Security Market Line and Systematic Risk Beta

Scenario

Under the Capital Asset Pricing Model (CAPM) framework, investors optimize terminal wealth using mean-variance utility. An asset's expected return is linearly related to its systematic risk covariance with the market portfolio $M$.

Your Task

1. Starting from an investor holding portfolio $p$ with weight $w$ in asset $i$ and $(1-w)$ in the tangency market portfolio $M$, take the marginal derivative of expected return and portfolio variance with respect to $w$ evaluated at $w=0$.
2. Prove that the market equilibrium condition requires: $\mathbb{E}[R_i] - R_f = \beta_i (\mathbb{E}[R_M] - R_f), \quad \beta_i = \frac{\text{Cov}(R_i, R_M)}{\text{Var}(R_M)}$
3. Given annualized risk-free rate $R_f = 4.0\%$, expected market return $\mathbb{E}[R_M] = 10.0\%$, $\sigma_M = 16.0\%$, asset volatility $\sigma_i = 24.0\%$, and correlation $\rho_{i, M} = 0.75$, compute $\beta_i$ and the asset's required expected return.

Constraints

• Homogeneous expectations, perfect liquidity, and borrowing/lending at $R_f$.

Examples If $\beta_i = 1.0$, $\mathbb{E}[R_i] = \mathbb{E}[R_M] = 10\%$. For $\beta_i > 1$, systematic risk is amplified.