Future Value of an Ordinary Annuity

Core idea

Overview

The Future Value of an Ordinary Annuity (FV_A) formula determines the total accumulated amount of a series of identical payments made at regular intervals, assuming these payments earn compound interest. An ordinary annuity means payments occur at the end of each period. This concept is fundamental in personal finance and investment planning, allowing individuals and businesses to project the growth of savings, retirement funds, or other periodic investments over time.

When to use: Apply this formula when you need to determine the total value of a series of regular, equal contributions (like monthly savings or retirement plan contributions) at a future point in time. It's essential for financial planning, projecting investment growth, and understanding the power of compound interest on periodic payments.

Why it matters: Understanding the future value of an annuity is vital for effective financial planning, enabling individuals to set realistic savings goals for retirement, education, or large purchases. For businesses, it helps in evaluating investment strategies, pension obligations, and long-term financial commitments, ensuring sound capital allocation and wealth accumulation.

Symbols

Variables

P = Payment per period, r = Interest rate per period, n = Number of periods, FV_A = Future Value of Annuity

P

Payment per period

£

r

Interest rate per period

decimal

n

Number of periods

periods

F V_{A}

Future Value of Annuity

£

Walkthrough

Derivation

Formula: Future Value of an Ordinary Annuity

Derives the formula for the total accumulated value of a series of equal, periodic payments made at the end of each period, earning compound interest.

Payments are equal in amount and made at regular intervals.
Payments occur at the end of each period (ordinary annuity).
The interest rate is constant over the entire period.
Interest is compounded at the same frequency as payments are made.

1

Future Value of Each Payment:

Each payment 'P' made at the end of a period earns compound interest until the end of the total 'n' periods. The first payment earns interest for n-1 periods, the second for n-2, and so on, until the last payment which earns no interest.

F V_{1} = P (1 + r)^{n - 1}, F V_{2} = P (1 + r)^{n - 2}, ..., F V_{n} = P (1 + r)^{0}

2

Sum of Future Values (Geometric Series):

The total future value of the annuity (FV_A) is the sum of the future values of all individual payments. This forms a geometric series.

F V_{A} = P (1 + r)^{n - 1} + P (1 + r)^{n - 2} + ... + P (1 + r)^{1} + P (1 + r)^{0}

3

Apply Geometric Series Sum Formula:

For a geometric series with first term 'a', common ratio 'R', and 'n' terms, the sum 'S' is given by this formula. In our annuity series (written in reverse: P + P(1+r) + ... + P(1+r)^(n-1)), the first term (a) is P, the common ratio (R) is (1+r), and there are 'n' terms.

S = a \frac{R ^{n} - 1}{R - 1}

4

Substitute and Simplify:

Substituting the values into the geometric series sum formula (with a=P and common ratio R=(1+r)) and simplifying the denominator yields the final formula for the Future Value of an Ordinary Annuity.

F V_{A} = P \frac{( 1 + r ) ^{n} - 1}{( 1 + r ) - 1} = P \frac{( 1 + r ) ^{n} - 1}{r}

Result

F V_{A} = P \frac{( 1 + r ) ^{n} - 1}{( 1 + r ) - 1} = P \frac{( 1 + r ) ^{n} - 1}{r}

Source: Brealey, Myers, Allen - Principles of Corporate Finance (Any edition)

Free formulas

Rearrangements

Solve for $P$

Future Value of an Ordinary Annuity: Make Payment per period (P) the subject

P = F V_{A} \times \frac{r}{( 1 + r ) ^{n} - 1}

To make the Payment per period (P) the subject of the Future Value of an Ordinary Annuity formula, divide the Future Value of Annuity (FV_A) by the annuity factor.

Difficulty: 2/5

Solve for $r$

Future Value of an Ordinary Annuity: Make Interest rate per period (r) the subject

(1 + r)^{n} - \frac{F V _{A}}{P} \cdot r - 1 = 0

Making the interest rate per period (r) the subject of the Future Value of an Ordinary Annuity formula requires numerical methods, as there is no direct algebraic solution.

Difficulty: 4/5

Solve for $n$

Future Value of an Ordinary Annuity: Make Number of periods (n) the subject

n = \frac{lo g ( \frac{F V _{A} \cdot r}{P} + 1 )}{lo g ( 1 + r )}

To make the Number of periods (n) the subject of the Future Value of an Ordinary Annuity formula, logarithmic properties are used after isolating the exponential term.

Difficulty: 4/5

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 because the future value is directly proportional to the payment amount. For a student of finance, this linear relationship means that doubling the payment amount will always result in exactly double the future value, regardless of the interest rate or time period. The most important feature of this curve is its constant slope, which demonstrates that the growth of the future value remains perfectly predictable as the payment amount increases.

Graph type: linear

Why it behaves this way

Intuition

Imagine a series of individual savings deposits, each growing independently with compound interest, like a snowball rolling down a hill, accumulating more snow (interest)

F V_{A}

The total accumulated value of all periodic payments and the interest earned on them at a future point in time.

This is the final lump sum you'll have from your regular savings, including all the interest that has compounded over time.

P

The constant amount of money contributed or received at the end of each period.

This is your regular, fixed payment or deposit, like a monthly contribution to a savings account.

r

The interest rate applied per compounding period, expressed as a decimal.

How quickly your money grows each period. A higher 'r' means faster accumulation.

n

The total number of periods over which payments are made and interest is compounded.

The total count of payments you make and the number of times interest is calculated and added.

Signs and relationships

(1+r)^n: This term represents the compound growth factor. The exponent 'n' signifies that interest is applied multiplicatively over 'n' periods, while '(1+r)' ensures the original principal and the periodic interest are included
-1: This subtraction is crucial for summing a geometric series. It effectively adjusts the future value factor to correctly account for a series of multiple payments rather than a single initial lump sum, ensuring each
/r: The division by 'r' normalizes the sum of the geometric series. It scales the accumulated growth to represent the future value per unit of periodic payment, effectively averaging the growth across all payments.

Free study cues

Insight

Canonical usage

Monetary values (FV_A, P) must be in the same currency, while the interest rate (r) and number of periods (n) must be consistent with the payment frequency and used as dimensionless decimals.

Common confusion

The most common mistake is failing to ensure that the interest rate (r) and the number of periods (n) are consistent with the payment period (e.g., using an annual rate with monthly payments, or using a percentage

Dimension note

The interest rate (r) and number of periods (n) are dimensionless quantities. The fraction ((1+r)^n - 1)/r is also dimensionless, ensuring that the future value (FV_A) has the same unit as the payment (P).

Unit systems

$F V_{A}$ Currency (e.g., USD, EUR, GBP) - The future value of the annuity, expressed in the chosen currency.

$P$ Currency (e.g., USD, EUR, GBP) - The regular payment amount per period, expressed in the same currency as FV_A.

$r$ dimensionless - The interest rate per period, expressed as a decimal (e.g., 0.05 for 5%). This rate must match the period of the payments and the number of periods (n).

$n$ dimensionless - The total number of periods. This number must correspond to the period of the payments and the interest rate (r).

One free problem

Practice Problem

You plan to deposit £100 at the end of each year into an account that pays 5% annual interest, compounded annually. What will be the future value of this ordinary annuity after 10 years?

Payment per period100 £

Interest rate per period0.05 decimal

Number of periods10 periods

Solve for: $F V_{A}$

Hint: Use the formula for the Future Value of an Ordinary Annuity directly.

The full worked solution stays in the interactive walkthrough.

Where it shows up

Real-World Context

In an economic or financial decision involving Future Value of an Ordinary Annuity, Future Value of an Ordinary Annuity is used to calculate Future Value of Annuity from Payment per period, Interest rate per period, and Number of periods. The result matters because it helps compare incentives, policy effects, market outcomes, or financial decisions in context.

Study smarter

Tips

Ensure 'r' (interest rate) and 'n' (number of periods) are consistent (e.g., if payments are monthly, 'r' should be monthly rate and 'n' should be total months).
This formula is for an *ordinary* annuity, where payments occur at the *end* of each period. For payments at the beginning, use the annuity due formula.
The interest rate 'r' must be expressed as a decimal (e.g., 5% = 0.05).
Compounding frequency must match payment frequency for 'r' and 'n'.

Avoid these traps

Common Mistakes

Using an annual interest rate 'r' with monthly periods 'n' without converting 'r' to a monthly rate.
Confusing ordinary annuity with annuity due (payments at the beginning of the period).
Incorrectly calculating the exponent (1+r)^n.

Keep going

Related Formulas

Common questions

Frequently Asked Questions

Derives the formula for the total accumulated value of a series of equal, periodic payments made at the end of each period, earning compound interest.

Apply this formula when you need to determine the total value of a series of regular, equal contributions (like monthly savings or retirement plan contributions) at a future point in time. It's essential for financial planning, projecting investment growth, and understanding the power of compound interest on periodic payments.

Understanding the future value of an annuity is vital for effective financial planning, enabling individuals to set realistic savings goals for retirement, education, or large purchases. For businesses, it helps in evaluating investment strategies, pension obligations, and long-term financial commitments, ensuring sound capital allocation and wealth accumulation.

Using an annual interest rate 'r' with monthly periods 'n' without converting 'r' to a monthly rate. Confusing ordinary annuity with annuity due (payments at the beginning of the period). Incorrectly calculating the exponent (1+r)^n.

In an economic or financial decision involving Future Value of an Ordinary Annuity, Future Value of an Ordinary Annuity is used to calculate Future Value of Annuity from Payment per period, Interest rate per period, and Number of periods. The result matters because it helps compare incentives, policy effects, market outcomes, or financial decisions in context.

Ensure 'r' (interest rate) and 'n' (number of periods) are consistent (e.g., if payments are monthly, 'r' should be monthly rate and 'n' should be total months). This formula is for an *ordinary* annuity, where payments occur at the *end* of each period. For payments at the beginning, use the annuity due formula. The interest rate 'r' must be expressed as a decimal (e.g., 5% = 0.05). Compounding frequency must match payment frequency for 'r' and 'n'.

Yes. Open the Future Value of an Ordinary Annuity equation in the Equation Encyclopedia app, then tap "Copy Excel Template" or "Copy Sheets Template".

References

Sources

Fundamentals of Financial Management by Brigham and Houston
Principles of Corporate Finance by Brealey, Myers, and Allen
Wikipedia: Annuity (finance)
Brealey, R. A., Myers, S. C., & Allen, F. (2020). Principles of Corporate Finance (14th ed.). McGraw-Hill Education.
Brigham, E. F., & Houston, J. F. (2020). Fundamentals of Financial Management (16th ed.). Cengage Learning.
Brealey, R. A., Myers, S. C., & Allen, F. (2020). Principles of Corporate Finance (13th ed.). McGraw-Hill Education.
Brigham, E. F., & Houston, J. F. (2019). Fundamentals of Financial Management (15th ed.). Cengage Learning. Chapter 4: Time Value of Money.
Brealey, Myers, Allen - Principles of Corporate Finance (Any edition)

Overview

Variables

Derivation

Future Value of Each Payment:

Sum of Future Values (Geometric Series):

Apply Geometric Series Sum Formula:

Substitute and Simplify:

Rearrangements

Graph

Intuition

Insight

Practice Problem

Real-World Context

Tips

Common Mistakes

Related Formulas

Future Value (FV)

Present Value of an Ordinary Annuity

Compound Interest

Frequently Asked Questions

Sources