2. Skew and tails: why shape matters more than width

Risk 1: Measuring risk

If the previous module left you with the feeling that standard deviation describes the width of the cloud but not its shape, this one turns that feeling into a money problem. We start with two series that a computer would declare identical twins and that nobody who has lived through them would ever confuse.

Two series that are twins in the numbers and opposites in real life

Consider two strategies, both with twelve months of track record:

• Strategy A. Eleven months of +1.5% and one month of -10.5%.
• Strategy B. Six months of +3.8% and six months of -2.8%.

Do the arithmetic. A's monthly mean is (11 x 1.5 - 10.5) / 12 = 0.5%; B's, (6 x 3.8 - 6 x 2.8) / 12 = 0.5%. Identical. In A the deviations from the mean are +1.0 eleven times and -11.0 once: the squares add up to 11 + 121 = 132, divided by twelve that is 11, and the square root is 3.3%. In B every deviation is worth 3.3 points in absolute terms, so its standard deviation is also 3.3%.

Same mean, same standard deviation, same Sharpe ratio. And if you compound the twelve months, A ends the year at +5.4% and B at +5.5%. Any ranking that looks only at those figures will put them side by side.

But look at A's record after eleven months: eleven hits, no misses, no losses, almost no volatility. A stratospheric Sharpe and a manager everyone wants to hand money to. All the relevant information sits in a single data point the sample does not yet contain.

What skew is and how to read its sign

Skew describes which way the distribution stretches.

• Negative skew: many small, positive results and a long tail to the left. You gain a little almost all the time and lose a lot once in a while; the median sits above the mean, which is to say the typical month is better than the average month.
• Positive skew: the other way round. You lose a little almost all the time and every so often you win a fortune. Most periods disappoint and even so the aggregate result can be excellent.

Strategy A is the pure case of negative skew, and it is no laboratory curiosity: it is the natural shape of anything that consists of collecting a recurring premium for taking on an infrequent event. Selling out-of-the-money options, insuring against defaults, funding long assets with short-term liabilities, buying high-yield debt in the middle of a boom. Every one of them takes in a reliable drip and has its bill concentrated at a single point in time.

Fat tails: width says nothing about how heavy the extreme is

Skew is about the side; fat tails are about the weight. A fat-tailed distribution assigns extreme events a probability far higher than a bell curve with the same standard deviation would give them.

Under the bell curve, a five-standard-deviation move should turn up once every several thousand years of trading sessions. In real markets it turns up every few years. On October 19, 1987 the S&P 500 fell around 20% in a single session: with the volatility of the time, more than twenty standard deviations, an event to which the normal model assigns a probability with no name in everyday language.

Benoit Mandelbrot had been pointing this out since the sixties: financial prices are not 'almost normal with the odd rare case', they belong to a different family of distributions in which extremes are a structural part of the phenomenon. Nassim Taleb added the practical consequence: if a handful of days explain most of the result, the mean and standard deviation of all the rest describe, above all, the irrelevant.

Why negative skew disguises itself as talent

For years on end, a negatively skewed strategy produces the very same signals that genuine talent would produce:

• Steady returns, higher than the market's.
• Low volatility, and therefore a sky-high Sharpe ratio.
• A small maximum drawdown, because the big fall has not happened yet.
• Low correlation with the stock market, until the day it becomes one.

The investor is paid for taking on a risk that has not yet materialized and books it as skill. Worse: success attracts capital, capital demands that the returns be kept up, and keeping them up forces more of the same to be taken on with more size. The tail fattens exactly when the most money is exposed to it.

Without any formulas, three questions clear away a lot of noise. What the economic mechanism behind the profit is, and whether it amounts to being paid to bear something that almost never happens. What would have to occur for this to lose a great deal all at once, and whether that scenario fits inside the period tested. And whether the worst month in the sample is a long way from the second worst, because the loss record of a negatively skewed series tends to be a data point with no company.

Case study: February 2018 and the products that sold calm

Through 2016 and 2017, exchange-traded products that bet on volatility staying low, by selling VIX futures, became popular. Their behavior was straight out of the textbook: constant rises, almost no setbacks, several hundred percent accumulated and a legion of retail investors convinced they had found a machine.

On February 5, 2018 the S&P 500 fell 4.1%, a bad session but not a historic one. The VIX, however, more than doubled on the day, the largest percentage jump on record. One of those products, Credit Suisse's XIV, lost around 90% of its value between that session and the next, and the issuer announced its early liquidation a few days later. Years of drip, one afternoon of bill.

Hold on to the practical conclusion: two strategies with the same mean and the same standard deviation can carry opposite risks, and the difference lies not in the width of the cloud but in where the tail is and how much it weighs.