optimization - Maximum Sharpe portfolio (no short selling restrictions)

Suppose we have $n$ assets whose expected return vector is $r$ and is positive, and whose covariance matrix is $\Sigma$. Is there a closed form or quasi closed form (like the eigenvector of a matrix etc) solution to the portfolio weights vector $w$ which maximises the Sharpe ratio $w^T r/\sqrt{w^T \Sigma w}$? (Assume no restriction on short selling.)

If in general its not possible, does a closed form solution exists for low dimensional cases like $n=2,3$? I remember being taught a closed form solution for $n=2$ in the class. The expression is a bit complicated and unfortunately isn't accompanied with a proof.

I tried differentiating wrt $w$ but the gradient is a mess. I don't think we can get anything useful out of it.

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Per the comment, the linked source claims that the maximiser is in the direction of $\Sigma^{-1}r$. Could someone explain or suggest otherwise? Thanks.

Answer

There are two cases, where short sales are allowed: With riskless lending and borrowing and without. As mentioned in the comments, you just have to solve a linear system.

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With riskless lending and borrowing

The existence of a riskless lending and borrowing rate $r_f$ implies that there is a single portfolio of risky assets, that is preferred to all other portfolios. You want to maximize the function

$$\theta = \frac{\bar{R}_P - R_f}{\sigma_p}$$

subject to the constraint $\sum_{i=1}^N X_i = 1$.

$\bar{R}_P$ denotes the average portfolio return of $N$ assets $i$ with weights $X_i$ and $\sigma_P$ is the standard deviation of portfolio returns. From the definition of portfolio return, $R_f = 1 \cdot R_f = \left( \sum_{i=1}^N X_i \right) \cdot R_f = \sum_{i=1}^N (X_iR_f)$, and their standard deviation, you get

$$\theta = \frac{\sum_{i=1}^N X_i(\bar{R}_i - R_f)}{\left[ \sum_{i=1}^N (X_i^2 \sigma_i^2) + \sum_{i=1}^N \sum_{\substack{j=1 \\ j\neq i}}^N X_iX_j\sigma_{ij} \right]^{\frac{1}{2}}}$$ where $\sigma{ij}$ is the covariance of asset returns $r_i$ and $r_j$.

It is a simple maximization problem, you take the derivative with respect to each variable and set it equal to zero. You get a system of equations:

1. $\frac{d\theta}{dX_1}=0$



2. $\frac{d\theta}{dX_2}=0$


3. $\frac{d\theta}{dX_i}=0$


4. ...

In general:

$$\frac{d\theta}{dX_i}=-(\lambda X_1\sigma_{1i}+\lambda X_2\sigma_{2i}+ ... + \lambda X_i\sigma_{i}^2+ ...+\lambda X_{N-1}\sigma_{N-1,i}+\lambda X_{N}\sigma_{Ni})+\bar{R}_i - R_f = 0$$

with

$$\lambda = \frac{\sum_{i=1}^N X_i(\bar{R}_i - R_f)}{ \sum_{i=1}^N (X_i^2 \sigma_i^2) + \sum_{i=1}^N \sum_{\substack{j=1 \\ j\neq i}}^N X_iX_j\sigma_{ij} }$$

After defining a new variable $Z_k = \lambda X_k$, the formulation simplifies to the system (lets call it A):

$$\bar{R}_1 - R_f = Z_1\sigma_1^2 + Z_2 \sigma_{12}+Z_3 \sigma_{13}+Z_N \sigma_{1N}$$ $$\bar{R}_2 - R_f = Z_1\sigma_{12} + Z_2 \sigma_2^2+Z_3 \sigma_{23}+Z_N \sigma_{2N}$$ $$...$$ $$\bar{R}_N - R_f = Z_1\sigma_{1N} + Z_2 \sigma_{2N}+Z_3 \sigma_{3N}+Z_N \sigma_N^2$$

The $Z_N$ are proportional to the optimum amount to invest in each security. First, solve the above system for $Z_N$, then the optimum weight $X_k$ for each asset $k$ is

$$X_k = \frac{Z_k}{\sum_{i=1}^N Z_i}$$

Without riskless lending and borrowing

If a riskless rate $R_f$ is not available, the solution above has to be modified. Assume that $R_f$ exists and find the optimum portfolio with the method above. Then assume a different $R_f$ and find the optimum portfolio that corresponds to this slightly changed riskless rate. Continue changing the assumed rate until the full efficient frontier is determined.

Consider again the linear system A. However, we do not have to substitute in a particular value of $R_f$. We can simply leave $R_f$ as a general parameter and solve for $Z_k$ in terms of $R_f$. This results in a solution of the form

$$Z_k = C_{0k} + C_{1k}R_f$$

where $C_{0k}$ and $C_{1k}$ are constants. They have a different value for each asset $k$, but that value does not change with changes in $R_f$. Once the $Z_k$ are determined as functions of $R_f$, we could vary $R_f$ to determine the amount to invest in each security at various points along the efficient frontier.

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Additional remark

Lintner(1965) has an alternative definition of short sales which is more realistic. He assumes correctly that when an investor sells stock short, cash is not received but rather is held as collateral. Furthermore, the investor must put up an additional amount of cash equal to the amount of stock he or she sells short. In summary, the constrain here becomes $\sum_{i=1}^N \left| X_i \right| = 1$.

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Reference

Elton/Gruber/Brown/Götzmann (2014), Modern Portfolio Theory and Investment Analysis, ed. 9, John Wiley & Sons.