Present Value (Single Sum) Equation, Derivation & Rearrangements

Core idea

Overview

The Present Value formula determines the current worth of a specific sum of money to be received in the future, adjusted for a specific rate of return. It reflects the fundamental principle that money available now is worth more than the same amount in the future due to its potential earning capacity.

When to use: Apply this formula when you need to evaluate the current value of a single future cash inflow or outflow. It is essential for comparing investment opportunities with different time horizons or assessing the impact of inflation and opportunity costs on future sums.

Why it matters: This concept is the bedrock of modern finance, enabling discounted cash flow analysis and net present value calculations. It allows individuals and businesses to make rational 'apples-to-apples' comparisons between immediate costs and distant rewards.

Symbols

Variables

PV = Present Value, FV = Future Value, r = Interest Rate, n = Periods

P V

Present Value

$

F V

Future Value

$

r

Interest Rate

V a r iab l e

n

Periods

V a r iab l e

Walkthrough

Derivation

Understanding Present Value (PV)

Present value calculates what a future sum is worth today by discounting it at a chosen rate of return.

The discount rate r is constant over the period.
The future value FV occurs at a known time n (end of period).

1

Start with Compounding to the Future:

If PV is invested at rate r for n periods, compounding gives the future value FV.

F V = P V (1 + r)^{n}

2

Rearrange to Discount Back to Today:

Divide by the compounding factor to convert a future cash amount into an equivalent value today (discounting).

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

Result

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

Source: Standard curriculum — A-Level Business / Finance

Free formulas

Rearrangements

Solve for $P V$

Make PV the subject

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

PV is already the subject of the formula.

Difficulty: 1/5

Solve for $F V$

Make FV the subject

F V = P V (1 + r)^{n}

Start with the Present Value (PV) formula. To make FV the subject, multiply both sides by the denominator, then swap the sides of the equation.

Difficulty: 2/5

Solve for $r$

Make r the subject

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

To make `r` the subject, first clear the denominator by multiplying by `(1+r)^n`, then divide by `PV`. Next, raise both sides to the power of `1/n` to remove the exponent, and finally subtract `1`.

Difficulty: 2/5

Solve for $n$

Make n the subject

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

To make $n$ (number of periods) the subject, first isolate the term containing $n$ , then take the natural logarithm of both sides, apply logarithm properties, and finally solve for $n$ .

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 because Present Value is directly proportional to Future Value. As Future Value increases, Present Value increases at a constant rate determined by the slope of one divided by the quantity one plus r raised to the power of n. For a student of finance, this linear relationship means that doubling the Future Value will always double the Present Value regardless of the starting amount. The most important feature is that the slope remains constant for any given in

Graph type: linear

Why it behaves this way

Intuition

A financial picture of money's value diminishing as it is brought backward from a future point to the present, like a value-decay curve.

PV

The equivalent value of a future sum of money expressed in today's terms.

What a future payment is worth to you right now, considering its earning potential.

FV

The specified amount of money at a future date.

The exact sum of money expected at a later point in time.

r

The periodic discount rate, representing the opportunity cost of capital or required rate of return.

The percentage return you could earn if you had the money today, or the cost of delaying its receipt. A higher 'r' means future money is less valuable today.

n

The number of compounding or discounting periods.

How many time intervals (e.g., years, quarters) separate the present from the future date. More periods mean a greater reduction in present value.

Signs and relationships

(1+r)^n in the denominator: This term acts as a discount factor. The positive 'r' and exponent 'n' mean that as either the discount rate or the number of periods increases, the denominator grows, thereby reducing the present value (PV).

Free study cues

Insight

Canonical usage

The present value (PV) and future value (FV) must be expressed in the same currency unit, while the discount rate (r) and number of periods (n) are dimensionless.

Common confusion

A common mistake is using a percentage value for 'r' directly in the formula instead of converting it to a decimal. Another is mismatching the time period of 'r' with the time period of 'n'.

Dimension note

The discount rate ('r') is a dimensionless ratio, and the number of periods ('n') is a dimensionless count.

Unit systems

$P V$ Currency (e.g., USD, EUR) · Represents the current worth of the future sum. Must be consistent with FV.

$F V$ Currency (e.g., USD, EUR) · Represents the future sum of money. Must be consistent with PV.

$r$ dimensionless · The discount rate or interest rate per period. Must be expressed as a decimal in the formula (e.g., 5% becomes 0.05). The period for 'r' must match the period for 'n'.

$n$ dimensionless · The total number of periods. The period for 'n' must match the period for 'r' (e.g., if 'r' is annual, 'n' must be in years).

One free problem

Practice Problem

An investor expects to receive a lump sum of $10,000 in 5 years. If the annual market interest rate is 6%, what is the value of this payment today?

Future Value10000 $

Interest Rate0.06

Periods5

Solve for: $P V$

Hint: Divide the future value by (1 + r) raised to the power of the number of years.

The full worked solution stays in the interactive walkthrough.

Where it shows up

Real-World Context

Deciding if $1000 today is better than $1100 in a year.

Study smarter

Tips

Ensure the discount rate (r) and the time periods (n) use the same time scale, usually annual.
A higher discount rate results in a lower present value, reflecting higher risk or opportunity cost.
This formula assumes a single lump-sum payment rather than a series of recurring payments.

Avoid these traps

Common Mistakes

Using integer percentage (5 instead of 0.05).
Confusing PV and FV.

Keep going

Related Formulas

Common questions

Frequently Asked Questions

Present value calculates what a future sum is worth today by discounting it at a chosen rate of return.

Apply this formula when you need to evaluate the current value of a single future cash inflow or outflow. It is essential for comparing investment opportunities with different time horizons or assessing the impact of inflation and opportunity costs on future sums.

This concept is the bedrock of modern finance, enabling discounted cash flow analysis and net present value calculations. It allows individuals and businesses to make rational 'apples-to-apples' comparisons between immediate costs and distant rewards.

Using integer percentage (5 instead of 0.05). Confusing PV and FV.

Deciding if $1000 today is better than $1100 in a year.

Ensure the discount rate (r) and the time periods (n) use the same time scale, usually annual. A higher discount rate results in a lower present value, reflecting higher risk or opportunity cost. This formula assumes a single lump-sum payment rather than a series of recurring payments.

References

Sources

Principles of Corporate Finance by Brealey, Myers, Allen
Wikipedia: Time value of money
Wikipedia: Present value
Brealey, R. A., Myers, S. C., & Allen, F. (2020). Principles of Corporate Finance (13th ed.). McGraw-Hill Education.
Brigham, E. F., & Ehrhardt, M. C. (2017). Financial Management: Theory and Practice (15th ed.). Cengage Learning.
Wikipedia: Time value of money (accessed 2023-10-27)
Standard curriculum — A-Level Business / Finance

Present Value (Single Sum)

Overview

Variables

Derivation

Start with Compounding to the Future:

Rearrange to Discount Back to Today:

Rearrangements

Graph

Intuition

Insight

Practice Problem

Real-World Context

Tips

Common Mistakes

Related Formulas

Future Value (Single Sum)

Frequently Asked Questions

Sources