📖 估值倍数：P/E 比率

Equity Investments

I. Lesson Focus

This lesson examines the most widely used relative valuation multiple — the price-to-earnings (P/E) ratio. Candidates must master the calculation of both trailing and leading P/E, derive the theoretical P/E from the Gordon growth model, adjust for non-recurring items, compare P/E across firms while considering differences in growth, risk, and accounting policies, and recognize when the P/E is inappropriate (e.g., negative earnings). The focus is on practical application and common CFA traps related to mis-specification of the earnings denominator and failure to normalize earnings.

II. The Problem

You are comparing two firms in the same industry. Company A trades at $50 with trailing EPS of $2.50. Company B trades at $36 with trailing EPS of $1.80. Which stock is cheaper? Raw share prices cannot be compared directly. The P/E ratio links price to earnings capacity, telling an investor how many dollars the market is willing to pay for each dollar of earnings. It is the quickest relative-valuation tool in equity analysis. However, failing to distinguish trailing from leading P/E, ignoring non-recurring items, or neglecting differences in expected growth and risk leads to completely wrong conclusions — a recurring CFA Level I testing point.

III. Basic Concepts and Classifications of P/E Ratios

The price-to-earnings ratio is defined as:

$$ P/E = \frac{P_0}{EPS} $$

It is classified into two types according to the time horizon of the earnings figure:

1. Trailing P/E
   Uses reported EPS over the past 12 months (trailing twelve months, TTM).
   Advantage: objective and verifiable.
   Disadvantage: reflects historical performance only and ignores future growth prospects.

2. Leading P/E (Forward P/E)
   Uses expected EPS for the next 12 months (usually consensus analyst forecasts).
   Advantage: forward-looking.
   Disadvantage: sensitive to forecast error.

Exam questions frequently require candidates to identify which P/E should be used given the data provided and to justify the choice.

IV. Theoretical Foundation — Derivation from the Gordon Growth Model

The constant-growth dividend discount model yields a theoretical expression for P/E that reveals its fundamental drivers.

From the Gordon model:

$$ P_0 = \frac{D_1}{r - g} = \frac{E_1 \times (1 - b)}{r - g} $$

where $b$ is the retention ratio and $1-b$ is the payout ratio.

Dividing both sides by next year’s expected earnings $E_1$ produces the leading P/E:

$$ \frac{P_0}{E_1} = \frac{1 - b}{r - g} $$

The trailing P/E can be expressed as:

$$ \frac{P_0}{E_0} = \frac{(1 - b)(1 + g)}{r - g} $$

Key insight: P/E increases with higher payout and higher sustainable growth, and decreases with higher required return (risk). Therefore, high-growth, high-payout, low-risk companies legitimately trade at higher P/E multiples. This framework is essential for judging whether a “high P/E” is necessarily expensive.

V. Practical Factors Affecting P/E Ratios

• Earnings sustainability: One-time gains, asset impairments, and non-recurring items distort trailing P/E; analysts must compute normalized (adjusted) EPS.
• Accounting policy differences: Inventory methods (FIFO vs. LIFO), depreciation policies, and R&D capitalization versus expensing affect reported EPS and therefore comparability of P/E.
• Growth prospects: High-growth sectors (e.g., technology) typically command higher P/E.
• Risk: Higher beta increases $r$, lowering justified P/E.
• Liquidity and size: Smaller-cap stocks often trade at a liquidity discount, resulting in lower P/E.

VI. Applications and Limitations of P/E

Applications: - Cross-sectional comparison within an industry - Assessment of overall market valuation (e.g., S&P 500 average P/E) - Screening for potentially undervalued stocks (low P/E combined with high growth)

Limitations: - Useless for loss-making companies (negative EPS produces meaningless P/E) - Subject to accounting manipulation - Ignores balance-sheet strength (should be used with P/B) - Highly volatile for cyclical industries across economic cycles

CFA curriculum stresses that P/E is most appropriate for mature, stable-earnings companies and least suitable for loss-making, high-growth start-ups, or deeply cyclical firms.

Worked Cases

Case 1: Calculating and Comparing Trailing and Leading P/E

A company reported EPS of $4.20 last year and is expected to earn $4.62 this year (10% growth). Current price is $78.54, payout ratio is 60%, required return is 9%, and perpetual growth is 4%.

Calculations: - Trailing P/E = 78.54 / 4.20 = 18.70 - Leading P/E = 78.54 / 4.62 ≈ 17.00

Theoretical check using the formula:
Leading P/E (justified) = (1 – 0.4) / (0.09 – 0.04) = 0.6 / 0.05 = 12.0.
The actual leading P/E is higher, implying the market is pricing in either faster growth or lower risk than assumed.

Case 2: Impact of Non-Recurring Items on P/E

Company X reported EPS of $2.80, including a one-time gain on asset sale of $0.60. Analysts estimate normalized EPS at $2.30. Share price is $45.

• Unadjusted trailing P/E = 45 / 2.80 ≈ 16.07
• Normalized trailing P/E = 45 / 2.30 ≈ 19.57

Conclusion: The stock appears cheaper on unadjusted numbers but is actually more expensive once earnings are normalized. Always use sustainable earnings.

Case 3: Cross-Company Comparison with Growth Adjustment (PEG)

Firm A: P/E = 25, expected growth = 15%.
Firm B: P/E = 18, expected growth = 9%.

PEG = P/E ÷ growth rate (in percent)
- A: 25 / 15 ≈ 1.67
- B: 18 / 9 = 2.00

Although A has a higher absolute P/E, its PEG is lower, indicating it is relatively more attractive once growth is considered. This illustrates the classic trap of judging valuation by P/E level alone.

Traps

Key Formulas

• $ \text{Trailing P/E} = \frac{P_0}{EPS_0} $
• $ \text{Leading P/E} = \frac{P_0}{EPS_1} $
• $ \frac{P_0}{E_1} = \frac{1-b}{r-g} $
• $ \frac{P_0}{E_0} = \frac{(1-b)(1+g)}{r-g} $
• PEG = $\frac{P/E}{\text{expected growth rate (\%)}}$
• Normalized EPS = reported EPS – after-tax non-recurring items

Practice Questions

Q1. A stock currently trades at $60. Trailing twelve-month EPS is $3.5 and expected next-year EPS is $4.0. The trailing and leading P/E ratios are respectively:
A. 17.14 and 15.00
B. 15.00 and 17.14
C. 17.14 and 17.14
D. 15.00 and 15.00

Q2. Holding other factors constant, an increase in the dividend payout ratio (decline in $b$) will cause the leading P/E to:
A. Decrease
B. Increase
C. Remain unchanged
D. First increase then decrease

Q3. A firm reports EPS of $2.0 that includes an after-tax restructuring charge of $0.5. Analysts estimate normalized EPS of $2.6. With a share price of $52, the most appropriate trailing P/E uses which EPS figure?
A. $2.0$
B. $2.6$
C. $2.5$
D. Cannot be determined

Q4. For which of the following companies is the P/E ratio least likely to be a useful valuation tool?
A. A mature consumer-goods manufacturer
B. A loss-making biotechnology firm in its high-growth phase
C. A stable utility company
D. A commercial bank

Q5. Firm A has P/E = 22 and expected EPS growth of 12%. Firm B has P/E = 16 and expected growth of 10%. Using the PEG ratio, which firm appears cheaper?
A. Firm A
B. Firm B
C. Both are equally valued
D. Insufficient information

Q6. If the required return $r$ rises from 10% to 12% while $g$ remains 5%, the leading P/E will:
A. Increase
B. Decrease
C. Stay the same
D. Change directionally ambiguous

Q7. Which statement about trailing P/E is most accurate?
A. It better reflects future growth than leading P/E
B. It is always superior to leading P/E because it uses realized data
C. It can be significantly distorted by major one-time restructuring charges
D. It is unaffected by accounting policy choices

Q8. A company trades at $85. Last year’s EPS was $5.0, this year’s expected EPS is $5.5, payout ratio is 50%, $r=10\%$, $g=6\%$. The justified leading P/E from the Gordon model is closest to:
A. 11.0
B. 12.5
C. 15.0
D. 17.0

Answers

Takeaways

• Always match the P/E label (trailing vs. leading) to the exact earnings period in the denominator
• The justified P/E formula $\frac{P_0}{E_1}=\frac{1-b}{r-g}$ explains why growth, payout, and risk drive multiples
• Normalize earnings by removing non-recurring items before computing trailing P/E
• P/E cannot be used for companies with negative earnings; switch to P/B or EV/EBITDA
• Never judge cheapness or richness by absolute P/E level alone — compare to peers, history, growth (PEG), and justified model value
• High-growth, high-payout, low-risk companies legitimately deserve higher P/E ratios