# Can we derive 5 year zero coupon interest rate by using 1, 2 and 3 year zero coupon interest rate?

Given that the 1 year zero coupon bond interest rate is 5%, 2 year zero coupon bond interest rate is 6% and 3 year zero coupon bond interest rate is 7%. 4 year coupon bond price and interest rate are unknown. How to derive for 5 year zero coupon bond interest rate ?

• You can extrapolate or use a more serious model (knowing the usual yield curve shape), but I don't think there is any precise 5 year rate that you can get from these data – MarinD Apr 23 '16 at 0:42

• 1 year forward rate is $7.01\% = (1+6\%)^2/(1+5\%)-1$
• 2 year forward rate is $9.03\% = (1+7\%)^3/(1+6\%)^2-1$
If you assume that forwards are flat after 2nd year: 3yFwd=4yFwd=2yFwd = $9.03\%$ then your 5y spot becomes $((1+5\%)*(1+7.01\%)*(1+9.03\%)^3)^{1/5} - 1 = 7.81\%$.
If you assume that forwards are linear then 3yFwd=$11.1\%$, 4yFwd=$13.1\%$ so your 5y spot becomes $((1+5\%)*(1+7.01\%)*(1+9.03\%)*(1+11.1\%)*(1+13.1\%)]^{1/5} - 1 = 9.01\%$