**Unformatted text preview: **Week 3: The Time Value of Money and Interest Rates Saving and Investment
Saving – Foregone consumption
Typically, people will not let their money sit
around in a box if they can do something with
it that provides them with a return. Such
opportunities exist.
Investment – Addition to the stock of capital
goods. Capital goods are goods that may be
used to produce other goods or services in the
future. Practice
Are each of the following saving or investment?
Bob deposits $10,000 in a money-market account.
Bob buys a delivery truck for his furniture moving business.
Bob pays his workers to go to trade school to learn about
furniture delivery.
Bob buys a used house as a rental property.
Bob purchases 1000 shares of IBM stock.
4–3 Saving and Investment
Savers are responsible for lending.
Savers are paid interest in exchange for loaning
their money to borrowers.
Borrowers are able to pay back the principal
and interest by investing in capital goods that
generate even greater returns. Financial Markets
Financial markets facilitate the lending of funds
from saving to those who wish to undertake
investments.
Those who wish to borrow to finance
investment projects sell IOUs to savers. The
various forms of IOUs are known as financial
instruments or securities.
These IOUs are claims that those who lend their
savings have on the future incomes of Primary and Secondary Markets - Review
Primary Market – Market in which newly
issued financial instruments are purchased
and sold. For instance, an Initial Public
Offering.
Secondary Market – Market in which
instruments that have not reached maturity
(the time the final interest and principal
payments are due) are traded. Money Markets
Money Markets – Markets for financial instruments
with short-tem maturities. High liquidity, lowrisk.
Treasury bills (U.S. government debt)
Commercial paper (short term debt by
corporations)
Certificates of deposit (deposits that cannot be
withdrawn without penalty). Capital Markets
Capital Markets – Markets for financial instruments
with intermediate and long-term maturities.
Equity (shares of ownership)
Common stock (entitles the shareholder to say in
operation of the company – common stock holders
are residual claimants)
Preferred stock (no say in the operation of the
company, but receive priority over common stock
holders) Capital Markets
Long-term treasury notes and bonds.
Corporate bonds
Municipal Bonds
Mortgage backed securities (Oh, will we talk about
these more later. Yes we will.)
Packaged consumer and commercial loans CRITICAL TAKEAWAY
In this lecture we will discuss the basics of
calculating of present and future value. Failure
to have a solid understanding of PV and FV
will have a strongly negative impact on your
performance in the course.
Besides, this is one of the things you can
honestly benefit from. 4–10 Calculating Interest Yield
Principal:
The amount of credit extended when one makes a loan or
purchases a bond.
Example: Assume I bought a house for $164,000, and borrowed
$161,000. That $161,000 is principal. Interest:
The price paid by a borrower for the use of an asset they do not own. Interest rate:
The percentage return, or percentage yield, earned by the lender. Why Pay Interest?
There is a market for loanable funds.
What determines a borrower’s maximum
willingness-to-pay for interest on a loan?
What determines a lender’s minimum
willingness-to-accept for interest on a loan? 4–12 Consumption How much house will you buy at 4.25%
interest? 14% interest?
How much will you finance a car at 0%
interest? 12.99% interest?
How much will you put on your credit card at
8.99% interest? 24.99% interest? 4–13 Saving/Investment
How much will you save if interest rates are 0.25%?
4.00%?
Will you be more or less likely to invest in
equipment upgrades if the cost of borrowing is 3%
or 10%?
How will you allocate your portfolio if savings
accounts pay 0.50% and stocks average 11%?
What if savings accounts pay 4.50% and stocks
average 7%?
4–14 Supply Curve
Slopes upwards because as
interest goes up, more people
want to lend more funds. Interest Rate
The interest rate is the cost of
borrowing and/or revenue
from lending. R*
Demand Curve
Slopes down because as interest goes
down, borrowing becomes cheaper. Q* Quantity of
Funds
4–15 Present and Future Value
We’ll start with the concept of present (discounted)
value.
This is based on the notion that $1.00 you receive
one year from now is less valuable to you than
$1.00 you receive today.
This is true for three reasons, that help determine
the supply of and demand for loanable funds:
1) Time preference
2) Foregone interest
3) Expected inflation. Time Preference
The premium a consumer places on present consumption
relative to future consumption.
Neoclassical theory: The interest rate determines the price of
present and future consumption, so the marginal rate of
substitution between present and future consumption must
equal the interest rate.
In plain(er) language: The highest interest rate you’re willing
to borrow at or the lowest interest rate you’re willing to lend at
should make you indifferent between borrowing and not
borrowing, or lending and not lending. You should be exactly
compensated for changing the timing of your consumption.
4–17 A Simple Loan
Lender provides borrower with principal.
A gives B $100 today
Borrower repays the principal at the maturity
date along with some additional interest
payments.
B pays A $100 + $10 one year from today
The simple interest rate equals interest
payments divided by the amount of the loan.
i = $10/$100 = 0.1 = 10%
4–18 A Simple Loan
If you had $100 to lend today at i = 10%, you receive later:
$100 (1 + i) = 100(1 + 0.1) = $110
If you relend $110 next year at i = 10%, you receive at the end
of the second year:
$100(1 + 0.1)(1 + 0.1) = $100(1 + 0.1) 2 = $121
….$100(1 + 0.1)3 = $133
Generalizing, at the end of n years, $100 will grow into $100(1
+ 0.1)n.
The PV of $133 in 2013 is $100.
Almost all finance formulas involve the compounding of
interest. You earn interest on past interest.
4–19 Compounding
“The most powerful force in the universe is compound
interest.” – attrib. to Albert Einstein
Imagine a checkerboard. We put one penny on the first
square, two pennies on the second square, four on the
third square. How much will the last square be worth?
That’s a 100% interest rate (doubling each period) for
64 periods (64 squares).
$0.01*(1+1.00)64
4–20 Present And Future Value
FV = PV * (1 + i)n <- Memorize this!
PV = FV / (1 + i)n <- This too!
PV – present value – what you’d have to invest today to
yield a given future return
FV – future value – what you’d receive in the future
given an investment today
i – simple interest rate
n – number of periods
PV of $200 to be paid in 4 years at i = 5%?
PV = $200/(1 + 0.05)4 = $164.50
4–21 The General Form
of the Present-Value Formula
FN
F1
F2 ... P
.
1
2
N
(1 i )
(1 i )
(1 i ) Each F is a payment in the future. The subscripts
1,2,N etc. represent the number of time periods
out from the present. Present Value of a Perpetuity
Perpetuity = financial security that never matures. It
pays interest forever, does not repay principal (eg. a
share of stock in a corporation)
Example: N = 1 year
F = $10,000
Perpetuity
Time
(years)
Payment 0 1 2 3 4 . . . F F F F . . . Perpetuities (cont’d)
Would you ever want to own a perpetuity?
Is the present value really infinite?
How much would you pay for it?
To calculate: P = F / i The present value of each successive
payment is less than the last…Therefore,
even the present value of a perpetuity is
finite. Perpetuities (cont’d)
Present value of
perpetuities are also
affected by the rate of
discount.
Compare the sensitivity of P
to i
where F = $1,000
i = .08
P = $12,500
i = .05
P = $20,000
i = .02
P = $50,000 Fixed Payment Securities
Fixed-payment security: The dollar payments are the
same every year so that the
principal is amortized
Amortization: The process of repaying a loan’s
principal gradually over time Using Present Value Formula
to Calculate Payments
Sometimes we already know present value, but are concerned with size of payments to be
made to the lender… Example: Mortgage loan
P = $100,000
monthly payments for 30 years (N = 360)
annual interest rate = 6% (therefore i = .
06/12 = 0.005) Using Present Value Formula
to Calculate Payments so Real World Applications
My car died over the summer after 10 years of
faithful service. So I went to buy a new one that
cost about $25,000. Customer incentives were
available.
0.0% financing for up to 60 months
$750 off the price of the car. 4–29 The Car Example
Which one’s better.
You can have a 0% interest rate for 60 months
or $750 off the price of the car. Assume your
credit qualifies you for 4% interest rate
normally.
I want to convince you that knowing present
and future value will actually help you outside
of coursework.
30 The Car Example
If you take the 0.0% rate, the car costs $25,000. Since
there is no interest, the final cost over 5 years is $25,000.
If you take the 4.0% rate, the car costs $24,250 after the
$750 cash back. Recall the formula to calculate payments. Over 60 months, the car would cost $26,793.41 (60
payments of $446.56). Thus, taking the 0.0% interest
would make more sense…or does it.
31 The Car Example
There are other considerations. For example, what if you had a down
payment. In fact, to be extreme, what if you had $25,000 and could buy
the car outright. It would cost $25,000 if you financed it at 0% and
$24,250 if you took the cash back option. You’d want the cheaper car,
right?
Well, imagine you had a savings account that paid 1.5% interest per
year, compounded annually. If you bought the car for $24,250, you’d
have $750 to put into that account. At the end of 5 years, you’d have: F The actual interest earned on the account if you put the $25,000 into it
and made payments at 0% interest for 60 months would be very hard
to calculate as $416.67 would be pulled out each month (the payment
on a $25,000 loan for 60 months at 0% interest ($25,000/60).
So let’s approximate. This will come in handy later.
32 The Car Example If you’re paying off zero interest loan at a constant rate, the
principal remaining looks like a right triangle. Or assume you took
$100 out of the ATM each morning and spent it at a constant rate
all day. The amount left in your pocket would again look like this
triangle.
So what’s the average amount you’d have over the course of the
day? $50. The height of the midpoint of the triangle. You’d have
more than $50 half the time and less than $50 half the time. 33 The Car Example So if the loan is $25,000, then on average, you owe $12,500 (half
the loan). To approximate, let’s assume that on average, you’d have
about $12,500 in your savings account (which would be true if you
didn’t reinvest your interest earned in that account).
Interest earned on $12,500 over 5 years at 1.5% = $966.05.
It’s still worth financing even if you could pay cash, because you
could hold onto your money and make more than $807.96
investing it into a savings account. 34 The Car Example
So when might you rather take the $750 cash back?
No profitable alternative option for your money. If you couldn’t earn interest on your money that would add up to
more than the $750 saved plus interest on it, you’d rather
take the $750 off.
You’re just debt averse. Economists are not all about
money. Say you owe your mother $1,000 at 0% interest
and a credit card company $1,000 at 20% interest. It
would clearly minimize your payments to pay down the
20% interest loan first, but if the guilt of borrowing from
your mother hurts worse than paying 20% interest, paying
your mother first is rational.
35 Practice Problem
Say you are interested in a Nissan Altima 3.5SR that
you’ve negotiated a price of $29,000 for (all taxes
and fees included).
You can get 0% financing for up to 60 months, or get
3% financing for up to 60 months and get $2,000
cash back. Assume there is no alternative
investment option and you can not make a down
payment.
Which is better?
36 Present Value of a Coupon Bond
Coupon Bond: Pays a regular interest
payment until maturity, when face value
is repaid (e.g. most corporate &
government bonds)
Time
(years) 0 Payments
Interest
Face Value 1 2 3 4 . . . N F F F F . . . F
V Present Value of a Coupon Bond (cont’d)
To calculate present value: interest payments are
fixed-payment security present value of
face value Here’s an Example
Three year bond. 5% interest. Face value of $10,000. Annual
coupon payments of $600.
To calculate the present value (price of the bond), we calculate
the present value of ALL PAYMENTS and sum them.
Payment 1: Coupon payment of $600 in one year. $600/1.05.
Payment 2: Coupon payment of $600 in two years. $600/1.05 2
Payment 3: Face value of $10,000 plus $600 coupon in three
years: $10,600/1.053
$571.43 + $544.22 + $9,156.68 = $10,272.33 4–39 Payments More Than Once Per Year
Many securities require payments more frequently
Semi-annually: Government & corporate bonds
Quarterly: Many stock dividends
Monthly: Consumer & business loans
Because of compounding, this frequency must be accounted for in calculating present value Payments More Than Once Per Year
Time period needs to be adjusted to account for payment frequency
Assume that interest compounds each
period and N = number of periods to
maturity
Example: 30 year mortgage at 9% N = 360 (12 months x 30 years) i = 0.0075 (0.09/12 months) Semi-annual coupons
$10,000 face value. 5% interest. Three years to maturity. $300
semi-annual coupon payment.
Payment
Payment
Payment
Payment
Payment
Payment 1:
2:
3:
4:
5:
6: $300 in 6 months
$300 in 1 year
$300 in 18 months
$300 in 2 years
$300 in 30 months
$10,300 in 3 years PV = $300/1.025
PV = $300/1.025 2
PV = $300/1.025 3
PV = $300/1.025 4
PV = $300/1.025 5
PV = $10,300/1.025 6 $292.68 + $285.54 + $278.58 + $271.79 + $265.16 + $8,881.66
= $10,275.41
Compare to $571.43 + $544.22 + $9,156.68 = $10,272.33
Why a little higher with the semi-annual coupons?
4–42 Present Value & Decision Making
Comparing alternative offers
A magazine subscription costs $50 for 1
year or $95 for 2 years. Which is better?
Comparing coupon bonds: use one as an alternative for the other; use the interest
rate on one bond as the rate of discount on
other bonds in the secondary market and
see if you get the same present value. Interest-Rate Risk
Why does the price of a security change when the market interest rate changes?
This uncertainty is interest-rate risk
Reflects a change in opportunities… suppose
you buy a bond paying 6% but market rates
rise to 8%. Does your bond price rise or fall? A Little Emphasis Prices of existing
bonds are
INVERSELY related
to changing market
interest rates.
4–45 Think about it
You purchase a three-year bond with a face value of $10,000 and a $500
annual coupon payment and the market interest rate is 5%.
PV = $500/1.05 + $500/1.052 + $10500/1.053
PV = $476.19 + $453.51 + $9070.30 = $10,000.00
Now, assume market interest rate falls to 3% immediately after purchase.
PV = $500/ 1.032 + $500/ 1.032 + $10500/ 1.033
PV = $485.44 + $471.30 + $9608.99 = $10,565.72
Why does the bond price increase? When the interest rate falls to 3%, an
asset with a locked-in interest rate of 5% looks more attractive. It would
take $10,565.72 invested now at 3% to yield the same present value as
that $10,000 face value bond with $500 coupons at 5%. 4–46 Think about it
You purchase a three-year bond with a face value of $10,000 and a $500
annual coupon payment and the market interest rate is 5%.
PV = $500/1.05 + $500/1.052 + $10500/1.053
PV = $476.19 + $453.51 + $9070.30 = $10,000.00
Assume market interest rate rises to 7% after year 1.
PV = $500/1.07 + $500/ 1.072 + $10500/ 1.073
PV = $467.29 + $436.72 + $8571.13 = $9,475.14
Why does the bond price decrease? Because after the first year, you have
this 5% bond, but you could be earning 7% elsewhere. It only takes
$9,475.14 invested today to earn the same as your $10,000 face value
bond at 5% with $500 coupons, so people aren’t willing to pay more and
you’re not willing to sell for less. That’s the price.
4–47 Interest-Rate Risk (cont’d)
Bond price = present value of bond
Present value of bond
inversely related to i
Bond price inversely
related to i Interest-Rate Risk (cont’d) Another Interest Rate Change Example
A $1000 bond matures in one year and pays
$50 interest. Other 1-year bonds also have
interest rates of 5%.
P = $1050/1.05 = $1000
What if market interest rate fell (just after you bought it) to 4%?
P = $1050/1.04 = $1009.62 Interest-Rate Risk (cont’d) Example (continued)
What if the market interest rate rose (just
after you bought it) to 10%?
P = $1050/1.10 = $954.55 Why did P change?
Market opportunity! Policy Insight: Annual Percentage Yield (APY)
Annual Percentage Yield = The annual interest rate that would give you the same amount you
would earn with more frequent compounding than
with the stated annual interest rate The U.S. government requires banks to report APY on savings. APY offers a way to compare investment with different periods of compounding. Example: Which is better to invest in?
A: 8.0% compounded annually
B: 7.95% compounded monthly Policy Insight:
Annual Percentage Yield (APY) (cont’d)
Example (cont.)
A: $1000 × 1.081 = $1080
B: $1000 × [1+(.0795/12)]12 = $1082.46 Option B is a better investment
To compare easily, define:
APY = [1 + (i/x)]x – 1
where compounding occurs x times per
year
APY(A) = .08; APY(B) = .08246. Different Concepts of Interest Yield
Coupon return:
A fixed interest return that a bond yields each
year.
Nominal yield:
The coupon return on a bond divided by the
bond’s face value; rn=C/F.
Current yield:
The coupon return on a bond divided by the
4–53 Example
Coupon return:
$600 annually on our 5%, $10,000 face value bond
with a price (PV) of $10,272.33.
Nominal yield:
The coupon return on a bond divided by the bond’s
face value; rn=C/F. $600/$10,000 = 6%
Current yield:
The coupon return on a bond divided by the bond’s
market price; rc=C/P. $600/$10,272.33 = 5.84%
4–54 Capital Gains
Capital gain:
An increase in the value of a financial instrument at
the time it is sold as compared with its market value
at the date it was purchased. I buy a house for $161,000. I sell it for $195,000 in
4 years. Capital gain? $34,000
Professor Price buys a house in 2006 in Reno for
$310,000. In 2009, it is worth $150,000. What is
the capital loss on sale?
4–55 Yield to Maturity
Yield to maturity:
The rate of return on a bond if it is held until it matures, which reflects the market price of the bond, the bond’s
coupon return, and any capital gain from holding the bond
to maturity. Perpetuity:
A bond with an infinite term to maturity.
Perpetuity price = C/r. Simple rule:
Prices of existing bonds are inversely related to changing market interest rates.
Higher interest rates causes bond prices to fall.
4–56 Back to Yield to Maturity
Interest rate that equates the PV of future cash flow
payments received on a debt instrument with its
value/price today.
Used to compare the returns on alternative credit
market/debt instruments.
What are these instruments?
In terms of the timing of their cash flow payments,
there are four basic types.
4–57 Simple Loan (already mentioned)
Lender provides the principal
Borrower repays principal plus interest at maturity.
Examples include commercial business loans
Example: P = $100. P + interest = $110 next year.
PV = FV / (1 + i)n
$100 = $110 / (1 + i)
i = 10%
4–58 Fixed-Payment Loan
Lender provides borrower with principal
Borrower repays some fixed amount every period
Periodic payments consist of both principal and interest.
Common examines: auto loans, mortgages
Example: Loan = $1000
Payments = $85.81 for 25 years
YTM sets PV of all future payments equal to the value of the
loan today.
$1000 = $85.81/(1 + i)1 + $85.81/(1+i)2 + … $85.51/(1+i)25
i = 7%
4–59 Coupon Bond
Borrower pays the owner of the bond a fixed interest payment
(coupon payment) every period (i...

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