IAI Actuarial Core Principles · Actuarial Mathematics for Modelling
Project Appraisal for IAI CM1: Chapter Guide
Project appraisal in CM1 means judging a project's cash flows using the equation of value. You find the net present value at a stated interest rate, or the internal rate of return where NPV is zero. You also check payback periods and how borrowing changes the result. Always state your assumptions.
What this chapter covers
Project appraisal asks one question: is a series of cash flows worth accepting? You take the money going out and coming in, put a time value on each payment, and compare. The tools are net present value (NPV), internal rate of return (IRR), payback period and discounted payback period.
The chapter builds on the theory of interest and the equation of value. If you can write NPV(i) = Σ cₜ vᵗ, with cₜ the net cash flow at time t and v = 1 ÷ (1 + i), you have the base for everything else. Cash flows may be discrete or continuous, so you may need to work with integrals as well as sums. For continuous cash flows, NPV = ∫ ρ(t) v(t) dt, where ρ(t) is the rate of cash flow at time t and v(t) = exp(-∫₀ᵗ δ(s) ds).
It also links to later parts of the paper. Borrowing at one rate and investing at another uses the same present value logic that you meet in loans and in pricing and reserving. Practice here sharpens your skill at setting up equations of value, which is useful across CM1.
Project appraisal draws on the equation of value topic (20% of the CM1 syllabus) and on the theory of interest rates (25%). No separate weighting is given for this chapter itself. Questions are usually short to set up but easy to lose marks on through sign errors, wrong time points or missing assumptions. Written questions reward clear working, so a student with a tidy method gets full marks while others lose them. The skills also help in MCQs and in Paper B, where you may solve for IRR numerically.
Project appraisal: topics in the order to study them
- 1Equation of Value and Net Present ValueEverything else uses it. NPV is the equation of value written as a single number at a chosen rate.
- 2Internal Rate of ReturnIRR is the rate that makes NPV zero, so you need NPV fluent first. Then you learn when it exists and when it is unique.
- 3Payback Period and Discounted Payback PeriodThese are simple time-based measures. They are easier once you can discount cash flows and compare them with NPV and IRR.
- 4Project Appraisal with Borrowing and Different Interest RatesThis is the hardest topic. It combines NPV with loans and separate borrowing and lending rates, so study it last.
How to prepare Project appraisal
Work mainly from problems. Each topic is a method, and you learn a method by using it on cash flow lines until setup is automatic.
- Revise discount factors, v, d, δ and the equation of value for both discrete and continuous cash flows.
- Draw a time line for every question. Mark each cash flow with its sign and its time before you write any equation.
- Practise NPV at several rates. Check that NPV falls as i rises for a project with outflow first, then inflows.
- Solve IRR by linear interpolation and by trial and error. Then solve simple cases algebraically, such as a quadratic in v.
- Compute payback and discounted payback from cumulative cash flows. State clearly whether you assume cash flows arrive at points or continuously.
- For borrowing questions, track the loan balance, interest at the borrowing rate and the investment at the lending rate. Then compute NPV or accumulated profit.
- Finish with timed past-paper questions. Write full working, state assumptions and compare your layout with the examiners' report.
Common mistakes in Project appraisal
Using the wrong sign or time for a cash flow
Fix: Draw a time line first. Net the cash flows at each time point and write the sign beside each.
Treating IRR as the only decision rule
Fix: Check that IRR exists and is unique. Compare projects with NPV at a common rate, and note that IRR can mislead for non-conventional cash flows.
Giving an interpolated IRR as exact
Fix: Describe the result as approximate. Use a closer bracketing pair of rates for better accuracy.
Confusing payback with discounted payback
Fix: For discounted payback, convert each cash flow to present value first, then accumulate.
Using one rate for borrowing and investing
Fix: List each rate and what it applies to. Accumulate the loan at the borrowing rate and surplus funds at the lending rate.
Leaving out assumptions
Fix: Write one line on the rate, timing of flows and any interpretation, such as payback measured at year end.
Last-day revision: Project appraisal
- NPV(i) = Σ cₜ vᵗ, where v = 1 ÷ (1 + i) and cₜ is the net cash flow at time t.
- Accept a project if NPV is positive at the chosen rate, assuming cash flows are fixed.
- IRR is the rate i at which NPV(i) = 0.
- IRR may fail to exist or be unique only when cash flows change sign more than once. With exactly one sign change, a unique IRR exists (with i > -100%).
- Linear interpolation gives only an estimate of IRR. Say so in your answer.
- Payback period is the time at which cumulative undiscounted cash flow first reaches zero. It may not exist if undiscounted inflows never recover the outflow.
- With an initial outflow followed only by inflows and i ≥ 0, discounted payback is never shorter than payback. It may not exist if NPV is negative over the project life.
- Payback ignores cash flows after the payback point.
- Net cash flow = inflows minus outflows at the same time point.
- With borrowing, interest is charged at the borrowing rate and investment earns the lending rate.
- Compare projects by NPV at a common rate, not by IRR alone.
- State your assumptions: rate, timing of cash flows, and whether flows are continuous.
Project appraisal practice questions
- A project requires an outlay of Rs 50,000 now and pays Rs 20,000 at the end of each half-year for 18 months (three payments). The cost of ca…
- A project has net cashflows of −₹100 at time 0, +₹230 at time 1 and −₹132 at time 2. Which statement about its IRR is correct?
- A project has cashflows of -₹100 at time 0, +₹230 at time 1 and -₹132 at time 2 (₹ lakh). Which statement about its internal rate of return …
- A project costs Rs 100,000 now and returns Rs 60,000 at the end of year 1 and Rs 60,000 at the end of year 2. Using an effective annual inte…
- A project requires an initial outlay of Rs 10,00,000 and returns Rs 3,00,000, Rs 4,00,000, Rs 5,00,000 and Rs 6,00,000 at the end of years 1…
- An investor pays Rs 10,000 now and receives Rs 5,500 at the end of year 1 and Rs 6,050 at the end of year 2. What is the internal rate of re…
- An investment requires Rs 10,000 now and pays Rs 5,500 at the end of year 1 and Rs 6,050 at the end of year 2. What is the internal rate of …
- A project requires an outlay of Rs 1,000,000 now and returns Rs 1,200,000 exactly two years from now. The firm borrows at an effective annua…
Project appraisal in other exams
The same ground in other exams, if you are preparing for more than one or want another angle on it.
Project appraisal: frequently asked questions
Is project appraisal examined in CM1 Paper A or Paper B?
Paper A is the 3 hour 15 minute written paper. Paper B is a 1 hour 45 minute computer-based exam. Check the current syllabus and past papers to see how this topic is tested in each. Learn the method by hand first, because the setup is the same either way.
How do I solve for IRR in the exam?
Write the equation of value with NPV set to zero. If it reduces to a quadratic in v, solve it directly. Otherwise try two rates around zero NPV and interpolate, and say the result is approximate.
When is IRR unreliable?
IRR may not exist, or may have more than one value, when the cash flows change sign more than once. In such cases use NPV at a stated rate to compare projects.
Which topic in this chapter should I spend the most time on?
Spend the most time on appraisal with borrowing and different interest rates. It combines several ideas and has the most room for errors. NPV and IRR are quicker once your time lines are neat.