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QUESTION

You wish to leave a fund which pays out a fixed amount $p$ each
year in perpetuity. If the interest rate is assumed to be $r$ from
now until the end of time, how much should the initial fund be? If
you wish to build up that fund by contributions over $n$ years,
how much should your annual payment be.


ANSWER

Want an amount $P$ paid out after $n$ years for all time. Interest
rate $=r$, assume annual compounding. Consider 2 sides: payments
in and payments out.

From mortgage calculation above over $n$ years with an annual
payment of $d$ you save a total of $\frac{d[(1+r)^n-1]}{r}$ (C)

If you want to pay out a sum $P$ for ever you need the following
amounts saved, assuming payment at the end of the year:

\begin{tabular}{ccccc}
Year 1&Year 2&Year 3&\ldots&Year n\\
\\
$\frac{P}{(1+r)}$&$\frac{P}{(1+r)^2}$&$\frac{P}{(1+r)^3}$&&$\frac{P}{(1+r)^n}$\\
\\
amount needed &amount needed &amount needed &&amount needed \\ at
start of&at start of&at start of&&at start of\\ year to pay&year
to pay&year to pay&&year to pay\\ $P$ in year 1&$P$ in year 2&$P$
in year 3&&$P$ in year $n$
\end{tabular}

The total required to pay $P$ at the end of each year in
perpetuity is:

$$\sum_{i=1}^\infty\frac{P}{(1+r)^i}=P\left[\frac{1}{1-\frac{1}{(1+r)}}\right]-P=\frac{P}{r}\
(D)$$

Thus (C) must equal (D). Hence

$$\frac{P}{r}=\frac{d[(1+r)^n-1]}{r}\Rightarrow
P=d[(1+r)^n-1]\Rightarrow d=\frac{P}{[(1+r)^n-1]}$$

Hence if you want to bequeath a prize of \pounds1000 per year in
35 years time and expect to get 5\% p.a. interest,

$$d=\frac{1000}{((1.05)^{35}-1)}=221.43 \textrm{ per year.}$$

(Note that with inflation the 1000 would be worth increasingly
less!)




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