# Minimum Downside Volatility

## How can a portfolio’s negative returns be minimized?

To minimize a portfolio’s volatility, investors can optimize the variance-covariance matrix of the stock returns in question. By doing so, both negative and positive deviations from the mean returns are treated as equally undesirable.

However, it’s important to point out that investors are interested in minimizing negative returns, whereas they do not worry about positive deviations. A more appropriate risk measure in line with investors’ preferences should therefore only consider returns that fall below a certain threshold.

In 1959, Harry Markowitz, the father of modern portfolio theory, pitched semi-variance as a smart alternative to variance. *“Analyses based on S tend to produce better portfolios than those based on V. Variance considers extremely high and extremely low returns equally undesirable. An analysis based on V seeks to eliminate both extremes. An analysis based on S _{E}, on the other hand, concentrates on reducing losses.”* (Harry Markowitz, 1959)

Inspired by this, we developed a new family of indices, the Minimum Downside Volatility Index Series, which is constructed around an optimization strategy seeking to minimize downside volatility (i.e. the square-root of the semi-variance) of baskets of stocks.

Specifically, to minimize a portfolio’s risk in terms of downside volatility, the semi-covariance matrix of asset returns as introduced by Estrada (2008) is calculated. Using this heuristic definition, the semi-covariance matrix is optimized and a closed form solution that minimizes the downside volatility of the portfolio is found. This results in an index that has minimum risk, as defined in a more intuitive way.

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