<Text-field style="Heading 1" layout="Heading 1">Introduction</Text-field>

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Optimising Asset Allocation by

Maximising the Sharpe Ratio

Samir Khan

Adept Scientific plc

<Text-field style="Heading 1" layout="Heading 1">Introduction</Text-field>

The Sharpe ratio quantifies how effectively a portfolio of risky assets utilises risk to maximise return. It is defined as follows.

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The expected portfolio return is predicted from historic data, the standard deviation of the asset mix is traditionally used as a proxy for risk (or volatility) and the risk free return is the return that can be expected from a zero-risk investment (i.e. the interest on US Treasury Bills or the redemption yield on UK gilts). A higher Sharpe Ratio essentially signifies a more risk efficient portfolio.

This application calculates the optimum asset mix for a portfolio of stocks by maximising its Sharpe ratio.

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<Text-field style="Heading 1" layout="Heading 1">Historic Returns and other Data</Text-field>

Number of stocks in the portfolio

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The following vectors contain historic returns of the stocks over 12 time periods.

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QyQ+SS5yZXR1cm5zU3RvY2syRzYiLUkkPHxncj5HSShfc3lzbGliR0YlNiMtSSQ8LD5HRig2LiQiJTY2ISIkJCIlOTZGLyQiJUI4Ri8kIiUsNUYvJCIlIlIiRi8kIiVCN0YvJCIlZzdGLyQiJTU3Ri8kIiVXNkYvJCIlNzlGLyQiJCgpKkYvJCIlYDpGLyEiIg==

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The risk free return.

QyQ+SS9yaXNrRnJlZVJldHVybkc2IiQiJVw1ISIkISIi

<Text-field style="Heading 1" layout="Heading 1">Sharpe Ratio and Investment Constraints</Text-field>

Define a vector that contains the mean returns.

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Define the covariance matrix.

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For the three stocks under consideration the Sharpe ratio is defined as follows, where the arguments to the function is the fraction of capital invested in each stock.

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Additionally, the fraction of capital invested in each stock must add up to 1.

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<Text-field style="Heading 1" layout="Heading 1">Maximising the Sharpe Ratio</Text-field>

The Sharpe Ratio is now maximised with respect to the constraint.

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Hence the optimised portfolio contains the three stocks in the following fractions.

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This asset mix has a Sharpe Ratio of

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A value of 1 or higher is traditionally regarded as good while a value over 2 is considered very good.

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