# 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

## 2 Answers

Who knows what the 5 year zero coupon rate is in that case, there could be an event 4.5 years out that will have serious interest rate implications that we don't know about. The only thing you can do with these three numbers is extrapolate and say the rate should 9%. You should be aware of what assumptions you're making when you do something like that, but I'll leave that up to you to ponder on.

The result depends on where you see 3y and 4y forward rates.

• 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 $((1+5\%)*(1+7.01\%)*(1+9.03\%)*(1+11.1\%)*(1+13.1\%)]^{1/5} - 1 = 9.01\%$

If you have any other view on 3y and 4y forwards you'll derive 5y spot accordingly. 