Following on from the prior article, we will now study the effect of leverage on the severity of drawdowns. This is done for a series with a positive mean daily return over the last decade — so it should produce profitable investments in expectation. We will investigate the effect of leverage on the maximum drawdown in the series by using the bootstrap to elucidate the mean relationship rather than looking at one specific realization.

Our approach is as before, with the exception that we pick the leverage for each trial at randome between zero and four. Four is the maximum leverage permitted to a Pattern Day Trader, and so that seems a sensible limit for our analysis.

The above chart shows the observed relationship between the leverage used and the maximum drawdown within the nine years of trading represented by each simulation. The curve fitted is to the expression below (and is done by non-linear least squares).

(Here M is the maximum drawdown and L is the leverage; I don't assert that this relationship is anything other than a convenient representation of the data.) We see that with a standard margin account (2× leverage permitted), we expect a maximum drawdown of around 75% of capital and the probability of a drawdown exceeding 50% of capital is of order unity. Near the higher levels of leverage, complete drawdown (maxmimum drawdown exceeding 99% of capital) is increasingly certain. We shall present an empirical model for these probabilities in the next post.

The Bootstrap is a technique for simulating the sampling distribution of a statistic invented by Bradley Efron. It is a technique that attempts to solve the following problem: the empirical p.d.f. of a dataset clearly rejects common parametrical representations or the statistic we are computing has a population distribution that is analytically difficult or impossible to compute; however, the statistic is useful and we need to estimate it's sampling distribution to place confidence limits on the observed value.

The method is discussed in many places, such as Efron's excellent little book The Jackknife, the Bootstrap, and Other Resampling Plans, but I will summarize it briefly: we simulate data drawn from the empirical distribution function of the data by resampling with replacement of the actual data. This is clearly not as good as sampling from the population distribution function, but there are strong theorems governing the convgence of the e.d.f. to the p.d.f. and it does allow us to produce monte-carlo simulations of data with all of the measured properties of the sample (although the procedure is a little more complicated in the presence of serially correlated data). It is important to note that the replacement is an important step — it means that the properties of the simulations we create do not exactly match the actual sample and that allows us to estimate quantities such as the bias of an estimator.

The above charts show our use of The Bootstrap to analyze the series of daily returns of the Compact Model Portfolio. The upper chart shows five simulated total return time series (black) and the actual total return time series (red). The returns are accumulated and the dispersion of the final states due to a fortunate run of returns is very evident. The histograms show the distributions of the maximum drawdown and Sharpe Ratio for each simulated series. This are both popular metrics for quant. traders and are examples of statistics with awkward sampling distributions that a traditional analysis only gives use one opportunity to compute from historical data. The maximum drawdown histogram is fitted to the Gamma Distribution, and the Sharpe Ratio histogram to the Student's t Distribution. We learn from these charts that the standard deviation of the Sharpe Ratio is approximately equal to it's sample value and that the probability of a maximum drawdown exceeding 25% of capital is close to unity. It would not be possible to obtain this information via other methods.

Finally, I would like to acknowledge Greg Laughlin for stimulating my interest in using the Bootstrap method.

I recently changed several things regarding the Compact Model Portfolio (in particular the hedging strategy). This change went live, at a new brokerage, on the 9th. of April. To check that things are ok, we need to verify whether the returns series from the traded portfolio are not significantly distinguishable from the index — i.e. we need to investigate whether the traded portfolio is accurately tracking the desired index performance.

We have a small data set, so far, of only 12 days, so we need to be careful about our inferences. However, we have three tools to use:

• we can look at the time series of the total returns of both series and see if they appear similar (an eyeball or ballpark test);
• we can perform a linear regression of the traded portfolio daily returns onto the index portfolio daily returns, and apply statistical tests to investigate the null hypothesis (α,β) = (0,1) i.e. perfect tracking
• we can use the two-sided Kolmogorov-Smirnov test to investigate whether the empirical distributions of the respective daily return series are consistent with eachother.

Before presenting our results, let's discuss our expectations for the alternate hypothesis (that the traded portfolio does not track the index portfolio). The traded portfolio has frictions — brokerage fees, financing fees etc. — that are not represented in the ideal portfolio. In addition, the treatment of dividends is different. The ideal portfolio receives the dividend on the ex-dividend date, which is done to prevent spurious returns due to the expected ex-dividend drop; whereas, the traded portfolio will experience the ex-dividend drop and then receive the dividend income on the settlement date. The former effect should lead to a linear-regression α of less than zero. The latter effect should be a depression of the β below unity.

For the brief data sample we have, we cannot reject the null hypothesis. However, we have now established the toolset needed to investigate this issue, and will return to it after more time has elapsed.

The following chart updates the actual daily performance of the Compact Model Portfolio. This is a nominal scale portfolio, but it is an actually traded portfolio. The gap in early April occurred as I made some modifications to the system. Since then we're taking more risk, and so far the performance has picked up noticably. I'll explain these changes in a later post.

In the prior post we discussed the Compact Model Portfolio, in terms of its anomalous covariance with the benchmark index that served it well for years. To provide a little more context, here we exhibit a chart of the cumulative daily performance of the portfolio index (as distinct from the value of a traded portfolio) and it's benchmark, the NASDAQ-100.

Stationarity in time-series analysis means that the expected value of (a property) of the series doesn't change with time — more precisely as the sequence of innovations is accumulated. Finance often studies the covariance of different assets, the β's, with the goal of using this information to extract relative value by hedging out the returns of common risk factors. For this to work our current estimate of the β, or the relative covariance of two factors, is not actually what we need. We need to forecast the future value of the β and we would, ideally, like those forecasts to be unbiased and efficient.

The above chart shows a time series of the β of the Compact Model Portfolio, as discussed in various posts on this blog, and the NASDAQ-100 index. When I was developing this system, which follows “hot stocks,” I had noticed that the index tracked the NASDAQ with a higher R² than either the DJIA or S&P 500. The points are the sample β, computed from the daily returns for each month independently and the line is a forecast, made from the data available at the end of the prior month.

We will discuss the forecasting technique later (apart from commenting that it is not a moving average), but let's first concentrate on the phenomenological description of the series. The covariance is apparently pretty stable, although it does exhibit some temporal trends and regions of instability (heteroskedasticity), up until the last few months of the Financial Crisis (as dated from the Dynamic Trading Risk Factor or from the cumulative kurtosis of Treasury Bill rates, both of which are discussed on this blog). At that point there is a dramatic departure from the prior region of apparent stability, which lasts for thirteen months, before the series returns to its prior region. A trader who used the apparent stability of the series for the most part of a decade to hedge their book would have been unpleasantly surprised during this period.

Many would assert, in response to this plot, that this late 2000's period was anomalous, due to the financial crisis, and that this period should just be ignored. But I believe that this is an example of Magical Thinking. I base this assertion on two planks: firstly, that these sorts of upsets happen in financial data all the time; and secondly, that a trader going into this period would have no way of predicting this mistrack until it was fully upon them.

I think the lesson to be learned from this data is that in finance nothing is stationary and that we need to build our analytical structures on that assumption and that, if we do so, we can do a better job at keeping things under control.