Pensions and annuities

Withdrawing a regular pension from a superannuation fund or annuity, finding the pension that runs the fund down to zero, and how long a fund will last.

The same mathematics as a loan

A fund paying a pension works exactly like a loan in reverse: the fund earns interest, and the pension $W$ is withdrawn each period. With $P$ in the fund and ratio $r$, the balance after $n$ withdrawals at the end of each period is $$A_n = Pr^n - W \times \frac{r^n - 1}{r - 1}.$$

Finding the pension

Example 1●●●

A retiree has $\$500\,000$ earning $5\%$ p.a. compounded annually. What pension can be withdrawn at the end of each year so the fund lasts exactly 20 years?

Set $A_{20} = 0$: $500\,000 \times 1.05^{20} = W \times \dfrac{1.05^{20} - 1}{0.05}$.

$W = \dfrac{500\,000 \times 0.05 \times 1.05^{20}}{1.05^{20} - 1} \approx \$40\,121.29$

How long will it last?

Example 2●●●

The same retiree withdraws $\$40\,000$ at the end of each year instead. How many full pensions can be paid?

$500\,000 \times 1.05^n = 40\,000 \times \dfrac{1.05^n - 1}{0.05} = 800\,000(1.05^n - 1)$

$300\,000 \times 1.05^n = 800\,000$, so $1.05^n = \tfrac83$.

$n = \dfrac{\ln \frac83}{\ln 1.05} \approx 20.1$, so $20$ full pensions, plus a small final payment.

Flashcards

Balance of a pension fund

$A_n = Pr^n - W\dfrac{r^n - 1}{r - 1}$

Fund runs out

$A_n = 0$

A fund that never runs out

Pension no more than the interest earned each year.

Practice

$\$500\,000$ at $5\%$ p.a. Find the yearly pension that lasts exactly 20 years, to the nearest cent.
The same fund pays $\$40\,000$ a year. How many full pensions can be paid?
What is the largest yearly pension that $\$500\,000$ at $5\%$ p.a. can pay forever?
Mathematically, a pension fund is most like:
$\$500\,000$ earns $5\%$ for one year and no pension has been paid. Find the balance.
A pension of $\$40\,000$ is then withdrawn. Find the balance.
A fund lasts for $n \approx 17.6$ years of full pensions. The number of full pensions is:
To make a fund last longer, the retiree could:
Withdrawing $W$ at the end of each period, the balance after 3 withdrawals is: