If you thought this was a post about how to generate outsize returns you are partially correct. We show a way of splitting a return distribution into a well-behaved bell curve component and a nasty outlier component where the money gets made or lost (and the Efficient Market Hypothesis gets kicked on the backside).

We use the S&P500 Index and work on daily log returns. Using the closing prices of the index, we generate the time series {log[P(t)/P(t-1)]} where {P(t)} is the time series of closing prices. A histogram of the returns is plotted in Fig1 on which we superpose a histogram of “normal” returns – i.e. returns generated from a Gaussian distribution with the same mean and standard deviation as the actual returns data.

As one would expect, the actual returns distribution is more concentrated in the middle (“leptokurtic”) and has more bars sticking out at the extremes (“fat tails”) than the Gaussian returns distribution. This makes it difficult to risk-manage mean reverting trading strategies (e.g. the theta-burn variety) constructed on the assumption of Gaussian returns: the probability of returns remaining within a given number of standard deviations for a given holding period is no longer as predicted by the Gaussian model. Similarly, break-out strategies are difficult to structure and risk-manage, given that large moves occur with a higher frequency than suggested by the Gaussian assumption.

It would be useful to be able to *truncate *the returns distribution – systematically take out outlier moves – so as to get a normal distribution. We could then view the actual returns distribution as a *mixture *of a normal distribution which holds x% of the time and another distribution which holds (100-x)% of the time. Using statistical hypothesis testing, we find x=90% for the S&P500 data used. This means that we expect the actual returns distribution to be Gaussian 90% of the time. We can then assert with a fair degree of confidence that 1 standard deviation of the truncated returns distribution represents 68% of the area under the (truncated) Gaussian curve. See Fig 2. Note that the 1sd of the truncated distribution is almost 40% lower than the 1sd of the actual distribution. This means that mean reversion strategies can be risk-managed more aggressively with distributional risk under tighter control.

We plot the outlier distribution in Fig 3. These are the “breakout” days – expected to occur 10% of the time for the S&P 500 for the period analyzed. This means we would expect a breakout move with a median absolute return of 2.53% once every fortnight. We thus have the ingredients in place to risk-manage a two week theta burn strategy.

We have chosen to fit a gamma distribution but other distributions could be deployed to tease out the key statistical characteristics of the breakout days. Higher frequency trading strategies would seek to identify the microstructure of the outlier days (opening location, type etc.) so as to be able to ride the wave or step aside in time.