📖 权益测试讲评

Equity Investments

I. Lesson Focus

This lesson provides a comprehensive review of equity valuation techniques, industry and company analysis, and index construction methods. It teaches the underlying domain knowledge with precise formulas, numerical applications, and conceptual distinctions required for the CFA Level I exam. Candidates will master when to apply dividend discount models versus free cash flow to equity models, how to normalize earnings for cyclical companies, the mechanics of different index weighting schemes, and DuPont analysis integration with valuation.

II. The Problem

A candidate in a mock exam confuses the applicability of the free cash flow to equity (FCFE) model with the dividend discount model (DDM) and incorrectly applies a current price-to-earnings (P/E) multiple to value a cyclical firm, resulting in a valuation error exceeding 15%. This lesson systematically reviews core equity investment concepts, formula derivations, real-world applications, and the most frequent conceptual and calculation traps so that candidates can both select the correct approach quickly and compute accurate values under exam pressure.

III. Core Logic and Applicability of Equity Valuation Models

Equity valuation is divided into absolute valuation and relative valuation approaches.

Absolute valuation focuses on a security’s intrinsic value by forecasting future cash flows and discounting them at the required rate of return. Primary models include: - Dividend Discount Models (DDM) - Free Cash Flow to Equity (FCFE) and Free Cash Flow to the Firm (FCFF) models - Residual Income Model

Relative valuation estimates value using multiples derived from comparable companies or precedent transactions. Common multiples are trailing and forward P/E, P/B, EV/EBITDA, and P/S.

The constant-growth DDM (Gordon Growth Model) is expressed as: $$P_0 = \frac{D_1}{r - g}$$ where $r$ is the required return and $g$ is the perpetual growth rate. The model requires $r > g$.

The single-stage FCFE model is: $$P_0 = \frac{FCFE_1}{r - g}$$

Key applicability rule: Use DDM when a company has a stable, predictable dividend policy. Use FCFE when dividend policy is unstable or unreliable but operating cash flows and capital structure are stable. A common trap is applying a two-stage DDM to a high-growth firm without verifying whether the stable-growth stage dividend payout is sustainable.

IV. Industry and Company Analysis Framework

Equity analysis typically follows a top-down or bottom-up process. Industry analysis examines the industry life cycle (introduction, growth, maturity, decline), Porter’s Five Forces, and how industry structure affects pricing power and profit margins.

Company analysis centers on economic moats and the sustainability of competitive advantage. DuPont analysis is a critical link between financial statement analysis and valuation: $$ROE = \frac{NI}{Sales} \times \frac{Sales}{Assets} \times \frac{Assets}{Equity}$$ or Profit Margin × Asset Turnover × Financial Leverage.

When valuing cyclical companies (steel, chemicals), current earnings are distorted by the economic cycle. Analysts must use normalized earnings or an average ROE over the cycle rather than trailing figures.

V. Index Construction and Passive Investment

CFA Level I requires knowledge of equity index weighting methods: - Price-weighted indices (e.g., Dow Jones Industrial Average): weights proportional to share price; stock splits require divisor adjustment. - Equal-weighted indices: require periodic rebalancing to eliminate size bias. - Market-capitalization-weighted indices (e.g., S&P 500): most common; automatically reflect company size but can suffer from concentration in large-cap names. - Fundamentally weighted indices: weights based on earnings, dividends, or book value.

Passive investing offers low costs, but tracking error must be monitored: $$Tracking\ Error = \sqrt{\frac{\sum (R_p - R_b)^2}{n-1}}$$

Worked Cases

Case 1: Gordon Growth Model and Two-Stage DDM

A stock currently pays $D_0 = 2.00$. Dividends are expected to grow at 25% for the next three years, then at a perpetual rate of 6%. The required return is 12%.

Step 1: High-growth dividends
$D_1 = 2.00 \times 1.25 = 2.50$
$D_2 = 2.50 \times 1.25 = 3.125$
$D_3 = 3.125 \times 1.25 = 3.90625$

Step 2: Terminal value at t=3
$D_4 = 3.90625 \times 1.06 = 4.140625$
$P_3 = \frac{4.140625}{0.12 - 0.06} = 69.01$

Step 3: Present values
$PV(D_1) = 2.50 / 1.12 = 2.23$
$PV(D_2) = 3.125 / 1.12^2 = 2.49$
$PV(D_3) = 3.90625 / 1.12^3 = 2.78$
$PV(P_3) = 69.01 / 1.12^3 = 49.12$

Intrinsic value = 2.23 + 2.49 + 2.78 + 49.12 = 56.62

Case 2: FCFE Valuation Combined with DuPont Analysis

A firm reports net income of 12 million, depreciation 4 million, capex 6.5 million, increase in working capital 1.5 million, and net borrowing 1 million. Equity required return is 10%, perpetual growth 4%, and 5 million shares outstanding.

FCFE calculation:
FCFE = NI + Depreciation – Capex – ΔWC + Net Borrowing
= 12 + 4 – 6.5 – 1.5 + 1 = 9 million

Equity value = $\frac{9 \times (1 + 0.04)}{0.10 - 0.04} = \frac{9.36}{0.06} = 156$ million
Per-share value = 156 / 5 = 31.20

DuPont analysis shows ROE = 18% versus an industry average of 12%, supporting the reasonableness of the 4% long-term growth assumption.

Case 3: Relative Valuation with Cyclical Adjustment

Company A reports EPS = 1.2 and trades at a P/E of 18. The industry is cyclical manufacturing. Normalized (cycle-average) EPS over the past five years is 2.1. Comparable firms trade at a normalized P/E of 14.

Incorrect approach: 1.2 × 18 = 21.6 (overvaluation at cycle peak)
Correct approach: Normalized EPS × industry normalized multiple = 2.1 × 14 = 29.4

This illustrates that cyclical firms must be valued on normalized earnings; using peak earnings produces severe overvaluation.

Traps

Key Formulas

• Gordon Growth Model: $P_0 = \frac{D_1}{r-g}$
• Single-stage FCFE: $P_0 = \frac{FCFE_1}{r-g}$
• Two-stage DDM: $P_0 = \sum_{t=1}^{n}\frac{D_t}{(1+r)^t} + \frac{P_n}{(1+r)^n}$
• DuPont ROE: $ROE = Profit\ Margin \times Asset\ Turnover \times Leverage$
• Residual Income: $RI_t = NI_t - (r \times BV_{t-1})$
• Price-weighted divisor adjustment: $New\ Divisor = \frac{Old\ Market\ Value}{New\ Sum\ of\ Prices}$
• Tracking Error: $\sqrt{\frac{\sum(R_p-R_b)^2}{n-1}}$

Practice Questions

Q1. Using the Gordon growth model with $r=11\%$, $g=5\%$, and $D_1=3.30$, the current fair price is closest to:
A. 30.00　B. 55.00　C. 60.00　D. 27.50

Q2. In which situation is the FCFE model most appropriate relative to DDM?
A. Stable and predictable dividend policy
B. Very low dividend payout but strong operating cash flow
C. Mature industry with stable ROE
D. Unprofitable high-tech start-up

Q3. A price-weighted index contains three stocks priced at 40, 80, and 120. If the stock priced at 80 undergoes a 1-for-2 split and the initial index level is 100, the new index level is closest to:
A. 90　B. 100　C. 110　D. 120

Q4. In DuPont analysis, if net profit margin increases, asset turnover decreases, and leverage is unchanged, ROE will:
A. Definitely increase　B. Definitely decrease　C. Cannot be determined　D. Remain unchanged

Q5. For a cyclical company at the peak of an economic expansion, the most appropriate valuation multiple is:
A. Current trailing P/E　B. Forward P/E　C. Normalized P/E　D. Current P/B

Q6. A firm’s 2023 FCFE is 5 million. Growth is expected to be 15% for five years, then 4% perpetually. Equity required return is 10%. The firm’s total equity value is closest to (in millions):
A. 8,500　B. 12,300　C. 15,800　D. 18,200

Q7. Which statement about market-capitalization-weighted indices is incorrect?
A. They automatically reflect company size
B. They do not require periodic rebalancing
C. Small-cap stocks have the same influence as large-cap stocks
D. They can exhibit growth bias

Q8. In the residual income model, if a firm’s ROE persistently equals its equity required return, intrinsic value is closest to:
A. Much higher than book value　B. Equal to book value　C. Much lower than book value　D. Cannot be determined

Answers

Takeaways

• Absolute valuation rests on discounted future cash flows; DDM and FCFE have distinct applicability conditions
• Cyclical companies require normalized earnings or average multiples
• Price-weighted indices need divisor adjustment after splits; market-cap-weighted indices do not
• DuPont analysis bridges financial ratios and valuation inputs
• The Gordon model is invalid if $r \leq g$
• In the residual income model, persistent ROE equal to the cost of equity implies value equals book value