Skill: Determine present value of an investment

To find the amount needed now to reach a future value, reverse the compound-growth formula:

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

Here:

• $P$ is the present value, or amount invested now.
• $A$ is the future value.
• $r$ is the annual interest rate written as a decimal.
• $n$ is the number of compounding periods per year.
• $t$ is the number of years.

Example

How much should be invested today to have 10{,}000$in 3 years if the account earns$6%$ interest compounded quarterly?

Step 1: Identify the values.

$$A=10{,}000,\qquad r=0.06,\qquad n=4,\qquad t=3$$

The rate is $0.06$ because $6\%=0.06$, and quarterly compounding means $4$ times per year.

Step 2: Substitute into the formula.

$$P=\frac{10{,}000}{\left(1+\frac{0.06}{4}\right)^{4(3)}}$$

Step 3: Simplify the exponent and rate per period.

$$P=\frac{10{,}000}{(1.015)^{12}}$$

There are $12$ total compounding periods because $4\cdot 3=12$.

Step 4: Calculate.

$$P\approx \frac{10{,}000}{1.1956}$$ $$P\approx 8{,}363.88$$

Therefore, about 8{,}363.88$ must be invested today.

The investment grows from 8{,}363.88$to$$10{,}000$ over 3 years. In general, for a fixed future value, a higher interest rate or a longer investment period means a smaller amount is needed today.