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I'm working on an unassessed course problem,

Consider a market with risk-free return $5\%$ and two risky investment $A$ and $B$. We are given the following data: \begin{matrix} \text{Investment} & \text{Expected return} & \text{Standard deviation of return} \\ \text{A} & 10\% & 10\% \\ \text{B} & 15\% & 20\% \end{matrix} We are also told that the correlation between investments A and B is $\rho=0.5$. We assume that the CAPM holds. Use the fact that the market portfolio is the unique portfolio that maximises $$\frac{r_P-r_C}{\sigma_P}$$ (where $C$ is a risk-free investment) as $P$ ranges over all investments consisting entirely of risky investments to find the market portfolio in the market described above.

I reason as follows.

Let $\Pi_t$ be the portfolio $tA+(1-t)B$ with $t\in[0,1]$; then \begin{align} & \; \mathbb{E}[\Pi_t] = tr_A+(1-t)r_B=0.1t+0.15(1-t)=0.15-0.05t, \hspace{2em} (1) \\[1em] & \begin{aligned}[t] \mathbb{V}[\Pi_t] & = (t\sigma_A)^2+2t(1-t)\text{Cov}(A,B)+((1-t)\sigma_B)^2 && (2.1) \\ & = (0.1t)^2 + 2t(1-t)\times0.5\times0.1\times0.2+(0.2(1-t))^2 && (2.2) \\ & = 0.01t^2+0.01(2t-2t^2)+0.04(1-2t+t^2) && (2.3) \\ & = 0.03t^2-0.06t+0.04 && (2.4) \end{aligned} \\[2em] \therefore \; & \frac{r_{\Pi_t}-r_C}{\sigma_{\Pi_t}} = \frac{(0.15-0.05t)-0.05}{\sqrt{0.03t^2-0.06t+0.04}} = \frac{0.1-0.05t}{\sqrt{0.03t^2-0.06t+0.04}} \hspace{2em} (3) \\[2em] \therefore \; & \begin{aligned}[t] \frac{\partial}{\partial t}\left(\frac{r_{\Pi_t}-r_C}{\sigma_{\Pi_t}}\right) & = \frac{-0.05}{\sqrt{0.03t^2-0.06t+0.04}} + \frac{(0.1-0.05t)(0.06t-0.06)}{-2(0.03t^2-0.06t+0.04)^\frac{3}{2}} && (4.1) \\[1em] & = \frac{0.1(0.03t^2-0.06t+0.04)+(0.1-0.05t)(0.06t-0.06)}{-2(0.03t^2-0.06t+0.04)^\frac{3}{2}} && (4.2) \\[1em] & = \frac{0.003t-0.008}{-2(0.03t^2-0.06t+0.04)^\frac{3}{2}} && (4.3) \end{aligned} \\[2em] \therefore \; & \frac{r_{\Pi_t}-r_C}{\sigma_{\Pi_t}} \text{ is maximised when } 0.003t-0.008 = 0 \; \therefore \; t = \frac{8}{3} \notin [0,1] \hspace{2em} (5). \end{align}

Would someone mind pointing out where I've gone wrong?

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1 Answer 1

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OP answering own question.

I think the answer is that $t$ doesn't need to be in $[0,1]$; rather the asset coefficients (namely $t,1-t$ in the two-asset case) need to sum to $1$, which doesn't depend on the size of $t$.

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