Training Financial Mathematics Simple & Compound Interest
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Simple & Compound Interest

23 min Financial Mathematics

Simple & Compound Interest

Simple Interest

$$I = Prt, \qquad A = P(1 + rt)$$

$P$ = principal, $r$ = annual rate (decimal), $t$ = time in years, $A$ = accumulated amount.

Compound Interest

$$A = P\left(1 + \frac{r}{n}\right)^{nt}$$

$n$ = compounding periods per year. As $n \to \infty$:

$$A = Pe^{rt}$$

Effective Annual Rate (EAR)

$$\text{EAR} = \left(1 + \frac{r}{n}\right)^n - 1$$

The true annual yield accounting for compounding.

Example 1

Find the simple interest on $\$5{,}000$ at $4\%$ for 3 years.

$I = 5000(0.04)(3) = \$600$. $A = \$5{,}600$.

Example 2

Find the compound amount of $\$2{,}000$ at $6\%$ compounded quarterly for 5 years.

$A = 2000(1 + 0.06/4)^{20} = 2000(1.015)^{20} \approx \$2{,}693.71$.

Example 3

Find the EAR for $8\%$ compounded monthly.

$\text{EAR} = (1 + 0.08/12)^{12} - 1 \approx 0.0830 = 8.30\%$.

Practice Problems

1. Simple interest on $\$3{,}000$ at $5\%$ for 2 years.
2. Compound amount: $\$1{,}000$, $4\%$, quarterly, 10 years.
3. Which is better: $5\%$ compounded monthly or $5.1\%$ simple?
4. EAR for $12\%$ compounded monthly?
5. Continuous compounding: $\$10{,}000$ at $3\%$ for 8 years.
6. How long to double at $6\%$ compounded annually?
7. Simple interest amount: $\$700$ at $3.5\%$ for 18 months.
8. $A = \$5{,}000$, $r = 4\%$, $n = 12$, $t = 3$. Find $P$ (present value).
9. Rule of 72: estimate doubling time at $9\%$.
10. Which earns more in 1 year on $\$1{,}000$: $6\%$ compounded semi-annually or $5.9\%$ compounded daily?
11. Find the interest earned: $\$4{,}000$ at $5\%$ compounded annually for 4 years.
12. EAR for $10\%$ compounded continuously?
Show Answer Key

1. $I = 3000(0.05)(2) = \$300$

2. $A = 1000(1.01)^{40} \approx \$1{,}488.86$

3. $5\%$ monthly: EAR = $5.12\%$ > $5.1\%$; compounded is better

4. $(1.01)^{12}-1 \approx 12.68\%$

5. $10000 e^{0.24} \approx \$12{,}712.49$

6. $2 = 1.06^t$; $t = \ln 2/\ln 1.06 \approx 11.9$ years

7. $t = 1.5$; $A = 700(1+0.035 \cdot 1.5) = \$736.75$

8. $P = 5000/(1+0.04/12)^{36} \approx \$4{,}435.27$

9. $72/9 = 8$ years

10. $6\%$ semi: EAR $= 6.09\%$; $5.9\%$ daily: EAR $\approx 6.08\%$; $6\%$ semi wins

11. $A = 4000(1.05)^4 \approx 4862.03$; interest $\approx \$862.03$

12. $e^{0.10}-1 \approx 10.52\%$

Present Value & Future Value