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Markowitz Portfolio Selection

Problem: Selecting assets to meet a desired return at minimum variance.

(In the March, 1952, issue of Journal of Finance, Harry M. Markowitz published an article called "Portfolio Selection". In it, he demonstrated how to reduce the standard deviation of returns on asset portfolios by selecting assets which don't move in exactly the same ways. At the same time, he laid down some basic principles for establishing an advantageous relationship between risk and return, and his work is still in use forty years later.)

You're considering investing in three assets and historical data reveal that the return from each asset has fluctuated over time. You want to reduce variability, or risk, by spreading your investment over the three stocks.

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From the historical data, you have calculated an expected return, the variance of the return rate, and the covariance of the return between the different assets. Variance is a measure of the fluctuation in the return - the higher the variance, the riskier the investment. The covariance is a measure of the correlation of return fluctuations of one stock with the fluctuations of another. High covariance indicates that an increase in one stock's return is likely to correspond to an increase in the other. A low covariance means the return rates are relatively independent and a negative covariance means that an increase in one stock's return is likely to correspond to a decrease in the other.

You have a target return of 15%. What percentages of your funds should you invest in each of the three assets to achieve this target and minimise the variance (or risk) of the portfolio? As an additional safety feature, you decide to invest no more than 75% in any single asset.

The objective is to determine the percent to invest in each asset while minimising risk of the entire portfolio.

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