Training Financial Mathematics Annuities
3 / 5

Annuities

23 min Financial Mathematics
An annuity is a series of equal payments made at regular intervals—mortgage payments, retirement withdrawals, and insurance premiums are all annuities. The present value of an ordinary annuity (payments at end of period) is PV = PMT × [(1 − (1 + r)⁻ⁿ) / r], and its future value is FV = PMT × [((1 + r)ⁿ − 1) / r]. An annuity due shifts each payment to the beginning of the period, multiplying PV and FV by (1 + r). Perpetuities—annuities that continue forever—simplify to PV = PMT / r. These formulas let you answer questions like 'How much must I save each month to retire with $1 million?' or 'What is the fair price of a bond that pays $50 every six months for 20 years?'

Annuities

Ordinary Annuity

Equal payments at the end of each period.

$$FV = R \cdot \frac{(1+i)^n - 1}{i}, \qquad PV = R \cdot \frac{1-(1+i)^{-n}}{i}$$

$R$ = payment, $i$ = rate per period, $n$ = number of periods.

Annuity Due

Equal payments at the beginning of each period. Multiply ordinary annuity formulas by $(1+i)$.

Example 1

Find the FV of $\$200$/month for 10 years at $6\%$ compounded monthly.

  1. $i = 0.005$, $n = 120$.
  2. $FV = 200 \cdot \frac{(1.005)^{120}-1}{0.005} \approx \$32{,}775.87$.
Example 2

PV of receiving $\$500$/quarter for 5 years, rate $8\%$ compounded quarterly.

  1. $i = 0.02$, $n = 20$.
  2. $PV = 500 \cdot \frac{1-(1.02)^{-20}}{0.02} \approx \$8{,}175.72$.
Example 3

How much must you save each month for 15 years at $5\%$ (monthly) to accumulate $\$100{,}000$?

  1. $R = \frac{FV \cdot i}{(1+i)^n - 1} = \frac{100000(0.004167)}{(1.004167)^{180}-1} \approx \$373.02$.

Practice Problems

1. FV: $\$100$/month, $4\%$ monthly, 20 years.
2. PV: $\$1{,}000$/year, $6\%$, 10 years.
3. Monthly payment to accumulate $\$50{,}000$ in 10 years at $5\%$ monthly.
4. FV of annuity due: $\$300$/month, $6\%$ monthly, 5 years.
5. PV of $\$2{,}000$/year for 15 years at $7\%$.
6. How long to save $\$25{,}000$ at $\$400$/month, $3\%$ monthly?
7. Compare: $\$10{,}000$ lump sum vs. $\$150$/month for 6 years at $4\%$.
8. Retirement: $\$500$/month for 30 years at $7\%$ monthly. FV?
9. PV of perpetuity: $\$100$/year at $5\%$.
10. Sinking fund: $\$200{,}000$ needed in 20 years, $4\%$ annually. Annual payment?
11. Annuity due PV: $\$500$/month at $6\%$ monthly, 3 years.
12. Which is larger: PV of ordinary annuity or annuity due (same R, i, n)?
Show Answer Key

1. $FV \approx \$36{,}677.46$

2. $PV \approx \$7{,}360.09$

3. $R \approx \$322.01$

4. Ordinary FV $\approx \$20{,}931.20$; due: $\times 1.005 \approx \$21{,}035.86$

5. $PV \approx \$18{,}225.87$

6. Solve: $25000 = 400 \cdot \frac{(1.0025)^n-1}{0.0025}$; $n \approx 58$ months

7. Lump: $FV = 10000(1.04/12)^{72}$... compare to annuity FV

8. $FV \approx \$609{,}985$

9. $PV = R/i = 100/0.05 = \$2{,}000$

10. $R = \frac{200000(0.04)}{(1.04)^{20}-1} \approx \$6{,}716.35$

11. Ordinary PV $\approx \$16{,}435.51$; due: $\times 1.005 \approx \$16{,}517.69$

12. Annuity due (payments arrive earlier)

💵 Annuity Calculator
FV (ordinary annuity)
PV (ordinary annuity)
Total contributed
Interest earned
Present Value & Future Value Loan Amortization