Present Value of a Single Sum Equation, Derivation & Rearrangements

Core idea

Overview

The present value of a single sum calculates what a future amount of money is worth today, considering that money has earning potential over time. This concept, known as the time value of money, acknowledges that a dollar today is worth more than a dollar in the future due to its capacity to be invested and earn a return. It involves 'discounting' a future value back to the present using an appropriate interest rate, also known as the discount rate.

When to use: This equation is applied when you need to determine the current worth of a single future payment or receipt. It is crucial for evaluating investment opportunities, comparing a future payout to a current investment, and making informed financial decisions. Additionally, it's used in financial planning for future goals like retirement or education, and for valuing assets, loans, or liabilities that involve a single future cash flow.

Why it matters: The Present Value of a Single Sum is a fundamental concept in finance because it allows for the comparison of cash flows occurring at different points in time, enabling rational financial decision-making. It quantifies the 'time value of money,' recognizing that money available today is more valuable than the same amount in the future due to factors like inflation, opportunity cost, and risk. Understanding this principle helps individuals and businesses make informed choices about investments, savings, and loans.

Symbols

Variables

PV = Present Value, FV = Future Value, r = Interest Rate (per period)

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

Walkthrough

Derivation

Derivation of Present Value of a Single Sum

This derivation shows how the formula for the Present Value of a Single Sum is derived from the fundamental concept of future value and compound interest, by rearranging the future value formula to solve for the present value.

Interest is compounded at the end of each period.
The interest rate (r) remains constant over all periods.
The number of periods (n) is known and fixed.
The future value (FV) is a single lump sum amount.

1

Start with the Future Value of a Single Sum Formula

The future value (FV) of a single present sum (PV) invested today at an interest rate (r) per period, compounded for 'n' periods, is given by this formula. This is the foundational equation for compound interest.

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

Note: This formula assumes compounding occurs at the end of each period.

2

Isolate Present Value (PV)

To find the present value, we need to rearrange the future value formula to solve for PV. This is done by dividing both sides of the equation by the compound interest factor, (1 + r)^n.

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

Note: This step effectively 'discounts' the future value back to the present.

3

Rewrite using Negative Exponent Notation

For mathematical elegance and ease of calculation, especially in financial calculators or spreadsheets, the division by (1 + r)^n is often expressed as multiplication by (1 + r) raised to the power of -n. This is a standard algebraic identity where 1/x^a = x^-a.

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

Note: The term (1 + r)^-n is known as the 'discount factor' or 'present value interest factor'.

Result

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

Source: Commonly found in introductory finance and economics textbooks (e.g., Principles of Corporate Finance by Brealey, Myers, and Allen; Fundamentals of Financial Management by Brigham and Houston)

Free formulas

Rearrangements

Solve for $F V$

Make FV the subject

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

Rearrange by dividing both sides by the discount factor.

Difficulty: 2/5

Solve for $r$

Make r the subject

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

Isolate r by dividing, exponentiating, and subtracting 1.

Difficulty: 4/5

Solve for $n$

Make n the subject

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

Isolate n using logarithms to solve for an 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

Imagine a timeline. At the far end of this timeline, you have a specific amount of money (FV). To find its Present Value (PV), you are essentially 'walking backward' along this timeline towards today. With each step backward (each period), you are removing the interest that money would have earned. It's like deflating a balloon: the Future Value is the fully inflated balloon, and the Present Value is the balloon after all the 'air' (interest) has been let out, revealing its true size today.

P V

Present Value

This is the 'worth today' of a single sum of money that you will receive or pay at some point in the future. It answers the question: 'How much would I need to invest *today* to have a specific amount in the future?' or 'What is that future amount *really* worth to me right now, considering I could invest money today?'

F V

Future Value

This is the specific amount of money you expect to have or receive at a designated point in the future. It's the target amount whose current equivalent value you are trying to determine.

r

Discount Rate / Interest Rate per Period

This represents the 'cost of money' or the 'opportunity cost' of not having the money today. It's the rate at which money grows over time (if invested) or the rate at which future money loses value when brought back to the present. A higher 'r' means future money is worth less today because you could have earned more by investing it.

n

Number of Compounding Periods

This is simply the number of time intervals (e.g., years, months, quarters) that separate the present from the future. The longer the time period, the more interest would have accrued (or is discounted), and thus the greater the difference between the Present Value and the Future Value.

Signs and relationships

\times: This indicates that the Future Value (FV) is being scaled down by the discount factor. The discount factor, `(1 + r)^-n`, is a multiplier that reduces the future amount to its equivalent value in today's terms.
+: Within the `(1 + r)` term, the `1` represents the original principal amount, and `r` represents the interest earned on that principal for one period. So, `(1 + r)` is the growth factor for a single period. If you invest $1, after one period you would have `$(1 + r)`.
-n: The negative exponent is the mathematical operation that performs 'discounting'. A positive exponent `(1 + r)^n` would compound money forward in time (calculating future value). A negative exponent `(1 + r)^-n` is equivalent to `1 / (1 + r)^n`, which effectively divides the future value by the compounding factor for 'n' periods. This division removes the interest that would have been earned over those periods, bringing the future amount back to its present-day worth.

One free problem

Practice Problem

What is the present value of $10,000 to be received in 5 years, if the discount rate is 8% compounded annually?

Present Value10000

Future Value0.08

Interest Rate (per period)5

Solve for: FV

Hint: Use the formula PV = FV / (1 + r)^n.

The full worked solution stays in the interactive walkthrough.

Where it shows up

Real-World Context

You want to set aside enough money today to have $500,000 in 18 years for a child's college education. If you can earn a 9% annual return on your investment, you would use this formula to calculate how much you need to invest today.

Study smarter

Tips

Ensure that the interest rate ('r') and the number of periods ('n') are consistent in their time units (e.g., if 'n' is in years, 'r' should be an annual rate).
Remember that a higher discount rate or a longer time period will result in a lower present value, reflecting the increased opportunity cost or risk.
This formula assumes a single, lump-sum payment in the future; for multiple payments or a series of equal payments, other time value of money formulas (like present value of an annuity) are more appropriate.

Avoid these traps

Common Mistakes

Using an inconsistent time unit for the interest rate and the number of periods (e.g., an annual rate with monthly periods without proper adjustment).
Failing to correctly adjust the interest rate and number of periods for compounding frequencies other than annual (e.g., semi-annual, quarterly, or monthly compounding).

Keep going

Related Formulas

Common questions

Frequently Asked Questions

This derivation shows how the formula for the Present Value of a Single Sum is derived from the fundamental concept of future value and compound interest, by rearranging the future value formula to solve for the present value.

This equation is applied when you need to determine the current worth of a single future payment or receipt. It is crucial for evaluating investment opportunities, comparing a future payout to a current investment, and making informed financial decisions. Additionally, it's used in financial planning for future goals like retirement or education, and for valuing assets, loans, or liabilities that involve a single future cash flow.

The Present Value of a Single Sum is a fundamental concept in finance because it allows for the comparison of cash flows occurring at different points in time, enabling rational financial decision-making. It quantifies the 'time value of money,' recognizing that money available today is more valuable than the same amount in the future due to factors like inflation, opportunity cost, and risk. Understanding this principle helps individuals and businesses make informed choices about investments, savings, and loans.

Using an inconsistent time unit for the interest rate and the number of periods (e.g., an annual rate with monthly periods without proper adjustment). Failing to correctly adjust the interest rate and number of periods for compounding frequencies other than annual (e.g., semi-annual, quarterly, or monthly compounding).

You want to set aside enough money today to have $500,000 in 18 years for a child's college education. If you can earn a 9% annual return on your investment, you would use this formula to calculate how much you need to invest today.

Ensure that the interest rate ('r') and the number of periods ('n') are consistent in their time units (e.g., if 'n' is in years, 'r' should be an annual rate). Remember that a higher discount rate or a longer time period will result in a lower present value, reflecting the increased opportunity cost or risk. This formula assumes a single, lump-sum payment in the future; for multiple payments or a series of equal payments, other time value of money formulas (like present value of an annuity) are more appropriate.

References

Sources

Brealey, R. A., Myers, S. C., & Allen, F. (Year). Principles of Corporate Finance. McGraw-Hill Education.
Corporate Finance Institute. (n.d.). Time Value of Money. Retrieved from Corporate Finance Institute website.
Finance Strategists. (2023, March 29). Present Value of a Single Amount. Retrieved from Finance Strategists website.
Standard Financial Mathematics Principles
Commonly found in introductory finance and economics textbooks (e.g., Principles of Corporate Finance by Brealey, Myers, and Allen; Fundamentals of Financial Management by Brigham and Houston)