Question

You observe the following prices of zero-coupon bonds. Assume semi-annual compounding throughout.

Time to Maturity in years	Zero-Coupon Bond Price
0.5	99.009901
1	97.066175
1.5	94.928528
2	94.218423
2.5	90.573081
3	87.502427

1) Compute the six-month forward curve, i.e. compute f(0,0.5,1.0), f(0,1.0,1.5), f(0,1.5,2.0), f(0,2.0,2.5), and f(0,2.5,3.0). Round to six digits after the decimal. Enter percentages in decimal form, i.e. enter 2.1234% as 0.021234.

2)  Compute the one-year forward rate in six months, i.e. compute f(0,0.5,1.5)

3) Compute the one-year forward rate in one year, i.e. compute f(0,1.0,2.0)

4)Compute the 1.5-year forward rate in six months, i.e. compute f(0,0.5,2.0)

Answer 1

Pt = 100 / (1 + Z0,t / 2)2t where Z(0, t) is the zero rate for the maturity t

Hence, Pt + 0.5 = 100 / (1 + Z0, t + 0.5 / 2)2(t + 0.5) = 100 / (1 + Z0, t + 0.5 / 2)2t + 1

Now, f(0,t, t + 0.5) is given by the equation:

(1 + Z0,t / 2)2t (1 + f / 2) = (1 + Z0, t + 0.5 / 2)2t + 1

Hence, (1 + f / 2) = [(1 + Z0, t + 0.5 / 2)2t + 1] /  [(1 + Z0,t / 2)2t] = Pt / Pt + 0.5

Hence, the forward rate = f(0,t, t + 0.5) = f = (Pt / Pt + 0.5 - 1) x 2

Part (1)

Set t = 0.5 to get f(0,0.5,1.0) = (P0.5 / P1 - 1) x 2 = (99.009901 / 97.066175 - 1) x 2 = 0.040050

Set t = 1 to get f(0,1.0,1.5) = (P1 / P1.5 - 1) x 2 = (97.066175 / 94.928528 - 1) x 2 = 0.045037

Set t = 1.5 to get f(0,1.5,2.0) = (P1.5 / P2 - 1) x 2 = (94.928528 / 94.218423 - 1) x 2 = 0.015074

Set t= 2.0 to get f(0,2.0,2.5) = (P2 / P2.5 - 1) x 2 = (94.218423 / 90.573081 - 1) x 2 = 0.080495

and set t = 2.5 to get f(0,2.5,3.0) = (P2.5 / P3 - 1) x 2 = (90.573081 / 87.502427 - 1) x 2 = 0.070184

Part (2)

f(0,0.5,1.5) will be given by:

(1 + Z0,0.5 / 2)(1 + f / 2)2 = (1 + Z0, 1.5 / 2)3

Hence, (1 + f / 2)2 = (1 + Z0, 1.5 / 2)3 / (1 + Z0,0.5 / 2) = P0.5 / P1.5

Hence, the desired forward rate = f(0,0.5,1.5) = f = [(P0.5 / P1.5)1/2 - 1] x 2 = [(99.009901 / 94.928528)1/2 - 1] x 2 = 0.042542

Part (3)

f(0,1.0,2.0) = [(P1 / P2)1/2 - 1] x 2 = [(97.066175 / 94.218423)1/2 -1] x 2 = 0.030000

Part (4)

f(0,0.5,2.0) = [(P0.5 / P2)1/3 - 1] x 2 = [(99.009901 / 94.218423)1/3 -1] x 2 = 0.033344