# 2 stocks, no shorting vs shorting. (concrete questions, mean-variance)

I'd appreciate help with the following questions.

Suppose there are two stocks $A$ and $B$ with expected returns $E_A, E_B >0$ and volatilities $v_A, v_B >0$, respectively . Also, suppose their correlation is $\rho_{AB} = \rho <0$. Given a dollar amount $D>0$ to invest without shorting, how should $D$ be invested in $A,B$ so that i) Expected return is maximized? ii) Overall volatility is minimized?

My second question: same questions (i) and (ii) but now with shorting allowed.

Intuitively, volatility is a standard deviation of a stock's price (or return) over a fixed period of time. Therefore (for a fixed period of time), in the case $E_A > E_B$ and $v_A>v_B$, I'd expect a 'middle-ground' determined by comparing the ratios $E_A/v_A$ with $E_B/v_B$.

Finally, how might Sharpe ratio play into these questions? (would it measure the 'strength' of a strategy?) Also, how approach this question for $n>2$ stocks $A_1,\ldots, A_n$? I would think to set $A = A_1$ and $B = A_2 +\cdots + A_n$

Concrete (mathematical) answers as well as general references to tackle these problems is appreciated.

• Welcome to Quant.SE! Are you familiar with Mean-Variance optimization? – Bob Jansen Jul 7 '14 at 10:40
• This seems like a mean-variance optimization for a portfolio of 2 stocks. "Pairs trading" in the title is misleading. – madilyn Jul 7 '14 at 12:02
• @BobJansen, thanks for the tip. Looked it up and I bet the answer to my question is contained in Ch. 6 of "Modern portfolio theory..." (9 ed) by Brown et al. – user9482 Jul 7 '14 at 16:49

## 2 Answers

The concrete (general) answer to part (ii) of my question seems to be contained in Equation 8 of the following link: http://www.columbia.edu/~ks20/FE-Notes/4700-07-Notes-portfolio-I.pdf

In particular, interpreting $\sigma$ as volatility, take for example $E_A=0.10,\sigma_A=0.15,E_B=0.25,\sigma_B=0.40$ and $\rho =−0.2$.

I get that about 83 percent of the money should be invested in $A$ and 17 percent for $B$. Namely, if $D = 1000$, then about 830 into $A$ and 170 into $B$. No shorting is required in this case since $\rho <0$.

The return, as calculated from Eq. (4) in the above pdf, in this case is about $+125$ in profit.

Update. Regarding (i). Please correct me if I'm wrong, but as for "maximizing return" it seems we want to maximize the following function: $R(a) = aE_A + (1-a)E_B$. Since $R(a)$ is linear in $a$, then we see that $\max R(a)$ subject to $0\leq a \leq 1$, occurs at $a = 1$ iff $E_A\geq E_B$ or either at $a = 0$ iff $E_A < E_B$. Certainly though, the approach should be different if we are maximizing return with respect to a specified level of risk or volatility in the desired portfolio (help on this last point would be appreciated).

Firstly, to answer your question for part (i), this part of the question makes no sense - your expected return is unbounded and is asymptotically linear with respect to risk.

Let ${\bf w}\in\mathbb{R}^{2}$ denote your vector of weights, $\Omega$ denote the covariance matrix and $\iota$ denote a unit exposure vector (defined by $\iota_{j}:=1\ \forall j, j \in \mathbb{Z}^{*}$). We have:

$\mu_{P}={\bf w}\cdot\mu=\sum w_{i}\mu_{i}$

$\sigma_{P}^{2}={\bf w}^{T}\Omega{\bf w}=\sum_{i\neq j}w_{i}w_{j}\sigma_{i}\sigma_{j}\rho_{ij}$

$n\in\mathbb{Z}^{*}$ equality constraints of form $g_{k}\left({\bf w}\right)=c,\ c\in\mathbb{R},k=1,...,n$ are imposed. We define the Lagrangian:

$\mathcal{L}\left({\bf w},k_{1},...,k_{n}\right):=\dfrac{1}{2}{\bf w}^{T}\Omega{\bf w}+k_{1}g_{k}\left({\bf w}\right)+...k_{2}g_{k}\left({\bf w}\right)$

For example, for a single constraint that you are fully invested, you solve for the Lagrange multipliers for:

$\mathcal{L}\left({\bf w},l\right)=\dfrac{1}{2}{\bf w}^{T}\Omega{\bf w}+l\left(1-{\bf w}\cdot\iota\right)$

Then, you can find the minimum variance (volatility) portfolio:

${\bf w}_{min}=l\Omega^{-1}\iota=\dfrac{\Omega^{-1}}{\iota^{T}\Omega^{-1}\iota}$

This answers your question for part (ii).

As for your remaining question about shorting, this could be expressed as an inequality constraint ${\bf w}\geq{\bf 0}$ for ${\bf w},{\bf 0}\in\mathbb{R}^{2}$. Then, you can formulate this as a classical nonlinear programming problem and solve the first order necessary conditions (Karush-Kuhn-Tucker conditions).

• Your first point seems quite odd. If the investment amount $D$ and the (average) returns $E_A, E_B$ are finite (implicit), how is it possible for a portfolio comprised solely of stocks $A$ and $B$ to have unbounded expected return? Also, what precisely do you mean by "risk" if the only input is $D, E_A, E_B, v_A, v_B$ (which are all finite)? – user9482 Jul 7 '14 at 12:44
• very concretely: suppose $D = 1000$, $E_A = 0.10, v_A=0.15, E_B = 0.25, v_B = 0.40$ and $\rho = -0.2$. How do you get an unbounded expected return from investing all of $D$ into some combination of $A$ and $B$ in this situation? – user9482 Jul 7 '14 at 12:55
• @user9482 Sure, if you have a budget constraint and no access to leverage, then the hyperbola will be cut off at some finite limit. But the answer that you are seeking is still at max $\sigma_{p}^2$ and should have no more than 2 trivial solutions - there seems no sensible purpose for this besides illustrating the basic tenet that "more risk, more reward". – madilyn Jul 7 '14 at 13:51
• Thanks, although the "budget constraint" of a fixed D>0 is already stated in the question. If you could revise your answer and solve the concrete case I mention (with D=1000), I'd be happy to accept your answer. The general theory you illustrate for (ii) is copacetic while your comment for (i) is simply wrong in this situation – user9482 Jul 7 '14 at 16:54