Skill: Calculate compound interest with varied compounding periods

Compound interest is calculated with

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

where:

• $P$ is the starting amount, or principal.
• $r$ is the nominal annual interest rate written as a decimal.
• $n$ is the number of compounding periods per year.
• $t$ is the time in years.
• $A$ is the accumulated value.

The periodic interest rate is $\frac{r}{n}$. The exponent $nt$ gives the total number of times interest is added.

Example

You invest 5{,}000$at a nominal annual rate of$6%$for$3$ years. Find the accumulated value if interest is compounded quarterly and monthly.

1. Quarterly compounding

Quarterly means interest is added $4$ times per year, so $n=4$.

Convert the annual rate to a decimal:

$$r=6\%=0.06$$

The interest rate per quarter is

$$\frac{r}{n}=\frac{0.06}{4}=0.015,$$

or $1.5\%$ per quarter.

The total number of compounding periods is

$$nt=4(3)=12.$$

Substitute into the formula:

$$A=5000\left(1+\frac{0.06}{4}\right)^{4(3)}$$ $$A=5000(1.015)^{12}$$ $$A\approx \$5{,}978.09.$$

The interest earned is

$$\$5{,}978.09-\$5{,}000=\boxed{\$978.09}.$$

2. Monthly compounding

Monthly means interest is added $12$ times per year, so $n=12$.

The interest rate per month is

$$\frac{r}{n}=\frac{0.06}{12}=0.005,$$

or $0.5\%$ per month.

The total number of compounding periods is

$$nt=12(3)=36.$$

Substitute into the formula:

$$A=5000\left(1+\frac{0.06}{12}\right)^{12(3)}$$ $$A=5000(1.005)^{36}$$ $$A\approx \$5{,}983.40.$$

The interest earned is

$$\$5{,}983.40-\$5{,}000=\boxed{\$983.40}.$$

Monthly compounding produces slightly more money than quarterly compounding:

$$\$5{,}983.40-\$5{,}978.09=\boxed{\$5.31}.$$

The nominal annual rate remains $6\%$ in both cases. However, monthly compounding uses a smaller periodic rate and adds interest more often. Each time interest is added, it becomes part of the balance and earns additional interest later. This repeated reinvestment is why more frequent compounding usually produces a larger accumulated value.