Midterm 1 (Practice 1)

Let us denote by \(r\) the rate of continuous compounding. Then we have \(e^r=\left(1+\frac{0.9}{365}\right)^{365}\approx 2.4568799491\) which implies that \(r=\ln\left(2.4568799491\right)\approx 0.8988922316=89.89\%\).

Let us denote by \(r\) the interest rate, \(F\) the face value of the bond, and by \(C_1\) and \(C_2\) the amounts of the coupon payments. The price of the bond is obtained from the equation: \[P_0=C_1e^{-r}+ C_2e^{-2r}+Fe^{-3r}\approx 89.26030747. \]

We first calculate \(\mu_{K}=0.02\cdot 0.3+0.22\cdot 0.7=0.16\) and \(\sigma_{K}=\sqrt{0.02^2\cdot 0.3+0.22^2\cdot 0.7-\mu_{K}^2}=9.165\%\).

• (a) The expected return is \(\mu_V=w_1\mu_1+w_2\mu_2+w_3\mu_3\approx 0.1268\). The covariance matrix is \[C=\left[\begin{array}{ccc} 0.0225&0.015&-0.01125\newline 0.015&0.0625&0.00625\newline -0.01125&0.00625&0.0625\end{array}\right].\] Therefore the variance of the return on the portfolio is \[\mbox{var }(K_V)=[w_1,w_2,w_3]C \left[\begin{array}{c}w_1\newline w_2\newline w_3\end{array}\right]\approx 0.0552705.\] Thus \(\sigma_V=\sqrt{\mbox{var }(K_V)}\approx 0.235096789\).

• (b) The minimal variance portfolio is given by \[w_0=\frac{uC^{-1}}{uC^{-1}u^T}=\left[0.6914893617, -0.0070921986, 0.3156028369\right]. \] The expected return of the minimal risk portfolio is \(\mu_0=w_0m^T=0.0961347518\) and the risk is \(\sigma_0=\sqrt{w_0Cw_0^T}=0.109094435\).

• (a) The minimal variance portfolio corresponds to the vertex of the hyperbola \(\sigma=\sqrt{0.29-4\mu+20\mu^2}\). The coordinates \((\sigma, \mu)\) of the vertex of the hyperbola satisfy \(\sigma^{\prime}(\mu)=0\). We find that \[\sigma^{\prime}=\frac{-4+40\mu}{2\sqrt{0.29-4\mu+20\mu^2}}.\] Solving the equation \(\sigma^{\prime}(\mu)=0\) gives us \(\mu=\frac1{10}\). Then we get \(\sigma\left(\frac1{10}\right)=0.3\).

• (b) Let us denote by \(\sigma_M\) and \(\mu_M\) the risk and the expected return of the market portfolio. The value \(\mu_M\) can be obtained as the number that maximizes the slope \(S(\mu)=\frac{\mu-R}{\sigma}\). We find it using the fMax function of the calculator, or by finding the solution of the equation \(S^{\prime}(\mu)=0\). Since \[S^{\prime}(\mu)=\frac{\left(0.29-4\mu+20\mu^2\right)-(\mu-0.07)(-2+20\mu)}{\sqrt{0.29-4\mu+20\mu^2}^3} =\frac{0.15-0.6\mu}{\sqrt{0.29-4\mu+20\mu^2}^3 } ,\] we obtain that the solution to \(S^{\prime}(\mu)=0\) is \(\mu=0.25\). Then we get \(\sigma(0.25)=\sqrt{0.29-1+1.25}=\sqrt{0.54}\).

The required portfolio must belong to the efficient frontier. Let us denote by \(\overrightarrow w_1\) and \(\overrightarrow w_2\) the weights of the portfolios \(V_1\) and \(V_2\), respectively.

According to the Two-Fund Theorem the weights \(\overrightarrow w\) of any portfolio on the minimal variance line must satisfy \(\overrightarrow w=\alpha\overrightarrow w_1+(1-\alpha)\overrightarrow w_2\). Thus we need to find \(\alpha\) such that the risk of the portfolio \(\overrightarrow w(\alpha)\) is \(3\) and the expected return is maximal.

Let us denote by \(\sigma=\sigma(\alpha)\) the risk of the portfolio with weights \(\overrightarrow w\). Then we have \(9=\sigma^2=\alpha^2\cdot 2+2\alpha(1-\alpha)\cdot 1+(1-\alpha)^2\cdot 1\). Solving for \(\alpha\) gives us \(\alpha=\pm \sqrt 8\). The return of the portfolio \(\overrightarrow w(-\sqrt 8)\) is \(\mu_{-\sqrt 8}=-\sqrt 8\cdot 3+(1+\sqrt 8)\cdot 10=10+7\sqrt 8\). Similarly, the return of the portfolio \(\overrightarrow w(\sqrt 8)\) is \(\mu_{\sqrt 8}=\sqrt 8\cdot 3+(1-\sqrt 8)\cdot 10=10-7\sqrt 8\). Thus, the maximal return is \(10+7\sqrt 8\approx 29.799\).