Questions: CENGAGE MINDTAP Ch. 6 7 eBook Problem Walk-Through A) Suppose you have two bonds that have a face value of 1,000 and pay a 10% annual coupon. Bond A matures in 15 years, while Bond B matures in 30 years. Assume that the yield to maturity (interest rate) is 5%, 7%, and 11%. Assume that only one more interest payment is to be made on Bond B at its maturity and that 15 more payments are to be made on Bond A. Round your answers to the nearest cent. Bond A: 1518.98 1273.24 928.70 Bond B: b) Why does the longer-term bond's price vary more than the price of the shorter-term bond when interest rates change? I. Long-term bonds have greater interest rate risk than do short-term bonds. II. The change in price due to a change in the interest rate is larger for bonds with greater maturity. III. The change in price due to a change in the required rate of return decreases as a bond's maturity increases. IV. Long-term bonds have smaller interest rate risk than do short-term bonds.

To solve the problem, we need to calculate the price of Bond B under different yield to maturity (YTM) scenarios and then address the conceptual question about interest rate risk.

Bond B has a face value of $1,000, a 10% annual coupon, and matures in 30 years. However, since only one more interest payment is to be made at its maturity, we treat it as a zero-coupon bond for the remaining period. We need to calculate its price for YTMs of 5%, 7%, and 11%.

The price of a bond is calculated using the present value formula:

\[ \text{Price} = \frac{C}{(1 + r)^n} + \frac{F}{(1 + r)^n} \]

Where:

• \( C \) is the annual coupon payment ($100 for a 10% coupon on $1,000 face value).
• \( F \) is the face value of the bond ($1,000).
• \( r \) is the yield to maturity (YTM).
• \( n \) is the number of years to maturity.

Since only one more payment is to be made, we consider \( n = 1 \).

1. YTM = 5%:

   \[ \text{Price} = \frac{100}{(1 + 0.05)^1} + \frac{1000}{(1 + 0.05)^1} = \frac{100}{1.05} + \frac{1000}{1.05} \]

   \[ \text{Price} = 95.24 + 952.38 = 1047.62 \]

2. YTM = 7%:

   \[ \text{Price} = \frac{100}{(1 + 0.07)^1} + \frac{1000}{(1 + 0.07)^1} = \frac{100}{1.07} + \frac{1000}{1.07} \]

   \[ \text{Price} = 93.46 + 934.58 = 1028.04 \]

3. YTM = 11%:

   \[ \text{Price} = \frac{100}{(1 + 0.11)^1} + \frac{1000}{(1 + 0.11)^1} = \frac{100}{1.11} + \frac{1000}{1.11} \]

   \[ \text{Price} = 90.09 + 900.90 = 990.99 \]

The question asks why the longer-term bond's price varies more than the price of the shorter-term bond when interest rates change. Let's evaluate the given statements:

I. Long-term bonds have greater interest rate risk than do short-term bonds.

• This statement is correct. Longer-term bonds are more sensitive to interest rate changes because the present value of their cash flows is affected more significantly over a longer period.

II. The change in price due to a change in the interest rate is larger for bonds with greater maturity.

• This statement is also correct. The longer the maturity, the more the bond's price will fluctuate with changes in interest rates.

III. The change in price due to a change in the required rate of return decreases as a bond's maturity increases.

• This statement is incorrect. The change in price actually increases with longer maturities.

IV. Long-term bonds have smaller interest rate risk than do short-term bonds.

• This statement is incorrect. Long-term bonds have greater interest rate risk.

• The calculated prices for Bond B are approximately $1047.62, $1028.04, and $990.99 for YTMs of 5%, 7%, and 11%, respectively.
• The correct reasons for the greater price variability of long-term bonds are statements I and II.