Determine present value of an investment

Present value is the amount that must be invested now to obtain a specified future value, found by reversing compound growth: $P=A/(1+r/n)^{nt}$ , where $r$ is the annual interest rate, $n$ the number of compounding periods per year, and $t$ the time in years. The concept connects exponential growth with discounting and emphasizes that more frequent compounding, higher rates, or longer periods reduce the present amount required; continuous compounding, variable rates, and complex cash-flow streams are excluded.

Detailed Explanation: 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.

Learn by doing: Determine present value of an investment

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