📖 估值综合复习

Equity Investments

I. Lesson Focus

This lesson provides a comprehensive review of equity valuation techniques required for the CFA Level I curriculum. Candidates will master the theoretical foundations, formulas, assumptions, and practical applications of absolute valuation models (DDM and DCF) and relative valuation multiples. Emphasis is placed on model selection criteria, justified multiples derived from fundamentals, terminal value calculations, and the interrelationships among growth, payout, ROE, and required return. The material directly supports both item-set and calculation-based exam questions on determining intrinsic value and identifying mispricing.

II. The Problem

An investor is evaluating whether shares of a stable-growth consumer goods company and a high-growth technology firm are attractively priced. Three primary approaches are available: dividend discount models (DDM), free cash flow to the firm or equity (FCFF/FCFE) discounted cash flow models, and market multiples (P/E, P/B, EV/EBITDA). These methods differ substantially in their cash flow definitions, growth assumptions, discount rates, and terminal value calculations. Incorrect model selection or flawed input parameters (especially perpetual growth rate or cost of equity) can produce valuation errors exceeding 30%. This lesson systematically reviews the formulas, assumptions, calculation sequences, linkages, and pitfalls of each approach so candidates can select the most appropriate model and compute intrinsic value accurately under exam pressure.

III. Primary Equity Valuation Frameworks

Equity valuation is divided into absolute valuation (intrinsic value) and relative valuation (multiples) approaches.

Absolute valuation estimates value as the present value of expected future cash flows or dividends. The two main families are: - Dividend Discount Models (DDM) - Discounted Cash Flow (DCF) models using Free Cash Flow to the Firm (FCFF) or Free Cash Flow to Equity (FCFE)

Relative valuation compares a stock to peer companies or its own history using price or enterprise value multiples.

1. Dividend Discount Models (DDM)

The intrinsic value of a share equals the present value of all expected future dividends.

Zero-growth model (special case of Gordon): $$ V_0 = \frac{D_1}{r} = \frac{D_0(1+g)}{r} \quad (g=0) $$

Constant-growth Gordon Growth Model (most frequently examined): $$ V_0 = \frac{D_1}{r-g} = \frac{D_0(1+g)}{r-g} $$ Requirements: $r > g$, and $g$ must be sustainable (typically close to long-run nominal GDP growth).

Two-stage DDM: $$ V_0 = \sum_{t=1}^{n} \frac{D_t}{(1+r)^t} + \frac{V_n}{(1+r)^n} $$ where terminal value $V_n = \frac{D_{n+1}}{r-g_n}$

Three-stage DDM adds a transitional period during which growth declines linearly toward the perpetual rate; it is appropriate when growth is expected to converge gradually.

2. Free Cash Flow Models

FCFF model values the entire firm, then subtracts net debt to obtain equity value.

$$ FCFF = EBIT(1-t) + Depreciation - FCInv - WCInv $$

Firm value: $$ V_{firm} = \sum_{t=1}^{n} \frac{FCFF_t}{(1+WACC)^t} + \frac{FCFF_{n+1}/(WACC-g)}{(1+WACC)^n} $$ Equity value = Firm value − Net debt

FCFE model values equity directly: $$ FCFE = NI + Depreciation - FCInv - WCInv + Net\ Borrowing $$ $$ V_{equity} = \sum_{t=1}^{n} \frac{FCFE_t}{(1+r)^t} + \frac{FCFE_{n+1}/(r-g)}{(1+r)^n} $$

3. Relative Valuation Using Multiples

Common multiples and their fundamental drivers: - P/E (price-to-earnings): Driven by growth, payout ratio, and cost of equity. Forward (leading) P/E uses next year’s expected EPS; trailing P/E uses current EPS. - P/B (price-to-book): Best for asset-heavy firms or those with stable ROE. Justified P/B rises with ROE. - P/S (price-to-sales): Useful for loss-making or volatile-profit firms. - EV/EBITDA: Eliminates differences in capital structure and depreciation policy; preferred for international or cross-industry comparisons.

Justified multiples are derived from fundamentals using payout, growth, and required return rather than simple peer medians.

IV. Model Selection and Parameter Estimation

• Growth stage analysis: Use single-stage Gordon or stable FCFE for mature firms; apply two- or three-stage models when high growth is expected to normalize.
• Cash flow choice: When dividend policy is unstable or payout is near zero, FCFE is superior to DDM.
• Terminal value: Gordon growth is most common, but $g$ must be realistic. An exit multiple can also be used.
• Cost of equity: CAPM is standard: $r = R_f + \beta(R_m - R_f)$. Add country risk premium or size premium when appropriate.
• Sensitivity: Value is highly sensitive to changes in $g$ and $r$; exam questions frequently test the impact of assumption revisions.

Worked Cases

Case 1: Constant-Growth Gordon DDM

ABC Corp pays a current dividend $D_0 = 2.00$, with perpetual growth $g = 4\%$ and required equity return $r = 9\%$. Calculate intrinsic value.

Solution: $D_1 = 2.00 \times 1.04 = 2.08$ $$ V_0 = \frac{2.08}{0.09-0.04} = \frac{2.08}{0.05} = 41.60 $$

At a market price of 38, the stock is undervalued.

Case 2: Two-Stage FCFE Model

XYZ Corp is expected to grow FCFE at 15% for three years, then at a perpetual 5%. Current $FCFE_0 = 5$, equity cost $r = 12\%$. Compute intrinsic value per share.

Solution: $FCFE_1 = 5 \times 1.15 = 5.75$ $FCFE_2 = 5.75 \times 1.15 = 6.6125$ $FCFE_3 = 6.6125 \times 1.15 = 7.604375$

Terminal value at t=3: $FCFE_4 = 7.604375 \times 1.05 = 7.9846$ $V_3 = 7.9846 / (0.12-0.05) = 114.066$

Present values: $PV_1 = 5.75 / 1.12 = 5.134$ $PV_2 = 6.6125 / 1.12^2 = 5.272$ $PV_3 = 7.604375 / 1.12^3 = 5.418$ $PV_{TV} = 114.066 / 1.12^3 = 81.23$

$V_0 = 5.134 + 5.272 + 5.418 + 81.23 = 97.05$

Case 3: Justified Leading P/E and Relative Valuation

Company A has expected EPS growth $g = 6\%$, payout ratio = 40%, and $r = 10\%$. Calculate its justified leading P/E. If peers trade at an average leading P/E of 18× and Company A currently trades at 16.5×, assess valuation.

Solution: Justified leading P/E = $\frac{(1-b)(1+g)}{r-g} = \frac{0.4 \times 1.06}{0.10-0.06} = 0.424 / 0.04 = 10.6$

Company A’s actual leading P/E (16.5) exceeds its justified multiple (10.6), indicating overvaluation on a fundamental basis despite appearing cheaper than peers at 18×. Caution is warranted.

Traps

Key Formulas

• Gordon DDM: $V_0 = \frac{D_1}{r-g}$
• Constant-growth FCFE: $V_0 = \frac{FCFE_1}{r-g}$
• Justified leading P/E: $\frac{(1-b)(1+g)}{r-g}$
• Justified trailing P/E: $\frac{(1-b)(1+g)(1+g)}{r-g}$
• Sustainable growth: $g = ROE \times b$ (retention ratio)
• $FCFF = FCFE + Int(1-t) - Net\ Borrowing$
• Justified P/B: $\frac{ROE - g}{r - g}$
• Enterprise value = Equity MV + Net Debt

Practice Questions

Q1. Which assumption is least likely when applying the Gordon growth model?
A. Growth rate equals the cost of equity
B. Dividend payout ratio remains constant
C. Perpetual growth rate is less than required return
D. The firm maintains a stable dividend policy

Q2. A company pays $D_0 = 1.5$, $g = 5\%$, $r = 11\%$. Its intrinsic value is closest to:
A. 15.75
B. 17.33
C. 26.25
D. 31.50

Q3. In which situation is the FCFE model most appropriate relative to DDM?
A. Stable 60% dividend payout
B. High-growth tech firm with zero dividend payout
C. Utility with stable cash flows
D. ROE equals cost of equity

Q4. Next year’s expected EPS is 4, payout = 40%, $g = 6\%$, $r = 10\%$. The justified leading P/E is:
A. 10.6
B. 11.24
C. 16.0
D. 17.0

Q5. In a two-stage model, terminal value at period $n$ should be calculated using:
A. Period-$n$ cash flow divided by $(r-g)$
B. Period-$(n+1)$ cash flow divided by $(r-g)$
C. Period-$n$ cash flow multiplied by $(1+g)$
D. Period-$n$ exit multiple only

Q6. When comparing firms with significantly different capital structures, the most appropriate multiple is:
A. P/E
B. P/B
C. EV/EBITDA
D. Dividend yield

Q7. ROE = 15%, cost of equity = 10%, sustainable $g = 6\%$. The justified P/B is closest to:
A. 0.6
B. 1.5
C. 2.25
D. 3.0

Q8. An analyst mistakenly uses current dividend of 2 as $D_1$, $g = 4\%$, $r = 9\%$ and obtains a value of 50. The correct intrinsic value is closest to:
A. 48
B. 52
C. 54
D. 47.92

Answers

Takeaways

• Absolute valuation rests on discounted future cash flows; DDM suits stable high-payout firms while FCFE is preferred when dividends are unreliable.
• The Gordon model demands $g < r$ and a realistic long-term growth rate; multistage models are the most frequently tested format.
• Justified multiples link fundamentals ($g$, payout, $r$, ROE) directly to price ratios and are essential for determining over- or undervaluation.
• Terminal value often represents more than 70% of total value and is extremely sensitive to $g$ and $r$ assumptions.
• Among relative multiples, EV/EBITDA best neutralizes differences in leverage and non-operating items.
• Model selection should be driven by the firm’s growth stage, dividend policy stability, and predictability of cash flows.