Estimating a Five-Year Zero-Coupon Rate from Shorter Rates
Summary
The discussion asks whether a five-year zero-coupon yield can be derived from one-, two-, and three-year zero-coupon rates when longer maturities are unknown. Its central lesson is that the available spot rates do not determine the five-year rate on their own: assumptions about future forward rates are required, and events or changing market expectations may alter them.
One answer illustrates two extrapolation approaches. It derives successive forward rates from the given spot rates, then assumes either that later forwards stay level or that they continue along a linear path, producing different five-year spot estimates. These are scenario calculations, not uniquely implied market prices or forecasts. The choice of forward-rate path is an additional view that should be stated clearly; other assumptions would lead to other estimates.
Key ideas
- Short-maturity zero-coupon rates alone do not uniquely determine a five-year spot rate.
- Forward rates can be inferred from adjacent spot rates using compounding relationships.
- A five-year estimate requires assumptions about forward rates beyond the observed maturities.
- Flat and linearly rising forward-rate assumptions produce different implied spot rates.
- Extrapolated yields are conditional scenarios rather than uniquely determined values.
Tags
Full text
# Can we derive 5 year zero coupon interest rate by using 1, 2 and 3 year zero coupon interest rate?
# 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 ?
## Answer by user2183336 (score 2)
https://quant.stackexchange.com/a/25574
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.
## Answer by Nicholas (score 0)
https://quant.stackexchange.com/a/25577
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.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.