Part II of the book consists of three chapters. The ultimate goal of this and the following two chapters is to establish a framework for modeling the dynamic distribution of portfolio returns. The methods we develop in Part II can also be used to model each asset in the portfolio separately. In Part III of the book we will consider multivariate models that can link the univariate asset return models together. If the risk manager only cares about risk measurement at the portfolio level then the univariate models in Part II will suffice.

We will proceed with the univariate models in two steps. The first step is to establish a forecasting model for dynamic portfolio variance and to introduce methods for evaluating the performance of these forecasts. The second step is to consider ways to model nonnormal aspects of the portfolio return—that is, aspects that are not captured by the dynamic variance.

The second step, allowing for nonnormal distributions, is covered in Chapter 6. The first step, volatility modeling, is analyzed in this chapter and in Chapter 5. Chapter 5 relies on intraday data to develop daily volatility forecasts. The present chapter focuses on modeling daily volatility when only daily return data are available. We proceed as follows:

The overall objective of this chapter is to develop a general class of models that can be used by risk managers to forecast daily portfolio volatility using daily return data.

We begin by establishing some notation and by laying out the underlying assumptions for this chapter. In Chapter 1, we defined the daily asset log return, R_{t + 1}, using the daily closing price, S_{t + 1}, as

We will use the notation R_{t + 1} to describe either an individual asset return or the aggregate return on a portfolio. The models in this chapter can be used for both.

We will also apply the finding from Chapter 1 that at short horizons such as daily, we can safely assume that the mean value of R_{t + 1} is zero since it is dominated by the standard deviation. Issues arising at longer horizons will be discussed in Chapter 8. Furthermore, we will assume that the innovation to asset return is normally distributed. We hasten to add that the normality assumption is not realistic, and it will be relaxed in Chapter 6. Normality is simply assumed for now, as it allows us to focus on modeling the conditional variance of the distribution.

Given the assumptions made, we can write the daily return as

where the abbreviation i.i.d. N(0, 1) stands for “independently and identically normally distributed with mean equal to zero and variance equal to 1.”

Together these assumptions imply that once we have established a model of the time-varying variance, we will know the entire distribution of the asset, and we can therefore easily calculate any desired risk measure. We are well aware from the stylized facts discussed in Chapter 1 that the assumption of conditional normality that is imposed here is not satisfied in actual data on speculative returns. However, as we will see later, for the purpose of variance modeling, we are allowed to assume normality even if it is strictly speaking not a correct assumption. This assumption conveniently allows us to postpone discussions of nonnormal distributions to a later chapter.

The focus of this chapter then is to establish a model for forecasting tomorrow's variance, We know from Chapter 1 that variance, as measured by squared returns, exhibits strong autocorrelation, so that if the recent period was one of high variance, then tomorrow is likely to be a high-variance day as well. The easiest way to capture this phenomenon is by letting tomorrow's variance be the simple average of the most recent m observations, as in

Notice that this is a proper forecast in the sense that the forecast for tomorrow's variance is immediately available at the end of today when the daily return is realized. However, the fact that the model puts equal weights (equal to 1/m) on the past m observations yields unwarranted results. An extreme return (either positive or negative) today will bump up variance by 1/m times the return squared for exactly m periods after which variance immediately will drop back down. Figure 4.1 illustrates this point for m = 25 days. The autocorrelation plot of squared returns in Chapter 1 suggests that a more gradual decline is warranted in the effect of past returns on today's variance. Even if we are content with the box patterns, it is not at all clear how m should be chosen. This is unfortunate as the choice of m is crucial in deciding the patters of σ_{t + 1}: A high m will lead to an excessively smoothly evolving σ_{t + 1}, and a low m will lead to an excessively jagged pattern of σ_{t + 1} over time.

JP Morgan's RiskMetrics system for market risk management considers the following model, where the weights on past squared returns decline exponentially as we move backward in time. The RiskMetrics variance model, or the exponential smoother as it is sometimes called, is written as

Separating from the sum the squared return term for τ = 1, where λ^{τ − 1} = λ⁰ = 1, we get

Applying the exponential smoothing definition again, we can write today's variance, as

so that tomorrow's variance can be written

The RiskMetrics model's forecast for tomorrow's volatility can thus be seen as a weighted average of today's volatility and today's squared return.

The RiskMetrics model has some clear advantages. First, it tracks variance changes in a way that is broadly consistent with observed returns. Recent returns matter more for tomorrow's variance than distant returns as λ is less than one and therefore the impact of the lagged squared return gets smaller when the lag, τ, gets bigger. Second, the model only contains one unknown parameter, namely, λ. When estimating λ on a large number of assets, RiskMetrics found that the estimates were quite similar across assets, and they therefore simply set λ = 0.94 for every asset for daily variance forecasting. In this case, no estimation is necessary, which is a huge advantage in large portfolios. Third, relatively little data need to be stored in order to calculate tomorrow's variance. The weight on today's squared returns is (1 − λ) = 0.06, and the weight is exponentially decaying to (1 − λ)λ⁹⁹ = 0.000131 on the 100th lag of squared return. After including 100 lags of squared returns, the cumulated weight is so that 99.8% of the weight has been included. Therefore it is only necessary to store about 100 daily lags of returns in order to calculate tomorrow's variance, .

Given all these advantages of the RiskMetrics model, why not simply end the discussion on variance forecasting here and move on to distribution modeling? Unfortunately, as we will see shortly, the RiskMetrics model does have certain shortcomings, which will motivate us to consider slightly more elaborate models. For example, it does not allow for a leverage effect, which we considered a stylized fact in Chapter 1 and it also provides counterfactual longer-horizon forecasts.

We now introduce a set of models that capture important features of returns data and that are flexible enough to accommodate specific aspects of individual assets. The downside of these models is that they require nonlinear parameter estimation, which will be discussed subsequently.

The simplest generalized autoregressive conditional heteroskedasticity (GARCH) model of dynamic variance can be written as

Notice that the RiskMetrics model can be viewed as a special case of the simple GARCH model if we force α = 1 − λ, β = λ, so that α + β = 1, and further ω = 0. Thus, the two models appear to be quite similar. However, there is an important difference: We can define the unconditional, or long-run average, variance, σ², to be

It is now clear that if α + β = 1 as is the case in the RiskMetrics model, then the long-run variance is not well defined in that model. Thus, an important quirk of the RiskMetrics model emerges: It ignores the fact that the long-run average variance tends to be relatively stable over time. The GARCH model, in turn, implicitly relies on σ². This can be seen by solving for ω in the long-run variance equation and substituting it into the dynamic variance equation. We get

Thus, tomorrow's variance is a weighted average of the long-run variance, today's squared return, and today's variance. Put differently, tomorrow's variance is the long-run average variance with something added (subtracted) if today's squared return is above (below) its long-run average, and something added (subtracted) if today's variance is above (below) its long-run average.

Our intuition might tell us that ignoring the long-run variance, as the RiskMetrics model does, is more important for longer-horizon forecasting than for forecasting simply one day ahead. This intuition is correct, as we will now see.

A key advantage of GARCH models for risk management is that the one-day forecast of variance, is given directly by the model by . Consider now forecasting the variance of the daily return k days ahead, using only information available at the end of today. In GARCH, the expected value of future variance at horizon k is

The conditional expectation, E_t[●], refers to taking the expectation using all the information available at the end of day t, which includes the squared return on day t itself.

We will refer to α + β as the persistence of the model. A high persistence—that is, an (α + β) close to 1—implies that shocks that push variance away from its long-run average will persist for a long time, but eventually the long-horizon forecast will be the long-run average variance, σ². Similar calculations for the RiskMetrics model reveal that

as α + β = 1 and σ² is undefined. Thus, persistence in this model is 1, which implies that a shock to variance persists forever: An increase in variance will push up the variance forecast by an identical amount for all future forecast horizons. This is another way of saying that the RiskMetrics model ignores the long-run variance when forecasting. If α + β is close to 1 as is typically the case, then the two models might yield similar predictions for short horizons, k, but their longer horizon implications are very different. If today is a high-variance day, then the RiskMetrics model predicts that all future days will be high-variance. The GARCH model more realistically assumes that eventually in the future variance will revert to the average value.

So far we have considered forecasting the variance of daily returns k days ahead. Of more immediate interest is probably the forecast of variance of K-day cumulative returns,

As we assume that returns have zero autocorrelation, the variance of the cumulative K-day returns is simply

So in the RiskMetrics model, we get

But in the GARCH model, we get

If the RiskMetrics and GARCH model have identical and if , then the GARCH variance forecast will be higher than the RiskMetrics forecast. Thus, assuming the RiskMetrics model if the data truly look more like GARCH will give risk managers a false sense of the calmness of the market in the future, when the market is calm today and . Figure 4.2 illustrates this crucial point. We plot for K = 1, 2, …, 250 for both the RiskMetrics and the GARCH model starting from a low and setting α = 0.05 and β = 0.90. The long-run daily variance in the figure is σ² = 0.000140.

The GARCH and RiskMetrics models share the inconvenience that the multiperiod distribution is unknown even if the one-day ahead distribution is assumed to be normal, as we do in this chapter. Thus, while it is easy to forecast longer-horizon variance in these models, it is not as easy to forecast the entire conditional distribution. We will return to this important issue in Chapter 8 since it is unfortunately often ignored in risk management.

In the previous section, we suggested a GARCH model that we argued should fit the data well, but it contains a number of unknown parameters that must be estimated. In doing so, we face the challenge that the conditional variance, is an unobserved variable, which must itself be implicitly estimated along with the parameters of the model, for example, α, β, and ω.

As we noted earlier, one of the distinct benefits of GARCH models is their flexibility. In this section, we explore this flexibility and present some of the models most useful for risk management.

Before we start using the variance model for risk management purposes, it is appropriate to run the estimated model through some diagnostic checks.

This chapter presented a range of variance models that are useful for risk management. Simple equally weighted and exponentially weighted models that require minimal effort in estimation were first introduced. Their shortcomings led us to consider more sophisticated but still simple models from the GARCH family. We highlighted the flexibility of GARCH as a virtue and considered various extensions to account for leverage effects, day-of-week effects, announcement effects, and so on. The powerful and flexible quasi-maximum likelihood estimation technique was presented and will be used again in coming chapters. Various model validation techniques were introduced subsequently. Most of the techniques suggested in this chapter are put to use in the empirical exercises that follow.

This chapter focused on models where daily returns are used to forecast daily volatility. In some situations the risk manager may have intraday returns available when forecasting daily volatility. In this case more accurate models are available. These models will be analyzed in the next chapter.

This appendix shows that the component GARCH model can be viewed as a GARCH(2,2) model. First define the lag operator, L, to be a function that transforms the current value of a time series to the lagged value, that is,

Recall that the component GARCH model is given by

Using the lag operator, the first equation can be written as

We can also rewrite the long-run variance equation using the lag operator

From this, the long run variance factor v_{t + 1} can be written as

Plugging this back into the equation for we get

Multiplying both sides by 1 − β_vL and simplifying we get

which is a GARCH(2,2) model of the form

where

While the component GARCH and the GARCH(2,2) models are theoretically equivalent, the component structure is typically easier to implement because it is easier to find good starting values for the parameters. Typically, α_σ < α_v ≈ 0.1 and . Furthermore, the interpretation of v_{t + 1} as a long-run variance when is convenient.

This appendix introduces the long-memory GARCH model of Davidson (2004), which he refers to as HYGARCH (where HY denotes hyperbolic). Consider first the standard GARCH(1,1) model where

Using the lag operator, L, from Appendix A the GARCH(1,1) model can be written as an ARMA-in-squares model

Solving for we can now rewrite the GARCH(1,1) model as a moving-average-in-squares model of the form

The long-memory model in Davidson (2004) is defined as a generalization of this expression to

where the long-memory parameter, d, defines fractional integration via

where denotes the gamma function. The long-memory parameter d enables the model to have squared return autocorrelations that go to zero at a slower (namely hyperbolic) rate than the exponential rate in the GARCH(1,1) case.

We can show that when , , , and we get the regular GARCH(1,1) model as a special case of the HYGARCH. When , , and we get the RiskMetrics model as a special case of HYGARCH.

The HYGARCH model can be estimated using MLE as well. Please refer to Davidson (2004) for more details on this model.

The literature on variance modeling has exploded during the past 30 years, and we only present a few examples of papers here. Andersen et al. (2007) contains a discussion of the use of volatility models in risk management. Evidence on the performance of GARCH models through the 2008–2009 financial crisis can be found in Brownlees et al. (2009).

Andersen et al. (2006) and Poon and Granger (2005) provide overviews of the various classes of volatility models. The exponential smoother variance model is studied in JP Morgan (1996). The exponential smoother has been used to forecast a wide range of variables and further discussion of it can be found in Granger and Newbold (1986). The basic GARCH model is introduced in Engle (1982) and Bollerslev (1986) and it is discussed further in Bollerslev et al. (1992) and Engle and Patton (2001). Engle (2002) introduces the GARCH model with exogenous variable. Bollerslev (2008) summarizes the many variations on the basic GARCH model.

Long memory including fractional integrated generalized autoregressive conditional heteroskedasticity (FIGARCH) models were introduced in Baillie et al. (1996) and Bollerslev and Mikkelsen (1999). The hyperbolic HYGARCH model in Appendix B is developed in Davidson (2004).

Component volatility models were introduced in Engle and Lee (1999). They have been applied to option valuation in Christoffersen et al., 2010 and Christoffersen et al., 2008. The Spline-GARCH model with a deterministic volatility component was introduced in Engle and Rangel (2008).

The leverage effect and other GARCH extensions are described in Ding et al. (1993), Glosten et al. (1993), Hentschel (1995) and Nelson (1990). Most GARCH models use squared return as the innovation to volatility. Forsberg and Ghysels (2007) analyze the benefits of using absolute returns instead.

Quasi maximum likelihood estimation of GARCH models is developed in Bollerslev and Wooldridge (1992). Francq and Zakoian (2009) survey more recent results. For practical issues involved in GARCH estimation see Zivot (2009). Hansen and Lunde (2005), Patton and Sheppard (2009) and Patton (2011) develop tools for volatility forecast evaluation and volatility forecast comparisons.

Taylor (1994) contains a very nice overview of a different class of variance models known as stochastic volatility models. This class of models was not included in this book due to the relative difficulty of estimating them.