Finance & Economics · Corporate Finance & Valuation · Valuation Models
Present Value of Annuity Calculator
Calculates the present value of a series of equal periodic payments discounted at a given interest rate, for both ordinary annuities and annuities due.
Calculator
Formula
PV is the present value of the annuity stream. PMT is the fixed periodic payment amount. r is the periodic interest rate (annual rate divided by number of periods per year). n is the total number of payment periods. For an ordinary annuity, payments occur at the end of each period; for an annuity due, payments occur at the beginning, which adds one period of compounding and is captured by multiplying by (1 + r).
Source: Brealey, Myers & Allen — Principles of Corporate Finance (McGraw-Hill, 13th ed.), Chapter 2: How to Calculate Present Values.
How it works
The core principle behind annuity valuation is the time value of money: a dollar received today is worth more than a dollar received in the future because today's dollar can be invested to earn a return. An annuity is a finite stream of equal, evenly-spaced cash flows. To find its present value, each future payment is individually discounted back to today and the results are summed. Because the payments are equal and evenly spaced, this sum simplifies to a compact closed-form formula.
The standard formula for an ordinary annuity — where payments occur at the end of each period — is: PV = PMT × [1 − (1 + r)−n] / r. Here, PMT is the fixed payment per period, r is the periodic interest rate (annual rate divided by compounding periods per year), and n is the total number of periods (years × periods per year). The bracketed factor is called the present value annuity factor (PVAF) or simply the annuity discount factor. For an annuity due, where payments fall at the beginning of each period, the entire PV is multiplied by (1 + r), reflecting the fact that each payment arrives one period earlier and therefore suffers one less period of discounting.
Practical applications span a broad range of fields. Insurance companies use PV annuity calculations to price structured settlements and life annuity products. Corporate finance teams apply the formula when evaluating lease obligations, bond coupon streams, and capital project cash flows under net present value (NPV) analysis. Pension actuaries rely on it to fund defined-benefit obligations. Even simple consumer decisions — such as whether to take a lottery lump sum or a 20-year payment plan — are governed directly by this formula.
Worked example
Scenario: You are evaluating a structured settlement that will pay you $1,000 per month for 10 years. The appropriate discount rate is 6% per year, compounded monthly. Payments are made at the end of each month (ordinary annuity). What is the present value of this settlement?
Step 1 — Identify inputs: PMT = $1,000; annual rate = 6%; periods per year = 12; years = 10; annuity type = ordinary.
Step 2 — Calculate periodic rate: r = 6% / 12 = 0.5% = 0.005 per month.
Step 3 — Calculate total periods: n = 10 × 12 = 120 months.
Step 4 — Apply the PV annuity formula:
PV = 1,000 × [1 − (1.005)−120] / 0.005
(1.005)−120 = 1 / (1.005)120 ≈ 1 / 1.8194 ≈ 0.54963
PV = 1,000 × (1 − 0.54963) / 0.005
PV = 1,000 × 0.45037 / 0.005
PV = 1,000 × 90.074
PV ≈ $90,073.45
Step 5 — Interpretation: The total undiscounted payments equal $1,000 × 120 = $120,000. After discounting at 6% annually, the present value is approximately $90,073. The discount of roughly $29,927 represents the cost of waiting — the erosion of value caused by receiving money in the future rather than today.
Annuity Due check: If the same payments were made at the beginning of each month (annuity due), the PV would be $90,073.45 × 1.005 ≈ $90,523.82 — about $450 more, because each payment arrives one month earlier.
Limitations & notes
This calculator assumes a constant periodic payment amount throughout the annuity's life. It cannot handle growing annuities (where PMT increases at a fixed rate) — use the Gordon Growth Model or growing annuity formula for those cases. It also assumes a fixed, flat discount rate across all periods; if the yield curve is steep or volatile, a more sophisticated DCF model with period-specific rates will produce a more accurate valuation. The formula further assumes exact periodicity: payments must arrive at equal intervals with no missed or irregular payments. For annuities with deferred start dates, you must first calculate the PV at the start date of payments and then discount that figure back to today. Finally, this tool does not account for taxes, inflation adjustments, or counterparty credit risk, all of which are material considerations when evaluating real-world annuity products such as insurance contracts or pension obligations.
Frequently asked questions
What is the difference between an ordinary annuity and an annuity due?
An ordinary annuity (also called an annuity in arrears) pays at the end of each period — for example, most bond coupons and mortgage payments. An annuity due pays at the beginning of each period — for example, rent and insurance premiums. Because annuity due payments arrive one period earlier, their present value is always higher by a factor of (1 + r). At a 6% annual rate compounded monthly, an annuity due is worth approximately 0.5% more than an otherwise identical ordinary annuity.
How does the interest rate affect the present value of an annuity?
Present value and interest rate (discount rate) are inversely related. As the discount rate rises, future payments are penalised more heavily, reducing the present value. For example, a $1,000/month ordinary annuity over 10 years is worth approximately $90,073 at 6% annually but only about $75,671 at 12% annually — a difference of nearly $14,400. This sensitivity is why annuity valuations are highly sensitive to the prevailing interest rate environment.
Can this formula be used to price mortgage or loan payments?
Yes — the same present value annuity formula underlies standard loan amortisation. If you know the loan amount (the PV), rate, and term, you can rearrange the formula to solve for the periodic payment: PMT = PV × r / [1 − (1 + r)<sup>−n</sup>]. Our Loan Payment Calculator applies exactly this rearrangement. Conversely, given a known payment stream, this calculator tells you its fair present value, which is useful for trading or buying structured loan receivables.
What discount rate should I use when valuing an annuity?
The appropriate discount rate depends on the risk profile of the cash flows and your opportunity cost. For a risk-free government-backed pension, you might use the current risk-free rate (e.g., 10-year Treasury yield). For a corporate annuity or structured settlement, a rate reflecting the creditworthiness of the payer is more appropriate. For personal financial planning, many analysts use the expected long-run return on a balanced portfolio (historically around 6–8% nominal). Always match the compounding frequency of the rate to the payment frequency.
How is present value of an annuity different from present value of a perpetuity?
A perpetuity is an infinite annuity — it pays the same amount forever. Its present value simplifies to PV = PMT / r. An annuity has a finite number of periods n, so its formula includes the additional term [1 − (1 + r)<sup>−n</sup>] which shrinks to zero as n grows very large, making the two converge. In practice, a long-dated annuity (e.g., 50 years at a 6% rate) is priced very close to a perpetuity, because distant cash flows are worth almost nothing in present value terms.
Last updated: 2025-01-15 · Formula verified against primary sources.