Actuarial Mathematics for Modelling · Project appraisal

Equation of Value and Net Present Value Explained

Updated 11 October 2026 · Fact-checked

An equation of value sets the present value of money coming in equal to the present value of money going out at a chosen date. Net present value (NPV) is the present value of all cash flows, inflows minus outflows, at a given interest rate. A positive NPV means the project earns more than that rate.

Understand Equation of Value and Net Present Value

Money received at different times is not directly comparable. ₹1,000 today is worth more than ₹1,000 in a year, because you can invest it and earn interest. To compare cash flows, you move each one to a single date using the interest rate.

The equation of value does this. You pick a date, called the valuation date. You value every payment in and every payment out at that date. Then you set the total value of the receipts equal to the total value of the payments. Any date gives the same answer if the interest rate is constant, so choose the date that makes the algebra simplest.

The net present value (NPV) uses the same idea for a project. You list the cash flows: usually an outlay at time 0, then net income at later times. You discount each to time 0 at the given rate of interest and add them up. NPV = PV of inflows − PV of outflows.

The decision rule is simple. If NPV at your rate is positive, the project returns more than that rate, so it is acceptable on this test. If NPV is negative, it returns less. If NPV is zero, the project earns exactly that rate. The rate where NPV equals zero is the internal rate of return (IRR).

The rate you use matters. It is often the cost of borrowing or the return you could earn elsewhere. NPV falls as the rate rises when the cash flows are outflows first and inflows later.

Key rules to remember

Discount factor (annual compound interest): v = 1 ÷ (1 + i), so the present value of 1 due at time t is vᵗ = (1 + i)⁻ᵗ; i is the effective annual rate. Time t is in years.
Net present value (discrete cash flows): NPV = Σ cₜ × vᵗ = Σ cₜ ÷ (1 + i)ᵗ; cₜ is the net cash flow at time t (inflow positive, outflow negative). Time 0 has factor 1.
Equation of value: PV of receipts = PV of payments at the chosen date; State the valuation date. Use any date; pick the one that simplifies the working.
Value at a later time T: Value at T = Σ cₜ × (1 + i)^(T − t); Accumulate payments before T and discount those after T.
Level annuity-immediate present value: aₙ = (1 − vⁿ) ÷ i; Payments of 1 at the end of each year for n years. Useful to shorten NPV sums.
Continuous cash flow NPV: NPV = ∫ ρ(t) × e^(−δt) dt; ρ(t) is the payment rate at time t and δ is the force of interest, taken over the project term.

How to solve Equation of Value and Net Present Value questions

Use this method for any equation of value or NPV question. Write each step so the examiner can follow your working.

1List all cash flows with their times in years. Mark outflows negative and inflows positive.
2Convert the rate to an effective annual rate if it is given as nominal or as a force of interest, and check the time unit matches.
3Choose the valuation date. For NPV it is time 0. For an unknown payment or time, choose the date that removes the unknown from the powers of v.
4Write the equation of value, or the NPV sum, using discount factors vᵗ for each flow.
5Simplify using annuity formulas where payments are level or regular.
6Solve for the unknown, or compute the NPV and state its sign.
7Interpret the result: say whether the project is acceptable at that rate, and give units in rupees.

Quickest way: Level annuity shortcut for NPV

When to use it: Use when the project has a single outlay and then level regular receipts, or a mix of a few one-off flows and a level stream.

Put the outlay at time 0 as it is.
Replace the level receipts by payment × aₙ at the given rate, using (1 − vⁿ) ÷ i.
If the stream starts later, multiply by vᵏ to discount it back k years further.
Add the single flows discounted individually.
Check the sign: a positive NPV means the project beats the rate.

Common mistakes in Equation of Value and Net Present Value

Discounting the time 0 outlay as if it were at time 1.
Students apply v to every cash flow without looking at its date.
Fix: The factor at time 0 is 1. Write the time beside each flow before you discount.
Using the wrong annuity timing, such as treating an annuity-immediate as an annuity-due.
Payments at the start or end of each year look similar in the wording.
Fix: Draw a timeline. aₙ starts one period after the valuation date, so the first payment is at time 1.
Mixing up the sign of inflows and outflows.
Students add all amounts as positives and then forget to subtract.
Fix: Label outflows negative from the start, then NPV is a plain sum.
Using a nominal rate directly as the annual effective rate.
The question says 8% per annum convertible half-yearly and students use 0.08.
Fix: Convert first: i = (1 + 0.08 ÷ 2)² − 1 = 8.16%. Or work in half-year periods.
Mixing valuation dates in one equation.
Some terms are discounted to time 0 and others accumulated to time 5 without adjusting.
Fix: State one valuation date and value every flow at that date.
Stating the decision without reference to the rate.
Students write 'accept' because NPV is positive but do not link it to the rate.
Fix: Say 'NPV is positive at the given rate, so the project earns more than that rate'.

Worked examples

Example 1

A project requires an outlay of ₹5,00,000 now. It returns ₹1,50,000 at the end of each year for 4 years. Calculate the NPV at 6% per annum effective, and state whether the project is acceptable.

Show the solution

Cash flows: −5,00,000 at time 0; +1,50,000 at times 1, 2, 3, 4.
v = 1 ÷ 1.06 = 0.943396.
v⁴ = 1 ÷ 1.06⁴ = 1 ÷ 1.262477 = 0.792094.
a₄ = (1 − 0.792094) ÷ 0.06 = 0.207906 ÷ 0.06 = 3.465106.
PV of receipts = 1,50,000 × 3.465106 = 5,19,766 (to nearest rupee).
NPV = 5,19,766 − 5,00,000 = 19,766.
NPV is positive, so the project earns more than 6% per annum.

Answer: NPV ≈ ₹19,766. The project is acceptable at 6%.

Example 2

You pay ₹20,000 now and ₹30,000 at the end of 3 years. In return you receive a single payment of X at the end of 5 years. Find X so that the transaction is fair at 5% per annum effective.

Show the solution

Choose the valuation date as time 5, so X has no discount factor.
Value of payments at time 5: 20,000 × 1.05⁵ + 30,000 × 1.05².
1.05⁵ = 1.276282, so 20,000 × 1.276282 = 25,525.63.
1.05² = 1.1025, so 30,000 × 1.1025 = 33,075.00.
Total = 25,525.63 + 33,075.00 = 58,600.63.
Equation of value at time 5: X = 58,600.63.

Answer: X ≈ ₹58,601 (to nearest rupee).

Exam tips

Always write the valuation date next to your equation of value. Examiners award marks for stating it.
Draw a timeline first. It prevents most timing errors and takes ten seconds.
Carry at least six decimal places in discount factors and round only at the end, or your final rupee figure may be off.
In computer-based Paper B, set out the cash flow vector, the discount vector and the NPV formula clearly in your sheet or code so the method is visible.
Finish with a sentence on what the NPV means for the decision. Many written questions allot a mark to it.

Practice questions from Project appraisal

Equation of Value and Net Present Value in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

Equation of Value and Net Present Value: frequently asked questions

What is the difference between the equation of value and NPV?

The equation of value sets the value of receipts equal to the value of payments at a chosen date, and you usually solve it for an unknown. NPV is the value of all net cash flows at time 0 at a given rate. NPV equal to zero is the equation of value for the IRR.

Does the valuation date affect the answer?

Not for the unknown, if the interest rate is constant over time. Every date gives the same solution. A well-chosen date just makes the algebra easier.

What does a positive NPV tell me?

It means the project earns more than the rate used for discounting. At that rate the project adds value, so it is acceptable on the NPV test. A negative NPV means it earns less.

How do I handle a nominal rate in an NPV question?

Convert it to an effective annual rate, or work in the payment period. For example, 8% per annum convertible half-yearly is 4% per half-year, or 8.16% effective per year.