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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.

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

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    $\begingroup$ (1+0.02)*(1+x) = (1+0.03)^2 Solve for x. $\endgroup$ – Alex C Feb 15 at 21:03
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    $\begingroup$ 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) $\endgroup$ – byouness Feb 16 at 20:33
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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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    $\begingroup$ thanks so much! now, let's say I have a two-year forward one-year rate, how would I find the three-year zero rate? I'm sorry for the questioning-- just a tad confused on this $\endgroup$ – Marie k Feb 16 at 19:41
  • $\begingroup$ 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 $(1+r_3)^3=(1+r_1)^1(1+f_{1,3})^{3-1}$ $\endgroup$ – Sanjay Feb 17 at 9:10
  • $\begingroup$ in this case, I'd have $f_{21}$, and are looking for the three-year zero rate. is that the same thing? $\endgroup$ – Marie k Feb 19 at 2:02
  • $\begingroup$ 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 $\endgroup$ – Sanjay Feb 23 at 10:13

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