Annuities and Perpetuities - TCE Mathematics Applications (Tasmania)

A retiree invests $\$50\,000$ at 0.5 percent per month and withdraws $\$1000$ each month.

Step 1: Identify the parameters

The initial balance is $A_0 = 50000$, the monthly interest rate is $r = 0.005$, and the monthly withdrawal is $d = 1000$. These values go directly into the annuity recurrence $A_{n+1} = A_n(1 + r) - d$, so it is worth checking them carefully before calculating.

Step 2: Apply the recurrence for the first two months

$A_1 = 50000(1.005) - 1000 = 50250 - 1000 = 49250$.
$A_2 = 49250(1.005) - 1000 = 49496.25 - 1000 = 48496.25$.

Step 3: Interpret the pattern

In month 1 the fund earns $\$250$ in interest, so only $\$750$ of the $\$1000$ withdrawal comes from principal. Because the withdrawal exceeds the interest earned, the balance falls a little each month. As the balance shrinks, the interest earned also falls, which means more and more principal is consumed with each subsequent withdrawal and the fund drains at an accelerating rate.

A university wants to pay a $\$3000$ scholarship every year forever from a fund earning 5 percent per annum. How much must be invested?

Step 1: State the perpetuity condition

For a fund to last forever, the annual payment must equal the interest earned on the principal. This means each year the interest replaces exactly what is withdrawn, so the balance never falls. The perpetuity formula $A = \frac{d}{r}$ comes directly from setting $d = A \times r$ and solving for the required lump sum $A$.

Step 2: Substitute the known values

$d = 3000$, $r = 0.05$.

Step 3: Verify the result makes sense

Investing $\$60\,000$ at 5 percent per annum earns exactly $0.05 \times 60000 = \$3000$ each year. That $\$3000$ is paid out as the scholarship, leaving the principal untouched at $\$60\,000$. Because the principal never changes, the interest earned each year is always $\$3000$, and the scholarship can continue indefinitely without any additional funding.

The retiree above withdraws $\$1000$ but month 1 interest is only $\$250$.

Reason about sustainability

Because $\$1000$ exceeds the $\$250$ interest, $\$750$ of principal is consumed in the first month. The withdrawal is well above the interest, so this is not a perpetuity and the fund will be exhausted after a finite number of months.

A $\$200\,000$ annuity earns $6\%$ p.a. compounding monthly and pays $\$1500$ per month.

Step 1: Convert the annual rate to a monthly rate

The annual rate of $6\%$ must be divided by 12 to match the monthly payment frequency. Monthly rate $i = \frac{0.06}{12} = 0.005$. Using the annual rate directly would overstate the interest earned each period and give an incorrect answer for $N$.

Step 2: Enter all inputs into the finance solver

Set present value $PV = 200000$ (the lump sum invested), payment $PMT = -1500$ (entered as negative because money is leaving the fund each month), and future value $FV = 0$ (the fund is fully drawn down at the end). The monthly rate $i = 0.005$ is also entered. With all other fields known, the solver can find $N$, the number of monthly payments.

Step 3: Read and interpret the solver output

The solver returns a value of $N$ near 248 months, so the annuity lasts about 20 years and 8 months. In the early months the fund earns interest of about $\$1000$ per month at the $0.5\%$ rate, which means only about $\$500$ of the $\$1500$ withdrawal comes from principal. Because so little principal is consumed at the start, the balance remains large for many years and the fund sustains payments for a considerable period before it is exhausted.