# Optimal Risky Portfolio and Shortfall Probability Given: - Asset 1 (A1): $\mu_1 = 8\%$, $\sigma_1 = 20\%$ - Asset 2 (A2): $\mu2 = \%$, $\sigma_2 = 10\%$ - Correlation: $\rho = .3$ - Risk-free rate: $r_f = 2\%$ - Target: Compute tangency (Markowitz) weights, then portfolio mean, standard deviation, and 1-year shortfall probability $P(R_p < -5\%)$. --- ## 1. Covariance Matrix \[ \Sigma = \begin{bmatrix} \sigma_1^2 & \rho \sigma_1 \sigma_2 \\ \rho \sigma_1 \sigma_2 & \sigma_2^2 \end{bmatrix} = \begin{bmatrix} .04 & .006 \\ .006 & .01 \end{bmatrix} \] --- ## 2. Excess Returns Vector \[ \mu - r_f \mathbf{1} = \begin{bmatrix} .08 - .02 \\ .05 - .02 \end{bmatrix} = \begin{bmatrix} .06 \\ .03 \end{bmatrix} \] --- ## 3. Compute $\Sigma^{-1}$ First, compute $\det(\Sigma)$: \[ \det(\Sigma) = (.04)(.01) - (.006)^2 = .0004 - .000036 = .000364 \] \[ \Sigma^{-1} = \frac{1}{.000364} \begin{bmatrix} .01 & -.006 \\ -.006 & .04 \end{bmatrix} = \begin{bmatrix} 27.4725 & -16.4835 \\ -16.4835 & 109.8901 \end{bmatrix} \] (Rounded to 4 decimal places) --- ## 4. Compute Unnormalized Tangency Weights \[ w \propto \Sigma^{-1} (\mu - r_f \mathbf{1}) \] \[ = \begin{bmatrix} 27.4725 & -16.4835 \\ -16.4835 & 109.8901 \end{bmatrix} \begin{bmatrix} .06 \\ .03 \end{bmatrix} \] Calculate each component: - First row: $27.4725 \times .06 + (-16.4835) \times .03 = 1.64835 - .494505 = 1.153845$ - Second row: $-16.4835 \times .06 + 109.8901 \times .03 = -.98901 + 3.296703 = 2.307693$ \[ w \propto \begin{bmatrix} 1.153845 \\ 2.307693 \end{bmatrix} \] --- ## 5. Normalize Weights Sum: $1.153845 + 2.307693 = 3.461538$ \[ _1 = \frac{1.153845}{3.461538} = .3333 \\[ w_2 = \frac{2.307693}{3.461538} = .666 \] So: - **Tangency weights:** - $w_1 = 33.33\$ in A1 - $w_2 = 66.67\%$ in A2 --- ## 6. Portfolio Mean and Standard Deviation \[ \mu_p = .333 \times .08 + .6667 \times .05 = .02667 + .03333 = .06 \text{ or } 6\% \] Portfolio variance: \[ \sigma_p^2 = w^T \Sigma w \] \[ = [.3333, .6667] \begin{bmatrix} .04 & .006 \\ .006 & .01 \end{bmatrix} \begin{bmatrix} .3333 \\ .6667 \end{bmatrix} \] Step by step: - $.3333 \times .04 + .6667 \times .006 = .01333 + .004 = .01733$ - $.3333 \times .006 + .6667 \times .01 = .002 + .006667 = .008667$ So: \[ \begin{bmatrix} .01733 & .008667 \end{bmatrix} \begin{bmatrix} .3333 \\ .6667 \end{bmatrix} = .01733 \times .3333 + .008667 \times .6667 = .005777 + .005778 = .01156 \] \[ \sigma_p = \sqrt{.01156} \approx .1075 \text{ or } 10.75\% \] --- ## 7. Shortfall Probability The probability $P(R_p < -5\%)$ is: \[ P(R_p < -.05) = \Phi\left( \frac{-.05 - .06}{.1075} \right) = \Phi\left( \frac{-.11}{.1075} \right) = \Phi(-1.023) \] From standard normal tables: \[ \Phi(-1.023) \approx .153 \] --- ## **Summary Table** | Asset | Weight | |-------|--------| | A1 | 33.3% | | A2 | 66.7% | - **Portfolio mean:** $6\%$ - **Portfolio std dev:** $10.75\%$ - **Shortfall probability ($R_p < -5\%$):** $\boxed{15.3\%}$ --- ## **Formulas Used** - **Weights:** $w \propto \Sigma^{-1}(\mu - r_f \mathbf{1})$ - **Portfolio mean:** $\mu_p = w^T \mu$ - **Portfolio variance:** $\sigma_p^2 = w^T \Sigma w$ - **Shortfall probability:** $P(R_p < x) = \Phi\left( \frac{x - \mu_p}{\sigma_p} \right)$ --- **Alt text:** *Table showing the tangency weights of two assets in the optimal risky portfolio, and the calculated probability of a shortfall below -5% annual return.*