Today, we bring you Part 2 of Dr. William Shadwick’s proposal to use “extreme value theory” in calculating value at risk (VaR) and its close cousin conditional value at risk (CVaR). (see Part 1).

Shadwick is a highly-regarded mathematician who crossed over into finance a decade ago and has since made his mark on the field of investment performance analysis, developing Omega Metrics® and winning a prestigious award from the Investment Management Consultants Association along the way. He is also the founder of Omega Analysis Limited, a quantitative research firm in London.

Special to AllAboutAlpha.com by: Dr. William Shadwick, Omega Analysis Limited

As I wrote two weeks ago, Extreme Value Theory Conditional Value at Risk (EVT CVaR) can be a very useful measure in portfolio analysis.

In a volatile market, EVT CVaR with a variable Value at Risk threshold (for example the worst loss in the past 250 days) is a useful method for tracking the evolving exposure of a portfolio position.  In fact, it provided accurate estimates of the declining fortunes of shares in companies such as AIG, Bank of America, Barclays Bank, Citigroup, HBOS, Lehman Brothers, Lloyds Bank Group and UBS over the past two years.

In each of these cases, the share price movements prior to severe loss events showed extremely fat tails.  Our implementation of EVT provided warning of the likelihood and severity of subsequent losses well in advance. Thus, while the weakness of these institutions may have come as a shock to their regulators, the prospect of large losses was quite apparent from their share price histories, through our CVaR estimates.

Citigroup’s Share Price Decline 2007-2009

By the end of 2006 the daily returns on Citigroup shares had tails too fat to be consistent with a normal distribution. The Tail Risk Level as measured by the “C-S Character” had been high for several months by the beginning of 2007. Those who followed this indicator after the publication of Omega Analysis’ Primer on Tail Risk on AllAboutAlpha.com in May 2007 received warning of this heightened risk well in advance of the impact of the credit crisis in the equity markets.

At the beginning of January 2007, the worst loss in the previous 250 days was the return of -4.69% on 20 January 2006. According to the normal model, the probability of a return less than or equal to -4.69% was one day in 78,000 years. By contrast, our tail estimate of the probability of such an event was one day in 588.

Table 1 shows Risk Assessment Levels and subsequent breaches, if any, at a monthly frequency.  In this table, we show a Monthly Risk Assessment report on Citigroup. Prior to the first trading day of each month we report: worst loss in sample, estimated probability of exceeding the worst loss, Conditional Value at Risk beyond the worst loss (based on the previous 250 days). The table shows the date and magnitude of any excess losses in the subsequent month.

In each case the probability estimates indicated that further losses would likely be observed within a few months at most. Each time the previous worst loss was breached, the probability of a recurrence steadily increased until the next event in the cascade of severe losses that Citigroup’s share price suffered.

Extracting critical risk estimates from Citigroup share price returns distribution is possible because of the high degree of information which the price movements contain. The same approach can be applied to hedge fund returns.

Daily returns for hedge fund indices are not as rich in information as share prices from major markets however the EVT CVaR estimates also gave clear warnings ahead of the flurry of losses in strategies such as Convertible Arbitrage in September and October of 2008.

Financial data distributions have fat tails. Extreme events are both inevitable and of critical importance EVT CVaR estimates provided ample warning of both the increasing likelihood and increasing severity of those losses before they occurred.

(By Ranjan Bhaduri, PhD, CFA, CAIA, Member, AllAboutAlpha.com Editorial Board)  Commodities have always had a reputation for risk, reward and volatility.  Managed futures - a set of strategies aimed at harnessing the return potential of commodities (and of other financial instruments upon which futures contracts are written), has long played a central role in the alternative investment industry.

As you might expect, managed futures have always been more volatile than many of the alternative investment cousins.  But does this mean they serve no role in a diversified portfolio?   Researcher and Harvard professor John Lintner addressed this question back in 1983 in a seminal paper presented at the Annual Conference of the Financial Analysts Federation in Toronto titled “The Potential Role of Managed Commodity-Financial Futures Accounts (and/or Funds) in Portfolios of Stocks and Bonds.”

Firstly, Lintner found the risk-adjusted return of a portfolio of managed futures to be higher than that of a traditional portfolio consisting of stocks and bonds.  But he also observed that portfolios of stocks and/or bonds combined with managed futures showed substantially less risk at every possible level of expected return than portfolios of stocks and/or bonds alone. The following passage from Lintner’s work furnishes good insight on his findings:

“…the improvements from holding efficiently selected portfolios of managed accounts or funds are so large - and the correlations between the returns on the futures portfolios and those on the stock and bond portfolios are surprisingly low (sometimes even negative) - that the return/risk trade-offs provided by augmented portfolios consisting partly of funds invested with appropriate groups of futures managers (or funds) combined with funds invested in portfolios of stocks alone (or in mixed portfolios of stocks and bonds), clearly dominate the trade-offs available from portfolios of stocks alone (or from portfolios of stocks and bonds). Moreover, they do so by very considerable margins.”

With managed futures offering such a compelling diversifier, it’s surprising that there still seems to be a  disconnect between many institutional investors and the managed futures space.  In a recent paper I co-wrote with Ryan Abrams of AlphaMetrix and Elizabeth Flores of the Chicago Mercantile Exchange, I paid homage to Lintner by comparing managed futures performance to that of equity markets and other hedge funds.

What we found was that managed futures may be more volatile than long/short equity or equity market neutral hedge funds, but not necessarily more “risky”.  Measuring risk by volatility is dangerous in the alternative investment space since the distributions are typically non-Gaussian.

Specifically, managed futures (proxied by the “BTOP 50 Index”) had a 10.44% annualized rate of return over the past 20 years vs. 11.77% for the HFRI.  Seeming to add insult to injury, managed futures had a 10.88% annualized standard deviation vs. only 7.13% for the HFRI.  This resulted in a Sharpe of 0.96 for managed futures and 1.65 for the HFRI.

So on the surface the prospect for managed futures looked pretty grim.  However, when we examined the skewness of the managed futures track record, a different picture emerged.

The BTOP 50 Index has a skew of +0.93.  In other words, it has a tendency to show “upside surprises”.  Meanwhile, the HFRI (like the equity market in general) had a negative skew (-0.79) meaning that it had a propensity for downside volatility.

Skewness describes the relative length of the tails or the degree of asymmetry of a distribution of outcomes. Positive skewness suggests that a number of relatively large positive deviations inflate the mean of the distribution, resulting in a fat right tail.  Conversely, negative skewness occurs when a number of relatively large negative deviations pull the mean down, resulting in a fat left tail.

Another way to look at the skewness of managed futures is to calculate its “Omega Ratio”.  Regular readers of this website will be familiar with this measure and with its co-creator, AllAboutAlpha.com contributor William Shadwick.  We found the Omega ratio (with a 3% threshold) of managed futures to be 1.74 compared to only 1.32 for the S&P 500.

In other words, a much higher proportion of the monthly returns of the HFRI occur to the left of 3% than the proportion of managed futures returns that occur to the left of 3%.  Again, this is the result of the negative outliers in S&P 500 returns and positive outliers in managed futures returns.

As the chart below shows, the out-performance of managed futures holds true at all return thresholds.  But a combination of managed futures and the S&P 500 provides diversification benefits and results in a higher Omega Ratio (as you can probably surmise then, the correlation between the BTOP 50 and S&P 500 is only -0.05):

Lintner did not have the benefit of the Omega tool during the time he conducted his work, and the Omega function encodes all the higher statistical moments and distinguishes between upside and downside volatility.

The Omega graph above indicates that for low thresholds, the combination of managed futures and a traditional portfolio is best, and for higher thresholds, a portfolio of managed futures is dominant.  Moreover, a traditional portfolio of stocks and bonds combined with managed futures is superior at every meaningful threshold (i.e., where any of the graphs have an Omega score of at least one - where I assume 1.0 is the lowest acceptable Omega Ratio).

In short, these Omega results yield a very compelling argument for the inclusion of managed futures in an institutional portfolio - a fact more institutional investors are now coming to realize. - R.B.

A lot of funds of hedge funds focus almost exclusively on smaller, newer hedge funds.  Whether its due to the backfill bias that gives young funds apparent superpowers or simply because their managers are hungrier, newer hedge funds seem to outperform their older compatriots.  Similarly, smaller funds (regardless of age) have generally performed better than larger ones.

Small Fry

Until now.  Data analytics firm Pertrac, recently found that 2008 was an anomalous year since smaller hedge funds actually underperformed larger ones.  As Pertrac’s Meredith Jones says in the company’s press release,

“Last year was a difficult one for hedge funds of all ages and sizes…However, when it comes to hedge fund performance as a function of fund size, we saw a reversal of the trend established from 1996 through 2007. During 2008, funds with the least assets actually performed the worst, while larger funds posted better returns.” (our emphasis)

This is what the company found: (note: standard deviations calculated over 1996-2008 time frame) More…

It is often argued that aggregate hedge fund performance data suffers from a near-fatal flaw: since it is voluntarily reported by the manager, hedge fund indices only include funds that the managers have deemed marketable.  In 2002, David Hsieh of Duke University and William Fung of London Business School wrote a seminal article on this issue called “Benchmarks of Hedge Fund Performance: Information Content and Measurement Biases.”

In contrast, regulations often require mutual funds to register with securities authorities before they can begin to assemble a track record.  As a result, mutual fund data is assumed to be free of such bias.

But as Alan Palmiter and Ahmed Taha, law professors at Wake Forest University write in a forthcoming article for the Vanderbilt Law Review called “Star Creation: The Manipulation of Mutual Fund Performance Through Incubation“, the requirement for a mutual fund to register does not eliminate the problems arising from so-called “mutual fund incubation.”

Observe the professors: More…

In August 2007, as quant hedge funds were swooning in an eerie precursor to the credit crunch, we reported on an academic study of “hedge fund contagion” (see post).  Researchers found “no systematic evidence of contagion from equity, fixed income, and currency markets to hedge fund indices”.  However, they did find that there was a contagion between hedge fund strategies themselves.

Now a new paper on this topic explores whether the correlation between hedge fund returns changes depending on market conditions and the overall performance of hedge funds.  Possible “asymmetric correlation” between hedge funds is of critical importance to funds of funds, and by extension, anyone hoping to benefit from hedge funds’ reputed lack of correlation with each other and with equity indices.

Evan Dudley and Mahendrarajah Nimalendran of the University of Florida focus their attention on the correlation between extreme hedge fund returns only - i.e., those that occur in the left and right tails of the return distribution.  Their hypothesis is that the correlation between hedge fund returns is somehow different during these extreme events than they are during “normal” times (a reasonable hypothesis given the often-cited hyperbole about how “all correlations go to one” in times of distress).

It turns out their hunch was correct.  However, the extent to which correlations increased in hard times was different across various hedge fund strategies.  Dudley and Nimalendran first examined the correlation between several hedge fund strategies and the S&P 500.  They divided the return distribution of each strategy into quantiles and compared each quantile to the S&P 500.  The chart below from the paper shows how two particular hedge fund strategies, Long/Short Equity and Event Driven stacked up: More…

Ever noticed that the term “decoupling” is almost always used to describe a situation where one asset tanks and the other manages to hold up okay?   A couple of years ago, many assumed that Asian economies had decoupled from the US economy - in other words, that a downdraft or stagnation in the US economy would not be commensurately felt in Asia.

But no one seems to have commonly referred to the dot-com bubble as a “decoupling” of Internet stocks from the bricks-and-mortar economy and few described last summer’s oil or potash bubble as a “decoupling” from other assets.

So far in 2009, hedge funds seem to be holding up (i.e. flat) while equity markets fall off their bar stools.  In a new report, Credit Suisse describes this year’s performance as “continuing January’s decoupling trend from equity markets.”

In fact, this year’s performance has been so “decoupled” from equity markets that return differences in January and February (2 months) are enough to push the 36-month rolling correlation of hedge funds down about 10%. More…

When conducting due diligence on a hedge fund, it’s appropriate to ask a manager for their AUM history along with their return history.  After all, studies (and intuition) say that assets start to flow into a fund only after it has put some solid numbers.  Naturally, young hedge funds have fewer assets than more mature ones.

But studies show that emerging managers also produce out-sized returns.  Skeptics of hedge fund indexes suggest that young funds only report their results publicly if they have achieved positive early results - leading to a “back-fill bias”.  Others say that its simply easier to produce good returns with less assets under management.  But whatever the reason, it seems that the golden years for many hedge funds happen early in life.

This is a useful observation for funds of funds and other hedge fund investors who are trying to tilt the playing field in their favour.  But it also has a profound implication on the returns actually experienced for the average hedge fund investor.  Essentially, those who invest in a fund after a hot start in life experience a much lower overall return than those lucky enough to have gotten in the ground floor.

Studies have long shown that the propensity to jump horses in mid-race leads to lower returns for equity investors than for the market as a whole.  Now a study on hedge funds shows that this axiom also applies to hedge fund investors. More…

Those new to the hedge fund industry often find the concept of “buying volatility” and “selling volatility” to be somewhat confusing.  Volatility, after all, is not a tangible thing that can be bought and sold (save for the VIX), but is rather a description of tangible assets (a “volatile stock” or a “low-volatility fund”).

Yet hedge funds are often accused of simply “selling” volatility to generate returns, resulting in them being “short volatility”.  A short position in a stock implies that you would gain if the stock went down and lose if the stock went up.  Similarly, a short position in volatility implies you would gain if volatility went down and lose if it went up.

So how could you construct a position that would gain when volatility goes down and lose when volatility goes up?  Sell (i.e. write) options contracts.

Not unlike writing an insurance contract, writing options generates a steady stream of premiums with no apparent cost - until the judgment day comes.  In other words, if volatility unexpectedly jumps (as it does in severe market downturns), then you’d have to pay the piper.  Obviously, this could erase all of the apparently risk-free returns you seemed to be receiving by writing options over the the years.

Some say that’s how hedge fund managers really make their money.  Last year, we told you about a paper by Wharton’s Dean Foster and Peyton Young of Oxford University that claimed, among other things, that hedge fund managers routinely “fake” their alpha by simply writing puts and collecting the premiums (see post).  Wrote Foster and Young: More…