Finance-Theory · Unit 21 · Video 2 · Interactive Practice

The Profitability Index: Right Sign, Wrong Size

IKey Formulas

FormulaNameWhat it says
PI=PV−CF0=PVI0\text{PI} = \dfrac{\text{PV}}{-CF_0} = \dfrac{\text{PV}}{I_0}Profitability indexGross PV of the future cash flows CF1,CF2,…CF_1, CF_2, \dots per dollar of initial investment
NPV=PV−I0=I0 (PI−1)\text{NPV} = \text{PV} - I_0 = I_0\,(\text{PI} - 1)Same signI0>0I_0 > 0, so PI>1  ⟺  NPV>0\text{PI} > 1 \iff \text{NPV} > 0: identical accept/reject for one independent project
PI−1=NPVI0\text{PI} - 1 = \dfrac{\text{NPV}}{I_0}NPV per dollarRanking by it maximizes total NPV only for divisible projects under a single budget limit

Key Insight: A ratio keeps the sign of NPV and drops its size. For accept/reject that loss is harmless; for choosing between projects, or for money a strategy cannot absorb, you need the dollar amount, which is NPV.

IIVisualization 1 — Why PI and NPV Always Agree on the Sign

Every project sits on the line NPV=I0(PI−1)\text{NPV} = I_0(\text{PI} - 1), which crosses zero exactly at PI=1\text{PI} = 1.

IIIVisualization 2 — NPV Is an Area: Width I0I_0, Height PI−1\text{PI} - 1

PI ranks the video's two projects by height; NPV ranks them by area, and scale is the width.

IVVisualization 3 — Capital Rationing: Ranking by PI Under a Budget

Filling a budget in PI order maximizes total NPV when projects are divisible, but not when they come whole.

💡 The PI order is exactly right only under one budget limit with divisible projects; with several limits (say, this year's and next year's budgets) a single ratio cannot rank the projects either.

VQuiz Questions

Problem 1 · Compute the Profitability Index

Given: an independent project costs $1,500 today and pays $700 at the end of year 1 and $1,200 at the end of year 2. The cost of capital is r=12%r = 12\% a year. Find the profitability index and the decision.

✅ Correct! PV ≈ $1,581.63, so PI ≈ 1.05 > 1, and NPV ≈ $81.63 > 0: the two rules agree, as they must.
❌ Not quite. 1,900/1,500≈1.271{,}900 / 1{,}500 \approx 1.27 adds up the cash flows without discounting them. The numerator of PI is the present value at 12%12\%.
❌ Not quite. 0.050.05 is NPV/I0=PI−1\text{NPV}/I_0 = \text{PI} - 1, the NPV per dollar. PI itself is PV/I0\text{PV}/I_0, which is one more than that, and it is above 1.
❌ Not quite. That is I0/PVI_0/\text{PV}, the ratio upside down. PI puts the gross PV of the future cash flows on top: PV/I0\text{PV}/I_0.
❌ Not quite. Discount each future cash flow at 12%12\%, add them for PV, then divide by I0=−CF0I_0 = -CF_0.
Show solution

Gross present value of the future cash flows (nothing subtracted):

PV=7001.12+1,2001.122≈625+956.63=1,581.63\text{PV} = \frac{700}{1.12} + \frac{1{,}200}{1.12^2} \approx 625 + 956.63 = 1{,}581.63

The initial investment is I0=−CF0=1,500I_0 = -CF_0 = 1{,}500, so

PI=PVI0≈1,581.631,500≈1.054\text{PI} = \frac{\text{PV}}{I_0} \approx \frac{1{,}581.63}{1{,}500} \approx 1.054

PI>1\text{PI} > 1, so accept. Check: NPV=PV−I0≈\text{NPV} = \text{PV} - I_0 \approx $81.63 >0> 0, and I0(PI−1)=I0⋅PVI0−I0=PV−I0I_0(\text{PI} - 1) = I_0 \cdot \frac{\text{PV}}{I_0} - I_0 = \text{PV} - I_0 is the same number.

Problem 2 · Two Mutually Exclusive Projects

Given: you can take only one of two projects; the cost of capital is r=12%r = 12\%. Project A costs $1,500 today and pays $2,240 in one year. Project B costs $4,000 today and pays $5,600 in one year. Which should you take, and why?

✅ Correct! When you can take only one, you want the one that adds more dollars of value. B is smaller per dollar but bigger in total.
❌ That is the PI rule's mistake. A has the higher ratio, but NPV=I0(PI−1)\text{NPV} = I_0(\text{PI} - 1): B's $4,000 at 0.250.25 per dollar is worth more than A's $1,500 at about 0.330.33 per dollar.
❌ Not quite. PI > 1 tells you each project is acceptable on its own, but the projects are mutually exclusive: you must choose one.
❌ Not quite. A rate of return is a ratio too, and it drops the scale just as PI does. Compare the dollar amounts of value created.
Show solution

Project A: PV=2,240/1.12=2,000\text{PV} = 2{,}240/1.12 = 2{,}000, so NPV=2,000−1,500=500\text{NPV} = 2{,}000 - 1{,}500 = 500 and PI=2,000/1,500≈1.33\text{PI} = 2{,}000/1{,}500 \approx 1.33.

Project B: PV=5,600/1.12=5,000\text{PV} = 5{,}600/1.12 = 5{,}000, so NPV=5,000−4,000=1,000\text{NPV} = 5{,}000 - 4{,}000 = 1{,}000 and PI=5,000/4,000=1.25\text{PI} = 5{,}000/4{,}000 = 1.25.

PI ranks A first; NPV ranks B first, by $500. Choose B. Another view: the extra money B needs, $2,500 today, brings an extra $3,360 next year, and 3,360/1.12−2,500=500>03{,}360/1.12 - 2{,}500 = 500 > 0, so the additional scale is itself worth taking.

Problem 3 · Capital Rationing With Whole Projects

Given: a $4,000 capital budget at r=12%r = 12\% and three independent projects, each taken whole or not at all: X needs $1,500 with PI=1.40\text{PI} = 1.40; Y needs $2,500 with PI=1.30\text{PI} = 1.30; Z needs $4,000 with PI=1.35\text{PI} = 1.35.

Filling the budget in PI order (skipping any project that no longer fits), what total NPV do you get?

Which choice gives the most total NPV within the budget?

✅ Correct! The PI order buys $1,350 of NPV; Z alone buys $1,400. With whole projects the highest ratios need not add up to the most dollars.
❌ Not quite. After X, $2,500 is left: Z does not fit, but you skip it and move on to Y, which does.
❌ Not quite. $1,475 takes X plus five-eighths of Z, which is what PI order gives if projects were divisible. Here each project comes whole.
❌ Not quite. X and Z together need $5,500, more than the $4,000 budget.
❌ That is the PI order's answer. X and Y give $600 + $750 = $1,350; another combination that fits the budget gives more.
❌ Not quite. X has the highest PI, but alone it uses $1,500 and leaves $2,500 idle, for an NPV of only $600.
❌ Not quite. Convert each PI to dollars with NPV=I0(PI−1)\text{NPV} = I_0(\text{PI} - 1), then add the NPVs of the projects the PI order takes.
❌ Not quite. Compare the total NPV of every combination that fits in $4,000.
Show solution

Dollar NPVs from NPV=I0(PI−1)\text{NPV} = I_0(\text{PI} - 1): X: 1,500×0.40=6001{,}500 \times 0.40 = 600; Y: 2,500×0.30=7502{,}500 \times 0.30 = 750; Z: 4,000×0.35=1,4004{,}000 \times 0.35 = 1{,}400.

PI order is X (1.401.40), Z (1.351.35), Y (1.301.30). Take X, leaving $2,500; Z needs $4,000, so skip it; Y needs $2,500, so take it. Total NPV: 600+750=1,350600 + 750 = 1{,}350.

Every combination that fits: X alone $600, Y alone $750, Z alone $1,400, X and Y $1,350 (X and Z, Y and Z, or all three exceed $4,000). The best is Z alone, with $1,400.

If the projects were divisible, PI order would be right: X, then $2,500 of Z at 0.350.35 per dollar, for 600+875=1,475600 + 875 = 1{,}475.

Problem 4 · The Assumption Behind a Ratio

Given: a team finds a one-year project in a narrow niche: invest $1,500 now and receive $2,016 in one year. At r=12%r = 12\%, PV is $1,800, so PI=1.2\text{PI} = 1.2 and NPV is $300. They now propose putting $4,000 into the same niche and report an NPV of $4,000 × 0.2 = $800. What does that $800 figure assume?

✅ Correct! Multiplying NPV per dollar by a bigger I0I_0 assumes the same cash per dollar at any size. A narrow niche may not absorb the extra money, and then the payoff per dollar falls.
❌ Not quite. That equivalence is an identity, true whenever I0>0I_0 > 0; it says nothing about how the payoff changes with size.
❌ Not quite. Treating them as alternatives would be a reason to compare their NPVs, not an assumption behind the $800.
❌ Not quite. The $800 keeps r=12%r = 12\% and the payoff per dollar unchanged; a rising cost of capital would lower the figure, not produce it.
Show solution

At $1,500: PV=2,016/1.12=1,800\text{PV} = 2{,}016/1.12 = 1{,}800, PI=1,800/1,500=1.2\text{PI} = 1{,}800/1{,}500 = 1.2, and NPV=300=1,500×0.2\text{NPV} = 300 = 1{,}500 \times 0.2.

The $800 multiplies the same NPV per dollar, PI−1=0.2\text{PI} - 1 = 0.2, by $4,000. That is right only if the niche still pays 2,016/1,500=1.3442{,}016/1{,}500 = 1.344 per dollar a year later when $4,000 goes in, that is, if scale does not matter. Expressing an opportunity as a ratio (PI, or a rate of return) builds in exactly this assumption. For a narrow, capacity-constrained niche it is likely false: the more money goes in, the less each dollar earns.

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