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Scenario Based Stochastic Programming Thinking for Asset Liability Problems Dr William T. Ziemba Alumni Professor of Financial Modeling and Stochastic Optimization University of British Columbia, Vancouver, Canada University of Zurich June 2003. Introduction.
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Scenario Based Stochastic Programming Thinking for Asset Liability Problems Dr William T. Ziemba Alumni Professor of Financial Modeling and Stochastic Optimization University of British Columbia, Vancouver, Canada University of Zurich June 2003
Introduction • All individuals and institutions regularly face asset liability decision making. • I will discuss an approach to model such decisions for pension funds, insurance companies, individuals, retirement, bank trading departments, hedge funds, etc. • It includes the essential problem elements: uncertainties, constraints, risks, transactions costs, liquidity, and preferences over time, to provide good results in normal times and avoid or limit disaster when extreme scenarios occur. • Strategic asset allocation is known to be the primary determinant of portfolio performance, namely, the correct balance of cash, stocks and bonds. • The stochastic programming approach while complex is a practical way to include key problem elements that other approaches are not able to model. • Other approaches (static mean variance, fixed mix, stochastic control, capital growth, continuous time finance etc.) are useful for the micro analysis of decisions and the SP approach is useful for the aggregated macro (overall) analysis of relevant decisions and activities.
Intro cont’d • Other approaches will yield good results most of the time but they frequently lead to the recipe for disaster: over-betting and not being truly diversified at a time when an extreme scenario occurs. • With derivative trading positions are changing constantly, and a non-overbet situation can become overbet very quickly. • The uncertainty of the random return and other parameters is modeled using discrete probability scenarios that approximate the true probability distributions. • The accuracy of the actual scenarios chosen and their probabilities contributes greatly to model success.
Intro cont’d • However, the scenario approach generally leads to superior investment performance even if there are errors in the estimations of both the actual scenario outcomes and their probabilities. • It is not possible to include all scenarios or even some that may actually occur. The modeling effort attempts to cover well the range of possible future evolution of the economic environment. • The predominant view is that such models do not exist, are impossible to successfully implement or they are prohibitively expensive. • I argue that give modern computer power, better large scale stochastic linear programming codes, and better modeling skills that such models can be widely used in many applications and are very cost effective.
Intro cont’d For academic reference:W T Ziemba and J M Mulvey, eds, Worldwide Asset and Liability Modeling, Cambridge University Press, 1998 + articles which is being updated in the Handbook of Asset Liability Management, Handbooks in Finance Series, North Holland edited by S. A. Zenios and W. T. Ziemba, forthcomng, 2004. For an MBA level practical tour of the areaW T Ziemba, The Stochastic Programming Approach to Asset and Liability Management, AIMR, in progress, out in 6 months. If you want to learn how to make and solve stochastic programming modelsS.W. Wallace and W.T. Ziemba, eds, Applications of Stochastic Programming, MPS SIAM, out in 6 months
“Most people still spend more time planning for their vacation than for their retirement” Citigroup “Half of the investors who hold company stock in their retirement accounts thought it carried the same or less risk than money market funds” Boston Research Group
The Pension Fund Situation • The stock market decline of 2000-3 has been very hard on pension funds in several ways: • If defined benefits then shortfalls • General Motors at start of 2002 • Obligations $76.4B • Assets 67.3B shortfall = $9.1B • Despite $2B in 2002, shortfall is larger now • Ford underfunding $6.5B Sept 30, 2002 • If defined contribution, image and employee morale problems • Worldwide shortfall of $2.5 trillion, Feb 2003
The Pension Fund Situation in Europe • Rapid ageing of the developed world’s populations - the retiree group, those 65 and older, will roughly double from about 20% to about 40% of compared to the worker group, those 15-64 • Better living conditions, more effective medical systems, a decline in fertility rates and low immigration into the Western world contribute to this ageing phenomenon. • By 2030 two workers will have to support each pensioner compared with four now. • Contribution rates will rise • Rules to make pensions less desirable will be made • UK discussing moving retirement age from 65 to 70 • Professors/teachers pension fund 16% underfunded
Key Points of the Lecture • Importance of means • Vulnerability from overbetting • Importance of true diversification • Role of other approaches • Correlations may change with a crisis • SP helps manage complexity • Scenarios are market forecasts, don’t worry about getting them exactly right • Role of computer technology • SP models in the future
Mean Percentage Cash Equivalent Loss Due to Errors in Inputs Conclusion: spend your money getting good mean estimates and use historical variances and covariances
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
Factor Model Best model uses 30 factors. All contribute to predictability.
Ranking of countries for mean returns One year return based on short term momentum, long-run mean reversion, and value. Source: Arrowstreet Capital (John Campbell) Model +b1(return)-1 - b2(10-year cumulative return) + b3(dividend yield)-1 Worst - Finland Poor- US, Japan Best - Asia ex Japan, including New Zealand, Australia
Points to Remember Point 2 Trouble arises when one overbets and a bad scenario occurs You must not overbet when there is any possibility of a bad scenario occurring
We are living in a dangerous, fat-tailed world! Extreme events are way underestimated by people and therefore by the models they build Look at this data on earthquake damage in California :
Data on earthquake damage • In this data some years have zero damage, some have five, etc. The highest is 129. • The question is how much earthquake damage occurred in California in the next year? Can you forecast the 1994 value?
The value was 2272.2!
Extreme events can occur that are beyond the range of all previous events. There may have been earthquakes in California 400 years ago that were bigger than Northridge's (greater Los Angeles) in 1994 but there were few people and buildings there then. So they could not destroy much • What we have is an outcome way beyond the range of all past data. Thirty-two insurance companies in the US declared bankruptcy in 1998 and 2001 was another difficult year post September 11 and 2002 is also terrible.
Events that were not supposed to happen in 1998 May 18 Indonesia's rupiah collapses, to 17,000 to the US dollar. Aug 17 Russia defaults on some debt; ruble collapses. Aug 31 The Dow plunges 512.61 points or 6.37% (on -1 day, strongest trading day of the month). July-Sept US banks suffer worst derivatives losses ever $445 million. Sept 24 Hedge fund Long-Term Capital Management is bailed out with $3.6 billion. Sept 27 Japan Leasing files for bankruptcy with $17.9 billion in liabilities; biggest financial failure since World War ll. Oct 5 30-year US treasury yields hits record 4.74% low. Oct 7 The US dollar plunges 7.8% against the yen, largest one-day loss in 12 years. Oct 8 China's yuan soars to an all-time high of 8.2777 to the US dollar. Oct 9: Japan's Nikkei index sinks to 11,542, lowest since 1984. Oct 13: London's FTSE-100 index soars a record 214.2 points. Nov 2 The US savings rate sinks to 0.2% Nov. 5 Some leading Western banks cut yen deposit rates to negative values. Nov. 11 Shares of theglobe.com skyrocket more than tenfold in first day of trading. Nov 30 US mortgage rates fall to 6.64%, the lowest since 1967. Dec 3 11 European countries cut interest rates simultaneously Dec 10 World oil prices slide below $10 a barrel, the lowest since 1986.
Cumulative probabilities of S&P500 returns Key: the probability distribution on day t given what is known up to day t- 1 1987 Crash 10-42 versus 0.4 -22% Oct 19 Stable distribution, Longin J. Bus 96 stock prices, 105 years data, Frechet Distribution in tail F(y(y)=exp(-y*) Source: Jackwert and Rubinstein, 1997
Asset structure of European Pension Funds in Percent, 1997 * European Federation for Retirement Provision (EFRP) (1996)
The trend is up but its quite bumpy. There have been three periods in the US markets where equities had essentially had essentially zero gains in nominal terms, 1899 to 1919, 1929 to 1954 and 1964 to 1981
Points to Remember cont’d Point 3 Trouble is exacerbated when the diversification does not hold in the scenario that occurs You must use scenario dependent correlation matrices.
Long Term Capital cont’d • Lessons: • must not overbet, it is too dangerous • must be aware of and consider extreme scenarios • must allow for extra illiquidity and contract defaults • must really diversify (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 (ij 1). Real correlations are scenario dependent
Some possible approaches to model situations with such events • Simulation too much output to understand but very useful as check • Mean Variance ok for one period but with constraints, etc • Expected Log very risky strategies that do not diversify well • fractional Kelly with downside constraints are excellent for risky investment betting • Stochastic Control bang-bang policies Brennan-Schwartz paper in ZMhow to constrain to be practical? • Stochastic Programming/Stochastic Control Mulvey does this with Decision Rules (eg Fixed Mix) • Stochastic Programming a very good approach • For a comparison of all these, see Introduction in ZM
Points to Remember cont’d • Point 4 • Other approaches, continuous time finance, capital growth theory, decision rule based SP, control theory, etc are useful for problem insights and theoretical results. • But in actual use, they may lead to disaster unless modified. • BS theory says you can hedge perfectly with LN assets and this can lead to overbetting. • But fat tails and jumps arise frequently and can occur without warning. The S&P opened limit down –60 or 6% when trading resumed after Sept 11 and it fell 14% that week • . • Be careful of the assumptions, including implicit ones, of theoretical models. Use the results with caution no matter how complex and elegant the math or how smart the author. • Remember you have to be very smart to lose millions and even smarter to lose billions.
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.
Points to Remember cont’d Point 5 When there is trouble in the stock market, the positive correlation between stocks and bond fails and they become negatively correlated When the mean of the stock market is negative, bonds are most attractive as is cash.
Between 1982 and 1999 the return of equities over bonds was more than 10% per year in EU countries During 2000 to 2002 bonds greatly outperformed equities
