# 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.

• (1+0.02)*(1+x) = (1+0.03)^2 Solve for x. Commented Feb 15, 2019 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) Commented Feb 16, 2019 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 $(1+r_3)^3=(1+r_1)^1(1+f_{1,3})^{3-1}$ Commented Feb 17, 2019 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? Commented Feb 19, 2019 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 Commented Feb 23, 2019 at 10:13