Perpetuity Present Value Equation, Derivation & Rearrangements

Core idea

Overview

The perpetuity present value formula calculates the current value of a stream of identical cash flows that continue indefinitely. This mathematical model assumes the first payment is received one period from today and that the discount rate remains constant over time.

When to use: Apply this formula when evaluating financial instruments with no maturity date, such as British Consols or perpetual preferred stock. It is also used in corporate finance to estimate the terminal value of a firm that has reached a stable, mature growth phase.

Why it matters: This formula simplifies the complex task of valuing infinite future payments into a single, manageable figure. It serves as a foundational tool for investors to determine if a permanent income stream is priced fairly relative to its risk-adjusted return.

Symbols

Variables

PV = Present Value, C = Cash Flow, r = Interest Rate

P V

Present Value

$

C

Cash Flow

$

r

Interest Rate

V a r iab l e

Walkthrough

Derivation

Derivation of Perpetuity Present Value

A perpetuity present value is the present value of an infinite stream of equal payments C, discounted at rate r.

Payments continue indefinitely.
Discount rate r is constant and r>0.
Payments occur at the end of each period.

1

Start from the Finite Annuity Formula:

This gives PV for n equal payments.

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

2

Take the Limit as n\to∞:

As n grows, the discount factor shrinks to zero because $(1 + r)^{- n}$ becomes very small.

n \to \infty lim (1 + r)^{- n} = 0

3

Obtain the Perpetuity Formula:

Substituting the limit into the annuity expression leaves $P V = C / r$ .

P V = \frac{C}{r}

Result

P V = \frac{C}{r}

Source: Standard curriculum — A-Level Accounting / Finance

Free formulas

Rearrangements

Solve for $P V$

Make PV the subject

P V = \frac{C}{r}

PV is already the subject of the formula.

Difficulty: 1/5

Solve for $C$

Perpetuity Present Value: Solve for C

C = P V r

Rearrange the Perpetuity Present Value formula to solve for the Cash Flow ( $C$ ). This involves multiplying both sides by the interest rate ( $r$ ) to isolate the numerator.

Difficulty: 2/5

Solve for $r$

Perpetuity Present Value: Make r the subject

r = \frac{C}{P V}

Rearrange the Perpetuity Present Value formula to solve for the interest rate, r.

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 with a slope of r, showing that Present Value increases at a constant rate as Cash Flow grows. For a finance student, this linear relationship means that doubling the Cash Flow will always exactly double the Present Value, regardless of the starting amount. Small values of Cash Flow result in a proportionally small Present Value, while large values lead to a proportionally large Present Value. The most important feature is the constant slope, which demonstrate

Graph type: linear

Why it behaves this way

Intuition

Imagine a never-ending waterfall of identical cash droplets, each losing value as it falls through time, and the present value is the single pool of money at the bottom that could generate this entire endless flow.

PV

The current equivalent value of an infinite series of future payments.

How much money you would need today to generate an endless stream of future payments, given the prevailing discount rate. A higher PV means the future payments are more valuable to you right now.

C

The constant amount of cash received or paid at regular intervals, continuing indefinitely.

The size of each individual payment in the endless stream. Larger payments naturally make the entire stream worth more.

r

The periodic interest rate used to discount future cash flows to their present value.

Represents the opportunity cost of capital or the rate of return available on alternative investments of similar risk. A higher 'r' makes future payments less valuable today because you could earn more elsewhere, or

Signs and relationships

r (in the denominator): The discount rate 'r' is in the denominator, signifying an inverse relationship: a higher discount rate reduces the present value of future payments, as money loses value more quickly over time or alternative investments

Free study cues

Insight

Canonical usage

Monetary values (PV, C) are expressed in a consistent currency unit, while the discount rate (r) is a rate per period, ensuring dimensional consistency.

Common confusion

The most common errors involve using percentage rates directly in the formula without converting to decimals, or using inconsistent time periods for the cash flow and the discount rate.

Unit systems

$P V$ currency unit (e.g., USD, EUR) · The present value of the perpetuity, representing a sum of money today.

$C$ currency unit per period (e.g., USD/year, EUR/month) · The constant cash flow received at the end of each period.

$r$ rate per period (e.g., %/year, decimal/month) · The discount rate or required rate of return per period. Must be consistent with the period of C.

One free problem

Practice Problem

A philanthropist wishes to establish a permanent university scholarship that pays out 15,000 dollars every year. If the annual interest rate is 6%, how much must the donor contribute today to fully fund this endowment?

Cash Flow15000 $

Interest Rate0.06

Solve for: $P V$

Hint: Divide the annual payment by the decimal interest rate.

The full worked solution stays in the interactive walkthrough.

Where it shows up

Real-World Context

Valuing a distinct bond (Consol) paying fixed interest forever.

Study smarter

Tips

Ensure the interest rate (r) and the cash flow (C) correspond to the same time period.
Convert percentage interest rates into decimal form before calculation.
This specific formula assumes payments occur at the end of each period.

Avoid these traps

Common Mistakes

Applying to finite streams.
Using growth rate incorrectly.

Keep going

Related Formulas

Common questions

Frequently Asked Questions

A perpetuity present value is the present value of an infinite stream of equal payments C, discounted at rate r.

Apply this formula when evaluating financial instruments with no maturity date, such as British Consols or perpetual preferred stock. It is also used in corporate finance to estimate the terminal value of a firm that has reached a stable, mature growth phase.

This formula simplifies the complex task of valuing infinite future payments into a single, manageable figure. It serves as a foundational tool for investors to determine if a permanent income stream is priced fairly relative to its risk-adjusted return.

Applying to finite streams. Using growth rate incorrectly.

Valuing a distinct bond (Consol) paying fixed interest forever.

Ensure the interest rate (r) and the cash flow (C) correspond to the same time period. Convert percentage interest rates into decimal form before calculation. This specific formula assumes payments occur at the end of each period.

References

Sources

Brealey, Richard A., Myers, Stewart C., and Allen, Franklin. Principles of Corporate Finance. McGraw-Hill Education.
Ross, Stephen A., Westerfield, Randolph W., and Jordan, Bradford D. Fundamentals of Corporate Finance. McGraw-Hill Education.
Wikipedia: Perpetuity (finance)
Perpetuity (finance) Wikipedia article
Brealey, R. A., Myers, S. C., & Allen, F. (2020). Principles of Corporate Finance (13th ed.). McGraw-Hill Education.
Wikipedia article 'Perpetuity' (finance)
Standard curriculum — A-Level Accounting / Finance

Perpetuity Present Value

Overview

Variables

Derivation

Start from the Finite Annuity Formula:

Take the Limit as n\to∞:

Obtain the Perpetuity Formula:

Rearrangements

Graph

Intuition

Insight

Practice Problem

Real-World Context

Tips

Common Mistakes

Related Formulas

Dividend Growth Model

Frequently Asked Questions

Sources