Nonparametric risk management and implied risk aversion

Abstract

Typical value-at-risk (VaR) calculations involve the probabilities of extreme dollar losses, based on the statistical distributions of market prices. Such quantities do not account for the fact that the same dollar loss can have two very different economic valuations, depending on business conditions. We propose a nonparametric VaR measure that incorporates economic valuation according to the state-price density associated with the underlying price processes. The state-price density yields VaR values that are adjusted for risk aversion, time preferences, and other variations in economic valuation. In the context of a representative agent equilibrium model, we construct an estimator of the risk-aversion coefficient that is implied by the joint observations on the cross-section of option prices and time-series of underlying assest values.

Introduction

One of the most pressing economic issues facing corporations today is the proper management of financial risks. In response to a series of recent financial catastrophes,¹ regulators, investment bankers, and chief executive officers have now embraced the notion of risk management as one of the primary fiduciary responsibilities of the corporate manager. Because financial risks often manifest themselves in subtle and nonlinear ways in corporate balance sheets and income statements, recent attention has focused on quantifying the fluctuations of market valuations in a statistical sense. These value-at-risk (VaR) measures lie at the heart of most current risk management systems and protocols. For example, JP Morgan's (1995) RiskMetrics system documentation describes VaR in the following way:

Value at Risk is an estimate, with a predefined confidence interval, of how much one can lose from holding a position over a set horizon. Potential horizons may be one day for typical trading activities or a month or longer for portfolio management. The methods described in our documentation use historical returns to forecast volatilities and correlations that are then used to estimate the market risk. These statistics can be applied across a set of asset classes covering products used by financial institutions, corporations, and institutional investors.

By modeling the price fluctuations of securities held in one's portfolio, an estimate and confidence interval of how much one can lose is readily derived from the basic principles of statistical inference. However, in this paper we argue that statistical notions of value-at-risk are, at best, incomplete measures of the true risks facing investors. In particular, while statistical measures do provide some information about the range of uncertainty that a portfolio exhibits, they have little to do with the economic valuation of such uncertainty. For example, a typical VaR statistic might indicate a 5% probability of a $15M loss for a $100M portfolio over the next month, which seems to be a substantial risk exposure at first glance. But if this 15% loss occurs only when other investments of similar characteristics suffer losses of 25% or more, such a risk may seem rather mild after all. This simplistic example suggests that a one-dollar loss is not always worth the same, and that circumstances surrounding the loss can affect its economic valuation, something that is completely ignored by purely statistical measures of risk.

In this paper, we propose an alternative to statistical VaR (henceforth S-VaR) that is based on economic valuations of value-at-risk, and which incorporates many other aspects of market risk that are central to the practice of risk management. Our alternative is based on the seminal ideas of Arrow (1964) and Debreu (1959), who first formalized the economics of uncertainty by introducing elementary securities each paying $1 in one specific state of nature and nothing in any other state. Now known as Arrow–Debreu securities, they are widely recognized as the fundamental building blocks of all modern financial asset-pricing theories, including the CAPM, the APT, and the Black and Scholes (1973) and Merton (1973) option-pricing models.

By construction, Arrow–Debreu prices have a probability-like interpretation – they are nonnegative and sum to unity – but since they are market prices determined in equilibrium by supply and demand, they contain much more information than statistical models of prices. Arrow–Debreu prices are determined by the combination of investors’ preferences, budget dynamics, information structure, and the imposition of market-clearing conditions, i.e., general equilibrium. Moreover, we shall show below that under certain special conditions, Arrow–Debreu prices reduce to the simple probabilities on which statistical VaR measures are based, hence the standard measures of value-at-risk are special cases of the Arrow–Debreu framework.

The fact that the market prices of these Arrow–Debreu securities need not be equal across states implies that a one-dollar gain need not be worth the same in every state of nature – indeed, the worth of a one-dollar gain in a given state is precisely the Arrow–Debreu price of that security. Therefore, we propose to use the prices of Arrow–Debreu securities to measure economic VaR (henceforth E-VaR).

Despite the fact that pure Arrow–Debreu securities are not yet traded on any organized exchange,² Arrow–Debreu prices can be estimated from the prices of traded financial securities using recently developed nonparametric techniques such as kernel regression, artificial neural networks, and implied binomial trees. Nonparametric techniques are particularly useful for value-at-risk calculations because departures from standard parametric assumptions, e.g., normality, can have dramatic consequences for tail probabilities³. Using such techniques, we compare the performance of S-VaR and E-VaR measures and develop robust statistical methods to gauge the magnitudes of their differences.

Moreover, to provide an economic interpretation for the differences between S-VaR and E-VaR, we show how to combine S-VaR and E-VaR to yield a measure of the aggregate risk aversion of the economy, i.e., the risk aversion of the representative investor in a standard dynamic asset-pricing model. We propose to extract (unobservable) aggregate risk-preferences, what we call implied risk aversion, from (observable) market prices of traded financial securities. In particular, we are inferring the aggregate preferences that are compatible with the pair of option and index values.

When applied to daily S&P 500 option prices and index levels from 1993, our nonparametric analysis uncovers substantial differences between S-VaR and E-VaR (see Fig. 2). A comparison of S-VaR and E-VaR densities shows that aggregate risk aversion is not constant across states or maturity dates, but changes in important nonlinear ways (see Fig. 4).

Of course, risk management is a complex process that is unlikely to be driven by any single risk measure, E-VaR or S-VaR. In particular, despite the fact that E-VaR provides information not contained in S-VaR, both measures belong in the arsenal of tools that risk managers can bring to bear on assessing and controlling risk. For certain purposes – regulatory reporting requirements, quick summaries of corporate exposure, or comparisons across companies and industries – S-VaR may be a simpler measure to compute and interpret. Nevertheless, E-VaR brings a new dimension into the risk management process and should be integrated into any complete risk management system.

In Section 2, we present a brief review of the theoretical underpinnings of Arrow–Debreu prices and their relation to dynamic equilibrium models of financial markets. In Section 3, we formally introduce the notion of economic value-at-risk, describe its implementation, and propose statistical inference procedures that can quantify its accuracy and relevance over statistical VaR. An explicit comparison of E-VaR with S-VaR, along with the appropriate statistical inference, is described and developed in Section 4. We construct an estimator of implied risk aversion in Section 5 and propose tests for risk neutrality and for specific preferences based on this estimator. To illustrate the empirical relevance of E-VaR, we apply our estimators to daily S&P 500 options data in Section 6. We conclude in Section 7 and collect our technical assumptions and results in the Appendix.