Actuarial Mathematics for Modelling · Equation of value

Applications of the Equation of Value: Project Appraisal and Loans

Updated 11 October 2026 · Fact-checked

Project appraisal uses the equation of value to compare cash flows at a chosen interest rate. NPV is the present value of inflows minus outflows. IRR is the rate that makes NPV zero. Discounted payback is the first time the accumulated discounted cash flow turns non-negative. A loan's effective rate is the IRR of its cash flows.

Understand Applications: Project Appraisal and Loans

An equation of value says that the present value of money in equals the present value of money out at a stated rate of interest. Project appraisal and loans are two uses of this one idea.

The net present value (NPV) of a project is the present value of all cash inflows minus the present value of all outflows, at a chosen rate i. Often this is your risk discount rate or cost of funds. If NPV > 0, the project earns more than i and is acceptable on that test. If NPV < 0, it earns less.

The internal rate of return (IRR) is the rate i at which NPV = 0. It is the project's own yield. You usually find it by trial and error, then linear interpolation between two rates with NPV of opposite signs. The result is approximate, because NPV is not a straight line in i.

The discounted payback period is the time it takes for the accumulated present value of the cash flows to become zero or positive. Ordinary payback ignores interest. Discounted payback uses it, so it is always at least as long as ordinary payback when i > 0 and all later cash flows are inflows.

A loan is a project seen from the lender's side. The borrower's effective rate of return, or cost, is the IRR of the loan cash flows once fees and charges are included. If a fee is taken at the start, the borrower receives less than the face amount but repays on the full amount. That pushes the true rate above the quoted rate.

Key rules to remember

Net present value: NPV(i) = Σ C(t) × v^t, where v = 1 ÷ (1 + i); C(t) is net cash flow at time t (inflow positive, outflow negative). For continuous flows use ∫ ρ(t) e^(-δt) dt.
Internal rate of return: NPV(i) = 0 solved for i; There may be no solution, or several, when cash flows change sign more than once.
Linear interpolation for IRR: i ≈ i₁ + (i₂ − i₁) × NPV(i₁) ÷ (NPV(i₁) − NPV(i₂)); Choose i₁ and i₂ close together with NPVs of opposite sign. The answer is an estimate.
Discounted payback period: smallest t such that Σ (s ≤ t) C(s) × v^s ≥ 0; Use the same rate i. State whether you give the time of the cash flow or an interpolated time.
Level annuity-immediate: a(n) = (1 − v^n) ÷ i; Useful to value level loan repayments or level project inflows.
Loan equation of value: Amount received = PV of repayments at the effective rate i; Solve for i to get the effective rate of return on the loan.
Effective rate from a monthly rate: (1 + i) = (1 + j)^12, where j is the effective monthly rate; Convert before comparing with annual rates.

How to solve Applications: Project Appraisal and Loans questions

Use this method for any question on NPV, IRR, payback or loan rates.

1List every cash flow with its time and sign. Outflows are negative, inflows positive. Include fees, tax and any residual value if the question gives them.
2Decide the unit of time and the rate. Convert nominal rates to the effective rate for your time unit.
3Write the equation of value. For NPV, discount all flows to time 0 at the given rate. For IRR, set the same sum equal to zero.
4Use annuity functions for level or regular flows. Do not add terms one by one if a pattern exists.
5For IRR, try two sensible rates, find NPV at each, then interpolate. Check by substituting your answer.
6For discounted payback, build a running total of discounted flows. Find the first period where it is non-negative. Interpolate only if asked.
7For a loan, treat the amount actually received as the inflow and all repayments and charges as outflows. Solve for the effective rate. Convert to the form asked for.
8State your decision or result with units and a one-line conclusion, such as accept or reject.

Quickest way: Anchor on one equation and use annuity functions

When to use it: Use this in timed written questions or MCQs where flows are level or follow a simple pattern.

Write the single equation of value first, with time 0 as the comparison date.
Group level flows into an annuity and any one-off flows separately.
For IRR, guess from the ratio of total inflow to outflow, then test two rates a few percentage points apart.
Interpolate once, then verify with a third NPV if time permits.
For payback, compute discounted flows one by one only until the running total turns positive.

Common mistakes in Applications: Project Appraisal and Loans

Treating IRR as the best measure whenever two projects are compared.
A higher yield looks better, but IRR ignores the size of the project and the reinvestment rate.
Fix: Compare NPVs at the same rate. Use IRR as a supporting measure, and check the incremental cash flows if the projects differ in scale.
Using ordinary payback instead of discounted payback.
Students add undiscounted cash flows because it is quicker.
Fix: Discount each flow at the stated rate before accumulating. Read the question for the word discounted.
Using the face value of a loan instead of the amount received.
Fees or charges deducted at the start are missed.
Fix: The inflow at time 0 is the face value less upfront fees. Repayments are based on the face value.
Quoting a monthly rate as the annual effective rate.
Students stop after solving the monthly equation.
Fix: Convert with (1 + i) = (1 + j)^12, or state the nominal rate convertible monthly if that is what is asked.
Assuming the IRR is unique.
Most textbook examples have one sign change.
Fix: Count sign changes in the cash flows. More than one change means there may be several IRRs, so say so and rely on NPV.
Interpolating between two rates that are far apart.
Students pick convenient rates such as 5% and 20%.
Fix: Pick rates within a few percentage points with NPVs of opposite sign, and treat the answer as approximate.

Worked examples

Example 1

A project costs ₹10,00,000 now and pays ₹3,00,000 at the end of each year for 5 years. (a) Find the NPV at 8% per year effective. (b) Find the discounted payback period in whole years.

Show the solution

Value of inflows = 3,00,000 × a(5) at 8%.
v = 1 ÷ 1.08 = 0.925926. v^5 = 0.680583.
a(5) = (1 − 0.680583) ÷ 0.08 = 0.319417 ÷ 0.08 = 3.992710.
PV of inflows = 3,00,000 × 3.992710 = ₹11,97,813 (to the nearest rupee).
NPV = 11,97,813 − 10,00,000 = ₹1,97,813.
For payback, discount each inflow: year 1 = 2,77,778; year 2 = 2,57,202; year 3 = 2,38,150; year 4 = 2,20,509; year 5 = 2,04,175.
Running total: after year 1 = 2,77,778; year 2 = 5,34,980; year 3 = 7,73,130; year 4 = 9,93,639; year 5 = 11,97,814.
Compare with 10,00,000. After year 4 the total is 9,93,639, which is below 10,00,000. After year 5 it is above.

Answer: NPV ≈ ₹1,97,813 (positive, so acceptable at 8%). Discounted payback occurs during year 5, so it is 5 years in whole years.

Example 2

A lender gives a borrower a loan of ₹1,00,000 but deducts an arrangement fee of ₹2,000 at the start. The borrower repays ₹1,05,000 at the end of one year. Find the effective annual rate of return to the lender.

Show the solution

Amount actually advanced = 1,00,000 − 2,000 = ₹98,000.
Amount received at time 1 = ₹1,05,000.
Equation of value: 98,000 × (1 + i) = 1,05,000.
1 + i = 1,05,000 ÷ 98,000 = 1.071429.
i = 0.071429, or 7.14% to two decimal places.

Answer: The effective annual rate of return is about 7.14%, higher than the 5% quoted because the fee reduces the amount actually lent.

Exam tips

Write the equation of value before any numbers. Marks are given for the correct equation even if arithmetic slips.
State the rate, time unit and comparison date explicitly, and say whether your IRR is exact or interpolated.
In loan questions, check what was actually received at time 0 and whether fees or charges are paid at the start or the end.
Give a short conclusion with each appraisal answer, such as accept or reject at the stated rate.
In computer-based papers, use a cash flow vector with discount factors, or a goal-seek or root-finding function for IRR, and note the result in your working.

Practice questions from Equation of value

Applications: Project Appraisal and Loans in other exams

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

ACCA Strategic ProfessionalNPV and IRR Investment Appraisal Rules

Applications: Project Appraisal and Loans: frequently asked questions

What is the difference between NPV and IRR?

NPV is a money amount: the present value of the flows at a chosen rate. IRR is a rate: the one at which NPV is zero. NPV tells you the value added, while IRR tells you the project's yield.

How do I calculate the discounted payback period?

Discount each cash flow at the given rate and add them up period by period. The discounted payback period is the first time the running total is zero or positive. If asked for an exact time, interpolate within that period.

How do I calculate the effective rate of return on a loan?

Write the cash flows from one party's view, using the amount actually received at the start and all repayments and charges. Set the present value of the flows to zero and solve for the rate i. Convert to an annual effective rate if the rate found is for a shorter period.

Can a project have more than one IRR?

Yes. If the cash flows change sign more than once, the equation NPV = 0 can have several solutions or none. In that case use NPV at the required rate to decide.