Question

Derive the probability distribution of the 1-year HPR on a 30-year U.S. Treasury bond with an    8% coupon if it is currently selling at par and the probability distribution of its yield to maturity a year from now is as follows:

State of the Economy	Probability	YTM
Boom	.20	11.0%
Normal growth	.50	8.0
Recession	.30	7.0

For simplicity, assume the entire 8% coupon is paid at the end of the year rather than every 6 months.

Answer 1

Holding period return (HPR) is the average rate of return earned by the investor from his investment holding. Holding period return will be affected by the changes in redemption value of the bond and bond purchase price.

State of the Economy	Probability	YTM
Boom	.20	11.0%
Normal Growth	.50	8.0%
Recession	.30	7.0%

HPR can be calculated by using formula given below:

A 30-year US Treasury bond at par is bought, so the price at the beginning is equal to the face value of the bond.

The ending price is the price of the bond in one year’s time. The price of a bond can be calculated by the following formula:

Where,

Assume that at the end of 1 year, there are still 29 years until maturity on the bond. With a coupon rate of 8%, the formula for price will be as shown below:

For YTM of 11.0%, the HPR can be calculated as shown below:

Therefore, holding period return is –17.95%.

For YTM of 8.0%, the HPR can be calculated as shown below:

Therefore, holding period return is 8%.

For YTM of 7.0%, the HPR can be calculated as shown below:

Therefore, holding period return is 20.3%.

Probability distribution of the 1-year HPR is summarized as shown below:

State of the Economy	Probability	Holding period return (HPR)
Boom	.20	–18.0%
Normal Growth	.50	8.0%
Recession	.30	20.3%