Calculating a One-Year Forward Rate from Zero Rates

Summary

The document derives a forward interest rate from two annual zero rates. It equates the growth from investing over the full longer maturity with the compounded growth from investing to the shorter maturity and then reinvesting at the forward rate for the remaining period. Rearranging this relationship gives the forward rate between the two dates.

For the example, the one-year and two-year spot rates are 2% and 3%, respectively; the implied rate from year one to year two is reported as approximately 4.01%. The calculation assumes annual compounding and zero-coupon spot rates. The thread does not address alternative day-count conventions, continuous compounding, discount-factor notation, or whether rates incorporate credit or liquidity effects, so conventions should be aligned with the instruments being analyzed.

Key ideas

• A forward rate can be derived by equating growth over the full maturity with growth over successive periods.
• The forward interval begins at the shorter maturity and ends at the longer maturity.
• Using the stated 2% and 3% annual zero rates gives a one-year forward rate of about 4.01% for year two.
• The formula shown uses annual compounding and assumes the inputs are zero-coupon spot rates.

Tags

# How to calculate one-year forward one-year rate?


# How to calculate one-year forward one-year rate?












I'm just a little lost on how to calculate forward rates. I know this is an easy question, but, if we are given a one-year and two-year zero rate (let's say, for the sake of the argument, 2% and 3% respectively), how do we calculate the one-year forward one-year rate?

I just am confused as to which formula to use.

## Answer by Sanjay (score 4)

https://quant.stackexchange.com/a/44083

Let $\{r_t\}_{t>0}$ be the spotrates and $f_{t,T}$ be the forward rate from time $t$ to $T$ for $t<T$. Then the general formula to compute $f_{t,T}$ is $$ (1+r_T)^T=(1+r_t)^t(1+f_{t,T})^{T-t} $$

Now you can solve for $f_{t,T}$ to obtain:

$f_{t,T}= \left( \frac{(1+r_T)^T}{(1+r_t)^t} \right) ^{1/(T-t)}-1$

In your example: Spot rates are given by the zero coupon bonds meaning $r_1=0.02$, $r_2=0.03$. So you can compute the forward from year $t=1$ to $T=2$ by plugging in the above equation and the result is:$f_{1,2}=0.040098$

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.