Derive the probability distribution of the 1-year HPR on a 30-year U.S. Treasury

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.) (Leave no cells blank - be certain to enter "0" wherever required. Negative values should be indicated by a minus sign. Do not round intermediate calculations. Round your answers to 2 decimal places.)

economy probabilty ytm (%) price capital gain coupon interest HPR boom .25 9 normal .50 7 recession .25 6

Explanation / Answer

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%

Boom Case

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

I/Y = YTM = 6%

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 = 66.0232 = 66.02. Ignore the negative sign if any

Capital gain = ending value - beginning value

=66.0232 - 100

= -33.9768 = -33.98

Coupon interest = 3.5%*100 = 3.5

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

= (-33.98+3.5)/100

= -30.48%

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