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.

In prior posts we have demonstrated relative skill of 18% in making out-of-sample forecasts of the Dynamic Trading Risk Factor, a factor series that we hypothesize is a driving factor for the returns of companies that make money by dynamically trading securities and earning a premium from by selling the synthetic options created by such trading activity to their investors. We have also shown that the returns of well known public  financial companies, such as Goldman Sachs and Berkshire Hathaway are well explained by this factor. By well explained we mean that a linear regression of the monthly returns to investors, including dividends if any, onto the factor has a significant β and a large R². Exhibited below is our up-to-date chart comparing the monthly returns of Goldman Sachs, on which the rest of this analysis will concentrate, with those of the Dynamic Trading Risk Factor.

We originally performed this analysis in February, 2009; so in the following I will treat the period 2001:01–2009:01 as in-sample and 2009:02–2010:03 as out-of-sample. Using the Jackknife procedure discussed earlier, we find a bias corrected skill of 4% ±14% relative to the forecasts computed from the in-sample average monthly returns. The Dynamic Trading Risk Factor based forecasts are obtained by using the α and β, as established by linear regression within the in-sample period, as the model coefficients and the out-of-sample conditional mean factor forecasts from the AR(1) model for the driving factor series.

Using the metrics established in Grinold & Kahn's Active Portfolio Management, this forecast has an Information Coefficient, or IC of 23% which would lead to a Sharpe Ratio of 0.8 if traded on a monthly basis (I used G&K's  “rule of thumb” SR≈IC√N to estimate the Sharpe Ratio).

Standing alone, this skill estimate is not statistically signficant, but we know we have skill in forecasting the factor series and we know that the out-of-sample α and β for Goldman Sachs are consistent with their in-sample estimates, so my best guess is that the skill is weak but real.

In an earlier post we examined the out-of-sample performance of a DJIA intraday strategy developed, and actively traded, here at Giller Investments.

That post provides some detail as to how the strategy was built and how it is operated. In addition to the DJIA system we also operate a similar system for the NASDAQ-100 futures.

In the chart above, as before, there are four panels. The two on the left hand side are regressions of the intraday price change onto the forecast. The upper panel is for all data, and the lower resticted for dates on which trades were done. Regressions are computed both for simple linear regression and when weighted with the forecast variance. The R² is ≅ 1.5%, which corresponds to a correlation coefficient of order 12%. According the the rule of thumb from Grinold & Kahn's Active Portfolio Management, if we could trade every day with negligible costs, this would give a Sharpe Ratio of approximately 1.9. Note how the linear relationship, although statistically well established, is difficult to observe by eye due to the low R². The t-Statistic for the fitted gradients is referenced relative to the null hypothesis value of 1 (not 0). This is because the null for this out-of-sample regression is that the system works as modelled (β = 1).

On the right hand side, there is a chart showing each day's forecast at trade time and how that compares to the trade entry barriers. This illustrates the variability of the scale of the alpha with the local volatility conditions. Finally, for a contextual reference, we present a chart showing the time series of the index level and the daily point volatility.

For the past few days it has been possible to subscribe to an on-line trade blotter which echoes futures trades done in one of Giller Investments proprietary trading accounts. Delayed feeds are also available on Twitter and as an RSS Feed at Feedburner .

The feeds are the trades, and a VWAP summary. The trades, which are actual confirmed trades as reported back by our brokerage, are scaled to take account of the relative sizes of the forecast, the level of volatility, and the account equity — which is currently around $100,000 for each contract type for this trading account. We are currently trading the Dow Jones Industrial Average e-mini contract (ticker symbol YMm, where m is the contract month) and the NASDAQ-100 Index e-mini contract (ticker symbol NQm).

The strategy is referred to at Giller Investments as the One-Shot strategy. The design goal is to forecast the intraday price change of the index based on the state of the markets in the morning, shortly after the opening session. If forecastable, a position is entered over several trades and the position held until the end of the day. Positions are never held overnight for this strategy. Although some risk management trades might be done, intraday, the strategy essentially has "one shot" at getting the trade right each day.

As one trade is done, and that trade is certain to be terminated, the trading strategy implemented is a straightforward barrier trading strategy. However, we will not discuss that specifically here. Here we will examine the forecasting power of the DJIA system.

This system was originally implemented at the end of September, 2006; and the data presented here was collected from that implementation to date — i.e. it is entirely out-of-sample.

In the chart above there are four panels. The two on the left hand side are regressions of the intraday price change onto the forecast. The upper panel is for all data, and the lower resticted for dates on which trades were done. Regressions are computed both for simple linear regression and when weighted with the forecast variance. The R² is ≅ 2%, which corresponds to a correlation coefficient of order 15%. According the the rule of thumb from Grinold & Kahn's Active Portfolio Management, if we could trade every day with negligible costs, this would give a Sharpe Ratio of approximately 2. Note how the linear relationship, although statistically well established, is difficult to observe by eye due to the low R².

On the right hand side, there is a chart showing each day's forecast at trade time and how that compares to the trade entry barriers. This illustrates the variability of the scale of the alpha with the local volatility conditions. Finally, for a contextual reference, we present a chart showing the time series of the index level and the daily point volatility.