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40

1050(1.05)-"(1.045)-" = 415.082.

Thus, die price of die bond is 528.334 + 415.082 = $943.42. The price is higher than in Example 7.3, as would be expected, since die yield rate is lower for die final five years in the term of the bond.

7.10 OTHER SECURITIES

The preceding sections have been limited to a discussion of redeemable bonds. This section is concerned with two other types of securities: (1) preferred stock and pe etual bonds, and (2) common stock.

Preferred stock and perpetual bonds

Preferred stock and perpetual bonds are similar in that they both are types of fixed-income securities without redemption dates. Thus, the. price must be equal to the present value of future dividends or coupons forever, i.e. the dividends or coupons form a perpetuity. The version of formula (7.1) for the situation would be

(7.31)

The reader should be aware that some preferred stock is issued with a redemption date. Such preferred stock can be handled exactly like a bond, as described in the preceding sections.

Common stock

Common stock presents a different problem, since it is not a fixed-income security, i.e. the dividends are not known in advance, nor are they level, hi practice, common stock prices fluctuate widely in the stock market, often for little apparent reason.

In theory, common stock prices should represent the present value of future dividends. Values computed in this fashion can be characterized as being based on the dividend discount model. Of course, this calculation should take into account projected changes in the dividend scale.

Consider a situation in which a corporation is planning to pay a dividend of D at the end of the current period. Assume that dividends are projected to change geometrically with common ratio \ + indefinitely and that the stock is purchased to yield i per period, where -1 < < i.

Then the theoretical price of the stock is obtained by taking the limit of formula (4.39) as n-*oo, which gives

p = z)

(7.32)

= $40.

.10-.05

Thus, die dieoretical price is 10 times current earnings.

The present value of dividends is

1 - .05

= $662/3 .

Thus, die tiieoietical price is 162/3 times current earnings.

The present value of dividends is

= $200.

.06 - .05

Thus, die dieoretical price is 50 times current earnings.

( Example 7.14 Rework Example 7.13 assuming that the rate of increase in earnings is 5% for the first five years, 2 1/2% for the second five years, and 0% thereafter. Assume an annual effective yield rate of 10%. Successive applications of formula (4.39) give

1.05 1.10

.10- .05

(1.05)5

1.025 1.10

(1.10)5 2(1.05)5(1.025)

.10-.025

(1.10)

= $25.72

A theoretical calculation of this type is illustrated in Example 7.13. The reader should note that the symbol D was also used in Section 6.4 to represent a sinking fund deposit.

It is probably unrealistic to project constant percentage increases in dividends indefinitely into the future. As corporations increase in size and become more mature, the rate of growth will generally slow down. An illustration of the dividend discount model applied under these conditions is given in Example 7.14.

In practice, the most common frequency of dividend payments on both preferred and common stock in the United States is quarterly. This is in contrast to the typical semiatmual frequency of coupon payments on bonds.

Example 7.13 A common stock is currently earning $4 per share and will pay $2 per share in dividends at the end of the current year. Assuming that the earnings of the corporation increase 5% per year indefinitely and that the corporation plans to continue to pay 50% of its earnings as dividends, find the theoretical price to earn an investor an annual effective yield rate of: (1) 10%,

m S%, and (3) 6%.

1. The present value of dividends is

The present value of the redemption value is



in calculating many of their liability values. The relationship between assets and liabilities is an important issue which is discussed further in Chapter 9.

The present value method is very sensitive to the choice of interest rate used in taking present values. This can be both an advantage and a disadvantage of the method. The method is flexible, but does have a degree of arbitrariness in connection with the choice of the interest rate to use in computing present values. The present value method can produce asset values which are significantly different from either market value or book value. Also the method is less easily understood than either market value or book value.

In summary, there is no valuation method for securities which is used in all situations. The reader should be careful when encountering asset values in practical situations to ascertain the mediod of valuation.

The choice of valuation metiiods for securities will affect computed yield rates. For example, in Section 5.5 it was noted tiiat formula (5.14)

(5.14)

I =.

A + B-I

is often used to compute die yield rate earned by an investment ftmd. The values of/4 and are dependent upon tiie asset valuation metiiods used. Also, the value of / may vary substantially depending upon whether or not capital gains are included in /.

EXERCISES

7.2 Types of securities

1. Find die price wWch should be paid for a zero coupon bond which matures for J $1000 in 10 years to yield:

a) 10% effective.

b) 9% effective.

c) Thus, a 10% reduction in die yield rate causes die price to increase by what \ percentage?

2. A 10-year accumulation bond widi an initial par value of $1000 earns interest of 8* J compounded semiannually. Find die price to yield an investor 10% effective.

3. A 26-week T-bill is bought for $9600 at issue and will mature for $10,000. Find the yield late computed as: l

a) A discount rate, using die typical mediod for counting days on a T-bill.

b) An annual effective rate of interest, assuming die investment period is exacdy I half a year.

4 T-bil!s of all maturities yield 8% computed on a discount basis. Find the ratio of the annual effective rate of interest earned on a 52-week T-bill to that earned on a 13-week T-bill. Use an approach which does not involve the counting of days.

7.3 Price of a bond

5. A 10-year $100 par value bond bearing a 10% coupon rate payable semiannually and redeemable at $105 is bought to yield 8% convertible semiannually. Find the price. Verify diat all four formulas produce die same answer.

6. For the bond in Example 7.3, determine the following:

a) Nominal yield, based on the par value.

b) Nominal yield, based on the redemption value.

c) Current yield.

d) Yield to maturity.

7. Two $100 par value bonds both with 8% coupon rates payabk semiannually are currendy selling at par. Bond A matures in 5 years at par, while Bond matures in 10 years at par. If prevailing market rates of interest suddenly go to 10% convertible semiannually, find the percentage change in the price of:

a) Bond A. IS b) Bond B.

-< c) Justify from general reasoning the relative magnitude of the answers to (a) and (*).

8. Two $1000 bonds redeemable at par at the end of the same period are bought to yield 4% convertible semiannually. One bond costs $1136.78 and has a coupon rate

s of 5% payable semiannually. The other bond has a coupon rate of 2 1/2% payable semiannually. Find the price of the second bond.

\ 9. a $1000 bond with a coupon rate of 9% payable semiannually is redeemable after an unspecified number of years at $1125. The bond is bought to yield 10% N convertible semiannually. If the present value of the redemption value is $225 at this yield rate, find the purchase price.

10. A $1000 par value n-year bond maturing at par with $100 annual coupons is purchased for $1110. If AT = 450, find die base amount G.

11. An investor owns a $1000 par value 10% bond with semiannual coupons. The bond will mature at par at the end of 10 years. The investor decides that an 8-year bond would be preferable. Current yield rates are 7% convertible semiannually. The investor uses die proceeds from die sale of die 10% bond to purchase a 6% bond

fi widi semiannual coupons, maturing at par at die end of 8 years. Find die par value of die 8-year bond.

12. An - $1000 par value bond matures at par and has a coupon rate of 12% convertible semiannually. It is bought at a price to yield 10% convertible



7.8 Serial bonds

36. A $10,000 serial bond is to be redeemed in $1000 instalbnents of principal per half-year over the next five years. Interest at the annual rate of 12% is paid semiannually on the balance outstanding. How much should an investor pay for this bond in order to produce a yield rate of 8% convertible semiannually?

37. A $10,000 serial bond is to be redeemed in $500 installments of principal at die end of die 6di dirough die 25th years from the date of issue. Interest at the rate of 6% is paid annually on die balance outstanding. What is the price to yield an investor 10% effective?

38. Find an expression for the present value of a $100,000 issue of serial bonds, if it is known that the yield rate is 125% of the coupon rate and diat the bonds are redeemable at par according to the following schedule:

End of Years Amount Redeemable

5, 8, 11 $10,000

14, 17 20,000

20 30,000

All rates are semiannual. Express your answer stricdy as a function of s for various values of n.

39. A $78,000 issue of serial bonds with annual coupons of 4% on the balance outstanding is to be redeemed in 12 annual installments beginning at the end of the Sth year. The amount redeemed at the end of the 5th year is $12,000, at the end of the 6th year $11,000, and so forth, until all die bonds are redeemed. Find an expression for the price to yield an investor 5% effective.

7.9 Some generalizations

40. Derive the following variations of formula (7.29): Fr

a) P=C +

P=K+ -3 {C-K).

41. Derive die following variations of formula (7.30):

a) = + ( - " \

b) P= Gs> + (C-Gs)v.

c) P = K+J-(C-K).

42. Find the price of a $1000 par value 10-year bond maturing at par which has coupons at 8% per annum payable quarterly and is bought to yield 6% per annum convertible semiannually.

43. The price of a $100 bond, which matures in n years for $105, has semiannual coupons of $4, and is bought to yield an effective rate /, can be expressed as

Av + B

,(2)

Find A and B.

A $1000 par value 20-year bond maUiring at par has annual coupons of 5% for die first 10 years and 4% for die second 10 years. Find an expression for die price of the bond bought to yield a rate of interest convertible quarterly.

A 10-year bond has annual coupons which vary 10, 9, 8,... ,1, and it matures for $1( . If die bond is bought to yield / effective, find expressions for:

a) The amount of the fifth coupon which is required interest.

b) The amount of the fifth coupon used to amortize die book value.

An investor buys a $1( par value 3% bond with semiannual coupons on which $50 matures in 9 years at $51 and die odier $50 matures in 10 years at $50. Show diat die price to yield an effective rate of interest / is

,(2)

+ (101 +5Ii- 1.55V°

7.10 Other securities

47. What would be die answer to Example 7.13 for a yield rate of 5% or less?

48. A preferred stock pays a $10 dividend at die end of die first year, widi each successive annual dividend being 5% greater than the preceding one. What level annual dividend would be equivalent if / = 12%?

; 49. A common stock pays annual dividends at the end of each year. The earnings per share in die year just ended were $6. Earnings are assumed to grow 8% per year in the ftiture. The percentage of earnings paid out as a dividend will be 0% for the next 5 years and 50% thereafter. Find the theoretical price of die stock to yield an investor 15% effective.

50. A common stock is purchased at a price equal to 10 times current earnings. During die next 6 years die stock pays no dividends, but earnings increase 60%. At die end of 6 years the stock is sold at a price equal to 15 times earnings. Find the effective annual yield rate earned on this investment.

51. Find an expression for the theoretical price of a common stock paying annual dividends at die end of each year. The earnings in die year just ending were E. It is assumed that the rate of growth in earnings for the rth year is k,, the yield rate



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