Module 2 · Portfolio Management

Portfolio Risk and Return — Part II (CAPM)

EN: CAPM, beta, Security Market Line, performance metrics.
VN: CAPM, beta, SML, các chỉ số đánh giá hiệu quả.

1. Capital Asset Pricing Model (CAPM) Core

\[ E(R_i) = R_f + \beta_i \cdot [E(R_m) - R_f] \]

Components

R_f Risk-free rate.
E(R_m) − R_f Equity risk premium / market risk premium.
β_i Beta — sensitivity of asset i to market.

Practice problem

R_f = 3%, E(R_m) = 10%, β = 1.4. Required return?

Show solution

E(R) = 3% + 1.4(10% − 3%) = 3% + 9.8%

= 12.8%

2. Beta Core

\[ \beta_i = \frac{\text{Cov}(R_i, R_m)}{\sigma_m^{2}} = \rho_{i,m} \cdot \frac{\sigma_i}{\sigma_m} \]

Interpretation

β = 1 Same risk as market.
β > 1 Aggressive — moves more than market.
β < 1 Defensive.
β = 0 Uncorrelated with market.
β < 0 Hedge against market.

Practice problem

Cov(Ri, Rm) = 0.012, σm² = 0.04. Compute β.

Show solution

β = 0.012/0.04

β = 0.30

3. Security Market Line (SML) Core

Graphical version of CAPM — relationship between expected return and beta. All correctly priced assets lie on the SML. Above SML = undervalued (positive alpha); Below = overvalued.

4. Performance Metrics Core

\[ \text{Sharpe} = \frac{R_p - R_f}{\sigma_p} \quad \text{(total risk)} \] \[ \text{Treynor} = \frac{R_p - R_f}{\beta_p} \quad \text{(systematic risk)} \] \[ \alpha_p = R_p - [R_f + \beta_p (R_m - R_f)] \quad \text{(Jensen's alpha)} \] \[ M^{2} = (R_p - R_f) \cdot \frac{\sigma_m}{\sigma_p} - (R_m - R_f) \quad \text{(M-squared)} \] \[ \text{Information ratio} = \frac{R_p - R_b}{\sigma(R_p - R_b)} \]

Sharpe and M² use total risk; Treynor and Jensen's α use systematic risk only — appropriate for well-diversified portfolios.

Practice problem

Rp = 12%, σp = 16%, βp = 1.1, Rm = 9%, Rf = 3%. Compute Sharpe, Treynor, Jensen's α.

Show solution

Sharpe = (12 − 3)/16 ≈ 0.563

Treynor = (12 − 3)/1.1 ≈ 8.18

α = 12 − [3 + 1.1(9 − 3)] = 12 − 9.6 = 2.4

Sharpe 0.563; Treynor 8.18; α = +2.4%