📖 回收期法（Payback Period）

Corporate Finance

I. Lesson Focus

This lesson explains the payback period method, including both the conventional (undiscounted) payback period and the discounted payback period. Candidates must be able to calculate both versions, apply the decision rule, recognize the method’s serious limitations, and understand why it is used only as a supplementary tool to NPV and IRR. The focus is on precise interpolation calculations, the impact of discounting, and common conceptual traps tested in CFA Level I.

II. The Problem

A manufacturing company is considering a new production line requiring an initial outlay of CNY 8 million. The project is expected to generate uneven cash inflows over the next five years. The CFO wants a quick, intuitive answer to “How long will it take to recover the initial investment?” without immediately performing a full NPV analysis that requires selecting an appropriate discount rate. The payback period directly addresses this liquidity and risk question, making it a common supplementary capital budgeting tool. However, its simplicity comes with significant theoretical weaknesses that candidates must fully understand.

III. Concept of the Payback Period

The payback period is the time required for the cumulative net cash flows from a project to equal the initial investment. It remains one of the oldest and simplest capital budgeting techniques.

Calculation Logic
Add the project’s net cash flows (NCF) sequentially until the cumulative total turns from negative to positive. The fractional year in the final period is found using linear interpolation.

Conventional Payback Period Formula: $$ \text{Payback Period} = \text{Years before full recovery} + \frac{\text{Unrecovered amount at start of final year}}{\text{Cash flow during final year}} $$

Advantages: - Simple to calculate and easy to communicate - Implicitly considers liquidity risk (shorter payback = faster cash recovery) - Useful when cash flow forecasts are uncertain or project life is short

Major Disadvantages (high-frequency exam points): - Ignores all cash flows after the payback period (ignores long-term profitability) - Conventional version ignores the time value of money - The maximum acceptable payback benchmark is arbitrary - Can lead to incorrect capital allocation by rejecting high-NPV, longer-term projects

IV. Discounted Payback Period

To address the failure to consider time value of money, the discounted payback period discounts each cash flow at the project’s required rate of return before accumulating.

Definition: The number of years needed to recover the initial investment using the present values of the project’s cash flows.

Calculation Steps: 1. Discount each period’s cash flow: $PV_t = \frac{CF_t}{(1+r)^t}$ 2. Accumulate the discounted cash flows 3. Identify the year in which the cumulative PV turns positive and interpolate the fractional year

Discounted Payback Period Formula: $$ \text{Discounted Payback Period} = \text{Years before full recovery} + \frac{\text{Unrecovered discounted amount}}{\text{Discounted cash flow in final year}} $$

The discounted payback period is always longer than (or equal to) the conventional payback period when the discount rate is positive.

V. Decision Rule

• Accept the project if its payback period is less than the firm’s maximum acceptable payback period.
• Reject if longer.
• When ranking mutually exclusive projects, prefer the one with the shorter payback (but NPV or IRR must still be the primary criterion).

Critical CFA Perspective: The payback method must not be used in isolation. It should supplement NPV and IRR. Exam questions frequently require candidates to explain why payback can lead to suboptimal decisions and why NPV is theoretically superior.

Worked Cases

Case 1: Conventional Payback — Even Cash Flows

Project requires CNY 10 million initial investment and generates CNY 3 million annually for 5 years. The firm’s maximum acceptable payback is 3 years.

Solution: Cumulative recovery: - End Year 1: 3m (remaining 7m) - End Year 2: 6m (remaining 4m) - End Year 3: 9m (remaining 1m) - Year 4 cash flow: 3m

$$ \text{Payback Period} = 3 + \frac{1}{3} = 3.33 \text{ years} $$

Decision: 3.33 > 3.0 → Reject the project.

Case 2: Conventional Payback — Uneven Cash Flows

Initial investment CNY 8 million. Cash flows (CNY millions): Year 1: 2.5, Year 2: 3.0, Year 3: 3.5.

$$ \text{Payback Period} = 2 + \frac{2.5}{3.5} = 2 + 0.714 = 2.714 \text{ years} $$

Case 3: Discounted Payback Period

Initial investment CNY 10 million, required return 10%. Annual cash flow CNY 4 million for 4 years.

$$ \text{Discounted Payback} = 3 + \frac{0.053}{2.732} \approx 3.019 \text{ years} $$

Note that the conventional payback would be exactly 2.5 years, illustrating how discounting lengthens the period.

Traps

Key Formulas

• Conventional Payback Period = Years before recovery + (Remaining amount / Next year’s cash flow)
• Discounted Payback Period = Years before recovery + (Remaining discounted amount / Next year’s discounted cash flow)
• Discounted Payback ≥ Conventional Payback (when r > 0)
• Payback method ignores all cash flows occurring after the payback date
• Decision rule: Accept if Payback < Management’s maximum acceptable payback period

Practice Questions

Q1. A project requires an initial investment of $500 and produces cash flows of $180 per year for the first three years and $150 in Year 4. The conventional payback period is closest to:
A. 2.78 years B. 3.00 years C. 3.22 years D. 3.56 years

Q2. Which statement about the discounted payback period is most accurate?
A. It is always shorter than the conventional payback period
B. It accounts for the time value of money
C. It includes all cash flows over the project’s life
D. It does not require a discount rate

Q3. A project has an initial outlay of $2,000 and a 12% discount rate. The cumulative discounted cash flows are –$180 at the end of Year 3 and the Year-4 discounted cash flow is $620. The discounted payback period is:
A. 3.19 years B. 3.29 years C. 3.45 years D. 4.00 years

Q4. The payback period method’s disadvantages do not include:
A. Failure to consider the time value of money
B. Ignoring cash flows after the payback period
C. Excessive computational complexity
D. Arbitrary maximum payback benchmark

Q5. Which statement regarding payback and NPV is most accurate?
A. Projects with higher NPV always have shorter paybacks
B. Payback can be used as a standalone decision criterion
C. Payback is useful as a supplement to NPV when liquidity risk is a concern
D. Discounted payback is always shorter than conventional payback

Q6. Project A requires $1,000,000 and has a conventional payback of 2.4 years. Project B requires $1,200,000 and has a payback of 2.1 years. The firm’s maximum acceptable payback is 2.3 years. The firm should:
A. Accept A and reject B
B. Accept B and reject A
C. Accept both
D. Reject both

Q7. At the end of Year 3 the cumulative discounted net cash flow is –$50. The discounted cash flow in Year 4 is $400. The fractional portion of Year 4 in the discounted payback calculation is:
A. 0.125 B. 0.875 C. 0.050 D. 0.400

Q8. Which situation best illustrates the limitation of the payback method?
A. Two projects have identical paybacks but very different NPVs
B. Cash flows are front-loaded
C. The discount rate changes during the project
D. Project life is shorter than the calculated payback

Answers

Takeaways

• Conventional payback ignores time value; discounted payback corrects this but remains imperfect
• Decision rule is “shorter than the firm’s cutoff” but payback must never be used alone
• Interpolation for the fractional year is a frequent calculation point
• The most serious flaw is complete omission of all cash flows occurring after the payback date
• Discounted payback is always longer than conventional payback when r > 0
• CFA curriculum stresses that payback is only a supplementary tool to NPV and IRR for assessing liquidity and risk