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QUESTION

Assuming an interest rate of $r$ and discrete annual payments,
what is the present value of a sum of money to be recieved in $T$
years' time, i.e., how much would you invest now to obtain the
desired sum in $T$ years? Hence what is the present value of
\pounds100 to be recieved in 10 years time when the interest rate
is 5\% and

\begin{description}

\item[(a)]
annual discounting is used;

\item[(b)]
semi-annual discounting is used;

\item[(c)]
continuous discounting.

\end{description}


ANSWER

Rate$=r$, $t$ years, $F=$amount to be received in future,
$P$=value of that money now.

It always helps to turn the question around: How much would I have
to invest now to receive $F$ in $T$ years.

\begin{description}

\item[(a)]
Annual compounding: If I invest $P$ now, in $T$ years I have

$$P(1+r)^T$$

Thus if this has to be $F$ we have

$$F+P(1+r)^T$$

or

$$P=\frac{F}{(1+r)^T}$$

the present value of $F$.

\item[(b)]
Semiannual compounding. Follows the same argument as (a), although
the interest factor is now $\left(1+\frac{r}{2}\right)^{2T}\
(m=2)$. Thus

$$P=\frac{F}{\left(1+\frac{r}{2}\right)^{2t}}$$

\item[(c)]
Continuous discounting: use $Pe^{rT}=F$

$$\Rightarrow P=Fe^{-rT}$$

Thus if $F=100,\ r=0.05,\ T=10$

$$P=\left\{\begin{array}{clcl} (a)&\frac{100}{(1.05)^10}&=&61.39\\
(b)&\frac{100}{(1.025)^20}&=&61.03\\
(c)&\frac{100}{e^{-0.5}}&=&60.65
\end{array}\right.$$

\end{description}





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