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.