Question

Pete plans to buy an 8 percent, $1,000 par bond that matures in three years and the interest is paid semiannually, and the bond’s YTM is 10 percent.

1. Calculate the bond’s duration.   

(b) Calculate the bond’s modified duration.   

(c) Assuming the bond’s YTM declines from 10 percent to 9.5 percent, calculate the bond’s price change. Explain your answer.

(d) Explain how changes in YTM affects the bond’s market price risk and reinvestment risk.

Answer 1

Part (a): Bond’s Duration= 2.71745 years

Part (b): Modified Duration= 2.58805 years

Calculations as below:

Part (c ):

As a general rule, for every change of 100 basis points in interest rate (YTM), bond price will change, in the opposite direction, by approximately 1% for every year of duration

Accordingly, Change in price (%)= Change in YTM (%)*Duration

In the given case,

Price = $949.24 (as in work sheet above)

Duration= 2.71745 years (as above)

Therefore, upon decline in YTM by 50 basis points,

Increase in price (%) = 0.5%*2.71745 = 1.35872%

New price= $949.24*(1+1.35872%) = $962.14

Part (d):

It is assumed that periodical interest (coupon) inflows are reinvested at the YTM. Hence when the yield (market interest rate) decreases, the subsequent inflows will have to be invested at lower interest rate. As a result, money available at the end of investment horizon will be less. This is the reinvestment risk posed by decline in YTM.

Price of the bond is the sum of present values (PVs) of all future cash flows. PVs are calculated by discounting the cash flows with the interest rate expected on reinvestment (YTM) as discount rate, for the time left till date of receipt of the cash flows. Hence, when the YTM (the discount rate) increases, amount of discount on future cash flows increases and as a result, the present values decreases. This result in decrease the bond price. (In cases where the investments are to be marked to market at the intervals as stipulated by the regulators, the reduction in market value of the securities will result in booking losses in the books of the investor). This is the price risk.

The reverse process takes place in case the market interest rates decrease after making the investment. In such a situation, price of the security increase when the market interest rate decreases.