Value-at-risk with info-gap uncertainty.

Abstract

Purpose--To study the effect of Knightian uncertainty--as opposed to statistical estimation error--in the evaluation of value-at-risk (VaR) of financial investments. To develop methods for augmenting existing VaR estimates to account for Knightian uncertainty.

Design/methodology/approach--The value at risk of a financial investment is assessed as the quantile of an estimated probability distribution of the returns. Estimating a VaR from historical data entails two distinct sorts of uncertainty: probabilistic uncertainty in the estimation of a probability density function (PDF) from historical data, and non-probabilistic Knightian info-gaps in the future size and shape of the lower tail of the PDF. A PDF is estimated from historical data, while a VaR is used to predict future risk. Knightian uncertainty arises from the structural changes, surprises, etc., which occur in the future and therefore are not manifested in historical data. This paper concentrates entirely on Knightian uncertainty and does not consider the statistical problem of estimating a PDF. Info-gap decision theory is used to study the robustness of a VaR to Knightian uncertainty in the distribution.

Findings--It is shown that VaRs, based on estimated PDFs, have no robusmess to Knightian errors in the PDF. An info-gap safety factor is derived that multiplies the estimated VaR in order to obtain a revised VaR with specified robustness to Knightian error in the PDF. A robustness premium is defined as a supplement to the incremental VaR for comparing portfolios.

Practical implications--The revised VaR and incremental VaR augment existing tools for evaluating financial risk.

Originality/value Info--gap theory, which underlies this paper, is a non-probabilistic quantification of uncertainty that is very suitable for representing Knightian uncertainty. This enables one to assess the robustness to future surprises, as distinct from existing statistical techniques for assessing estimation error resulting from randomness of historical data.

Keywords Risk management, Uncertainty management, Information theory, Mathematical modelling, Statistical methods

Paper type Research paper

Introduction

The physicist Nils Bohr was fond of the Danish aphorism that "Prediction is always difficult, especially of the future" (Moore, 1989). Bohr lived in tumultuous times and he knew what he was talking about. The aphorism applies to financial risk whether or not Bohr had that in mind. For instance, Hendricks (1996) finds extensive empirical evidence for:

 
   ... two well-known characteristics of daily financial market data. 
   First, extreme outcomes occur more often and are larger than 
   predicted by the normal distribution (fat tails). Second, the size 
   of market movements is not constant over time (conditional 
   volatility). 

Just as Bohr said.

The dispute about predictability of market outcomes has raged for a long time and is here to stay. Fama (1965, p. 34) argued that "[T]he series of price changes has no memory, that is, the past cannot be used to predict the future in any meaningful way" (Bohr again). More recently (Fama and French, 1988, p. 247), we read about "the mounting evidence that stock returns are predictable" (Bohr, applied to econometrics).

Of course, "predictable" does not mean "infinitely reliable":

 
   Virtually all of the approaches [to assessing value-at-risk] produce 
   accurate 95th percentile risk measures. The 99th percentile risk 
   measures, however, are somewhat less reliable and generally cover 
   only between 98.2 percent and 98.5 percent of the outcomes 
   (Hendricks, 1996). 

VaRs are evaluated from estimated probability density functions (PDFs); estimates are based on history. This limits the accuracy of estimated VaRs as predictions of future risk for reasons that we will divide into two categories. The first category is estimation uncertainty, while the second category we will refer to broadly as Knightian uncertainty.

The evaluation and management of estimation uncertainty is in the province of statistical analysis. Numerous powerful methods are available for estimating quantiles and for evaluating the error of those estimates. Standard errors of quantile estimates can be evaluated (Kendall et al., 1987, [section]10.9). Confidence intervals can be constructed for quantile values (DeGroot, 1986, p. 563). Sign tests use order statistics for testing hypotheses on the value of an estimated quantile (Kendall et al., 1979, [section]32.2-10). Kernel smoothing methods provide a rich array of methods for estimating a PDF (Hardle, 1990). The starting point of the current paper is the assumption that a PDF of returns has been competently estimated, and the reliability of this estimate, vis-a-vis the historical data, has been established using appropriate statistical tools. We are not concerned with the statistical task of estimating a PDF.

This paper is concerned with the second class of factors that limit the accuracy of VaRs: Knightian uncertainty about the future. The systematic errors in VaRs mentioned earlier suggest that something more than random estimation error is involved. The most troublesome source of error in predicting future VaRs from historical data is that things can change. The true PDF in the future can differ substantially from the true realization in the past. Changes in a market or in its economic, social, and political environment can cause significant changes in the actual shape of a PDF. Environmental change arising well outside the specific market in question is virtually impossible to predict, and experience has shown many examples of profound and sometimes sudden alterations. This source of error can be neither assessed nor rectified by statistical methods since future innovation has no manifestation in historical data. Unpredictable changes are ones which cannot be modeled probabilistically, and against which one can in no way insure in any actuarial sense. This uncertainty is what Frank Knight called a "true uncertainty" for which "there is no objective measure of …