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.