Question

Robert Campbell and Carol Morris are senior vice presidents of the Mutual of Chicago Insurance Company. They are codirectors of the company’s pension fund management division, with Campbell having responsibility for fixed income securities (primarily bonds) and Morris being responsible for equity investments. A major new client, the California League of Cities, has requested that Mutual of Chicago present an investment seminar to the mayors of the represented cities. Campbell and Morris, who will make the actual presentation, have asked you to help them by answering the following questions:

a. What is the value of a one-year, $1,000 par value bond with a 10 percent annual coupon if its required rate of return is 10 percent? (1 answer)

b. What is the value of a similar 10-year bond? (1 answer)

c. What would be the value of the bond (both the one-year and 10-year) described in part (a&b) if, just after it had been issued, the expected inflation rate rose by 3 percentage points, causing investors to require a 13 percent return? Is the security now a discount bond or a premium bond? (2 answers)

d. What would happen to the bond’s value (both the one-year and the 10-year) if inflation fell, and rd declined to 7 percent? Would it now be a premium bond or a discount bond? (2 answers)

e. Redo part “c” assuming the bonds have semiannual rather than annual coupons (2 answers)

f. Redo part “d” assuming the bonds have semiannual rather than annual coupons (2 answers)

Answer 1

Solution to A

Since the coupon rate is equal to the required rate of return, the price of the bond will be par, that is, $1000

Solution to B

It does not matter how much time is left to maturity. If the coupon rate is equal to the required rate of return, the price of the bond will be its par value, $1000

Solution to C

To calculate the price of the bonds, we will have to discount future cash flows from the bond, using the expected rate of return.

The price of the 1-year bond = (10% *$1000)/ 1.13 + $1000/1.13 = $973.45

The price of the 10-year bond = {(10% *$1000)/ 1.13} + {(10% *$1000)/ 1.13²} + {(10% *$1000)/ 1.13³} + {(10% *$1000)/ 1.13⁴} + {(10% *$1000)/ 1.13⁵} + {(10% *$1000)/ 1.13⁶} + {(10% *$1000)/ 1.13⁷} + {(10% *$1000)/ 1.13⁸} + {(10% *$1000)/ 1.13⁹} + {(10% *$1000)/ 1.13¹⁰} + {$1000/(1.13¹⁰) = $837.21

Since both bonds are trading below their respective par values, both bonds are discount bonds.

Solution to D

The price of the 1-year bond, with a 7% expected rate of return = (10% *$1000)/ 1.07 + $1000/1.07 = $1,028.04

The price of the 10-year bond = {(10% *$1000)/ 1.07} + {(10% *$1000)/ 1.07²} + {(10% *$1000)/ 1.07³} + {(10% *$1000)/ 1.07⁴} + {(10% *$1000)/ 1.07⁵} + {(10% *$1000)/ 1.07⁶} + {(10% *$1000)/ 1.07⁷} + {(10% *$1000)/ 1.07⁸} + {(10% *$1000)/ 1.07⁹} + {(10% *$1000)/ 1.07¹⁰} + {$1000/(1.07¹⁰) = $1,210.71

Since both the 1- and 10-year bonds are priced above their respective par values they are now both premium bonds.

Solution to E

The price of the 1-year bond considering semi-annual coupon payments = {($100/2)/(1 + 0.13/2)} + {$1,000 +(100/2)}/(1 + 0.13/2)² = $972.69

Similarly the price of the 10-year bond will be {($100/2)/(1 + 0.13/2)} + {($100/2)/(1 + 0.13/2)²} + {($100/2)/(1 + 0.13/2)³} and so on till + {$1,000 +(100/2)}/(1 + 0.13/2)¹⁰ = $834.72

Since both bonds are trading below their respective par values, both bonds are discount bonds.

Solution to F

The price of the 1-year bond considering semi-annual coupon payments with the discount rate of 7% is

{($100/2)/(1 + 0.07/2)} + {$1,000 +(100/2)}/(1 + 0.07/2)² = $1,028.50

The price of the 10-year bond will be {($100/2)/(1 + 0.07/2)} + {($100/2)/(1 + 0.07/2)²} + {($100/2)/(1 + 0.07/2)³} and so on till + {$1,000 +(100/2)}/(1 + 0.07/2)¹⁰ = $1,213.19

Since both the 1- and 10-year bonds are priced above their respective par values they are now both premium bonds.