Question

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5. The following is a list of prices for zero coupon bonds with different maturities and par value of $1,000. Maturity (Years) Price $943.40 $881.68 $808.88 $742.09 a. What is, according to the expectations theory, the expected one-year forward rate in the third year? (i.e. borrow two years from now, repay three years from now) b. Suppose you have entered a forward contract, promising to lend customer A $1 million two years from now, who will repay you three years from now at the forward rate in part a). How would you structure your holdings of zero coupon bonds so that you can exactly match the future cash flow of this loan? What is the duration of a perpetuity with a yield of 8%?!

Answer 1

According to Pure Expectations Theory, forward rate today is an unbiased estimate of the expected spot rate in the future.

a) "borrow after two years, repay after three years" means one year forward rate after 2 years

f2,3 = (1+ r03)3/(1+r02)2 - 1

= (1.073)3/(1.065)2 - 1

= 8.9% [ans]

Working note:

r is the spot rate calculate with the proce given so r02 = (1000/881.68)1/2 -1 and r03= (1000/808.88)1/3 - 1

b) The future cash flow of the loan = 1,000,000*1.089= $1,089,000

Thus to match this outflow following amount should be invested in three year maturity ZCB = 1089000/1.0733 = $881,513

Price of one bond = $808.88

No. of bonds to be purchased = 881513/808.88 = 1090 bonds

Duration of perpetuity with 8% yield

D = (1+ YTM) / YTM

= 1.08/0.08

= 13.5