CMA Final · Strategic Financial Management

Asset Pricing Theories: formula sheet

Full chapter guide

Key formulas

CAPM expected return: E(Ri) = Rf + βi × (Rm − Rf); Gives the required return for the security. Use the same basis (annual, in %) for all inputs.
Market risk premium: Market risk premium = Rm − Rf; If the question gives the premium directly, do not subtract Rf again.
Beta: β = Cov(Ri, Rm) ÷ σm² = ρim × σi ÷ σm; Use when covariance, correlation or standard deviations are given.
Portfolio beta: βp = Σ (wi × βi); Weights are the proportion of portfolio value in each security and should add to 1.
Security Market Line test: Expected return > CAPM return: underpriced (buy). Expected return < CAPM return: overpriced (sell).; Compare the return you expect from the security with the CAPM required return.
Alpha (excess return): α = Expected return − CAPM required return; A positive alpha means the security plots above the SML.
Capital Market Line: E(Rp) = Rf + [(E(Rm) − Rf) ÷ σm] × σp; σp is the standard deviation of the efficient portfolio; σm is the market's standard deviation.
Slope of CML: (E(Rm) − Rf) ÷ σm; Market price of risk per unit of total risk.
Security Market Line (CAPM): E(Ri) = Rf + βi × (E(Rm) − Rf); Gives the required return for a security with beta βi.
Market risk premium: E(Rm) − Rf; This is the slope of the SML.
Alpha / valuation test: α = Expected return − Required return (from SML); α > 0: undervalued (buy). α < 0: overvalued (sell). α = 0: fairly priced.
Beta: βi = Cov(Ri, Rm) ÷ σm² = ρim × σi ÷ σm; Measures systematic risk.
Beta from covariance: β = Cov(Rs, Rm) ÷ σm²; σm² is the variance of market returns. Use the same type of variance (sample or population) in both numerator and denominator.
Beta from correlation: β = r(s,m) × σs ÷ σm; Use when correlation and standard deviations are given instead of covariance.
Regression (characteristic line): Rs = α + β × Rm + e; β is the slope. The error term e represents unsystematic return.
Beta from paired data: β = [nΣXY − ΣXΣY] ÷ [nΣX² − (ΣX)²]; X = market return, Y = stock return. Do not swap them.
Systematic and unsystematic variance: Systematic variance = β² × σm²; Unsystematic variance = σs² − β² × σm²; Also systematic share of total risk = r² (coefficient of determination).
Portfolio beta: βp = Σ (wi × βi); Weights are based on market value invested and must add up to 1. Include a risk-free asset with β = 0 if held.
CAPM expected return: E(Ri) = Rf + βi × (Rm − Rf); Gives the required return for a given beta.
APT expected return (multi-factor): E(Rᵢ) = Rf + β₁ × RP₁ + β₂ × RP₂ + … + βₙ × RPₙ; Rf is the risk-free rate. RPₖ is the risk premium of factor k, that is, the expected return on a portfolio with beta 1 to factor k and 0 to others, minus Rf. Use the premium directly if given.
APT using factor expected returns: E(Rᵢ) = Rf + Σ βₖ × (E(Fₖ) − Rf); Use when the question gives the expected return of each factor portfolio, not the premium. Subtract Rf from each factor return first.
Actual return under factor model: Rᵢ = E(Rᵢ) + β₁ × (F₁ − E(F₁)) + β₂ × (F₂ − E(F₂)) + … + ε; Surprises in factors move the actual return away from expected. ε is the firm-specific term, which is assumed to be zero for a well-diversified portfolio.
Portfolio beta and return: βₚ,ₖ = Σ wᵢ × βᵢ,ₖ ; E(Rₚ) = Σ wᵢ × E(Rᵢ); Weights wᵢ sum to 1. Portfolio factor betas are weighted averages of security betas.
Single-factor APT: E(Rᵢ) = Rf + βᵢ × (E(Rm) − Rf); With one factor being the market, APT looks like CAPM. This is a special case.
Arbitrage test: Mispricing = Given expected return − APT expected return; Positive means underpriced: buy. Negative means overpriced: sell or short.
Random walk: Pt = Pt-1 + expected change + random error; The error is unpredictable and independent of past changes. Past price changes cannot predict the next change.
Abnormal return: Abnormal return = Actual return − Expected (required) return; EMH tests ask whether abnormal returns can be earned repeatedly using a given information set.
Nested information sets: Weak (past prices) ⊂ Semi-strong (all public) ⊂ Strong (all public + private); Use this to decide what each form rules out.
Fama-French three-factor model: E(Ri) = Rf + βm × (Rm − Rf) + βs × SMB + βh × HML; Rm − Rf is the market risk premium. SMB and HML are the average factor premiums. βs and βh are the stock's loadings.
General multifactor model: E(Ri) = Rf + Σ βk × λk; βk is the loading on factor k and λk is the risk premium of factor k.
CAPM (single-factor case): E(Ri) = Rf + β × (Rm − Rf); Use this for comparison. It is the Fama-French model with the size and value loadings set to zero.
Factor definitions: SMB = R(small) − R(big); HML = R(high B/M) − R(low B/M); Both are long-short portfolio returns. B/M is book value to market value.
Alpha (realised version): α = Actual return − Expected return from the model; Positive alpha suggests the stock earned more than its factor risks justify.

Quick revision

CAPM: required return = Rf + β × (Rm − Rf).
(Rm − Rf) is the market risk premium.
Beta measures systematic risk only; diversification removes unsystematic risk.
Market beta is 1; a risk-free asset has beta 0.
Portfolio beta is the weighted average of the betas of its securities.
Beta of a security = Cov(security, market) ÷ Variance of market.
SML plots required return against beta and applies to individual securities and portfolios.
CML plots return against total risk (standard deviation) and applies to efficient portfolios only.
A share above the SML is undervalued, and one below it is overvalued.
APT lets several factors drive returns and rests on no-arbitrage; it does not name the factors.
EMH has three forms: weak, semi-strong and strong.
Fama-French adds size and value factors to the market factor.

Common mistakes

Multiplying beta by Rm instead of by (Rm − Rf). Fix: Always compute the premium first. Then check: beta 1 must give Rm.
Subtracting Rf again when the question already gives the market risk premium. Fix: Underline the words 'premium' or 'excess return' in the question. A premium is already Rm − Rf.
Using beta on the CML or standard deviation on the SML. Fix: Remember: CML uses σ (total risk); SML uses β (systematic risk).
Using the market return instead of the market risk premium when multiplying by beta. Fix: Always compute Rm − Rf first, then multiply by beta.
Dividing covariance by market standard deviation instead of variance. Fix: Square σm first. Beta = Cov ÷ σm². Check that the units work: % squared over % squared.
Using equal weights in portfolio beta. Fix: Weight each beta by the amount invested in that security divided by total portfolio value.
Using the factor's expected return as the premium without subtracting the risk-free rate. Fix: Read whether the figure is a premium or a total return. If it is a total return on a factor portfolio, subtract Rf first.
Forgetting to add the risk-free rate at the end. Fix: Write the formula first, with Rf at the front, and tick it off after the final addition.
Saying that weak form means fundamental analysis cannot work. Fix: Weak form rules out only technical analysis based on past prices and volumes. Fundamental analysis is ruled out only under semi-strong form.
Saying that random walk means prices are random or have no link to value. Fix: Say that price changes are unpredictable because they follow new information, which is itself unpredictable. Prices still reflect value.

Exam tips

In Section A, expect a direct substitution question or a test of the meaning of beta. Do the arithmetic carefully because each MCQ carries 2 marks and there is no negative marking.
In the 14-mark questions CAPM is usually a part of a larger problem such as cost of equity, share valuation or a buy-sell decision. Show the CAPM line clearly, since the later steps depend on it.
Write the formula before substituting. Step marks are given for the formula and the premium.
For theory parts, prepare five assumptions and five limitations in crisp one-line points with reasons.
Always end a security evaluation with a clear recommendation: buy if underpriced, sell or avoid if overpriced.
Write the axis of each line (σ for CML, β for SML) in the first line of any theory answer. It earns easy marks.
In a difference question, give three points: risk measure, assets covered, and the equation behind each line.
In numericals, show the required return calculation separately and then state the buy or sell recommendation.