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.