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Summary The definition of risk can take various forms. One of the most used is the standard deviation or portfolio volatility. The evolution of the conditional variance may be parameterized by many different specifications. Here, I consider three models: the rolling window approach, the JPMorgan’s RiskMetrics and the GARCH(1,1). The rolling window and the RiskMetrics approach are methods that share similar features and the same drawback: they don’t account for the fact that volatility is a stationary process. GARCH (1,1) is a better method since it takes into account today’s variance as a starting point, but then unconditional variance in the far long run. In my previous research , I claimed that choosing the optimal portfolio strategy is critical in order to achieve extra return, and I provided the reader with a review of existing strategies, from the most naïve ones, such as 1/N, to the most sophisticated, such as the Bayesian strategies. In addition to portfolio construction, risk management is another essential topic that should be discussed. Indeed, risk is ubiquitous, and the intelligent investor has to be able to manage it. The definition of risk can take various forms. One of the most used is the standard deviation or portfolio volatility, which measures the spread of the distribution of returns around its mean. Volatility has different characteristics: it is not directly observable, it evolves over time in a continuous manner, it reacts differently to positive and negative price changes, and last but not least: Volatility is a stationary process. Bear in mind this last feature, as it would be critical in the analysis below. Conditional and unconditional volatility A key distinction is between the conditional and unconditional volatility. The unconditional volatility (σ) is just the standard measure of the volatility, whereas the conditional volatility (h t 1/2 ) is the measure of uncertainty about a variable given a model and an information set. Consider the return (r t ) at time t decomposed in its location and scale representation as follows. μ t is the conditional mean of r t and may be parameterized by a time series model like an ARMA(p,q) while ε t might be defined as follows: Where: is the conditional variance (volatility 2 ) of r t depending on the information set F available at time t-1, and is the unconditional variance (volatility 2 ) of r t , which does not depend on previous information. The focus of this research is on the time varying volatility and risk and therefore on the conditional variance. The evolution of the conditional variance may be parameterized by many different specifications. Note that with the word “evolution”, I mean how the conditional volatility evolves over time, as new information becomes available. Here, I consider three models: the rolling window approach, the JPMorgan’s RiskMetrics and the GARCH(1,1). The rolling window approach The rolling window approach relies on a particular stylized fact: the best guess of future volatility is based on an equally weighted average of the volatility of past m periods. To capture this feature, let tomorrow’s variance be equal to the sample variance computed over the last m observations: This specification implies that if volatility is high today, it is also likely to be high tomorrow. Naturally, the choice of m is critical: If it is too high, h t+1 results excessively smooth and slow evolving (exhibit 1) If it is too low, h t+1 presents excessively jagged patterns over time (exhibit 2) (click to enlarge) Exhibit 1 – Rolling window approach (m=120) (click to enlarge) Exhibit 2 – Rolling window approach (m=20) Note that in both cases, the forecasted volatility is constant over time, and it depends on today’s volatility: if volatility is high today, it is also likely to be high tomorrow, but how we will understand later, it is not always the case. In addition, the farer past m period has the same weight as the most recent. JPMorgan’s RiskMetrics The RiskMetrics approach can be seen as a generalization of the rolling window. All we have to do is: Replace the equal weights 1/m with exponentially decaying weights λτ-1 Replace the averaging over the past m period with an infinite summation The result is as follows: Or equivalently: According to the RiskMetrics, the forecast for tomorrow’s volatility is a weighted average of today’s volatility h t and today’s squared residual ε t. This method is slightly better than the rolling window since it gives more importance to recent observations rather than older ones. In other words, it doesn’t use an equally weighted average of the observations of past m periods, but exponentially decaying weights. However, it shares the same drawback: the forecasted volatility is constant over time and the unconditional volatility is completely ignored, as the graph below shows: (click to enlarge) Exhibit 3 – RiskMetrics If today is a low (high) variance day, RiskMetrics predicts low (high) variance for all future days. This will give a false sense of calmness (activity) of the market in the future. GARCH(1,1) Compared to the last two methods, the GARCH model represents the best way to estimate the future conditional volatility. In particular: Where ω> 0,α j ≥0,β j ≥0. Given the unconditional volatility: solving for ω and substituting in the GARCH equation, we obtain: meaning that the future variance (volatility 2 ) is a weighted average of: • The long-run variance (unconditional variance) • Today’s squared innovation • Today’s variance The more you forecast volatility ahead in the future, the more it depends on the long-run variance rather than today’s variance while the latter matters if you forecast volatility in the near future. In other words, if today is a low (high) variance day, the GARCH(1,1) predicts low (high) variance in the near future, and the long-run variance far in the future. In order to grasp the meaning of these words, exhibit 4 shows the results of GARCH. (click to enlarge) Exhibit 4 – GARCH(1,1) As the reader may understand, GARCH accounts for the fact that volatility is a stationary process, whereas the last two processes consider the process non-stationary. Thus, it is reasonable that tomorrow’s variance is similar to yesterday’s variance, but the volatility far in the future cannot be constant (like the rolling window and RiskMetrics predict) and it will stick to its mean or to the unconditional (long-run) variance. At the end, volatility remains a stationary process. Conclusions The rolling window and the RiskMetrics approach are methods that share similar features and the same drawback: they don’t account for the fact that volatility is a stationary process. Hence, the forecasted volatility is constant and it depends too heavily on today’s volatility. GARCH (1,1) is a better method since it takes in account today’s variance as a starting point, but then unconditional variance in the far long run. Hence, after having selected the best portfolio strategy or a combination of strategies, think about your risk management approach, and if you use volatility as a measure of risk, remember that, among the three models examined here, GARCH(1,1) is the best to forecast volatility. If you would like to read more about GARCH, I suggest you reading Bollerslev (1986).