Compounding issues

In reality interest rates underlying fixed income securities, bank deposits, bank loans etc. are compounded using a variety of ways. They could be compounded annually, semiannually, daily and so on. Here we are going to use continuously compounded interest rates, just because they help us derive closed form solutions, not only for forwards and futures, but for a whole set of derivative securities.

Consider an investment $A$ deposited for $n$ years at an interest rate $R$, which is compounded annually. The terminal value of the investment will be

$$A\left( 1+R\right) ^{n}$$.

If it is compounded $m$ times per year, the terminal value will be

$$A\left( 1+\frac{R}{m}\right) ^{mn}$$.

If compounding occurs continuously, or equivalently if $m\rightarrow \infty$ in the above expression, then

$$\lim_{m\rightarrow \infty }A\left( 1+\frac{R}{m}\right) ^{mn}=Ae^{Rn}$$.

Discounting an amount $A$ with an interest rate $R$ which is compounded continuously will give a present value

$$Ae^{-Rn}$$.

Suppose that $R_{c}$ is the interest rate which is compounded continuously and $R_{m}$ the equivalent interest rate compounded $m$ times per year. Equivalent here means that they offer the same payoff for the same investment $A$, or that

$$A\left( 1+\frac{R_{m}}{m}\right) ^{mn}=Ae^{R_{c}n}$$.

Solving for the two interest rates will give us the conversion formulae

$$R_{c} = m\ln \left( 1+\frac{R_{m}}{m}\right)$$

$$R_{m} = m\left( e^{\frac{R_{c}}{m}}-1\right)$$