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 Portfolio Construction Based on Management of Tail Risk
 2 Risk Measures
The most common and widely adopted measure of risk is volatility, which forms the basis of Modern Portfolio Theory and other “traditional" approaches, including portfolio metrics such as the Sharpe ratio, and the Black-Scholes model for options. The volatility is another word for the standard deviation, which is a measure of the size of typical (“standard") deviations from an expected rate of return, which are manifested in the day-to-day fluctuations of a portfolio’s value. As it turns out, the reason for its wide spread use is its simplicity rather than its accuracy in quantifying risk, as we now explain.
2.1 The Problem with Volatility
We believe risk is fundamentally about extremes, not averages. Since volatility is inherently a measure of averages, it does not offer information about the size of the large, atypical deviations due to events such as the current Covid-19 pandemic that have an outsized impact on the portfolio.
At the heart of all of these standard-deviation-driven techniques is the Gaussian distribution, a probabilistic model that is completely described by its mean and standard deviation. The Gaussian is ubiquitous because it is mathematically simple and one of the few probabilistic models that can be used without the power of modern computing. However, the Gaussian is a poor model of financial returns. Some important features of real financial data that cannot be described by the Gaussian include skewness and “fat-tailedness," both of which are represented in Figure 1 and described below:
• Skewness: Financial returns often exhibit "stairs up, elevator down" behavior, i.e., fluctuations down tend to be bigger than fluctuations up. The Gaussian is symmetric and cannot model this skew. Furthermore, portfolio selection techniques that penalize the standard deviation implicitly penalize unexpected upward movements just as much as downward movements, making it difficult to disentangle risk from reward.
• Fat-tailedness: While Gaussian data rarely lie more than three standard deviations from the mean (happening only .1% of the time), financial returns undergo these extreme “three sigma"1 (and sometimes much larger) movements far more frequently. When returns are plotted in terms of their probability density, these events make the “tails" of the distribution appear far thicker than those of a Gaussian.
Several alternative measures of risk have been proposed to tackle these problems. For example, the
1Sigma, written σ, is the Greek letter typically used to denote the standard deviation.
Figure 1: A Gaussian distribution is compared to a distribution exhibiting left-skew and fat tails. Despite both distributions exhibiting similar scale, the fat-tailed distribution has much higher probability density in the left tail, corresponding to more realized outcomes with extremely negative returns.
“downside volatility" addresses skewness by isolating only the downside contribution to the volatility, lead- ing to “Postmodern Portfolio Theory," which applies Markowitz’s ideas to this modified risk metric. While this approach solves part of the problem, it still only describes “typical" downside fluctuations and is unable to capture extremes. Any risk measure that restricts its focus to these typical fluctuations is completely blind to what we argue is the far more important source of risk: fat-tailed losses.
2.2 Tail Risk
Risk measures that focus squarely on tail losses, such as the Value at Risk (VaR), the Conditional Value at Risk (CVaR), and the Expected Max Drawdown (EMDD), tell us specifically about extreme negative outcomes rather than describing typical behaviors. We describe these measures below and discuss some of their strengths and weaknesses.
Following the 2008 financial crisis and the enact- ment of Dodd–Frank, the Basel Committee on Banking Supervision (BCBS) adopted VaRas the standard mea- sure to determine minimum capital requirements for market risk in order to overcome the substantial and pertinent shortcomings of volatility-based measures. The Value at Risk at level α, written VaRα, is the maxi- mum loss (smallest return) such that the probability of exceeding that loss is at least α, where we simply define loss to be the negative of a return. For example, if a portfolio has a one-day VaRα of $1 million with α = .01, this means that there is an estimated 1% chance that the portfolio will lose more than $1 million on any given day. We expect to experience a loss exceeding the one-
day 1% VaRroughly once in any 100 day period (since 100 = 1/.01). In this sense, theVaRrepresents a

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