# Comparison between Optimization Strategies

All results are based on the CVaR dispersion measure (except for the equally weighted portfolio used as a standard benchmark) for portfolios containing the following instruments: GLD, TLT, XLV, IHI, VGT. Furthermore, the optimized portfolios are compared to a sample of 2000 portfolios with random weights summing up to 100%.

An annualized targeted expected portfolio return of 10% was chosed for the strategies Risk and Diverse.

The gray histogram and curve of following graph show the density of the CVaR -95% for a daily horizon of the 2000 randomly chosen portfolios, meaning that most of them have an expected shortfall between -1.7% and -2.5%. In other words, choosing a random portfolio with the above instruments will lead to a portfolio with an expected average loss of around -2% for 5% of the worst performing days.

The vertical lines correspond each to an optimization strategy explained above. Note that the minimum risk strategy finds the combination of underlying instruments leading to the portfolio with the lowest CVaR, followed by the strategies trying to get the most diverse portfolio under the constraint of minimal risk.

The two strategies targeting an expected portfolio return (Risk, Diverse) have a slighlty higher risk whereas the strategy maximizing the Sharpe ratio has the highest risk. Also shown is the CVaR of the equally weighted portfolio which has the same risk as most of the random portfolios as it is located at the maximum .

The following graph shows the same results as above but as a risk-return scatter plot. Each of the 2000 random portfolios is shown as a gray dot while the optimized portfolios are shown as larger colored dots.

The x-axis corresponds to the CVaR-95% with a horizon of a day and the y-axis the annualized expected return.

The minimum risk portfolio (MinRisk) is the portfolio at the most left part of the graph whereas the MaxDiverse portfolio trades a bit of risk for more diversification, albeit with a slightly higher expected return.

Since a targeted portfolio return of 10% was chosen, it is not surprising that the 2 strategies Risk and Diverse manage to come close to a 10% annualized expected return. Note that the strategy Diverse is riskier, yet doesn't expected a higher return.

The strategy with the highest expected return is Sharpe. Furthermore, the equally weighted portfolio is in the middle of the possible outcomes and thus not on the efficient frontier.

The following graph shows the Sharpe ratio where each of the 2000 random portfolios is shown as a gray dot while the optimized portfolios are shown as larger colored dots.

The x-axis corresponds to the annualized expected returns [%] with a horizon of a day and the y-axis the Sharpe ratio.

The strategy Sharpe is unsurprisingly the portfolio with the highest Sharpe ratio. Furthermore, choosing the strategies Diverse and Risk also lead to a good Sharpe ratio. On the other hand, the 2 strategies solely minimizing CVaR (MinRisk, MaxDiverse) lead to poor Sharpe ratios.