Present Value (Single Sum, Discrete Compounding) Equation, Derivation & Rearrangements

Core idea

Overview

This equation represents the foundational concept of discounted cash flow analysis, determining what a future payment is worth in today's currency. By dividing the future value by the compounding growth factor, it accounts for the opportunity cost of capital and the time value of money. It assumes that interest is credited at discrete intervals, such as annually or monthly.

When to use: Apply this when you need to determine the current worth of a specific cash inflow or outflow expected at a known future date given a constant discount rate.

Why it matters: It is the core mechanism behind bond pricing, investment appraisal, and capital budgeting, allowing investors to compare the value of money across different time horizons.

Symbols

Variables

PV = Present Value, FV = Future Value, r = Interest Rate per Period, n = Number of Periods

P V

Present Value

V a r iab l e

F V

Future Value

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 (Single Sum, Discrete Compounding)

This derivation demonstrates how the present value of a future sum is determined by rearranging the compound interest formula to isolate the initial principal.

The interest rate (r) remains constant over the investment period.
Compounding occurs at discrete, fixed intervals (n).
Cash flows are known with certainty and there is no risk of default.

1

Defining Future Value with Compound Interest

The future value (FV) of an investment is equal to the present value (PV) multiplied by the growth factor (1 + r) compounded over n periods.

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

Note: This is the foundational formula for discrete time-value-of-money calculations.

2

Isolating the Present Value

To solve for the present value (PV), divide both sides of the equation by the compounding factor (1 + r)^n.

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

Note: Ensure the interest rate r is expressed as a decimal (e.g., 5% = 0.05).

3

Final Form

After canceling terms on the right side, we arrive at the standard formula for discounting a single future sum to its present value.

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

Note: This process is known as 'discounting', where (1+r)^-n is called the Present Value Interest Factor (PVIF).

Result

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

Source: Brealey, R. A., Myers, S. C., & Allen, F. (2020). Principles of Corporate Finance.

Free formulas

Rearrangements

Solve for $P V$

Make PV the subject

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

The formula is already solved for the present value (PV).

Difficulty: 1/5

Solve for $F V$

Make FV the subject

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

Rearrange to solve for the future value by multiplying both sides by the discount factor.

Difficulty: 2/5

Solve for $r$

Make r the subject

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

Solve for the interest rate by isolating the (1+r) term and using the nth root.

Difficulty: 4/5

Solve for $n$

Make n the subject

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

Solve for the number of periods using logarithms to isolate the exponent.

Difficulty: 5/5

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

Visual intuition

Graph

Graph preview unavailable for this formula.

Why it behaves this way

Intuition

Think of this as a 'reverse zoom' lens. If a future amount is a large object at the end of a long hallway (the future), discounting is the process of walking backward toward the present. As you move backward through each time period (n), the object appears smaller and smaller until it reaches its true, compact size at the starting point (the present).

P V

Present Value

The 'price' of money today; what a future sum is worth if you held it in your hand right now.

F V

Future Value

The 'destination' amount; the larger lump sum waiting for you at a specific point in the future.

r

Discount Rate

The 'gravity' or 'tax' of time; it represents the opportunity cost of not having the money today.

n

Number of periods

The 'distance' of the time horizon; the number of compounding cycles the money must traverse.

Signs and relationships

(1+r): The 1 represents the principal, and the r represents the growth factor. Adding them together creates the multiplier for a single period of growth.
Exponent n: Positive exponentiation signifies geometric growth (compounding) over time; the division by this term effectively reverses that growth to strip away interest earned.
Division (/): Division serves as the inverse operation to compounding. By dividing FV by (1+r)^n, you are 'peeling back' the layers of interest accumulated over time.

One free problem

Practice Problem

What is the present value of $5,000 received in 3 years at an annual interest rate of 5%?

Future Value5000

Interest Rate per Period0.05

Number of Periods3

Solve for: PV

Hint: Divide 5000 by 1.05 raised to the power of 3.

The full worked solution stays in the interactive walkthrough.

Where it shows up

Real-World Context

An investor wants to know how much they should invest today in a guaranteed bond that pays $1,000 in exactly 5 years, assuming an annual interest rate of 4%.

Study smarter

Tips

Ensure the interest rate and the number of periods match the compounding frequency (e.g., if compounding is monthly, use a monthly rate and total months).
The discount rate should reflect the risk-free rate plus a risk premium appropriate for the cash flow.
Remember that higher discount rates will always result in a lower present value.

Avoid these traps

Common Mistakes

Using an annual interest rate when compounding is performed on a sub-annual basis without adjusting the period count.
Confusing the nominal interest rate with the effective periodic rate.
Neglecting to adjust the number of periods when the compounding frequency changes.

Keep going

Related Formulas

Common questions

Frequently Asked Questions

This derivation demonstrates how the present value of a future sum is determined by rearranging the compound interest formula to isolate the initial principal.

Apply this when you need to determine the current worth of a specific cash inflow or outflow expected at a known future date given a constant discount rate.

It is the core mechanism behind bond pricing, investment appraisal, and capital budgeting, allowing investors to compare the value of money across different time horizons.

Using an annual interest rate when compounding is performed on a sub-annual basis without adjusting the period count. Confusing the nominal interest rate with the effective periodic rate. Neglecting to adjust the number of periods when the compounding frequency changes.