Portfolio Risk and Expected Return with Short Positions

# Standard deviation formula with Short selling- Markowitz model


# Standard deviation formula with Short selling- Markowitz model












I have 2 fast quastions.

Before I begin I want to show you that I found minus before SD of bills in the book Principles of corporate finance(1.screen). I know SD of bills is zero and minus in this concrete example isnt important. But I want to know if I should calculate with minus when I'm selling short risky asset.

(1) For example if I sell short JNJ(-1) and buy 2x portfolio T, will SD of this short sold asset reduce risk(SD)of portfolio(2xT) like this?

(2) If suppose interest rate for short selling 2% can I set E (Rp) for this combination (2xT-1JNJ) like this:

In the end I want to apologize for my English, I am not a native speaker.

## Answer by Pontus Hultkrantz (score 1, accepted)

https://quant.stackexchange.com/a/59679

#### Why the formula in the book is wrong

The "formula" in the book for standard deviation of two assets is severely misleading, even wrong. You cannot add together SDs like that, you must add variances.

For two independent random variables $X_1$ and $X_2$ with weights $w_1$ and $w_2$ $$ \mathbb{V}[w_1X_1 + w_2X_2] = w_1^2\sigma_1^2 + w_2^2\sigma_2^2 $$ $$\implies$$ $$ \text{SD}[w_1X_1 + w_2X_2] = \sqrt{\mathbb{V}[w_1X_1 + w_2X_2]} = \sqrt{w_1^2\sigma_1^2 + w_2^2\sigma_2^2 }.$$

Also, you can see that it does not matter if any of the weights $w$ are negative, since they are squared.

#### How to combine two independent assets (uncorrelated)

If $A$ denotes an asset with return denoted by the random variable $R_A$, then the expected return is $\mu_A=\mathbb{E}[R_A]$, return variance is $\sigma^2_A=\mathbb{V}[R_A] \;\implies \; \text{SD}_A=\sigma_A=\sqrt{\mathbb{V}[R_A]}$

So now you have two assets

- Asset (portfolio) $S$ with $\mu_S = 0.15, \;\sigma_S=0.16$.

- Treasury Bill $B$ with $\mu_B = r_f = 0.05, \;\sigma_B=0.0$.

You create a new combined portfolio $P$ with some relative proportion of each asset. Let $w_S$ and $w_B$ denote these weights. E.g. 25% in asset $S$ and remaining 75% in B means that $w_S=0.25$, $w_B=1-w_S=0.75$. The random return for your portfolio can then be written as

$$ R_P = w_S R_S + w_B R_B = w_S R_S + (1-w_S) R_B$$ which has expected return

\begin{align} \mu_P=\mathbb{E}[R_P] &= w_S \mathbb{E}[R_S] + (1-w_S) \mathbb{E}[R_B] \\ &= w_S \cdot 0.15 + (1-w_S) \cdot 0.05 \\ &= 0.1\cdot w_S + 0.05, \end{align}

and variance \begin{align} \sigma_P^2 = \mathbb{V}[R_P] &= w_S^2 \mathbb{V}[R_S] + (1-w_S)^2 \mathbb{V}[R_B] \\ &= w_S^2 \cdot \sigma_S^2 + (1-w_S)^2 \cdot \sigma_B^2 \\ &= w_S^2 \cdot 0.16^2 + (1-w_S)^2 \cdot 0^2 \\ &= w_S^2 \cdot 0.16^2. \end{align} This means that standard deviation is given by \begin{align} \sigma_P= \text{SD}[R_P] &= \sqrt{\sigma_p^2} \\ &= 0.16 \cdot |w_S| \\ &= 0.16 \cdot w_S, \quad 0 \leq w_S \\ &= -0.16 \cdot w_S, \quad w_S \leq 0. \end{align}

Now isolate $w_S$ from this equation and substitute in into the expression for the expected return

- $w_S \leq 0$. \begin{align} \mu_P &= 0.1\cdot w_S + 0.05 = 0.1 \cdot \left(\tfrac{\sigma_P}{-0.16} \right) + 0.05 = -\frac{5}{8} \sigma_P + 0.05 \end{align}

- $w_S \geq 0$. \begin{align} \mu_P &= 0.1\cdot w_S + 0.05 = 0.1 \cdot \left(\tfrac{\sigma_P}{+0.16} \right) + 0.05 = +\frac{5}{8} \sigma_P + 0.05. \end{align}

Hence your combined portfolio lies on any of these two lines (depending on the weights). These two lines are the plotted green straight lines in your answer.

#### Anwers to your questions

> (1) For example if I sell short JNJ(-1) and buy 2x portfolio T, will SD of this short sold asset reduce risk(SD)of portfolio(2xT) like this?

Do the same as I did above, but now you have two risky assets $$\mu = w_{\text{JNJ}}\mu_{\text{JNJ}} + (1-w_{\text{T}})\mu_{\text{T}}$$

$$\sigma = \sqrt{w_{\text{JNJ}}^2 \sigma_{\text{JNJ}}^2+ (1-w_{\text{JNJ}})^2\sigma_{T}^2}$$

(2) If suppose interest rate for short selling 2% can I set E (Rp) for this combination (2xT-1JNJ) like this:

The expected return changes by 2% units, only in the case then your are short. So you substract 0.02 from expected return when the weight is negative. If the rate applies when going short either of the two assets, then you can substract it whenever $w_{\text{JNJ}}$ is negative or larger than 1.

## Answer by Miroslav Holub (score 0)

https://quant.stackexchange.com/a/59673

Personally if SD of JNJ is with plus it makes me more sense but I'll be very happy if someone confirms or refutes it.