#### How “Tail Risk” Changes Over the Market Cycle via First Quadrant

Investors have been more conscious of “tail risk” since the financial crisis of 2008. The term refers to the frequency distribution of stock market returns having fatter “tails” than the normal distribution (the traditional “bell-shaped curve”) which has been used for years to model market returns and evaluate risk. The fatter tails mean that the probability of a significant positive or negative event is larger than the normal distribution would lead us to believe. Since many risk management and asset allocation tools use the normal distribution as a starting assumption, the risk of a large or catastrophic event is often understated. The on-line financial dictionary Investopedia defines “tail risk” as

“A form of portfolio risk that arises when the possibility that an investment will move more than three standard deviations from the mean is greater than what is shown by a normal distribution.”

Investors have also become more accepting of the existence of “volatility regimes.” That is, the market can be defined by periods of persistently higher or lower than average volatility. Volatility is usually defined as annualized standard deviation. But when investors discuss a “three-standard-deviation event” they are likely speaking of the average standard deviation over time rather than recent standard deviation. Investors typically use 15% annualized standard deviation for the stock market as a rule of thumb since that fi ts the long-term standard deviation of many large cap indices such as the S&P 500 or the MSCI equity indices. So a “three-standard-deviation event” in monthly terms would be a decline of approximately -13%.

We can postulate that whether annualized volatility of the above indices is running at a 10% or 25% rate in the recent past, investors will still be referring to a -13% or greater monthly decline when they are describing “tail risk.” For many, a two-standard-deviation decline of -8.66% would also be considered a “fat-tail event.”

But is anchoring onto what would statistically be referred to as the “population standard deviation” (which assumes tail risk is constant) reasonable if we have significant volatility regimes? Previous work (Peters (2009)) has found that the market can be defined by periods of persistently higher or lower than median volatility with market characteristics quite different in the two regimes. These periods are partially explained by looking at the VIX, an index of the implied volatility of stock index options. Since the VIX is the implied volatility of options that falls out of the Black-Scholes option pricing formula (Black and Sholes (1973)), many confuse this number with a forecast of realized volatility.

Unfortunately the VIX is a poor predictor of actual volatility. Instead, it is a better indicator of a level of market, or even macroeconomic, uncertainty. That is, it is a signal that the markets are entering a period of exceptional high or low uncertainty with changing market characteristics. What is not widely appreciated is that the VIX can also be a signal of increasing tail risk as well as a sign that the market is prone to “negative skew,” meaning that there will be more large down-market than large up-market moves.

In this paper, we will document that for many sectors of the stock and bond markets, tail risk and negative skew are conditional upon volatility regime. Tail risk is not constant. We will first see this through a graphical representation of the distribution of returns. Then we will look at this phenomenon through new statistics we call conditional kurtosis and conditional skewness. That is, given we are in a high or low uncertainty environment, what is the probability of a “fat-tail event”? We will find that the risk of a large move and a negative skew in market returns is much higher when the VIX is above its median than below. So traditional statistics both understate and overstate tail risk. The actual methodology for calculating conditional kurtosis and skewness is simple and intuitive though it has not appeared in the literature as far as we know. It is also different than the conditional statistics discussed in the literature about GARCH models.

This research has significant implications for asset and security allocation and is also one more indicator of weakness in standard capital market theory such as the Sharpe-Litner-Mossin Capital Asset Pricing Model (CAPM) which remains the primary model for long-term strategic asset allocation (Sharpe (1964), Litner (1965), Mossin (1966)).

The Distribution of Market Returns Quantitative finance has long assumed that market returns are normally distributed, or follow the familiar bell-shaped curve. This was necessary in order for many mathematical techniques to have validity (Mandelbrot (1964) and Peters (1994) among others). However, it has been well known since the 1950s that market returns are not normally distributed and have a high peak at the mean and fatter tails than the normal distribution (Osborne (1959)).

Exhibit 01 illustrates the distribution of monthly market returns (the MSCI World Index in local currency in excess of 3-month LIBOR, “MSCI”) for 24 years ending 2013. The normal distribution is also shown using the mean and standard deviation of the MSCI.

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