📖 估值测试讲评

Equity Investments

I. Lesson Focus

This lesson provides a comprehensive review of equity valuation techniques required for the CFA Level I curriculum. It integrates absolute valuation models (DDM, FCFE, residual income) with relative valuation multiples, model selection criteria, and common calculation pitfalls. Candidates will master formula application, multi-stage modeling, justified multiples, and the critical $g < r$ constraint through detailed instruction and realistic scenarios.

II. The Problem

An analyst is valuing three stocks: Company A is in a high-growth phase best suited for a two-stage DDM, Company B is a mature firm appropriate for the Gordon growth model, and Company C has negative free cash flow with an ROE significantly above its sustainable growth rate. Which valuation model should be selected in each case? How is intrinsic value calculated, and how do we determine whether the current market price represents over- or undervaluation? CFA exams frequently test model selection errors, use of a perpetual growth rate exceeding the required return, confusion between $r$ and $g$, or mixing $D_0$ with $D_1$. This lesson delivers the underlying domain knowledge with formulas, three fully worked numerical cases, a targeted trap table, and eight rigorous practice questions to build mastery.

III. Core Equity Valuation Approaches

Equity valuation is divided into absolute valuation (intrinsic value based on discounted cash flows) and relative valuation (multiples compared with peers). Absolute methods include dividend discount models (DDM), free cash flow to equity (FCFE) models, and residual income (RI) models. Relative methods use ratios such as P/E, P/B, and EV/EBITDA.

Dividend Discount Models (DDM)
Dividends represent the cash flows actually received by shareholders. The Gordon growth model (constant growth) is used for firms with stable perpetual growth:
$$V_0 = \frac{D_1}{r - g}$$
where $D_1 = D_0(1+g)$, $r$ is the required return calculated via CAPM ($r = r_f + \beta(r_m - r_f)$), and $g$ is the perpetual growth rate, typically estimated as $g = ROE \times b$ ($b =$ retention ratio).

The two-stage DDM is appropriate when a firm experiences a high-growth period followed by stable growth:
$$V_0 = \sum_{t=1}^{n} \frac{D_t}{(1+r)^t} + \frac{V_n}{(1+r)^n}$$
where terminal value $V_n = \frac{D_{n+1}}{r - g_s}$ and $g_s$ is the stable growth rate.

Free Cash Flow Models
When dividends are absent or unstable, use FCFE:
$$FCFE = FCFF - Int(1-t) + Net\ Borrowing$$
Stable-growth FCFE valuation: $V_0 = \frac{FCFE_1}{r - g}$.

Residual Income Model
$$V_0 = B_0 + \sum_{t=1}^{\infty} \frac{RI_t}{(1+r)^t}$$
where $RI_t = NI_t - r \times B_{t-1}$. This model is useful when ROE differs from $r$.

Relative Valuation
Common multiples include:
- Forward P/E = $P_0 / EPS_1$
- P/B = $P_0 / BV_0$ (justified P/B = $\frac{ROE - g}{r - g}$)
- PEG = $(P/E) / g$

IV. Model Selection Logic

• Growth profile: high and unstable growth → multi-stage DDM or FCFE; stable mature growth → Gordon model.
• Dividend policy: stable high payout → DDM; low or zero payout → FCFE.
• Data availability: reliable dividend forecasts favor DDM; detailed financial projections favor DCF.
• Critical constraint: $g$ must be less than $r$ in perpetuity; violation renders the model invalid (the most frequent exam trap).

V. Key Relationships and Derivations

• Sustainable growth rate: $g = ROE \times (1 - Dividend\ Payout\ Ratio)$.
• Required return $r$: must be computed via the CAPM formula.
• P/B and ROE linkage: when $ROE > r$, P/B > 1, indicating value creation.
• In the Gordon model, an increase in $g$ raises both the numerator ($D_1$) and denominator ($r-g$); net impact depends on whether ROE exceeds $r$.

Worked Cases

Case 1: Constant-Growth Gordon Model

Company A pays a current dividend $D_0 = \$2.00$, with a perpetual growth rate $g = 5\%$. Beta = 1.1, risk-free rate = 4%, equity risk premium = 6%. Current market price is $45. Calculate intrinsic value and determine whether the stock is overvalued.

Solution Steps:
1. $r = 4\% + 1.1 \times 6\% = 10.6\%$
2. $D_1 = 2.00 \times 1.05 = 2.10$
3. $V_0 = \frac{2.10}{0.106 - 0.05} = \frac{2.10}{0.056} = \$37.50$

Conclusion: Intrinsic value of $37.50 is below the market price of $45; the stock is overvalued and should be sold.

Case 2: Two-Stage DDM

Company B will grow dividends at 20% for the next three years, then transition to a stable 5% growth rate. Current $D_0 = \$1.50$, $r = 12\%$. Calculate current intrinsic value.

Solution Steps:
$D_1 = 1.50 \times 1.20 = 1.80$
$D_2 = 1.80 \times 1.20 = 2.16$
$D_3 = 2.16 \times 1.20 = 2.592$
$D_4 = 2.592 \times 1.05 = 2.7216$ (stable phase begins)
Terminal value at t=3: $V_3 = \frac{2.7216}{0.12 - 0.05} = \$38.88$

$$V_0 = \frac{1.80}{1.12} + \frac{2.16}{1.12^2} + \frac{2.592 + 38.88}{1.12^3} = 1.607 + 1.724 + 29.812 = \$33.143$$

Case 3: Justified P/B and Residual Income Consistency

Company C has book value $BV_0 = \$20$, ROE = 18%, $r = 12\%$, sustainable $g = 6\%$, and trades at a P/B of 2.2. Is the valuation reasonable?

Justified P/B = $\frac{ROE - g}{r - g} = \frac{0.18 - 0.06}{0.12 - 0.06} = \frac{0.12}{0.06} = 2.0$
Observed P/B of 2.2 exceeds the justified multiple, indicating slight overvaluation. The residual income model yields the same conclusion because the excess ROE generates positive RI, but the current price already incorporates part of that value.

Traps

Key Formulas

• Gordon growth: $V_0 = \frac{D_1}{r-g}$
• Sustainable growth: $g = ROE \times (1 - \frac{D}{NI})$
• CAPM: $r = r_f + \beta \times ERP$
• Two-stage terminal value: $V_n = \frac{D_{n+1}}{r-g_s}$
• Justified P/B: $\frac{ROE - g}{r - g}$
• PEG: $\frac{(P_0/EPS_1)}{g \times 100}$
• Residual income: $RI_t = NI_t - (r \times B_{t-1})$

Practice Questions

Q1. Given $r=10\%$, $g=6\%$, and $D_1=\$3.0$, the stock’s intrinsic value is closest to:
A. $30.00　B. $75.00　C. $50.00　D. $37.50

Q2. Which situation is most appropriate for the Gordon growth model?
A. High-growth start-up　B. Stable mature utility company　C. Highly cyclical manufacturer　D. Zero-dividend high-ROE technology firm

Q3. A firm has $\beta=1.2$, $r_f=3.5\%$, equity risk premium $=5.5\%$, $D_0=\$1.8$, $g=4\%$. Its intrinsic value is closest to:
A. $32.4　B. $37.1　C. $41.8　D. $28.5

Q4. In a two-stage DDM, the terminal value after the high-growth phase should be calculated using:
A. The high-growth rate　B. The stable growth rate　C. Zero growth　D. Industry average growth

Q5. With ROE = 16%, $r = 11\%$, $g = 5\%$, the justified P/B ratio is closest to:
A. 1.22　B. 2.20　C. 1.83　D. 0.91

Q6. Which statement about the residual income model is incorrect?
A. It is suitable when ROE ≠ $r$　B. $V_0 = B_0 +$ PV of all future RI
C. When ROE = $r$, value equals book value　D. Future dividends must be forecasted to apply it

Q7. A stock trades at $40 with expected EPS₁ = $4 and $g=8\%$. Its PEG ratio is:
A. 1.25　B. 0.80　C. 5.00　D. 0.125

Q8. When a firm pays no dividends but generates positive FCFE, the most appropriate model is:
A. Gordon DDM　B. FCFE discount model　C. P/E multiple　D. Wait until dividends begin

Answers

Takeaways

• The Gordon model is invalid if $g$ is not permanently below $r$.
• Two-stage models require clear separation of high-growth dividends and stable-growth terminal value.
• Justified P/B is driven by ROE, $r$, and $g$; ROE > $r$ implies P/B > 1.
• Zero or unstable dividend firms should use the FCFE model.
• Always compute $r$ via CAPM using the three inputs: $r_f$, $\beta$, and ERP.
• The most tested traps are $g > r$ violations and confusion between $D_0$ and $D_1$.