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Risk management of hedge funds using stochastic programming asset-liability models William T Ziemba Alumni Professor of Financial Modeling and Stochastic Optimization (Emeritus), UBC, Vancouver, BC, Canada.

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Risk management of hedge funds using stochastic programming asset-liability modelsWilliam T ZiembaAlumni Professor of Financial Modeling and Stochastic Optimization (Emeritus), UBC, Vancouver, BC, Canada

Avoiding the recipe for disaster: overbetting, non-truly diversifying and then being hit by a bad scenario

Do not overbet.


The recipe for disaster

It is clear that hedge funds got into trouble by overbetting and not being truly diversified, and vulnerable, they then got caught by low probability but plausible disaster scenarios that occurred.

It is exactly then - when you are in trouble - that you need access to new cash and since that is usually not available, it makes more sense to plan ahead for such contingencies by not overbetting and by being truly diversified in advance.


Markets are understandable most (95%+) of the time. However real asset prices have fat tails because extreme events occur much more than lognormal or normal distributions indicate.

      • Keim-Ziemba (2000) Security Market Imperfections in Worldwide Equity Markets, Cambridge University Press, much of asset returns are NOT predictable.
      • Must have way to use conventional models, options pricing, etc and the irrational unexplainable aspects once in a while.
      • Whether the extreme events are predictable or not is not the key issue - what is crucial is that you consider that they can happen in various levels with various chances.
  • How much should one bet on a favorable investment situation?
  • It’s clear that hedge funds got into trouble by overbetting and having plausible but low probability disastrous scenarios occur.
  • It is exactly then - when you are in trouble - that you need access to new cash.


Hedge Funds: Some Directional Strategies

  • Macro - an attempt to capitalize on country, regional and/or economic change affecting securities, commodities, interest rates and currency rates. Asset allocation can be aggressive, using leverage and derivatives. The method and degree of hedging can vary significantly.
  • Long – a “growth”, “value”, or other model approach to investing in equities with no shorting or hedging to minimize market risk. These funds mainly invest in emerging markets where there may be restrictions on short sales.
  • Long Bias – similar to equity convergence but a net long exposure.
  • Short – selling short over-valued securities attempting to repurchasing them in the future at a lower price.


Long Term Capital Management

That’s about 1% of all the world’s derivative positions


The bad scenario


Kelly and fractional Kelly - explaining the overbetting that led to the LTCM disaster


If returns are lognormal then fractional Kelly is just the negative power coefficient otherwise approximate

Ref MacLean, Ziemba and Li, Time to Wealth (2005)

Fractional Kelly has less growth and less variability, it lowers the bet to give a smoother wealth path.


What is the optimal fractional Kelly? -w, >0, what  is good?

  • MacLean, Sanegre, Zhao and I, JEDC (2004) solve this in a continuous time model where you check discretely and use a Var type criterion on the wealth path.
  • With Kelly, the better the bet is the more you bet and thus the more you lose when you lose so it’s very hard to stay above a wealth path.
  • Theory developed under restrictive assumptions; calculations: algorithm exists but computations lengthy.


Long run exponential growth is equivalent to maximizing the expected log of one period’s returns


Thus the criterion of maximizing the long run exponential rate of asset growth is equivalent to maximizing the one period expected logarithm of wealth. So an optimal policy is myopic.

  • Max G(f) = p log (1+f) + q log (1-f)  f* = p-q
  • The optimal fraction to bet is the edge p-q


Slew O’ Gold, 1984 Breeders Cup Classic

f*=64% for place/show; suggests fractional Kelly.


Classic Breiman Results


Kelly and half Kelly simulations


Medium time simulations

Notice 166 times wealth more than 100 times initial wealth fail with full Kelly but only once with half Kelly

But probability of being ahead is higher with half Kelly, 870 vs 954

Min wealth is 18

700 bets all independent with a 14% edge, result you still lose over 98% of your fortune with bad scenarios


You really do need to use scenario dependent correlation matrices and consider extreme scenarios. LTCM was not subject to VAR regulation but still used it.

  • Do not overbet
  • Be aware of and consider extreme scenarios.
  • Allow for extra illiquidity and contract defaults. LTCM also suffered because of the copycat firms which put on similar positions and unwound them at the same time in August/September 1998.
  • Really diversify (to quote Soros, ``we risked 10% of our funds in Russia and lost it, $2 billion, but we are still up 21% in 1998").
  • Historical correlations work when you do not need them and fail when you need them in a crisis when they approach one. Real correlations are scenario dependent. Sorry to be repetitive, but this is crucial.

Good information on the demise of LTCM and the subsequent $3.5 billion bailout by major brokerage firms organized by the FED are in a Harvard Business School case by Andre Perold (1998), and articles by Philippe Jorion (2000) and Franklin Edwards (1999).

Eventually the positions converged and the bailout team was able to emerge with a profit on their investment.

Geyer, Ziemba et al (2005) InnoALM for scenario dependent correlation matrices or see Ziemba (2003) The stochastic programming approach to asset liability and wealth management, AIMR

Ziemba and Ziemba (2007)


The 2 inch dinner

  • The currency devaluation of some two thirds was no surprise to me.
  • In 1992 my family and I were the guests in St. Petersburg of Professor Zari Rachev, an expert in stable and heavy-tail distributions and editor of the first handbook in North Holland's Series on Finance (Rachev, 2003) of which I am the series editor.
  • As we arrived I gave him a $100 bill and he gave me four inches of 25 Ruble notes.
  • Our dinner out cost two inches for the four of us; and drinks were extra in hard currency.
  • So I am in the Soros camp; make bets in Russia if you have an edge but not risking too much of your wealth.


Where was the money lost? Everywhere!

The bad scenario was confidence which hit every financial market.

  • The score card according to Dunbar (2000) was a loss of $4.6 billion.
  • Emerging market trades such as those similar to my buy Italy, sell Florence lost 430 million.
  • Directional, macro trades lost 371 million.
  • Equity pairs trading lost 306 million.
  • Short long term equity options, long short term equity lost 1.314 billion.
  • Fixed income arbitrage lost 1.628 billion.
  • The bad scenario of investor confidence that led to much higher interest rates for lower quality debt and much higher implied equity volatility had a serious effect on all the trades.
  • The long-short equity options trades, largely in the CAC40 and Dax equity indices, were based on a historical volatility of about 15% versus implieds of about 22%.
  • Unfortunately, in the bad scenario, the implieds reached 30% and then 40%.
  • With smaller positions, the fund could have waited it out but with such huge levered positions, it could not.
  • Equity implieds can reach 70% or higher as Japan's Nikkei did in 1990/1991 and stay there for many months.


The imported crash of October 27 and 28, 1997

  • A currency crisis developed in various Asian countries in mid 1997. It started in Thailand and moved all across the region.
  • The problem was lack of foreign reserves that occurred because spending and expectations that led to borrowing were too high and Japan, the main driver of these economies, was facing a consumer slowdown so its imports dropped.
  • Also loans were denominated in what was then considered a weak currency, the US dollar.
  • So that effectively these countries were long yen and short dollars. A large increase in the US currency in yen terms exacerbated the crisis.
  • The countries devalued their currencies, interest rates rose and stock prices fell.
  • A well-known hedge fund failure in 1997 was Victor Niederhoffer's fund which had an excellent previous record with only modest drawdowns.
  • A large long bet on cheap Thai stocks that became cheaper and cheaper turned $120 million into $70 million.
  • Buying on dips added to losses. Then the fund created a large short position in out-of-the-money S&P futures index puts.
  • A typical position was November 830's trading for about $4-6 at various times around August-September 1997.


The crisis spreads

  • The crisis devastated the small economies of Malaysia, Singapore, Indonesia, etc.
  • Finally it spread to Hong Kong.
  • There, the currency was pegged to the US dollar at around 7.8.
  • The peg was useful for Hong Kong's trade and was to be defended at all costs.
  • The weapon used was higher interest rates which almost always lead to a stock market crash but with a lag.
  • See the discussion in chapter 2 of my AIMR book (Ziemba, 2003) for the US and Japan and other countries.
  • The US S&P500 was not in the danger zone in October 1997 by my models and I presume by others and the trade with Hong Kong and Asia was substantial but only a small part of the US trade. US investors thought that this Asian currency crisis was a small problem because it did not affect Japan very much. In fact, Japan caused a lot of it.


A tempest in a teapot

  • The week of October 20-25 was a difficult one with the Hang Seng dropping sharply.
  • The S&P was also shaky so the November 830 puts were 60 cents on Monday, Tuesday and Wednesday but rose to 1.20 Thursday and 2.40 on Friday.
  • The Hang Seng dropped over 20% in a short period including a 10% drop on Friday, October 25.
  • The S&P 500 was at 976 way above 830 as of Friday's close.
  • A further 5% drop on Monday, October 27 in Hong Kong led to a panic in the S&P500 futures later on Monday in the US.
  • The fall was 7% from 976 to 906 which was still considerably above 830.
  • On Tuesday morning there was a further fall of 3% to 876 still keeping the 830 puts out of the money.
  • The full fall in the S&P500 was then 10%.


Volatility exploded

  • But the volatility exploded and the 830's were in the $16 area.
  • Refco called in Niederhoffer's puts mid morning. They took a loss of about $20 million.
  • So Niederhoffer's $70 million fund was bankrupt and actually in the red since the large position in these puts and other instruments turned 70 million into minus 20 million.
  • The S&P500 bottomed out around the 876 area and moved violently in a narrow range then settled and then moved up by the end of the week right back to the 976 area. So it really was a tempest in a teapot like the cartoon depicts.
  • The November 830 puts expired worthless. Investors who were short equity November 830 puts were required to put up so much margin that that forced them to have small positions and they weathered the storm and their $4-$6, while temporarily behind at $16 did eventually go to zero.

volatility cont d
Volatility (cont’d)
  • So did the futures puts, but futures shorters are not required to post as much margin so if they did not have adequate margin because they had too many positions. They could have easily been forced to cover at a large loss.
  • I argue that futures margins, at least for equity index products, do not fully capture the real risk inherent in these positions. I follow closely the academic studies on risk measures and none of the papers I know deals with this issue properly.
  • When in doubt, always bet less. Niederhoffer is back in business having profited by this experience.

Whoops, maybe not!!

  • He still overbets: makes a lot, losses a lot and then tries to recover (ch 22 in Ziemba and Ziemba, 2007)

the economist nov 1 7 1997
The Economist, Nov 1-7, 1997



  • The lessons for hedge funds are much as with LTCM. Do not overbet, do diversify, watch out for extreme scenarios.
  • Even short term measures to keep one out of potentially large falls did not work in October 1997. That was an imported fear-induced crash not really based on US economics.
  • My experience is that most crashes occur when interest rates relative to price earnings ratios are too high. In that case there almost always is a crash, see Ziemba (2003) for the 1987 US, the 1990 Japan, and the US in 2000 are the leading examples as in US in 2001, which predicted the over 20% fall in the S&P500 in 2002.
  • Interestingly the measure moved out of the danger zone then in mid to late 2001 it become even more in the danger zone than in 1999.
  • There is an effect of the time to unwind - that’s why having enough capital to withstand the effect of crashes is one of the most important aspects of risk capital.
  • One of my Vancouver neighbors, I learned later, lost $16 million in one account and $4 million in another account.
  • The difference was the time given to liquidate.

the symmetric downside risk sharpe ratio
The Symmetric Downside-Risk Sharpe Ratio
  • The Sharpe ratio is a very useful measure of investment performance.
  • However, it is based on mean-variance theory and thus is basically valid only for quadratic preferences or normal distributions.
  • Hence skewed investment returns can lead to misleading conclusions.
  • This is especially true for superior investors such as Warren Buffett and others with a large number of high returns.
  • Many of these superior investors use capital growth wagering ideas to implement their strategies which leads to higher growth rates but also higher variability of wealth.
  • A simple modification of the Sharpe ratio to assume that the upside deviation is identical to the downside risk provides a useful modification that gives more realistic results.


The wealth levels from December 1985 to April 2000 for the Windsor Fund of George Neff, the Ford Foundation, the Tiger Fund of Julian Robertson, the Quantum Fund of George Soros and Berkshire Hathaway, the fund run by Warren Buffett, as well as the S&P500 total return index.

some superior mutual and hedge fund managers
Some superior mutual and hedge fund managers
  • The means, standard deviations and Sharpe (1966, 1994) ratios of these six funds, based on monthly, quarterly and yearly net arithmetic and geometric total return data are shown in Table 1.
  • Also shown here is data from the Harvard endowment (quarterly) plus that of US Treasuries, T-bills and US inflation and the number of negative months and quarters.
  • Hence by the Sharpe ratio, the Harvard endowment and the Ford Foundation had the best performance, followed by the Tiger Fund then the \sp500 total return index, Berkshire Hathaway Quantum and Windsor.
  • The basic conclusions are the same with monthly or quarterly data and with arithmetic and geometric means. Because of data smoothing, the Sharpe ratios with yearly data usually exceed those with quarterly data which in turn exceed the monthly calculations.
  • The reason for this ranking is that the Ford Foundation and the Harvard endowment, while they had less growth, they also had much less variability.
  • Indeed, these funds have different purposes, different investors, different portfolio managers, and different fees, so such differences are not surprising.

ford foundation and harvard investment corporation returns quarterly data june 1977 to march 2000
Ford Foundation and Harvard Investment Corporation Returns, quarterly data, June 1977 to March 2000

arithmetic vs geometric means

Typically the Sharpe ratio is computed using arithmetic returns since most academic theory is based on arithmetic returns.

  • However, for asset returns over time, the geometric mean is a more accurate measure of average performance since the arithmetic mean is biased upwards.
  • The geometric mean helps mitigate the autocorrelated and time varying mean and other statistical properties of returns that are not iid.
  • For example, if one has returns of +50% and -50% in two periods, then the arithmetic mean is zero which does not correctly reflect the fact that 100 became 150 and then 75.
  • The geometric mean which is -13.7% is the correct measure to use.
  • For investment returns in the 10-15% range, the arithmetic returns are about 2% above the geometric returns.
  • But for higher returns this approximation is not accurate.
  • Hence, geometric means as well as more typical arithmetic means are used in this paper.
  • Lo (2002) points out that care must be used in Sharpe ratio estimations when the investment returns are not iid, which they are for the investors discussed here.
  • For dependent but stationary returns he derives a correction of the Sharpe ratios that deflates artificially high values back to correct values using an estimation of the correlation of serial returns.
  • The Sharpe ratios are almost always lower when geometric means are used rather than arithmetic means with the difference between these two measures a function of return volatility.
Arithmetic vs geometric means


Fund return data: yearly means, standard deviations and Sharpe ratios, Dec 1985 to April 2000

Increase in per share book value of Berkshire Hathaway versus returns on the S&P500 with dividends included, 1965-2004 in percent

$15 --> $87,000 in October 2005

the chest fund 1927 1945 keynes
The Chest Fund, 1927-1945 (Keynes)

-w-0.25 (80% Kelly, 20% cash), see Ziemba (2003)

using the sharpe ratio
Using the Sharpe ratio

using a modified sharpe ratio that does not penalize gains
Using a modified Sharpe ratio that does not penalize gains

Summary over funds of negative observations and arithmetic and geometric means

the symmetric downside sharpe ratio performance measure
The symmetric downside Sharpe ratio performance measure
  • we want to determine if Warren Buffett really is a better investor than the rather good but lesser funds mentioned here, especially the Ford Foundation and the Harvard endowment, in some fair way.
  • The idea is presented in a Figure below where we have plotted the Berkshire Hathaway and Ford Foundation monthly returns as a histogram and show the losing months and the winning months in a smooth curve. We want to penalize Warren for losing but not for winning. So define the downside risk as
  • This is the downside variance measured from zero, not the mean, so it is more precisely the downside risk.
  • To get the total variance we use twice the downside variance

comparison of ordinary and symmetric downside sharpe yearly performance measures
Comparison of ordinary and symmetric downside Sharpe yearly performance measures

Buffett: still does not beat the Ford Foundation - and Harvard is also better than Buffett but not Ford with the quarterly data

Why? Tails still too fat

Thorp (1997) shows that Buffett is essentially a full Kelly bettor.

berkshire hathaway versus ford foundation monthly returns distribution january 1977 to april 2000
Berkshire Hathaway versus Ford Foundation, monthly returns distribution, January 1977 to April 2000

return distributions of all the funds quarterly returns distribution december 1985 to march 2000
Return distributions of all the funds, quarterly returns distribution, December 1985 to March 2000


The record of Princeton Newport Partners, LP, cumulative results, Nov 1968- Dec 1998 (Thorp) DSSR=13.8 the highest I have seen, 3 monthly losses in 20 years

the record of bill benter the world s greatest racetrack bettor
The record of Bill Benter, the world’s greatest racetrack bettor.

Efficiency of Racetrack Betting Markets, Academic Press (1994)

He made 400M+ in a market with >15% track take.

But he had advantages such as computer betting into the pools an very poor bettors in opposition to him.


Disasters will occur as traders greatly overbet, do not diversify and a bad scenario occurs, see column on Amaranth by RESZ and WTZ in Wilmott and our Wiley book

The incentive structure is set up to encourage good traders to expand their positions and this can easily lead to rogue trading.

What’s needed is careful risk control of strategies that are well diversified and have clearly definable edges.

Managers who do this have been very successful both in hedge funds and, for example, in university endowments, see our column on the Yale endowment