Miller Corporation has a premium bond making semiannual payments. The bond pays an 8 percent
coupon, has a YTM of 6 percent, and has 13 years to maturity. The Modigliani Company has a discount
bond making semiannual payments. The bond pays a 6 percent coupon and has a YTM of 8 percent, and also has a 13 years maturity. Assume a face value of $1,000 for both bonds.
(a) If interest rates remain unchanged, what do you expect the price of these bonds to be 1 year from
now? One day before maturity?
(b) Suppose the YTM increases 1 percent for each of the bonds (7 percent and 9 percent, respectively).
Calculate the Holding Period Yield of the one-year investment for each of the bonds (from today
to one year from today).
Note: Holding Period Yield is the total effective annual return for the investor that buys the bond today
and sells the bond in one year, given the change in interest rates. The return can be decomposed in the income component (current yield) and the capital gain component (capital gain yield).
To calculate the price of the bonds 1 year from now and 1 day before maturity, we need to understand the relationship between bond prices and interest rates.
Bond prices and interest rates have an inverse relationship. When interest rates rise, bond prices fall, and when interest rates fall, bond prices rise. This is because when interest rates rise, newly issued bonds pay higher interest, making existing bonds with lower interest rates less valuable. Conversely, when interest rates fall, existing bonds with higher interest rates become more valuable.
(a) 1 year from now:
If interest rates remain unchanged, the prices of the bonds will remain the same. This is because the yield to maturity (YTM) of the bonds is equal to their coupon rates, so there is no capital gain or loss. The price of the premium bond will be $1,000, and the price of the discount bond will also be $1,000.
1 day before maturity:
At maturity, all bonds are redeemed at face value ($1,000). Therefore, one day before maturity, both bonds will have a price of $1,000.
(b) To calculate the Holding Period Yield (HPY) of the one-year investment for each bond, given a change in interest rates, we need to calculate the current yield and the capital gain yield.
For the premium bond:
1. Calculate the current yield:
Current Yield = Annual Coupon Payment / Current Price
Annual Coupon Payment = Coupon Rate * Face Value / 2 (since semiannual payments)
Current Price = Market Price (which could change due to the change in interest rates)
2. Calculate the capital gain yield:
Capital Gain Yield = (Ending Price - Beginning Price) / Beginning Price
For the discount bond, the calculations are the same.
Using the new YTMs of 7% and 9%, respectively, we can perform these calculations to find the HPY for each bond.
Step 1: Calculate the current yield
Premium bond:
Annual Coupon Payment = 0.08 * $1,000 / 2 = $40
Current Price = Market Price (which could change due to the change in interest rates)
Discount bond:
Annual Coupon Payment = 0.06 * $1,000 / 2 = $30
Current Price = Market Price (which could change due to the change in interest rates)
Step 2: Calculate the capital gain yield
Premium bond:
Capital Gain Yield = (Ending Price - Beginning Price) / Beginning Price
Ending Price = Face Value ($1,000) since we are calculating HPY from today to one year from today
Beginning Price = Current Price (from step 1)
Discount bond:
Capital Gain Yield = (Ending Price - Beginning Price) / Beginning Price
Ending Price = Face Value ($1,000) since we are calculating HPY from today to one year from today
Beginning Price = Current Price (from step 1)
With the new YTMs, you can calculate the current yield and capital gain yield for each bond to find the HPY of the one-year investment.