Question

Clifford Clark is a recent retiree who is interested in investing some of his savings in corporate bonds. His financial planner has suggested the following bonds:

o Bond A has a 7% annual coupon, matures in 12 years, and has a $1,000 face value.

o Bond B has a 9% annual coupon, matures in 12 years, and has a $1,000 face value.

o Bond C has an 11% annual coupon, matures in 12 years, and has a $1,000 face value.

Each bond has a yield to maturity of 9%.

a. Before calculating the prices of the bonds, indicate whether each bond is trading at a premium, at a discount, or at par.

b. Calculate the price of each of the three bonds.

c. Calculate the current yield for each of the three bonds. (Hint: Refer to Footnote 7 for the definition of the current yield and to Table 7.1.)

d. If the yield to maturity for each bond remains at 9%, what will be the price of each bond 1 year from now? What is the expected capital gains yield for each bond? What is the expected total return for each bond?

e. Mr. Clark is considering another bond, Bond D. It has an 8% semiannual coupon and a $1,000 face value (i.e., it pays a $40 coupon every 6 months). Bond D is scheduled to mature in 9 years and has a price of $1,150. It is also callable in 5 years at a call price of $1,040.

i. What is the bond’s nominal yield to maturity?

ii. What is the bond’s nominal yield to call?

iii. If Mr. Clark were to purchase this bond, would he be more likely to receive the yield to maturity or yield to call? Explain your answer.

f. Explain briefly the difference between price risk and reinvestment risk. Which of the following bonds has the most price risk? Which has the most reinvestment risk?

i. A 1-year bond with a 9% annual coupon

ii. A 5-year bond with a 9% annual coupon

iii. A 5-year bond with a zero coupon

iv. A 10-year bond with a 9% annual coupon

v. A 10-year bond with a zero coupon

Answer 1

a) YTM = yield to maturity = 9%

Bond A : coupon rate = 7%

since coupon rate < YTM , Bond A is trading at a discount

for Bond B : coupon rate = 9%

since coupon rate = YTM , bond B is trading at par

for Bond C : coupon rate = 11%

since coupon rate > YTM , Bond C is trading at a premium

b)

(I) BOND A

YTM = 9% = 0.09

coupon value, C = coupon rate* par value = 0.07*1000 = 70

price of bond = present value of coupons + present value of maturity amount

Present value of coupons = C*PVIFA( 9% , 12 years)

PVIFA( 9% , 12 years) = present value interest rate factor of annuity

= [((1+YTM)ⁿ - 1)/((1+YTM)ⁿ*YTM)] = [((1.09)¹² - 1)/((1.09)¹²*0.09)] = 7.16072528

Present value(PV) of coupons = C*PVIFA( 9% , 12 years) = 70*7.16072528 = 501.25076936

PV of maturity amount = par value/(1+YTM)ⁿ = 1000/(1.09)¹² = 355.53472510

Price of bond when YTM is 9%, p0 = 501.25076936 + 355.53472510 = $856.785494 OR $856.79 ( ROUNDING OFF TO 2 DECIMAL PLACES)

(II) BOND B

Since bond B trades at par , its price = par value, p0 = $1000

(III) Bond C

YTM = 9% = 0.09

coupon value, C = coupon rate* par value = 0.11*1000 = 110

price of bond = present value of coupons + present value of maturity amount

Present value of coupons = C*PVIFA( 9% , 12 years)

PVIFA( 9% , 12 years) = present value interest rate factor of annuity

= [((1+YTM)ⁿ - 1)/((1+YTM)ⁿ*YTM)] = [((1.09)¹² - 1)/((1.09)¹²*0.09)] = 7.16072528

Present value(PV) of coupons = C*PVIFA( 9% , 12 years) = 110*7.16072528 = 787.67978043

PV of maturity amount = par value/(1+YTM)ⁿ = 1000/(1.09)¹² = 355.53472510

Price of bond when YTM is 9%, p0 = 787.67978043 + 355.53472510 = $1143.214506 OR $1143.21 ( ROUNDING OFF TO 2 DECIMAL PLACES)

c)

taking the coupon value and price values from previous part of the question

(I) Current yield for Bond A = Annual coupon value/price = 70/856.785494 = 0.08170073 or 8.17% ( rounding off to 2 decimal places)

(II) Current yield for Bond B = Annual coupon value/price = 90/1000 = 0.09 or 9.00% ( rounding off to 2 decimal places)

(III) Current yield for Bond C = Annual coupon value/price = 110/1143.214506 = 0.096219913 or 9.62% ( rounding off to 2 decimal places)

d) maturity of each bond now will be = 11 years

(I) BOND A

YTM = 9% = 0.09

coupon value, C = coupon rate* par value = 0.07*1000 = 70

price of bond = present value of coupons + present value of maturity amount

Present value of coupons = C*PVIFA( 9% , 11 years)

PVIFA( 9% , 11 years) = present value interest rate factor of annuity

= [((1+YTM)ⁿ - 1)/((1+YTM)ⁿ*YTM)] = [((1.09)¹¹ - 1)/((1.09)¹¹*0.09)] = 6.80519055

Present value(PV) of coupons = C*PVIFA( 9% , 11 years) = 70*6.80519055 = 476.36333861

PV of maturity amount = par value/(1+YTM)ⁿ = 1000/(1.09)¹¹ = 387.53285036

Price of bond when YTM is 9% , p1= 476.36333861 + 387.53285036 = $863.896189 OR $863.90 ( ROUNDING OFF TO 2 DECIMAL PLACES)

Expected capital gains yield = (p1-p0)/p0 = (863.896189 - 856.78549447)/856.78549447 = 0.00829927 or 0.829927% or 0.83% ( rounding off to 2 decimal places)

expected total return = current yield + expected capital gains yield = 0.08170073 +  0.00829927 = 0.09 or 9%

(II) BOND B

YTM = 9% = 0.09

coupon value, C = coupon rate* par value = 0.09*1000 = 90

price of bond = present value of coupons + present value of maturity amount

Present value of coupons = C*PVIFA( 9% , 11 years)

PVIFA( 9% , 11 years) = present value interest rate factor of annuity

= [((1+YTM)ⁿ - 1)/((1+YTM)ⁿ*YTM)] = [((1.09)¹¹ - 1)/((1.09)¹¹*0.09)] = 6.80519055

Present value(PV) of coupons = C*PVIFA( 9% , 11 years) = 90*6.80519055 = 612.46714964

PV of maturity amount = par value/(1+YTM)ⁿ = 1000/(1.09)¹¹ = 387.53285036

Price of bond when YTM is 9% , p1= 612.46714964 + 387.53285036 = $1000 OR $1000 ( ROUNDING OFF TO 2 DECIMAL PLACES)

Expected capital gains yield = (p1-p0)/p0 = (1000 - 1000)/1000 = 0

expected total return = current yield + expected capital gains yield = 0.09 +  0 = 0.09 or 9%

(III) Bond C

YTM = 9% = 0.09

coupon value, C = coupon rate* par value = 0.11*1000 = 110

price of bond = present value of coupons + present value of maturity amount

Present value of coupons = C*PVIFA( 9% , 11 years)

PVIFA( 9% , 11 years) = present value interest rate factor of annuity

= [((1+YTM)ⁿ - 1)/((1+YTM)ⁿ*YTM)] = [((1.09)¹¹ - 1)/((1.09)¹¹*0.09)] = 6.80519055

Present value(PV) of coupons = C*PVIFA( 9% , 11 years) = 110*6.80519055 = 748.57096067

PV of maturity amount = par value/(1+YTM)ⁿ = 1000/(1.09)¹¹ = 387.53285036

Price of bond when YTM is 9%, p1 = 748.57096067 + 387.53285036 = $1136.103811 OR $1136.10 ( ROUNDING OFF TO 2 DECIMAL PLACES)

Expected capital gains yield = (p1-p0)/p0 = (1136.103811 - 1143.2145055)/1143.2145055 = -0.006219913 or -0.6219913% or -0.62% ( rounding off to 2 decimal places)

expected total return = current yield + expected capital gains yield =0.09621991  -0.006219913 = 0.09 or 9%