3.1 Risk, Return & CAPM

Measuring Expected Return and Risk

The expected return on an asset is the probability-weighted average of its possible returns: E(R) = sum of (probability x return) across all outcomes. If a stock returns 20% with 30% probability, 10% with 50% probability, and -5% with 20% probability, E(R) = (0.30 x 20) + (0.50 x 10) + (0.20 x -5) = 6 + 5 - 1 = 10%.

Risk is the dispersion of returns around that expected value. Variance is the probability-weighted average of squared deviations from the mean, and standard deviation is its square root. Standard deviation is the workhorse measure of total return risk because it is stated in the same percentage units as return.

Variance and Standard Deviation

• Variance = sum of [probability x (return - expected return)^2]
• Standard deviation (sigma) = square root of variance
• A higher standard deviation means a wider, riskier spread of outcomes
• The coefficient of variation = standard deviation / expected return measures risk per unit of return, useful for comparing assets of different scales

Portfolio Expected Return

A portfolio's expected return is simply the weighted average of its assets' expected returns: E(Rp) = sum of (weight x expected return). If 60% of a portfolio is in a stock returning 12% and 40% in a bond returning 5%, E(Rp) = 0.60 x 12% + 0.40 x 5% = 7.2% + 2.0% = 9.2%. Portfolio risk, however, is not a simple weighted average of standard deviations, because returns are imperfectly correlated. The lower the correlation between two assets, the more risk diversification removes, which is the mathematical engine behind the diversification benefit described below.

Systematic vs. Unsystematic Risk

Total risk has two components. Unsystematic risk (also called firm-specific, diversifiable, or idiosyncratic risk) comes from events affecting one company or industry: a lawsuit, a product recall, a strike. Systematic risk (market risk, non-diversifiable risk) comes from economy-wide forces: interest rates, inflation, recessions, and is shared by all assets.

Diversification is the central insight: holding many imperfectly correlated assets averages away unsystematic risk because one firm's bad news is offset by another's good news. As the portfolio grows, firm-specific shocks cancel out, but systematic risk remains no matter how many stocks you add.

Why Only Systematic Risk Is Priced

Because investors can eliminate unsystematic risk for free through diversification, the market does not reward bearing it. Investors are compensated only for systematic risk they cannot escape. This principle drives the entire pricing model that follows: the relevant risk measure for a well-diversified investor is not total standard deviation but the asset's contribution to portfolio risk, captured by beta.

Beta: Measuring Systematic Risk

Beta measures how much an asset's return moves with the overall market, capturing only systematic risk. The market portfolio has a beta of exactly 1.0.

• Beta > 1.0: more volatile than the market (aggressive, e.g., a tech stock at 1.5 moves 1.5% for each 1% market move)
• Beta = 1.0: moves in lockstep with the market
• Beta < 1.0: less volatile than the market (defensive, e.g., a utility at 0.6)
• Beta = 0: return is uncorrelated with the market (e.g., a risk-free Treasury)

A portfolio's beta is the weighted average of the betas of its holdings.

The Capital Asset Pricing Model (CAPM)

The Capital Asset Pricing Model (CAPM) translates beta into a required rate of return:

Re = Rf + Beta x (Rm - Rf)

where Rf is the risk-free rate (typically a Treasury yield), Rm is the expected market return, and (Rm - Rf) is the market risk premium, the extra return investors demand over the risk-free rate for bearing average market risk.

Worked Example

Suppose the risk-free rate is 4%, the expected market return is 11%, and a stock's beta is 1.3. The market risk premium is 11% - 4% = 7%. Then:

Re = 4% + 1.3 x (11% - 4%) = 4% + 1.3 x 7% = 4% + 9.1% = 13.1%.

This 13.1% is the return investors require to hold this stock given its systematic risk. If the firm cannot earn at least 13.1%, the investment destroys value. CAPM cost of equity feeds directly into the WACC calculation in Section 3.2.

The Security Market Line

The Security Market Line (SML) is the graph of CAPM: required return on the vertical axis, beta on the horizontal axis. Its intercept is the risk-free rate and its slope is the market risk premium.

• Assets plotting above the SML offer more return than their beta requires, so they are underpriced (buy)
• Assets plotting below the SML offer too little return for their risk, so they are overpriced (sell)
• In equilibrium, all correctly priced assets lie exactly on the line

Trap: the SML uses beta (systematic risk), while the Capital Market Line uses total standard deviation. Confusing the two is a classic exam error.

CAPM Assumptions and Limits

CAPM rests on simplifying assumptions: investors are rational and diversified, can borrow and lend at the risk-free rate, face no taxes or transaction costs, and share the same expectations. Its main practical weaknesses are the difficulty of estimating a forward-looking beta and market risk premium, and the empirical finding that beta alone does not fully explain returns. Despite these limits, CAPM remains the standard tool tested on the exam for deriving the cost of equity.