Question

The one, two, three and five year zero rates are 3.0%, 4.5% , 5.0% and 5.5% respectively. The five year spot rate is not given, but the forward price at year five for a five year zero coupon bond is known to be $74.73 per $100 par value. What is the price of a ten year zero coupon bond?

Answer 1

Given to us,

5 year zero rate r5 = 5.5%

the forward price at year five for a five year zero coupon bond is known to be $74.73 per $100 par value

So, 5 year forward rate for 5 years from now based on the zerocoupon bond is (FV/PV)^(1/n) - 1

5 year forward rate for 5 years from now 5f5 = (100/74.73)^(1/5) - 1 = 6.00%

using r5 and 5f5, 10 year zero rate today is (((1+r5)^5)*((1+5f5)^5))^(1/10) - 1

So, r10 = ((1.055^5)*(1.06^5))^(1/10) - 1 = 5.75%

So, price of a ten year zero coupon bond today with par value of 100 is

Price = 100/(1+r5)^10 = 100/1.0575^10 = $57.18