RISK AND VOLATILITY: ECONOMETRIC MODELS AND FINANCIAL PRACTICE
Nobel Lecture, December 8, 20031
University, Department of Finance ( ), 44 West Fourth Street, , NY 10012-1126, USA.
INTRODUCTION
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The advantage of knowing about risks is that we can change our behavior to avoid them. Of course, it is easily observed that to avoid all risks would be im- possible; it might entail no flying, no driving, no walking, eating and drinking only healthy foods and never being touched by sunshine. Even a bath could be dangerous. I could not receive this prize if I sought to avoid all risks. There are some risks we choose to take because the benefits from taking them ex- ceed the possible costs. Optimal behavior takes risks that are worthwhile. This is the central paradigm of finance; we must take risks to achieve rewards but not all risks are equally rewarded. Both the risks and the rewards are in the fu- ture, so it is the expectation of loss that is balanced against the expectation of reward. Thus we optimize our behavior, and in particular our portfolio, to maximize rewards and minimize risks.
This simple concept has a long history in economics and in Nobel cita- tions. Markowitz (1952) and Tobin (1958) associated risk with the variance in the value of a portfolio. From the avoidance of risk they derived optimizing portfolio and banking behavior. Sharpe (1964) developed the implications when all investors follow the same objectives with the same information. This theory is called the Capital Asset Pricing Model or CAPM, and shows that there is a natural relation between expected returns and variance. These con- tributions were recognized by Nobel prizes in 1981 and 1990.
Black and Scholes (1972) and Merton (1973) developed a model to evalu- ate the pricing of options. While the theory is based on option replication ar- guments through dynamic trading strategies, it is also consistent with the CAPM. Put options give the owner the right to sell an asset at a particular
1 This paper is the result of more than two decades of research and collaboration with many many people. I would particularly like to thank the audiences in B.I.S., Stockholm, Uppsala, Cornell and the University de Savoie for listening as this talk developed. , , and erstenberg provided detailed suggestions. Nevertheless, all lacunas remain my responsibility.
price at a time in the future. Thus these options can be thought of as insur- ance. By purchasing such put options, the risk of the portfolio can be com- pletely eliminated. But what does this insurance cost? The price of protection depends upon the risks and these risks are measured by the variance of the as- set returns. This contribution was recognized by a 1997 Nobel prize.
When practitioners implemented these financial strategies, they required estimates of the variances. Typically the square root of the variance, called the volatility, was reported. They immediately recognized that the volatilities were changing over time. They found different answers for different time periods. A simple approach, sometimes called historical volatility, was and remains wide- ly used. In this method, the volatility is estimated by the sample standard de- viation of returns over a short period. But, what is the right period to use? If it is too long, then it will not be so relevant for today and if it is too short, it will be very noisy. Furthermore, it is really the volatility over a future period that should be considered the risk, hence a forecast of volatility is needed as well as a measure for today. This raises the possibility that the forecast of the average volatility over the next week might be different from the forecast over a year or a decade. Historical volatility had no solution for these problems.
On a more fundamental level, it is logically inconsistent to assume, for ex- ample, that the variance is constant for a period such as one year ending to- day and also that it is constant for the year ending on the previous day but with a different value. A theory of dynamic volatilities is needed; this is the role that is filled by the ARCH models and their many extensions that we dis- cuss today.
In the next section, I will describe the genesis of the ARCH model, and then discuss some of its many generalizations and widespread empirical sup- port. In subsequent sections, I will show how this dynamic model can be used to forecast volatility and risk over a long horizon and how it can be used to value options.
THE BIRTH OF THE ARCH MODEL
The ARCH model was invented while I was on sabbatical at the London School of Economics in 1979. Lunch in the Senior Common Room with , , and many leading econometri- cians provided a stimulating environment. I was looking for a model that could assess the validity of a conjecture of (1977) that the unpredictability of inflation was a primary cause of business cycles. He hy- pothesized that the level of inflation was not a problem; it was the uncertain- ty about future costs and prices that would prevent entrepreneurs from in- vesting and lead to a recession. This could only be plausible if the uncertainty were changing over time so this was my goal. Econometricians call this het- eroskedasticity. I had recently worked extensively with the and knew that a likelihood function could be decomposed into the sum of its pre- dictive or conditional densities. Finally, my colleague with whom I share this prize, had recently developed a test for bilinear time series
models based on the dependence over time of squared residuals. That is, squared residuals often were autocorrelated even though the residuals them- selves were not. This test was frequently significant in economic data; I sus- pected that it was detecting something besides bilinearity but I didn’t know what.
The solution was autoregressive conditional heteroskedasticity or ARCH, a name invented by . The ARCH model described the forecast variance in terms of current observables. Instead of using short or long sample stan- dard deviations, the ARCH model proposed taking weighted averages of past squared forecast errors, a type of weighted variance. These weights could give more influence to recent information and less to the distant past. Clearly the ARCH model was a simple generalization of the sample variance.
The big advance was that the weights could be estimated from historical da- ta even though the true volatility was never observed. Here is how this works. Forecasts can be calculated every day or every period. By examining these forecasts for different weights, the set of weights can be found that make the forecasts closest to the variance of the next return. This procedure, based on Maximum Likelihood, gives a systematic approach to the estimation of the optimal weights. Once the weights are determined, this dynamic model of time varying volatility can be used to measure the volatility at any time and to forecast it into the near and distant future. Granger’s test for bilinearity turned out to be the optimal or test for ARCH and is widely used today.
There are many benefits to formulating an explicit dynamic model of volatility. As mentioned above, the optimal parameters can be estimated by Maximum Likelihood. Tests of the adequacy and accuracy of a volatility mod- el can be used to verify the procedure. One-step and multi-step forecasts can be constructed using these parameters. The unconditional distributions can be established mathematically and are generally realistic. Inserting the rele- vant variables into the model can test economic models that seek to deter- mine the causes of volatility. Incorporating additional endogenous variables and equations can similarly test economic models about the consequences of volatility. Several applications will be mentioned below.
’s associate, wrote the first ARCH program. The application that appeared in Engle (1982) was to inflation in the U.K. since this was Friedman’s conjecture. While there was plenty of evidence that the uncertainty in inflation forecasts was time varying, it did not correspond to the U.K. business cycle. Similar tests for U.S. inflation data, reported in Engle (1983), confirmed the finding of ARCH but found no business cycle effect. While the trade-off between risk and return is an important part of macro- economic theory, the empirical implications are often difficult to detect as they are disguised by other dominating effects, and obscured by the reliance on relatively low frequency data. In finance, the risk/return effects are of pri- mary importance and data on daily or even intra-daily frequencies are readi- ly available to form accurate volatility forecasts. Thus finance is the field in which the great richness and variety of ARCH models developed.
GENERALIZING THE ARCH MODEL
Generalizations to different weighting schemes can be estimated and tested. The very important development by my outstanding student (1986), called Generalized Autoregressive Conditional Heteroskedasticity or GARCH, is today the most widely used model. This essentially generalizes the purely autoregressive ARCH model to an autoregressive moving average mod- el. The weights on past squared residuals are assumed to decline geometri- cally at a rate to be estimated from the data. An intuitively appealing inter- pretation of the GARCH (1,1) model is easy to understand. The GARCH forecast variance is a weighted average of three different variance forecasts. One is a constant variance that corresponds to the long run average. The se- cond is the forecast that was made in previous period. The third is the new in- formation that was not available when the previous forecast was made. This could be viewed as a variance forecast based on one period of information. The weights on these three forecasts determine how fast the variance changes with new information and how fast it reverts to its long run mean.
A second enormously important generalization was the Exponential GARCH or EGARCH model of (1992) who prematurely passed away in 1995 to the great loss of our profession as eulogized by Bollerslev and Rossi (1995). In his short academic career, his contributions were extremely influential. He recognized that volatility could respond asymmetrically to past forecast errors. In a financial context, negative returns seemed to be more important predictors of volatility than positive returns. Large price declines forecast greater volatility than similarly large price increases. This is an eco- nomically interesting effect that has wide ranging implications to be discussed below.
Further generalizations have been proposed by many researchers. There is now an alphabet soup of ARCH models that include: AARCH, APARCH, FI- GARCH, FIEGARCH, STARCH, SWARCH, GJR-GARCH, TARCH, MARCH, NARCH, SNPARCH, SPARCH, SQGARCH, CESGARCH, Component ARCH, Asymmetric Component ARCH, Taylor-Schwert, Student-t-ARCH, GED- ARCH, and many others that I have regrettably overlooked. Many of these models were surveyed in Bollerslev, Chou and Kroner (1992), Bollerslev (1994), Engle (2002b), and Engle and Ishida (2002). These models recog- nize that there may be important non-linearity, asymmetry and long memory properties of volatility and that returns can be non-normal with a variety of parametric and non-parametric distributions.
A closely related but econometrically distinct class of volatility models called Stochastic Volatility or SV models have also seen dramatic development. See for example, Clark (1973), Taylor (1986), Harvey, Ruiz and Shephard (1994), Taylor (1994). These models have a different data generating process which makes them more convenient for some purposes but more difficult to esti- mate. In a linear framework, these models would simply be different repre- sentations of the same process; but in this non-linear setting, the alternative specifications are not equivalent, although they are close approximations.
MODELING FINANCIAL RETURNS
The success of the ARCH family of models is attributable in large measure to the applications in finance. While the models have applicability for many sta- tistical problems with time series data, they find particular value for financial time series. This is partly because of the importance of the previously dis- cussed trade-off between risk and return for financial markets, and partly be- cause of three ubiquitous characteristics of financial returns from holding a risky asset. Returns are almost unpredictable, they have surprisingly large numbers of extreme values and both the extremes and quiet periods are clus- tered in time. These features are often described as unpredictability, fat tails and volatility clustering. These are precisely the characteristics for which an ARCH model is designed. When volatility is high, it is likely to remain high, and when it is low it is likely to remain low. However, these periods are time limited so that the forecast is sure to eventually revert to less extreme volatili- ties. An ARCH process produces dynamic, mean reverting patterns in volatil- ity that can be forecast. It also produces a greater number of extremes than would be expected from a standard normal distribution, since the extreme values during the high volatility period are greater than could be anticipated from a constant volatility process.
The GARCH (1,1) specification is the workhorse of financial applications. It is remarkable that one model can be used to describe the volatility dynam- ics of almost any financial return series. This applies not only to US stocks but also to stocks traded in most developed markets, to most stocks traded in emerging markets, and to most indices of equity returns. It applies to ex- change rates, bond returns and commodity returns. In many cases, a slightly better model can be found in the list of models above, but GARCH is gener- ally a very good starting point.
The widespread success of GARCH (1,1) begs to be understood. What the- ory can explain why volatility dynamics are similar in so many different fi- nancial markets? In developing such a theory, we must first understand why asset prices change. Financial assets are purchased and owned because of the future payments that can be expected. Because these payments are uncertain and depend upon unknowable future developments, the fair price of the as- set will require forecasts of the distribution of these payments based on our best information today. As time goes by, we get more information on these fu- ture events and re-value the asset. So at a basic level, financial price volatility is due to the arrival of new information. Volatility clustering is simply cluster- ing of information arrivals. The fact that this is common to so many assets is simply a statement that news is typically clustered in time.
To see why it is natural for news to be clustered in time, we must be more specific about the information flow. Consider an event such as an invention that will increase the value of a firm because it will improve future earnings and dividends. The effect on stock prices of this event will depend on eco- nomic conditions in the economy and in the firm. If the firm is near bank- ruptcy, the effect can be very large and if it is already operating at full cap-
acity, it may be small. If the economy has low interest rates and surplus labor, it may be easy to develop this new product. With everything else equal, the re- sponse will be greater in a recession than in a boom period. Hence we are not surprised to find higher volatility in economic downturns even if the arrival rate of new inventions is constant. This is a slow moving type of volatility clus- tering that can give cycles of several years or longer.
The same invention will also give rise to a high frequency volatility cluster- ing. When the invention is announced, the market will not immediately be able to estimate its value on the stock price. Agents may disagree but be suffi- ciently unsure of their valuations that they pay attention to how others value the firm. If an investor buys until the price reaches his estimate of the new value, he may revise his estimate after he sees others continue to buy at successively higher prices. He may suspect they have better information or models and consequently raise his valuation. Of course, if the others are selling, then he may revise his price downward. This process is generally called price discovery and has been modeled theoretically and empirically in market microstructure. It leads to volatility clustering at a much higher frequency than we have seen before. This process could last a few days or a few minutes.
But to understand volatility we must think of more than one invention. While the arrival rate of inventions may not have clear patterns, other types of news surely do. The news intensity is generally high during wars and eco- nomic distress. During important global summits, congressional or regulato- ry hearings, elections or central bank board meetings, there are likely to be many news events. These episodes are likely to be of medium duration, last- ing weeks or months.
The empirical volatility patterns we observe are composed of all three of these types of events. Thus we expect to see rather elaborate volatility dynam- ics and often rely on long time series to give accurate models of the different time constants.
MODELING THE CAUSES AND CONSEQUENCES OF FINANCIAL VOLATILITY
Once a model has been developed to measure volatility, it is natural to at- tempt to explain the causes of volatility and the effects of volatility on the economy. There is now a large literature examining aspects of these ques- tions. I will only give a discussion of some of the more limited findings for fi- nancial markets.
In financial markets, the consequences of volatility are easy to describe al- though perhaps difficult to measure. In an economy with one risky asset, a rise in volatility should lead investors to sell some of the asset. If there is a fixed supply, the price must fall sufficiently so that buyers take the other side. At this new lower price, the expected return is higher by just enough to com- pensate investors for the increased risk. In equilibrium, high volatility should correspond to high expected returns. Merton (1980) formulated this theo- retical model in continuous time, and Engle, Lilien and Robins (1987) pro-
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SP500 SPRETURNS
Figure 1. S&P 500 Daily Prices and Returns from January 1963 to November 2003.
posed a discrete time model. If the price of risk were constant over time, then rising conditional variances would translate linearly into rising expected re- turns. Thus the mean of the return equation would no longer be estimated as zero, it would depend upon the past squared returns exactly in the same way that the conditional variance depends on past squared returns. This very strong coefficient restriction can be tested and used to estimate the price of risk. It can also be used to measure the coefficient of relative risk aversion of the representative agent under the same assumptions.
Empirical evidence on this measurement has been mixed. While Engle et al. (1987) find a positive and significant effect, Chou, Engle and Kane (1992), and Glosten, Jagannathan and Runkle (1993), find a relationship that varies over time and may be negative because of omitted variables. French, Schwert and Stambaugh (1987) showed that a positive volatility surprise should and does have a negative effect on asset prices. There is not simply one risky asset in the economy and the price of risk
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