PVP = $8,000 / .09
PVP = $88,888.89
So, the difference in the present values is:
Difference = $88,888.89 82,189.23
Difference = $6,699.66
There is another common way to answer this question. We need to recognize that the difference in
the cash flows is a perpetuity of $8,000 beginning 31 years from now. We can find the present value
of this perpetuity and the solution will be the difference in the cash flows. So, we can find the
present value of this perpetuity as:
PVP = C / r
PVP = $8,000 / .09
PVP = $88,888.89
This is the present value 30 years from now, one period before the first cash flows. We can now find
the present value of this lump sum as:
PV = FV / (1 + r)t
PV = $88,888.89 / (1 + .09)30
PV = $6,699.66
This is the same answer we calculated before.
48. Here we need to find the present value of an annuity at several different times. The annuity has
semiannual payments, so we need the semiannual interest rate. The semiannual interest rate is:
Semiannual rate = 0.11/2
Semiannual rate = .055
Now, we can use the present value of an annuity equation. Doing so, we get:
PVA = C({1 [1/(1 + r)]t } / r)
PVA = $9,000{[1 (1 / 1.055)10] / .055}
PVA = $67,838.63
CHAPTER 5 B-91
This is the present value one period before the first payment. The first payment occurs nine and onehalf years from now, so this is the value of the annuity nine years from now. Since the interest rate is
semiannual, we must also be careful to use the number of semiannual periods. The value of the
annuity five years from now is:
PV = FV / (1 + r)t
PV = $67,838.63 / (1 + .055)8
PV = $44,203.58
And the value of the annuity three years from now is:
PV = FV / (1 + r)t
PV = $67,838.63 / (1 + .055)12
PV = $35,681.87
And the value of the annuity today is:
PV = FV / (1 + r)t
PV = $67,838.63 / (1 + .055)18
PV = $25,878.13
49. Since the first payment is received six years form today and the last payment is received 20 years
from now, there are 15 payments. We can use the present value of an annuity formula, which will
give us the present value four years from now, one period before the first payment. So, the present
value of the annuity in four years is:
PVA = C({1 [1/(1 + r)]t } / r)
PVA = $1,450{[1 (1/1.09)15] / .09}
PVA = $11,688.00
And using the present value equation for a lump sum, the present value of the annuity today is:
PV = FV / (1 + r)t
PV = $11,688.00 / (1 + .09)5
PV = $7,596.40
50. Here, we have an annuity with two different interest rates. To answer this question, we simply need
to find the present value in multiple steps. The present value of the last six years payments at an
eight percent interest rate is:
PVA = C({1 [1/(1 + r)]t} / r)
PVA = $2,500[{1 1 / [1 + (.08/12)]72} / (.08/12)]
PVA = $142,586.31
CHAPTER 5 B-92
We can now discount this value back to time zero. We must be sure to use the number of months as
the periods since interest is compounded monthly. We also need to use the interest rate that applies
during the first four years. Doing so, we find:
PV = FV / (1 + r)t
PV = $142,586.31 / (1 + .11/12)48
PV = $92,015.03
Now we can find the present value of the annuity payments for the first four years. The present value
of these payments is:
PVA = C({1 [1/(1 + r)]t } / r)
PVA = $2,500[{1 1 / [1 + (.11/12)]48} / (.11/12)]
PVA = $96,728.55
So, the total present value of the cash flows is:
PV = $92,015.03 + 96,728.55
PV = $188,743.58
51. To answer this question we need to find the future value of the annuity, and then find the present
value that makes the lump sum investment equivalent. We also need to make sure to use the number
of months as the number of periods. So, the future value of the annuity is:
FVA = C{[(1 + r)t 1] / r}
FVA = $1,200{[(1 + .07/12)120 1] / (.07/12)}
FVA = $207,701.77
Now we can find the present value that would permit the lump sum investment to be equal to this
future value. This investment has annual compounding, so the number of periods is the number of
years. So, the present value we would need to deposit is:
PV = FV / (1 + r)t
PV = $207,701.77 / (1 + .09)10
PV = $87,735.47
52. Here we need to find the present value of a perpetuity at a date before the perpetuity begins. We will
begin by find the present value of the perpetuity. Doing so, we find:
PVP = C / r
PVP = $2,500 / .0545
PVP = $45,871.56
CHAPTER 5 B-93
This is the present value of the perpetuity at year 19, one period before the payments begin. So,
using the present value of a lump sum equation to find the value at year 7, we find:
PV = FV / (1 + r)t
PV = $45,871.56 / (1 + .0545)12
PV = $24,265.23
53. Here we are given the PVA, number of periods, and the amount of the annuity. We need to solve for
the interest rate. We need must be careful to use the cash flows of the loan. Using the present value
of an annuity equation, we find:
PVA = C({1 [1/(1 + r)]t } / r)
$20,000 = $1,916.67[{1 [1 / (1 + r)]12}/ r]
To find the interest rate, we need to solve this equation on a financial calculator, using a spreadsheet,
or by trial and error. If you use trial and error, remember that increasing the interest rate lowers the
PVA, and increasing the interest rate decreases the PVA. Using a spreadsheet, we find:
r = .02219 or 2.219%
This is the monthly interest rate. To find the APR with a monthly interest rate, we simply multiply
the monthly rate by 12, so the APR is:
APR = .02219 12
APR = .2662 or 26.62%
And the EAR is:
EAR = [1 + (APR / m)]m 1
EAR = [1 + .02219]12 1
EAR = .3012 or 30.12%
54. To solve this problem, we must find the FV of each cash flow and add them. To find the FV of a
lump sum, we use:
FV = PV(1 + r)t
FV = $25,000(1.094)3 + $45,000(1.094)2 + $65,000
FV = $151,591.08
Notice, since we are finding the value at Year 5, the cash flow at Year 5 is simply added to the FV of
the other cash flows. In other words, we do not need to compound this cash flow. To find the value
in Year 10, we simply need to find the future value of this lump sum. Doing so, we find:
FV = PV(1 + r)t
FV = $151,591.08(1.094)5
FV = $237,552.86
CHAPTER 5 B-94
55. The payment for a loan repaid with equal payments is the annuity payment with the loan value as the
PV of the annuity. So, the loan payment will be:
PVA = C({1 [1/(1 + r)]t } / r)
$75,000 = C{[1 1 / (1 + .09)5] / .09}
C = $29,629.11
The interest payment is the beginning balance times the interest rate for the period, and the principal
payment is the total payment minus the interest payment. The ending balance is the beginning
balance minus the principal payment. The ending balance for a period is the beginning balance for
the next period. The amortization table for an equal payment is:
Year
1
2
3
Beginning
Bal
ance
$75,000.00
52,120.89
27,182.67
Total
Payment
$29,629.11
29,629.11
29,629.11
Interest
Payment
$6,750.00
4,690.88
2,446.44
Principal
Payment
$22,879.11
24,938.23
27,182.67
Ending
Balance
$52,120.89
27,182.67
0
In the third year, $2,446.44 of interest is paid.
Total interest over life of the loan = $6,750.00 + 4,690.88 + 2,446.44
Total interest over life of the loan = $13,887.32
56. This amortization table calls for equal principal payments of $15000 per year. The interest payment is
the beginning balance times the interest rate for the period, and the total payment is the principal
payment plus the interest payment. The ending balance for a period is the beginning balance for the
next period. The amortization table for an equal principal reduction is:
Year
1
2
3
Beginning
Bal
ance
$75,000.00
50,000.00
25,000.00
Total
Payment
$31,750.00
29,500.00
27,250.00
Interest
Payment
$6,750.00
4,500.00
2,250.00
Principal
Payment
$25,000.00
25,000.00
25,000.00
Ending
Balance
$50,000.00
25,000.00
0
In the third year, $2,250 of interest is paid.
Total interest over life of the loan = $6,750 + 4,500 + 2,250
Total interest over life of the loan = $13,500
Notice that the total payments for the equal principal reduction loan are lower. This is because more
principal is repaid early in the loan, which reduces the total interest expense over the life of the loan.
CHAPTER 5 B-95
Challenge
57. To find the APR and EAR, we need to use the actual cash flows of the loan. In other words, the
interest rate quoted in the problem is only relevant to determine the total interest under the terms
given. The cash flows of the loan are the $12,000 you must repay in one year, and the $10,680 you
borrow today. The interest rate of the loan is:
$12,000 = $10,440(1 + r)
r = ($12,000 / 10,440) 1
r = .1494 or 14.94%
Because of the discount, you only get the use of $10,440, and the interest you pay on that amount is
14.94%, not 11%.
58. Here we have cash flows that would have occurred in the past and cash flows that would occur in the
future. We need to bring both cash flows to today. Before we calculate the value of the cash flows
today, we must adjust the interest rate so we have the effective monthly interest rate. Finding the
APR with monthly compounding and dividing by 12 will give us the effective monthly rate. The
APR with monthly compounding is:
APR = 12[(1.09)1/12 1] = 8.65%
To find the value today of the back pay from two years ago, we will find the FV of the annuity, and
then find the FV of the lump sum. Doing so gives us:
FVA = ($44,000/12) [{[ 1 + (.0865/12)]12 1} / (.0865/12)] = $45,786.76
FV = $45,786.76(1.09) = $49,907.57
Notice we found the FV of the annuity with the effective monthly rate, and then found the FV of the
lump sum with the EAR. Alternatively, we could have found the FV of the lump sum with the
effective monthly rate as long as we used 12 periods. The answer would be the same either way.
Now, we need to find the value today of last years back pay:
FVA = ($46,000/12) [{[ 1 + (.0865/12)]12 1} / (.0865/12)] = $47,867.98
Next, we find the value today of the five years future salary:
PVA = ($49,000/12){[{1 {1 / [1 + (.0865/12)]12(5)}] / (.0865/12)}= $198,332.55
The value today of the jury award is the sum of salaries, plus the compensation for pain and
suffering, and court costs. The award should be for the amount of:
Award = $49,907.57 + 47,867.98 + 198,332.55 + 100,000 + 20,000 = $416,108.10
CHAPTER 5 B-96
As the plaintiff, you would prefer a lower interest rate. In this problem, we are calculating both the
PV and FV of annuities. A lower interest rate will decrease the FVA, but increase the PVA. So, by a
lower interest rate, we are lowering the value of the back pay. But, we are also increasing the PV of
the future salary. Since the future salary is larger and has a longer time, this is the more important
cash flow to the plaintiff.
59. Again, to find the interest rate of a loan, we need to look at the cash flows of the loan. Since this loan
is in the form of a lump sum, the amount you will repay is the FV of the principal amount, which
will be:
Loan repayment amount = $10,000(1.09) = $10,900
The amount you will receive today is the principal amount of the loan times one minus the points.
Amount received = $10,000(1 .03) = $9,700
Now, we simply find the interest rate for this PV and FV.
$10,900 = $9,700(1 + r)
r = ($10,900 / $9,700) 1 = .1237 or 12.37%
60. We need to find the FV of the premiums to compare with the cash payment promised at age 65. We
have to find the value of the premiums at year 6 first since the interest rate changes at that time. So:
FV1 = $800(1.11)5 = $1,384.05
FV2 = $800(1.11)4 = $1,214.46
FV3 = $900(1.11)3 = $1,230.87
FV4 = $900(1.11)2 = $1,108.89
FV5 = $1,000(1.11)1 = $1,110.00
Value at year six = $1,384.05 + 1,214.46 + 1,230.87 + 1,108.89 + 1,110.00 + 1,000 = $7,012.26
Finding the FV of this lump sum at the childs 65th birthday:
FV = $7,012.26(1.07)59 = $379,752.76
The policy is not worth buying; the future value of the deposits is $379,752.76, but the policy
contract will pay off $350,000. The premiums are worth $29,752.76 more than the policy payoff.
Note, we could also compare the PV of the two cash flows. The PV of the premiums is:
PV = $800/1.11 + $800/1.112 + $900/1.113 + $900/1.114 + $1,000/1.115 + $1,000/1.116
PV = $3,749.04
CHAPTER 5 B-97
And the value today of the $350,000 at age 65 is:
PV = $350,000/1.0759 = $6,462.87
PV = $6,462.87/1.116 = $3,455.31
The premiums still have the higher cash flow. At time zero, the difference is $2,148.25. Whenever
you are comparing two or more cash flow streams, the cash flow with the highest value at one time
will have the highest value at any other time.
Here is a question for you: Suppose you invest $293.73, the difference in the cash flows at time zero,
for six years at an 11 percent interest rate, and then for 59 years at a seven percent interest rate. How
much will it be worth? Without doing calculations, you know it will be worth $29,752.76, the
difference in the cash flows at time 65!
Calculator Solutions
1.
CFo
$0
C01
$800
F01
1
C02
$500
F02
1
C03
$1,300
F03
1
C04
$1,480
F04
1
I = 10
NPV CPT
$3,128.07
2.
Enter
9
N
CFo
$0
C01
$800
F01
1
C02
$500
F02
1
C03
$1,300
F03
1
C04
$1,480
F04
1
I = 18
NPV CPT
$2,591.65
6%
I/Y
PV
CFo
$0
C01
$800
F01
1
C02
$500
F02
1
C03
$1,300
F03
1
C04
$1,480
F04
1
I = 24
NPV CPT
$2,278.18
$5,500
PM
FV
T
Solve for
Enter
$37,409.31
5
N
6%
I/Y
PV
$8,000
PM
FV
T
Solve for
Enter
$33,698.91
9
N
22%
I/Y
PV
$5,500
PM
FV
T
CHAPTER 5 B-98
Solve for
$20,824.57
CHAPTER 5 B-99
Enter
5
N
22%
I/Y
PV
$8,000
PM
FV
T
Solve for
$22,909.12
3.
CFo
$0
C01
$700
F01
1
C02
$900
F02
1
C03
$1,400
F03
1
C04
$2,000
F04
1
I = 10
NFV CPT
$5,443.56
4.
Enter
15
N
CFo
$0
C01
$700
F01
1
C02
$900
F02
1
C03
$1,400
F03
1
C04
$2,000
F04
1
I = 18
NFV CPT
$5,620.23
9%
I/Y
PV
CFo
$0
C01
$700
F01
1
C02
$900
F02
1
C03
$1,400
F03
1
C04
$2,000
F04
1
I = 24
NFV CPT
$6.454.48
$7,000
PM
FV
T
Solve for
Enter
$56,424.82
40
N
9%
I/Y
PV
$7,000
PM
FV
T
Solve for
Enter
$75,301.52
75
N
9%
I/Y
PV
$7,000
PM
FV
T
Solve for
5.
Enter
$77,656.48
12
N
8.5%
I/Y
PM
T
$2,723.06
Solve for
6.
Enter
$20,000
PV
9
8.25%
$50,000
FV
CHAPTER 5 B-100
N
I/Y
PV
PM
FV
T
Solve for
$309,123.20
CHAPTER 5 B-101
7.
Enter
20
N
9.5%
I/Y
PV
$3,000
PM
FV
T
Solve for
Enter
$162,366.70
40
N
9.5%
I/Y
PV
$3,000
PM
FV
T
Solve for
8.
Enter
$1,159,559.98
8
N
5.25%
I/Y
PV
7
N
8%
I/Y
$50,000
PV
9
NO
Solve for
Enter
13%
NO
Solve for
4
C/Y
EFF
12
C/Y
13.80%
16%
NO
Solve for
Enter
EFF
9.31%
Solve for
Enter
PM
T
$9,603.62
Solve for
12.
Enter
$30,000
FV
T
$3,113.68
Solve for
9.
Enter
PM
EFF
365
C/Y
17.35%
19%
NO
EFF
19.90%
2
C/Y
FV
CHAPTER 5 B-102
13.
Enter
NO
Solve for
9.76%
10%
EFF
2
C/Y
CHAPTER 5 B-103
Enter
NO
Solve for
NO
NO
14.
Enter
12.4%
NO
12.7%
NO
365
C/Y
EFF
12
C/Y
EFF
2
C/Y
13.10%
15.
Enter
NO
16.
Enter
16%
EFF
13.13%
Solve for
Solve for
52
C/Y
14.85%
Solve for
Enter
9%
EFF
8.62%
Enter
Solve for
12
C/Y
13.17%
Enter
Solve for
14%
EFF
18%
EFF
365
C/Y
16.56%
13 2 =
N
11% / 2 =
I/Y
$1,280
PV
PM
FV
T
Solve for
17.
Enter
$5,149.61
5 365 =
N
4.5% / 365 =
I/Y
$5,000
PV
PM
FV
T
Solve for
$6,261.53
CHAPTER 5 B-104
Enter
10 365 =
N
4.5% / 365 =
I/Y
$5,000
PV
PM
FV
T
Solve for
$7,841.34
CHAPTER 5 B-105
Enter
20 365 =
N
4.5% / 365 =
I/Y
$5,000
PV
PM
FV
T
Solve for
18.
Enter
$12,297.33
6 365 =
N
12% / 365 =
I/Y
PV
$75,000
FV
PM
T
Solve for
$36,510.74
19. APR = 12(20%) = 240%
Enter
240%
NO
Solve for
20.
Enter
EFF
12
C/Y
791.61%
60
N
8.6% / 12 =
I/Y
$69,500
PV
FV
T
$1,429.25
Solve for
Enter
PM
8.6%
NO
Solve for
EFF
12
C/Y
8.95%
21.
Enter
N
1.1%
I/Y
$13,850
PV
$500
PM
FV
T
Solve for
22.
Enter
33.22
1
N
I/Y
$4
PV
$5
FV
PM
T
Solve for
25.00%
APR = 52(25.00%) = 1,300.00%
Enter
1,300%
52
CHAPTER 5 B-106
NO
Solve for
EFF
10,947,544%
C/Y
CHAPTER 5 B-107
24.
Enter
30 12 =
N
12% / 12 =
I/Y
PV
$300
PM
FV
T
Solve for
25.
Enter
$1,048,489.24
12%
NO
Solve for
Enter
EFF
12
C/Y
12.68%
30
12.68%
I/Y
N
PV
$3,600
PM
FV
T
Solve for
26.
Enter
$992,065.28
4 4=
N
.90%
I/Y
PV
$2,200
PM
FV
T
Solve for
$32,646.61
27.
CFo
$0
C01
$300
F01
1
C02
$900
F02
1
C03
$700
F03
1
C04
$600
F04
1
I = 9.50
NPV CPT
$1,975.08
28.
CFo
C01
F01
C02
F02
C03
F03
C04
$0
$2,400
1
$3,200
1
$6,800
1
$8,100
CHAPTER 5 B-108
F04
1
I = 8.16
NPV CPT
$16,247.04
CHAPTER 5 B-109
29. First Simple: $100(.07) = $7; 10 year investment = $100 + 10($7) = $170
Enter
10
N
$100
PV
I/Y
$170
FV
PM
T
Solve for
5.45%
30. 2nd BGN 2nd SET
Enter
60
N
7.45% / 12 =
I/Y
$48,000
PV
PM
FV
T
Solve for
31.
Enter
$954.75
6
N
1.50% / 12 =
I/Y
$10,000
PV
PM
FV
T
Solve for
Enter
$10,075.23
6
N
18% / 12 =
I/Y
$10,075.23
PV
PM
FV
T
Solve for
$11,016.70
Interest = $11,016.70 10,000
Interest = $1,016.70
32. First: $89,000 (.05) = $4,450 per year
($175,000 89,000) / $4,450 = 19.33 years
Second:
Enter
N
5% / 12 =
I/Y
$89,000
PV
$175,000
FV
PM
T
Solve for
162.61
162.61 / 12 = 13.55 years
33.
Enter
12
N
1.05%
I/Y
$1
PV
PM
FV
T
Solve for
$1.13
CHAPTER 5 B-110
Enter
24
N
1.05%
I/Y
$1
PV
PM
FV
T
Solve for
$1.28
CHAPTER 5 B-111
34.
Enter
31
N
I/Y
440
PV
60
PM
FV
T
Solve for
35.
Enter
13.36%
24
N
7% /12 =
I/Y
PV
$7,500
PM
FV
T
Solve for
Enter
$167,513.24
24
N
7% / 12 =
I/Y
PV
$6,000
PM
FV
T
Solve for
$134,010.60
$134,010.60 + 30,000 = $164,010.60
36.
Enter
20
N
13%
I/Y
Solve for
37.
Enter
PV
$15,000
PM
T
FV
$105,371.27
6
N
I/Y
$75,000
PV
$120,000
FV
PM
T
Solve for
Enter
8.63%
13
N
I/Y
$75,000
PV
$220,000
FV
PM
T
Solve for
38.
Enter
8.15%
10
N
10%
I/Y
Solve for
Enter
PV
$10,000
PM
T
FV
$61,445.67
10
N
5%
I/Y
PV
$10,000
PM
FV
CHAPTER 5 B-112
T
Solve for
Enter
$77,217.35
10
N
Solve for
15%
I/Y
PV
$50,187.69
$10,000
PM
T
FV
CHAPTER 5 B-113
39.
Enter
N
10% / 12 =
I/Y
PV
$250
PM
$50,000
FV
T
Solve for
40.
Enter
118.19
60
N
$75,000
PV
I/Y
$1,600
PM
FV
T
Solve for
.848%
APR = .848%(12) = 10.18%
42.
CFo
$24,000,000
C01
$24,000,000
F01
3
C02
$27,000,000
F02
1
C03
$25,000,000
F03
1
C04
$26,000,000
F04
4
C05
F05
C06
F06
C07
F07
I = 11%
NPV CPT
$163,141,086.06
43.
Enter
30 12 =
N
Solve for
I/Y
CFo $16,000,000
C01 $18,500,000
F01 1
C02 $21,000,000
F02 1
C03 $23,500,000
F03 1
C04 $23,000,000
F04 1
C05 $20,000,000
F05 1
C06 $18,000,000
F06 1
C07 $20,000,000
F07 1
I = 11%
NPV CPT
$113,170,277.46
.80($2,500,000)=
PV
.589%
APR = 0.589%(12) = 7.07%
Enter
Solve for
7.07%
NO
EFF
7.30%
12
C/Y
$13,400
PM
FV
CHAPTER 5 B-114
CHAPTER 5 B-115
44. Future value of bond account:
Enter
10
7.5%
I/Y
N
$150,000
PV
$9,000
PM
FV
T
Solve for
$436,478.52
Future value of stock account:
Enter
10
N
11.5%
I/Y
$450,000
PV
PM
FV
T
Solve for
Future value of retirement account:
FV = $436,478.52 + 1,366,476.07
FV = $1,772,954.59
$1,336,476.07
Annual withdrawal amount:
Enter
25
N
6.75%
I/Y
$1,772,954.59
PV
PM
FV
T
$148,727.69
Solve for
45. 2nd BGN 2nd SET
Enter
60
N
8.15% / 12 =
I/Y
$61,000
PV
FV
T
$1,232.87
Solve for
46.
Enter
PM
5
N
11%
I/Y
Solve for
PV
$10,000
PM
T
FV
$10,000
PM
T
FV
$36,958.97
2nd BGN 2nd SET
Enter
5
N
11%
I/Y
Solve for
Enter
PV
$41,024.46
5
N
11%
I/Y
PV
$10,000
PM
FV
CHAPTER 5 B-116
T
Solve for
$62,278.01
CHAPTER 5 B-117
2nd BGN 2nd SET
Enter
5
N
11%
I/Y
PV
$10,000
PM
T
Solve for
FV
$69,128.60
47. Present value of annuity:
Enter
30
N
9%
I/Y
PV
$8,000
PM
FV
T
Solve for
$82,189.23
And the present value of the perpetuity is:
PVP = C / r
PVP = $8,000 / .09
PVP = $88,888.89
So the difference in the present values is:
Difference = $88,888.89 82,189.23
Difference = $6,699.66
48. Value at t = 9
Enter
10
N
11% / 2 =
I/Y
PV
$9,000
PM
FV
T
Solve for
$67,838.63
Value at t = 5
Enter
4 2=
N
11% / 2 =
I/Y
PV
$67,838.63
FV
PM
T
Solve for
$44,203.58
Value at t = 3
Enter
6 2=
N
11% / 2 =
I/Y
PV
$67,838.63
FV
PM
T
Solve for
Value today
$35,681.87
CHAPTER 5 B-118
Enter
9 2=
N
11% / 2 =
I/Y
PV
$67,838.63
FV
PM
T
Solve for
$25,878.13
CHAPTER 5 B-119
49. Value at t = 5
Enter
15
N
9%
I/Y
PV
$1,450
PM
FV
T
Solve for
$11,688.00
Value today
Enter
5
N
9%
I/Y
PV
$11,688.00
FV
PM
T
Solve for
$7,596.40
50. Value at t = 4
Enter
6 12 =
N
8% / 12 =
I/Y
PV
$2,500
PM
FV
T
Solve for
$142,586.31
Value today
Enter
4 12 =
N
11% / 12 =
I/Y
PV
$2,500
PM
$142,586.31
FV
T
Solve for
$188,743.58
51. FV of A
Enter
10 12 =
N
7% / 12 =
I/Y
PV
$1,200
PM
FV
T
Solve for
$207,701.77
Value to invest in B
Enter
10
N
9%
I/Y
PV
$207,701.77
FV
PM
T
Solve for
53.
Enter
$87,735.47
12
N
Solve for
I/Y
2.219%
$20,000
PV
$1,916.67
PM
T
FV
CHAPTER 5 B-120
APR = 2.219%(12) = 26.62%
Enter
Solve for
26.62%
NO
EFF
30.12%
12
C/Y
CHAPTER 5 B-121
CFo
$0
C01
$0
F01
1
C02
$25,000
F02
1
C03
$45,000
F03
1
C04
$0
F04
1
C05
$65,000
F05
1
I = 9.4%
NPV NFV
$151,591.08
Enter
5
N
I/Y
$151,591.08
PV
PM
FV
T
Solve for
57.
Enter
9.4
1
N
I/Y
$237,552.86
$10,440
PV
$12,000
FV
PM
T
Solve for
14.94
58.
Enter
NO
Solve for
Enter
9%
EFF
12
C/Y
8.65%
12
N
8.65% / 12 =
I/Y
PV
$44,000 / 12 =
PM
T
Solve for
Enter
FV
$45,786.76
1
N
9%
I/Y
$45,786.76
PV
PM
FV
T
Solve for
Enter
$49,907.57
12
8.65% / 12 =
$46,000 / 12 =
CHAPTER 5 B-122
N
I/Y
PV
PM
FV
T
Solve for
$47,867.98
CHAPTER 5 B-123
Enter
60
N
8.65% / 12 =
I/Y
Solve for
PV
$49,000 / 12=
PM
T
FV
$198,332.55
Award = $49,907.57 + 47,867.98 + 198,332.55 + 100,000 + 20,000 = $416,108.10
59.
Enter
1
N
I/Y
$9,700
PV
$10,900
FV
PM
T
Solve for
12.37%
60. Value at Year 6:
Enter
5
N
11%
I/Y
$800
PV
PM
FV
T
Solve for
Enter
$1,348.05
4
N
11%
I/Y
$800
PV
PM
FV
T
Solve for
Enter
$1,214.46
3
N
11%
I/Y
$900
PV
PM
FV
T
Solve for
Enter
$1,230.87
2
N
11%
I/Y
$900
PV
PM
FV
T
Solve for
Enter
$1,108.89
1
N
11%
I/Y
$1,000
PV
PM
FV
T
Solve for
$1,110
So, at Year 5, the value is: $1,348.05 + 1,214.46 + 1,230.87 + 1,108.89 + 1,100
CHAPTER 5 B-124
+ 1,000 = $7,012.26
CHAPTER 5 B-125
At Year 65, the value is:
Enter
59
N
7%
I/Y
$7,012.26
PV
PM
FV
T
Solve for
$379,752.76
The policy is not worth buying; the future value of the deposits is $379,752.76 but the policy
contract will pay off $350,000.
CHAPTER 5 B-126
CHAPTER 6
INTEREST
RATES
VALUTION
AND
BOND
Answers to Concepts Review and Critical Thinking Questions
1.
No. As interest rates fluctuate, the value of a Treasury security will fluctuate. Long-term
Treasury securities have substantial interest rate risk.
2.
All else the same, the Treasury security will have lower coupons because of its lower default
risk, so it will have greater interest rate risk.
3.
No. If the bid were higher than the ask, the implication would be that a dealer was willing to
sell a bond and immediately buy it back at a higher price. How many such transactions
would you like to do?
4.
Prices and yields move in opposite directions. Since the bid price must be lower, the bid
yield must be higher.
5.
There are two benefits. First, the company can take advantage of interest rate declines by
calling in an issue and replacing it with a lower coupon issue. Second, a company might
wish to eliminate a covenant for some reason. Calling the issue does this. The cost to the
company is a higher coupon. A put provision is desirable from an investors standpoint, so it
helps the company by reducing the coupon rate on the bond. The cost to the company is that
it may have to buy back the bond at an unattractive price.
6.
Bond issuers look at outstanding bonds of similar maturity and risk. The yields on such
bonds are used to establish the coupon rate necessary for a particular issue to initially sell for
par value. Bond issuers also simply ask potential purchasers what coupon rate would be
necessary to attract them. The coupon rate is fixed and simply determines what the bonds
coupon payments will be. The required return is what investors actually demand on the
issue, and it will fluctuate through time. The coupon rate and required return are equal only
if the bond sells for exactly par.
7.
Yes. Some investors have obligations that are denominated in dollars; i.e., they are nominal.
Their primary concern is that an investment provide the needed nominal dollar amounts.
Pension funds, for example, often must plan for pension payments many years in the future.
If those payments are fixed in dollar terms, then it is the nominal return on an investment
that is important.
8.
Companies pay to have their bonds rated simply because unrated bonds can be difficult to
sell; many large investors are prohibited from investing in unrated issues.
CHAPTER 5 B-127
9.
Treasury bonds have no credit risk, so a rating is not necessary. Junk bonds often are not
rated because there would no point in an issuer paying a rating agency to assign its bonds a
low rating (its like paying someone to kick you!).
CHAPTER 5 B-128
10. Bond ratings have a subjective factor to them. Split ratings reflect a difference of opinion
among credit agencies.
11. As a general constitutional principle, the federal government cannot tax the states without
their consent if doing so would interfere with state government functions. At one time, this
principle was thought to provide for the tax-exempt status of municipal interest payments.
However, modern court rulings make it clear that Congress can revoke the municipal
exemption, so the only basis now appears to be historical precedent. The fact that the states
and the federal government do not tax each others securities is referred to as reciprocal
immunity.
12. One measure of liquidity is the bid-ask spread. Liquid instruments have relatively small
spreads. Looking at Figure 6.4, the bellwether bond has a spread of one tick; it is one of the
most liquid of all investments. Generally, liquidity declines after a bond is issued. Some
older bonds, including some of the callable issues, have spreads as wide as six ticks.
13. Companies charge that bond rating agencies are pressuring them to pay for bond ratings.
When a company pays for a rating, it has the opportunity to make its case for a particular
rating. With an unsolicited rating, the company has no input.
14. A 100-year bond looks like a share of preferred stock. In particular, it is a loan with a life
that almost certainly exceeds the life of the lender, assuming that the lender is an individual.
With a junk bond, the credit risk can be so high that the borrower is almost certain to default,
meaning that the creditors are very likely to end up as part owners of the business. In both
cases, the equity in disguise has a significant tax advantage.
15. a.
b.
c.
The bond price is the present value when discounting the future cash flows from a
bond; YTM is the interest rate used in discounting the future cash flows (coupon
payments and principal) back to their present values.
If the coupon rate is higher than the required return on a bond, the bond will sell at a
premium, since it provides periodic income in the form of coupon payments in excess
of that required by investors on other similar bonds. If the coupon rate is lower than the
required return on a bond, the bond will sell at a discount, since it provides insufficient
coupon payments compared to that required by investors on other similar bonds. For
premium bonds, the coupon rate exceeds the YTM; for discount bonds, the YTM
exceeds the coupon rate, and for bonds selling at par, the YTM is equal to the coupon
rate.
Current yield is defined as the annual coupon payment divided by the current bond
price. For premium bonds, the current yield exceeds the YTM, for discount bonds the
current yield is less than the YTM, and for bonds selling at par value, the current yield
is equal to the YTM. In all cases, the current yield plus the expected one-period capital
gains yield of the bond must be equal to the required return.
CHAPTER 5 B-129
Solutions to Questions and Problems
NOTE: All end-of-chapter problems were solved using a spreadsheet. Many problems require
multiple steps. Due to space and readability constraints, when these intermediate steps are
included in this solutions manual, rounding may appear to have occurred. However, the final
answer for each problem is found without rounding during any step in the problem.
Basic
1.
The yield to maturity is the required rate of return on a bond expressed as a nominal annual
interest rate. For noncallable bonds, the yield to maturity and required rate of return are
interchangeable terms. Unlike YTM and required return, the coupon rate is not a return used
as the interest rate in bond cash flow valuation, but is a fixed percentage of par over the life
of the bond used to set the coupon payment amount. For the example given, the coupon rate
on the bond is still 10 percent, and the YTM is 8 percent.
2.
Price and yield move in opposite directions; if interest rates rise, the price of the bond will
fall. This is because the fixed coupon payments determined by the fixed coupon rate are not
as valuable when interest rates risehence, the price of the bond decreases.
NOTE: Most problems do not explicitly list a par value for bonds. Even though a bond can
have any par value, in general, corporate bonds in the United States will have a par value of
$1,000. We will use this par value in all problems unless a different par value is explicitly
stated.
3.
The price of any bond is the PV of the interest payment, plus the PV of the par value. Notice
this problem assumes an annual coupon. The price of the bond will be:
P = $90({1 [1/(1 + .07)]8} / .07) + $1,000[1 / (1 + .07)8]
P = $1,119.43
We would like to introduce shorthand notation here. Rather than write (or type, as the case
may be) the entire equation for the PV of a lump sum, or the PVA equation, it is common to
abbreviate the equations as:
PVIFR,t = 1 / (1 + r)t
which stands for Present Value Interest Factor
PVIFAR,t = ({1 [1/(1 + r)]t } / r )
which stands for Present Value Interest Factor of an Annuity
These abbreviations are shorthand notation for the equations in which the interest rate and
the number of periods are substituted into the equation and solved. We will use this
shorthand notation in the remainder of the solutions key. The bond price equation for this
problem would be:
P = $90(PVIFA7%,8) + $1,000(PVIF7%,8)
P = $1,119.43
CHAPTER 5 B-130
4.
Here, we need to find the YTM of a bond. The equation for the bond price is:
P = $1,047.50 = $80(PVIFAR%,9) + $1,000(PVIFR%,9)
Notice the equation cannot be solved directly for R. Using a spreadsheet, a financial
calculator, or trial and error, we find:
R = YTM = 7.26%
If you are using trial and error to find the YTM of the bond, you might be wondering how to
pick an interest rate to start the process. First, we know the YTM has to be lower than the
coupon rate since the bond is a premium bond. That still leaves a lot of interest rates to
check. One way to get a starting point is to use the following equation, which will give you
an approximation of the YTM:
Approximate YTM = [Annual interest payment + (Price difference from par / Years to
maturity)] /
[(Price + Par value) / 2]
Solving for this problem, we get:
Approximate YTM = [$80 + ($47.50 / 9)] / [($1,047.50 + 1,000) / 2]
Approximate YTM = .0833 or 8.33%
This is not the exact YTM, but it is close, and it will give you a place to start.
5.
Here we need to find the coupon rate of the bond. All we need to do is to set up the bond
pricing equation and solve for the coupon payment as follows:
P = $1,051 = C(PVIFA6.8%,16) + $1,000(PVIF6.8%,16)
Solving for the coupon payment, we get:
C = $73.33
The coupon payment is the coupon rate times par value. Using this relationship, we get:
Coupon rate = $73.33 / $1,000
Coupon rate = .0733 or 7.33%
6.
To find the price of this bond, we need to realize that the maturity of the bond is 14 years.
The bond was issued one year ago, with 15 years to maturity, so there are 14 years left on
the bond. Also, the coupons are semiannual, so we need to use the semiannual interest rate
and the number of semiannual periods. The price of the bond is:
P = $42.50(PVIFA3.95%,28) + $1,000(PVIF3.95%,28)
P = $1,050.28
CHAPTER 5 B-131
7.
Here, we are finding the YTM of a semiannual coupon bond. The bond price equation is:
P = $960 = $42(PVIFAR%,26) + $1,000(PVIFR%,26)
Since we cannot solve the equation directly for R, using a spreadsheet, a financial calculator,
or trial and error, we find:
R = 4.463%
Since the coupon payments are semiannual, this is the semiannual interest rate. The YTM is
the APR of the bond, so:
YTM = 2 4.463%
YTM = 8.93%
8.
Here, we need to find the coupon rate of the bond. All we need to do is to set up the bond
pricing equation and solve for the coupon payment as follows:
P = $1,070 = C(PVIFA3.45%,21) + $1,000(PVIF3.45%,21)
Solving for the coupon payment, we get:
C = $39.24
Since this is the semiannual payment, the annual coupon payment is:
2 $39.24 = $78.48
And the coupon rate is the coupon rate divided by par value, so:
Coupon rate = $78.48 / $1,000
Coupon rate = .0785 or 7.85%
9.
The approximate relationship between nominal interest rates (R), real interest rates (r), and
inflation (h), is:
R=r+h
Approximate r = .08 .047
Approximate r =.033 or 3.30%
The Fisher equation, which shows the exact relationship between nominal interest rates, real
interest rates, and inflation, is:
(1 + R) = (1 + r)(1 + h)
(1 + .08) = (1 + r)(1 + .047)
Exact r = [(1 + .08) / (1 + .047)] 1
Exact r = .0315 or 3.15%
CHAPTER 5 B-132
10. The Fisher equation, which shows the exact relationship between nominal interest rates, real
interest rates, and inflation, is:
(1 + R) = (1 + r)(1 + h)
R = (1 + .032)(1 + .043) 1
R = .0764 or 7.64%
11. The Fisher equation, which shows the exact relationship between nominal interest rates, real
interest rates, and inflation, is:
(1 + R) = (1 + r)(1 + h)
h = [(1 + .14) / (1 + .09)] 1
h = .0459 or 4.59%
12. The Fisher equation, which shows the exact relationship between nominal interest rates, real
interest rates, and inflation, is:
(1 + R) = (1 + r)(1 + h)
r = [(1 + .13) / (1.041)] 1
r = .0855 or 8.55%
13. This is a note. The lower case n beside the maturity denotes it as such. The coupon rate,
located in the first column of the quote is 4%. The bid price is:
Bid price = 97:29 = 97 29/32
Bid price = 97.90625% $1,000
Bid price = $979.0625
The previous days ask price is found by:
Previous days asked price = Todays asked price Change
Previous days asked price = 97 90/32 (1/32)
Previous days asked price = 97 31/32
The previous days price in dollars was:
Previous days dollar price = 97.96875% $1,000
Previous days dollar price = $979.6875
CHAPTER 5 B-133
14. This is a premium bond because it sells for more than 100% of face value. The current yield
is based on the asked price, so the current yield is:
Current yield = Annual coupon payment / Price
Current yield = $61.25/$1,135.625
Current yield = .0539 or 5.39%
The YTM is located under the ASK YLD column, so the YTM is 4.78%.
The bid-ask spread is the difference between the bid price and the ask price, so:
Bid-Ask spread = 113:18 113:17
Bid-Ask spread = 1/32
Intermediate
15. Here, we are finding the YTM of semiannual coupon bonds for various maturity lengths.
The bond price equation is:
P = C(PVIFAR%,t) + $1,000(PVIFR%,t)
X:
Y:
P0
P1
P3
P8
P12
P13
P0
P1
P3
P8
P12
P13
= $90(PVIFA7%,13) + $1,000(PVIF7%,13)
= $90(PVIFA7%,12) + $1,000(PVIF7%,12)
= $90(PVIFA7%,10) + $1,000(PVIF7%,10)
= $90(PVIFA7%,5) + $1,000(PVIF7%,5)
= $90(PVIFA7%,1) + $1,000(PVIF7%,1)
= $1,167.15
= $1,158.85
= $1,140.47
= $1,082.00
= $1,018.69
= $1,000
= $70(PVIFA9%,13) + $1,000(PVIF9%,13) = $850.26
= $70(PVIFA9%,12) + $1,000(PVIF9%,12) = $856.79
= $70(PVIFA9%,10) + $1,000(PVIF9%,10) = $871.65
= $70(PVIFA9%,5) + $1,000(PVIF9%,5) = $922.21
= $70(PVIFA9%,1) + $1,000(PVIF9%,1) = $981.65
= $1,000
All else held equal, the premium over par value for a premium bond declines as maturity
approaches, and the discount from par value for a discount bond declines as maturity
approaches. This is called pull to par. In both cases, the largest percentage price changes
occur at the shortest maturity lengths.
Also, notice that the price of each bond when no time is left to maturity is the par value,
even though the purchaser would receive the par value plus the coupon payment
immediately. This is because we calculate the clean price of the bond.
CHAPTER 5 B-134
16. Any bond that sells at par has a YTM equal to the coupon rate. Both bonds sell at par, so the
initial YTM on both bonds is the coupon rate, 7 percent. If the YTM suddenly rises to 9
percent:
PBill
= $40(PVIFA5%,6) + $1,000(PVIF5%,6)
= $949.24
PTed
= $40(PVIFA5%,40) + $1,000(PVIF5%,40) = $828.41
The percentage change in price is calculated as:
Percentage change in price = (New price Original price) / Original price
PBill%
= ($949.24 1,000) / $1,000 = 0.0508 or 5.08%
PTed%
= ($828.41 1,000) / $1,000 = 0.1716 or 17.16%
If the YTM suddenly falls to 5 percent:
PBill
= $40(PVIFA3%,6) + $1,000(PVIF3%,6)
= $1,054.17
PTed
= $40(PVIFA3%,40) + $1,000(PVIF3%,40) = $1,231.15
PBill%
= ($1,054.17 1,000) / $1,000 = 0.0542 or +5.42%
PTed%
= ($1,231.15 1,000) / $1,000 = 0.2311 or +23.11%
All else the same, the longer the maturity of a bond, the greater is its price sensitivity to
changes in interest rates.
17. Initially, at a YTM of 7 percent, the prices of the two bonds are:
PJ
= $20(PVIFA3.5%,16) + $1,000(PVIF3.5%,16) = $818.59
PS
= $50(PVIFA3.5%,16) + $1,000(PVIF3.5%,16) = $1,181.41
If the YTM rises from 7 percent to 9 percent:
PJ
= $20(PVIFA4.5%,16) + $1,000(PVIF4.5%,16) = $719.15
PS
= $50(PVIFA4.5%,16) + $1,000(PVIF4.5%,16) = $1,056.17
The percentage change in price is calculated as:
Percentage change in price = (New price Original price) / Original price
PJ% = ($719.15 818.59) / $818.59
= 0.1215 or 12.15%
PS% = ($1,056.17 1,181.41) / $1,181.41 = 0.1060 or 10.60%
CHAPTER 5 B-135
If the YTM declines from 7 percent to 5 percent:
PJ
= $20(PVIFA2.5%,16) + $1,000(PVIF2.5%,16) = $934.72
PS
= $50(PVIFA2.5%,16) + $1,000(PVIF2.5%,16) = $1,326.38
PJ% = ($934.72 818.59) / $818.59
= 0.1419 or +14.19%
PS% = ($1,326.38 1,181.41) / $1,181.41 = 0.1227 or +12.27%
All else the same, the lower the coupon rate on a bond, the greater is its price sensitivity to
changes in interest rates.
18. The current yield is:
Current yield = Annual coupon payment / Price
Current yield = $80 / $930
Current yield = 0.0860 or 8.60%
The bond price equation for this bond is:
P0 = $930 = $40(PVIFAR%,36) + $1,000(PVIFR%,36)
Using a spreadsheet, financial calculator, or trial and error we find:
R = 4.39%
This is the semiannual interest rate, so the YTM is:
YTM = 2 4.39%
YTM = 8.78%
The effective annual yield is the same as the EAR, so using the EAR equation from the
previous chapter:
Effective annual yield = (1 + 0.0439)2 1
Effective annual yield = 8.97%
19. The company should set the coupon rate on its new bonds equal to the required return. The
required return can be observed in the market by finding the YTM on outstanding bonds of
the company. So, the YTM on the bonds currently sold in the market is:
P = $1,062 = $37.50(PVIFAR%,40) + $1,000(PVIFR%,40)
Using a spreadsheet, financial calculator, or trial and error, we find:
R = 3.46%
CHAPTER 5 B-136
This is the semiannual interest rate, so the YTM is:
YTM = 2 3.46%
YTM = 6.92%
20. Accrued interest is the coupon payment for the period times the fraction of the period that
has passed since the last coupon payment. Since we have a semiannual coupon bond, the
coupon payment per six months is one-half of the annual coupon payment. There are four
months until the next coupon payment, so one month has passed since the last coupon
payment. The accrued interest for the bond is:
Accrued interest = $86/2 5/6
Accrued interest = $7.17
And we calculate the clean price as:
Clean price = Dirty price Accrued interest
Clean price = $1,090 7.17
Clean price = $1,082.83
21. Accrued interest is the coupon payment for the period times the fraction of the period that
has passed since the last coupon payment. Since we have a semiannual coupon bond, the
coupon payment per six months is one-half of the annual coupon payment. There are three
months until the next coupon payment, so three months have passed since the last coupon
payment. The accrued interest for the bond is:
Accrued interest = $75/2 3/6
Accrued interest = $18.75
And we calculate the dirty price as:
Dirty price = Clean price + Accrued interest
Dirty price = $865 + 18.75
Dirty price = $883.75
22. The bond has 10 years to maturity, so the bond price equation is:
P = $871.55 = $41.25(PVIFAR%,20) + $1,000(PVIFR%,20)
Using a spreadsheet, financial calculator, or trial and error, we find:
R = 5.17%
This is the semiannual interest rate, so the YTM is:
YTM = 2 5.17%
YTM = 10.34%
CHAPTER 5 B-137
The current yield is the annual coupon payment divided by the bond price, so:
Current yield = $82.50 / $871.55
Current yield = 0.0947 or 9.47%
23. a.
The coupon bonds have an 8% coupon which matches the 8% required return, so they
will sell at par. The number of bonds that must be sold is the amount needed divided by
the bond price, so:
Number of coupon bonds to sell = $30,000,000 / $1,000 = 30,000
The number of zero coupon bonds to sell would be:
Price of zero coupon bonds = $1,000/1.0820 = $214.55
Number of zero coupon bonds to sell = $30,000,000 / $214.55 = 139,829
b.
The repayment of the coupon bond will be the par value plus the last coupon payment
times the number of bonds issued. So:
Coupon bonds repayment = 30,000($1,080) = $32,400,000
The repayment of the zero coupon bond will be the par value times the number of
bonds issued, so:
Zeroes: repayment = 139,829($1,000) = $139,829,714
c.
The total coupon payment for the coupon bonds will be the number bonds times the
coupon payment. For the cash flow of the coupon bonds, we need to account for the tax
deductibility of the interest payments. To do this, we will multiply the total coupon
payment times one minus the tax rate. So:
Coupon bonds: (30,000)($80)(1 .35) = $1,560,000 cash outflow
Note that this is cash outflow since the company is making the interest payment.
For the zero coupon bonds, the first year interest payment is the difference in the price
of the zero at the end of the year and the beginning of the year. The price of the zeroes
in one year will be:
P1 = $1,000/1.0819 = $231.71
The year 1 interest deduction per bond will be this price minus the price at the
beginning of the year, which we found in part b, so:
CHAPTER 5 B-138
Year 1 interest deduction per bond = $231.71 214.55 = $17.16
The total cash flow for the zeroes will be the interest deduction for the year times the
number of zeroes sold, times the tax rate. The cash flow for the zeroes in year 1 will be:
Cash flows for zeroes in Year 1 = (139,829)($17.16)(.35) = $840,000
Notice the cash flow for the zeroes is a cash inflow. This is because of the tax
deductibility of the imputed interest expense. That is, the company gets to write off the
interest expense for the year, even though the company did not have a cash flow for the
interest expense. This reduces the companys tax liability, which is a cash inflow.
During the life of the bond, the zero generates cash inflows to the firm in the form of
the interest tax shield of debt. We should note an important point here: If you find the
PV of the cash flows from the coupon bond and the zero coupon bond, they will be the
same. This is because of the much larger repayment amount for the zeroes.
24. The maturity is indeterminate. A bond selling at par can have any length of maturity.
25. The bond asked price is 108:14, so the dollar price is:
Percentage price = 108 14/32 = 108.4375%
Dollar price = 108.4375% $1,000
Dollar price = $1,084.375
So the bond price equation is:
P = $1,084.375 = $32.25(PVIFAR%,30) + $1,000(PVIFR%,30)
Using a spreadsheet, financial calculator, or trial and error, we find:
R = 2.805%
This is the semiannual interest rate, so the YTM is:
YTM = 2 2.805%
YTM = 5.61%
26. The coupon rate of the bond is 5.375 percent and the bond matures in 25 years. The bond
coupon payments are semiannual, so the asked price is:
P = $26.875(PVIFA3.62%,50) + $1,000(PVIF3.62%,50)
P = $864.10
CHAPTER 5 B-139
The bid-ask spread is two ticks. Each tick is 1/32, or .03125 percent of par. We also know
the bid price must be less than the asked price, so the bid price is:
Bid price = $864.10 2(.03125)(10)
Bid price = $863.48
27. Here, we need to find the coupon rate of the bond. The price of the bond is:
Percentage price = 106 17/32 = 106.53125%
Dollar price = 106.53125% $1,000
Dollar price = $1,065.3125
So the bond price equation is:
P = $1,065.3125 = C(PVIFA2.48%,14) + $1,000(PVIF2.48%,14)
Solving for the coupon payment, we get:
C = $30.38
Since this is the semiannual payment, the annual coupon payment is:
2 $30.38 = $60.76
And the coupon rate is the coupon rate divided by par value, so:
Coupon rate = $60.76 / $1,000
Coupon rate = .0608 or 6.08%
28. Here we need to find the yield to maturity. The dollar price of the bond is:
Dollar price = 94.183% $1,000
Dollar price = $941.83
So, the bond price equation is:
P = $941.83 = $27(PVIFAR%,24) + $1,000(PVIFR%,24)
Using a spreadsheet, financial calculator, or trial and error, we find:
R = 3.045%
This is the semiannual interest rate, so the YTM is:
YTM = 2 3.045%
YTM = 6.09%
CHAPTER 5 B-140
29. The bond price equation is:
P = $35.625(PVIFA3.01%,18) + $1,000(PVIF3.01%,18)
P = $1,075.92
The current yield is the annual coupon payment divided by the bond price, so:
Current yield = $71.25 / $1,075.92
Current yield = .0662 or 6.62%
30. Here, we need to find the coupon rate of the bond. The dollar price of the bond is:
Dollar price = 94.375% $1,000
Dollar price = $943.75
Now, we need to do is to set up the bond pricing equation and solve for the coupon payment
as follows:
P = $943.75 = C(PVIFA3.425%,36) + $1,000(PVIF3.425%,36)
Solving for the coupon payment, we get:
C = $31.68
Since this is the semiannual payment, the annual coupon payment is:
2 $31.68 = $63.37
And the coupon rate is the coupon rate divided by par value, so:
Coupon rate = $63.37 / $1,000
Coupon rate = .0634 or 6.34%
Challenge
31. To find the capital gains yield and the current yield, we need to find the price of the bond.
The current price of Bond P and the price of Bond P in one year is:
P:
P0 = $90(PVIFA7%,5) + $1,000(PVIF7%,5) = $1,082.00
P1 = $90(PVIFA7%,4) + $1,000(PVIF7%,4) = $1,067.74
Current yield = $90 / $1,082.00 = .0832 or 8.32%
CHAPTER 5 B-141
The capital gains yield is:
Capital gains yield = (New price Original price) / Original price
Capital gains yield = ($1,067.74 1,082.00) / $1,082.00 = .0132 or 1.32%
The current price of Bond D and the price of Bond D in one year is:
D:
P0 = $50(PVIFA7%,5) + $1,000(PVIF7%,5) = $918.00
P1 = $50(PVIFA7%,4) + $1,000(PVIF7%,4) = $932.26
Current yield = $50 / $918.00 = .0545 or 5.45%
Capital gains yield = ($932.26 918.00) / $918.00 = +.0155 or +1.55%
All else held constant, premium bonds pay high current income while having price
depreciation as maturity nears; discount bonds do not pay high current income but have
price appreciation as maturity nears. For either bond, the total return is still 7%, but this
return is distributed differently between current income and capital gains.
32. a.
The rate of return you expect to earn if you purchase a bond and hold it until maturity is
the YTM. The bond price equation for this bond is:
P0 = $1,105 = $80(PVIFAR%,10) + $1,000(PVIF R%,10)
Using a spreadsheet, financial calculator, or trial and error we find:
R = YTM = 6.54%
b.
To find our HPY, we need to find the price of the bond in two years. The price of the
bond in two years, at the new interest rate, will be:
P2 = $80(PVIFA7.54%,8) + $1,000(PVIF7.54%,8) = $1,155.80
To calculate the HPY, we need to find the interest rate that equates the price we paid
for the bond with the cash flows we received. The cash flows we received were $80
each year for two years, and the price of the bond when we sold it. The equation to find
our HPY is:
P0 = $1,105 = $80(PVIFAR%,2) + $1,155.80(PVIFR%,2)
Solving for R, we get:
R = HPY = 9.43%
The realized HPY is greater than the expected YTM when the bond was bought because
interest rates dropped by 1 percent; bond prices rise when yields fall.
CHAPTER 5 B-142
Calculator Solutions
3.
Enter
8
$90
7%
N
I/Y
PV
$1,000
PM
FV
T
Solvefor
4.
Enter
$1,119.43
9
$1,047.50
N
I/Y
PV
$80
$1,000
PM
FV
T
Solvefor
5.
Enter
7.26%
16
6.8%
N
I/Y
$1,051
PV
$1,000
PM
FV
T
Solvefor
$73.33
Coupon rate = $73.33 / $1,000
Coupon rate = .0733 or 7.33%
6.
Enter
14 2 =
N
$85 / 2 =
7.90% / 2 =
I/Y
PV
PM
$1,000
FV
T
Solvefor
7.
Enter
$1,050.28
13 2 =
N
$960
I/Y
PV
$84 / 2 =
$1,000
PM
FV
T
Solvefor
4.463%
YTM = 4.463% 2
YTM = 8.93%
8.
Enter
10.5 2 =
N
6.9% / 2 =
I/Y
$1,070
PV
$1,000
PM
FV
T
Solvefor
$39.24
CHAPTER 5 B-143
Annual coupon = $39.24 2
Annual coupon = $78.48
Coupon rate = $78.48 / $1,000
Coupon rate = 7.85%
CHAPTER 5 B-144
15.
Enter
Bond X
13
$90
7%
N
I/Y
PV
$1,000
PM
FV
T
Solvefor
Enter
$1,167.15
12
$90
7%
N
I/Y
PV
$1,000
PM
FV
T
Solvefor
Enter
$1,158.85
10
$90
7%
N
I/Y
PV
$1,000
PM
FV
T
Solvefor
Enter
$1,140.47
5
$90
7%
N
I/Y
PV
$1,000
PM
FV
T
Solvefor
Enter
$1,082.00
1
$90
7%
N
I/Y
PV
$1,000
PM
FV
T
Solvefor
Enter
$1,018.69
Bond Y
13
$70
9%
N
I/Y
PV
$1,000
PM
FV
T
Solvefor
Enter
$850.26
12
$70
9%
N
I/Y
PV
$1,000
PM
FV
T
Solvefor
$856.79
CHAPTER 5 B-145
Enter
10
$70
9%
N
I/Y
PV
$1,000
PM
FV
T
Solvefor
Enter
$871.65
5
$70
9%
N
I/Y
PV
$1,000
PM
FV
T
Solvefor
$922.21
CHAPTER 5 B-146
Enter
1
$70
9%
N
I/Y
PV
$1,000
PM
FV
T
Solvefor
$981.65
16. If both bonds sell at par, the initial YTM on both bonds is the coupon rate, 8 percent. If the
YTM suddenly rises to 10 percent:
PBill
6
5%
$40
$1,000
Enter
N
I/Y
PV
PM
FV
T
Solvefor
$949.24
PTed
Enter
40
$40
5%
N
I/Y
PV
$1,000
PM
FV
T
Solvefor
$828.41
PBill% = ($949.24 1000) / $1,000 = 5.08%
PTed% = ($828.41 1000) / $1,000 = 17.16%
If the YTM suddenly falls to 6 percent:
PBill
6
3%
Enter
N
I/Y
$40
PV
$1,000
PM
FV
T
Solvefor
$1,054.17
PTed
Enter
40
$40
3%
N
I/Y
PV
$1,000
PM
FV
T
Solvefor
$1,231.15
PBill% = ($1,054.17 1000) / $1,000 = +5.42%
PTed% = ($1,231.15 1000) / $1,000 = +23.11%
All else the same, the longer the maturity of a bond, the greater is its price sensitivity to
changes in interest rates.
17. Initially, at a YTM of 7 percent, the prices of the two bonds are:
CHAPTER 5 B-147
PJ
Enter
16
$20
3.5%
N
I/Y
PV
$1,000
PM
FV
T
Solvefor
$818.59
CHAPTER 5 B-148
PS
Enter
16
$50
3.5%
N
I/Y
PV
$1,000
PM
FV
T
Solvefor
$1,181.41
If the YTM rises from 7 percent to 9 percent:
PJ
16
4.5%
Enter
N
I/Y
$20
PV
$1,000
PM
FV
T
Solvefor
$719.15
PS
Enter
16
$50
4.5%
N
I/Y
PV
$1,000
PM
FV
T
Solvefor
$1,056.17
PJ% = ($719.15 818.59) / $818.59 = 12.15%
PS% = ($1,056.17 1,181.41) / $1,181.41 = 10.60%
If the YTM declines from 7 percent to 5 percent:
PJ
16
2.5%
Enter
N
I/Y
PV
$20
$1,000
PM
FV
T
Solvefor
$934.72
PS
Enter
16
$50
2.5%
N
I/Y
PV
$1,000
PM
FV
T
Solvefor
$1,326.38
PJ% = ($934.72 818.59) / $818.59 = +14.19%
PS% = ($1,326.38 1,181.41) / $1,181.41 = +12.27%
All else the same, the lower the coupon rate on a bond, the greater is its price sensitivity to
changes in interest rates.
18.
CHAPTER 5 B-149
Enter
18 2 =
N
$930
I/Y
PV
$80 / 2 =
$1,000
PM
FV
T
Solvefor
YTM = 2 4.39%
YTM = 8.78%
4.39%
CHAPTER 5 B-150
Effective annual yield:
Enter
8.78%
NO
Solve for
EFF
2
C/Y
8.97%
19. The company should set the coupon rate on its new bonds equal to the required return; the
required return can be observed in the market by finding the YTM on outstanding bonds of
the company.
Enter
20 2 =
N
$1,062
$75 / 2 =
$1,000
PV
PM
FV
I/Y
T
Solvefor
3.46%
YTM = 2 3.46%
YTM = 6.92%
22.
Enter
10 2 =
N
$871.55
I/Y
PV
$82.50 / 2 =
PM
$1,000
FV
T
Solvefor
5.17%
YTM = 2 5.17%
YTM = 10.34%
23. a. The coupon bonds have an 8% coupon which matches the 8% required return, so they
will sell at par. For the zeroes, the price is:
Enter
20
$1,000
8%
N
I/Y
PV
PM
FV
T
Solvefor
c.
Enter
$214.55
The price of the zeroes in one year will be:
19
$1,000
8%
N
I/Y
PV
PM
FV
T
Solvefor
25.
Enter
$231.71
15 2 =
$1084.375
$64.50 / 2
$1,000
CHAPTER 5 B-151
N
I/Y
PV
PM
FV
T
Solvefor
YTM = 2 2.805%
YTM = 5.61%
2.805%
CHAPTER 5 B-152
26.
Enter
25 2 =
N
$53.75 / 2 =
6.48% / 2 =
I/Y
PV
PM
$1,000
FV
T
Solvefor
27.
Enter
$864.10
72=
N
4.96% / 2 =
I/Y
$1,065.3125
PV
$1,000
PM
FV
T
Solvefor
$30.38
Annual coupon = $30.38 2
Annual coupon = $60.76
Coupon rate = $60.76 / $1,000
Coupon rate = .0608 or 6.08%
28.
Enter
12 2
N
$941.83
I/Y
PV
$54 / 2
$1,000
PM
FV
T
Solvefor
3.045%
YTM = 2 3.045%
YTM = .0609 or 6.09%
29.
Enter
92
N
6.02% / 2
I/Y
$71.25 / 2
PV
PM
$1,000
FV
T
Solvefor
30.
Enter
$1,075.92
18 2
N
6.85% / 2
I/Y
$947.35
PV
$1,000
PM
FV
T
Solvefor
Annual coupon = $31.68 2
Annual coupon = $63.37
Coupon rate = $63.37 / $1,000
Coupon rate = .0634 or 6.34%
$31.68
CHAPTER 5 B-153
CHAPTER 5 B-154
31.
Bond P
P0
Enter
5
N
7%
I/Y
PV
$90
PM
$1,000
FV
T
Solve for
P1
Enter
$1,082.00
4
N
7%
I/Y
PV
$90
PM
$1,000
FV
T
Solve for
$1,067.64
Current yield = $90 / $1,082.00 = 8.32%
Capital gains yield = ($1,067.64 1,082.00) / $1,082.00 = 1.32%
Bond D
P0
Enter
5
N
7%
I/Y
PV
$50
PM
$1,000
FV
T
Solve for
P1
Enter
$918.00
4
N
7%
I/Y
PV
$50
PM
$1,000
FV
T
Solve for
$932.26
Current yield = $50 / $918.00 = 5.45%
Capital gains yield = ($932.26 918.00) / $918.00 = +1.55%
All else held constant, premium bonds pay high current income while having price
depreciation as maturity nears; discount bonds do not pay high current income but have
price appreciation as maturity nears. For either bond, the total return is still 8%, but this
return is distributed differently between current income and capital gains.
32.
a.
Enter
10
N
I/Y
$1,105
PV
$80
PM
$1,000
FV
T
Solve for
6.54%
This is the rate of return you expect to earn on your investment when you purchase the bond.
CHAPTER 5 B-155
b.
Enter
8
N
5.54%
I/Y
PV
$80
PM
$1,000
FV
T
Solve for
$1,155.80
The HPY is:
Enter
2
N
I/Y
$1,105
PV
$80
PM
$1,155.80
FV
T
Solve for
9.43%
The realized HPY is greater than the expected YTM when the bond was bought because
interest rates dropped by 1 percent; bond prices rise when yields fall.

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DatabasePracticeAccounting 307PART I ACCESS QUIZ PRACTICENote: You should work through the entire Query Practice document BEFORE looking at thisdocument. If you use this practice quiz as your primary source of studying, you will likely failthe quiz.

S.F. State - ACCT - 307

CHAPTER 17 Instigations and CommitmentsSeveral events usually occur before an economic transaction occurs:1. Inquiry2. Promise3. ExchangeWe can expand the basic REA pattern to include these additional events:InstigationfulfillsCommitmentexecutes

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CHAPTER 15 Building and Using the DatabaseTerminologyFormaltermCommon namesSQL commandentitytupleattributeCardinalities See PowerpointBuilding a database from an ER diagram (ERD)insidepart.1(1,1)Resource -Event (0,N)INVENTORYstockflow1(

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Chapter 15: Introduction to READata models depict user requirements for data which is stored in a databaseData is stored in tables that look like spreadsheetsWe can use queries to retrieve data from multiple tables at once to answer a questiono For ex

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CHAPTER 8 OPERATION AND INFORMATION CONTROLSOperation ControlsThe ERM framework details that internal controls should be employed to ensure thatmanagements objectives and policies are being met. Therefore, many internal controls are usedto make sure t

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CHAPTER 8 SECURITY CONTROLSSegregation of duties no single employee should be in a position to perpetrate and concealfraud, errors, or other kinds of systems failures.Segregation of Accounting DutiesFour functions should be separated1.2.3.4.Segre

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CHAPTER 5Internal controls why are they important?Sarbox o Increases responsibility and accountability of CEOs and directors Section 302:Section 404:o Increases white-collar crime penaltieso Auditors and consultingWhat does this do to the accounti

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CHAPTER 7 OUTLINE - ERMCOSO Framework Enterprise Risk ManagementCOSO Internal Control Framework (ICF) The ICF had a major impact on guiding internal control frameworksCOSO 1992Internal ControlFrameworkObjectives : efficiency ,effectiveness , r ep

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CHAPTER 5 DATABASE MANAGEMENT SYSTEMSTwo approaches to storing business event informationApplications approach DATAUser 1TransactionProgram 1A,B,CUser 2TransactionProgram 2X,B,YUser 3TransactionProgram 3L,B,Mo Disadvantageso AdvantagesDa

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Chapter 2 Business ProcessesScope of the AIS Figure 2-1THE ENTREPRENEURIAL SCRIPTDualities Entrepreneurial script The Entrepreneurial Script (extension of Figure 2-2)FINANCINGCASHRECEIPTCASHPAYMENT$$PROCUREMENT$PAYMENTTRANSPORTATIONLOGIS

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CHAPTER 4 OUTLINE DFDs AND SYSTEM FLOWCHARTSDATA FLOW DIAGRAMSData flow diagrams (DFD) Where are DFDs used?DFDs consist of only four symbols:Bubble Arrow External entity Data store The basic concepts of a data flow diagram: explode and balanceo

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CHAPTER 3: E-BUSINESS SYSTEMSE-business:Types of E-business B2B:B2C:Three barriers to e-business1. Displacing paper records with electronic ones2. Technological barriers to linking two companies together3. Securing B2B and B2C transactionsHow doe

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CHAPTER 2 ENTERPRISE SYSTEMSEnterprise systems vs. ERPsERPsERP modules:What are some major modules in SAP R/3?Enterprise application integrationCustomizationWho are the major ERP vendors?EPMrk t Sa Etimte - F 2 0R a e hre s a s Y 0 6Su e A R os

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CHAPTER 1The Changing Role of the AccountantHow has technology changed the role of the accountant?The Accountants Job TodayPublic accountingo Tax accounting and advising (TAX)oPrepare tax formsTax planning and consulting (individuals, small compan

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Cardinalities Practice SolutionsSEMANTIC PROBLEMSALLRESTAURANTSIN SANFRANCISCO(1,1)(0,1)ALLRESTAURANTSIN CALIFORNIA(1,1)(0,1)ALLSFSU EMAILADDRESS1.SFSU STUDENT2.By policy, each SFSU student gets one and only one SFSU email account. Howe

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CARDINILATIES PRACTICESEMANTIC PROBLEMS1.ALLRESTAURANTSIN SANFRANCISCO(_,_)( _,_)ALLRESTAURANTSIN CALIFORNIA( _,_)( _,_)ALLSFSU EMAILADDRESSES(_,_)( _,_)FINANCEMAJORS2.SFSU STUDENT3.ACCOUNTINGMAJORSNOTE: Some students may double-

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An REA Modeled EnterpriseElm S s Inc.os hoeMonday AMroiniste Loanr Our he borrows $100 fromtheSSharkAgentinside part . 1Resource -Event stockflow1Agentoutside part . 1GivedualityTakeAgentinside part . 2Resource +Event +Agentstockfl

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VALUECHAINANALYSISAnenterprisemodelFirst,getsomemoneyFINANCINGFirst,getsomemoneyFINANCINGCASHPAYMENTCASHRECEIPT$$(LATER)$$Now,letsbuylaborFINANCINGCASHRECEIPTCASHPAYMENT$HUMANRESROUCESBuyinglabor$$HUMANRESOURCESLABORACQUIREP

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Problem2problem3problem4problem 4problem 5

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Problem 32nd attempt3rd Attempt

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2nd attempt

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2nd attempt

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2nd attempt

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Problem 72nd

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Problem 3 1st Attempt2nd attempt3rd attempt1st attempt 2nd2nd attempt

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Problem 8

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Oklahoma State - FPST - 2023

IntroductiontoOccupationalSafetyQ.Wang,PhDOklahomaStateUniversity11.WhatisOccupationalSafety Protectthesafetyandhealthofpeopleengagedinworkoremployment,andfosterasafeworkenvironment. Interactionsamongmanysubjectareas:IndustrialHygiene,SafetyEngi

Oklahoma State - FPST - 2023

Chapter1SafetyThroughDesignDr.QingshengWangOklahomaStateUniversity11.Introduction DefiningsafetythroughdesignIntegratinghazardanalysisandriskassessmentmethodsearlyinthedesignandredesignprocessandtakingtheactionsnecessarysothattherisksofinjuryord

Oklahoma State - FPST - 2023

Chapter2Dr. Qingsheng WangOklahoma State UniversityRemovinghazardsratherthanaddingprotectiveequipmentMajorconsiderationsa.b.c.d.DesignofworkplaceCompliancewithCodesandStandardsSize,shape,andtypeoffacilitiesSafetyprocedureandfireprotectionstan

Oklahoma State - FPST - 2023

ConstructionofFacilitiesChapter3ClicktoeditMastersubtitlestyleDr.QingshengWangOklahomaStateUniversityOutline1.2.3.4.5.6.SafetyinConstructionElementsofaSafetyPlanConstructionProcessInsuranceContractorSelectionContractCloseout1.SafetyinCon

Oklahoma State - FPST - 2023

Chapter4MaintenanceofFacilitiesDr.QingshengWangOklahomaStateUniversity1Outline1. FacilityMaintenance1. GroundsMaintenance1. MaintenanceCrews21.FacilityMaintenance Caretoinsurelongtermlifeofcompanyassets Routinecaretomaintainuninterruptedserv

Oklahoma State - FPST - 2023

Chapter6SafeguardingDr.QingshengWangAssistantProfessorClicktoeditMastersubtitlestyleFireProtectionandSafetyTechnologyOklahomaStateUniversityOutline1.2.3.4.DefinitionsMechanical HazardsSafeguardsControl of Hazardous EnergySource1. Definitio

Oklahoma State - FPST - 2023

Name: _FPST 2023 Lab 1High cost of poultry processingLab ReportPlease write a one-page double space lab report summarizing what you have seen andwhat you have learned during this video watching. The report should relate with your coursestudy in occu

Oklahoma State - FPST - 2023

Name: _FPST 2023 Lab 3Piper Alpha DisasterLab ReportPlease write a one-page double space lab report summarizing what you have seen andwhat you have learned during this video watching. The report should relate with your coursestudy in occupational sa

Oklahoma State - FPST - 2243

FPST 2243 EXAM 1 FALL 2000The exam is open NFPA 13, but no other notes or materials are permitted. The point value of eachquestion is as indicated. It is NOT necessary to give the section of the standard from which your answerscome.(4)1.In which ear

Oklahoma State - FPST - 2243

F PST 2243 Exam 1 Fall 2002(5) 1. Give the names for the indicated piping components of the typical sprinkler sprinkler system shownbelow.A. Branch line; B. Riser Nipple; C. Cross main; D. Feed main; E: Riser(10) 2.Draw simple schematic r iser diagra

Oklahoma State - FPST - 2243

F PST2243 EXA M 1 SPR I NG 1998Each question is worth 5 points. I t is NOT necessary to give the section reference.1. According to NFPA 13, what is a small room?(3.3.17) light hazard/unobstructed construction/floor area < 800 sq ft.2. What is the maxi

Oklahoma State - FPST - 2243

F PST 2243 Exam 1 Spring 2004Respond as directed to the following questions. The point value of each is indicated. IT ISN OT NECESSARY TO L IST CODE SECTIONS.(10) 1. Give the complete occupancy classification of the following:a.Underground parking ga