Training Financial Mathematics Annuities
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Annuities

23 min Financial Mathematics

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.

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

Example 2

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

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

Example 3

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

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

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)

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