# An Intuitive Total Risk-Adjusted Performance Measure and Characteristics Matrix

How can anyone really know how to determine a good investment? People—including the overwhelming majority of financial advisors—are quick to resort to stories, demonstrably misguided heuristics, and arbitrary ancillary qualitative considerations in attempts to determine the quality of an investment, but these considerations are highly subjective and have proven time and again to be grossly unreliable, if not completely misleading. (Page 2)

Also in 1952, Arthur D. Roy laid the groundwork for measuring the relationship between mean (expected return) and variance (risk) of an asset or portfolio, which was formalized in 1966 by William Sharpe as the Sharpe ratio (S): (Page 2)

Beta measures how much the price of an asset or portfolio moves relative to the market: a beta of 1 means the price moves exactly with the market; a beta of 2 means the price moves exactly double that of the market; a beta of -1 means the price moves precisely opposite of the market, and a beta of 0 means price movements have no correlation to the market. (Page 3)

Beta is calculated by taking the covariance between the price movements of the asset or portfolio in question and that of the market divided by the variance of the market: (Page 3)

While popular, alpha, beta, C A PM , Sharpe ratio, and M 2 are all second-order statistical measures (variance) whose accuracy is dependent upon a normal, or Gaussian, distribution (bell curve) of returns, which is a grossly inaccurate assumption for investment behavior. (Page 4)

In the 1960s, Benoit Mandelbrot discovered that price changes in financial markets follow Lévy stable distributions, which have infinite variance. In other words, investments and financial markets simply do not follow the normal distribution. To capture the true total risk characteristics of an investment, all four statistical moments must be taken into account: mean, variance, skewness and kurtosis: (Page 4)

Mean is simply another term for average. For investment returns, which compound, geometric mean is typically used in lieu of its arithmetic counterpart. (Page 4)

Skewness measures the symmetry between losses and gains. For an example of positive skewness, on December 26, 2022, you could buy a call option on Lloyds Banking Group PLC (NYSE: LYG) for \$0.35 that expired seven months later with a strike price of \$2, which was \$0.25 below the share price of \$2.25 at the time. The most you could lose was \$0.35, but the share price of Lloyds could have gone up without limit before expiration. Kurtosis measures the thickness of the tails, or the outer regions, of a data set. With investments, tail events such as significant market corrections and crises tend to occur much more frequently (tail risk is higher) than what is predicted by the normal distribution. For example, the normal distribution predicts daily S&P 500 price movements greater than 4.86% to occur only once about every 6,900 years. In reality, such moves have occurred over 50 times in the last 72 years, including losses of 20.47% on October 19, 1987, 11.98% on March 16, 2020, and 9.04% on October 15, 2008. (Page 4)

In 2014, Michalis Kapsos, Nicos Christofides, and Berc Rustem simplified the Omega ratio into a function of expected values, which is much more useful as a portfolio optimization tool: (Page 5)

This new calculation, the Summers Total Risk-Adjusted Performance Measure ( ), represents the only true total SΩ risk-adjusted performance measure expressed as a scaled percentage against the market, which provides a user-friendly, intuitive, comprehensive performance measure: (Page 5)

To compare two assets or portfolios with , SΩ = ∞ M 2 return relative to the market. For an asset or portfolio that experiences no volatility where becomes the next best performance measure as it communicates volatility-adjusted expected M 2 = ∞ , E(Ri) implies a risk-free rate. (Page 6)

For risk-adjusted performance measures to have any meaning whatsoever, the timeframe of the data set being analyzed must encompass a minimum of one complete market cycle and ideally capture every secular macroeconomic environment. If you don’t know how an asset or portfolio performs across a true market bottom, you know nothing of its downside exposure, and by extension, any of its risk-adjusted performance characteristics. (Page 6)

What about black swan events (BSEs): events that can happen outside the limits of previous experience? It’s true that by definition BSEs cannot be accounted or planned for, but we can calculate the probability of a BSE occurring. It’s a surprisingly simple calculation equal to the inverse of the number of data points being analyzed expressed as a percentage: (Page 6)

Nassim Taleb’s solution is what he coins the barbell: hold 90% of a portfolio in risk-free assets with the remaining 10% in super high-yield, high-risk investments like long deep out-of-the-money puts. The problems with this approach are 1) there is no such thing as a truly risk-free asset immune to the effects of a black swan event, 2) counterparty risk, and 3) investment time horizon. (Page 7)

On the high-yield end of the barbell, investments that bank on everything crashing have significant counterparty risk as we saw in 2008 with the likes of the UK branch of Lehman Brothers or the dozens of hedge funds that failed. A perfectly designed investment has no value if its contractual obligations can’t be met. (Page 7)

Maximum drawdown (MDD) is the measure of the greatest loss peak-to-trough experienced by an asset or portfolio. Maximum drawdown duration (MDDD) is commonly defined as the time elapsed between the high point used to determine maximum drawdown and the point a new high point was reached (Page 7)

For liquidity, MDDDA is insufficient. The time required to get back to zero is not helpful. If a return to zero was sufficient, the investor would simply hold cash avoiding additional risk. It’s important to know how much time is required to recover losses plus accrued expected return ( MDDDS (Page 8)

The difference between MDDDS and MDDDA can be substantial. In figure 1, we see the S&P 500’s performance across a timeframe of 38 years (December 1984 - December 2022). is approximately EA(Ri) (yellow line) is 6 years 9 months (June 2000 - March 2007). 8.35% (blue line). MDDDA line) is 33 years (June 1987 - June 2020). MDDDS (red (Page 8)

Of the risk-adjusted performance measures, is the only SΩ one that captures all four statistical moments while providing a user-friendly output as a percentage relative to the market. SIC paints an accurate, comprehensive picture of the risk, return, and liquidity characteristics of any asset or portfolio regardless of asset class—from equities to real property—and its predictive value can be easily measured. (Page 10)