# Chapter 7

THE VALUATION AND CHARACTERISTICS OF BONDS

PROBLEMS

Assume all bonds pay interest semiannually.

Finding the Price of a Bond – Example 7.1 (page 306)

1. The Altoona Company issued a 25-year bond 5 years ago with a face value of $1,000. The bond pays interest semiannually at a 10% annual rate. a. What is the bond's price today if the interest rate on comparable new issues is 12%? b. What is the price today if the interest rate is 8%?

c. Explain the results of parts a and b in terms of opportunities available to investors. d. What is the price today if the interest rate is 10%? e. Comment on the answer to part d.

SOLUTION:

PB = PMT [PVFAk,n] + FV [PVFk,n]

a. n = 20 ( 2 = 40 k = 12/2 = 6 PMT = $1,000 ( .10/2 = $50 FV = $1,000 PB = $50 [PVFA6,40] + $1,000 [PVF6,40]

= $50 (15.0463) + $1,000 (.0972)

= $849.52

b. k = 8/2 = 4

PB = $50 [PVFA4,40] + $1,000 [PVF4,40]

= $50 (19.7928) + $1,000 (.2083)

= $1,197.94

c. In part a the interest rate has risen above the coupon rate. Therefore an investment equal to the bond's face value would earn more if placed in newly issued bonds. That means the bond's price has to decrease below face value to keep its yield competitive with new issues. In part b the bond offers more than new issues costing $1,000. Therefore, its price can increase above $1,000 and still remain competitive. d. $1,000.

e. A bond will always sell at par value when the interest rate is equal to its coupon rate.

2.Calculate the market price of a $1,000 face value bond under the following conditions.

Coupon Time Until Current

Rate Maturity Market Rate

a. 12% 15 yrs 10%

b. 7% 5 yrs 12%

c. 9% 25 yrs 6%

d. 14% 30 yrs 9%

e. 5% 6 yrs 8%

SOLUTION: PB = PMT [PVFAk,n] + FV [PVFk,n]

a. PB = $60 [PVFA5,30] + $1,000 [PVF5,30]

= $60 (15.3725) + $1,000 (.2314)

= $1,153.75

b. PB = $35 [PVFA6,10] + $1,000 [PVF6,10]

= $35 (7.3601) + $1,000 (.5584)

= $816.00

c. PB = $45 [PVFA3,50] + $1,000 [PVF3,50]

= $45 (25.7298) + $1,000 (.2281)

= $1,385.94

d. PB = $70 [PVFA4.5,60] + $1,000 [PVF4.5,60]

= $70 (20.638) + $1,000 (.0713)

= $1,515.96

e. PB = $25 [PVFA4,12] + $1,000 [PVF4,12]

= $25 (9.3851) + $1,000 (.6246)

= $859.23

3.What is the current yield on each of the bonds in the previous problem.

SOLUTION:

a.$120.00/$1,153.72 = .104 = 10.4%

b.$70.00/$816.00 = .086 = 8.6%

c.$90.00/$1,385.95 = .065 = 6.5%

d.$140.00/$1,515.95 = .092 = 9.2%

e.$50.00/$859.23 = .058 = 5.8%

4.The Sampson Company issued a $1,000 bond 5 years ago with an initial term of 25 years and a coupon rate of 6%. Today’s interest rate is 10%. a. What is the bond’s current price if interest is paid semiannually as it is on most bonds? b. What is the price if the bond’s interest is paid annually? Comment on the difference between a and b.

c. What would the price be if interest was paid semiannually and the bond was issued at a face value of $1,500?

SOLUTION:

a. PB = PMT[PVFAk,n] + FV[PVFk,n]

= $30[PVFA5,40] + $1,000[PVF5,40]

= $30(17.1591) + $1,000(.1420)

= $514.77 + $142.00

= $656.77

b. PB = PMT[PVFAk,n] + FV[PVFk,n]

= $60[PVFA10,20] + $1,000[PVF10,20]

= $60(8.5136) + $1,000(.1486)

= $510.82 + $148.60

= $659.42

In most cases semiannual rather than annual compounding makes only small difference in the price of a bond. c. PB = PMT[PVFAk,n] + FV[PVFk,n]

= $45[PVFA5,40] + $1,500[PVF5,40]

= $45(17.1591) + $1,500(.1420)

= $772.16 + $213.00

= $985.16

5.Fix-It Inc. recently issued 10-year, $1,000 par value bonds at an...

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