Skill: Determine present and future values

To determine present value or future value, first make sure the interest rate and number of periods match the compounding frequency.

For compound interest:

$$F=P(1+i)^n$$

where:

• $F$ is the future value
• $P$ is the present value
• $i$ is the interest rate per compounding period
• $n$ is the total number of compounding periods

Example

An account earns $6\%$ annual interest compounded quarterly.

1. Find the future value of 2{,}000$ after 3 years.
2. Find the present value of 2{,}500$ due in 3 years.

Step 1: Convert the rate and time

Because interest is compounded quarterly, there are 4 periods per year.

The rate per quarter is:

$$i=\frac{0.06}{4}=0.015$$

The total number of quarters is:

$$n=3(4)=12$$

Part 1: Find the future value

Use:

$$F=P(1+i)^n$$

Substitute the known values:

$$F=2000(1.015)^{12}$$

Calculate:

$$F\approx \$2{,}391.24$$

So, 2{,}000$will grow to approximately **$$2{,}391.24$** after 3 years.

Part 2: Find the present value

To find the amount needed today for a future amount, rearrange the formula:

$$P=\frac{F}{(1+i)^n}$$

Substitute the values:

$$P=\frac{2500}{(1.015)^{12}}$$

Calculate:

$$P\approx \$2{,}090.61$$

Therefore, the present value of 2{,}500$due in 3 years is approximately **$$2{,}090.61$**.

Remember: use multiplication to move money forward to its future value, and division to move money backward to its present value.