Question

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.)

	Economy Probability YTM Price Capital Gain Coupon Interest HPR Boom 0.30 9.0 % % Normal Growth 0.50 7.0 % % Recession 0.20 5.0 % %

Answer 1

You will need to use a financial calculator

BOOM CASE:

N = 29 years(After 1 year, it will be a bond with 29 years remaining for maturity

I/Y = YTM = 9%

PMT = 3.5%*100 = 3.5 (the coupon payment)

FV = 100

To find the price, you will need to calculate the Present Value using a calculator. Plugging in these values, we get PV = 43.90944. Ignore the negative sign if any

Capital gain = ending value - beginning value

= 43.909 - 100

= -56.09056

Coupon interest = 3.5%*100 = 3.5

HPR = (Capital Gain + Coupon interest)/Beginning value

= (-56.09056+3.5)/100

= -52.59%

NORMAL CASE:

N = 29 years(After 1 year, it will be a bond with 29 years remaining for maturity)

I/Y = YTM = 7%

PMT = 3.5%*100 = 3.5 (the coupon payment)

FV = 100

To find the price, you will need to calculate the Present Value using a calculator. Plugging in these values, we get PV = 57.02814 = 57.03. Ignore the negative sign if any

Capital gain = ending value - beginning value

= 57.02814 - 100

= -42.97

Coupon interest = 3.5%*100 = 3.5

HPR = (Capital Gain + Coupon interest)/Beginning value

= (-42.97+3.5)/100

= -39.47%

RECESSION CASE:

N = 29 years(After 1 year, it will be a bond with 29 years remaining for maturity)

I/Y = YTM = 5%

PMT = 3.5%*100 = 3.5 (the coupon payment)

FV = 100

To find the price, you will need to calculate the Present Value using a calculator. Plugging in these values, we get PV = 77.2887 = 77.30. Ignore the negative sign if any

Capital gain = ending value - beginning value

=77.2887- 100

= -22.7112 = -22.70

Coupon interest = 3.5%*100 = 3.5

HPR = (Capital Gain + Coupon interest)/Beginning value

= (-22.70+3.5)/100

= -19.21%.