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**Unformatted text preview: **R. 1. R. 2. R. 3. R. 4. R. 5. R. 6. CHAPTER 5 REVIEW QUESTIONS An annuity is a series of equal cash flows through time. A perpetuity is a
series of equal cash flows though time that continue forever. Therefore, a
perpetuity is a special type of annuity. Pure discount securities are securities that pay a single cash flow. The
return on the pure discount security is comprised of the difference between
the purchase price and the security’s value at maturity or when sold. Treasury bills are discount securities issued by the US. Treasury because
they offer no interim cash flow payments but instead are issued at a price
below its par or maturity value. Bond prices are inversely related to changes in market interest rates, that
is, rising interest rates reduce the value of outstanding bonds, and falling
interest rates increase the value of outstanding bonds. When market interest
rates rise, investors comparing already issued bonds with new bonds will place
a higher value on the newly issued bonds. In this scenario, the price of the
already issued bonds must fall to keep competitive with new bonds. In a
similar way, the price of already issued bonds will rise when market interest
rates fall because these bonds become more attractive when compared with newly
issued bonds. Bonds paying interest on a semiannual basis will place dollars in the hands of
the bond investor sooner than if interest were paid on an annual basis. Given
the time value of money, paying semiannual interest makes a bond more valuable
than it would be with annual coupon payments. A bond’s yield to maturity is the interest rate that discounts the promised
future cash flows to a present value equal to the bond’s current price. The
yield to maturity can be thought of as the average rate of return that would
be earned on a bond each year if purchased at the current price and held to
maturity. Determining the value of a bond means finding its price given the set of cash
flows and the bond’s discount rate or required rate of return. Bonds, known
also as fixed payment securities, offer a fixed stream of cash flows that are
known to any bond investor. In contrast, the set of cash flows to be received
by common stock investors are not fixed but rather are represented by the cash
flow left over after the claims of all other people who contract through the
corporation are satisfied. These residual cash ﬂow amounts, when compared
with a bond’s fixed payments, are more difficult to estimate. 22 R. 7. R. 8. R. 9. Related to question 6 above, the value of common stock requires estimates of
residual cash flows. One simple model of common stock valuation is the no
growth in dividends model. This particular model assumes that the current
dividend represents the residual cash flow and this amount never changes. In
this case, the value of the stock is the present value of the current dividend
received at the end of each year forever, or the present value of a
perpetuity. This formula would be useful for a firm that never expected to grow. In common stock valuation, the interest rate (small r in the formula) is an
estimate of the rate of return investors expect to receive when investing in a
share of stock. The dividend cash flow model can be used to show that the
interest rate is composed of a dividend yield and a capital gain yield. The equity value model requires estimates of the cash ﬂow stream (see
question 7 above) and the required rate of return (see question 8 above). The biggest shortcoming of the equity value model is the difficulty of estimating
both residual cash flows and the required rate of return. 23 CHAPTER 5 PROBLEMS 1. Using lump sum present value method: $103.67 2. $797.19. The price of the two year bond is higher since they offer the same
amount of cash, the two year bond’s cash ﬂow is received sooner and there is a
positive time value of money. 3. Bond A: $712.99 Bond B: $926.39 Bond C: $620.17
4. $1,000. A bond sells for par when its yield equals its coupon. 5. $961.10. It is irrelevant when the bond was issued. Simply enter five as the
number of periods, $80 as the payment, $1,000 as the future value and 9% as the
discount rate and solve using the calculator or formula methods. 6. Price of 5% Coupon Bond = >$947.51
Price of 7% Coupon Bond = >$949.37
Difference in Prices = >$ 1.86 7. a. Price of bond with three years remaining = $925.39
b. Because the current required rate of return exceeds the bond’s coupon. 8. The interest rate must be divided by two and the number of years must be
multiplied by two in order to reﬂect semi—annual compounding. Therefore, using
two as the number of periods and 5% as the interest rate, price=$981.41. 9. Using $950 as the present value and the other variables as given ($1,000 is
future value, n= 3 years, payment=$100), the yield is 12.08%. 10. $1,000 is future value: payment = coupon rate x face value = $75.
Inputting variables as in previous problem, yield to maturity = 9.03%. 11. Noting that payment = coupon rate x face value = $80, this problem is
solved as in previous two problems. Yield = 14.08%. 12. The same answer can be generated for any face value. We will use $100 as
the face value. As above, payment = $9, present value = price = $105.04, etc.
Yield = 8.24% 13. Yield = 8.00%. 14. $20 15. $100 16. $18.75 17. $150.00 18. $5.88 24 19. Since no perpetual growth rate is given, but rather the end of year price is
given, we use a single period discount: ($50 + $2.11)/1.08 = $48.25. 20. Using the same approach as problem #19:
a. Price with 7% required rate of return = $37.00
b. Price with 6% required rate of return = $37.35
Therefore the price would rise by $0.35. 21. $90.06 This problem requires a two period dividend/stock discount model
and is analogous to a two period bond. The stock offers two periods of payments
of $1.50 each and a future value in year two of $100. The present value can be
found in one step as $90.06 or by summing the present value of the first dividend ($1.402) with the present value of the combined dividend and $100 price
($88654). 22. Each stock offers a total of three cashflows that must be present
valued: dividends for two years and then a third dividend and $100 in the third
year. Present values may be performed as follows: Cashﬂows in Dollars Present Values in Dollars
Stock Year 1 Year 2 Year 3 Year 1 Year 2 Year 3 Price
A $4.00 $4.00 $104.00 $3.76 $3.53 $86.10 $93.38
B $4.40 $4.84 $105.32 $4.13 $4.27 $87.19 $95.59 The dividends for Stock B are found by multiplying each year’s previous dividend
by 1.1 to reflect 10% growth. All present values are found using lump sum present
value discounting. The prices are the sum of the present values of the individual
cashﬂows. Especially Stock A could have been solved using annuity shortcuts. 23. Each stock offers a total of two cashflows that must be present
valued: a dividend for one year and then a second dividend and $100 in the second
year. Present values may be performed as follows: Cashﬂows Present Values
Stock Year 1 Year 2 Year 1 Year 2 Price
C $2.00 $102.00 $1.85 $87.45 $89.30
D $1.00 $103.00 $0.93 $88.31 $89.23
Difference in Price: ' $0.07 Although both stocks offer the same total cashflows, Stock C offers $1 in
dividends one year sooner and therefore is worth approximately $0.07 more
today. 24. $40. This is found using the perpetuity model (dividing the dividend by the
required rate of return. 25 25. $250 using the perpetuity model as above. 26. $33.00. The answer is found using the perpetual growth model. The difficulty
is realizing that the numerator of the model requires the next dividend rather than
the current dividend. Since dividends are growing at 5%, next year’s dividend of
$1.155 can be found by multiplying the current dividend ($1.10) by 1.05. The
price of the stock may be found by dividing the next dividend by the required rate
of return (.085) minus the growth rate (0.05), in other words, 0.035. 27. 10%. The solution is found using the perpetual growth model. The three
known variables are inserted and the equation is factored to determine the
unknown variable: 10% = (Div/P) + g = ($1.50/$30.00) + 5%. 28. The solutions are found using the perpetual growth mode. The three known
variables are inserted into the model for each stock and the equation is factored
to determine the required rates of return. The difficulty is that current dividends
are given rather than next period’s dividends. The next year’s dividends can be
found by multiplying the current dividends by the expected growth rates (plus one). Since each stock has the same expected growth rates, the next dividends
are the same: Next dividend = $1.00 x 1.05 = $1.05 Required rate of Return = (Div/P) + g
Stock E: 15.500% = ($1.05/$10.00) + 5%
Stock F: 14.375% = ($1.05/$11.20) + 5% 29. This problem is solved using the perpetual growth model. First, the current price is found by from knowing that the dividend is 2% of the stock price and that
the dividend is $0.25: Dividend = 0.02x Price
$0.25 = 0.02 X Price
Price = $12.50 The next dividend is found by multiplying the current dividend by one plus the
growth rate: Next Dividend Current Dividend x (1+g) = $0.25 x (1.05) = $02625 Next, the three known variables are inserted into the perpetual growth model and
the equation is factored to find the required rate of return. Required Rate of Return = (Div/P) + g = ($0.2625/$12.50) + 5% = 7.1%. 26 CHAPTER 5 DISCUSSION QUESTIONS 1. The present value of a perpetuity is the sum of the present value of all of its
future cashﬂows. Although a perpetuity offers an infinite number of future
cashflows and therefore an infinite amount of money, the present values of the
cashflows that are far off into the future are worth very little. For example, the
present value of $1 to be received in 50 years at 10% interest is less than one
cent. The present value of $1 to be received in 200 years at 10% interest is less
than one millionth of one cent. Thus, the total present value of a perpetuity differs
very little from the combined present values of its first 50 or so years of cashflows. 2. In general, when a bond is issued the amount of cash that it offers to investors
is fixed. For example, a thirty year $1,000 bond with a coupon rate of 8% offers
$80 per year in interest or coupon payments regardless of what happens to
interest rates in the future. If interest rates rise above 8%, people will demand a
higher rate of return on a $1,000 investment than simply $80 per year. Since the
$80 payment if fixed, the price of the bond must drop in order to provide
new investors in the bond the same rate that can be earned elsewhere. When interest rates fall, the opposite occurs. People will bid up the price of the bond since the fixed (e.g., $80) coupon is becoming more and more attractive
relative to the coupons being issued on new bonds. 3. The price of a common stock is still the present value of the future dividends
even if the stock has never paid a dividend and has no short term plan of paying
a dividend. This is because the market is valuing the dividends that it anticipates
in the medium to long term future. A reasonably good analogy is found in dairy
farming. The price of dairy cows is primarily determined by the present value of
the net proceeds from their future milk production. Baby cows (heifers) have
never produced any milk and only can be hoped to produce milk when they mature.
Nevertheless, these cows are worth money since farmers recognize the potential
for milk. Similarly, immature companies have value and this value is derived from the cash that they will eventually be able to produce for their shareholders when
they mature. 4. a. $0. There is no value to holding a stock or any other financial asset that
has absolutely no probability of producing cash for its holder. We assume that
voting rights to one share are worth zero and that any type of distribution is
included in the arrangement. b. $0. Since the person who would end up with the stock would pay $0 since it would
be worth nothing to them as well.
c. $0. No rational person would buy this stock from its holder since they would know that they would be unable to find a rational person to buy it from
them. 27 This problem is very difficult for some students to understand ("But you can sell
the stock certificate at a profit as its price goes up!"). An excellent method of
convincing them of its truth is to repeat the problem numerous times using
different assets. The first time, solve the problem changing the financial asset from a share of stock
to a normal bond. Thus, the orginal holder of the bond gets all of the bonds
interest and principal payments while the new holder of the bond receives only the
privilege of holding the bond certificate. It is relatively obvious that the person
who gets all of the bond’s cash flows receives everything and that the person left
holding the peice of paper gets nothing. This is especially clear because the
bond’s cashflows are known and the bond has a fixed maturity date. The second time, repeat the problem assuming that the financial asset is a consol
bond. In other words, the bond pays a perpetual interest payment and therefore
has an infinite maturity. It is relatively easy to understand that this arrangement
would produce no value to the person holding the piece of paper (the bond
certificate) and all of its value to the person receiving the cash flows. Thus, it
doesn’t matter that the financial asset has no maturity. The third time, repeat the problem assuming that the financial asset is a bond of
fixed maturity but with a coupon that rises by a small amount each year. Thus the
bond may pay 4% interest the first year, 4.5% interest the second year and so
forth through the bond’s finite maturity. Similar results are obtained. Thus it
doesn’t matter that the financial asset has growing cash flows (and even a growing
price in certain circumstances). The final time, repeat the problem assuming that the financial asset has both of
the features of the second and third financial assets. In other words, it is a
bond of infinite maturity and with growing coupon payments. Similar results can
be obtained and can be shown using the perpetual growth model. Finally, students should then be able to see that the infinite maturity and growing dividends of a common stock do not change the results even though dividend
payments are more uncertain than coupon payments. 28 ...

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