The cash futures price of a 3-month zero coupon bond with a face value of $100 f

Question

The cash futures price of a 3-month zero coupon bond with a face value of $100 for delivery in B1 months from now is B2 dollars. Suppose that the current spot interest rate for a term of B3 months is B4 per cent per annum. Assume continuous compounding to answer the following:

9. The forward rate of interest for period B1 months to B1+3 months.

10. The spot rate of interest for period 0 to B1 months.

11. The current fair value of a B1-month zero coupon bond with $100 face.

Shiw all work

Explanation / Answer

9. By the definition of the futures price of a 3 month zero coupon bond to be delivered in 6.3 months, we can calculate the fwd rate if interest for 3 months 6.3 monhts from now as follows:

P = 100 * exp (-rT)

where r = rate you want to calculate

P = 97.85

T = 3 months

=> r = ln(100/P) / T = ln(100/97.85) / (3/12) = 8.69% (Answer 9)

10. Using no arbitrage principle, we can calculate the spot rate for 6.3 months as follows:

exp(s1*T1) * exp(f*T) = exp(s2*T2)

where, s1 = spot rate at 6.3m = T1, f = fwd rate for 3m = T at 6.3 months from now, s2 = spot rate at 9.3m = T2

=> s1 = (s2*T2 - f*T)/T1 = ((9.6%*9.3) - (8.69%*3))/6.3 = 10.03% (Answer 10)

11. Using P = 100 * exp (-rT)

=> P = 100 * exp(-1*10.03%*6.3/12) = $94.87 (Answer 11)

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