A Stress Test to Incorporate Correlation Breakdown

Feb 1, 2000

We introduce the broken arrow stress test, in which we estimate correlation levels in stress situations using a mixture of normal distributions and apply these correlation levels to arrive at the expected loss for the peripheral assets.

Furthermore, joint distributions estimated over periods without panics will misestimate the degree of correlation between asset returns during panics. Under these circumstances, fear and disengagement by investors often result in simultaneous declines in the values of private obligations, as investors no longer realistically differentiate among degrees of risk and liquidity, and increases in the values of riskless government securities. Consequently, the benefits of portfolio diversification will tend to be overestimated when the rare panic periods are not taken into account.

Remarks by Federal Reserve Board Chairman Alan Greenspan (1999)

Introduction

Stress tests are a common counterpart to the objective models used for day-to-day risk monitoring. One example of the objective models is Value-at-Risk (VaR). The objective models typically forecast worst case losses conditional on markets behaving generally as they have in the recent past. To make accurate forecasts, these models rely on a relatively short (one year at most) history of market factor returns. While certain models will extrapolate from these returns and forecast losses greater than those observed in the historical period, the loss forecasts are always restricted by the historical returns. Stress tests – point estimates of portfolio losses based on market factor returns that have never occurred, or that occurred outside of the relevant historical period for the model – complement the objective model forecasts by providing a notion of losses deemed implausible by the model, but which certainly could occur.

The Basle Committee on Banking Supervision (1996)requires banks that use the internal models approach for market risk capital requirements to implement a rigorous and comprehensive stress testing framework. Further, the BIS and the Technical Committee of the International Organization of Securities Commissions (1998)oblige institutions with significant trading activities to execute stress tests on a regular basis, using a variety of assumptions which cover both hypothetical and historical events. The President's Working Group on Financial Markets (1999), after the bail-out of Long Term Capital Management, emphasizes that, in addition to routine risk management, banks need to stress test credit, as well as market, risk profiles in order to evaluate the potential impact of adverse market conditions on cash flows and asset/collateral values supporting trading transactions.

Generically, stress tests involve the specification of adverse market moves (scenarios) and the revaluation of the portfolio under these moves (Laubsch (1999)). To specify scenarios, it is first necessary to select which market factors (the core assets) to stress, then define the amount by which to stress them and the time period over which the stress move will take place. For the remaining (peripheral) assets, there are a number of methods to specify the moves that would coincide with moves in the core assets.

Table 1

Assumptions on Peripheral Assets in Alternative Stress Tests

The simplest specification for peripheral asset moves is to simply assume no change (call this the "zeroed-out" stress test). A second specification (the predictive stress test) utilizes current estimates of volatility and correlation to estimate the conditional expectation of peripheral asset moves given the stress moves in the core assets. See Kupiec (1998) for more details. A third specification (historical stress test) applies the moves in the peripheral assets that have coincided with large moves in the core assets historically. We summarize the three stress test methodologies in Table 1.

Of the three methods, the predictive stress test appears most attractive, if only we can justify the contention that standard volatility and correlation estimates will produce good stress forecasts of the peripheral asset moves. Though Kupiec (1998) does show that this contention tends to hold up in practice, the sentiment expressed by Chairman Greenspan remains and is supported by a number of empirical observations of "correlation breakdown." See Longin (1999) for an example. With this breakdown in mind, it is tempting to filter the data and compute correlations based only on days which, in retrospect, appear most volatile. Boyer (1999) issue an explicit warning against this practice, citing the bias it introduces, and state in their conclusion:

In order to carry out a valid test, we argue that it is necessary that the researcher begin with a data-coherent model of the data generating process that builds in the possibility of structural changes, estimate the model's parameters, and only then decide whether the estimated parameters imply changing correlations.

In this paper, we introduce the broken arrow stress test, in which we apply the predictive stress methodology, but using an estimate of correlation in stress situations. For one core and one peripheral asset, we will specify the joint distribution as a mixture of bivariate normals. For the majority of cases (quiet days), the asset returns are drawn from a normal distribution with lower volatility and one level of correlation; on rarer hectic days, the asset returns are drawn from a different normal distribution with higher levels of volatility and a second correlation level. Using historical data on the core asset, we identify the ex post conditional probabilities that a given day's returns were drawn from a quiet or hectic distribution. With these conditional probabilities, we compute the correlation levels for the quiet and hectic periods and test for the significance of the difference of these two levels. We finish by applying the predictive stress testing methodology with our estimates of hectic correlation levels.