Net Present Value (NPV) — 1 period | Equation, Derivation and Rearrangements

Core idea

Overview

The single-period Net Present Value calculates the difference between the current value of an expected future cash inflow and the initial investment cost. By discounting the future amount, it accounts for the time value of money and the opportunity cost of capital over a specific timeframe.

When to use: This formula is used for short-term financial decisions where a single investment leads to one payout exactly one period later. It is common in bridge loans, short-term inventory flips, or basic capital budgeting scenarios where the discount rate is known.

Why it matters: It provides a clear 'yes or no' signal for investments: a positive result means the project adds value, while a negative result suggests the investment will lose value relative to other opportunities. It is the gold standard for assessing wealth creation in finance.

Symbols

Variables

NPV = Net Present Value, $C_{1}$ = Cash Flow Year 1, r = Discount Rate, $C_{0}$ = Initial Cost

NPV

Net Present Value

$

C_{1}

Cash Flow Year 1

$

r

Discount Rate

Variable

C_{0}

Initial Cost

$

Walkthrough

Derivation

Understanding Net Present Value (NPV)

NPV compares the present value of an investment’s cash inflows to its initial cost. For one period, it is the discounted inflow minus the initial outlay.

Cash flows are estimated reliably.
Discount rate r reflects the relevant required return (cost of capital/risk).
Cash flow timing is end-of-period for the inflow C1.

1

State the One-Period NPV:

Initial cost $C_{0}$ is an outflow at time 0, and $C_{1}$ is the inflow at the end of period 1 discounted back to time 0.

N P V = - C_{0} + \frac{C _{1}}{1 + r}

Note: For multiple periods: $N P V = - C_{0} + \sum_{t = 1}^{n} \frac{C _{t}}{( 1 + r ) ^{t}}$ . If NPV>0, the project adds value at rate r.

Result

N P V = - C_{0} + \frac{C _{1}}{1 + r}

Source: AQA A-Level Business — Financial Decision Making

Free formulas

Rearrangements

Solve for $C_{1}$

Make C1 the subject

C_{1} = (N P V + C_{0}) (1 + r)

Start from the Net Present Value (NPV) formula for one period. To make $C_{1}$ the subject, first add $C_{0}$ to both sides, then multiply by $(1 + r)$ , and finally swap the sides.

Difficulty: 2/5

Solve for $C_{0}$

Make C0 the subject

C_{0} = \frac{C _{1}}{( 1 + r )} - N P V

To make C0 the subject, start with the Net Present Value (NPV) formula for one period. Add C0 to both sides, then subtract NPV from both sides.

Difficulty: 2/5

Solve for $r$

Make r the subject

r = \frac{C _{1}}{N P V + C _{0}} - 1

To make 'r' (Discount Rate) the subject, first isolate the fraction by adding ' $C_{0}$ '. Then, clear the denominator by multiplying by '(1+r)', divide by '(NPV + $C_{0}$ )', and finally subtract '1'.

Difficulty: 3/5

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

Visual intuition

Graph

The graph is a hyperbola where the discount rate r appears in the denominator, causing the Net Present Value to decrease rapidly as the rate increases toward a horizontal asymptote. For a finance student, this shape illustrates that higher discount rates significantly erode the profitability of an investment, as larger values of r represent a greater reduction in the present value of future cash flows. The most critical feature of this curve is the inverse relationship between the discount rate and the Net Present Value, which demonstrates that even small changes in the rate have a disproportionately large impact on the investment outcome when the rate is low.

Graph type: hyperbolic

Why it behaves this way

Intuition

Imagine a financial timeline where an initial cost is an outflow at time zero, and a future cash inflow is pulled back to time zero, shrinking in value based on the discount rate, before being compared to the initial

NPV

The net increase or decrease in wealth resulting from an investment, expressed in today's dollars.

A positive NPV indicates the investment is financially attractive, as it generates more value than its cost, considering the time value of money.

C_{1}

The cash flow expected to be received at the end of the first period.

This is the future financial benefit or return that the investment is projected to yield.

C_{0}

The initial cash outflow (cost) required to undertake the investment at the start (time zero).

This represents the upfront capital commitment or the immediate cost of the project.

r

The discount rate, representing the opportunity cost of capital or the required rate of return for an investment of similar risk.

It quantifies how much future money is worth less today; a higher 'r' means future cash flows are discounted more heavily, reflecting higher risk or better alternative investment opportunities.

Signs and relationships

- C_0: The initial investment $C_{0}$ is a cash outflow, meaning it reduces the investor's cash balance, hence it is subtracted from the present value of future inflows.
/(1+r): This term discounts the future cash flow $C_{1}$ to its equivalent value today. It accounts for the time value of money, reflecting that money available now is worth more than the same amount in the future due to its earning

Free study cues

Insight

Canonical usage

All monetary values (NPV, $C_{0}$ , $C_{1}$ ) must be expressed in the same currency, and the discount rate (r) must be used as a dimensionless decimal.

Common confusion

The most common mistake is failing to convert the discount rate (r) from a percentage to a decimal before performing the calculation, or mixing different currency units for cash flows.

Dimension note

The discount rate (r) is a ratio representing a percentage return or cost, making it dimensionless when expressed as a decimal.

Unit systems

NPVAny consistent currency (e.g., USD, EUR, GBP) - Represents the net monetary value added or lost by the investment.

$C_{1}$ Same consistent currency as C_0 - The cash inflow expected at the end of the period.

$C_{0}$ Same consistent currency as C_1 - The initial investment cost or cash outflow at time zero.

$r$ Dimensionless - The discount rate or cost of capital, which must be expressed as a decimal (e.g., 0.05 for 5%).

One free problem

Practice Problem

A business owner invests 10,000 today to upgrade equipment. They expect the upgrade to generate an additional 11,500 in revenue at the end of next year. If the business requires an 8% return on investment, what is the Net Present Value of this upgrade?

Initial Cost10000 $

Cash Flow Year 111500 $

Discount Rate0.08

Solve for: NPV

Hint: Divide the future cash flow (C1) by 1 plus the discount rate, then subtract the initial cost (C0).

The full worked solution stays in the interactive walkthrough.

Where it shows up

Real-World Context

In deciding to buy a machine based on future profits, Net Present Value (NPV) — 1 period is used to calculate Net Present Value from Cash Flow Year 1, Discount Rate, and Initial Cost. The result matters because it helps compare incentives, policy effects, market outcomes, or financial decisions in context.

Study smarter

Tips

Always express the discount rate 'r' as a decimal, for example, 5% is 0.05.
C0 represents the cash outflow at the start, so it is subtracted from the discounted inflow.
If the NPV is exactly zero, the investment perfectly meets the required rate of return without adding extra value.

Avoid these traps

Common Mistakes

Ignoring the initial outflow.
Wrong discount rate.

Keep going

Related Formulas

Common questions

Frequently Asked Questions

NPV compares the present value of an investment’s cash inflows to its initial cost. For one period, it is the discounted inflow minus the initial outlay.

This formula is used for short-term financial decisions where a single investment leads to one payout exactly one period later. It is common in bridge loans, short-term inventory flips, or basic capital budgeting scenarios where the discount rate is known.

It provides a clear 'yes or no' signal for investments: a positive result means the project adds value, while a negative result suggests the investment will lose value relative to other opportunities. It is the gold standard for assessing wealth creation in finance.

Ignoring the initial outflow. Wrong discount rate.