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 impossible; 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 exceed 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 future, 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.

In the aftermath of the financial crisis, institutions have been asked to reduce leverage in order to reduce risk. To address the effectiveness of this measure, we build a model of equity volatility that accounts for leverage. Our approach blends Merton’s insights on capital structure with traditional time-series models of volatility. We estimate that precautionary capital needs for the entire financial sector reached $2 trillion during the crisis. We also investigate the long-standing observation that equity volatility asymmetrically responds to positive and negative news. Volatility asymmetry is mostly explained by exposure to the aggregate market, not a mechanical leverage effect.

[Twenty-five years of volatility research has left the macroeconomic environment playing a minor role. This paper proposes modeling equity volatilities as a combination of macroeconomic effects and time series dynamics. High-frequency return volatility is specified to be the product of a slow-moving component, represented by an exponential spline, and a unit GARCH. This slow-moving component is the low-frequency volatility, which in this model coincides with the unconditional volatility. This component is estimated for nearly 50 countries over various sample periods of daily data. Low-frequency volatility is then modeled as a function of macroeconomic and financial variables in an unbalanced panel with a variety of dependence structures. It is found to vary over time and across countries. The low-frequency component of volatility is greater when the macroeconomic factors of GDP, inflation, and short-term interest rates are more volatile or when inflation is high and output growth is low. Volatility is higher not only for emerging markets and markets with small numbers of listed companies and market capitalization relative to GDP, but also for large economies. The model allows long horizon forecasts of volatility to depend on macroeconomic developments, and delivers estimates of the volatility to be anticipated in a newly opened market.]

I am pleased to offer a comment on this very interesting article by an author who is always in the forefront of the research on empirical financial models. This article presents data analysis that estabishes the stylized facts about stock market volatility around market crashes. He concludes that volatility is high during periods of stock market decline and that it gradually returns to more normal levels. In the case of 1987, the peak was higher than usual and the decline was more rapid. The article uses 28,000 daily observations but does not really estimate a usable model; instead, it explores the data by estimating highly overparameterized models that reveal important features of the data. I suggest that this be considered an exploratory investigation and that in the face of more parsimonious models, rather interesting and somewhat different conclusions are revealed. The basic model estimated by Schwert is a 22-order autoregression of daily returns with a heteroskedastic error standard deviation which is itself assumed to be a 22-order autoregression in the absolute errors. Even with 28,000 observations, there is apparently a lot of noise in the coefficients. To allow for a risk premium, the mean is related to the variance, and in this case it is therefore related to 22 lagged absolute residuals. This part of the model uses 66 parameters. An alternative model is a first-order generalized autoregressive conditionally heteroskedastic model with variance influencing the mean [GARCH (l, l)-m], with a first-order moving average to correct for non-synchronous trading as used in Engle, Lilien, and Robins (1987), French, Schwert, and Stambaugh (1987), or Chou (1988), following the earlier work of Engle (1982). This requires only four coefficients! In the context of the parsimonious model, the parameter regulating the risk-return trade-off can be interpreted as the median agent’s taste for risk or his coefficient of relative-risk aversion. One naturally asks whether this parameter is constant over time, and we then recognize that the Schwert parameterization cannot answer the question.

Ultra-high-frequency data is defined to be a full record of transactions and their associated characteristics. The transaction arrival times and accompanying measures can be analyzed as marked point processes. The ACD point process developed by Engle and Russell (1998) is applied to IBM transactions arrival times to develop semiparametric hazard estimates and conditional intensities. Combining these intensities with a GARCH model of prices produces ultra-high-frequency measures of volatility. Both returns and variances are found to be negatively influenced by long durations as suggested by asymmetric information models of market micro-structure.

Traditional econometric models assume a constant one-period forecast variance. To generalize this implausible assumption, a new class of stochastic processes called autoregressive conditional heteroscedastic (ARCH) processes are introduced in this paper. These are mean zero, serially uncorrelated processes with nonconstant variances conditional on the past, but constant unconditional variances. For such processes, the recent past gives information about the one-period forecast variance. A regression model is then introduced with disturbances following an ARCH process. Maximum likelihood estimators are described and a simple scoring iteration formulated. Ordinary least squares maintains its optimality properties in this set-up, but maximum likelihood is more efficient. The relative efficiency is calculated and can be infinite. To test whether the disturbances follow an ARCH process, the Lagrange multiplier procedure is employed. The test is based simply on the autocorrelation of the squared OLS residuals. This model is used to estimate the means and variances of inflation in the U.K. The ARCH effect is found to be significant and the estimated variances increase substantially during the chaotic seventies.

[A set of standard dynamic disaggregated price equations are estimated to examine the relationship between changes in input prices and output prices. The equations perform satisfactorily by conventional criteria; however, when disaggregated by frequency, it is found that the high and low frequency components appear to satisfy different models. The differences are generally significant suggesting that the model is misspecified and that another lag distribution should be used. In particular, the sum of the lag coefficients for labor inputs is substantially larger when estimated with the low frequency component than the high. Therefore, such a price equation estimated during a regime of continued wage inflation would exhibit a much larger long run output price elasticity with respect to wages, than would one estimated during a period of stable or randomly fluctuating wages.]