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Formula Sheet

Investment Theory

  1. If a return r equals r_s in state s with probability p_s, its expected return and variance are \operatorname{E}(r) = \sum_{s} p_s r_s \quad \text{and} \quad \operatorname{V}(r) = \sum_{s} p_s \left(r_s - \operatorname{E}(r)\right)^2 For two returns r_A and r_B, the covariance is \operatorname{Cov}(r_A, r_B) = \sum_{s} p_s \left(r_{A,s} - \operatorname{E}(r_A)\right) \left(r_{B,s} - \operatorname{E}(r_B)\right)

  2. For a portfolio between two securities, the return, expected return and variance are \begin{aligned} r & = w_A r_A + w_B r_B \\ \operatorname{E}(r) & = w_A \operatorname{E}(r_A) + w_B \operatorname{E}(r_B) \\ \operatorname{V}(r) & = w_A^2 \operatorname{V}(r_A) + w_B^2 \operatorname{V}(r_B) + 2 w_A w_B \operatorname{Cov}(r_A, r_B) \end{aligned}

  3. The covariance between the returns of two securities can be expressed as \operatorname{Cov}(r_A, r_B) = \sigma_{AB} = \sigma_A \sigma_B \rho_{AB}

  4. Given two risky assets A and B, the weights of the minimum variance portfolio are w_{A} = \frac{\sigma_{B}^{2} - \sigma_{AB}}{\sigma_{A}^{2} + \sigma_{B}^{2} - 2 \sigma_{AB}} \quad \text{and} \quad w_{B} = \frac{\sigma_{A}^{2} - \sigma_{AB}}{\sigma_{A}^{2} + \sigma_{B}^{2} - 2 \sigma_{AB}}

  5. Given a risk-free asset and a risky asset with expected return \mu_Q and variance \sigma_Q^2, a portfolio that invests a fraction w of wealth in the risky asset has \mu = r_f + w (\mu_Q - r_f) \quad \text{and} \quad \sigma = w \sigma_Q The optimal fraction of wealth to invest in the risky asset for an investor with utility function U = \mu - \frac{1}{2} A \sigma^2 is w^* = \frac{\mu_Q - r_f}{A \sigma_Q^2}

  6. In the CAPM, the expected return of a security is \operatorname{E}(r) = r_f + \beta (\operatorname{E}(r_M) - r_f) \quad \text{where} \quad \beta = \frac{\operatorname{Cov}(r, r_M)}{\operatorname{V}(r_M)} = \frac{\rho_{rM} \, \sigma(r)}{\sigma_M}

  7. The Sharpe ratio is given by \text{Sharpe ratio} = \frac{\operatorname{E}(r) - r_{f}}{\sigma(r)}

  8. In the index model, the excess returns R_i = r_i - r_f of security i are modeled as R_i = \alpha_i + \beta_i R_M + e_i \quad \text{where} \quad \sigma_i^2 = \beta_i^2 \sigma_M^2 + \sigma^2(e_i) The alpha and the R-square of the regression are \alpha_i = \operatorname{E}(R_i) - \beta_i \operatorname{E}(R_M) \quad \text{and} \quad R^2 = \frac{\beta_i^2 \sigma_M^2}{\sigma_i^2} For two distinct securities i \neq j, the covariance of returns is \operatorname{Cov}(R_i, R_j) = \beta_i \beta_j \sigma_M^2 A portfolio with weights w_i has \alpha_P = \sum_{i} w_i \alpha_i, \quad \beta_P = \sum_{i} w_i \beta_i \quad \text{and} \quad \sigma^2(e_P) = \sum_{i} w_i^2 \sigma^2(e_i)