Question

Three investors invest in the same 10-year 8% annual coupon bond. They bought the bond at the same price ($85.503075 for a par value of $100) and at the same time. A is a buy-and-hold investor (hold till maturity), B will sell the bond after four years, and C will sell the bond after seven years.

1. What is the yield to maturity of this bond?
2. For each of these three investors, find the total cash flow (in dollar amount) at the time of maturity (for A) and at the time of sale (for B and C).
3. After the bond is purchased by the three investors and before the first coupon is received, interest rate go up to 11.4%. What happens to the realized yield of these investors?
4. The Macaulay duration of this bond is: 7.0029 years. The difference between the Macaulay duration of a bond and the investment horizon is called the duration gap. For each of these three investors, find their respective duration gap.
5. Combine answers from ABCD, what are the relations between duration gap, interest rate risk, and reinvestment risk?

Answer 1

A)

By extracting the information:

Par value of the bond (FV) = $100

Selling price = $85.503075

Time to maturity (N) = 10 years

Coupon rate (C_r) = 8%

Compounding annually (m) = 1

Calculate the yield to maturity:

The formula for calculating the return is given below:

Bond value = FV{1/(1+r/m)^mN + (C_r/r) (1-(1+r/m)^-mN)}

By substituting the values:

$85.505075 = $100 {1/(1+r/1)^1x10 + (0.08/r) (1-(1+r/1)^-1x10)}

        r = 0.104

Thus, the yield to maturity is 10.4%

B)

A’s cash flow = $100 + ($100x 8%) = $108

B’s cash flow:

By extracting the information:

Par value of the bond (FV) = $100

Yield to maturity = 10.4%

Time to maturity (N) = 6 years(10-4)

Coupon rate = 8%

Compounding annually (m) = 1

Calculate price of the bond:

The formula for calculating the price of the bond is given below:

Bond value = FV{1/(1+r/m)^mN + (C_r/r) (1-(1+r/m)^-mN)}

By substituting the values:

Bond value = $100 {1/(1+0.104/1)^1x6 + (0.08/0.104) (1-(1+0.104/1)^-1x6)}

                   = $89.668770

Thus, the price of the bond is $89.67

B’s cash flow = $89.67 + ($100x8%) = $97.67

C’s cash flow:

By extracting the information:

Par value of the bond (FV) = $100

Yield to maturity = 10.4%

Time to maturity (N) = 3 years (10-7)

Coupon rate = 8%

Compounding annually (m) = 1

Calculate price of the bond:

The formula for calculating the price of the bond is given below:

Bond value = FV{1/(1+r/m)^mN + (C_r/r) (1-(1+r/m)^-mN)}

By substituting the values:

Bond value = $100 {1/(1+0.104/1)^1x3 + (0.08/0.104) (1-(1+0.104/1)^-1x3)}

                   = $ 94.0733

Thus, the price of the bond is $ 94.07

C’s cash flow = $94.07 + ($100x8%) = $102.07

C)

By extracting the information:

Par value of the bond (FV) = $100

Yield to maturity = 11.4%

Time to maturity (N) = 10 years

Coupon rate = 8%

Compounding annually (m) = 1

Calculate price of the bond:

The formula for calculating the price of the bond is given below:

Bond value = FV{1/(1+r/m)^mN + (C_r/r) (1-(1+r/m)^-mN)}

By substituting the values:

Bond value = $100 {1/(1+0.114/1)^1x10 + (0.08/0.114) (1-(1+0.114/1)^-1x10)}

                   = $ 80.30806

Thus, the price of the bond is $ 80.30806