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