<p>OP answering own question.</p>
<p>I think the answer is that <span class="math-container">$t$</span> doesn't need to be in <span class="math-container">$[0,1]$</span>; rather the asset coefficients (namely <span class="math-container">$t,1-t$</span> in the two-asset case) need to sum to <span class="math-container">$1$</span>, which doesn't depend on the size of <span class="math-container">$t$</span>.</p>

CC BY-SA 4.0
2023-11-14T15:24:04.417Z
2023-11-14T15:24:04.417Z
mjc
<p>I'm working on an unassessed course problem,</p>
<blockquote>
<p>Consider a market with risk-free return <span class="math-container">$5\%$</span> and two risky investment <span class="math-container">$A$</span> and <span class="math-container">$B$</span>. We are given the following data: <span class="math-container">\begin{matrix}
\text{Investment} &amp; \text{Expected return} &amp; \text{Standard deviation of return} \\
\text{A} &amp; 10\% &amp; 10\% \\
\text{B} &amp; 15\% &amp; 20\%
\end{matrix}</span> We are also told that the correlation between investments A and B is <span class="math-container">$\rho=0.5$</span>. We assume that the CAPM holds. Use the fact that the market portfolio is the unique portfolio that maximises <span class="math-container">$$\frac{r_P-r_C}{\sigma_P}$$</span> (where <span class="math-container">$C$</span> is a risk-free investment) as <span class="math-container">$P$</span> ranges over all investments consisting entirely of risky investments to find the market portfolio in the market described above.</p>
</blockquote>
<p>I reason as follows.</p>
<blockquote>
<p>Let <span class="math-container">$\Pi_t$</span> be the portfolio <span class="math-container">$tA+(1-t)B$</span> with <span class="math-container">$t\in[0,1]$</span>; then <span class="math-container">\begin{align}
&amp; \; \mathbb{E}[\Pi_t] = tr_A+(1-t)r_B=0.1t+0.15(1-t)=0.15-0.05t, \hspace{2em} (1) \\[1em]
&amp; \begin{aligned}[t]
\mathbb{V}[\Pi_t]
&amp; = (t\sigma_A)^2+2t(1-t)\text{Cov}(A,B)+((1-t)\sigma_B)^2 &amp;&amp; (2.1) \\
&amp; = (0.1t)^2 + 2t(1-t)\times0.5\times0.1\times0.2+(0.2(1-t))^2 &amp;&amp; (2.2) \\
&amp; = 0.01t^2+0.01(2t-2t^2)+0.04(1-2t+t^2) &amp;&amp; (2.3) \\
&amp; = 0.03t^2-0.06t+0.04 &amp;&amp; (2.4)
\end{aligned} \\[2em]
\therefore \; &amp; \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 \; &amp; \begin{aligned}[t]
\frac{\partial}{\partial t}\left(\frac{r_{\Pi_t}-r_C}{\sigma_{\Pi_t}}\right)
&amp; = \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}} &amp;&amp; (4.1) \\[1em]
&amp; = \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}} &amp;&amp; (4.2) \\[1em]
&amp; = \frac{0.003t-0.008}{-2(0.03t^2-0.06t+0.04)^\frac{3}{2}} &amp;&amp; (4.3)
\end{aligned} \\[2em]
\therefore \; &amp; \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}</span></p>
</blockquote>
<p>Would someone mind pointing out where I've gone wrong?</p>

<markets><asset-pricing><price-theory><efficient-markets>
Finding the market portfolio in a two-asset market under CAPM
mjc
# Finding the market portfolio in a two-asset market under CAPM

(Asked by: username_0 on 2023-11-08)

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?

**Tags:** markets, asset-pricing, price-theory, efficient-markets

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## 1 answers

### Answer by username_0 on 2023-11-14. Score: 0 | Accepted answer

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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