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

Compute the 1-year forward rate in 2 years, i.e. compute f(0,2.0,3.0)

Compute the 2-year forward rate in 6 months, i.e. compute f(0,0.5,2.5)

Compute the 1.5-year forward rate in 1 year, i.e. compute f(0,1.0,2.5)

Compute the 2.5-year forward rate in 6 months, i.e. compute f(0,0.5,3.0)

Compute the 2-year forward rate in 1 year, i.e. compute f(0,1.0,3.0)

Compute the 1.5-year forward rate in 1.5 years, i.e. compute f(0,1.5,3.0)

Answer 1

First we find the spot rates for each term using the compound interest formula as shown below:

Maturity Compounding Periods Price 99.009901 97.066175 94.928528 94.218423 90.573081 87.502427 Spot rate = (100/Price) periods) - 1)* 2 2.00% 3.00% 3.50% 3.00% 4.00% 4.50%

Next we find each required forward rate using the forward rate formula as shown below:

In case of semi-annual compounding and r & t expressed in annual figures, the forward rate equation can be solved as:
f(0, t1 , t2)=	=(((1+ r2/2)^(2*t2) / (1 + r1/2)^(2*t1))^(1/(2*(t2 - t1))) - 1) * 2
	=(((1+ r2/2)^(t2) / (1 + r1/2)^(t1))^(1/(t2 - t1)) - 1) * 2
This rate will be such rate that investing firstly in a t1-year zero bond & than at this forward rate should give same effective result as investing in t2 year zero bond.

So for each case, the calculations will be as shown below:

f(0,0.5,2.5):
r1	2.00%
r2	4.00%
t1	0.5
t2	2.5
f(0,0.5,2.5)	=(((1+ r2/2)^(t2) / (1 + r1/2)^(t1))^(1/(t2 - t1)) - 1)*2
	0.045031
f(0,1.0,2.5):
r1	3.00%
r2	4.00%
t1	1
t2	2.5
f(0,1.0,2.5)	=(((1+ r2/2)^(t2) / (1 + r1/2)^(t1))^(1/(t2 - t1)) - 1)*2
	0.046694
f(0,0.5,3.0):
r1	2.00%
r2	4.50%
t1	0.5
t2	3
f(0,0.5,3.0)	=(((1+ r2/2)^(t2) / (1 + r1/2)^(t1))^(1/(t2 - t1)) - 1)*2
	0.050037
f(0,1,3.0):
r1	3.00%
r2	4.50%
t1	1
t2	3
f(0,1,3.0)	=(((1+ r2/2)^(t2) / (1 + r1/2)^(t1))^(1/(t2 - t1)) - 1)*2
	0.052542
f(0,1.5,3.0):
r1	3.50%
r2	4.50%
t1	1.5
t2	3
f(0,1.5,3.0)	=(((1+ r2/2)^(t2) / (1 + r1/2)^(t1))^(1/(t2 - t1)) - 1)*2
	0.055049
f(0,0.5,1.5):
r1	2.00%
r2	3.50%
t1	0.5
t2	1.5
f(0,0.5,1.5)	=(((1+ r2/2)^(t2) / (1 + r1/2)^(t1))^(1/(t2 - t1)) - 1)*2
	0.042542
f(0,1.0,2.0):
r1	3.00%
r2	3.00%
t1	1
t2	2
f(0,1.0,2.0)	=(((1+ r2/2)^(t2) / (1 + r1/2)^(t1))^(1/(t2 - t1)) - 1)*2
	0.030000
f(0,0.5,2.0):
r1	2.00%
r2	3.00%
t1	0.5
t2	2
f(0,0.5,2.0)	=(((1+ r2/2)^(t2) / (1 + r1/2)^(t1))^(1/(t2 - t1)) - 1)*2
	0.033344

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