FinanceTime Value of MoneyUniversity

Present Value of an Ordinary Annuity

Calculates the current value of a series of equal payments made or received at the end of each period for a specified number of periods.

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P V A = P M T \times \frac{1 - ( 1 + r ) ^{- n}}{r}

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Core idea

Overview

This formula is a fundamental concept in finance, allowing for the valuation of future cash flows in today's terms. It discounts each future payment back to its present value and sums them up, reflecting the time value of money where a dollar today is worth more than a dollar in the future due to its earning potential. It's crucial for understanding the true worth of a stream of regular payments.

When to use: This equation is used to: 1. Value a stream of regular income, such as pension payments or lottery winnings paid out over time. 2. Calculate the present cost of a loan or mortgage, where payments are made periodically. 3. Determine the amount needed today to fund a future series of equal withdrawals, like retirement planning. 4. Evaluate investment opportunities that promise a series of fixed, periodic returns.

Why it matters: The Present Value of an Ordinary Annuity is a cornerstone of financial mathematics, underpinning critical decisions in investment appraisal, retirement planning, and loan amortization. It helps in comparing different financial products or investment strategies that involve periodic payments, ensuring that decisions are based on the true economic value. Understanding this concept is vital for personal financial planning, corporate finance, and investment analysis, as it quantifies the impact of interest rates and time on money.

Symbols

Variables

PVA = Present Value of Annuity, PMT = Payment per Period, r = Interest Rate per Period, n = Number of Periods

P V A

Present Value of Annuity

V a r iab l e

P M T

Payment per Period

V a r iab l e

r

Interest Rate per Period

V a r iab l e

n

Number of Periods

V a r iab l e

Walkthrough

Derivation

Derivation of Present Value of an Ordinary Annuity

This derivation shows how the formula for the present value of an ordinary annuity is obtained by summing the present values of a series of equal payments, which forms a finite geometric series.

Payments (PMT) are equal in amount.
Payments are made at the end of each period (ordinary annuity).
The interest rate (r) is constant per period.
The number of periods (n) is fixed.

Present Value of Individual Payments

The present value of a single payment (PMT) received at the end of period 'k' is calculated by discounting it back 'k' periods using the per-period interest rate 'r'. For an ordinary annuity, payments occur at the end of each period, so the first payment is discounted for 1 period, the second for 2 periods, and so on, up to the nth payment discounted for n periods.

P V_{k} = P M T \times (1 + r)^{- k}

Note: Remember that (1 + r)^(-k) is equivalent to 1 / (1 + r)^k.

Sum of Present Values

The total Present Value of an Ordinary Annuity (PVA) is the sum of the present values of all individual payments. We can factor out the constant payment amount (PMT) from the summation.

P V A = k = 1 \sum n P M T \times (1 + r)^{- k} P V A = P M T \times [(1 + r)^{- 1} + (1 + r)^{- 2} + \dots + (1 + r)^{- n}]

Note: This step clearly shows the series we need to sum.

Identify the Geometric Series

The expression inside the square brackets is a finite geometric series. We can identify its key components: the first term (a), the common ratio (x), and the number of terms (n).

S_{n} = (1 + r)^{- 1} + (1 + r)^{- 2} + \dots + (1 + r)^{- n}

Note: A geometric series is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio.

Apply the Geometric Series Sum Formula

Substitute the identified first term, common ratio, and number of terms into the formula for the sum of a finite geometric series. This formula allows us to express the sum in a more compact form.

First term (a) = (1 + r)^{- 1} Common ratio (x) = (1 + r)^{- 1} Number of terms (n) = n S_{n} = a \times \frac{1 - x ^{n}}{1 - x} S_{n} = (1 + r)^{- 1} \times \frac{1 - (( 1 + r ) ^{- 1} ) ^{n}}{1 - ( 1 + r ) ^{- 1}} S_{n} = (1 + r)^{- 1} \times \frac{1 - ( 1 + r ) ^{- n}}{1 - ( 1 + r ) ^{- 1}}

Note: The formula for the sum of a finite geometric series is a fundamental tool in finance.

Simplify the Expression

Perform algebraic simplification. First, simplify the denominator of the geometric series sum. Then, substitute this simplified denominator back into the expression for $S_{n}$ and cancel out common terms. Finally, multiply by PMT to arrive at the full Present Value of an Ordinary Annuity formula.

Simplify the denominator: 1 - (1 + r)^{- 1} = 1 - \frac{1}{1 + r} = \frac{1 + r - 1}{1 + r} = \frac{r}{1 + r} Substitute back into S_{n} : S_{n} = (1 + r)^{- 1} \times \frac{1 - ( 1 + r ) ^{- n}}{\frac{r}{1 + r}} S_{n} = \frac{1}{1 + r} \times \frac{1 - ( 1 + r ) ^{- n}}{1} \times \frac{1 + r}{r} S_{n} = \frac{1 - ( 1 + r ) ^{- n}}{r} Finally, substitute S_{n} back into the PVA equation: P V A = P M T \times \frac{1 - ( 1 + r ) ^{- n}}{r}

Note: Careful algebraic manipulation is crucial here to reach the final, simplified form.

Result

Simplify the denominator: 1 - (1 + r)^{- 1} = 1 - \frac{1}{1 + r} = \frac{1 + r - 1}{1 + r} = \frac{r}{1 + r} Substitute back into S_{n} : S_{n} = (1 + r)^{- 1} \times \frac{1 - ( 1 + r ) ^{- n}}{\frac{r}{1 + r}} S_{n} = \frac{1}{1 + r} \times \frac{1 - ( 1 + r ) ^{- n}}{1} \times \frac{1 + r}{r} S_{n} = \frac{1 - ( 1 + r ) ^{- n}}{r} Finally, substitute S_{n} back into the PVA equation: P V A = P M T \times \frac{1 - ( 1 + r ) ^{- n}}{r}

Source: Commonly found in introductory finance and financial mathematics textbooks.

Free formulas

Rearrangements

Solve for $P M T$

Make PMT the subject

P M T = \frac{P V A \times r}{1 - ( 1 + r ) ^{- n}}

Isolate the payment amount by dividing the present value by the annuity factor.

Difficulty: 2/5

Solve for $r$

Make r the subject

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

Isolate the interest rate by solving the transcendental annuity equation.

Difficulty: 5/5

Solve for $n$

Make n the subject

n = - \frac{ln ( 1 - \frac{P V A \cdot r}{P M T} )}{ln ( 1 + r )}

Isolate the number of periods using logarithms.

Difficulty: 4/5

Solve for $P V A$

Make PVA the subject

P V A = P M T \times \frac{1 - ( 1 + r ) ^{- n}}{r}

PVA is already the subject of the original formula.

Difficulty: 1/5

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

Why it behaves this way

Intuition

Imagine a series of identical, equally spaced cash payments arriving at your doorstep in the future. The Present Value of an Ordinary Annuity (PVA) is like asking: 'How much money would I need to put into a single savings account *today* (at time zero) to be able to withdraw exactly those future payments as they become due, leaving the account empty after the last payment?' It's collapsing all those future payments back to a single, equivalent lump sum at the beginning of the timeline, considering the interest rate as the cost of waiting.

PVA

Present Value of an Ordinary Annuity

This is the single, lump-sum amount of money *today* that is financially equivalent to receiving a series of equal payments in the future. It's what those future payments are 'worth' right now, after accounting for the time value of money.

PMT

Payment per period

This is the fixed, constant amount of money that is paid or received at the end of each period. Think of it as the size of each individual 'installment' in the series of payments.

Interest rate per period

This is the rate at which money grows or is discounted over each period. It represents the opportunity cost of money or the return you could earn on an investment. A higher 'r' means future money is worth less today, and vice-versa.

Number of periods

This is the total count of payments or periods over which the annuity extends. It tells you how many individual 'PMT's are included in the annuity stream.

Signs and relationships

(1 + r)^-n: This term calculates the present value of a single dollar received 'n' periods from now. The negative exponent indicates that we are 'discounting' a future value back to the present, meaning we are reducing its value because money today is worth more than money tomorrow.
1 - (1 + r)^-n: This part of the formula is derived from the sum of a geometric series. It effectively represents the present value of an annuity that pays $1 f o r^{'} n^{'} p er i o d s . T h e^{'} 1^{'} r e p r ese n t s t h e ini t ia l v a l u e, an d s u b t r a c t in g t h e d i sco u n t e d v a l u eo f a f u t u r e$ 1 helps to isolate the value of the stream of payments.
/ r: Dividing by 'r' scales the present value of the $1 ann u i t y f a c t or t or e f l ec tt h e a c t u a l in t er es t r a t e . I t^{'} s a cr u c ia l p a r t o f t h e g eo m e t r i cser i ess u m f or m u l a f or ann u i t i es, e f f ec t i v e l y co n v er t in g t h es u m o f d i sco u n t e d$ 1 payments into a factor that, when multiplied by PMT, gives the total present value.
PMT × ...: The entire fraction `(1 - (1 + r)^-n) / r` calculates the present value of an annuity that pays $1 per period. To find the present value of an annuity that pays `PMT` per period, we simply multiply this factor by `PMT`. It scales the 'per dollar' value to the 'per PMT' value.

One free problem

Practice Problem

You are offered an investment that pays $500 at the end of each year for the next 10 years. If the discount rate is 6% per year, what is the present value of this annuity?

Payment per Period500

Interest Rate per Period0.06

Number of Periods10

Solve for: $P V A$

Hint: Use the direct formula for Present Value of an Ordinary Annuity with the given payment, rate, and number of periods.

The full worked solution stays in the interactive walkthrough.

Where it shows up

Real-World Context

Imagine you win a lottery that offers you $50,000 at the end of each year for the next 20 years. If the prevailing interest rate is 5% per year, you can use the Present Value of an Ordinary Annuity formula to determine the lump sum amount you would need today to generate those future payments, or, conversely, the equivalent cash value of your lottery winnings today if you chose a one-time payout instead of annual installments.

Study smarter

Tips

Ensure that the interest rate ('r') and the number of periods ('n') are consistent in their time units (e.g., if payments are monthly, 'r' should be the monthly rate and 'n' the total number of months).
Remember that this formula is for an *ordinary* annuity, meaning payments occur at the *end* of each period. For payments at the beginning of each period (annuity due), a slight modification is needed (multiply the ordinary annuity formula by (1 + r)).
Use a financial calculator or spreadsheet software for complex calculations to minimize rounding errors and save time, especially when solving for 'r' or 'n'.

Avoid these traps

Common Mistakes

Confusing Ordinary Annuity with Annuity Due: Incorrectly applying the ordinary annuity formula when payments are made at the beginning of the period, leading to an underestimation of the present value.
Inconsistent Time Units: Using an annual interest rate with monthly payments or vice-versa without proper conversion, resulting in significant calculation errors.
Misinterpreting 'n': Confusing the number of years with the total number of payment periods, especially when payments are more frequent than annual.

Common questions

Frequently Asked Questions

This derivation shows how the formula for the present value of an ordinary annuity is obtained by summing the present values of a series of equal payments, which forms a finite geometric series.

This equation is used to: 1. Value a stream of regular income, such as pension payments or lottery winnings paid out over time. 2. Calculate the present cost of a loan or mortgage, where payments are made periodically. 3. Determine the amount needed today to fund a future series of equal withdrawals, like retirement planning. 4. Evaluate investment opportunities that promise a series of fixed, periodic returns.

The Present Value of an Ordinary Annuity is a cornerstone of financial mathematics, underpinning critical decisions in investment appraisal, retirement planning, and loan amortization. It helps in comparing different financial products or investment strategies that involve periodic payments, ensuring that decisions are based on the true economic value. Understanding this concept is vital for personal financial planning, corporate finance, and investment analysis, as it quantifies the impact of interest rates and time on money.

Confusing Ordinary Annuity with Annuity Due: Incorrectly applying the ordinary annuity formula when payments are made at the beginning of the period, leading to an underestimation of the present value. Inconsistent Time Units: Using an annual interest rate with monthly payments or vice-versa without proper conversion, resulting in significant calculation errors. Misinterpreting 'n': Confusing the number of years with the total number of payment periods, especially when payments are more frequent than annual.

Ensure that the interest rate ('r') and the number of periods ('n') are consistent in their time units (e.g., if payments are monthly, 'r' should be the monthly rate and 'n' the total number of months). Remember that this formula is for an *ordinary* annuity, meaning payments occur at the *end* of each period. For payments at the beginning of each period (annuity due), a slight modification is needed (multiply the ordinary annuity formula by (1 + r)). Use a financial calculator or spreadsheet software for complex calculations to minimize rounding errors and save time, especially when solving for 'r' or 'n'.

References

Sources

Brealey, Richard A., Stewart C. Myers, and Franklin Allen. Principles of Corporate Finance. McGraw-Hill Education.
Ross, Stephen A., Randolph W. Westerfield, and Bradford D. Jordan. Corporate Finance. McGraw-Hill Education.
Brealey, R. A., Myers, S. C., & Allen, F. (2020). Principles of Corporate Finance (13th ed.). McGraw-Hill Education.
Brigham, E. F., & Houston, J. F. (2020). Fundamentals of Financial Management (16th ed.). Cengage Learning.
Commonly found in introductory finance and financial mathematics textbooks.

Present Value of an Ordinary Annuity

Overview

Variables

Derivation

Present Value of Individual Payments

Sum of Present Values

Identify the Geometric Series

Apply the Geometric Series Sum Formula

Simplify the Expression

Rearrangements

Intuition

Practice Problem

Real-World Context

Tips

Common Mistakes

Related Formulas

Present Value of a Single Sum

Present Value of an Annuity Due

Frequently Asked Questions

Sources