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Time Value of Money

Present and future value.

A dollar today is worth more than a dollar next year, because today's dollar can be invested to earn interest. The time value of money makes this precise. Future value grows a present amount forward: $FV = PV(1 + r)^n$ . Present value discounts a future amount back: $PV = \frac{FV}{(1 + r)^n}$ .

Discounting lets you compare cash flows that arrive at different times, which is the core of investment decisions. Net present value adds up the present values of all future cash flows and subtracts the cost; a positive NPV means the project earns more than the discount rate.

Key terms

Present value: Discount a future amount back to today.
$PV = \frac{FV}{(1+r)^t}$

Key ideas

$FV = PV(1 + r)^n$ and $PV = \frac{FV}{(1 + r)^n}$ .
Annuity present value: $PV = PMT \cdot \frac{1 - (1 + r)^{-n}}{r}$ .
NPV $= \sum \frac{CF_t}{(1 + r)^t} - \text{initial cost}$ .
Higher discount rates and longer waits shrink present values.

Worked example

Is it better to receive $10,000 in 5 years or $7,500 today, if money earns 6% a year?

$PV = \frac{10{,}000}{1.06^5}$ .
$1.06^5 \approx 1.3382$ .
$PV \approx 7472.58$ .
$7,500 today is worth slightly more.

Answer: Take $7,500 today (the future sum is worth about $7,472.58 now)

Common mistakes

Comparing amounts at different dates without discounting.
Multiplying by $(1 + r)^n$ when discounting.
Using an annual rate with monthly periods without converting.

Practice questions

Try each one, then open the answer. Numerical answers on this page are verified by an automated calculation check.

1.Future value of $1,000 at 5% for 3 years.
Show answer
$1,157.63
2.Present value of $2,000 due in 2 years at 10%, to two decimals.
Show answer
$1,652.89
3.Present value of $1,000 per year for 3 years at 8%, to two decimals.
Show answer
$2,577.10

Key terms

Key ideas

Worked example

Common mistakes

Practice questions

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