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ssrn.com/abstract=1767338

AN ALGORITHM FOR FINDING A PORTFOLIO

WITH THE HIGHEST SHARPE RATIO

Valentyn Khokhlov, MBA

Corporate Finance and Investments Consultant

Tel: +380-50-4779316

E-mail: val.khokhlov@gmail.com

Abstract:

An optimal portfolio with the highest possible Sharpe ratio plays an important role for capital allocation

and performance evaluation. This paper introduces a simple algorithm for finding the Sharpe-optimal

portfolio without solving a non-linear problem. The results are tested on S&P 100 components in year

2010. They also address the issue of using arithmetic means or actual returns as the optimization inputs.

Keywords:

Portfolio optimization; Sharpe ratio; mean-variance optimization; capital allocation line.

Asset allocation is one of the key tasks of any portfolio manager. The modern portfolio

theory starting with Markowitz [1952] provided theoretical basis for asset allocation using the

mean-variance optimization (MVO). MVO involves solving a quadratic programming problem,

and one of the most known algorithms was proposed by William Sharpe (see Sharpe [1987]).

The Sharpe algorithm allows finding assets weights for the optimal portfolio given the investor’s

risk tolerance. While it works perfectly well for individual investors, for whom the risk tolerance

can be specified, its usage is questionable in case of funds or generic portfolios.

Practitioners often adopt some indicators of investments performance, and one of the

most popular is the Sharpe ratio, which is the ratio of the return in excess of the risk-free rate to

the standard deviation of investment returns. Fund and portfolio managers are commonly

evaluated using this indicator. So, maximization of the Sharpe ratio can be an important criterion

for asset allocation. Although the original Sharpe algorithm can’t directly solve the MVO

problem to maximize the Sharpe ratio of the optimal portfolio, with a minor modification

proposed in this paper it can altered to find the Sharpe-optimal portfolio.

Another usage of the Sharpe-optimal portfolio is to find an optimal capital allocation in

presence of a riskless asset. It can be shown that the optimal capital allocation line for the given

set of risky assets and a riskless asset is tangent to the efficient frontier of optimal portfolios at

the point that has the highest Sharpe ratio. Hence, finding the Sharpe-optimal portfolio allows

building the capital allocation line (CAL) for the given set of assets.

The Sharpe algorithm

The algorithm developed by William Sharpe is a quadratic programming problem solver,

which finds the assets weights that maximize the following objective function: