<p>The Sharpe ratio tells us the amount of excess return we get for taking on each additional unit of portfolio standard deviation. 
$$\frac{\mu_p - r_f}{\sigma_p }$$</p>

<p>We are looking for the combination of the two risky assets with the highest Sharpe ratio ($P^*$). Once we do that, we can take linear combinations of that portfolio and the risk-less asset and form the Capital-Market Line. This is usually solved for numerically rather than analytically but it is possible to do so analytically, particularly in the two asset case. </p>

<p>A portfolio $p$ has an expected return of:
$$\mu_p(W_A) = w_A \cdot \mu_A + (1 - w_A) \cdot \mu_B  $$</p>

<p>and a standard deviation of:
$$\sigma_p(W_A) = \sqrt(w^2_A \cdot \sigma^2_A + (1 - w_A)^2 \cdot \sigma^2_B + 2(1-W_A)W_A \sigma_{AB}) $$
where $\sigma^2_A$ is the variance of asset $A$, $\sigma^2_B$ is the variance of asset $B$, and $\sigma_{AB}$ is their covariance. It therefore has a Sharpe Ratio of:</p>

<p>$$\frac{\mu_p(W_A) - r_f}{\sigma_p(W_A) } = \frac{w_A \cdot \mu_A + (1 - w_A) \cdot \mu_B}{+\sqrt(w^2_A \cdot \sigma^2_A + (1 - w_A)^2 \cdot \sigma^2_B + 2(1-W_A)W_A \sigma_{AB})}$$ </p>

<p>To maximize this you'll want to solve:
$$ \frac{d}{dW_A} \frac{\mu_p(W_A) - r_f}{\sigma_p(W_A) } = 0 $$
$\Rightarrow W^{*}_A$ s.t. $P(W^{*}_A)=P^*$
and check that the second order condition is met:
$$ \frac{d^2}{dW^2_A} \frac{\mu_p(W_A) - r_f}{\sigma_p(W_A) } &lt; 0$$</p>

<p>The algebra is a bit hairy but there is nothing tricky from here on out. </p>

CC BY-SA 3.0
2016-01-19T14:01:14.223Z
2016-01-19T14:09:06.517Z
2016-01-19T14:09:06.517Z
BKay
<p>I practice with some excercises about the Markowitz theory.
If we have a portfolio with two stocks A and B, with given return $r_A$ and $r_B$, the expected return can be computed as: $r_P= w_A \cdot r_A + (1-w_A) \cdot r_B$, where $w_A$ is the weight invested in stock A.</p>

<p>For a given risk free rate ($r_f$), I now have to compute the return of the one fund of the one fund theorem. What I am exactly supposed to do here? I know about the one fund theorem, but I don't know what to do. Any hints?</p>

<microeconomics><portfolio-theory>
Two asset Markowitz Portfolio Optimization and Capital-Market Line construction for a Given Risk Free Rate
clubkli
# Two asset Markowitz Portfolio Optimization and Capital-Market Line construction for a Given Risk Free Rate

(Asked by: username_0 on 2016-01-18)

I practice with some excercises about the Markowitz theory.
If we have a portfolio with two stocks A and B, with given return $r\_A$ and $r\_B$, the expected return can be computed as: $r\_P= w\_A \cdot r\_A + (1-w\_A) \cdot r\_B$, where $w\_A$ is the weight invested in stock A.

For a given risk free rate ($r\_f$), I now have to compute the return of the one fund of the one fund theorem. What I am exactly supposed to do here? I know about the one fund theorem, but I don't know what to do. Any hints?

**Tags:** microeconomics, portfolio-theory

---

## 1 answers

### Answer by username_1 on 2016-01-19. Score: 2 | Accepted answer

The Sharpe ratio tells us the amount of excess return we get for taking on each additional unit of portfolio standard deviation.
$$\frac{\mu\_p - r\_f}{\sigma\_p }$$

We are looking for the combination of the two risky assets with the highest Sharpe ratio ($P^\*$). Once we do that, we can take linear combinations of that portfolio and the risk-less asset and form the Capital-Market Line. This is usually solved for numerically rather than analytically but it is possible to do so analytically, particularly in the two asset case.

A portfolio $p$ has an expected return of:
$$\mu\_p(W\_A) = w\_A \cdot \mu\_A + (1 - w\_A) \cdot \mu\_B $$

and a standard deviation of:
$$\sigma\_p(W\_A) = \sqrt(w^2\_A \cdot \sigma^2\_A + (1 - w\_A)^2 \cdot \sigma^2\_B + 2(1-W\_A)W\_A \sigma\_{AB}) $$
where $\sigma^2\_A$ is the variance of asset $A$, $\sigma^2\_B$ is the variance of asset $B$, and $\sigma\_{AB}$ is their covariance. It therefore has a Sharpe Ratio of:

$$\frac{\mu\_p(W\_A) - r\_f}{\sigma\_p(W\_A) } = \frac{w\_A \cdot \mu\_A + (1 - w\_A) \cdot \mu\_B}{+\sqrt(w^2\_A \cdot \sigma^2\_A + (1 - w\_A)^2 \cdot \sigma^2\_B + 2(1-W\_A)W\_A \sigma\_{AB})}$$

To maximize this you'll want to solve:
$$ \frac{d}{dW\_A} \frac{\mu\_p(W\_A) - r\_f}{\sigma\_p(W\_A) } = 0 $$
$\Rightarrow W^{\*}\_A$ s.t. $P(W^{\*}\_A)=P^\*$
and check that the second order condition is met:
$$ \frac{d^2}{dW^2\_A} \frac{\mu\_p(W\_A) - r\_f}{\sigma\_p(W\_A) } < 0$$

The algebra is a bit hairy but there is nothing tricky from here on out.


---

