Annuity Present Value

Core idea

Overview

The Annuity Present Value formula calculates the current lump-sum value of a series of future equal payments made at regular intervals. It applies the concept of discounting to account for the time value of money, assuming a constant interest rate and fixed payment amounts.

When to use: This equation is used when evaluating 'ordinary annuities' where equal payments occur at the end of each period. It is essential for determining the initial value of loans, mortgages, or fixed income streams where the interest rate and payment periods are consistent.

Why it matters: Understanding present value allows individuals and firms to compare immediate cash totals against future payment streams. It is a fundamental tool for retirement planning, bond valuation, and calculating the true cost of borrowing.

Symbols

Variables

PV = Present Value, P = Payment/Period, r = Rate per Period, n = Num Periods

PV

Present Value

$

P

Payment/Period

$

r

Rate per Period

Variable

n

Num Periods

Variable

Walkthrough

Derivation

Derivation of Annuity Present Value

An annuity present value is the total present value of a fixed payment C received each period for n periods (ordinary annuity: payments at the end of each period).

Payments C are equal each period.
Discount rate r is constant.
Payments occur at the end of each period (ordinary annuity).

1

Write the Sum of Discounted Payments:

Each cash flow is discounted back to today, then added to get total PV.

P V = \frac{C}{1 + r} + \frac{C}{( 1 + r ) ^{2}} + \dots + \frac{C}{( 1 + r ) ^{n}}

2

Recognise a Geometric Series:

Factoring out C leaves a geometric series with ratio $(1 + r)^{- 1}$ , which sums to the standard annuity PV formula.

P V = C [\frac{1 - ( 1 + r ) ^{- n}}{r}]

Result

P V = C [\frac{1 - ( 1 + r ) ^{- n}}{r}]

Source: Standard curriculum — A-Level Accounting / Finance

Free formulas

Rearrangements

Solve for $P$

Make P the subject

P = P V \frac{r}{1 - ( 1 + r ) ^{- n}}

To make P (Payment per Period) the subject of the Annuity Present Value formula, first multiply both sides by r (Rate per Period), then divide by the term 1 - (1+r)^-n.

Difficulty: 2/5

Solve for $n$

Annuity Present Value: Solve for Number of Periods (n)

n = - \frac{ln ( 1 - \frac{P V r}{P} )}{ln ( 1 + r )}

To solve for 'n' (number of periods) in the Annuity Present Value formula, first isolate the term containing 'n', then take the natural logarithm of both sides, and finally rearrange to solve for 'n'.

Difficulty: 3/5

Solve for $r$

Annuity Present Value: Make r the subject

Solve P V = P \frac{1 - ( 1 + r ) ^{- n}}{r}

The Annuity Present Value formula relates present value, payment, rate, and number of periods. Solving for the rate per period (r) algebraically in a closed form is not possible.

Difficulty: 2/5

The static page shows the finished rearrangements. The app keeps the full worked algebra walkthrough.

Visual intuition

Graph

The graph is a straight line passing through the origin, showing that the present value increases at a constant rate as the payment amount increases. For a student of finance, this linear relationship means that doubling the payment amount will always result in exactly doubling the present value. Because the line passes through the origin, a payment of zero results in a present value of zero, highlighting that the total value is directly proportional to the size of the periodic payment.

Graph type: linear

Why it behaves this way

Intuition

Imagine a timeline where each future payment is individually discounted back to time zero, and the present value is the sum of all these discounted individual payments.

PV

The current equivalent lump-sum value of a future stream of equal payments.

How much a series of future payments is worth *today*, considering the time value of money.

P

The constant amount of each payment in the annuity.

The size of each individual payment in the recurring series.

r

The interest rate or discount rate applied per period.

The rate at which future money is discounted to its present value; a higher 'r' means future payments are worth less today.

n

The total number of payment periods over which the annuity occurs.

The duration or total count of payments in the series.

Signs and relationships

(1+r)^-n: The negative exponent signifies discounting. It reduces the value of future payments to their present equivalent, reflecting that money received later is worth less than money received now due to the opportunity cost of the relevant quantity.

Free study cues

Insight

Canonical usage

Monetary values (PV and P) must be expressed in the same currency, while the interest rate (r) and number of periods (n) are dimensionless.

Common confusion

A common mistake is using an annual interest rate 'r' when payments 'P' are made more frequently (e.g., monthly or quarterly), or failing to convert percentage rates to decimals before calculation.

Unit systems

PVcurrency unit (e.g., USD, EUR) - The present value of the annuity.

$P$ currency unit (e.g., USD, EUR) - The payment made per period.

$r$ dimensionless (decimal) - The interest rate per period, expressed as a decimal (e.g., 0.05 for 5%).

$n$ dimensionless (count) - The total number of payment periods.

One free problem

Practice Problem

A retiree is offered a pension that pays 5,000 dollars at the end of every year for the next 20 years. If the annual discount rate is 4 percent, what is the present value of this pension?

Payment/Period5000 $

Rate per Period0.04

Num Periods20

Solve for: PV

Hint: Use the annual interest rate as a decimal (0.04) and ensure n represents the total number of years.

The full worked solution stays in the interactive walkthrough.

Where it shows up

Real-World Context

In the loan amount affordable with monthly payments, Annuity Present Value is used to calculate Present Value from Payment/Period, Rate per Period, and Num Periods. The result matters because it helps compare incentives, policy effects, market outcomes, or financial decisions in context.

Study smarter

Tips

Ensure the interest rate (r) and number of periods (n) use the same time units (e.g., monthly rate for monthly payments).
Convert percentages to decimals (e.g., 5% becomes 0.05) before calculation.
This specific formula assumes the first payment occurs at the end of the first period.
A higher interest rate will result in a lower present value for the same payment stream.

Avoid these traps

Common Mistakes

Using annual rate for monthly payments.
Confusing annuity due.

Keep going

Related Formulas

Common questions

Frequently Asked Questions

An annuity present value is the total present value of a fixed payment C received each period for n periods (ordinary annuity: payments at the end of each period).

This equation is used when evaluating 'ordinary annuities' where equal payments occur at the end of each period. It is essential for determining the initial value of loans, mortgages, or fixed income streams where the interest rate and payment periods are consistent.

Understanding present value allows individuals and firms to compare immediate cash totals against future payment streams. It is a fundamental tool for retirement planning, bond valuation, and calculating the true cost of borrowing.

Using annual rate for monthly payments. Confusing annuity due.

In the loan amount affordable with monthly payments, Annuity Present Value is used to calculate Present Value from Payment/Period, Rate per Period, and Num Periods. The result matters because it helps compare incentives, policy effects, market outcomes, or financial decisions in context.

Ensure the interest rate (r) and number of periods (n) use the same time units (e.g., monthly rate for monthly payments). Convert percentages to decimals (e.g., 5% becomes 0.05) before calculation. This specific formula assumes the first payment occurs at the end of the first period. A higher interest rate will result in a lower present value for the same payment stream.

Yes. Open the Annuity Present Value equation in the Equation Encyclopedia app, then tap "Copy Excel Template" or "Copy Sheets Template". The corresponding spreadsheet function is: =PV(rate, nper, -pmt) | =RATE(nper, -pmt, pv). Note: Use =PV(r, n, -P) to find present value, or =RATE(n, -P, PV) to find the periodic interest rate. Enter payment as negative (cash out).

References

Sources

Corporate Finance by Stephen A. Ross, Randolph W. Westerfield, Jeffrey F. Jaffe
Principles of Corporate Finance by Richard A. Brealey, Stewart C. Myers, Franklin Allen
Wikipedia: Present value of an annuity
Fundamentals of Financial Management (15th ed.) by Brigham, E. F., & Houston, J. F.
Brealey, Richard A., Stewart C. Myers, and Franklin Allen. Principles of Corporate Finance. 13th ed. McGraw-Hill Education, 2020.
Ross, Stephen A., Randolph W. Westerfield, and Jeffrey Jaffe. Corporate Finance. 12th ed. McGraw-Hill Education, 2019.
Wikipedia: Annuity (finance)
Standard curriculum — A-Level Accounting / Finance