Chapter 7, Problem 19SP

a.

Summary Introduction

To identify:

Discount:

Discount refers to a situation where price issued for the bond is below the par value of the bond.

Premium:

Premium refers to a situation where price issued for the bond is above the par value of the bond.

Par value of bonds:

Par value of bond also mentioned as the face value of the bond is the original price printed on the bond certificate. A bond is considered to be issued at par when yield to maturity of a bond is equal to coupon rate of the bond.

Explanation of Solution

Yield to maturity is 9%.

Bond A has 7% annual coupon rate.

Bond B has 9% annual coupon rate.

Bond C has 11% annual coupon rate.

Bond A has an annual coupon rate of 7% which is less than the required return of 9%, it means that the bond is being traded at below the par value or at discount.

Bond B has an annual coupon rate of 9% which is equal to the required return of 9%, it means that the bond is being traded at par value.

Bond C has an annual coupon rate of 11% which is more than the required return of 9%, it means that the bond is being traded at above the par value or at a premium.

b.

Summary Introduction

To compute: Price of bonds.

Bonds:

Bonds are a financial instrument, generally issued to raise debt generally for activities which require a significant amount of funds, with an undertaking to repay the amount with appropriate interest.

Explanation of Solution

Bond A

Given,

The coupon rate is 7% or 0.07.

Par value is $1,000

Yield to maturity is 9%

Number of periods is 12

PVIF is 0.35553

PVIFA is 7.1607

The formula to compute the price of bonds:

$\text{Price of bond}=\left[\begin{array}{c}\left[\text{Par value of the bond}\times \text{PVIF}\left(i,n\right)\right]\\ +\left[\text{Interest to be paid each year}\times \text{PVIFA}\left(i,n\right)\right]\end{array}\right]$

Where,

i is the interest rate

n is number of time period

PVIFA is Present Value Interest Factor of Annuity

PVIF is Present Value Interest Factor

Substitute $1,000 for the par value of the bond, 0.35553 for PVIF (i,n), $70 for interest to be paid each year and 7.1607 for PVIFA (i,n)

$\begin{array}{c}\text{Price of bond}=\left[\text{\$1,000}\times 0.35553\right]+\left[\text{\$70}\times 7.1607\right]\\ =\$355.53+501.25\\ =\$856.78\end{array}$

Bond B

Given,

The coupon rate is 9% or 0.09.

Par value is $1,000

Yield to maturity is 9%

Number of periods is 12

PVIF is 0.35553

PVIFA is 7.1607

Since bond B is issued at par, the price of the bond will be its value $1,000.

Bond C

Given,

The coupon rate is 11% or 0.11.

Par value is $1,000

Yield to maturity is 9%

Number of periods is 12

PVIF is 0.35553

PVIFA is 7.1607

The formula to compute the price of bonds:

$\text{Price of bond}=\left[\begin{array}{c}\left[\text{Par value of the bond}\times \text{PVIF}\left(i,n\right)\right]\\ +\left[\text{Interest to be paid each year}\times \text{PVIFA}\left(i,n\right)\right]\end{array}\right]$

Where,

i is the interest rate

n is number of time period

PVIFA is Present Value Interest Factor of Annuity

PVIF is Present Value Interest Factor

Substitute $1,000 for the par value of the bond, 0.35553 for PVIF (i,n), $110 for interest to be paid each year and 7.1607 for PVIFA (i,n)

$\begin{array}{c}\text{Price of bond}=\left[\text{\$1,000}\times 0.35553\right]+\left[\text{\$110}\times 7.1607\right]\\ =\$355.53+787.68\\ =\$1,143.21\end{array}$

Working Note:

Bond A

Calculation of interest to be paid each year:

$\begin{array}{c}\text{Interest payment}=\text{Coupon rate}\times \text{Par value}\\ =0.07\times \$1,000\\ =\$70\end{array}$

Bond B

Calculation of interest to be paid each year:

$\begin{array}{c}\text{Interest payment}=\text{Coupon rate}\times \text{Par value}\\ =0.09\times \$1,000\\ =\$90\end{array}$

Bond C

Calculation of interest to be paid each year:

$\begin{array}{c}\text{Interest payment}=\text{Coupon rate}\times \text{Par value}\\ =0.011\times \$1,000\\ =\$110\end{array}$

Conclusion

Hence, the price of the bond A, B and C are computed to be $856.78, $1,000 and $1,143.21.

c.

Summary Introduction

To compute: Current yield.

Current Yield:

Current yield is the anticipated rate of return on the basis of annual coupon payment and present market price of a bond.

The formula for current yield:

$\text{Current yield}=\frac{\text{Annual coupon payment}}{\text{Current price}}$

Explanation of Solution

Bond A

Given,

Annual coupon payment as computed is $70.

Current price as computed is $856.78.

The formula to calculate the current yield of Bond A:

$\text{Current yield}=\frac{\text{Annual coupon payment}}{\text{Current price}}$

Substitute $70 for annual coupon payment and $856.78 for the current price,

$\begin{array}{c}\text{Current yield}=\frac{\text{\$70}}{\text{\$856}\text{.78}}\\ =8.17\%\end{array}$

Bond B

Given,

Annual coupon payment as computed is $90.

Current price as computed is $1,000.

The formula to calculate the current yield of Bond B:

$\text{Current yield}=\frac{\text{Annual coupon payment}}{\text{Current price}}$

Substitute $90 for annual coupon payment and $1,000 for the current price,

$\begin{array}{c}\text{Current yield}=\frac{\text{\$90}}{\text{\$1,000}}\\ =9\%\end{array}$

Bond C

Given,

Annual coupon payment as computed is $110.

Current price as computed is $1,143.21.

The formula to calculate the current yield of Bond C:

$\text{Current yield}=\frac{\text{Annual coupon payment}}{\text{Current price}}$

Substitute $110 for annual coupon payment and $1,143.21 for the current price,

$\begin{array}{c}\text{Current yield}=\frac{\text{\$110}}{\text{\$1,143}\text{.21}}\\ =9.62\%\end{array}$

Conclusion

Hence, the current yield of Bond A, B and C are computed to be $8.17%, 9.00%, and 9.62%.

d.

Summary Introduction

To compute: Price of each bond 1 year from now. Expected capital gains yield for each bond. Expected total return for each bond.

Bonds:

Bonds are a financial instrument, generally issued to raise debt generally for activities which require a significant amount of funds, with an undertaking to repay the amount with appropriate interest.

Explanation of Solution

Price of each bond one year from now:

Bond A

Given,

The coupon rate is 7% or 0.07.

Par value is $1,000

Yield to maturity is 9%

Number of periods is 11

PVIF is 0.3875

PVIFA is 6.8052

The formula to compute the price of bonds:

$\text{Price of bond}=\left[\begin{array}{c}\left[\text{Par value of the bond}\times \text{PVIF}\left(i,n\right)\right]\\ +\left[\text{Interest to be paid each year}\times \text{PVIFA}\left(i,n\right)\right]\end{array}\right]$

Where,

i is the interest rate

n is number of time period

PVIFA is Present Value Interest Factor of Annuity

PVIF is Present Value Interest Factor

Substitute $1,000 for the par value of the bond, 0.3875 for PVIF (i,n), $70 for interest to be paid each year and 6.8052 for PVIFA (i,n)

$\begin{array}{c}\text{Price of bond}=\left[\text{\$1,000}\times 0.3875\right]+\left[\text{\$70}\times 6.8052\right]\\ =\$387.50+476.36\\ =\$863.86\end{array}$

Bond B

Given,

The coupon rate is 9% or 0.09.

Par value is $1,000

Yield to maturity is 9%

A number of periods is 11.

PVIF is 0.3875

PVIFA is 6.8052

Since bond B is issued at par, the price of the bond will be its value $1,000.

Bond C

Given,

The coupon rate is 11% or 0.11.

Par value is $1,000

Yield to maturity is 9%

Number of periods is 11.

PVIF is 0.3875

PVIFA is 6.8052

The formula to compute the price of bonds:

$\text{Price of bond}=\left[\begin{array}{c}\left[\text{Par value of the bond}\times \text{PVIF}\left(i,n\right)\right]\\ +\left[\text{Interest to be paid each year}\times \text{PVIFA}\left(i,n\right)\right]\end{array}\right]$

Where,

i is the interest rate

n is number of time period

PVIFA is Present Value Interest Factor of Annuity

PVIF is Present Value Interest Factor

Substitute $1,000 for the par value of the bond, 0.3875 for PVIF (i,n), $110 for interest to be paid each year and 6.8052 for PVIFA (i,n)

$\begin{array}{c}\text{Price of bond}=\left[\text{\$1,000}\times 0.3875\right]+\left[\text{\$110}\times 6.8052\right]\\ =\$387.50+748.57\\ =\$1,136.07\end{array}$

Expected total return for each bond:

Expected total return for each bond is equal to YTM which is 9%.

Expected capital gains yield for each bond:

Bond A

Given,

Expected total return for bond A, B and C is 9%.

The current yield of bond A as computed is 8.17%.

The formula to calculate capital gain yield for Bond A:

$\text{Capital gain yield}=\text{Total return}-\text{Current yield}$

Substitute 9% for total return and 8.17% for current yield,

$\begin{array}{c}\text{Capital gain yield}=\text{9\%}-8.17\%\\ =0.83\%\end{array}$

Bond B

Given,

Expected total return for bond A, B and C is 9%.

The current yield of bond A as computed is 9%.

The formula to calculate capital gain yield for Bond B:

$\text{Capital gain yield}=\text{Total return}-\text{Current yield}$

Substitute 9% for total return and 9% for current yield,

$\begin{array}{c}\text{Capital gain yield}=\text{9\%}-9\%\\ =0\%\end{array}$

Bond C

Given,

Expected total return for bond A, B and C is 9%.

The current yield of bond C as computed is 9.62%.

The formula to calculate capital gain yield for Bond A:

$\text{Capital gain yield}=\text{Total return}-\text{Current yield}$

Substitute 9% for total return and 9.62% for current yield,

$\begin{array}{c}\text{Capital gain yield}=\text{9\%}-9.62\%\\ =-0.62\%\end{array}$

Working Note:

Bond A

Calculation of interest to be paid each year:

$\begin{array}{c}\text{Interest payment}=\text{Coupon rate}\times \text{Par value}\\ =0.07\times \$1,000\\ =\$70\end{array}$

Bond B

Calculation of interest to be paid each year:

$\begin{array}{c}\text{Interest payment}=\text{Coupon rate}\times \text{Par value}\\ =0.09\times \$1,000\\ =\$90\end{array}$

Bond C

Calculation of interest to be paid each year:

$\begin{array}{c}\text{Interest payment}=\text{Coupon rate}\times \text{Par value}\\ =0.011\times \$1,000\\ =\$110\end{array}$

Conclusion

Hence, the price of the bond A, B and C are computed to be $863.86, $1,000 and $1,136.07 respectively. The capital gain yield of Bond A, B and C are computed to be 0.83%, 0% and -0.62% respectively. Expected total return for each bond is computed to be 9%.

e.1.

Summary Introduction

To compute: Bond’s normal yield to maturity.

Explanation of Solution

Bond D

Given,

The semi-annual coupon rate is 8% or 0.08.

Par value is $1,000

Number of periods is 18 $\left(9\text{ semi annual years}\right)$

Bond price is $1,150.

The formula to compute bond’s nominal yield to maturity:

$\text{Yield to maturity}=\frac{\text{C}+\frac{\text{FV}-\text{P}}{\text{n}}}{\frac{\text{FV}+\text{P}}{2}}$

Where,

C is coupon value

FV is face value

P is the price of the bond

n is number of periods

Substituting $80 for C, $1,000 for FV, $1,150 for P and 18 months for n,

$\begin{array}{c}\text{Yield to maturity}=\frac{40+\frac{1,000-1,150}{18}}{\frac{1,000+1,150}{2}}\\ =\frac{40-8.33}{1,075}\\ =\frac{31.67}{1,075}\\ =2.94\%\text{ or 5}\text{.88\% annually}\end{array}$

Working note:

Calculation of semiannual rate:

$\text{Semi annual contract rate=}\frac{\text{Annual contract rate}}{2}$

$\begin{array}{c}\text{Semi annual contract rate=}\frac{\text{8\%}}{2}\\ =4\%\end{array}$

Interest is $\$40\left(\$1,000\times 4\%\right)$

Conclusion

Hence, yield to maturity is computed to be 5.88%

2.

Summary Introduction

To compute: Yield to call

Explanation of Solution

Given,

Semi-annual coupon rate is 8% or 0.08.

Par value is $1,000

Number of periods is 10 $\left(\text{5 semi annual years}\right)$

The call price is $1,040

The formula to compute bond’s nominal yield to maturity:

$\text{Yield to maturity}=\frac{\text{C}+\frac{\text{FV}-\text{P}}{\text{n}}}{\frac{\text{FV}+\text{P}}{2}}$

Where,

C is coupon value

FV is face value

P is the price of the bond

n is number of periods

Substituting $80 for C, $1,150 for FV, $1,040 for P and 10 months for n,

$\begin{array}{c}\text{Yield to maturity}=\frac{40+\frac{1,150-1,040}{10}}{\frac{1,150+1,040}{2}}\\ =\frac{40-11}{1,095}\\ =\frac{29}{1,095}\\ =2.64\%\text{ or 5}\text{.29\% annually}\end{array}$

Conclusion

Hence, yield to maturity is computed to be 5.29.

3.

Summary Introduction

To identify: Decision to choose between yield to maturity or yield to call.

Answer to Problem 19SP

Mr. C will earn Yield to call in the given case.

Explanation of Solution

Since the bonds are trading at a premium, it indicates that interest rates have fallen.

In case interest rates remain to be constant at present level, Mr. C should anticipate the bond to be called.

As a result, he will earn Yield to call.

Conclusion

Hence, Mr. C will earn Yield to call in the given case.

f.

Summary Introduction

To identify: Difference between price risk and reinvestment risk. Bonds which have highest reinvestment risk.

Explanation of Solution

Price risk

Price risk is the possibility of the fall in the price of bonds due to rising in the interest rates.

Price risk is higher on bonds having longer maturity period as it gives sufficient time to bondholder to replace the bond.

Reinvestment risk

Reinvestment risk is the possibility of fall in the interest rates which will subsequently result in fall in income from the bond portfolio.

Reinvestment risk is higher on short-term bonds as less high old coupon bonds will be replaced with a new low-coupon bond.

Bonds have been ranked in order from the most interest rate risk to the least interest rate risk:

18 year bond with a 9% annual coupon

A 10-year bond with a zero coupon

A 10-year bond with a 9% annual coupon

A 5-year bond with a zero coupon

A 5-year bond with a 9% annual coupon

Conclusion

Hence, bonds have been ranked above from the most interest rate risk to the least interest rate risk.

g.1.

Summary Introduction

To compute: Expected interest rate for each bond in each year.

Explanation of Solution

Expected interest yield or current yield for each bond in each year:

N	Bond A	Bond B	Bond C
12	8.17%	9.00%	9.62%
11	8.10%	9.00%	9.68%
10	8.03%	9.00%	9.75%
9	7.95%	9.00%	9.82%
8	7.87%	9.00%	9.90%
7	7.78%	9.00%	9.99%
6	7.69%	9.00%	10.09%
5	7.59%	9.00%	10.21%
4	7.48%	9.00%	10.33%
3	7.37%	9.00%	10.47%
2	7.26%	9.00%	10.63%
1	7.13%	9.00%	10.80%

Conclusion

Hence, above table shows the expected interest yield or current yield for each bond in each year.

2.

Summary Introduction

To compute: Expected capital gains yield for each bond in each year.

Explanation of Solution

Expected capital gains yield for each bond in each year:

N	Bond A	Bond B	Bond C
12	0.83%	0.00%	-0.62%
11	0.90%	0.00%	-0.68%
10	0.97%	0.00%	-0.75%
9	1.05%	0.00%	-0.82%
8	1.13%	0.00%	-0.90%
7	1.22%	0.00%	-0.99%
6	1.31%	0.00%	-1.09%
5	1.41%	0.00%	-1.21%
4	1.52%	0.00%	-1.33%
3	1.63%	0.00%	-1.47%
2	1.74%	0.00%	-1.63%
1	1.87%	0.00%	-1.80%

Conclusion

Hence, above table shows the expected capital gains yield for each bond in each year.

3.

Summary Introduction

To compute: Total return for each bond in each year.

Explanation of Solution

Given,

Expected total return for bond A, B and C is 9%.

Total return for each bond in each year:

N	Bond A	Bond B	Bond C
12	9.00%	9.00%	9.00%
11	9.00%	9.00%	9.00%
10	9.00%	9.00%	9.00%
9	9.00%	9.00%	9.00%
8	9.00%	9.00%	9.00%
7	9.00%	9.00%	9.00%
6	9.00%	9.00%	9.00%
5	9.00%	9.00%	9.00%
4	9.00%	9.00%	9.00%
3	9.00%	9.00%	9.00%
2	9.00%	9.00%	9.00%
1	9.00%	9.00%	9.00%

Conclusion

Hence, above table shows the total return for each bond in each year.