Financial Turbulence

In Chapter 12, we introduced financial turbulence, which is based on a statistical quantity called the Mahalanobis distance.¹ In a single number, financial turbulence characterizes the degree of unusualness in a cross section of asset class returns. It captures not only extreme price moves but also unusual correlations that affect a portfolio's diversification. The turbulence score for any given month , equals:

(16.1)

In Equation (16.1), is a vector of monthly returns across asset classes, is a vector of average returns for each asset class over the full 40‐year sample, and is the inverse of the covariance matrix computed from the 40‐year sample. We multiply by so that the expected value of turbulence equals 1. The inverse covariance matrix puts the monthly returns in context, and this step is critically important. For example, stock prices may regularly move by 5 or 10 percent per month, but for bonds these returns would be extreme. Likewise, the divergence of two asset classes is surprising if they are positively correlated, but not if they are negatively correlated. The inverse covariance matrix embeds all of the expected relationships across asset classes, based on the full sample of data.

We apply this formula to the six major asset classes in our empirical example (excluding cash equivalents): U.S. equities, foreign developed market equities, emerging market equities, Treasury bonds, corporate bonds, and commodities. Figure 16.2 shows historical monthly turbulence since 1976. It clearly spikes during well‐known events such as the Black Monday stock market crash of 1987, the Russian debt default in 1998, and the global financial crisis in 2008. It also flags some less obvious periods of stress. These events carry important consequences for portfolio risk, even if they do not make newspaper headlines.

Portfolio Construction with Conditional Risk Estimates

Regime variables are informative if they describe differences in investment performance. Financial turbulence passes this test. Table 16.1 shows asset class standard deviations and correlations corresponding to the 10 percent most turbulent months, which we call the turbulent sample, and the remaining 90 percent of months, which we call the nonturbulent sample.

Standard deviations are much higher during turbulent periods, as we should expect. Most of the correlations also increase when conditions are turbulent; hence, there is less potential for diversification. These conditions usually cause traditional portfolios to perform poorly. Should we optimize our portfolio for turbulent regimes instead? We can, but we face the following trade‐off. The more we engineer a portfolio to protect it from financial turbulence, the more we sacrifice optimality in nonturbulent periods. On the other hand, if we choose a portfolio that is optimized for nonturbulent times, we should brace for discomfort when turbulence occurs. Perhaps we should compromise, by tilting our portfolio toward asset classes that are more resilient to turbulence. Ideally, the portfolio will perform well when turbulence prevails without underperforming significantly during nonturbulent periods.

To implement this strategy, we optimize a portfolio based on a covariance matrix that blends covariances from both regimes. In our example, we use a simple average of the turbulent and nonturbulent covariance matrices. Because the turbulent sample represents only 10 percent of history, a 50 percent weight constitutes a substantial tilt toward the estimates that prevailed during turbulent regimes. Table 16.2 shows the moderate portfolio from Chapter 2 alongside a new portfolio based on the turbulence‐conditioned covariance matrix. The allocations are meaningfully different, even though both portfolios target a 7.5 percent return and use the same expected return assumptions. The blended‐covariance solution favors emerging market equities compared to U.S. equities, and it substitutes cash equivalents for corporate bonds. The full‐sample standard deviation increases from 10.8 percent to 11.5 percent, and the standard deviation in nonturbulent periods increases from 8.8 percent to 9.8 percent. However, the new portfolio reduces the standard deviation during turbulent regimes from 18.8 percent to 17.0 percent. This trade‐off may or may not be worthwhile. The benefit this portfolio provides in turbulent regimes is modest. Moreover, we receive this benefit only 10 percent of the time, while we experience the cost of higher volatility 90 percent of the time. This result underscores the limitation of a static asset mix.

Identifying a Predictive Signal

A good predictive signal is necessary, but not sufficient, to create a profitable tactical asset allocation strategy. Some investors might prefer to rely on their qualitative judgment, but we focus here on data‐driven signals, in part because they are naturally more transparent and replicable. Let's continue with our analysis of turbulence. As we soon show, turbulence is predictive by virtue of its persistence.

There are, of course, other signals that might add value. Economic indicators such as growth and inflation might be valuable. Another indicator that has gained widespread traction in recent years is the absorption ratio, introduced by Kritzman, Li, Page, and Rigobon (2011). Like turbulence, it is derived statistically and based on asset prices. Whereas turbulence represents extreme dislocations in asset prices, the absorption ratio reveals whether the risk within an asset universe is highly concentrated or diffuse. When a small set of factors explains a large fraction of total risk, portfolios composed from those assets are vulnerable to large drawdowns because shocks tend to propagate quickly and broadly in such a setting. When risk is broadly distributed across many factors, negative shocks are more likely to have a localized and contained effect on portfolios. The absorption ratio is based on Principal Components Analysis (described in detail in Chapter 18). It equals the fraction of total variation in returns that is explained—or absorbed—by a subset of the most important statistical factors. Statistical factors are nothing more than sets of portfolio weights across a universe of assets, chosen in such a way that they are all uncorrelated and explain successively smaller fractions of total risk. Statistical factors do not have descriptive names, but they are convenient in that they always explain variation in the data as efficiently as possible.

We do not include empirical tests of economic indicators or the absorption ratio here, but the technique we are about to describe using financial turbulence applies as well to economic variables and the absorption ratio.

Detecting Regimes: Hidden Markov Models

We begin by defining a regime as a period in which an indicator, in this case financial turbulence, is above or below a particular threshold. We might devise a trading rule that positions a portfolio more defensively when turbulence is elevated and more aggressively when it is not. Even a simple rule like this might add value. However, simple thresholds have three important limitations. First, they fail to recognize persistence, which can lead to excessive trading. Second, by focusing only on the level of the indicator, this rule fails to account for shifts in its volatility, which may offer additional insight. Third, such a rule is sensitive to the choice of the threshold, which in many cases is an arbitrary choice.

Hidden Markov models offer a more sophisticated alternative. These models presume that the data we observe emanates from multiple regimes, each of which has a degree of persistence from one period to the next. For example, if we are in regime A this month, regime A might be more likely to prevail next month as well. But there is also a chance that the regime will shift abruptly from A to B. The term “Markov” refers to the assumption that the underlying regime variable follows a Markov process in that next period's regime probability depends only on the regime we are in today. Unfortunately, we never know for certain which regime we are in, even with the benefit of hindsight. As their name suggests, the regimes in hidden Markov models are indeed hidden variables (sometimes called latent variables). We rely on a statistical method called Maximum Likelihood Estimation (MLE) to infer from the data which regime is most likely to explain each observation, given all the data that preceded and followed it. The solution is obtained by using a well‐known and highly reliable search method called the Baum‐Welch algorithm. (See the Appendix to this chapter for more detail.) By allowing for the presence of multiple regimes, we are better able to explain the complex behavior of many real‐world variables. Recall from Chapter 13 that the composite mixture of two normal distributions does not yield another normal distribution, but a fat‐tailed one.

Kritzman, Page, and Turkington (2012) use hidden Markov models to solve for the two normal distributions and regime transition probabilities that best explain historical data for U.S. gross national product, U.S. consumer price index, and market turbulence in U.S. equities and currencies. In an intentionally contrived example in which data is drawn from known regimes that switch over time, the authors show that the true underlying behavior is much more likely to be recovered by a hidden Markov model than a simple threshold. They also provide empirical evidence that economic data conforms to regimes. The authors demonstrate that relatively simple regime‐switching strategies across risk premiums and asset classes outperform their static investment alternatives.

Let's explore a case study to make these concepts more tangible. We use financial turbulence as our regime indicator. The only parameters we need to specify are the number of regimes and the type of distribution we envision for our variable of interest in each regime. The model determines everything else. For this example, we ask the model to identify three distinct regimes that best explain historical turbulence.² It returns to us a transition matrix stating the probability of occurrence for each regime in the next period as a condition of the regime that prevails in the current period. The top panel of Table 16.3 shows the results of the model. We see that regime persistence, which comes from the diagonal entries of the transition matrix, is above 50 percent in all cases, which indicates predictive potential. We also see that it varies across regimes. The first regime, which happens to be the most persistent, has low average turbulence and low dispersion in outcomes. We label it “calm.” We label the second regime “moderate” because it is characterized by a higher average return and a higher standard deviation. Finally, the third regime represents “turbulent” conditions. It is the least stable, yet still predictive. Keep in mind that we do not predetermine how the persistence, averages, and standard deviations align across regimes. The fact that regimes with worse investment conditions on average are also more uncertain and more fleeting is a feature of markets that turbulence and the hidden Markov model have jointly uncovered. If we tried to fit the same model on random data, it would either fail to find a solution or it would produce three indistinguishable regimes.

Now that we know more about the dynamics of turbulence, we link this information to asset class returns. The middle panel of Table 16.3 shows the average annualized return of our seven asset classes in each regime, and the bottom panel presents standard deviations. Again, recall that even with hindsight we do not know for certain which regime corresponds to each month in history. For this analysis, we simply assign the most likely regime to each month. Our findings are intuitive: Calm periods are kind to investments, moderate periods have markedly elevated volatility despite sometimes‐higher mean returns, and turbulent periods have very high risk and typically negative average returns. Figure 16.3 shows the probabilities associated with each regime through time.

Testing out of Sample

To test the efficacy of our tactical asset allocation rule, we perform an out‐of‐sample backtest in which tactical decisions rely only on data available at each point in time. Our original turbulence calculation uses information spanning the entire 40‐year sample, so we must recalculate it. Each month, we compute turbulence using the formula shown previously, but now we estimate means and covariances using the prior 60 months of returns. This process produces turbulence data starting in 1980. Next, we calibrate the hidden Markov model using historical data only. Starting in December 1990, we calibrate the model using the prior 10 years of turbulence, and we append new data each month to form a growing lookback window on which we recalibrate the model. Figure 16.4 shows the probability forecast of each regime per month, out of sample. Each month, the model supplies an updated probability that the current regime is calm, moderate, or turbulent. Next, we forecast the probability that each regime will prevail next month by using the transition matrix to project the current regime probabilities forward one period. For example, the probability that regime A will prevail next month is equal to the probability that we are currently in regime A times the probability of remaining in regime A, plus the probability that we are currently in regime B times the probability of transitioning from B to A, plus the probability that we are currently in regime C times the probability of transitioning from C to A. The next‐period probability of each regime, , is described succinctly by the product of the transition matrix of conditional regime probabilities and the current regime probabilities, :

(16.2)

Figure 16.4 shows the probability forecast of each forthcoming regime per month, out of sample.

There are many ways to specify portfolio weights using the regime signals in Figure 16.4. For illustration, we use the conservative, moderate, and aggressive portfolios from Chapter 2 and blend them in proportion to the regime probabilities at the end of each month. We then record the returns of each portfolio as well as the tactically blended portfolio over the following month. This rule depends on very few parameters, and it smoothly transitions the portfolio's composition among a conservative, moderate, and aggressive asset mix as regime probabilities shift. Table 16.4 and Figure 16.5 show that the tactical strategy produced the largest cumulative return, with a standard deviation substantially below that of the aggressive portfolio. It delivered a higher Sharpe ratio than the conservative portfolio at a return level 2.6 percent higher. We have little doubt that we could produce performance that is more compelling by devising more sophisticated strategies involving additional indicators, conditional return forecasts, and reoptimized conditional allocations. Also, though we have not adjusted our results for transaction costs, the amount of trading involved here is certainly reasonable and can be dialed up or down using trading thresholds, minimum holding periods, and other techniques.