Future Value and Present Value

A dollar today is worth more than a dollar in the future, because money on hand can earn interest. This idea is called the time value of money, and it is the foundation behind the future value formula and the present value formula. Both formulas answer the same question from opposite directions: future value tells you what money today will grow into later, and present value tells you what money in the future is worth right now.

Underneath the finance vocabulary, both formulas are just exponential functions, the kind you already work with in graphing exponential functions. If you understand how a base raised to a growing exponent behaves, you already understand most of what makes these formulas work.

The future value formula measures how a lump sum grows when it earns a fixed interest rate, compounded once per period, for a number of periods:

\( FV = P(1 + r)^t \)

• \(FV\) is the future value, the amount the money grows to
• \(P\) is the principal, the starting amount invested
• \(r\) is the interest rate per period, written as a decimal
• \(t\) is the number of compounding periods

Notice the structure is exactly \(a \cdot b^x\), the same exponential form covered in exponential growth and decay by a factor. The base \((1 + r)\) is the growth factor per period, and applying the exponent rules from order of operations with exponents keeps the calculation straightforward as long as you raise the whole quantity \((1 + r)\) to the power \(t\), not just the \(r\) part.

Worked Example: Future Value

Suppose \(P = \$1000\) is invested at an annual interest rate of \(5\%\), compounded yearly, for \(t = 10\) years.

\( FV = 1000(1 + 0.05)^{10} = 1000(1.05)^{10} \approx \$1628.89 \)

The graph below shows this same $1000 principal growing at \(5\%\) per year over a longer stretch of time. The curve bends upward because it is exponential growth, not a straight line.

Present value asks the reverse question: given a known amount in the future, what is it worth today? You get the present value formula by solving the future value formula for \(P\):

\( PV = \dfrac{FV}{(1 + r)^t} \)

Here \(FV\) is the known future amount, \(r\) is the discount rate (the same idea as the interest rate, viewed from today's perspective), and \(t\) is the number of periods until that amount is received. Because the denominator grows as \(t\) increases, present value shrinks the farther into the future the payment is, which is exactly the behavior of exponential decay.

Worked Example: Present Value

Suppose you will receive \(\$5000\) in \(8\) years, and the discount rate is \(4\%\) per year.

\( PV = \dfrac{5000}{(1.04)^8} \approx \$3653.04 \)

This means \(\$3653.04\) invested today at \(4\%\) would grow into \(\$5000\) after \(8\) years, the two amounts are equivalent once time and interest are taken into account.

Many financial situations, like loan payments or retirement savings, involve a series of equal payments made at regular intervals, called an annuity, rather than a single lump sum. The formulas extend naturally:

Future value of an annuity: \( FV = Pmt \cdot \dfrac{(1 + r)^n - 1}{r} \)

Present value of an annuity: \( PV = Pmt \cdot \dfrac{1 - (1 + r)^{-n}}{r} \)

Here \(Pmt\) is the fixed payment made each period, \(r\) is the interest rate per period, and \(n\) is the total number of payments. Each formula is really the single-payment formula applied repeatedly and summed, since every individual payment has its own future or present value.

Net present value combines several present value calculations to judge whether an investment is worth its cost. It takes the present value of every future cash flow the investment produces, adds them together, and subtracts the initial cost \(C_0\):

\( NPV = \displaystyle\sum \dfrac{CF_t}{(1 + r)^t} - C_0 \)

If the \(NPV\) is positive, the investment is expected to be worth more than it costs once the time value of money is taken into account. If it is negative, the future cash flows are not enough to justify the upfront cost at that discount rate.

Both the future value formula and the present value formula are variations of \(y = a \cdot b^x\). Future value uses growth, with a base greater than \(1\), while present value effectively divides by that same growing base, which behaves like decay. If the exponential shapes involved feel unfamiliar, it can help to review graphing exponential functions before working through more finance problems, since every technique for reading and sketching those curves applies here too.

Related lessons

• Algebra Help