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 Portfolio Construction Based on Management of Tail Risk
boundary between typical (with probability 1-α) and atypical (with probability α) loss events. However, the VaR fails to capture how large we expect those atypical and extreme losses to actually be.
The Conditional Value at Risk (CVar), also known as the Expected Shortfall or the Expected Tail Loss, directly addresses this issue. CVaRα is the expected loss under the condition that VaRα is exceeded. For example, the 1% daily CVaR is the loss we expect over the worst 1% of days. The distinction between the CVaR and the VaR may seem subtle, but CVaRα can be far greater than VARα, particularly in the presence of fat tails, as is depicted in Figure 2. Its greater sensitivity to fat-tailedness (as well as its enjoyment of a few extra desirable mathematical properties such as coherence) make the CVaR a considerable improvement over the VaR. In fact, in order to more sufficiently account for tail risk, CVaR replaced VaR as the BCBS standard measure of market risk in 2016.
Figure 2: VaR and CVaR at α = .1 for a toy model. By defi- nition, 10% of losses are higher than the VaR, and the CVaR represents the average loss exceeding this threshold. Due to the loss distribution’s fat upper tail, the CVaR 6.4% is much larger than the VaR 2.7%. Note that the x-axis is in percent loss, the negative of return.
While both the VaR and CVaR are able to capture important aspects of tail risk, they can only describe extreme losses over short, fixed time-windows, such as days or weeks. One reason for this is that, without access to hundreds of years of historical data, it is difficult to estimate the severity of events that occur on average only once every one-hundred years. Furthermore, neither measure can account for the tendency of extreme losses to cluster: sometimes it is a bad day or week, but sometimes it is a bad month or quarter. Max Drawdown (MDD), on the other hand, measures peak-to-trough losses, no matter how quickly or slowly these losses accumulate. Expected Max Drawdown (EMDD) is the MDD that we expect over a wide time-horizon such as a quarter or a year. While the 1% daily CVaR tells us the average worst one-day loss that occurs
roughly once every one-hundred days, the one-hundred day EMDD tells us the average worst drawdown over a one
one-hundred day period, whether it be due to a single bad day or a string of many bad days.
3 Probabilistic Model Comparisons
The CVaR and EMDD are fantastic single-number measures of tail risk, but they are somewhat odd in that they describe “average extremes". What if we want to know about the extremes of our extremes? This motivates us to shift away from point estimates of extreme risk and towards full probabilistic models. For example, instead of asking, “what is the average quarterly MDD?", we can ask, “what is the probability that the quarterly MDD exceeds 5%? 10%? 20%?"
The simplest approach to estimating these probabilities is empirical. If we want to know the probability that the quarterly MDD exceeds 10%, we can simply count how many past quarters have seen such an extreme MDD and divide that number by the total number of quarters on record. This approach has several substantial blind-spots, however:
• Nonstationarity: The distribution of quarterly MDDs may change with time, behaving differently in the future from how they behaved in the past.
• Serial dependence: The MDD next quarter may depend on the MDD this quarter. For example, the 2008 financial crisis lasted over a year. This serial dependence is information that can be leveraged to predict risk more effectively, but it can also be confounding to models that assume that each period on record is independent of every other.
• Undersampled tails: How might we estimate the probability of an MDD greater than 10% if we have yet to observe such an extreme? Surely the probability is greater than zero! Because tail events are rare by definition, they tend to be under- represented in historical data, causing naive inferences about tail risks to vary widely and to underestimate reality.
In this report, we focus on overcoming the final of these three limitations, undersampled tails, but the importance of the first
two cannot be understated. Detecting nonstationarity and estimating short and long- term changes in risk constitute essential components of our risk-management approach.
3.1 Extreme Value Theory
In order to estimate the probabilities (and the expected values) of extreme events, it is crucial to shift away from the empirical perspective and to choose a suitable model with a few relevant parameters. Then, instead of having to separately estimate the probabilities of exceeding 5%, 10%, 20%, and so on, we simply estimate our model parameters, and our model will answer these questions for us. Having fewer parameters greatly reduces the variability of our estimates. But we have to be careful to choose a model class that is consistent with our problem so we do not introduce unnecessary statistical bias. The Central Limit Theorem states that the mean value of a large collection of random variables tends to a Gaussian distribution – provided the random vari-

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