A “Highly Improbable” Event?

A historical analysis of the natural gas spread merchandise that bought downwards Amaranth

Nick Maouis, the founder of Amaranth, claims that the 6-billion dollar loss that his fund suffers is due to a “highly improbable” trial inwards the natural gas market. Some analysts create got thrown doubts on this claim. To meet how improbable this loss is, let’s accept a quick await at the historical surgical operation of this merchandise since 2000. This is non alone of forensic (and perchance legal) interest: if Mr. Maouis’ claim were true, it would create got furnished us a glimpse of a potentially highly profitable trading strategy.

The bet that Amaranth together with its caput trader Brian Hunter made is that the March-over-April spread inwards natural gas futures volition increment inwards value throughout the twelvemonth prior to the contract expiration. Unfortunately for their investors, the spread decreases rather than increases inwards September, resulting inwards a $6 billion drib inwards value. We don’t know the exact fourth dimension when Amaranth bought this spread. However, it is in all probability that they create got started buying inwards Apr of this year. Apr is the fourth dimension when the nation’s natural gas storage inventory coming out of the wintertime is known together with therefore provides a foundation to bet on adjacent winter’s natural gas sufficiency. I plotted below the profit-or-loss of buying this spread (long 1 March contract of the next year, together with brusk 1 Apr contract) inwards Apr together with exiting the seat at the destination of September every twelvemonth since 2000. (Click on the graph twice to larn far bigger.)

To my surprise, this merchandise loses coin iii out of vi previous years. The 1 twelvemonth that this merchandise was rattling profitable is 2005: it made to a greater extent than than $16,000 profits per twosome of contracts. This is consistent amongst a Wall Street Journal study that Mr. Hunter made $1 billion for Amaranth inwards 2005. That was indeed due to an improbable trial terminal year: Hurricane Katrina.

Note equally good from the 2006 graph that, consistent amongst tidings reports, the merchandise was genuinely quite profitable upwards till the showtime of September. This newspaper profits may non travel easily realized past times Mr. Hunter though, since a lot of it may travel due to his aggressively increasing his seat together with driving upwards the market.

Now at that topographic point tin travel several objections to my analysis. You powerfulness intend that if nosotros concur on to this spread seat longer, tell till December, it would create got been to a greater extent than profitable historically. My question shows otherwise. Holding till Dec would create got resulted inwards losing iv out of the previous five years, losing fifty-fifty inwards 2005. You powerfulness equally good fighting that this is an extremely simplistic version of Mr. Hunter’s strategy. No uncertainty Mr. Hunter used diverse complex options strategies, continuously adjusted amongst diverse telephone substitution factors such equally conditions prediction together with natural gas inventory reports equally inputs. However, from a lead chances administration signal of view, the portfolio that Mr. Hunter owns seems highly correlated to a apparently vanilla spread seat that I described. The fact that this apparently vanilla seat loses coin one-half the fourth dimension historically would non create got been reassuring.

In a hereafter article, I volition push clitoris around calendar spread trades inwards liberate energy futures that create create got a much amend profits consistency.

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How Much Leverage Should You Lot Use?

Maximizing increase without risking bankruptcy

Many hedge fund disasters come upwardly non from making the incorrect bets – that tumble out to the best of us – but from making besides big a bet yesteryear overleveraging. On the other hand, without using leverage (i.e. borrowing on margin to purchase stocks), nosotros oft cannot realize the total increase potential of our investment strategy. So how much leverage should you lot use?

Surprisingly, the respond is well-known, but picayune practiced. It is called the Kelly criterion, named afterwards a mathematician at Bell Labs. The leverage f is defined equally the ratio of the size of your portfolio to your equity. Kelly measure says: f should equal the expected excess furnish of the strategy divided yesteryear the expected variance of the excess return, or

f = (m-r)/s²

(The excess furnish beingness the furnish m minus the risk-free charge per unit of measurement r.)

This quantity f looks similar the familiar Sharpe ratio, but it is not, since the denominator is s², non s equally inward the Sharpe ratio. However, if you lot tin approximate the Sharpe ratio, say, from or together with thus backtest results of a strategy, you lot tin also approximate f simply equally easily. Suppose I convey a strategy alongside expected furnish of 12% over a flow alongside risk-free charge per unit of measurement beingness 4%. Also, let’s tell the expected Sharpe ratio is 1. It is slow to calculate f, which comes out to hold upwardly 12.5.

This is a shocking number. This is telling you lot that for this strategy, you lot should hold upwardly leveraging your equity 12.5 times! If you lot convey $100,000 inward cash to invest, together with if you lot actually believe the expected values of your returns together with Sharpe ratio, you lot should borrow coin to merchandise a $1.2 1000000 portfolio!

Of course, estimates of expected returns together with Sharpe ratio are notoriously over-optimistic, what alongside the inevitable data-snooping bias together with other commons pitfalls inward backtesting strategies. The mutual recommendation is that you lot should halve your expected returns estimated from backtests when calculating f. This is oft called the half-Kelly criterion. Still, inward our example, the recommended leverage comes to 6.25 afterwards halving the expected returns.

Fixing the leverage of a portfolio is non equally slow or intuitive equally it sounds. Back to our $100,000 example. Say you lot followed the (half-) Kelly measure together with bought a portfolio worth $625,000 alongside or together with thus borrowed money. The side yesteryear side day, disaster struck, together with you lot lost 5%, or $31,250, of the value of your portfolio. So at nowadays your portfolio is worth solely $593,750, together with your equity is at nowadays solely $68,750. What should you lot do? Most people I know volition simply stick to their guns together with produce nothing, hoping that the strategy volition “recover”. But that’s non what the Kelly measure would prescribe. Kelly says, if you lot desire to avoid eventual bankruptcy (i.e. your equity going to null or negative), you lot should at nowadays farther cut back the size of your portfolio to $429,688. Why? Because the recommended leverage, 6.25, times your electrical current equity, $68,750, is close $429,688.

Thus Kelly measure requires you lot to sell into a loss (assuming you lot convey a long-only portfolio here), together with buys into a profits – something that requires steely champaign of report to achieve. It also runs counter to the commons mean-reversion expectation. But fifty-fifty if you lot strongly believe inward mean-reversion, equally no incertitude many of the ruined hedge funds did, you lot postulate to visit protecting you lot together with your investors from the possibility of bankruptcy earlier the marketplace position reverts.

Besides helping you lot to avoid bankruptcy, the Kelly measure has or together with thus other of import mathematically proven property: it is a “growth-optimal” strategy. I.e. if your destination is to maximize your wealth (which equals your initial equity times the maximum increase charge per unit of measurement possible using your strategy), Kelly measure is the way.

Notice this destination is non the same equally many hedge managers’ or their investors’ goal. They oft desire to maximize their Sharpe ratio, non increase rate, for the argue that their investors desire to hold upwardly able to redeem their shares at whatever fourth dimension together with hold upwardly reasonably for sure that they volition redeem at a profit. Kelly measure is non for such investors. If you lot adopt the Kelly criterion, at that topographic point may hold upwardly long periods of drawdown, highly volatile returns, depression Sharpe ratio, together with and thus forth. The solely affair that Kelly guarantees (to an exponentially high flat of certainty), is that you lot volition maximize the increase potential of your strategy inward the long run, together with you lot volition non hold upwardly bankrupt inward the interim because of the inevitable short-term marketplace position fluctuations.

For farther reading:

Poundstone, William. (2005). Fortune’s Formula. New York: Hill together with Wang.

Thorp, Edward O. (1997; revised 1998). The Kelly Criterion inward Blackjack, Sports Betting, together with the Stock Market. www.bjmath.com/bjmath/thorp/paper.htm

Maximizing Compounded Charge Per Unit Of Measurement Of Return

A elementary formula that few traders utilize

Here is a fiddling puzzle that may stymie many a professional person trader. Suppose a sure enough stock exhibits a truthful (geometric) random walk, yesteryear which I hateful at that topographic point is a 50-50 hazard that the stock is going upward 1% or downwardly 1% every minute. If you lot purchase this stock, are you lot nearly likely, inward the long run, to brand money, lose money, or endure flat?

Most traders volition blurt out the respond “Flat!”, in addition to that is wrong. The right respond is you lot volition lose money, at the charge per unit of measurement of 0.5% every minute! That is because for a geometric random walk, the average compounded charge per unit of measurement of furnish is non the short-term (or one-period) furnish m (1% here), only is m – s²/2, where s (also 1% here) is the measure departure of the short-term return. This is consistent alongside the fact that the geometric hateful of a laid of numbers is ever smaller than the arithmetics hateful (unless the numbers are identical, inward which illustration the 2 agency are the same). When nosotros assume, equally I did, that the arithmetics hateful of the returns is zero, the geometric mean, which gives the average compounded charge per unit of measurement of return, must endure negative.

This quantity m – s²/2 holds the cardinal to selecting a maximum increase strategy. In a previous article (“How much leverage should you lot use?”), I described a scheme to maximize the long-run increase charge per unit of measurement of a given investment strategy (i.e., a strategy alongside a fixed m in addition to s) yesteryear leveraging. However, frequently nosotros are faced alongside a alternative of dissimilar strategies alongside dissimilar expected returns in addition to risk. How produce nosotros direct betwixt them? Many traders retrieve that nosotros should pick the 1 alongside the highest Sharpe ratio. This is reasonable if a trader ready each of his or her bet to convey a constant size. But if you lot are a trader interested inward maximizing long-run wealth (like the Kelly investor I mentioned inward the previous article), the bet size should ever endure proportional to the compounded return. Maximizing Sharpe ratio does non guarantee maximal increase for multi-period returns. Maximizing m – s²/2 does.

For farther reading:

Miller, Stephen J. The Arithmetic in addition to Geometric Mean Inequality. ArithMeanGeoMean.pdf

Sharpe, William. Multi-period Returns. http://www.stanford.edu/ wfsharpe/mia/rr/mia_rr3.htm

Poundstone, William. (2005). Fortune’s Formula. New York: Hill in addition to Wang.

An Update On The Issue Energy Stocks Vs Futures Arbitrage Trade

I argued earlier inwards the outset of Oct ("An arbitrage merchandise betwixt issue energy stocks as well as futures") that issue energy stocks are overvalued relative to issue energy futures. At that time, a portfolio of long one forepart calendar month QM (crude oil Emini time to come contract) as well as curt 640 shares of XLE (energy stocks ETF) has a value of -$2,584. Where is it now? As of the closed of Oct 31, Dec QM is at $58.725, piece XLE is at $55.73 a share. The portfolio is straightaway at -$6,305 (the multiplier for QM is 500). The spread has clearly widened: it is straightaway at a 3-year low.

We are straightaway faced alongside the park arbitrage trader's quandary. Is this an unprecendented profits chance to double upwardly on this trade, or was this a colossal blunder on my part? I came across this New York Times article most the earnings reports from Exxon as well as Shell that gave me approximately comfort. While both issue energy companies posted huge profits, the article quoted Fadel Gheit, a senior issue energy analyst at Oppenheimer & Company, that for the 4th quarter, "“the enquiry is non if earnings volition decline, the enquiry is past times how much.” According to the article, analysts enjoin that for every dollar the toll of a barrel of unsmooth oil drops, Exxon forgoes $500 1000000 inwards profit.

So yes, alongside my fingers crossed, I am nevertheless waiting for the hateful solar daytime when this spread closes up.

Gold Vs. Gold-Miners: Roughly Other Arbitrage Opportunity?

Recently in that place is mounting involvement inwards buying golden (for example, encounter this before why I believe discover energy futures in addition to discover energy companies ETF are “cointegrated”, i.e. when their spread wanders far from a hateful value, in that place is a high probability that they volition revert to the mean. The same analysis tin last made well-nigh other pairs of commodity futures in addition to ETF’s. Therefore I apply this to gold.

Looking or in addition to then for ETF’s that agree golden miners, I constitute GDX. It started trading on May 23, 2006 in addition to thence has a relatively brusk history for us to analyze. We could own got paired it against the front-month golden futures contract GC, exactly this may last inconvenient because i has to rollover the contracts monthly. So instead, nosotros duad it against an ETF that holds golden every bit a commodity. GLD is i such example. (So is IAU, exactly GLD is far to a greater extent than liquid.) Using the same Matlab cointegration parcel that I mentioned inwards the previous article, I produce upward one's heed that fifty-fifty alongside the brusk history, GLD cointegrates alongside GDX alongside a 90% probability. Also, the parcel tells us the proper combination is sixty shares of GLD vs. 100 shares of GDX. So if nosotros bird a duad yesteryear buying sixty shares of GLD in addition to shorting 100 shares of GDX, nosotros tin plot the value over fourth dimension here:

There were indeed numerous instances of reversion to the mean. I was able to accept wages of the high or in addition to then mid-July in addition to shorted this spread profitably, in addition to I likewise bought the spread or in addition to then the depression inwards early on September in addition to exited my positions profitably or in addition to then mid-September. As of the 1st of November, the spread is in i trial once again inwards sufficiently negative territory to warrant attention.

There are some caveats alongside trading this spread. First, it is non ever slowly to borrow GDX or GLD to short. It depends on if your broker has a expert securities lending desk. Secondly, the history of GDX is short. So whatsoever analysis must last taken alongside a grain of salt. To overcome this brusk history, I could own got constructed my ain handbasket of golden mining stocks in addition to plot the toll of this handbasket against the golden futures GC. If you lot intend to invest heavily into this spread, I would definitely recommend doing this slice of difficult work.

Cointegration Is Non The Same Every Minute Correlation

A reader asked me latterly why I believe that liberate energy stock prices (e.g. XLE) are correlated amongst unsmooth crude futures front-month contract (QM). Actually I don’t believe they are necessarily correlated – I solely recollect they are “cointegrated”.

What is the departure betwixt correlation as well as cointegration? If XLE as well as QM were actually correlated, when XLE goes upwards i day, QM would probable acquire upwards too on the same day, as well as vice versa. Their daily (or weekly, or monthly) returns would accept risen or fallen inwards synchrony. But that’s non what my analysis was about. I claim that XLE as well as QM are cointegrated, pregnant that the 2 toll serial cannot wander off inwards reverse directions for real long without coming dorsum to a hateful distance eventually. But it doesn’t hateful that on a daily terra firma the 2 prices accept to motion inwards synchrony at all.

Two hypothetical graphs illustrate the differences. In the outset graph, stock Influenza A virus subtype H5N1 as well as stock B are correlated. You tin encounter that their prices motion inwards the same management almost everyday.

Now catch stock Influenza A virus subtype H5N1 as well as stock C.

Stock C clearly doesn’t motion inwards whatsoever correlated fashion amongst stock A: some days they motion inwards same direction, other days opposite. Most days stock C doesn’t motion at all! But respect that the spread inwards stock prices betwixt C as well as Influenza A virus subtype H5N1 e'er provide to nearly $1 later on a while. This is a manifestation of cointegration betwixt Influenza A virus subtype H5N1 as well as C. In this instance, a profitable merchandise would move to purchase Influenza A virus subtype H5N1 as well as brusk C at around twenty-four hours 10, as well as thence acquire out both positions at around twenty-four hours 19. Another profitable merchandise would move to purchase C as well as brusk Influenza A virus subtype H5N1 at around twenty-four hours 31, as well as thence closing out the positions around twenty-four hours 40.

Cointegration is the foundation upon which distich trading (“statistical arbitrage”) is built. If 2 stocks only motion inwards a correlated manner, in that place may never move whatsoever widening of the spread. Without a temporary widening of the spread inwards either direction, in that place is no chance to brusk (or buy) the spread, as well as no argue to await the spread to revert to the hateful either.

For farther reading:

Alexander, Carol (2001). Market Models: Influenza A virus subtype H5N1 Guide to Financial Data Analysis. John Wiley & Sons.