A remedial lesson in statistics

Tuesday, May 08, 2007

You can always tell when financial market volumes start to dry up because brokers start organizing a seemingly endless sequence of ancillary ways to add value. Economist meetings, golf outings, and dinners tend to feature prominently on this list. As such, Macro Man has spent a fair few evenings over the past several weeks at roundtable discussion meetings, wherein traders and investors gather together to exchange views and grumble about the market.

And grumble we have. It has become rather evident at these discussions that Macro Man is not the only person in the market to be frustrated by the triumph of beta over alpha. Indeed, a common theme has been frustration over the low level of risk premia, the failure of market timing, and the continued success of what most consider to be relatively naïve, beta like strategies across markets. Far from being the greedy coupon-clippers that are often portrayed in the popular press, the people with whom Macro Man has spoken have felt underinvested and unhappy for much of the past few months, if not longer.

One of the most commonly cited factors causing dissatisfaction has been the relatively short-term time horizon of many investors. Now, fund-of-funds investors have always had an itchy trigger finger, and that will probably never change. In that industry, “what have you done for me lately” isn’t just an investment strategy, it’s a way of life. However other investors, both in hedge funds and in long-only products, are also increasingly demanding a steady stream of positive performance with shallow and infrequent drawdowns.

Now, there are of course exceptions to every rule, and it would be foolhardy to suggest that no one out there has the wherewithal to look beyond a couple of weeks when taking an investment decision. Yet surely where there is smoke there’s fire, and traders and fund managers are being pushed by natural selection towards strategies delivering steady positive returns. This, of course, is the virtual definition of a volatility selling or carry strategy, which clips coupons every day until the black swan event wipes out the bulk of accumulated profits in one fell swoop. To date, the black swan hasn’t really made an appearance, so these strategies continue to fare well and attract new assets. Investors, it would appear, want all of the upside of taking risk, with none of the downside.
Now if a strategy could generate the return distribution above in perpetuity, it would of course be worth pursuing whole-hog. The problem, of course, is that strategies that generate such returns at any point in time usually have an extremely fat tail lurking on the left side of the distribution somewhere in their future. Macro Man thought it would be worthwhile to review basic principles of statistics to review just how often even skilled managers can generate negative returns when not pursuing a strategy designed to maximize the probability of positive return at any particular point in time.

Imagine we have an investor whose true skill is known to deliver a Sharpe ratio of 1- say an annualized excess return of 10% per year, with a volatility of returns of 10%. What are the chances in any given time period that this investor makes money with his trading activities (ignoring, for the purposes of this exercise, any cash component of return, as well as any management or performance fees charged)? While Nassim Nicholas Taleb, among others, have pointed out that high-frequency returns are mostly the result of noise, it is worth looking at graphic representations of return distributions to illustrate just how powerful this effect is.

How likely is it that a Sharpe ratio 1 strategy will lose money on any given day? 47.5%, as it turns out- a little less that you’d get from flipping a coin.

Extending the horizon to weekly does little to alter the likelihood of success. A Sharpe ratio 1 manager will still lose money during 44.5% of the weeks he is running the strategy.

How about monthly? Our manager still loses money just under two out of every five months.

Extending the frequency to quarterly improves the chances of success, but not by as much as you’d expect. Our Sharpe ratio 1 manager still loses money in just over 30% of the quarters in which he runs the strategy. By this point, some itchy trigger investors may already be thinking of heading for the exits, potentially missing out on the manager’s skill in coming quarters.

Even when measured on an annual basis, a Sharpe ratio 1 manager will still lose money one out of every six years. Unfortunately, in the current climate, many investors will not be along for the ride in the other five.
Now, all of these probabilities are based on normal distributions, which we all know do not accurately describe financial market returns. Yet Macro Man’s perception is that many underperforming managers have done so through buying, rather than selling, risk premia. If so, it is probably reasonable to expect a degree of outperformance during fat tail events.

By way of disclosure, Macro Man does not have any particular axe to grind here. He has not suffered any sort of redemption in his real job, and his performance this year is positive- though as with the blog portfolio, perhaps not as positive as it could be had he focused exclusively on beta strategies. However, observing the degree of frustration amongst fellow professional investors has been eye-opening. When he sees things like structured product selling of 15 year S&P 500 vol and 10 year EUR/USD vol, when these implied volatilities are at all time lows, he begins to wonder if the apotheosis of vol-selling, carry driven strategies has finally run its course, and that the time is finally ripe for buying value and/or tail risk. When the 10 year straddle in EUR/USD is given at less than 12.5 big figures, the risk/reward of taking the other side of vol selling begins to look awfully attractive.

Posted by Macro Man at 11:07 AM  


Just taking a quick look, but I´m afraid where you say the manager will "lose" money he will actually be just underperforming... If not, then please delete this comment.

Best regards.

avinash goldfish

Anonymous said...
1:38 PM  

Yes, good point. There is no cash return embedded in this data, nor is there any sort of fee structure either.

The focus here is entirely on the manager's trading returns, where underperformance means he has indeed lost money trading.

Macro Man said...
1:47 PM  

Perhaps using the risk-free rate is no longer the most appropriate way of benchmarking excess return in the Sharpe ratio. A beta-adjusted benchmark may be more appropriate in this day and age?

max_drawdown said...
2:13 PM  

Hmmm. That's the whole basis of how real money managers are evaluated. It sounds great when markets are roaring (remember 1999?), but funnily enough loses its luster when risk premia rise.

The absolute return industry got a huge boost in 2001-2002, when pension funds and the like got tired of allocating to long only guys who beat the benchmark- and thus were down "only" 18%.

I suspect we are moving towards a world where traders are internally benchmarked against beta, however. Investors, though, will always want the benchmark to be the max of (cash, beta strategy return.)

Macro Man said...
2:30 PM  

i completely agree, but at the same time those lows in vol could have been recognized as a buy, buy, buy signal several times over the past few (2-3) years, to anyone's detriment. that's the beguilding aspect of markets. trades on such trends can seem so obvious, bu the timing is so tricky. it's not just currency vol: equities, treasuries, swaps, etc. take your pick. i think calling this low in vol is very, very difficult because of the role of voldemort et al. they are essentially selling vol at any cost, and so common sense must be set aside.


Anonymous said...
3:29 PM  

I agree that calling absolute lows is for suckers, but by the same token there comes a time when things get so cheap that you can afford to start buying lottery tickets and stuffing them in a drawer. I am just getting a sense that the time is rapidly approaching when a wingnut long term vol lottery ticket overlay on beta coupon clipping startegies is going to be a no-brainer.

It's true that Voldemort runs the show and sells vol like no tomorrow. But it's also true that ten years ago, Voldemort and co. were about to teete on the brink of going out of business. A lot can change in a decade.

Two things that we do know are that Voldemort is changing something- moving some assets to a more profit maximizing (and presumably marking-to-market) entity- and that the (US) political will to crimp his style is growing.

Maybe these things will have no impact, BCA is right, and EUR/JPY goes to 180. If so, then beta strategies should fare very, very well. But if you can hedge your entire VAR risk (assuming that vols rise) at a minimal upfront premium cost and with relatively little decay...well, that makes a whole lotta sense to me.

Macro Man said...
3:50 PM  

When buying those lottery tickets, do not forget to choose the counterparty rather carefully. This is NOT in the statistics books...not even in the footnotes.

"Cassandra" said...
4:00 PM  

The problem, of course, is knowing who will remain solvent during a six sigma event, assuming system risk becomes a serious issue...

Macro Man said...
4:12 PM  

Now they are eating like chicken, but the time will surely come...

On the black swan - Taleb argued that out of the money options are very mispriced, but with vols at record level you won't even need a very sharp move to make a killing.

An interesting thing will be to watch how trading systems will behave when the 6 sigma comes - it took only 3 percent slide to send some system down not long ago.

What will happen if all hedgers and sellers rush to the same door...

flipper said...
5:58 PM  

BTW Goldman Sachs bottom-up analysts in Japan are using 145 Euro-yen for their next FY forecasts (mar07-mar08).

Flipper, Mr Taleb's own investment well dried-up by crisis-hunting in cheap, long vol, and subsequently waiting for Godot.

Reminds me of the slot machines with infrequent payout settings that forlorn gamblers will feed incessantly until they are bled dry. Someone will, of course, reap a percentage of what the unlucky have fed into the beast, but it most often is NOT the person who fed the beast in the first instance. Now, there ARE people who ply the casinos watching for precisely such an occurance (a sort of second derivative crisis-hunter, not in the crisis, but in the people crisis-hunting), and in my misspent youth have seen fights break out where an unfortunate soul, having fed the machine lots, runs out of coins and needs change, but refuses to leave the machine, or let another carpetbagging scum play. Which one is Taleb?

"Cassandra" said...
6:51 PM  

Taleb certainly does himself no favours by using such condescending tones in discussing other (empirically more successful) investment managers. And the systematic purchase of lottery tickets is clearly not a winning long-run strategy, as lottery tickets are intentionally priced to generate a negative expected value for the purchaser.

That, of course, does not mean that lottery tickets should never be purchased. This particularly holds when today's six-sigma, market-shattering move is yesterday's (and perhaps tomorrow's?) bog standard one standard deviation event.

Timing, of course, is everything. I may not know much, but I do know that I don't know when the blowup (if there is to be one) will come. This is why very long term vol looks so appealing- implieds are at record lows, and thanks to the square-root-of-time principle of option pricing, one can extend an exposure almost indefinitely at very little marginal cost, thus avoiding the risk of getting 'Talebed'.

Macro Man said...
7:40 PM  

Super excellent post, MM.

Apparently, you're not alone in your nervousness. The VIX is stalled at March 21 levels despite what has ensued.

Taleb seems to suffer under the illusion that what's good for the goose has no bearing whatsoever on the gander. The returns, though, from playing the long tail are not normally distributed either - as much as he would deny even implying such. Nobody walks on water.

Charles Butler said...
7:55 PM  

With thanks to "Junk Charts" for finding my way to Columbia Univ statistician Dr Andrew Gelman's interesting site:

Gelman on Taleb's Black Swans

Talebs reply to Gelman's Comments

"Cassandra" said...
10:18 PM  

I've been watching this storm brew from a mtg trading desk..it looks a whole lot like Columbus day 1998 to me...w/ the exception of swap apreads ..and incredible support of Asia and the ME.

So the real question isn't 'Is risk premia too low ?'..the real question is "what can u buy vs the dollar ?"

What's wrong w/ this logic ?
We import Oil and Junk and pay for it in dollars ..the dollars are repatriated by unsophistaced investors w/ few non-dollar choices ..

They get to choose :

Negative real rates
Negative OAS on IR derivs (little implied vol)
Negative Default adjusted credit spreads (adjusted for sanity not the last 3 hours worth of defaults)

Colmubus day may not come until global trade levels out the production arb ..and that's longer than anyone can stay liquid ..

In the end the dollar returns are brutal ..which really means that on a fwd basis imports are well below spot px ..and the dollar market is best left to those who either don't get it or have no choice ..so what to do w/ $10 or $12 tril that wants to be converted ?

Anonymous said...
7:07 AM  

I am really beginning to wonder if we are entering the beginning of the end. While yesterday's currency manipulation hearings didn't generate many headlines, legislation is still being prepared behind the scenes. And the closer we get to campaign season in the US, the more protectonist the US will become.

The last five years have been about maximizing gross utility from globalization....where the returns to owners of Western capital and Eastern labour have outweighed the losses to Western labour.

However, there are more Western labourers than owners of capital, and most western countries are democracies. Nicolas Sarkozy swept into the Elysee Palace over the weekend...but as a reformer, or as a guardian of France's vested interest? Probably a bit of both.

In 2004, the average per capita income of the lowest 20% of US households was more or less the same as the national average in China. Of course, two and a bit thousand dollars per person gets you a hell of a lot further in Beijing than it does in New Orleans. In roughly a decade, per capita income in China will be roughly the same as that of the 21-40% quintile in the US at current income growht rates. That would represent 130 or so million Americans who average less income than your average Chinese, living in what in all likelihood will still be a higher cost economy.

It seems to me that political opposition to this trend continuing unchecked can only grow over time.


Macro Man said...
1:25 PM  

This paper http://www.brookings.edu/papers/2007/1114_hedge_fund_young.aspx is germane. Not only is it difficult to tell whether a manager is truly skilled or not by looking at his or her returns, but it is _impossible_ to devise a compensation scheme that adequately rewards good managers while screening out bad managers.

Anonymous said...
5:42 PM  

I think it is probably possible to tell who is god and who is bad. The problem is that you need a statistically significant sample size; to wit, returns from a full market cycle, which could take from 7-10 years to compile.

I'd concur, however, that it is difficult to provide an incentive scheme that doesn't encourage "shoot the moon" or "put it all on red" behaviour from unskilled managers, but which at the same time rewards skilled managers for their performance on a feasible timescale.

Frankly, the same dilemma holds true for managers of companies as well.

Macro Man said...
9:32 PM  

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