Time Value of Money: PV, FV, Annuities, and Perpetuities — Complete Guide With Examples

The Core Concept: Why Time Matters

If someone offers you $1,000 today or $1,000 in three years, rational analysis always chooses today — not because of inflation risk or credit risk, but simply because the $1,000 received today can be invested for three years to grow into more than $1,000. This is the time value of money: the economic principle that the timing of a cash flow affects its value.

The discount rate (r) is the rate of return used to compare cash flows at different points in time. It reflects the opportunity cost of money — what you give up by not having the cash available to invest. In capital budgeting, the discount rate is typically WACC. In personal finance, it might be a mortgage rate. In bond pricing, it is the market yield. All TVM problems use one equation family with the same variables: present value (PV), future value (FV), interest/discount rate (r), and number of periods (n).

Future Value: How Money Grows

Future value answers: if I invest $X today at rate r for n periods, what will it be worth?

Example 1: You invest $5,000 today at 7% per year for 10 years.
FV = $5,000 × (1.07)¹⁰ = $5,000 × 1.9672 = $9,836

Example 2 — Compounding frequency matters: $5,000 at 7% for 10 years, compounded quarterly (rate per period = 7% ÷ 4 = 1.75%; periods = 10 × 4 = 40).
FV = $5,000 × (1.0175)⁴⁰ = $5,000 × 2.0016 = $10,008

More frequent compounding produces a higher future value because interest is being earned on interest more frequently. For continuous compounding: FV = PV × eʳⁿ (where e ≈ 2.71828).

Present Value: What Future Cash Is Worth Today

Present value answers: what is a future cash flow worth in today's dollars?

Example: You will receive $20,000 in 5 years. Your required rate of return is 8%.
PV = $20,000 ÷ (1.08)⁵ = $20,000 ÷ 1.4693 = $13,612

This means receiving $20,000 in 5 years is equivalent to receiving $13,612 today, given an 8% return opportunity. If someone offers to sell you the right to receive $20,000 in 5 years for $12,000, the deal is better than fair — you would be getting something worth $13,612 for $12,000. This is positive NPV thinking applied at the most basic level.

Future Value of an Annuity

An annuity is a series of equal cash flows made at regular intervals. The future value of an annuity asks: if I deposit $PMT at the end of each period for n periods at rate r, what will the total grow to?

Example: You deposit $3,000 at the end of each year for 6 years at 6%.
FVA = $3,000 × [(1.06)⁶ − 1] ÷ 0.06 = $3,000 × [1.4185 − 1] ÷ 0.06 = $3,000 × 6.9753 = $20,926

Total deposits: $3,000 × 6 = $18,000. The extra $2,926 is interest earned on interest — the power of compounding.

Present Value of an Annuity

The present value of an annuity answers: what is the lump sum today equivalent to receiving $PMT per period for n periods at rate r?

Example 1 — Loan payment: You borrow $200,000 at 5% annual rate for 20 years. What is the annual payment?
$200,000 = PMT × [1 − (1.05)⁻²⁰] ÷ 0.05 = PMT × 12.4622
PMT = $200,000 ÷ 12.4622 = $16,049 per year

Example 2 — Bond pricing: A bond pays $60 annually for 5 years and $1,000 at maturity. Market rate is 7%.
PV of coupons = $60 × [1 − (1.07)⁻⁵] ÷ 0.07 = $60 × 4.1002 = $246
PV of face value = $1,000 ÷ (1.07)⁵ = $1,000 ÷ 1.4026 = $713
Bond price = $246 + $713 = $959 (discount because coupon rate 6% < market rate 7%)

This same calculation appears in bond accounting — the issue price of a bond is calculated by finding the PV of all its future cash flows.

Annuity Due vs Ordinary Annuity

An ordinary annuity has payments at the end of each period (most common — used for loans, bonds). An annuity due has payments at the beginning of each period (used for leases, insurance). Because annuity due payments occur one period earlier, each payment has one more period to compound (for FV) or one fewer period to discount (for PV).

For the $3,000 annual deposit example above as an annuity due: FV = $20,926 × 1.06 = $22,182. The first payment is made at t=0 rather than t=1, so it earns one extra year of interest.

Perpetuities

A perpetuity is an annuity that pays forever — infinite equal payments. The present value formula simplifies elegantly because the series converges:

Example: A preferred stock pays a $5 annual dividend in perpetuity. If your required return is 8%: PV = $5 ÷ 0.08 = $62.50. This is also the Gordon Growth Model formula for valuing a stock that grows dividends at a constant rate g: P₀ = D₁ ÷ (r − g).

Real-World Applications