Risk and Return
Risk and Return
The exam tests measuring expected return and risk, the mechanics of diversification, beta, the CAPM, and risk-adjusted performance evaluation.
Core concepts
- Expected return = Σ p_i r_i, the probability-weighted average of outcomes.
- Variance = Σ p_i (r_i − E(r))^2; standard deviation is its square root. Coefficient of variation = σ/E(r) measures risk per unit of return.
- Portfolio expected return is the weighted average of component returns, but portfolio standard deviation depends on covariance or correlation, not just weights.
- Diversification eliminates unsystematic (firm-specific) risk; systematic (market) risk remains and is the only risk priced.
- Beta = Cov(ri, rm)/σm^2; it measures sensitivity to market movements, and a portfolio beta is the weighted average of component betas.
- CAPM: E(r) = rf + β(E(rm) − rf). The security market line plots required return against beta.
- Alpha = expected return − CAPM required return; positive alpha suggests undervaluation.
- Sharpe ratio = (E(r) − rf)/σ, a total-risk-adjusted performance measure.
Worked example
A portfolio holds 60% in stock A (expected return 10%, beta 0.8) and 40% in stock B (expected return 14%, beta 1.4). Portfolio expected return = 0.6(10) + 0.4(14) = 11.6%. Portfolio beta = 0.6(0.8) + 0.4(1.4) = 0.48 + 0.56 = 1.04. With rf = 4% and a market risk premium of 6%, required return = 4 + 1.04(6) = 10.24%. Alpha = 11.6 − 10.24 = +1.36%, so the portfolio has earned more than its systematic risk requires.
Traps
- Treating standard deviation as the risk that matters in a diversified portfolio; only beta is compensated under CAPM.
- Forgetting that correlation of +1 gives no diversification benefit, while any ρ < 1 reduces portfolio σ.
- Plugging the total market return into CAPM instead of the market risk premium.
- Confusing the SML (beta, all securities) with the CML (total risk, efficient portfolios only).
- Weighting betas by cost or book value rather than market value.