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QUESTION

Show that if you compound interest $m$ times a year, with a quotes
rate of $f$, then an amount $P$ invested at the beginning of the
year grows to $P(1+\frac{r}{m})^m$ by the end of that year. Hence
show that after $T$ years the investment has grown to
$P(1+\frac{r}{m})^{mT}$. Hence show that in the limit of
continuous compounding the investment grows as $P \exp(rT)$.

(Hint: $\lim_{n\to\infty}(1+\frac{1}{n})^n=e$.)


ANSWER

The annual quotes rate is $r$ but you pay interest $m$ times a
year in installments of $\frac{r}{m}$. However you also compound
$m$ times a year. So after 1 year you have

$$P\underbrace{\left(1+\frac{r}{m}\right)
\times\left(1+\frac{r}{m}\right)\times \ldots \times
\left(1+\frac{r}{m}\right)}_{m\textrm{
times}}=P(1+\frac{r}{m})^m$$

Thus after $T$ years you have

$$P\underbrace{\left(1+\frac{r}{m}\right)^m}_{\textrm{year 1}}
\times \underbrace{\left(1+\frac{r}{m}\right)^m}_{\textrm{year 2}}
\times \underbrace{\left(1+\frac{r}{m}\right)^m}_{\textrm{year 3}}
\times\ldots
\underbrace{\left(1+\frac{r}{m}\right)^m}_{\textrm{year }T}
\times=P\left(1+\frac{r}{m}\right)^{mT}$$

Is it more than annual compounding ? Expand

\begin{eqnarray*}
\left(1+\frac{r}{m}\right)^{mT}&=&1+m\frac{Tr}{m}+mT(mT-1)\frac{r^2}{2m^2}+\ldots\\
&=&\underbrace{1+rT}_{\textrm{simple}}+\frac{T^2r^2}{2}-\frac{Tr^2}{2m}
\end{eqnarray*}

cf. annual compounding

\begin{eqnarray*}
(1+r)^T&=&1+rT+T(T-1)\frac{r^2}{2}\\
&=&1+rT+\frac{T^2r^2}{2}-\frac{Tr^2}{2}
\end{eqnarray*}

Therefore annual compounding pays less since $m>1$.

Let $m\to\infty$. This is the limit of continuous compounding,
i.e. paying interest at each and every time instant. The
investment grows as

\begin{equation}
\lim_{m\to\infty}P\left(1+\frac{r}{m}\right)^{mT}
\end{equation}

Now set $n=\frac{m}{r}$ with $r$ fixed. Hence as $m\to\infty$ so
does $n$. Thus (1) becomes

\begin{eqnarray*}
\lim_{n\to\infty}P\left(1+\frac{1}{n}\right)^{nrT}&=&P\left[\lim_{n\to\infty}\left(1+\frac{1}{n}\right)^n\right]^{rt}\
(r,t\textrm{ fixed})\\ &=&P[e]^{rt}\textrm{ by the hint given }\\
&=&Pe^{rt}
\end{eqnarray*}

$r$ is here called the  \lq\lq spot rate''

Note that the exponentials of positive numbers $rT$ grows faster
than any power of $(rT)$ and hence this pays more than any
discrete compounding which pays more than simple interest. that's
why banks normally only pay annual interest, or reduced their
rates for quarterly etc. payments.




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