## Ch. 13: Return, Risk, and the Security Market Line

Expected Returns

• Expected returns are based on the probabilities of possible outcomes.
• `Expected’ means average if the process is repeated many times.
• $E(R) = \sum_{t=1}^T p_t R_t$

Realized Returns

• Realized returns are generally not equal to expected returns.
• Realized returns are comprised of an expected component and an unexpected component.
• Are all news and announcements about stocks unexpected?
• Systematic vs. Unsystematic Risk
• Systematic risk - risk that influences many securities (macroeconomic factors)
• Unsystematic risk - company-, industry-, or asset-specific risks

Variance using Probabilities

• Weighted average of squared deviations
• $\sigma^2 = \sum_{t=1}^T p_t (R_t - E(R))^2$

Example

• There are three possible states of the world: Boom, Normal, and Recession. The Normal state occurs half of the time, while the Boom state occurs 30 percent of the time.
• Consider 2 stocks: C and T. C returns 15%, 10%, and 2% in the Boom state, the Normal state, and the Recession state, respectively. Similarly, T returns 25%, 20%, and 1%.
• What is the expected return of each stock? The variance?

Portfolios

• A portfolio is a collection of assets.
• A portfolio’s risk and return are directly linked to the risk and return of the assets that comprise the portfolio.
• $E(R_p) = \sum_{i=1}^N w_i E(R_i)$
• $Var(R_p) = \sum_{t=1}^T p_t (R_t - E(R_p))^2$
• Note: $Var(R_p) \neq \sum_{i=1}^N w_i Var(R_i)$

Diversification

• Diversification benefits drive the difference between these two expressions.
• Reduction in risk arises because worse-than-expected returns from one asset are offset by better-than-expected returns from another.
• Unsystematic risk can essentially be diversified away. Systematic risk cannot.
• For a diversified portfolio, only systematic risk is present, and the portfolio’s total risk will be roughly equal to it’s systematic risk.
• See Figure 13.1

Systematic Risk

• If investors can diversify unsystematic risk away, should the market reward that risk?
• Thus, expected returns depend only on the asset’s systematic risk.
• How do we measure systematic risk?
• The market as a whole is a portfolio of assets with no unsystematic risks.
• By looking at how closely returns on an individual asset match the returns of the market as a whole, we can gauge the systematic risk of that asset.
• Use the asset’s beta. A beta of 1 implies the asset has the same systematic risk as the overall market.
• Example
• Consider two securities: C and K. Security C has a standard deviation of 20% and a beta of 1.25. Security K has a standard deviation of 30% and a beta of .95.
• Which security is more risky?
• Which security should have a higher expected return?
• $\beta_p = \sum_{i=1}^N w_i \beta_i$

The Security Market Line

• What if we plot expected returns against beta?
• Where does the plot intersect the y-axis?
• What happens if a stock lies above this line? Below this line?
• See Figure 13.3
• Therefore, every asset must be on this line.
• How can we characterize this line? We need a point and a slope.

The Capital Asset Pricing Model

• $E(R_i) = R_f + \beta_i (E(R_m)-R_f)$
• Expected return depends on three things:
• Pure TVM
• The Reward for Bearing Systematic Risk (Market Risk Premium)
• The Amount of Systematic Risk