# Calculating spot rates from forward rates

I am working on a problem where I am trying to calculate the forward rates from two different spot rates. I have the following:

1 Year Spot Rate = 1%

2 Year Spot Rate = 2%


Specifically, I would like to find the forward rate between the first and second year. (Using semiannual compounding).

My thoughts are to use the following:

Forward Rate = $$(1 + r_a)^{ta} \over (1 + r_b)^{tb}$$ - 1

Which in my case would look like:

Forward Rate = $$(1 + .01)^{1} \over (1 + .02)^{2}$$ - 1

Is this the correct approach? It seems like using this method might not account for semiannual compounding., Any thoughts or advice would be greatly appreciated.

## 2 Answers

Forward Rate = $$\frac {(1+(0.5) 2\%)^{2 * 2}} {(1+(0.5) 1\%)^{2 *1}} -1$$

The above works fine when the day count convention is 30/360.

General formula -

$$F(t,t+1,t+2)= \frac {P(t,t+1) - P(t,t+2)} {\tau P(t,t+2)}$$

where $$F(t,t+1,t+2)$$ is the forward rate between $$t+1$$ and $$t+2$$, as seen at $$t$$

$$P(t,t+1)$$ is the price of zero-coupon bond with maturity $$t+1$$, as seen at $$t$$

$$\tau$$ is the accrual fraction between $$t+1$$ and $$t+2$$

The formula is easy to understand by reading it as -

$$F(t,t+1,t+2){\tau P(t,t+2)} = {P(t,t+1) - P(t,t+2)}$$

so that

on LHS: $$F(t,t+1,t+2)\tau$$ is the interest rate earned by \$1 between $$t+1$$ and $$t+2$$, and this interest discounted to $$t$$ by multiplying by $$P(t,t+2)$$ on RHS: the difference between the two zero-coupon bonds, which return \$1 at $$t+1$$ and $$t+2$$ respectively is the discounted value of interest earned by re-investing maturity proceeds of "$$t+1$$"-maturity bond (i.e. \\$1) between $$t+1$$ and $$t+2$$ at the forward rate.

With numbers in the question above:

(assuming 30/360 day count convention)

$$P(t,t+1) = (1+\frac {1\%} {2})^{-2}$$

$$P(t,t+2) = (1+\frac {2\%} {2})^{-4}$$

$$\tau = yearfrac(t+1,t+2) = 1$$

• I am not sure that I follow the math here.
– QFII
May 13, 2019 at 14:26
• ok - I added further explanation. May 14, 2019 at 3:00

If you want to calculate the forward rate given semi-annual compounding then the answer should be:

$$$$F(0,t_a,t_b)=\Bigg(\sqrt[2*(t_b-t_a)]{\frac{(1 + \frac{r_b}{2})^{2*t_b}}{(1 + \frac{r_a}{2})^{2*t_a}}}-1\Bigg)*2$$$$

This is derived by the fact that : $$$$\Bigg(1+\frac{r_b}{2}\Bigg)^{2*t_b} = \Bigg(1+\frac{r_a}{2}\Bigg)^{2*t_a}*\Bigg(1+\frac{F(0,t_a,t_b)}{2}\Bigg)^{2*(t_b-t_a)}$$$$

If you rearrange the terms in the last formula then you get the first equation. Thank you for the question, I hope this helps.