- Commodities (DB Liquid Commoties Index)
- U.S. Stocks (Fama French top 30% by market capitalization)
- European Stocks (Stoxx 350 Index)
- Japanese Stocks (MSCI Japan)
- Emerging Market Stocks (MSCI EM)
- U.S. REITs (Dow Jones U.S. Real Estate Index)
- International REITs (Dow Jones Int’l Real Estate Index)
- Intermediate Treasuries (Barclays 7-10 Year Treasury Index)
- Long Treasuries (Barclays 20+ Year Treasury Index)
- Sharpe Ratio- Created by William Sharpe, it is a popular risk adjusted ratio that measures excess return per unit of volatility, according to the following formula:
- Omega Ratio- The Omega Ratio is a measure that takes into account all the different moments of the distribution. It separates return above and below a given threshold before calculating a ratio between the two means of the two vectors.
- Sortino Ratio- The Sortino Ratio is a modified version of the Sharpe ratio. While the Sharpe ratio takes into account both upside and downside volatility the Sortino ratio only uses the downside semi deviation in the denominator. The intuition behind this measure is that investors don’t penalize upside volatility, so the risk measure should only focus on downside risk.
- Calmar Ratio (MAR)- Published by the Managed Accounts Reports, this risk return measure uses the largest drawdown as its proxy for risk. It is widely used to gauge an investment’s annualized return to its maximum drawdown.
- DVR- This ratio, which was formalized by David Varadi, is concerned with both returns per unit of risk, and the linear fit of the price trajectory. Technically, it is the product of the Sharpe ratio and the coefficient of determination, or R-squared measure, fit to the price trajectory through time.
- Value at Risk- Used widely in the financial industry for measuring firm wide risk and exposure, the Value at Risk calculates, for a given confidence level, the magnitude of expected loss. We took this risk measure and incorporated the mean return to derive a risk return ratio.
- Conditional Value at Risk- A variation of the VaR, the Conditional Value at Risk is simply the median value of all return observations between the maximum loss and the VaR for a given confidence level. Theoretically, it does a better job of capturing the true tail risk. Similar to the above, we adjusted it for the mean return for a risk return ratio.
- Return to Max Loss Ratio- The max loss ratio uses the worst daily return as a proxy for risk.
- Return to Average Drawdown Ratio- The Average drawdown ratio simply takes return and adjust it for the average drawdown over the entire period.
- High Low Differential- More sophisticated in calculation, the High Low Differential takes in to account the current position relative to the highest and lowest prices in a prescribed period.
- Ulcer Index- Created by Peter Martin in 1987, the index takes into account risk from drawdowns as oppose to traditional volatility. It is derived from deviations from the most recent highs. The following is the pseudocode for computing the Ulcer Index (UI), the following is the equation employing the UI to derive the UPI.
MaxValue = 0
for T = 1 to NumOfPeriods do
else SumSq = SumSq + sqr(100 * ((Value[T] / MaxValue) – 1))
- Gain to Pain Ratio- Popularized in the book Hedge Fund Market Wizards by Jack Schwager, this ratio takes the sum of all positive periods divided by the sum of all the negative periods.
- Fractal Efficiency- The most efficient line segment between two points is a straight line. Essentially, fractal efficiency is the ratio between the straight-line magnitude of price change over the period divided by the distance the price actually traveled on its path. The equation below should help with the intuition.
Consistent with our previous post’s analytical framework, we show the distribution of performance statistics across all of the 44 universe/concentration combinations for each risk adjusted momentum method.
Below we plot the percentile performance of each system’s CAGR, Sharpe, and Maximum Drawdown, paying special attention to scores at the 5th percentile, because this quantile is standard for interpreting statistical significance. Whereas the instantaneous slope measure proved to be the most consistent performer at the 5th percentile among raw momentum metrics in Article 1, the Gain to Pain ratio seems to deliver the most consistent performance of all the risk adjusted momentum indicators.
Charts 55 through 57 show the average performance of each methodology with portfolio concentrations of 2 holdings through 5 holdings across all 11 asset universes tested.
Comparing the index to all the constituent systems from Table 1., we observe a material reduction in volatility and drawdown. Among the improved performance statistics include Sharpe, Maximum Drawdown, and rolling positive 12 month periods. Clearly the different measures of risk capture slightly different information, which offers diversification at more critical times.
In upcoming posts, we will introduce a variety of sizing algorithms. We will incorporate both traditional optimization procedures and heuristic methods to identify the optimal sizing combination. Stay tuned.