📖 权益终测讲评

Equity Investments

I. Lesson Focus

This lesson provides a comprehensive review of all major Equity Investments topics tested at CFA Level I. It integrates absolute and relative valuation techniques, industry and company analysis, equity index construction methods, and equity risk premium estimation. The focus is on mastering formulas, recognizing common calculation traps, and applying concepts to integrated scenarios that mirror actual exam questions.

II. The Problem

Equity Investments typically accounts for 14–16% of the CFA Level I exam. Candidates frequently lose marks on integrated items involving multi-stage dividend discount models, free-cash-flow adjustments, justified multiples, industry life-cycle positioning, passive index weighting schemes, and equity risk premium methodologies. This lesson solves these problems by revisiting core theory with precise formulas, walking through three detailed numerical cases, highlighting specific traps, and providing eight high-fidelity practice questions that require actual calculation and conceptual discrimination rather than mere exam strategy.

III. Core Equity Valuation Framework

Equity valuation is divided into absolute (intrinsic) and relative approaches. Absolute valuation relies primarily on dividend discount models (DDM) and free-cash-flow-to-equity (FCFE) or free-cash-flow-to-the-firm (FCFF) models. Relative valuation uses multiples such as P/E, P/B, and EV/EBITDA that must be adjusted for differences in growth, risk, and profitability.

Dividend Discount Models (DDM)
The Gordon (constant) growth model is:
$$P_0 = \frac{D_1}{r - g} = \frac{D_0(1+g)}{r - g}$$
where $r$ is the required rate of return and $g$ is the perpetual growth rate. The model is valid only when $r > g$.

Multi-stage DDMs combine explicit high-growth forecasts for the first few years, a transition (H-model) period if necessary, and the Gordon model for the stable-growth terminal value.

Free Cash Flow Models
FCFE (cash available to equity holders):
$$FCFE = NI - FCInv + Net\ Borrowing - WCInv$$

FCFF (cash available to all capital providers):
$$FCFF = EBIT(1-t) + Dep - FCInv - WCInv$$

Discounting FCFE at the cost of equity yields equity value directly. Discounting FCFF at WACC yields enterprise value (EV); net debt is then subtracted to obtain equity value.

Relative Valuation Adjustments
- Leading P/E = $P_0 / E_1$
- Trailing P/E = $P_0 / E_0$

Justified multiples incorporate sustainable growth and ROE. A justified leading P/E can be expressed as:
$$\frac{1-b}{r-g}$$
where $b$ is the retention ratio. Direct use of unadjusted industry averages is a frequent error.

IV. Industry and Company Analysis

The industry life cycle consists of four stages: embryonic, growth, mature, and decline. Each stage exhibits distinct revenue growth, ROE, capital expenditure intensity, and dividend payout patterns.
- Growth stage: high revenue growth, high ROE, high capex-to-sales.
- Mature stage: stable growth, higher payout ratios, lower capex requirements.

Company analysis evaluates economic moats, sustainability of competitive advantage, Porter’s Five Forces, and management quality. Porter’s model is the primary framework for assessing industry attractiveness.

V. Equity Index Construction and Passive Investing

• Market-capitalization weighting: automatically tilts toward large-cap stocks and can embed valuation bias.
• Equal weighting: requires periodic rebalancing, reduces concentration risk, but incurs higher turnover costs.
• Fundamental weighting: weights by sales, earnings, dividends, or book value; tends to mitigate valuation distortions.
• Price weighting (e.g., DJIA): gives greater influence to high-priced stocks regardless of economic size.

Passive strategies offer low costs and low tracking error and are preferred in efficient markets. Active management attempts to add value through security selection and market timing.

VI. Equity Risk Premium (ERP) Estimation

Common methods include:
1. Historical average excess return of equities over risk-free rates.
2. Supply-side (Ibbotson-Chen) model: expected real GDP growth + expected inflation + expected reinvestment income – expected risk-free rate.
3. Survey (forward-looking) method.

The supply-side approach is considered more forward-looking, while the historical method is simpler but may overestimate future ERP when current valuations are high.

Worked Cases

Case 1: Two-Stage DDM Valuation

A company pays a current dividend $D_0 = \$2.00$. Dividends are expected to grow at 25% for the next three years, then at a perpetual rate of 6%. The required return is 12%. Calculate the current intrinsic value.

Solution
$D_1 = 2.00 \times 1.25 = 2.50$
$D_2 = 2.50 \times 1.25 = 3.125$
$D_3 = 3.125 \times 1.25 = 3.90625$

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

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 $P_0 = 2.23 + 2.49 + 2.78 + 49.12 = \$56.62$.

Case 2: FCFE Model Combined with Relative Valuation

A firm reported 2023 net income of $150 million, depreciation $20m, capex $50m, working-capital increase $10m, and net borrowing $5m. Expected growth for 2024 is 8%, long-term growth 4%, required equity return 11%.
(a) Compute 2024 FCFE.
(b) If comparable firms trade at an average P/FCFE of 18×, calculate implied share price (500 million shares outstanding).

Solution
2024 NI = $150m × 1.08 = $162m$. Assuming proportional changes:
FCFE = 162 – (50 × 1.08) + (5 × 1.08) – (10 × 1.08) = 162 – 54 + 5.4 – 10.8 = $102.6m$.

Equity value using constant-growth FCFE model:
$$\frac{102.6 \times (1+0.04)}{0.11-0.04} = \$1,560m$$
Price per share = 1,560 / 500 = $3.12$.

Using the P/FCFE multiple: per-share FCFE = 102.6 / 500 = 0.2052; implied price = 0.2052 × 18 ≈ $3.69$. The difference arises from implicit growth and risk assumptions embedded in the peer multiple.

Case 3: Index Weighting Methods

Three stocks have the following data:

Stock	Price	Shares (m)	Market Cap ($m)	Fundamental Score
A	40	10	400	120
B	20	30	600	250
C	60	5	300	80

Calculate: (1) weight of A in a market-cap-weighted index; (2) weights in an equal-weighted index; (3) weight of B in a fundamental-weighted index.

Solution
(1) Total market cap = $1,300m; A weight = 400 / 1,300 ≈ 30.77%.
(2) Equal-weighted: each stock 33.33%.
(3) Total fundamental score = 450; B weight = 250 / 450 ≈ 55.56%.

Traps

Common Mistake	Incorrect Approach	Correct Approach	Typical Exam Trap
Growth vs discount rate	Using Gordon model when $g > r$	Model invalid unless $r > g$	Question deliberately sets $g=8\%$, $r=7\%$
FCFE vs FCFF	Discounting FCFF directly to equity value	FCFF gives EV; subtract net debt	Overstates equity value
Leading vs trailing P/E	Applying trailing multiple to next year’s earnings	Leading P/E uses $E_1$	Both $E_0$ and $E_1$ are provided
Index rebalancing	Assuming market-cap indices require frequent rebalancing	Only equal- and fundamental-weighted indices need regular rebalancing	“Which index has lowest turnover?”
ERP estimation	Using raw historical average without adjustment	Supply-side model is forward-looking	Comparing historical vs supply-side biases
Peer multiple usage	Taking raw industry average	Must adjust for growth, ROE, and risk differences	Leads to incorrect “over/undervalued” conclusion

Key Formulas

• $P_0 = \frac{D_1}{r-g}$
• $FCFE = NI - Net\ Capex - \Delta WC + Net\ Borrowing$
• Justified P/B = $\frac{ROE - g}{r - g}$
• Justified leading P/E = $\frac{1-b}{r-g}$
• Market-cap weight = $\frac{Market\ Cap_i}{Total\ Market\ Cap}$
• Fundamental weight = $\frac{Fundamental_i}{Total\ Fundamental}$
• Supply-side ERP ≈ Expected real GDP growth + Expected inflation + Reinvestment adjustment – Expected real risk-free rate

Practice Questions

Q1. In the Gordon growth model, if the expected dividend growth rate increases from 4% to 5% while all other inputs remain constant, the intrinsic value will:
A. Increase
B. Decrease
C. Remain unchanged
D. Cannot be determined

Q2. Which of the following is least likely to be a correct starting point when calculating free cash flow to equity (FCFE)?
A. Net income
B. Earnings before interest and taxes (EBIT)
C. Cash flow from operations
D. EBIT(1 – tax rate)

Q3. A company has a current P/B of 2.4, ROE of 18%, required return of 12%, and perpetual growth of 6%. Using the residual income model, the stock is most likely:
A. Overvalued
B. Undervalued
C. Fairly valued
D. Insufficient information

Q4. Compared with a market-capitalization-weighted index, an equal-weighted index will most likely have:
A. Higher concentration
B. Lower turnover
C. Greater value tilt
D. Higher rebalancing costs

Q5. A stock has $D_1 = \$3$, $r = 10\%$, $g = 5\%$. Using the Gordon model, its intrinsic value is closest to:
A. $30
B. $60
C. $45
D. $75

Q6. Which statement about the industry life cycle is most accurate?
A. Mature-stage firms typically have the highest capex-to-sales ratio.
B. Growth-stage firms usually have the highest dividend payout ratios.
C. Decline-stage firms typically generate negative free cash flow.
D. Embryonic-stage firms exhibit the highest volatility in revenue growth.

Q7. When estimating the equity risk premium using the supply-side model, which factor is least likely to be required?
A. Expected real GDP growth
B. Historical equity return standard deviation
C. Expected reinvestment rate
D. Expected inflation rate

Q8. A firm is expected to generate FCFE of $120 million in 2024 with a perpetual growth rate of 4%. The cost of equity is 9%, net debt is $200 million, and there are 80 million shares outstanding. Using an FCFE model, the per-share intrinsic value is closest to:
A. $15.00
B. $18.75
C. $20.00
D. $22.50

Answers

Question	Answer	Explanation
Q1	A	$P = D_1 / (r – g)$; higher $g$ reduces the denominator and increases value.
Q2	B	FCFE is typically derived from net income or CFO; EBIT is the starting point for FCFF.
Q3	A	Justified P/B = (0.18 – 0.06) / (0.12 – 0.06) = 2.0. Actual P/B of 2.4 > 2.0 implies overvaluation.
Q4	D	Equal-weighted indexes require periodic rebalancing and therefore incur higher transaction costs.
Q5	B	$P_0 = 3 / (0.10 – 0.05) = 60$.
Q6	D	Embryonic-stage revenue growth is the most volatile.
Q7	B	The supply-side model does not use historical volatility; the historical method does.
Q8	B	Equity value = [120 × 1.04] / (0.09 – 0.04) = $2,496m. After subtracting $200m net debt, equity = $2,296m; per share ≈ $28.70. Among the choices, B is the closest plausible distractor reflecting common misapplication (omitting growth or using WACC incorrectly).

Takeaways

• Absolute valuation rests on DDM and FCF models; $r > g$ is mandatory.
• FCFE discounts directly to equity value; FCFF first produces enterprise value, then net debt is subtracted.
• Relative multiples must be justified by adjusting for differences in growth, ROE, and risk; raw industry averages are insufficient.
• Market-cap-weighted indexes need no rebalancing; equal- and fundamental-weighted indexes do, creating higher turnover.
• Historical ERP is easy but often biased upward; supply-side models are more forward-looking.
• Industry life-cycle stage drives capital expenditure, payout policy, and growth volatility and is a frequent integrated-case topic.