a.
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.
a.
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.
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.
b.
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:
Where,
i is the interest rate
n is number of time period
PVIFA is Present Value Interest Factor of
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)
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:
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)
Working Note:
Bond A
Calculation of interest to be paid each year:
Bond B
Calculation of interest to be paid each year:
Bond C
Calculation of interest to be paid each year:
Hence, the price of the bond A, B and C are computed to be $856.78, $1,000 and $1,143.21.
c.
To compute: Current yield.
Current Yield:
Current yield is the anticipated
The formula for current yield:
c.
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:
Substitute $70 for annual coupon payment and $856.78 for the current price,
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:
Substitute $90 for annual coupon payment and $1,000 for the current price,
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:
Substitute $110 for annual coupon payment and $1,143.21 for the current price,
Hence, the current yield of Bond A, B and C are computed to be $8.17%, 9.00%, and 9.62%.
d.
To compute: Price of each bond 1 year from now. Expected
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.
d.
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:
Where,
i is the interest rate
n is number of time period
PVIFA is Present Value Interest Factor of
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)
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:
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)
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:
Substitute 9% for total return and 8.17% for current yield,
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:
Substitute 9% for total return and 9% for current yield,
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:
Substitute 9% for total return and 9.62% for current yield,
Working Note:
Bond A
Calculation of interest to be paid each year:
Bond B
Calculation of interest to be paid each year:
Bond C
Calculation of interest to be paid each year:
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.
To compute: Bond’s normal yield to maturity.
e.1.
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
Bond price is $1,150.
The formula to compute bond’s nominal yield to maturity:
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,
Working note:
Calculation of semiannual rate:
Interest is
Hence, yield to maturity is computed to be 5.88%
2.
To compute: Yield to call
2.
Explanation of Solution
Given,
Semi-annual coupon rate is 8% or 0.08.
Par value is $1,000
Number of periods is 10
The call price is $1,040
The formula to compute bond’s nominal yield to maturity:
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,
Hence, yield to maturity is computed to be 5.29.
3.
To identify: Decision to choose between yield to maturity or yield to call.
3.
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.
Hence, Mr. C will earn Yield to call in the given case.
f.
To identify: Difference between price risk and reinvestment risk. Bonds which have highest reinvestment risk.
f.
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
Hence, bonds have been ranked above from the most interest rate risk to the least interest rate risk.
g.1.
To compute: Expected interest rate for each bond in each year.
g.1.
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% |
Hence, above table shows the expected interest yield or current yield for each bond in each year.
2.
To compute: Expected capital gains yield for each bond in each year.
2.
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% |
Hence, above table shows the expected capital gains yield for each bond in each year.
3.
To compute: Total return for each bond in each year.
3.
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% |
Hence, above table shows the total return for each bond in each year.
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Chapter 7 Solutions
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