Learning Module 12 of Equities 11–14% of the exam

The CAPM, the Market Model and Factor Models

3 lessons. About 19 minutes in cram mode, 90 minutes in full.

Start with lesson 1 Learn with Sensei Read the module summary Practice this module

Lessons

1
Estimating the cost of equity with the CAPM and the market model
calculate 7 min cram, 32 min full
2
Industry betas, the market risk premium and country risk
calculate 7 min cram, 32 min full
3
Arbitrage pricing theory and multi-factor models
describe 5 min cram, 26 min full

What the exam expects you to do

Explain the use of the CAPM and the market model to estimate a company’s required return on equity.Estimating the cost of equity with the CAPM and the market model
Describe the arbitrage pricing theory and the importance of multi-factor models used in equity investing.Arbitrage pricing theory and multi-factor models

Module summary

The whole module on one sheet, organised by lesson: what to memorize, how the ideas fit together, the formulas and the key terms. No paragraphs, just what you need to revise.

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How deeply to learn each part

Know the idea: Understand it in general. It frames other topics, but you will not calculate with it.
Use it: Understand it well enough to solve a question with it. Recognize when it applies.
Memorize: Learn it by heart. The exam will not give it to you.

Estimating the cost of equity with the CAPM and the market model

Known

MemorizeFrom market data to a cost of equity

Step	Formula
Beta	$\beta_i = \rho_{i,m}\,\sigma_i / \sigma_m$
Correlation from beta	$\rho_{i,m} = \beta_i\,\sigma_m / \sigma_i$
Adjusted beta	$w_1 \beta_{raw} + (1 - w_1) \times 1.0$ , often ⅔ and ⅓
Cost of equity	$r_f + \beta_i(E[r_m] - r_f)$

Memory hook Beta = how closely × how wildly.

Know it cold Same country, same date: only beta differs. A higher $r_f$ with $E[r_m]$ fixed lowers the cost of equity when $\beta > 1$ .

MemorizeChoices behind a beta

Choice	What to watch
Regression	Excess returns, or the market model (no risk-free data, similar beta)
Index	Broad and liquid; not a narrow index the stock dominates
Noise	Low $R^2$ , high standard error = imprecise
Horizon	About 100 weekly returns over 2 years is common; fewer points, more noise

Memory hook Index, noise, horizon, adjust.

Know it cold State the method; average the estimates if they disagree.

Key terms

cost of equity Memorize
capital asset pricing model (CAPM) Memorize
security market line (SML) Know the idea
market risk premium (MRP) Memorize
beta Memorize
high-beta stock Memorize
low-beta stock Memorize
excess-return regression Know the idea
market model Memorize
raw beta Know the idea
adjusted beta Memorize
mean reversion of beta Know the idea
market portfolio proxy Know the idea
R-squared (R²) Know the idea
standard error of beta Know the idea
horizon and return frequency Know the idea

Choices behind a beta

Estimating betastate the method you used

Market index Know the idea broad and liquid; avoid a narrow index the stock dominates
Regression form
- Excess returns Know the idea subtract a short-term risk-free rate from both series
- Market model Memorize raw returns, no risk-free data needed
Noise Know the idea R-squared (R²) and the standard error of beta
Horizon and frequency Know the idea about 100 weekly returns over 2 years is a common default
Adjustment Memorize w₁ × raw + (1 − w₁) × 1.0

Formulas

Capital asset pricing model (CAPM) Memorize

E[r_i] = r_f + \beta_i\left(E[r_m] - r_f\right)

Quick example

Risk-free rate 2.0%, expected market return 8.0%, beta 1.25. The premium is 8.0 − 2.0 = 6.0%, so the cost of equity is 2.0 + 1.25 × 6.0 = 9.50%.

On your calculator

8−2=×1.25+2= 9.5

8ENTER2−1.25×2+ 9.5

ENTER enter (on the HP, puts the number on the stack)

Beta Memorize

\beta_i = \dfrac{\sigma_{i,m}}{\sigma_m^2} = \dfrac{\rho_{i,m}\,\sigma_i\,\sigma_m}{\sigma_m^2} = \rho_{i,m}\,\dfrac{\sigma_i}{\sigma_m}

Quick example

Correlation 0.80, stock volatility 24%, market volatility 16%. Beta = 0.80 × 24 ÷ 16 = 1.20.

On your calculator

0.8×24÷16= 1.2

0.8ENTER24×16÷ 1.2

ENTER enter (on the HP, puts the number on the stack)

Correlation from beta Memorize

\rho_{i,m} = \beta_i\,\dfrac{\sigma_m}{\sigma_i}

Quick example

Beta 0.85, stock volatility 34%, market volatility 20%. Correlation = 0.85 × 20 ÷ 34 = 0.50.

On your calculator

0.85×20÷34= 0.5

0.85ENTER20×34÷ 0.5

ENTER enter (on the HP, puts the number on the stack)

Excess-return regression Know the idea

r_{i,t} - r_{f,t} = \alpha_i + \beta_i\left(r_{m,t} - r_{f,t}\right) + \varepsilon_t

Quick example

With α = 0 and β = 1.2, a week in which the market beats the risk-free rate by 1.0% predicts a stock excess return of 1.2 × 1.0 = 1.2%.

Market model Memorize

r_{i,t} = \alpha_i + \beta_i\,r_{m,t} + \varepsilon_{i,t}

Quick example

α = 0.05% a week, β = 1.10, market return 0.80% in the week. The fitted stock return is 0.05 + 1.10 × 0.80 = 0.93%.

On your calculator

1.1×0.8+0.05= 0.93

1.1ENTER0.8×0.05+ 0.93

ENTER enter (on the HP, puts the number on the stack)

Adjusted beta Memorize

\text{Adjusted beta} = w_1 \times \text{Raw beta} + w_2 \times 1.0, \quad w_2 = 1 - w_1

Quick example

Raw beta 1.60 with weights of two-thirds and one-third: (2/3)(1.60) + (1/3)(1.0) = 1.40.

On your calculator

1.6×2+1=÷3= 1.4, two parts raw plus one part 1.0, over three

1.6ENTER2×1+3÷ 1.4

ENTER enter (on the HP, puts the number on the stack)

Industry betas, the market risk premium and country risk

Known

MemorizeIndustry beta in two steps

Step	Formula	Use
Unlever	$\beta_u = \beta_{L,Comps} / [1 + (1 - t)D/MVE]$	The comparables' averages
Relever	$\beta_{L,firm} = \beta_u[1 + (1 - t)D_{firm}/MVE_{firm}]$	The company's own ratio and tax rate

Memory hook Strip the peers' debt, add your own.

Know it cold Debt over the market value of equity, never D/(D + E). A levered beta is never below the unlevered one. The peers must share the company's asset risk.

MemorizePremiums for the CAPM

Premium	How
Historical MRP	Index mean − government mean; geometric < arithmetic; against bonds < against bills; use long-term bonds for valuation
Implied MRP	$Div_{1,m}/P_{0,m} + g_{l,m} - r_f$
Country risk premium	Spread × equity σ ÷ bond σ; $r_f + \beta_i(MRP_{developed} + CRP)$

Memory hook History looks back, implied looks forward, country adds on.

Know it cold Rising market: the historical MRP goes up, the implied MRP goes down.

Key terms

industry beta Use it
asset risk Know the idea
levered beta Memorize
unlevered beta Memorize
relevered beta Memorize
debt to market value of equity (D/MVE) Memorize
historical market risk premium Memorize
implied market risk premium Memorize
sensitivity analysis Know the idea
country risk premium (CRP) Memorize
sovereign bond yield spread Memorize
equity-to-bond volatility ratio Memorize

Three CAPM inputs to estimate

Cost of equity, CAPMr_f + β × (premium)

Beta
- Own regression if it is precise
- Industry beta Memorize unlever the comparables, relever at the company's own D/MVE
Market risk premium
- Historical Memorize geometric below arithmetic; against bonds below against bills
- Implied Memorize Div₁ ÷ P₀ + g − r_f
Country risk premium Memorize spread × equity σ ÷ bond σ; added to the developed premium before beta

Formulas

Unlevered beta Memorize

\beta_u = \dfrac{\beta_{L,Comps}}{1 + (1 - t)\,\dfrac{D}{MVE}}

Quick example

Comparables average a levered beta of 1.25, a 22% tax rate and D/MVE of 0.35. β_u = 1.25 ÷ [1 + 0.78 × 0.35] = 1.25 ÷ 1.273 = 0.98 (0.9819).

On your calculator

0.78×0.35+1=STO1 1.273
1.25÷RCL1= 0.9819

STO store · RCL recall a stored number

1.25ENTER0.78ENTER0.35×1+÷ 0.9819

ENTER enter (on the HP, puts the number on the stack)

Relevered beta Memorize

\beta_{L,firm} = \beta_u\left[1 + (1 - t)\,\dfrac{D_{firm}}{MVE_{firm}}\right]

Quick example

Unlevered beta 0.95; the company has a 30% tax rate and D/MVE of 0.60. β_L,firm = 0.95 × [1 + 0.70 × 0.60] = 0.95 × 1.42 = 1.35 (1.349).

On your calculator

0.7×0.6+1=×0.95= 1.349

0.95ENTER0.7ENTER0.6×1+× 1.349

ENTER enter (on the HP, puts the number on the stack)

Historical market risk premium Memorize

MRP = \bar{r}_{index} - \bar{r}_{government}

Quick example

Arithmetic means: stocks 9.8%, long-term government bonds 5.1%, bills 3.2%. Against bonds the premium is 9.8 − 5.1 = 4.7%; against bills it would be 6.6%.

On your calculator

9.8−5.1= 4.7

9.8ENTER5.1− 4.7

ENTER enter (on the HP, puts the number on the stack)

Implied market risk premium Memorize

E[r_m] = \dfrac{Div_{1,m}}{P_{0,m}} + g_{l,m}, \qquad \text{Implied MRP} = E[r_m] - r_f

Quick example

An index stands at 3,000 with expected dividends of 69 next year, long-run dividend growth of 5.0% and a government yield of 3.8%. E[r_m] = 69 ÷ 3,000 + 5.0% = 7.3%; implied MRP = 7.3 − 3.8 = 3.5%.

On your calculator

69÷3000+0.05−0.038= 0.035

69ENTER3000÷0.05+0.038− 0.035

ENTER enter (on the HP, puts the number on the stack)

CAPM with a country risk premium Memorize

E[r_i] = r_f + \beta_i\left(MRP_{developed} + CRP\right)

Quick example

Risk-free rate 3.2%, beta 1.15, developed MRP 5.0%, CRP 3.6% (a 2.0% spread × 1.8). 3.2 + 1.15 × (5.0 + 3.6) = 3.2 + 9.89 = 13.09%.

On your calculator

2×1.8+5=×1.15+3.2= 13.09

2ENTER1.8×5+1.15×3.2+ 13.09

ENTER enter (on the HP, puts the number on the stack)

Country risk premium Memorize

CRP = \text{Sovereign bond yield spread} \times \dfrac{\text{Annualized equity market standard deviation}}{\text{Annualized bond market standard deviation}}

Quick example

Spread 2.0%, equity market 1.8 times as volatile as the bond market: CRP = 2.0 × 1.8 = 3.6%.

On your calculator

2×1.8= 3.6

2ENTER1.8× 3.6

ENTER enter (on the HP, puts the number on the stack)

Arbitrage pricing theory and multi-factor models

Known

MemorizeCAPM, APT and the four factors

Model or factor	Key point
CAPM	One systematic risk: the market
APT	$r_f + \sum_k \beta_{i,k}E(f_k)$ ; factors and their number not specified; the market is optional
SMB, small minus big (size)	Long small caps, short large caps
HML, high minus low (value)	Long high book-to-market (value), short low (growth)
UMD, up minus down (momentum)	Long recent winners, short recent losers

Memory hook Small Minus Big, High Minus Low, Up Minus Down: always long the first, short the second.

Know it cold Average betas: market 1, size 0, value 0, momentum 0. Monthly premiums × 12.

MemorizeReading factor betas

Beta	Positive	Negative
Size	Acts small	Acts large
Value	Acts like value	Acts like growth
Momentum	Rides recent winners	Lags when winners keep winning

Memory hook The sign tells the style; the size of the number tells how strongly.

Know it cold Market beta is read against 1.0, the others against 0.

Key terms

single-factor model Know the idea
systematic risk Know the idea
arbitrage pricing theory (APT) Memorize
factor exposure Use it
factor risk premium Use it
multi-factor model Memorize
sources of factors Know the idea
data mining Know the idea
four-factor model Memorize
size factor, small minus big (SMB) Memorize
value factor, high minus low (HML) Memorize
momentum factor, up minus down (UMD) Memorize
zero-investment portfolio Memorize
value stock Use it
growth stock Use it
annualizing monthly premiums Use it
factor investing Know the idea

The four factors

Four-factor modelmarket plus three long-short portfolios

Market Memorize excess market return; average beta 1
Size: small minus big (SMB) Memorize long small, short big; average beta 0
Value: high minus low (HML) Memorize long high book-to-market, short low; average beta 0
Momentum: up minus down (UMD) Memorize long recent winners, short losers; average beta 0

Formulas

Arbitrage pricing theory (APT) Memorize

E[r_i] = r_f + \beta_{i,1}E(f_1) + \beta_{i,2}E(f_2) + \cdots + \beta_{i,k}E(f_k)

Quick example

Risk-free rate 2.5%; exposure 1.1 to a factor with a 4.0% premium and −0.6 to a factor with a 3.0% premium. 2.5 + 1.1 × 4.0 − 0.6 × 3.0 = 2.5 + 4.4 − 1.8 = 5.1%.

On your calculator

1.1×4−(0.6×3)+2.5= 5.1

1.1ENTER4×0.6ENTER3×−2.5+ 5.1

ENTER enter (on the HP, puts the number on the stack)

Four-factor model Memorize

E(r_i) = r_f + \beta_{i,Market}E(r_m - r_f) + \beta_{i,Size}E(SMB) + \beta_{i,Value}E(HML) + \beta_{i,Momentum}E(UMD)

Quick example

Risk-free rate 2.5%; betas 1.05, 0.30, −0.25, 0.15; annual premiums 5.2%, 1.5%, 3.0%, 4.0%. 2.5 + 5.46 + 0.45 − 0.75 + 0.60 = 8.26%.

On your calculator

1.05×5.2+(0.3×1.5)−(0.25×3)+(0.15×4)+2.5= 8.26

1.05ENTER5.2×0.3ENTER1.5×+0.25ENTER3×−0.15ENTER4×+2.5+ 8.26

ENTER enter (on the HP, puts the number on the stack)

Multi-factor cost of equity Use it

E(r_i) = r_f + \sum_{k}\beta_{i,k}\,E(f_k), \qquad E(f_k)_{annual} = 12 \times E(f_k)_{monthly}

Quick example

Monthly premiums 0.55%, 0.20%, 0.30%, 0.50% become 6.6%, 2.4%, 3.6%, 6.0%. With a 4.0% risk-free rate and betas 1.30, −0.30, −0.80, 0.10: 4.0 + 8.58 − 0.72 − 2.88 + 0.60 = 9.58%.

On your calculator

1.3×6.6−(0.3×2.4)−(0.8×3.6)+(0.1×6)+4= 9.58

1.3ENTER6.6×0.3ENTER2.4×−0.8ENTER3.6×−0.1ENTER6×+4+ 9.58

ENTER enter (on the HP, puts the number on the stack)