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Great investors and hedge fund managers: their methods and evaluation

Prof William T ZiembaAlumni Professor of Financial Modeling and Stochastic Optimization

(Emeritus)Mathematical Institute, Oxford University

ICMA Centre, University of ReadingWilliam T Ziemba Investment Management Inc, Vancouver, BC

Dr Z Investments Inc, San Luis Obispo, CA andPrivate International Wealth Management, BC Capital Group, Nassau

CARISMA Seminar, June 25, 2007

Session 1: What it takes to win and have smooth wealth paths

1. Some great investors: Buffett, Keynes, Benter, Thorp, the Harvard Endowment and the Ford Foundation

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Keynes as a Hedge Fund Manager

• Unofficial, private hedge funds have been run for centuries using futures, equities, and other financial instruments

• Futures in rice trading in Japan date from the 1700s• Futures trading in Chicago was active in the mid-1800s• The Chest Fund at King’s College, Cambridge: an early hedge fund • Managed by the famous economist John Maynard Keynes from 1927–1945.

• November 1919 - Keynes appointed second bursar• Prior to 1919 King’s College invested only in fixed-income trustee securities

and the college’s own land and buildings. • June 1920 - Keynes convinced the college to start a separate fund containing

stocks, currency, and commodity futures. • 1924 Keynes became first bursar and thus had final authority on investment

decisions

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Keynes (cont’d)

Keynes (1938) emphasized three principles of successful investing:

(1) a careful selection of a few investments (or a few types of investments), having regard to their cheapness in relation to their probable actual and potential intrinsic value over a period of years ahead and in relation to alternative investments at the time;

(2) a steadfast holding of these investments in fairly large units through thick and thin, perhaps for several years, until either they have fulfilled their promise or it is evident that they were purchased by mistake;

(3) a balanced investment position (i.e., a variety of risks in spite of large individual holdings and, if possible, opposed risks).

Keynes did not believe in market timing:We have not proved able to take much advantage of a general systematic movement out of and into ordinary shares as a whole at different phases of the trade cycle dots. . .As a result of these experiences I am clear that the idea of wholesale shifts is for various reasons impracticable and indeed undesirable. Most of those who attempt to sell too late and buy too late, and do both too often, incurring heavy expenses and developing too unsettled and speculative a state of mind, which, if it is widespread, has besides the grave social disadvantage of aggravating the scale of the fluctuations. (Keynes, 1938)

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Keynes (cont’d)

• By emphasizing value, large holdings, and patience, Keynes’ ideas overlap those used by Warren Buffett in his Berkshire Hathaway Fund.

• Buffett has also added effective side businesses, such as insurance, and a greater level of involvement in the management of his holdings.

• Keynes’ approach of having a small number of positions even partially hedged naturally led to fairly high volatility

• Details of the holdings over time, as with hedge funds, are not known, but it is known that in 1937, the fund had 130 separate positions.

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The record of The Chest Fund, King’s College, Cambridge, 1927-1945 (Keynes)

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

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Absolute and relative performance of the Chest Fund managed by J.M.Keynes, 1927-1945 Source: Updated from Chua and Woodward (1983)

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Keynes (cont’d)

• The returns do not include dividends and interest. • The income, which is not public information, was spent on modernizing and

refurbishing King’s College. • The index’s dividend yield was under 3 percent. • A capital asset pricing model (CAPM) analysis suggests that Keynes was an

aggressive investor with a beta of 1.78 versus the benchmark U.K. market return, a Sharpe ratio of 0.385, and geometric mean returns of 9.12 percent a year versus –0.89 percent for the benchmark. Keynes had a yearly standard deviation of 29.28 percent versus 12.55 percent for the benchmark.

• The drawdown from 100 in 1927 to 49.6 in 1931, with losses of 32.4 percent in 1930 and 24.6 percent in 1931, was more than 50 percent, but this drawdown occurred during the depression, when the index fell 20.3 percent and 25.0 percent, respectively.

• The College’s patience with Keynes was rewarded in the substantial rise in the index from 1932 to 1937 to 315.4.

• Keynes’ aggressive approach caught up with him as Britain prepared for World War II in 1938–1940, when the index fell to 179.9. Then, during the war, 1941–1945, Keynes had a strong record.

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Keynes (cont’d)

• Keynes’ turning 100 into 480.3 in 18 years, versus 85.2 for the index, plus dividends and interest given to the College, gives Keynes a good record as a hedge fund manager.

• Keynes’ aggressive style (which is similar to the Kelly betting I discuss later) led to great returns—and some losses and embarrassments.

• When a grain contract was not covered in time, Keynes had to take delivery and fill up the famous King’s College chapel, which was fortunately large enough to store the grain safely until it could be sold.

• The Sharpe ratios and CAPM results are based on normality, and a better way to calculate Keynes’ B’s is through Leland’s (1999) B’s which applies for the fat tails that Keynes had.

• For example, Keynes’ aggressive style is close to log utility, so a g relative risk aversion of 1.25 approximates his behavior, which compares to 1/g = 80 percent in the log optimal portfolio and 20 percent in cash.

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Gamblers as Hedge Fund Managers

• I have consulted for seven individuals who used investment market anomalies and imperfections and hedge funds ideas to turn a humble beginning with essentially zero wealth into hundreds of millions of dollars

• Five are billionaires.

• They shared several common characteristics

• a gambling background usually obtained by playing blackjack or other sports betting professionally or a knowledge of the mathematics of gambling;

• a focused, fully researched, computerized system for asset-position selection;

• careful attention to the possibility of loss

• focusing more on not losing than on winning.

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Gamblers (cont’d)

• Four were relative value long–short managers who consistently eked out small edges.

• One was a futures trader who made bets on a large number of liquid financial assets and some short term trading on a daily or almost instantaneous basis based on favorable trends (interest rates, bonds, and currencies were the best then but not now). He is now a multi-billionaire.

• The sixth was a Hong Kong horserace bettor (see Benter in Hausch, Lo, and Ziemba, 1994).

• The seventh does all sorts of trading and investing in superior hedge funds to make billions.

Their gambling backgrounds led them to conservative investment behavior and excellent results, both absolute and risk adjusted.

• They had their losses, but rarely did they overbet or lack sufficient diversification.

• Good systems for diversification and bet size were crucial in avoiding major blowouts.

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Ed Thorp

• Edward Thorp, a mathematician with a Ph.D. from the University of California at Los Angeles (UCLA), became famous in 1960 after devising a simple-to-use card-counting system for use in the card game blackjack.

• In 1966, Beat the Market, a follow-up to his book Beat the Dealer, outlined a system for obtaining edges in warrant markets.

• Thorp was close to finding the Black–Scholes formula for pricing options, at least from an approximate, empirical, and discrete-time point of view,

• Some of his ideas were used in his hedge fund Princeton Newport Partners (PNP).

• He ran this fund, with offices in Newport Beach, California, and Princeton, New Jersey, from 1969 to 1988 and used a variety of strategies, many of which can be classified as convergence or long–short strategies.

• A Nikkei put warrant risk-arbitrage trade that Thorp and I jointly executed based on my ideas gives some idea of the approach. The basic idea is to sell A at an expensive price, hedge it with the much cheaper close substitute A’ and then wait until the prices converge and cash out, remembering not to overbet.

• Thorp wrote the foreword to my book, Beat the Racetrack written with Donald Hausch. By using probabilities estimated from the simpler win market, we provided a simple-to-use winning system for racetrack betting based on weak market inefficiencies in the complex place and show markets.

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Thorp (cont’d)

• See PNP’s performance record - actual trades and positions used by Thorp and his colleagues are not public information

• PNP gained 15.1 percent, net of fees (which were about 4 percent given the 20 percent profit fee structure), versus 10.2 percent for the S&P 500 and 8.1 percent for T-bills.

• An initial index of 100 on 1 November 1969 became 148,200 at the end of December 1988 versus 64,500 for the S&P 500 and 44,500 for T-bills.

• But what is impressive is that the risk control, by using various stochastic optimization procedures, led to no years with losses and a high Sharpe ratio, which, based on monthly data, approached 3.0.

• Of course, Thorp had an easier market to deal with than Keynes. For example, the S&P 500 had negative returns only in 1973, 1974, and 1976.

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The record of Princeton Newport Partners, LP, cumulative results, Nov 1968- Dec 1998 (Thorp)

This is ideal: 20 years, 3 losing months, no losing quarters, very smooth path.

The best smooth wealth path

Downside Sharpe measure = 13.8

It is this WTZIMI tries to achieve.

See WTZ’s JPM (2005) paper on this

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PNP: 15.1% net vs 10.2% for the S&P500

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Growth of assets for various high-performing funds, Dec 1985 - June 2000

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Buffett’s Berkshire Hathaway

• Warren Buffett’s Berkshire Hathaway is a closed-end fund which is run like a hedge fund with an emphasis on value investing, insurance businesses, and other ventures.

• A comparison with Keynes’ and Thorp’s performance,shows his brilliant record from December 1985 to [June 2000

• Buffett also had a great record from 1977 to 1985, turning the index value of 100 into 1,429.87 in 1985 and 65,852.40 in April 2000 and about 110,000 now.

• Also shown are the records of George Soros’ Quantum Fund, John Neff’s Windsor Fund, Julian Robertsons’ Tiger Fund, and the Ford Foundation.

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Capital growth

• Capital growth or Kelly criterion fans, such as Thorp and I, prefer Buffett’s long-term growth record.

• Indeed, Thorp (2006) has argued that Buffett largely uses Kelly-like strategies. Keynes and Buffett have many common characteristics in addition to their aggressive style, namely, an emphasis on value, large holdings, and patience.

• From January 1980 to March 2000, Buffett’s Sharpe ratio was 0.786, whereas the Ford Foundation’s was 0.818.

• So, by the Sharpe ratio measure, the Ford Foundation beat Berkshire Hathaway because it had good growth with little variation.

• But much of Berkshire Hathaway’s variation was on gains, and the total wealth at the horizon was much greater. Using the symmetric downside risk measure provides a fairer evaluation—with Berkshire Hathaway improved but not dominating, see §22.

• Starting from July 1977, Berkshire Hathaway’s Sharpe ratio was 0.850 versus Ford’s 0.765, and the S&P’s was 0.676, and geometric mean returns were 32.07 percent (Berkshire Hathaway), 14.88 percent (Ford), and 16.71 percent (S&P 500).

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Buffett’s 39 Year Record: only one decline, $15 in 1965 --> $110,000 in June 2007

Increase in per share book value of Berkshire Hathaway versus returns on the S&P500 with

dividends included, 1965-2003, in percent. In Feb 2007 it’s about $110,000.

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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.

He had advantages such as computer betting into the pools, very poor bettors in opposition and data updates every 12 seconds.

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A+ Performance: Harvard an in house hedge fund

Harvard boasts the largest endowment of any university in the nation,in part because of the outstanding performance turned in by its in-house portfolio managers...

Annual Returns**

School 1-Year,% 10-Year, % Size (bil)Harvard 21.1 15.9 $22.6Yale 19.4 16.8 12.7Stanford 18.0 15.1 10.0Princeton 16.8 15.5 9.9Columbia 16.9 11.6 4.525 LargestEndowments*

17.1 12.8 --

*Median return is shownSources: Harvard; Yale; Princeton; Stanford; Columbia

Barron’s, January 31, 2005

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...who have racked up impressive gains in many investment sectors.

Harvard's Holdings Annual Returns** Weight InBy Sector 1-Year, % 10-Year, % Endowment, %Domestic Equities 22.8 17.8 15Foreign Equities 36.1 8.5 10Emerging Markets 36.6 9.7 5Private Equity 20.8 31.5 13Hedge Funds 15.7 n.a 12High Yield 12.4 9.7 5Commodities 19.7 10.9 13Real Estate 16.0 15.0 10Domestic Bonds 9.2 14.9 11Foreign Bonds 17.4 16.9 5Inflation-indexedBonds

4.2 n.a. 6

Total Endowment 21.1 15.9 105%*** **Period ended Ju ne 2004, ***E quals 105 because of slight leverage.Sources: Harvard; Yale; Princeton; Stanford; Columbia

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The top 25 US university endowment funds, 2003-4, 2004-5 and 2005-6

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Yale’s endowment, 1950-2006 YEC (2006)

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Yale’s endowment, asset classes (YEC, 2006)

Changing allocations, 1985-2006

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Alternative asset returns exhibit significant dispersion (YEC, 2005)

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Yale growth, 1905-2005 (YEC, 2006)

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University revenue by source, 1905-2005 (YEC, 2006)

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2. The Importance of getting the mean right. The mean dominates if the two distributions cross only once.

Thm: Hanoch and Levy (1969)

• If X~F( ) and Y~G( ) have CDF’s that cross only once, but are otherwise arbitrary, then F dominates G for all concave u.

• The mean of F must be at least as large as the mean of G to have dominance.

• Variance and other moments are unimportant. Only the means count.

• With normal distributions X and Y will cross only once iff the variance of X does not exceed that of Y

• That’s the basic equivalence of Mean-Variance analysis and Expected Utility Analysis via second order (concave, non-decreasing) stochastic dominance.

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Errors in Means, Variances and Covariances

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Mean Percentage Cash Equivalent Loss Due to Errors in Inputs

When RA is very low such as with log u, then the errors in means become 100 times as important.

Conclusion: spend your money getting good mean estimates and use historical variances and covariances

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Average turnover: percentage of portfolio sold (or bought) relative to preceding allocation

• Moving to (or staying at) a near-optimal portfolio may be preferable to incurring the transaction costs of moving to the optimal portfolio

• High-turnover strategies are justified only by dramatically different forecasts

• There are a large number of near-optimal portfolios

• Portfolios with similar risk and return characteristics can be very different in composition

In practice (Frank Russell for example) only change portfolio weights when they change considerably 10, 20 or 30%.

Tests show that leads to superior performance, see Turner-Hensel paper in ZM (1998).

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Effect of the Risk Premium: Differing Future Equity Mean Returns: InnoALM

Siemens Austria Pension fund Multiperiod Model

• mean of US stocks 5-15%.

• mean of European stocks constrained to be the ratio of US/European

• mean bond returns same

• case NM (normal distribution and mixing correlations).

• As expected, [Chopra and Ziemba (1993)], the results are very sensitive to the choice of the mean return.

• If the mean return for US stocks is assumed to equal the long run mean of 12% as estimated by Dimson et al. (2002), the model yields an optimal weight for equities of 100%.

• a mean return for US stocks of 9% implies less than 30% optimal weight for equities

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Optimal Asset Weights at Stage 1 for Varying Levels of US Equity Means

Observe extreme sensitivity to mean estimates

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The Effects of State Dependent Correlations (Geyer and Ziemba, 2007)

Optimal Weights Conditional on Quintiles of Portfolio Wealth at Stage 2 and 5

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Different approaches to estimating means

• Econometrics

• Factor models

• Technical analysis

• Crash models

• History

• Mean reversionJames-Stein means shrink the historic mean towards the grand meanBayes-Stein means shrink the historic mean towards the min Var portfolio

• Truncation estimatorsFoster, MacLean, Ziemba, 2002, 2005

If you get the mean right and do not overbet and truly diversify, you should win.Truly means: in ALL scenarios.

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Zenit Stockholm Multi-Billion Hedge Fund - data

Getting the right mean/hedge ratio -0.60 all in US

Two of WTZ crash models signaling downturn in 2002 tilt scenarios

If you do the most important thing right and watch your risk control, you can win big time.

They were long when models said to be long, then went short when models said to be short.

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Zenit graph: 35%/year 1996-2003

Crash models +then -

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

Do not overbet.

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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.

The recipe for disaster

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• 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.

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• 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.

Hedge Funds: Some Directional Strategies

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Long Term Capital Management

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

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The bad scenario

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

More on these strategies in section 6

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

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

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

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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.

• Better to penalize with a convex penalty for falling below the pathimplemented for a gold hedge fund in London

• The more you are below the target, the more you are penalized as in Russell Yasuda Kasai and Vienna models; see papers on this and slide 64