📖 估值周测（15 题）

Equity Investments

I. Lesson Focus

This lesson consolidates all major equity valuation techniques required for the CFA Level I curriculum. Candidates must master the selection and application of absolute valuation models (DDM, FCFE, Residual Income) and relative valuation multiples (P/E, P/B, PEG), understand the economic relationships between ROE, growth, payout, and required return, and be able to identify and correct common modeling errors under exam pressure. The focus is on rigorous calculation, justified model choice, and sensitivity to input assumptions.

II. The Problem

An investor is evaluating three listed companies: a high-growth technology firm, a stable dividend-paying consumer staples company, and a cyclical manufacturing firm. The investor must select the most appropriate valuation model (DDM, FCFE, or relative valuation) and determine whether each stock is overvalued or undervalued at its current market price. Relying solely on raw P/E or P/B ratios without adjusting for differences in growth, risk, or earnings normalization frequently leads to incorrect conclusions. This weekly quiz reviews the underlying theory, formulas, and practical application of equity valuation models through 15 targeted questions, ensuring candidates can defend their valuation choices with quantitative rigor.

III. Overview of Equity Valuation Approaches

Equity valuation is divided into absolute valuation (intrinsic value based on discounted cash flows or earnings) and relative valuation (market multiples compared with peers).

Absolute valuation models include: - Dividend Discount Models (DDM) - Free Cash Flow to Equity (FCFE) models - Residual Income (RI) models

Relative valuation uses multiples such as trailing and leading P/E, P/B, EV/EBITDA, and PEG ratios. Adjustments for differences in growth, risk, and accounting policies are essential.

1. Dividend Discount Models (DDM)

The constant-growth Gordon Growth Model is expressed as: $$P_0 = \frac{D_1}{r - g}$$ where $D_1$ is the expected dividend next year, $r$ is the required equity return (often from CAPM), and $g$ is the perpetual growth rate (must be less than $r$ and typically capped near long-term nominal GDP growth).

Multi-stage DDM is used when a company is expected to experience a high-growth phase followed by stable growth.

2. Free Cash Flow to Equity (FCFE) Model

The general form is: $$P_0 = \sum_{t=1}^{n} \frac{FCFE_t}{(1+r)^t} + \frac{FCFE_{n+1}/(r-g)}{(1+r)^n}$$ A simplified calculation of FCFE is:
FCFE = Net Income − Net Capital Expenditure − Change in Working Capital + Net Borrowing.

3. Residual Income Valuation

$$V_0 = B_0 + \sum_{t=1}^{\infty} \frac{RI_t}{(1+r)^t}$$ where $RI_t =$ Net Income$t − (r × Book\ Value{t-1})$.

4. Key Relationships in Relative Valuation

• Leading P/E is generally preferred to trailing P/E.
• PEG = (P/E) / expected growth rate (in percent). A PEG below 1.0 may indicate undervaluation, but industry context matters.
• P/B is particularly useful for asset-heavy or financial firms; theoretical P/B is linked to ROE and growth via $P/B = \frac{ROE - g}{r - g}$.

IV. Model Selection Framework

• Stable, high-payout, mature companies → Gordon DDM is often most appropriate.
• High-growth firms with unstable or zero dividends → Multi-stage DDM or FCFE.
• Cyclical industries → Normalized earnings + relative valuation.
• Financial institutions → Residual income or P/B models perform well because of stable book values.

V. Critical Input Estimation

• Required return $r$: CAPM = Risk-free rate + β × Equity risk premium (+ country risk premium when applicable).
• Perpetual growth $g$: Usually estimated as long-term inflation + real GDP growth; $g$ must be < $r$.
• Sustainable growth: $g = ROE × Retention\ ratio = ROE × (1 − Payout\ ratio)$.

Worked Cases

Case 1: Single-Stage Gordon Growth Model

Company A pays a current dividend $D_0 = 2.00$. Expected perpetual growth is 5%, and the required return is 11%. The stock trades at 28. Calculate intrinsic value and determine whether the stock is mispriced.

Solution: $D_1 = 2.00 × 1.05 = 2.10$
$P_0 = 2.10 / (0.11 − 0.05) = 2.10 / 0.06 = 35.00$

The intrinsic value (35) exceeds the market price (28), so the stock is undervalued by 7.

Case 2: Two-Stage DDM

Company B is expected to grow earnings and dividends at 20% for three years with a 30% payout ratio, then transition to a stable 6% growth rate with a 60% payout. Current EPS = 5.00, $r = 12\%$.

High-growth phase: - Year 1: EPS = 6.00, D1 = 1.80 - Year 2: EPS = 7.20, D2 = 2.16 - Year 3: EPS = 8.64, D3 = 2.592

Terminal value at t=3: D4 = 8.64 × 1.06 × 0.60 = 5.50
$P_3 = 5.50 / (0.12 − 0.06) = 91.67$

Present value: PV of D1–D3 + PV of P3 ≈ 75.84

Intrinsic value is approximately 75.84.

Case 3: Residual Income and P/B Linkage

Company C has book value per share $B_0 = 20$, ROE = 18%, $r = 12\%$, $g = 6\%$.

Calculation: RI1 = (0.18 − 0.12) × 20 = 1.20
$V_0 = 20 + 1.20 / (0.12 − 0.06) = 20 + 20 = 40$

Theoretical P/B = 40 / 20 = 2.0, which matches the formula $P/B = 1 + (ROE − r)/(r − g)$.

Traps

Key Formulas

• Gordon Growth: $P_0 = \frac{D_1}{r-g}$
• Sustainable growth: $g = ROE × (1 − \text{Payout Ratio})$
• Residual Income value: $V_0 = B_0 + \frac{(ROE - r)B_0}{r - g}$
• Justified P/B: $P_0/B_0 = \frac{ROE - g}{r - g}$
• Justified leading P/E (constant growth): $\frac{P_0}{E_1} = \frac{(1-b)(1+g)}{r-g}$
• FCFE relation: FCFE = FCFF − Interest(1−t) + Net Borrowing

Practice Questions

Q1. Using the Gordon model with $D_1 = 3.0$, $r = 10\%$, $g = 4\%$, the intrinsic value is closest to:
A. 45 B. 50 C. 75 D. 30

Q2. Which company is most suitable for a single-stage dividend discount model?
A. High-growth biotech firm B. Stable high-dividend utility C. Highly cyclical steel producer D. Non-dividend-paying internet firm

Q3. A firm has ROE = 16%, $r = 10\%$, $g = 5\%$. Using the residual income model, its justified P/B is closest to:
A. 1.22 B. 1.83 C. 2.20 D. 3.00

Q4. In a two-stage FCFE model, the most likely analyst error is:
A. Using a higher discount rate in the stable phase B. Using a lower beta in the high-growth phase C. Assuming a lower payout ratio in the stable phase than in the high-growth phase D. Using historical rather than forward-looking beta

Q5. A stock has a P/E of 18 and expected earnings growth of 12%. The industry average PEG is 1.6. The stock is most likely:
A. Significantly overvalued B. Undervalued C. Fairly valued D. Cannot be determined

Q6. The residual income model is most appropriate for which type of company?
A. Asset-light, high-R&D technology firm B. Asset-heavy bank with stable ROE C. Highly cyclical commodity producer D. Zero-dividend high-growth company

Q7. Given EPS0 = 4, retention ratio b = 0.4, ROE = 15%, $r = 11\%$, the justified leading P/E under constant growth is closest to:
A. 9.09 B. 10.91 C. 12.73 D. 15.00

Q8. Which statement about relative valuation is least accurate?
A. The method implicitly assumes comparable risk and growth profiles.
B. EV/EBITDA is independent of capital structure.
C. P/B can be greater than 1 even if ROE < required return.
D. For cyclical industries, current (un-normalized) EPS should be used.

Answers

Takeaways

• Absolute valuation ultimately rests on discounted cash flows or economic profit; DDM, FCFE, and residual income models are mathematically linked and should give consistent values when inputs are aligned.
• The Gordon model strictly requires $g < r$; perpetual growth is typically 3–6%.
• Residual income and justified P/B are tightly connected through the formula $P/B = 1 + (ROE − r)/(r − g)$.
• Relative multiples must be adjusted for differences in growth, risk, and accounting quality; mechanical application without adjustments is a major trap.
• Cyclical companies demand normalized rather than reported earnings to avoid valuation at cyclical extremes.
• Model choice depends on dividend policy, growth trajectory, and industry characteristics; candidates should be prepared to justify the selected approach and cross-verify results across methods.