📖 获利指数（PI）

Corporate Finance

I. Lesson Focus

This lesson explains the profitability index (PI), its calculation, interpretation, and application in capital budgeting decisions. Particular emphasis is placed on its relationship with NPV, its advantages under capital rationing, and the ranking conflicts that can arise when projects differ in scale or when investment capital is limited. Candidates must be able to compute PI, select projects under hard capital constraints, and recognize when PI and NPV give conflicting recommendations.

II. The Problem

A company has five mutually exclusive projects but only CNY 8 million left in this year’s capital budget. Project A requires CNY 6 million upfront and has a present value of future cash flows of CNY 7.5 million. Project B requires CNY 3 million and has a present value of CNY 4.2 million. Although Project A has a higher absolute NPV, accepting it prevents taking B plus two smaller projects that together may create more total value. The profitability index (PI) solves exactly this real-world problem: when capital is scarce, how do we rank projects by the value created per unit of scarce capital to maximize total shareholder wealth?

III. Basic Concept of the Profitability Index

The profitability index (PI), also called the present value index, is defined as:

$$PI = \frac{\text{Present value of future cash flows}}{\text{Initial investment}}$$

An equivalent and often more convenient formula is:

$$PI = 1 + \frac{NPV}{\text{Initial investment}}$$

Decision rules for independent projects: - PI > 1: accept (NPV > 0) - PI = 1: indifferent (NPV = 0) - PI < 1: reject (NPV < 0)

The key strength of PI is that it is a relative measure. It tells us how many dollars of discounted cash inflow each dollar invested generates. Under capital rationing, PI allows managers to rank projects by “bang for the buck” and select the combination that maximizes total NPV within the spending limit.

IV. Relationship Between PI and NPV

NPV is an absolute dollar amount; PI is a ratio. For independent projects with no capital constraint, the two metrics always give identical accept/reject decisions. Conflicts arise in two common situations examined on the CFA exam:

1. Projects of different sizes (different initial outlays)
2. Hard capital rationing, forcing choice among multiple positive-NPV projects

In these cases, the project with the highest PI does not necessarily have the highest NPV, but it delivers more value per dollar invested. CFA questions frequently test the conflict between “ranking by PI” versus “selecting the highest-NPV feasible combination.”

V. Calculation Steps for PI

1. Discount all future cash flows to present using the project’s required rate of return (risk-adjusted discount rate).
2. Divide that present value by the initial net cash outflow (absolute value of the time-zero outflow).
3. If additional capital outlays occur later, discount those outflows and add their present value to the denominator.

Important: The denominator must reflect only the initial net cash outflow, not arbitrary later expenditures.

VI. Applying PI under Capital Rationing

Under a hard spending ceiling: - Compute PI for every project. - Rank projects in descending order of PI. - Select projects sequentially until the budget is exhausted. - Because the last dollar may not fund an entire project, always verify total NPV across feasible combinations; mechanical ranking can miss the globally optimal mix.

The curriculum stresses that PI ranking is a useful first screen but must be followed by explicit comparison of total NPV of feasible bundles.

Worked Cases

Case 1: Independent Project Decision (Basic Calculation)

A project requires an initial investment of CNY 5 million and is expected to generate CNY 2.2 million per year for three years. The required return is 10%.

Present value of inflows = 2.2 × (1/1.1 + 1/1.1² + 1/1.1³) = 2.2 × 2.48685 ≈ CNY 5.471 million.

$$PI = 5.471 / 5 = 1.0942 > 1$$ → accept.

NPV = 5.471 – 5 = 0.471 million.
Cross-check: PI = 1 + (0.471 / 5) = 1.0942, consistent.

Case 2: Project Ranking under Capital Rationing

A firm has a CNY 10 million capital budget and four independent projects (figures in millions):

Ranking by PI: A and B tie at 1.30, then C (1.24), then D (1.20).

A + B + C = CNY 12 million (over budget). A + B + D = CNY 13 million (also over).
Feasible combinations within CNY 10 million include: - A + B = CNY 7 million, total NPV = CNY 2.1 million - A + C = CNY 9 million, total NPV = CNY 2.4 million (highest) - B + C = CNY 8 million, total NPV = CNY 2.1 million

Although pure PI ranking favors A and B, the combination A + C actually delivers the highest total NPV. This illustrates that mechanical PI ranking must be validated against total NPV of feasible bundles.

Case 3: Scale Conflict between Mutually Exclusive Projects

Project X: Initial outlay CNY 1 million, PV of inflows CNY 1.6 million → NPV = 0.6 million, PI = 1.60
Project Y: Initial outlay CNY 8 million, PV of inflows CNY 11.2 million → NPV = 3.2 million, PI = 1.40

With unlimited capital, choose Y because of higher NPV. With a CNY 5 million budget, only X is feasible. Here PI correctly signals that X creates more value per scarce dollar (extra CNY 0.60 vs CNY 0.40 per dollar invested).

Traps

Key Formulas

• $PI = \frac{\text{PV of future cash flows}}{\text{Initial Investment}}$
• $PI = 1 + \frac{NPV}{\text{Initial Investment}}$
• Accept independent project if PI > 1
• Under capital rationing: rank by descending PI, then verify total NPV of feasible combinations
• NPV and PI give identical accept/reject for independent projects with no capital constraint
• PI is especially useful for comparing projects of different scales

Practice Questions

Q1. If a project’s PI equals 1.18, the ratio of its NPV to initial investment equals:
A. 0.18
B. 1.18
C. 0.82
D. Cannot be determined

Q2. In which situation may PI and NPV give conflicting accept/reject decisions?
A. Independent projects with no capital limit
B. Mutually exclusive projects with capital rationing
C. All projects have identical initial investments
D. Projects have conventional cash-flow patterns

Q3. Project A: initial outlay CNY 2 million, PV of cash flows CNY 2.8 million. Project B: initial outlay CNY 6 million, PV of cash flows CNY 7.8 million. With a CNY 4 million capital budget, which project should be chosen first?
A. Project A
B. Project B
C. Both
D. Neither

Q4. A project has an initial outflow of CNY 8 million and annual inflows of CNY 2.5 million for five years. Discount rate is 12%. Its PI is closest to:
A. 1.13
B. 1.27
C. 0.89
D. 1.41

Q5. The main advantage of using PI to rank projects under capital rationing is that it:
A. Guarantees selection of the single project with highest NPV
B. Maximizes value created per unit of scarce capital
C. Automatically resolves mutually exclusive conflicts
D. Eliminates the need to calculate NPV

Q6. If two projects have identical PI but different sizes and capital is limited, the analyst should:
A. Always choose the larger project
B. Always choose the smaller project
C. Compare total NPV of feasible combinations
D. Choose randomly

Q7. Which statement about PI is least accurate?
A. PI is a relative measure
B. PI > 1 is equivalent to NPV > 0
C. PI is useful for comparing projects of different sizes
D. The denominator of PI is always the initial investment and is independent of the discount rate

Q8. With a CNY 5 million budget, projects are available: P (outlay 3m, PI=1.45), Q (outlay 2.5m, PI=1.38), R (outlay 4m, PI=1.22). The best combination is:
A. P and Q
B. Only R
C. P and R
D. Q and R

Answers

Takeaways

• PI = PV(future cash flows) / Initial outlay = 1 + NPV / Initial outlay
• PI > 1 is equivalent to NPV > 0 for independent projects
• Under capital rationing, rank by PI but always validate with total NPV of feasible bundles
• PI is particularly useful when comparing projects of different investment sizes
• Always apply the risk-adjusted discount rate specific to the project
• Mechanical ranking can miss the globally optimal combination; enumerate feasible sets when necessary