Finding Expression for Optimal Markowitz Weights

So there are two assets with return rates $r_1$ and $r_2$ which have identical variances and a correlation coefficient $p$. The risk free rate is $r_f$.

I need to find an expression for the optimal Markowitz weights for the two assets.

The books says that the answer is ($s_1 - p s_2$)/[($s_1-s_2$)*($1-p$)], but I'm not sure how this makes any sense as I don't know what the s's mean.

Thank you

• what do you mean by "optimal"? are you trying to find the minimum variance portfolio? – Mark Joshi Dec 7 '14 at 22:53

Let $s_1 = r_1 -r_f$ and $s_2 =r_2-r_f$. Then, this is the maximization problem: \begin{align*} & \ \max_{w_1, w_2} SR = \frac{\mu_p}{\sigma_p}, \, \mbox{ subject to}\\ \mu_p = & \ w_1 s_1 + w_2 s_2,\\ \sigma_p^2 = & \ \sigma^2\big(w_1^2 + w_2^2 + 2 w_1 w_2 \rho\big),\\ 1 = & \ w_1+w_2. \end{align*} By certain substitution, we convert the problem to the following \begin{align*} \max_{w_1} \frac{w_1(s_1-s_2)+s_2}{\sqrt{w_1^2 + (1-w_1)^2+2 w_1(1-w_1)\rho}} = \max_{w_1} \frac{w_1(s_1-s_2)+s_2}{\sqrt{2\big(w_1-w_1^2\big)(\rho-1) +1}}. \end{align*} From the first order condition, \begin{align*} (s_1-s_2)\big[2(w_1-w_1^2)(\rho-1)+1\big] -\big[w_1(s_1-s_2) + s_2\big](1-2w_1)(\rho-1)=0, \end{align*} we obtain that \begin{align*} (s_1-s_2) + w_1 (s_1-s_2) (\rho-1) - s_2(1-2w_1)(\rho-1)=0, \end{align*} and, consequently, \begin{align*} w_1 &= \frac{s_1-\rho s_2}{(s_1+s_2)(1-\rho)},\\ w_2 &= 1- w_1. \end{align*}

• I would think that you have a constrained convex problem and would need to solve it through lagrange multiplier method? – emcor Jun 25 '15 at 11:38
• @emcor, if you have a constrained convex problem, then the Karush-Kuhn-Tucker (KKT) theorem can be applied. For this question, the Lagrange multiplier method can also be applied, however, it will be much neater to deal with a single variable problem. – Gordon Jun 25 '15 at 12:49

I'm sorry for the late answer. I hope you passed the exam anyway!

TO answer your question, $s_2 = r_2-r_f$, that is the excess return over the risk free rate/asset.

However, there seems to be a typo in your formula, I believe it should be

$w_1 = \frac{s_1-ps_2}{(s_1+s_2)(1-p)}$, i.e. plus in the denominator.

$w_1$ is the weight for asset 1 and $w_2 = 1-w_1$ the weight for asset 2 that maximize the Sharpe ratio.

Ohh, and Hi Mark Joshi! :) (in the comments)

• Could you show us the derivation of this result? Thanks! – Ric Jan 19 '15 at 7:14

This is the general solution (where $C$ is the covariance matrix of returns): 