Present Value of an Annuity

What it is

The present value (PV) of an annuity is the current worth of a series of future, equal payments, discounted at a specified rate (the discount rate). It answers the question: how much money today is equivalent to receiving a fixed payment each period in the future?

Why it matters

• Reflects the time value of money: a dollar today can be invested and earn a return, so it is worth more than the same dollar received later.
• Helps compare a lump-sum payment versus a stream of payments (e.g., pension payout options).
• Informs decisions about loans, leases, pensions, and investment trade-offs.

Key factors

• PMT: the periodic payment amount.
• r: the discount rate (periodic interest rate or opportunity cost of capital).
• n: number of periods.

Choosing the discount rate

The discount rate should reflect your opportunity cost of capital—what you could reasonably earn by investing the money elsewhere. Common benchmarks include expected returns on comparable investments or the risk-free rate (e.g., Treasury yields) plus a risk premium.

Formulas

• Present value of an ordinary annuity (payments at period end):

P = PMT × (1 − (1 + r)^−n) / r

• Present value of an annuity due (payments at period beginning):

P_due = P_ordinary × (1 + r)

Example

You can receive $50,000 per year for 25 years. Using a 6% discount rate, compare the annuity to a $650,000 lump sum.

1. Ordinary annuity PV:
   P = 50,000 × (1 − (1 + 0.06)^−25) / 0.06 ≈ $639,168

Since $639,168 < $650,000, the lump-sum is better by $10,832.

1. Annuity due PV:
   P_due = $639,168 × (1 + 0.06) ≈ $677,518

In this case, the annuity due exceeds $650,000 by $27,518, so the annuity due would be preferable.

Ordinary annuity vs. annuity due

• Ordinary annuity: payments at the end of each period (common for loans and many pensions).
• Annuity due: payments at the beginning of each period (common for rent, some immediate annuities).
• All else equal, an annuity due has a higher present value because payments occur sooner.

Practical tips

• Use the same compounding period for r and payments (annual rate with annual payments, monthly rate with monthly payments, etc.).
• When comparing options, adjust for taxes, fees, inflation, and personal liquidity needs—these can change which choice is preferable.
• For uneven cash flows, compute the PV of each payment individually and sum them.

Bottom line

Present value converts a future stream of equal payments into a single current value using a chosen discount rate. It’s a fundamental tool for comparing lump-sum payments with periodic payouts and for evaluating financial choices that span time.