Annuity Math — Lump Sums and Payments

An annuity is a series of equal payments made at regular intervals. The math of annuities answers two related questions: given a lump sum invested today, what regular payment does it produce? And given a desired payment, how large a lump sum is required?

The Formulas

Start with a lump sum $A$ invested at interest rate $r$ per period. If that sum is paid out in equal installments over $t$ periods, the payment amount $P$ is:

$$P = A \cdot \frac{r}{1 - (1+r)^{-t}}$$

Rearranged to find the lump sum required to fund a given payment:

$$A = P \cdot \frac{1 - (1+r)^{-t}}{r}$$

In both formulas, $r$ is the rate per period and $t$ is the number of periods. For monthly payments at an annual rate of 6%, use $r = 0.06/12 = 0.005$ and $t = \text{years} \times 12$. Keep the units consistent.

Example 1 — Find the payment

Mary invests a lump sum of $139,581 in an annuity paying 6% annually. She wants monthly payments for 20 years. What is her monthly payment?

Here $r = 0.06/12 = 0.005$ and $t = 20 \times 12 = 240$:

$$P = 139581 \cdot \frac{0.005}{1 - (1.005)^{-240}} = 139581 \cdot \frac{0.005}{1 - 0.3021} \approx 139581 \cdot 0.007164 \approx $1,000$$

Over 20 years she receives $240 \times $1,000 = $240,000$ in total — considerably more than the $139,581 she put in. The difference is the interest earned over time.

Example 2 — Find the lump sum

What lump sum is needed today to fund payments of $500 per month for 10 years at 4% annual interest?

$r = 0.04/12 \approx 0.003333$, $t = 120$:

$$A = 500 \cdot \frac{1 - (1.003333)^{-120}}{0.003333} = 500 \cdot \frac{1 - 0.6710}{0.003333} \approx 500 \times 98.77 \approx $49,385$$

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Practice Problems

1. A $200,000 annuity earns 5% annually and pays monthly for 15 years. What is the monthly payment? Show answer$r = 0.05/12 \approx 0.004167$, $t = 180$. $P = 200000 \times \frac{0.004167}{1-(1.004167)^{-180}} \approx 200000 \times 0.007908 \approx $1,582/\text{month}$.

2. You want to receive $2,000 per month for 10 years from an annuity earning 4.8% annually. How large a lump sum is required? Show answer$r = 0.048/12 = 0.004$, $t = 120$. $A = 2000 \times \frac{1-(1.004)^{-120}}{0.004} \approx 2000 \times 94.28 \approx $188,560$.

3. An annuity pays $1,200/month for 20 years at 6% annual interest. What is the total amount paid out, and how does that compare to the lump sum required? Show answerTotal paid out: $1200 \times 240 = $288,000$. Lump sum: $A = 1200 \times \frac{1-(1.005)^{-240}}{0.005} \approx 1200 \times 139.58 \approx $167,500$. The annuity pays out $120,500 more than was invested — the excess is interest earned over the 20 years.