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Index Futures Trades on Twitter --- A Web 2.0 Experiment Part II

Published earlier on blog.gillerinvestments.com

This has taken a while, due to teething issues, but my Web 2.0 CTA experiment advances one step further today with a delayed Twitter feed and a real-time Twitter feed. Like the RSS feeds, these Twitter feeds provide a trade blotter for my index futures intraday strategy. There are details on how to subscribe to the real-time feeds on my main company web site.

IKOS Equity Hedge Fund --- Data Update

Published earlier on blog.gillerinvestments.com

With another month of data for the dynamic trading risk factor available, we can look again at how various funds' and companies' performance compares to this factor. As we do not have a great deal more data, and nothing very dramatic has happened since this analysis was last performed, it is unlikely that the pro articulum parameter estimates will have changed very much, so I won't report the regression analysis in depth.

Continued…

Hedge Fund Factor for March, 2009

Published earlier on blog.gillerinvestments.com

I've updated the factor regressions for the Dynamic Trading Risk Premium model, as described earlier. This updates the dynamic trading risk factor data series and also updates the dynamic trading risk factor chart (including the version linked to in earlier posts).

Continued…

A Monte-Carlo of I.I.D. Normal Innovations

Published earlier on blog.gillerinvestments.com

When talking about the SPX data, I glibly asserted that the data was evidently not I.I.D. normal. I then proceeded to show how the Generalized Error Distribution can be used to describe the data quite well and to reject the hypothesis that the data is I.I.D. Normal with a reasonable degree of confidence.

Continued…

If Not Normal then What

Published earlier on blog.gillerinvestments.com

In the previous post we illustrated the evident abnormality of financial data by examining the longitudenal returns of the S&P 500 Index.

I used the Generalized Error Distribution as it possesses the ability to be smoothly transformed from a Normal Distribution into a leptokurtotic distribution and that allowed me to use the Maximum Likelihood Ratio Test to distinguish between the null hypothesis (that the data is I.I.D. Normal) and the alternate hypothesis (that it is not).

I subscribe to the theory that if something is right you should be able to draw the same conclusions via various methods and data sets. So I am going to look again at the likely models for the innovations of financial data (we're taking a GARCH(1,1) model as given); but, this time, I decided to look at the S&P Goldman Sachs Commodity Index and to use a test based on Pearson's χ² Test. (In the following the data is actually based on the first deliverable contract on the GSCI traded at the CME.)

Before that, however, we should discuss what the possible options are the for the PDF of the process innovations. The candidates are:

• The Normal Distribution


• Levy Flight


• The Generalized Error Distribution


• Student's t Distribution


• something else…

Continued…

Why use the Generalized Error Distribution?

Published earlier on blog.gillerinvestments.com

This post is to address the question why use the Generalized Error Distribution? The subject, the evident abnormality of financial data, should be very familiar to the intended audience of this blog; but I'm going to summarize some basic facts here as there have been requests as to why the GED should be used.

Firstly, longitudenal returns of financial asset prices are evidently not described by the Normal Distribution. Many statements one hears along the lines of "a once in a hundred years"event are made in the context on comparing the scale of a realized event with its expected rate under the normal distribution. However, financial data are so clearly non-normal (more specifically not identically and independently distributed, or I.I.D., normal) that only a naive analyst would even start off an argument by discussing that hypothesis.

Even without doing any statistical tests, a cursory analysis of the time series of daily S&P 500 Index returns (the upper panel in the above figure) would suggest that the returns are not homoskedastic — or constant in variance.

The lower panel shows the best fit of the normal distribution form to a histogram of daily index returns. The fit is clearly poor, and the data shows the pattern typical of leptokurtotic data. There is a deficit of events in the sides of the distribution (in the region around ±1σ) and an excess in the centre and in the tails.

Since the data seems heteroskedastic, and since there seem to be episodes of heteroskedasticity, this data is clearly a candidate to try to fit a GARCH model to. It's possible to specify a GARCH model with normally distributed innovations, but which would give rise to the leptokurtotic distribution we observe in the histogram, so we should test for that.

I'm interested in specifying the process distribution correctly because it directly affects the relative weighting of the various data periods in any regression analysis we do. Ordinary least squares is only the correct estimation procedure when the underlying data are i.i.d. normal. This procedure assumes that deviations at the level of 3σ–5σ, or more, are highly significant and will cause the estimated parameters to be chosen to explain these particular realizations more than those in the lower range.

In the case of the data above, the regression will listen strongly to the current period, although the process realization now many not be that characteristic of the entire period. One might argue that we should just replace OLS with generalized least squares which, if we weight with the appropriate covariance matrix, is equivalent to maximum likelihood estimation which is a very powerful technique. However, this does not circumvent the problem of estimation based on the normal distribution treating 3σ–5σ residuals as very very significant whereas, under a leptokurtotic distribution, they are not particularly so.

The GED is useful because it can be smoothely transformed from a Normal distribution into a leptokurtotic distribution ("fat tails") or even into a platykurtotic distribution ("thin tails"). This allows us to use the maximum likelihood ratio test to test the hypothesis as to whether the GARCH process innovations are IID normal.

This test convincingly rejects the null hypothesis that the GARCH process innovations are normally distributed (shape=1). The estimated shape parameter, which controls the kurtosis of the distribution, is also approximately 6σ from the null hypothesis value.

In another post I will go into more depth about the various distributional choices that are available once one rejects the Normal.

Rounding --- An Implicit Buy-High, Sell-Low Strategy

Published earlier on blog.gillerinvestments.com

Last year, before the crash of the emerging markets – pro articulum in general – Prof. Jeremy Siegel was featured in an advert played regularly on CNBC for Wisdom Tree, talking about the inherent "buy high, sell low" strategy embedded in cap. weighted indices.

The basic problem is that when the price of a subset of the index increases then their weight relative to the rest of the index also increases. The index tracking investor is then required to buy more of those components, at their new higher price. If their prices should subsequently decline, then the index tracking investor will be required to sell a little of the investment, for the same reasoning as before, at the new lower price.

Unfortunately, stocks do regularly go up and down relative to each other and so the logic embedded in the previous paragraph represents an embedded buy high – sell low strategy which is overlaid over the basic strategy represented by the index. This is one of the defects of cap. weighted indices and will lead a fund manager that attempts to track such an index to underperform through no fault of their own.

The Markowitz Portfolio is constructed to be Mean-Variance efficient and weights components so that the expected risk-adjusted profit from each position is equal. However, cap. weighting doesn't follow any utility driven formalism and it explicitly contradicts known facts about the market (it overweights large cap. stocks whereas academic reasarch by Fama and French indicates that small cap. stocks consistently outperform).

The adverts. caught my attention because I had just tackled a similar buy high – sell low defect in the basket I own to track the Compact Model Portfolio. The portfolio that tracks the CMP Index is equally weighted, meaning that we allocate the same fraction of the overall equity to each individual investment.

Now equal weighting also has an embedded strategy, but in this case it is reversion rather than momentum. With an equal weighted basket, every time returns occur we need to reduce the position in the stocks that outperformed and increase the position in the stocks that underperformed, in order that we maintain the equal weighting. This is an embedded sell high – buy low strategy.

I was aware of this, but as I watched my basket I realized that I kept repeating the opposite. On the daily rebalance, the strategy would buy some more of a stock that went up at the end of the day and then, then next day, if it lost money, it would sell at a loss. This was repeated again and again.

I finally realized that this was because I was rounding my position into round lots, of a given size. The conventional algorithm for rounding positive numbers is to add one half and then truncate to an integer. The number of lots to hold in a given company is the fraction of the capital allocated to that company divided by the product of the price and the lot size. Following conventional ½ rounding we tend to round up after we've made money and round down after we've lost money. This is an embedded buy high – sell low strategy.

I solved this by rounding against it. I round up on a losing day and round down on a winning day. i.e.

shares=lotsize×⌊capital/(lotsize×price)−½sign δprice⌋.

This seems to work.

A Brief Summary of the Compact Model Portfolio

In the very first post on this blog, I referred to the Compact Model Portfolio, which is a strategy I have researched and traded for a while.

The premise of the strategy is that the stock market is a voting mechanism for trading ideas (not a radical theory, I admit.) However, in this case we specifically assume that the market is able to pick the winning stocks but that it is not very good at trading them. i.e. That we can infer from market activity what the stocks we should own are, but that the market is not sufficiently efficient to eliminate the excess return that accrues to the owner of those stocks. I call this a semi-efficient markets approach.

The strategy is described in detail in the document linked to above and I do trade based upon this strategy. I find it appealing because it is a different way of looking at the market to the paradigm followed by traditional alpha trading. Basically, we examine the dollar volume of each stock in the market and use this to create a ranking based upon the markets' interest in each company. We then cherry pick this ranking for a subset of stocks to hold in a portfolio.

Technically we are assuming that the ordinal ranks are efficiently expressed but that the cardinal ranks are not — that the market can pick the stocks to own but that it doesn't do a good job of trading them.

The purpose of this post is not to recommend this strategy as an investment vehicle for the general public. It is to highlight a different way of looking at things. Quantitative traders can get trapped into thought ruts — particularly if their methodology leads to some success for them. It's to answer the question "what does the market think I should hold?" This way, the analyst has a base portfolio to compare their holdings too that is not wedded to implicit biases and scales in the way the major market indices (as currently composed) are.

This portfolio might also not be a suitable starting point for many investors. Based on it's current composition, the market is currently interested in holding SDS and SKF, which are ultrashort (i.e. two times leveraged) exchange traded funds. As a result of this holding, the index is currently profitable for this year. I recently added a link to the historical holdings of the Compact Model Portfolio. This allows one to follow the market's preferences as they change (it should be noted that the rankings involve a time scale of several months, so members will not change rapidly — they will drift up and down).

UPDATE (14:30 PM EDT): Of course, the on day I post this SKF and SDS are having a terrible day (at the time of writing SKF is down 21% and SDS down 11%). I hope this underscores the point that this post and the data associated with it should not be taken as blind investment calls — you should always verify that an investment is right for you.

Refreshed Hedge Fund Data

We have a new month and new data for the dynamic trading risk factor. Last month's forecast was for a profit of 96 bp; however, the realization was a loss of 96 bp.

The data series estimates update slightly to a sample mean of 43 bp/month drift (which has a t-Statistic of 2.30 and a p-value of 0.024). The sample standard deviation is 187 bp/month and the simple Sharpe ratio (the t-Statistic times the square root of 12) is 0.80.

I don't want to read a lot into the estimated form for the fit of a Generalized Error Distribution, after all there are only 98 data points in total, but we note that it has a spectral index of approximately 0.5 which is indicative of a platykurtotic distribution — i.e. one with censored tails. This should be viewed skeptically as it is inconsistent with the sample excess kurtosis of 3.73.

The forecast for March'09 is a loss of 9 bp.

Penny Stocks and Index Bias

The SEC defines a penny stock as one in a small and illiquid company that trades for less than $5 per share. Note that this is not a stock who's price is less than $1, it is one who's price is less than $5 (with other conditions). Penny shares have typically been associated with disreputable or highly speculative corporations that the SEC felt it needed to have special rules to protect ordinary investors from.

Today, a large number of very prominent companies trade below $5, and this is exposing some biases and flaws in processes that we have previously taken for granted.

The NYSE has suspended it's rule that all shares must trade above $1 or face mandatory delisting; the market capitalization of a substantial number of Dow Jones Industrial Average constituents have declined dramatically over the past months; some suggest that many S&P 500 Index constituents would no longer be selected to be included at this stage (the S&P index construction methodology is outlined here).

We need to remember, when examining the performance during this current financial crisis (per articulum) of indices composed before the crisis (pro articulum), that these indices are not updated frequently and are composed using arbitary rules that are not scale free.

For example, Standards and Poors currently requires that a company have a market capitalization of at least $3 Billion to be included in the index — Citigroup's is currently just $5.6B; General Motor's is just $1.1B; and both are index constituents.

Of course, this problem occurs because the selection is not done in a scale free fashion. Instead of starting with all domestic companies who exceed this arbitary threshold, they should rank the capitalization of all domestic companies and then use those ranked 1–500 for the index. That is a scale free method that is not purturbed by a sudden downdraught in the market.

Another bias in the indices is that people (meaning both the users and composers) don't like the index constituents to change all the time — yet, any grouping of items by a random number relative to a threshold will create a jitter in the membership among those close to the threshold. This is impossible to remove for any criterion that has a "hard" cutoff — although it can be damped by appropriate filtering techniques. The dislike of this phenomenon causes members to be kept in (or out) of the index for too long as the index committee tries to decide whether the fluctuation that carrier a single stock over the barrier will persist or not.

NASDAQ-100 Volatility — Difficult to Assess, but Not Extreme

Below is presented our analysis of the daily point volatility of the NASDAQ-100 Index. Although I normally like to talk about daily point volatilies, because there's no ambiguity about what that means, I'm presenting this series as an annualized returns volatility in percent because the substantial excursions of the NASDAQ indices over the past 15 years make it difficulte to make a concrete statement about what the "normal" level of volatility is, and removing the scale does seem to help somewhat. My guess would be that "normal" means about 25% per annum and, although the current level is in the 40's, it is nowhere near the extreme levels of either the late 90's or of the end of last year.

The innovations are driven by a Generalized Error Distribution and fit nicely to this with a spectral index of ≅ 1.2 — which is not that far from the Normal Distribution (which corresponds to 1.0 in the parameterization in use). I'm actually a bit suprised at the niceness of this distribution.

Some Volatility Data

As I was building a summary page about the volatility of the NASDAQ-100 Index, as we looked at the Dow Jones Industrial Average and the S&P 500 Index in earlier posts, I thought that volatility data is actually a fairly scarce commodity on the internet. Due to the heteroskedasticity of financial markets, using dynamically forecast volatility is critical to investment decisions and to simple analysis, such as linear regressions, which should be variance weighted (making the common least-squares regression actually equivalent to the more general maximum likelihood estimation method).

Making a volatility forecast that is reasonably good is actually not that hard, and simple GARCH models are easy to fit and provide fairly good out-of-sample forecasting ability. So, without further ado, here are links to simple volatility models for the three major market indices: the Dow Jones Industrial Average; the S&P 500 Index; and, the NASDAQ-100 Index. This data is computed from publically available information that is believed but not guaranteed to be correct. The data is a statistically derived estimate and should be correct out-of-sample on average. It is updated daily and each estimate applies only for the day indicated in the series. For each date the annualized relative volatility (i.e. of returns) in percent, the daily point volatility, and the day's actual index point change are presented. All are from prior-close to close.

S&P 500 Volatility — Now Barely Normal

All the major market indices had massive spikes in volatility at the end of last year. I use a common paradigm to model all of them. A simple GARCH model to forecast daily price volatility. These models were all developed on prior data and are running out of sample.

We looked at the volatility of the DJIA relative to its history in a recent post. Presented here is the same chart built for the S&P 500 Index, which is popular with institutional fund managers but has an inbuilt large-cap bias and does not represent a mimumum variance portfolio. Although the member selection method is less arbitary than that for the Dow, it is still not 100% mechanical.

We see the S&P volatility has also reduced and is at the upper end of the "normal" range. Like the Dow analysis, the volatility model is presented in terms of daily point move (for clarity of exposition) and is fitted with driving innovations that are intrinsically leptokurtotic — i.e. we model the fat tails as an intrinsic property of the driving process and not solely attributable to the composite nature of GARCH type process. The
generalized error distribution is used to model the i.i.d. innovations.

Autometric Part II — How is the Compact Model Portfolio Doing?

When I started this blog, I mentioned a system I call Compact Model Portfolio.

This is a portfolio selection system in which econometric methods are applied to the time series of daily dollar volume for stocks traded on U.S. exchanges. The goal is to answer the question: which stocks are market participants most interested in, using dollar value traded as a metric of interest. Using this data we select a small portfolio which represents the stocks voted by the market as those most likely to outperform.

I call this a "semi-efficient markets" approach because we accept the hypothesis that the market is a voting method which possesses the ability to efficiently select the best stocks; however, we do not accept the hypothesis that all information about these companies is fully and efficiently incorporated into their current prices.

I select these stocks daily, although the turnover is low, and a
representative portfolio is available from my website. Historical regression analysis shows that this portfolios' next day returns are well correlated with the NASDAQ-100 index, but that it outperforms this benchmark over the long run.

I did this analysis before the current work on dynamic trading risk factors; however, since this is a dynamically selected portfolio, it is interesting to ask whether there is a covariance between this system and what, we have found to be, is a common factor behind the returns of many large hedge funds.

If this system is well characterized by the null hypothesis (α,β)=(0,1), then we have a discovered a simple procedure that replicates what we have discovered to be an explanatory factor for the returns of several large hedge funds — this is a very interesting outcome!

The chart shows a comparison of the monthly returns accruing to the Compact Model Portfolio when hedged by allocating one third of the assets to a long position in the ProShares UltraShort QQQ ETF (AMEX:QID).

The results of this regression shows an insignificant but positive alpha of (1.04±0.73)%/month and a beta onto the dynamic trading risk factor of 0.84±0.32, which is not significantly different from unity. Overall, the R² is 20%.

This analysis is restricted to the period for which QID traded. For a longer period we have to look at hedging with a short position in QQQQ.

Dow Volatility Back to "Normal" Levels

I thought that now would be a suitable time to take an aside and look at the long term volatility of major market indices. Many market participants use the term "volatility" to mean "large losses" and so, in current times, we are hearing the term frequently.

I use a simple GARCH model to forecast volatility for the Dow Jones Industrial Average. Although many professional money managers dismiss the Dow, I like to look at it because: a, it is what the media and public talk about when they talk about "the market;" and b, it is equal weighted rather than "cap. weighted" so it represents a more efficient variance reduction than cap. weighting (which over emphasizes the largest companies and so represents the economy and not the market).

The chart above shows the level, and volatility of the Dow, since 1995. The volatility model was fitted on data from 2000 to 2003 and is out-of-sample prior to 2000 and from 2003 to date. (For clarity of exposition, I'm presenting the volatility as a daily point volatility.) We see that the volatility has fallen precipitously from the extreme levels at the end of the prior year.

The innovations are well described by the generalized error distribution, and no severe shocks seem to have occurred since the beginning of 2007 (which was associated by the Jerome Kervial panic liquidation by Societe Generale).

Forecasts for February Returns of Berkshire Hathaway

In the post discussing Warren Buffett's Berkshire Hathaway vehicle, I omitted to forecast returns for February, 2009 (which was done for all the other funds and companies studied here).

So, briefly, based on the whole data sample linear regression we forcast a return of 0.75% for BRK A shares. The prior number is for the least squares estimator, which is equivalent to assuming that the innovations are i.i.d. Normal. In the prior post, we raised the issue as to whether a robust regression might provide a more accurate result. I repeated the regression using least absolute deviations, which is equivalent to assuming that the innovations are i.i.d. Laplacian (i.e. of the form exp -|x|). This tempered the forecast to 0.27%.

UPDATE: This forecast is based on January data and regressions up to the end of January. February is 2/3 over at this point, and BRK A is down 15% on the month (from $90,000 to $76,900 per share). At this point, it seems unlikely that the return for the rest of the month will be sufficient to put BRK into the black, as the model predicts.

Is Berkshire Hathway a Hedge Fund?

I was listening to Dylan Ratigan's Fast Money TV show in my car this evening and was interested by the panel discussing the fact that Warren Buffett sold half of his position in JNJ. A memorable comment was "Maybe Warren's finally become a trader?"

I recalled that there was much discussion several months ago around the fact that Berkshire Hathaway had sold $40 billion in at the money index puts, receiving $5 billion in premium income. Of course, these puts were now heavily in the money, leaving Berkshire with a substantial liability on it's books.

This is an odd strategy for one who called derivative securities "weapons of financial mass destruction." Selling index puts is a hedge fund/investment bank strategy, not that of a long term value investor.

So this brings us to the question: is Berkshire Hathaway a hedge fund?

We can answer this question, as far as the equity investor is concerned, as before by comparing the monthly returns of Berkshire Hathaway to the returns accruing to dynamic trading. For this regression we have a strong prior, which differs to that for pure play investment banks such as Morgan Stanley or Goldman Sachs. We expect a significant positive alpha and zero beta, indicating that Berkshire makes money in a way entirely independant of trading risk premia.

The charts above show the Value Added Monthly Index for both Berkshire and the dynamic trading risk factor and a longitudenal regression of the monthly returns. This is for the entire dataset, from 2001 to date.

The regression shows that, over almost the entire previous decade, the monthly returns of Berkshire Hathaway common stock have a beta of 0.70±0.24 onto the dynamic trading risk factor, with a significance level (p-Value) of 0.005. The alpha is positive, but not significant, at 0.10±0.46.

Again we can break down the analysis into the pro articulum and per articulum parts; and, from this division, we see that this result is not driven by the current financial crisis.

As a final note, the appearance of the scatter plot suggests that a larger beta might be a more suitable estimate, which could be established with a robust regression procedure.

Omitted Wells Fargo -- They're Interesting

When I compile my original list of "banking" stocks, for the analysis presented in a prior post regressing common stock returns onto the dynamic trading risk factor, I omitted to include Wells Fargo. This is my fault, and probably represents an East Coast Bias of my own. The list was not compiled via a rigorous procedure --- it was entirely ad hoc.

That notwithstanding, the regressions for WFC are actually very interesting when compared to those for other banks.

Wells shows no significant covariance with the factor and no significant alpha with respect to it either.

On this basis, Wells Fargo is quite a different animal to the other banks studied previously.

How do the Parameters Change, and What Could it Mean?

In the previous post we exhibited regressions of the returns of various banking companies onto the dynamic trading risk factor. Two distinct regression periods we used, and we made general comments about how the parameter estimates had changed.

In this post we're going to try to delve a little more into those changes. We're going to assume that the changes represent an actual change of behaviour on behalf of the institution concerned rather than that they represent statistical fluctuations about a common "true" value. With such a small sample, and such large errors relative to the estimates, this is a dubious exercise, but we will press on as it is entertaining.

These changes are represented by the vector flows on the "tadpole" chart below. The vector is from the "thin end" to the "head" (and is represented as such because Excel can't draw arrows).

So, overinterpreting to the best of our abilities, we see that: MS and GS have moved towards eachother - adopting similar behavioural profiles; JPM has essentially abandoned its hedge fund like trading business; MER (which were "rescued" by BAC), BAC, and C have started winding down their trading businesses at considerable expense; and LEH and BSC traded more desperately as they failed.

The above is, of course, entirely unrigourous and barely supported by the data. Don't place too much faith in it.

Common Stock Regressions for All of the Usual Suspects

The table below shows the estimated parameters, alpha and beta, for a linear regression of the monthly adjusted returns of the common stock of a well known group of companies onto the dynamic trading risk factor series. These regressions are done separately for the pro articulus (01/2001 until 12/2006) and per articulus (01/2007 to date) periods. If the company still exists as an independent entity, a forecast is given for the return for 02/2009 (i.e. this month) based on a "whole dataset" regression (01/2001 to date). The companies studied are: Goldman Sachs; Morgan Stanley; Citigroup; Bank of America; Merrill Lynch; Lehman Brothers; JP Morgan; and, Bear Stearns. (Of course MER, LEH, and BSC terminated at some point within the latter period. For these companies, the regression used data upto the "end" of the company and not for the later trading of "stub" equity in the remnants of the company, if any.)

What is there to conclude from this? Starting with the innocent days of the pro articulus period, we see that all of these firms, with the exception of BAC, have an alpha estimated to be of order -1%/month to -2%/month and a beta of approximately 3 to 4 onto the risk factor. An plausible explanation is that, with the exeption of BAC, these firms all were in the business of trading and the negative alpha represents the high costs of financing this activity. Interestingly, the damage done due to the fiscal crisis, at least as far as the parameter estimates for the per articulus period go, was done idiosyncratically (i.e. it is expressed through the alpha) and not as a result of highly leveraged exposure to dynamic trading.

Is Morgan Stanley a Hedge Fund Too?

Same analysis, different stock. For full disclosure: I used to work in a quantitative trading group at Morgan Stanley. I traded Eurodollar Futures based on quantitative models, until I left in January 2000 (which is before this analysis starts, not that I had any significant impact on their top or bottom lines). I held MS options and stock until October, 2007. Since then I have traded MS both long and short.

Morgan Stanley's regressions tell a similiar story to those for Goldman Sachs; however, in this case the alpha is -2%/month and the beta is 4. These numbers are consistent both pro and per crisis, as they are for Goldman. The regression results are here, and charts below.

The forecast return for February, 2009, is 1.56%.

Is Goldman Sachs a Hedge Fund?

It's common for people to quip "Goldman Sachs is just a large hedge fund." Armed with our series of the returns due to the risk premium associated with dynamic trading we can attempt to quantitatively answer that question by asking whether the monthly returns of Goldman Sachs Group Inc. is explained by the dynamic trading risk factor or whether it contains a significant idiosyncratic element, which would indicate that Goldman is more than just a hedge fund.

Our procedure is straightforward. We regress the monthly returns of GS common stock, adjusted for distributions, onto the series of factor returns. Our null hypothesis is that the alpha is non-zero and the beta zero, indicating that GS is not like a typical hedge fund making money by selling risk premia. The pure alternate hypothesis, that alpha is zero and beta is significant and positive, is that GS is just like a typical hedge fund. Of course, the real answer may lie somewhere in between.

(Just to be clear, I don't work for Goldman and have never worked for Goldman. I currently have no interest in them either positive or negative and don't currently have any exposure to their stock, although I have previously traded it.)

So, we follow our by now typical linear regression analysis.

In this analysis the data is divided into two wholly independent periods. Until the end of 2006 and from the start of 2007 to date (pro articulus and per articulus, so to speak). We see that the results of both periods are consistent with the alternate hypothesis -- that Goldman's monthly returns to investors (which are distinct to their actual return on equity for these periods) are wholly explained by the trading risk factor. The R-Squared's here are large (of order 40% and 50% respectively) and in both periods alphas are negative, but insignificantly different from zero, and the betas are over three and approximately 4 and 5 s.d. from unity; and the estimates from these independent periods are consistent with each other within the errors.

So in conclusion, this analysis supports the hypothesis that, as far as investors in its common stock are concerned, the returns of Goldman Sachs are similar to those of a typical hedge fund leveraged by three times more than the norm. Using the whole sample, we forecast a return for 02/2009 of 1.83%.

A VAMI chart is included below.

Fitting a GARCH Model to the Hedge Fund Risk Factor

In an earlier post we examined our estimated series of the dynamic trading risk factor returns for trending in the context of building an ARMA model. This was done via a Box-Jenkins estimation procedure and our methodology assumed the innovations where homoskedastic.

The obvious counterpoint, particularly in the context of financial data series, is whether the data is actually homoskedastic or whether it is more accurately modelled by an ARCH or GARCH type structure (and GARCH(1,1) should really be our null hypothesis for financial time series).

There are several tests for heteroskedasticity in the literature. A common method is to divide the data into two (or more) groups and examine the ratio of the subsample variances (this is the Goldfield-Quandt test for two groups, or the likelihood ratio test) or to perform an regression of the squared residuals onto lagged regressors and their cross products (White's test).

This discussion notwithstanding, in financial data the evidence of heteroskedasticity is often so compelling that little need of a formal test exists outside the desired to publish. We do know, whatever the underlying process, that for a homoskedastic process the residual sum of squares should accumumulate linearly with the sample size. The chart below is a nice exposition of this for our dynamic trading risk factor data. Plotted is the ratio of the cumulative residual sum of squares (CRSS) to the total residual sum of squares (TRSS) --- i.e. the variance x (T-1) where T is the sample size. Also plotted are the expected linear accumulation and a monte-carlo sample path for an IID N(0,1) process. The vertical bar represents the location of the maximum absolute difference between the CRSS and the expectation.
(Clearly this analysis is motivated by the Kolmogorov-Smirnov test, although I am not aware of the distribution of the test statistic d(T) = max |CRSS(t)/CRSS(T)-t/T|.)

With this graph cheering us on, for it does appear to indicate a notable departure from linear growth, I fitted the data to an AR(1)xGARCH(1,1) model with IID innovations drawn from the Generalized Error Distribution. The results, are shown below.

The regression prefers the inclusion of the GARCH terms, although from the 96 datapoints we have the significance is not overwhemling. The result is a slight downweighting of the AR(1) term in the process vs. the estimate for the homoskedastic model. Since the significance is low (greater than 1% of rejecting the null by chance), I will stick with the prior process specificiation for the time being.

General Data Appeal

I've been having a lot of fun with the hedge fund factor analysis; so, for a limited time, if anybody out there would like to provide me with data on a particular fund, I'll do the regression for them. Just email it to blog@gillerinvestments.com and I'll see what I can do.

I'd also be happy to share the data I have with any of the folks at the funds we've analyzed here or maybe you could fill in some of the data holes and we can get a complete picture out there (I know you're reading this stuff, thanks to Google Analytics).

Clarium LP

Another fund of interest is Clarium LP. This is a notable global macro fund, run by Peter Thiel. Global macro, as noted in an earlier post, regresses poorly onto the VIX-GARCH variance spread.

The alpha for this fund is large and borderline in significance (1.9 +/- 0.9) %/month, a p-Value of just 0.03. And although it is borderline, this alpha is nevertheless quite large and represents a cumulative return, over the 76 months the fund has been in operation, of 318%. The beta for this fund is 0.79 and statistically indistinct from unity. Based on this data, we can predict a return of 2.7% for February, 2009. The regression results are below.

The Value Added chart below differs from the others previously exhibited in that, due to the huge alpha, I had to present the Clarium series and the Hedge Fund Risk Factor series on different scales (left and right axes, respectively).

I think the next year will be critical for Clarium, as we will find out whether their alpha persists or is anomalous.

February Predicted Returns For Hedge Funds

Based on the data accumulated and the regression models fitted to them (not to extrapolations), we can compute expected returns for the four funds studied for the month of February, 2009.

Fund                                      Return
================================================
Citadel Kensington Global Strategies Fund +1.15%
IKOS Equity Hedge Fund                    +0.78%
Renaissance Institutional Equity Fund     +1.25%
Millenium International Ltd.              +1.06%
------------------------------------------------
Hedge Fund Trading Risk Factor            +0.93%

Note that these are predictions, not measurements, so we'll have to wait and see how they actually do.

UPDATE: The Barclay Hedge data is all in for the month of January, and the numbers above changed slightly.

Millenium International Regression and Extrapolation

Another regression result follows, this one for Millenium International Ltd. The data I have access to for Milleniuim ends in October, 2007. Our standard regression, of the monthly returns of the fund onto the hedge fund trading risk factor, shows an alpha of 0.81 +/- 0.10 %/month and a beta of 0.26 +/- 0.06. Both estimates are highly significantly different from the null hypothesis of (0,1). Full regression results are below.

Again, my dataset excludes the Hedge Fund Crash of 2008, and we know that Millenium suffered along with its compatriots. In the cumulative performance chart (below) I've added three extropolations (green lines) which show the expected performance of Millenium had it: a, continued just as before; b, decreased it's alpha by a factor of 2 and increased it's beta by the same factor; c, as b with a factor of 3. Based on available public information, I think case (b) is a likely match. If anybody has real data, please send it to blog@gillerinvestments.com and I'll update these charts to reflect it.

Divergent Moments in Financial Data - Blowups Happen

In an earlier post I described noted that hedge fund trading had a propensity to blow up but provided no justification for that assertion. This post is about the reasons why I believe this is so.

Risk management is contigent on understanding the second central moment of the distribution of portfolio returns i.e. the variance. The variance is a statistic, meaning a random number derived from observable data, and as such it has a sampling error which represents the uncertainty of the measured value of the statistic about the true population value. The sampling error of the variance is discussed in The Econometrics of Financial Markets and is shown there to be proportional to the kurtosis of the distribution.

Everybody knows that financial data is leptokurtotic, but I'm not sure how many people appreciate that knowledge of the kurtosis is critical to the entire practice of risk management -- and not just as far as it affects long-tail events. In one of my favourite books Fractals and Scaling In Finance, by Benoit Mandelbrot, there is an extensive discussion of the possibility that moments of the distributions of financial data may not exist (meaning that their population expected values do not converge). The sample value of a moment is always finite, for it just a function of the data sample, so an interesting question is how do you measure a moment in such a way that you can accept the hypothesis that the population value is divergent?

Mandelbrot suggests looking at the cumulative sample moment, i.e. to compute the sample moment for every sub-period starting with the beginning of the data we have and ending on each successive date. If the population moment is finite then this will converge on the true value, following the Law of Large Numbers. If the population moment is divergent then the cumulative sample moment will not show signs of convergence.

The chart below shows the cumulative kurtosis and cumulative standard deviation for the daily change in the yield of three month treasury bills (the data is from the Federal Reserve Bank of St. Louis available here).

This chart shows the first two cumulative even moments from what is perhaps the principal risk factor for the entire U.S. economy, if not the world. It is annotated with the terms of the Chairman of the Board of Governors of the Federal Reserve System who was responsible for interest rate management during their terms.

I think one could argue that it is suggestive that the kurtosis is not converging to any finite value. In fact, it would appear that the entire term of Alan Greenspan, and that of Ben Bernanke until the recent financial crisis forced an abrupt policy change directed at curtailing interest rate risk, contributed to steady divergence in the kurtosis of the distribution of changes of interest rates. I also include the cumulative standard deviation, which steadily declines during the Greenspan period.

The lack of clear convergence is worrying because it requires us to consider the possibility that the kurtosis is undefined, which implies that the variance is not reliable measurable, and (to be deliberately provocative) that our risk management infrastruture is a self-deluding fantasy -- i.e. that blowups will happen and not "once in a hundred years."

Renaissance Institutional Equities Fund

For another big name, let's look at the Renaissance Technologies "Renaissance Institutional Equities Fund." This is interesting because Renaissance is a 100% quant/model shop and so they actually present simulated history for the period during which the fund was not trading live. (And once again, I have no axe to grind regarding Jim Simons or Renaissance Technologies. They are a very large, very notable, and very successful manager -- probably the best in the world -- and that's why they're analysed here.)

So, taking the monthly data, we ploughed ahead and did our, by now, standard regression onto the dynamic trading risk factor.

The results where quite a suprise to me. Coming from Process Driven Trading at Morgan Stanley, I feel I have a little more understanding of what's behind this kind of model. To see an insignificant alpha and a beta statistically indistinguishable from unity (which is my null hypothesis for a hedge fund engaging in typical trading patterns) was not what I expected. On the other hand, Renaissance's strongest claim about this fund was that it was massively scalable -- so perhaps that's where they invested their technological edge. The cumulative performance is presented below.

Regression of IKOS Equity Hedge Fund onto Risk Factor

Here's a another hedge fund regression for comparison. The fund is the IKOS Equity Hedge Fund, which has monthly performance data here. Briefly: the alpha is 0.60 +/- 0.24 and the beta onto the trading risk factor is 0.19 +/- 0.12 which is not significantly distant from zero but is very significantly distant from unity. There's a chart below.

Again, I'm not trying to promote or attack anybody, I'm just using publicly available data for well known funds.

Hedge Fund Forecast for February, 2009

In the post Do Hedge Fund Returns Trend? we showed evidence for first lag autocorrelation in the returns of our derived hedge fund risk factor series and predicted a return for that factor of 54 bps in January, 2009.

Partial results are in, and the factor for January is currently estimated at 188 bps. The forecast for February, 2009, is currently 101 bps. I'll update this as more data arrives.

The full data is here. As before, this data is based on the Barclay Hedge index data published on their website.

Regression of Citadel Kensington onto Trading Risk Factor

I thought it would be interesting to see how a very well known "flagship" hedge fund regressed onto our derived series of the risk premium accruing to the takers of trading risk. Following is a regression analysis of data for the Citadel Investments' Kensington Global Strategies Fund, Ltd. This data was taken from various public sources and omits most of the really interesting period (i.e. from December, 2007 until August, 2008 and October 2008 -- I did find data advertised for the "Wellington" fund for October, 2008, but decided to exclude that).
If somebody has data for this missing period, I'd be happy to receive it at blog@gillerinvestments.com.

I should point out that this is not any kind of attack on Ken Griffin or Citadel in any shape or form. (Mr. Griffin has created a huge and successful business out of nothing and is clearly a lot more prosperous and successful than I am and his companies have created great wealth for their investors.) It is a study of academic interest on a fund that is very much in the public eye -- nothing more than that.

Anyway, on to the data. Citadel Kensington displays a high beta and a very significant regression onto the factor (unlike the previous analysis, and since this concerns just one fund, I did not allow the regression procedure to fit the factor this time). If we exclude the Hedge Fund Crash of recent history the beta is about half as large but still significant. We should also note that Citadel has a significant alpha which is essentially eliminated by the recent history.

Now the data I've omitted I know includes more negative returns so the picture shown here probably overestimates the alpha and underestimates the beta. The cumulative performance is shown in the chart below.

From the chart we can see the effect of the positive beta, even before the drawdown at the end of the series.

As I wrote below that I do not believe in censoring data on the theory that there are some data that "don't fit" and so we should remove them. I believe that these strategies always had the capacity for such blow ups, we just hadn't wandered into that region of phase space yet.

UPDATE: I found the data for October, 2008, on Bloomberg, and have updated the charts and reports. This datum of -22% effectively kills the alpha of the fund, leaving it a 20% leveraged bed on the general factor (which, let's not forget, has a non-zero mean).

Statistician Measure Thyself

After producing a factor model to explain every body else's returns, I though honesty demanded that I regress my own trading profits onto the factor. First of is the monthly returns of the commodity pool I ran from 2000 to the end of 2003 (it was closed after a drawdown which coincided with personal committments that I could not neglect). The regression results are presented below.

During the period where the data coincides, my data is not well explained by the factor series. (I admit that this data is now a little ancient, but it is real.)

To follow: regressions for the Compact Model Portfolio.

Is the Trending of the Hedge Fund Factor a Recent Phenomenon?

In the last post, we found evidence that there is month-to-month trending and that there is a non-zero mean in the time series of the principal factor explaining hedge fund returns.

However, we don't have many data points (96) and all time series analysis is subject to temporal bias and financial time series seem to be particularly prone to not smoothely exploring their available phase space. And we know that recently, hedge funds just suck! So an important question is whether the positive autocorrelation we are finding in the data is, in fact, driven by just our recent experience in which there have been a few months of consecutive terrible returns.

The only real way to find out is to wait for more data to roll in and see if the best estimate, or a Bayesian adjustment to a lower value, is the better predictor. But that's going to take time, so an alternative is to pry into the data and see if it looks like the estimate is driven by recent history.

The chart shows the confidence bounds for estimating the AR(1) parameter for every sub-interval ending on dates between December 2001 and December 2008 (the sample always starts on January 2001, so the estimates are not independent of each other).

It's clear that the recent problems in the market have kicked up the estimated autocorrelation to a higher level than prior history but also that there was definitely a prior autocorrelation which was also significant.

So what value should we use? I guess in my heart I'm a frequentist so my instinct is to use all the data rather than make a bet about which subset is more accurate. We should learn from recent experience that these trends are possible rather than dismissing them as anomalous or "once in a hundred years floods."

My strongest preference is that confidence intervals are more reliable estimates than point estimates, so I'd bet that the true parameter lies somewhere between the green lines on the right-hand most edge of the chart.

Do Hedge Fund Returns Trend?

In previous posts we've suggested that the returns of dynamic traders, such as hedge funds, are equivalent to the profits that arise from writing options on a trading risk premium. We've shown data from the spread between the VIX and a model of S&P 500 volatility which explains some of the variance of hedge fund returns, and which represents the profitability of option writing. We've also estimated a hedge fund index returns factor and shown that the data (the indices from Barclay Hedge) are consistent with the hypothesis that most of the profits arise from exposure to the dynamic trading risk factor, but that a small style alpha does exists.

Of course, an extremely interesting question is then whether the dynamic trading risk factor is entirely random or whether it can be conditionally forecast. (We can easily see that it has a non-zero mean!) To put it simply: do hedge fund returns trend; do they revert; or, are they conditionally random?

I use a very nice time series analysis program called RATS. This program has a command specifically designed to perform a Box-Jenkins ARIMA analysis. Below are the results of fitting a parsimonous AR(1) model to the estimated factor returns data. The results of fitting an AR(1) model are shown below.

The answer is "yes, hedge fund returns trend strongly." The autocorrelation function, with the expected ACF for an AR(1) model, is shown below and suggests that this month-on-month trending is a sufficient model for the data.

This allows us to forecast a factor return of 54 b.p. for January, 2009.

Estimating a Factor Model for Hedge Funds

We're going to go a head and fit a single factor model to the Barclay Hedge Fund Index data. To do this, I'm going to exclude all the funds that are clearly composite (Fund of Funds, Multi-Strategy Funds, and the the full Barclay Hedge Fund Index). Let's follow the structure suggested by the Fama-French equity factor models and fit the following model to the returns.

In this model r is the individual index return; R is the single global factor return; and, we assume the innovations are i.i.d. (I'm going to start of with least-squares, which is equivalent to assuming they are i.i.d. normal -- but that assumption is likely to prove false). The alpha, beta, and the entire factor return series R are to be estimated by our procedure. This is a slightly more complicated problem since the model is a little bit bi-linear in free parameters. (I say "a little bit bi-linear" because the factor at any time is common for all individuals, and the alpha and beta for any individual are common for all time, so providing the data is not too pathological, we should be ok.)

The approach is to start with an initial condition of the alpha and beta set to their estimates from the prior regression onto the VIX-GARCH variance spread, and the factor set to the returns of the overall aggregate index, and follow a simplex optimization with the factor fixed. I then free the factor and complete the simplex optimization. Finally, I used the simplex estimates as a starting point for a BFGS Steepest Descent optimization. This latter step is useful to confirm the minimum identified by the simplex method and also to build a Hessian matrix to allow error estimation. In all there are 1440 data points and 126 free variables (1314 d.o.f.), so we should not expect too much from the data.

The regression did actually converge. I won't present the full output, because there are so many free variables. Some interesting charts follow. The first is the series of the estimated factor returns.

The upper chart shows the time series of the estimated factor returns and the lower chart is a histogram of those returns. The histogram has been fitted to the Generalized Error Distribution. This is a p.d.f. with an adjustable kurtosis that I find quite suitable for modelling financial data.

Secondly we have two histograms of the estimated values of the alphas and betas from the regressions.

These parameters estimate the idiosyncratic drift of an investment style and it's exposure to the general factor (the risk premium that accrues to traders). With such a small dataset, we can't really compare the alphas and betas to the null hypothesis (that alpha=0 and beta=1 i.e. that style makes no difference and all hedge fund managers just expose themselves to the general risk factor) in a manner that gives us statistically meaningful statement. However, the data do seem consistent with the hypothesis that the style index returns are mostly due to the common factor with a little bit of positive alpha.

Finally let's examine the relationship between the factor series we just estimated and the VIX-GARCH premium. In this case I will use the simple premium (literally just VIX-GARCH), since this will make the regression coefficient dimensionless.

Here we do find a statistically significant covariance, at a significance level of 0.003.

Regression Results for the Entire Barclay Hedge Universe

I knuckled down and did the regressions for every sub-index tracked by Barclay (now known as Barclay Hedge). You can find the raw data published on their website here.

Firstly, here is a chart of the cumulative returns for all of the indices.

This was prepared from the data that Barclay Hedge make available on their website.

The following table

shows the results of all of these regressions. The method is a simple linear regression onto the VIX-GARCH variance spread.

The table shows a range of responses, and some fairly high (as much as 37%) R-Squareds as well as some fairly low ones. The largest R-Squared is for Convertible Aribitrage, which is not surprising as this is most purely a delta-hedging strategy and so should correlate very strongly with the available risk premium expressed via the VIX-GARCH spread.

However, I'm quite suprised that Distressed Securities is the investment style with the second strongest regression. Perhaps this indicates that DS traders are implementing their strategies via options or perhaps it indicates that distressed securities could be thought of as binary call options on the profitability of a company.

The third strongest is Fixed Income Arbitrage which is, I'm hypothesising, not actually "Arbitrage" but dominated by convexity spread plays. In this scenario, it would also be a fairly pure delta hedging strategy -- If so this might indicate that the profitability of interest rate option trading and that of equity option trading are closely linked; which is the kind of hypothesis we originally advocated in our original post.

Of course, the observable similarity of the returns in these series is crying out for the establishment of a proper factor model, which we will examine in the next post.

Regression Results for Convertible Arbitrage

Since Convertible Arbitrage is a delta hedging strategy (although I would have thought that the recent problems with that strategy were more to do with credit risk than market risk), it seems likely that there would be a strong correlation between the performance of this strategy and the VIX-GARCH spread. The results are presented here. As expected, the regression is strong with positive correlation and a p-Value of 0.00065.

The Returns of Traders as a Risk Premium

This article is a refinement of the theory backing the previous one (Can the Spread of the VIX Over a GARCH Model Predict Hedge Fund Returns). I was trying to clarify the logical steps to permit the regression to be meaningful --- i.e. to establish the causality of the link between the risk premium acquired by writing S&P 500 options and the profits made by traders.

We assert that hedge funds, by selling shares in the profitability of a trading strategy, are essentially writing option contracts which must be hedged by executing their trading strategy and the income they receive from their clients is the risk premium embedded in the spread between the option selling price and it's fair value (which is the value realized by the dynamic hedging strategy).

The question is how is that risk premium valued and what is the theoretical link to the VIX-GARCH spread? I will introduce an additional hypothesis, which is a more concrete argument than that made previously.

Each trading strategy is different, but there are commonalities in the general changes in the price of risk. So we can model the risk premium for any given strategy very much as we model the returns of common stocks. We represent it as a linear combination of an idiosyncratic risk premium and a systematic risk premium, with a "beta" to the systematic premium. Additionally, we will assume that the risk premium beta is likely to be positive and significantly different from zero.

In this framework, every strategies premium income is correlated with each other, just as every stock's returns are correlated with each others. It's also quite straightforward to see that the profitability of a put writing strategy is explicitly dependent on the spread between VIX and an empirically accurate model of the actual volatility of the S&P 500, which we model with a simple GARCH(1,1) variance process. Therefore, we expect the risk premium income of a hedge fund strategy to be positively correlated with the risk premium income of put writing, which the "risk beta" to be empirically established. This the the theoretical construct we need to make our regression a reasonable operation.

However, using this framework we can now reason that some particular strategies might have a stronger exposure to the systematic component of risk premia than others. So it makes sense to look at regressions between the hedge fund sector index returns and the VIX-GARCH spread. I will present the results of these regressions, as I do them, in future posts.

Can the Spread of the VIX Over a GARCH Model Predict Hedge Fund Returns

In the post VIX vs GARCH: Results from a New Region of Phase Space I exhibited a chart of the longitudenal relationship between the VIX index of volatility and a simple GARCH(1,1) model of the volatility of the S&P 500 index. I pointed out that, for most of the history, the VIX traded at a premium to the GARCH predicted volatility; however, during the current financial crisis this relationship had reversed.

It can be asserted that all dynamic trading strategies can be replicated by some kind of option (in much the same way that options are replicated by dynamic hedging strategies) and thus the profits accruing to traders can be mapped into the risk premium income they receive by writing "their" particular kind of options (this comment was made to me by Pete Kyle).

One could then suggest that the general price of financial risk might be related in some way to the spread between the VIX, which is the market price of market risk, and a forecast of the actual level of market risk, which we can approximate with a simple GARCH model. (Actually I would look at the spread between the squares of these quantities since "standard deviation" is a slightly artificial number from a statistical point of view -- variance is the real process that occurs.)

With this in mind I looked at the relationship between the
Barclay Trading Group's index of hedge fund returns and the lagged spread between the square of the VIX and the GARCH variance process (the Barclay data is monthly, so I looked at the variance spread on the final date of the prior month). If the assertion and inference is correct, then there should be positive correlation between these quantities.

A simple linear regression between these quantities is presented here (the notation {1} means "first lag" in the regression program I'm using). The regression indicates that this correlation exists is significant at the 3% level; although, from a physicist's point of view should be regarded on the weaker side.

Are Returns a Portable Metric for Comparing Price Changes - Part I

In finance we are accustomed to talking about, and thinking about, and predicting, and measuring, returns. This is the first of a small set of articles that asks a very basic question: are returns a portable metric for comparing price changes?

What I mean is: does it make sense to say IBM went up 3.6% and CSCO went up 3.7% so IBM's performance was comparable to that of CSCO. Does taking a price change (IBM went up $3.04) and dividing it by the prior price (IBM's was $84.15/share) produce a quantity (3.61%) that it is meaningful to compare to that for CSCO (up $0.60 from $16.30/share implies a change of 3.68%) and to then conclude "they changed by the same amount."

The first reaction to this musing is: of course it makes sense --- you're telling me that making an equal investment in each company produced an equal return. But investment performance and price dynamics are not the same thing and this comparison is a little more subtle that it first appears. Tabulated below is the daily standard deviation for both stocks from approximately 800 business days.

Ticker St.Dev.
------ -------
CSCO    2.35 %
IBM     1.57 %

This shows that we should not expect both stocks to typically have the same scale of move on any given day. It casts our data (which is for today, 01/02/2009) into a different light. Statistically speaking, i.e. relative to the typical scale of daily moves, IBM moved more than CSCO (2.3 s.d. vs 1.6 s.d.).

Our comparison of the volatilities of the stocks has revealed a hidden assumption that exists when we compare returns: we assume that the volatility scales with the price of the stock in a linear fashion. i.e. We assume that because IBM costs over five times as much as CSCO per share that IBM's daily move should be of order five times as much as that of CSCO.

At this point, one could argue that all we've discovered is that volatility contains an idiosyncratic element and different stocks have different daily volatilities. i.e. That in the basic stochastic drift equation dS/S = m dt + k dX, all we need to say is that k is idiosyncratic. However, look at the equation: we've blythely included a scaling factor (1/S) to render our stochastic process model dimensionless. Ought we not ask, at least once, whether this assumption is empirically justified? Do we have any evidence that volatility scales with price level? After all, we know that volatility differs from stock to stock and we also know that volatility differs from day to day for the same stock. How certain are we that IBM's daily volatility at $84/share is twice what it was at $42/share?

Given that we know that although volatility varies, it varies somewhat slowly, let's take a readily available datum that scales with (the average of the) volatility for a given day, the absolute value of the daily price change, and see how this varies with price level.

This data, for recent history, is presented in
Scaling of Volatility with Price Level for U.S. Tech Stocks. This data set is a good candidate for analysis because it contains a substantial price excursion, and so is not confined solely to the current region of phase space. We see that not only do the regression lines differ substantially from a unit gradient, there is no consistency between the two stocks.

But, perhaps, this analysis is too idiosyncratic and too focused on recent history, so we should look at the behaviour of market aggregates over a substantial period of time (even though, for the operation of comparing returns to be useful, it is necessary that it be useful idiosyncratically). The second chart, Scaling of Volatility with Price Level for U.S. Stock Indices shows the same analysis for the DJIA and the S&P500 for daily data since 1928. In this chart we see a much smaller departure from unity, with an undeniable statistical and practical significance. Although the spectral index for the S&P500 is just 0.9 vs. 1.0, it is over 12 s.d. away based on the standard regression errors, and the effect of this scaling law vs. the null hypothesis, when aggregated daily for eighty years, will be quite substantial.

Again, one can poke holes in the methodology of (and very historical data for) both of these indices, I think that it's fair to conclude that this elementary conjecture is worthy of further study, which is what we shall do in Part II.