Question

Question 2 A 4-year zero coupon bond has a face value of $100 and a price of $82.270. The table below contains prices, coupons and the time to maturity for 3 annual coupon paying bonds that have a face value of $100. Please use annual compounding and annual discounting in your calculations and 3 decimal places in your workings. Time to Maturity (Years) Annual Coupon Price 1 3.0% $98.095 2 6.5% $102.790 3 5.8% $102.179 a) What is the price of a 3-year bond paying annual coupons of 5% and a face value of $100? b) If you had $1 million invested in the 3-year 5% coupon bond, how many of the 4-year zerocoupon bonds would you need to short-sell to hedge your interest rate risk? c) Without doing any calculations, discuss what actions you would need to take with your short position in the zero coupon to ensure your interest rate risk remained hedged until the maturity of the 3-year coupon paying bond.

Answer 1

The 4 year spot rate s4 is given by the four year zero coupon bond as

s4 = (100/82.27)^(1/4)-1 = 0.050001 or 5.000%

Similarly,

From the 1 year coupon bond, 1 year spot rate, s1 = (100+3)/98.095 -1 = 0.050003 or 5.000%

From the 2 year coupon bond, 2 year spot rate s2 is given by

6.5/1.05 + 106.5/(1+s2)^2 =  102.79

s2= 0.049995 or 5.000%

From the 3 year coupon bond, 3 year spot rate s3 is given by

5.8/1.05+5.8/1.05^2+105.8/(1+s3)^3 = 102.179

=>s3 = 0.049998 or 5.000%

So price of a 3 year annual coupon bond paying 3%

= 5/1.05+ 5/1.05^2+105/1.05^3

= $100

b) Duration of the 3 year bond = (1*5/1.05+ 2*5/1.05^2+3*105/1.05^3)/100 = 2.85941 years

Duration of 4 year zero coupon bond = 4 years

If $X worth of 4 year Zero coupon bonds are shorted

X*4 = 2.85941 * 1 million

X = $0.714853 million or $718453

So, one need to short $718453 worth of Zero coupon bonds

c) After every coupon payment, the Duration of the coupon bond will have to be matched with the Zero coupon bond as above and Zero coupon bonds will have to be bought again. Finally all zero coupon bonds would have to be bought again at the end of 3 years.