June 20, 2017
In 2010, a paper published by the Financial Analysts Journal proposed a measure of financial turbulence based on the Mahalanobis Distance. The latter measures the daily surprise move of all financial assets compared to historical volatilities and correlations.
The Mahalanobis Distance is expressed as followed:
R is the vector of daily returns for each asset, ? is the vector average rolling return for each asset, ? is the historical covariance matrix. To filter noise, the rolling period is set at 3 years of daily returns (750 days).
For a diagonal covariance matrix where all correlations would be zero, the distance becomes:
In the case of uncorrelated markets, the distance is the square root of the quadratic sum of the standardized magnitude of the moves measured in units of standard deviations. Therefore, one can also derive the normalized Distance by dividing by the number of assets composing the investment universe:
To provide an intuition of the impact of correlations beyond the magnitude of the move, we measure the Distance D of a two-asset portfolio for various correlation scenarios (0, +0.9 or -0.9). As shown in Figure 1, the Distance « overshoots » if the realized moves deviate from the predicted pattern reflecting the long-run co-movement. This difference is deemed a « correlation surprise » in the distance. For instance, a correlation surprise occurs whenever two positively (negatively) correlated assets experience opposing (similar) moves. By analogy, all other things being equal, there is no such surprise if the same two assets move in lockstep. In other words, a correlation surprise exists if the realized moves deviate from the predicted pattern taking long-run co-movement into account. Moreover, the curve iso-Distance is an ellipsoid (see Figure 2 for an example of a two-asset portfolio) which is skewed to the sign of correlation. Hence, the distance of expected joint moves is relatively lower and conversely higher for unexpected joint moves.
For a multi-asset universe, the Distance is a single measure of financial turbulence taking into account the entire covariance structure. It aggregates the magnitude and correlation surprise into a single measure. Figure 3 presents the Distance for a basket of 15 assets comprising equity index, commodity and bond futures. As the Distance is highly volatile, it is averaged over a time horizon of 60 days (3 months). Most strikingly, each major crisis is characterized by a peak in the Distance irrespective of its nature, e.g. surging yields (1994), the Taper Tantrum (2013), 9/11 Attacks (2001), Subprime Credit Crisis (2007), Lehman Crash (2008) or Euro Crisis (2010-2012).
Finally, the Turbulence Distance is compared to Riskelia’s own proprietary Risk Aversion metric as shown in Figures 4 and 5. The Riskelia Risk Aversion is calculated from implied risk premia (implied volatilities, CDS, credit spreads) rather than historical returns and risk measures (for a detailed description see Figure 4). Both indicators are strongly correlated (+0.65). They also share the same asymmetry property to the upside conveying that fear is more pronounced than relief. Meanwhile, to compensate for the asymmetry, relief is more persistent than fear.
We then evaluate how well both measures actually predict financial turmoil. For this purpose, we measure the statistical properties of the S&P 500, i.e. its monthly return and volatility, conditional on the indicators at the end of the previous month. Our results presented in Figure 6 show that Riskelia’s Risk Aversion best characterizes the volatility regime by splitting financial markets into two distinct categories:
When the Risk Aversion stands above the median, future monthly returns of S&P 500 are 40% more volatile and furthermore less attractive, thus displaying a strong negative skew in a high Risk Aversion regime.
When the Risk Aversion stands below the median, future monthly returns are less volatile and more attractive.
The Turbulence Distance has an ability to discriminate high and low volatility regimes but fails to predict returns. This important shortcoming makes the Distance measure thus inferior to the Riskelia Risk Aversion indicator. In our view, this is probably because timely implied risk premia hold an informational advantage over lagging historical risk data.
Therefore, Riskelia’s Risk Aversion wins the match. Easily.
Figure 1: Calculation of the Mahalanobis Distance for a two assets universe with various scenarios of correlation, magnitude and direction of moves.
Figure 2: Iso-Distance Curves in a Two-Asset Portfolio for Varying Correlations (rho=0,+0.5,-0.5). The Assumed Volatility if Asset 1 (Asset 2) is 10% (20%).
2.a. No Correlation (rho=0). Iso-Distance Ellipse (D=2, D=3, D=4).
2.c. Negative Correlation (rho=-0.5). Iso-Distance Ellipse (D=2, D=3, D=4).
Figure 3: Turbulence Distance for 15 Assets (5 Equity Indices, 5 Commodities and Currencies, 5 Sovereign Bonds).
Figure 4: Description and Chart of Riskelia’s Risk Aversion Indicator.
For a given asset class, the risk aversion indicator rates the reward market participants require for risk taking (implied volatilities on equities indices, currencies and commodities, Banks’ CDS, Sovereigns’ CDS, Corporate Credit Spreads, Emerging Spread). The scores are expressed in numbers of standard deviations to a set of moving averages (from 3 months to 2 years). They are averaged into a Global Risk Indicator characterizing the liquidity regime of financial markets.
Figure 5: Comparison of the Turbulence Distance and Riskelia’s Proprietary Risk Aversion Indicator.
Figure 6: Future S&P 500 Return and Volatility Conditional on Preceding Risk Aversion and Turbulence Distance (as of the Previous Month).
Published on Riskelia’s Blog