Question

Problem 5-8

Derive the probability distribution of the 1-year HPR on a 30-year U.S. Treasury bond with a coupon of 3.5% if it is currently selling at par and the probability distribution of its yield to maturity a year from now is as shown in the table below. (Assume the entire 3.5% coupon is paid at the end of the year rather than every 6 months. Assume a par value of $100.) (Leave no cells blank - be certain to enter "O" wherever required. Negative values should be indicated by a minus sign. Do not round intermediate calculations. Round your answers to 2 decimal places.)
Economy
Probability
Boom
YTM
0.25
Price
Capital Gain
Coupon Interest
HPR
Normal Growth
0.55
0.20
6.0 %
4.0 %
%
%
8.0 %
Recession
%

Answer 1

To derive the probability distribution of the 1-year HPR on a 30-year U.S. Treasury bond with a coupon of 3.5%, you need to calculate the price, capital gain, and coupon interest. The HPR is the sum of the capital gain and coupon interest. Then, you can calculate the probability distribution of the HPR by multiplying the probability of each scenario by their respective HPR values.

Step-by-step explanation:

The probability distribution of the 1-year Holding Period Return (HPR) on a 30-year U.S. Treasury bond with a coupon of 3.5% can be derived using the given information. To do this, we first calculate the different components that contribute to the HPR. The components are the price, capital gain, and coupon interest. The probability distribution of each component is multiplied by their respective values and summed up to calculate the probability distribution of the HPR.

Step 1: Calculate the price of the bond. Since the bond is currently selling at par, the price is equal to the par value, which is $100.

Step 2: Calculate the capital gain. The capital gain is the difference between the price of the bond at the end of the year and the initial price. Since the bond is selling at par and the price remains constant, the capital gain is zero.

Step 3: Calculate the coupon interest. The coupon interest is the annual coupon payment, which is 3.5% of the par value. So, the coupon interest is $3.50.

Step 4: Calculate the HPR. The HPR is the sum of the capital gain and the coupon interest. In this case, since the capital gain is zero, the HPR is equal to the coupon interest.

Step 5: Calculate the probability distribution of the HPR. Multiply the probability of each scenario (Boom and Normal Growth) by their respective HPR values and sum them up.

Probability distribution of the 1-year HPR:
Boom: 0.20 x 3.50% = 0.70%
Normal Growth: 0.55 x 3.50% = 1.93%

Answer 2

To derive the probability distribution of the 1-year HPR on a 30-year U.S. Treasury bond with a coupon of 3.5%, calculate the HPR for each possible YTM outcome. Use the probability distribution of the YTM to determine the probability of each outcome. Calculate the price, capital gain, and HPR for each YTM outcome.

Step-by-step explanation:

To derive the probability distribution of the 1-year Holding Period Return (HPR) on a 30-year U.S. Treasury bond with a coupon of 3.5%, we need to calculate the HPR for each possible yield to maturity (YTM) outcome. The HPR is defined as the sum of the capital gain and the coupon interest received. The probability distribution of the YTM from the table can be used to determine the probability of each outcome. The price can be calculated as the present value of the bond's cash flows. Given a par value of $100 and a coupon interest of 3.5%, you can determine the price for each YTM outcome. The capital gain can be found by subtracting the price from the par value. Finally, the HPR is calculated as the sum of the capital gain and the coupon interest received. Gather the data from the table and apply these calculations to find the probability distribution of the 1-year HPR.