# How to calculate one-year forward one-year rate? [closed]

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.

## closed as off-topic by LocalVolatility, skoestlmeier, Lliane, ZRH, HelinFeb 25 at 1:05

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• (1+0.02)*(1+x) = (1+0.03)^2 Solve for x. – Alex C Feb 15 at 21:03
• Buy USD $1$ in zero bonds today with a maturity of 1Y, in 1Y you'll have USD $(1 + r(1))$, then buy invest all this amount in zero bonds with a maturity of 1Y, you'll get $(1 + r(1)) \times (1 + f(1,1))$ where $r(1)$ is the 1Y zero rate, and $f(1,1)$ is the 1Y forward rate in 1Y. By absence of arbitrage, this investment should be equivalent to investing USD $1$ in 2Y zero bonds. So: $(1 + r(2))^2 = (1 + r(1)) \times (1 + f(1,1))$. This way you can deduce the value of the forward rate from the zero rates. (I assumed annual compounding for the sake of the example) – byouness Feb 16 at 20:33

Let $$\{r_t\}_{t>0}$$ be the spotrates and $$f_{t,T}$$ be the forward rate from time $$t$$ to $$T$$ for $$t. Then the general formula to compute $$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$$
• What is two-year forward one-year rate? Do you men two-year forward AND one-year rate. In that case you have $r_1$ and $f_{1,3}$. For $t=1$ and $T=3$ you can solve for $r_3$ in $(1+r_3)^3=(1+r_1)^1(1+f_{1,3})^{3-1}$ – Sanjay Feb 17 at 9:10
• in this case, I'd have $f_{21}$, and are looking for the three-year zero rate. is that the same thing? – Marie k Feb 19 at 2:02
• I am not sure what you are looking for and what $f_{2,1}$ is. Consider accepting the answer if your orignial questions has been answered – Sanjay Feb 23 at 10:13