THE INTERNAL RATE OF RETURN IS THE DISCOUNT RATE AT WHICH THE NET PRESENT VALUE OF A PROJECT EQUALS ZERO. Σ Cₜ / (1 + IRR)ᵗ = 0 EQUIVALENTLY: the rate at which the present worth of the benefits exactly equals the present worth of the costs. IT IS THE RATE OF RETURN THE PROJECT ITSELF EARNS ON THE CAPITAL STILL INVESTED IN IT — hence "internal": it depends only on the project's own cash flows and on no external rate. THERE IS NO CLOSED-FORM SOLUTION for more than a few periods, because the equation is a polynomial of degree n. IRR IS FOUND BY TRIAL AND ERROR WITH INTERPOLATION, and that procedure is what an exam expects to see: STEP 1 — Guess a rate and compute NPV. STEP 2 — If NPV > 0 the rate is too low; increase it. If NPV < 0 it is too high; decrease it. STEP 3 — Find two rates that BRACKET zero: one with positive NPV, one with negative. STEP 4 — INTERPOLATE LINEARLY: IRR ≈ i₁ + (i₂ − i₁) × NPV₁ / (NPV₁ − NPV₂) where NPV₁ is positive at i₁ and NPV₂ negative at i₂. WORKED — the running example: 100,000 invested, 30,000/yr for 5 years. at i = 10%: NPV = −100,000 + 30,000(3.7908) = +13,724 at i = 20%: NPV = −100,000 + 30,000(2.9906) = −10,282 Both bracket zero, so interpolate: IRR ≈ 10 + (20 − 10) × 13,724 / (13,724 + 10,282) = 10 + 10 × 0.5717 = 15.72% THE EXACT VALUE IS 15.24%. The interpolation overestimates by about half a point BECAUSE THE NPV CURVE IS CONVEX, NOT STRAIGHT — a linear interpolation across a 10-point gap cuts the corner. NARROWING THE BRACKET IMPROVES IT: at 15%: NPV = +565; at 16%: NPV = −1,771 IRR ≈ 15 + 1 × 565/(565 + 1,771) = 15.24% ✓ THE LESSON FOR THE EXAM: BRACKET TIGHTLY. Two rates one or two points apart give a nearly exact answer; a wide bracket does not. THE DECISION RULE: IRR > MARR → ACCEPT IRR = MARR → indifferent IRR < MARR → REJECT

THE MINIMUM ATTRACTIVE RATE OF RETURN (MARR), also called the hurdle rate or cut-off rate, IS THE LOWEST RETURN AN ORGANISATION WILL ACCEPT ON AN INVESTMENT. IT IS SET BY, in ascending order of what it must cover: 1. THE COST OF CAPITAL — what the money costs to raise, whether borrowed or supplied by shareholders (the WACC). THIS IS THE FLOOR: earning less than the capital costs destroys value. 2. A RISK PREMIUM matched to the project's uncertainty. A proven process gets a small premium; a new technology in a new market gets a large one. 3. THE OPPORTUNITY COST — the return on the best available alternative that will be forgone. 4. CAPITAL RATIONING. When there is more good work than money, MARR is raised until the acceptable projects fit the budget. IN THAT SITUATION MARR IS NOT REALLY A COST AT ALL — it is a rationing device. TYPICAL VALUES: 8–12% for low-risk utility investment, 15–20% for industrial projects, 25%+ for high-risk ventures. Development banks commonly use 10–12% for infrastructure in developing economies. THE THREE TRAPS IN USING IRR — each is a standard exam question. ── TRAP 1: MULTIPLE IRRs ──────────────────────────────────── DESCARTES' RULE OF SIGNS: a cash flow series can have AS MANY IRRs AS IT HAS SIGN CHANGES. A CONVENTIONAL project (one outflow followed by inflows) has ONE sign change and therefore ONE IRR — the usual case. A NON-CONVENTIONAL project (a later outflow — a mid-life overhaul, a mine's restoration cost, a nuclear plant's decommissioning) has more. WORKED: cash flows −100,000, +300,000, −220,000. Setting x = 1/(1+r), the equation is −100,000 + 300,000x − 220,000x² = 0 whose two roots give IRR = 27.64% AND IRR = 72.36% BOTH ARE MATHEMATICALLY CORRECT AND NEITHER IS MEANINGFUL. The NPV is negative between them and positive outside, which inverts the usual rule entirely: at 10%: NPV = −9,091 at 25%: NPV = −800 at 30%: NPV = +592 IRR CANNOT BE USED HERE AT ALL. Use NPV, which gives one unambiguous answer at the actual MARR. ── TRAP 2: THE REINVESTMENT ASSUMPTION ────────────────────── IRR IMPLICITLY ASSUMES INTERMEDIATE CASH FLOWS ARE REINVESTED AT THE IRR ITSELF. For a project with a 40% IRR, that assumes every rupee it returns can immediately be reinvested at 40% — WHICH IS USUALLY FALSE, since such opportunities are rare by definition. NPV, by contrast, assumes reinvestment at the DISCOUNT RATE, which is precisely the return actually available elsewhere. NPV'S ASSUMPTION IS THE REALISTIC ONE, and this is the main theoretical argument for preferring it. THE FIX, where IRR must be used: the MODIFIED IRR (MIRR), which compounds the inflows forward at the MARR and then solves for the single rate linking that future value to the investment. ── TRAP 3: RANKING MUTUALLY EXCLUSIVE PROJECTS ────────────── THE MOST IMPORTANT TRAP, because the error looks entirely reasonable. PROJECT A: cost 100,000, returns 30,000/yr for 5 yrs IRR = 15.24%, NPV at 10% = +13,724 PROJECT B: cost 150,000, returns 42,000/yr for 5 yrs IRR = 12.38%, NPV at 10% = +9,213 Here both criteria happen to prefer A. BUT CHANGE THE MARR TO 5% AND THEY DISAGREE: NPV at 5%: A = +29,884 B = +31,838 → B is better IRR still says A (15.24% > 12.38%) → A is better THE CONFLICT ARISES BECAUSE IRR IS A RATE AND IGNORES SCALE. A high percentage on a small investment can create less total value than a lower percentage on a larger one. THE RESOLUTION — INCREMENTAL ANALYSIS, and this is the technique the question is really testing: DO NOT COMPARE IRRs DIRECTLY. Instead, ask whether the EXTRA investment earns its keep. 1. Rank the alternatives by increasing initial cost. 2. Take the cheapest acceptable one as the current champion. 3. Compute the INCREMENTAL cash flow (challenger − champion) and its IRR. 4. IF THE INCREMENTAL IRR > MARR, the extra investment is justified: the challenger becomes champion. Otherwise keep the champion. 5. Repeat against the next alternative. APPLIED TO A AND B: incremental (B − A): −50,000 now, then +12,000/yr for 5 incremental IRR = 6.40% AT MARR = 10%: 6.40% < 10% → REJECT the increment → CHOOSE A. (NPV agrees: A 13,724 > B 9,213) AT MARR = 5%: 6.40% > 5% → ACCEPT the increment → CHOOSE B. (NPV agrees: B 31,838 > A 29,884) INCREMENTAL ANALYSIS AND NPV NOW AGREE AT EVERY RATE. That agreement is the point: 6.40% IS THE CROSSOVER RATE at which the two NPV profiles intersect, and it is exactly the IRR of the difference between them. THE COMPARISON: NPV IRR ─────────────────────────────────────────────────────────── result rupees of value a PERCENTAGE reinvestment assumed at the DISCOUNT RATE the IRR itself (realistic) (often unrealistic) multiple answers? NEVER possible, if signs change ranks mutually exclusive projects? YES, directly ONLY via incremental analysis value additive? YES NO needs MARR beforehand? YES no (only to judge the result) intuitive? less MORE — a percentage needs no context ─────────────────────────────────────────────────────────── THE PROFESSIONAL PRACTICE: DECIDE WITH NPV, REPORT THE IRR. IRR's one genuine advantage is the last row — it does not require the MARR to be fixed before the analysis, so it can be presented to an audience that disagrees about the appropriate hurdle rate.