The Power of Volatility in Evolutionary Finance

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1 Deloitte LLP Risk & Regulation MAY 30, 2012 CASS BUSINESS SCHOOL Financial Engineering Workshop

2 Outline 1 Volatility and Growth Growth generated from Volatility Jensen s Inequality in Action 2 Log Optimal Portfolios Setting Log Optimality in Continuous time 3 Early Research The Basic Assumptions of the Model The Model 4 Preliminary Results New Results Time Varying Portfolio Strategies 5 Extend Current Research Conclusion

3 Growth generated from Volatility Jensen s Inequality in Action Trading Puzzle (Dempster et al., 2007) Question: Suppose some asset prices fluctuate randomly, forming a stationary stochastic process. Consider a fixed-mix self-financing investment strategy prescribing rebalancing one s portfolio at each time so as to keep equal investment proportions of wealth in all the assets. What will happen with the portfolio value in the long run? What will be its tendency?: to decrease; to increase; or to fluctuate randomly, converging to a stationary (statistical properties will not change over time) process

4 The Conventional thinking Growth generated from Volatility Jensen s Inequality in Action Stationary market seems to suggest that the portfolio value for a constant proportions strategy must converge to a stationary process The self-financing property seems to exclude possibilities of growth This must be the case since in the deterministic case both the security prices and the portfolio value are constant

5 The Counterintuitive Result Growth generated from Volatility Jensen s Inequality in Action 50 Simulation of Wealth Dynamics Logarithm of Wealth Asset 1 Asset 2 Asset Mixture Number of coin flips

6 An Important Theorem Growth generated from Volatility Jensen s Inequality in Action Dempster et al. (2006): When asset returns are stationary ergodic (roughly speaking: statistical properties will not change over time and can be deduced from a sufficiently long sample), their volatility, together with any fixed-mix trading strategy, generates a portfolio growth rate in excess of the individual asset growth rates E [ln (R t λ)] }{{} 1 lim t t t j=1 ln( K k=1 Rk j λk ) > K λ k E [ ln ( Rt k )] }{{} ρ k k=1 where St k = S0 krk 1... Rk t is stationary and ergodic and λ is a fixed constant proportions strategy λ = ( λ 1,..., λ i ) K

7 Volatility Induced Growth Growth generated from Volatility Jensen s Inequality in Action Therefore, even if the growth rates of the individual securities all have mean zero, the value of a fixed-mix portfolio rule is positive in the long run This growth is financially engineered by making use of Jensen s Inequality In case of stationary prices, this results is more surprising since growth seems to emerge from nothing In case of stationary returns, volatility causes the acceleration of growth

8 Volatility and Risk Growth generated from Volatility Jensen s Inequality in Action Modern portfolio theory views volatility as an obstacle to financial growth This view is shared by the majority of investors who dislike volatile markets However, some Hedge funds use mathematical techniques to capture the short-term volatility of stocks And thus, even some quite naive techniques can achieve remarkably high information ratios

9 Volatility & Statistical Arbitrage Growth generated from Volatility Jensen s Inequality in Action Equal-weighted portfolios are dynamic portfolios in which all the stocks have the same constant weights It is empirically showed that more frequent trading allows the portfolio to capture more volatility and thus more profits (Fernholz & Maguire, 2007) In the Fernholz model (Continuous time), superior diversification (related to a term called excess growth rate ) not only reduces risk, but also increases the growth rate of a portfolio

10 Volatility viewed as energy Growth generated from Volatility Jensen s Inequality in Action Analogy: It is tempting to say that Volatility is like energy!!! When constructing a mixed portfolio, it converts into growth and therefore decreases The greater the volatility reduction, the higher the growth acceleration It is possible that a mixed portfolio may have a greater volatility than each of the assets from which is constructed In general, this energy interpretation of volatility is not valid. Results on Volatility induced growth remain valid under small transactions costs

11 Optimal Growth Portfolios Log Optimal Portfolios Setting Log Optimality in Continuous time Log Optimal Strategy: long-term expected capital growth can be maximized by selecting a strategy that maximizes the expected logarithm of return at each trial For stocks, the log optimal strategy pumps money between volatile stocks by keeping fixed proportions of capital in each stock rebalancing each period The log optimal approach is mathematically more tractable in a continuous-time framework because explicit formulas can be derived for the log-optimal strategy and the resulting expected growth rate Terms such volatility pumping and excess growth have been used to describe enhanced returns due to the volatility effect

12 Optimal Growth Portfolios Log Optimal Portfolios Setting Log Optimality in Continuous time The continuous-time version reveals better how volatility pumping works: d d log S i (t) = γ i (t)dt + ξ iν (t)dw ν (t), t [0, ) ν=1 where ξ iν measures the ith stock sensitivity to the νth source of uncertainty and γ i denotes the growth rate process. Thus, d ds i (t) = α i (t)s i (t)dt + S i (t) ξ iν (t)dw ν (t) α i (t) = γ i (t) ν=1 d ξiν(t) 2, t [0, ), a.s. ν=1 } {{ } σi 2(t)

13 Optimal Growth Portfolios Log Optimal Portfolios Setting Log Optimality in Continuous time For a portfolio of n assets we have a self-financing portfolio: dv p (t) n V p (t) = w i (t) ds i(t), t [0, ) S i (t) i=1 where V p (t) represent the value of a portfolio with positive weights that sum to unity and after applying Ito s rule we can express this as n d log V p (t) = γp(t)dt + w i (t)d log S i (t), t [0, ), a.s. i=1 where γp(t) = 1 n w i (t)σ ii (t) 2 i=1 n w i (t)σ ij (t)w j (t) i,j=1

14 Optimal Growth Portfolios Log Optimal Portfolios Setting Log Optimality in Continuous time γ p(t) is called the excess growth rate process (Fernholz, 2002) and is always positive for a portfolio with no short sales and more than one stock. Then the portfolio growth rate γ p (t) will satisfy: γ p (t) = γ p(t) + n w i (t)γ i (t), t [0, ), a.s. i=1 So at any given time our portfolio will have a higher growth rate than the weighted average of the growth rates of its component stocks. Thus, superior diversification not only reduces risk, but also increases the growth rate of a portfolio.

15 Log Optimal Portfolios Setting Log Optimality in Continuous time The Portfolio of Maximum Growth Rate We can find the Portfolio of Maximum Growth Rate by maximising the following: n max w i α i 1 2 i=1 }{{} α p n w i σ ij w j i,j=1 }{{} σp 2 s.b. n w i = 1 i=1

16 Log Optimal Portfolios Setting Log Optimality in Continuous time Reducing Portfolio Risk: A Simple Example The growth rate for the portfolio level is γ p = α p 1 2 σ2 p Assume that we have n assets uncorrelated, all with same mean and variance so γ = α 1 2 σ2. Include n stocks in the portfolio with equal weight of 1 n so γ p = α 1 2n σ2 So the pumping effect reduces the effect on the volatility term increasing thus the growth rate. This effect is larger when the original variance is high.

17 Optimal Growth Portfolios Log Optimal Portfolios Setting Log Optimality in Continuous time It is well-known that diversification can lower portfolio volatility but it is less known that diversification also affects the portfolio growth rate Optimal portfolio growth can be applied with any rebalancing period a year, a month, a week or a day. In the limit of very short time periods we consider continuous rebalancing. See Stochastic Portfolio Theory models (Fernholz 2002) The Optimal Growth Portfolio implies that the relation between risk and return is quadratic rather than linear as in the CAPM model. Empirical evidence tends to support this quadratic relationship

18 Optimal Growth Strategies Log Optimal Portfolios Setting Log Optimality in Continuous time Several studies: Hakansson (1970) Thorp (1971) Algoet & Cover (1988) Karantzas & Shreve (1998) explored an approach where the determination of the growth optimal portfolio based on an exogenous return process requires the solution of a stochastic non-linear programming problem with no closed form solution in general. Maximum Growth Portfolio Problem Intractable for incomplete markets

19 Betting your Beliefs Early Research The Basic Assumptions of the Model The Model Kelly (1951): (Kelly Rule)/Criterion by maximising the expected logarithm of final wealth in every period, one would have good final wealth Breiman (1961): Betting your Beliefs proved that this strategy would give more wealth than any other strategy in the long run Hens & Schenk-Hoppé (2002) proved that if there was one Kelly rule better and a finite number of other betters all competing for the same wealth, then the Kelly Rule better not only get the most wealth but would get all of it in the end

20 Betting your Beliefs Early Research The Basic Assumptions of the Model The Model Best Strategy is to divide wealth between assets (bets) proportionally to their Expected Returns Consider Arrow Securities then outcomes zero or one (win or loose). Then the Best Strategy is to divide wealth according to the probabilities of success Equivalent of maximizing the Expected logarithm of the growth rate of returns

21 Equilibrium Considerations Early Research The Basic Assumptions of the Model The Model Questions: Does demand raise the price of attractive Bets? Do higher prices of attractive bets (assets) lower their returns? Is it possible, considering demand that less attractive bets offer superior returns?

22 Earlier Results Volatility and Growth Early Research The Basic Assumptions of the Model The Model Answer: NO Blume and Easley (1992) & (2006), Sandroni (2000): Best Strategy is Still Betting you Beliefs but their model assumed complete markets Evstigneev, et al. (2006): Betting your beliefs Divide wealth proportionally to the expected payoffs of the securities (Incomplete markets)

23 Evolutionary Portfolio Theory Early Research The Basic Assumptions of the Model The Model Evstigneev, Hens, and Schenk-Hoppe (2006) Initial idea based on Evolutionary Game Theory and Evolutionary Stable Strategy (ESS) Darwinian paradigm: Strategies fight for capital This theory has the mathematical structure of a random dynamical system and portfolio rules compete for market wealth The success of a rule is determined by the wealth share that a rule is able to collect based on a market selection process

24 Evolutionary Portfolio Theory Early Research The Basic Assumptions of the Model The Model Eventually, evolutionary pressure will determine a single surviving portfolio rule which is called the Evolutionary Portfolio Rule Evolutionary Portfolio Rule is the equivalent of ESS Evolutionary Portfolio Rule is based on fundamentals In its simple form this rule attributes wealth proportionately according to the expected relative dividends of the assets

25 The Basic Assumptions of the Model Early Research The Basic Assumptions of the Model The Model There are K long-lived assets with index k = 1, 2,..., K each paying an uncertain dividend Dt k 0 (Total Dividend) The portfolio rule of investor i is a fixed constant proportions strategy λ i = ( λ i 0, λi 1,..., ) λi K with λ i k [0, 1] for all k = 1, 2,..., K and K k=0 λi k = 1 Investment strategies are distinct across investors for i = 1,..., I Here λ i 0 is the fraction of wealth that each investors holds on cash (Assumed the same for all investment strategies) θt,k i denotes the units of assets held by investor i, w t i is the wealth of the investor i and pt k denotes the market clearing price of asset k in period t

26 The Basic Assumptions of the Model Early Research The Basic Assumptions of the Model The Model The price for cash is normalized so p 0 t = 1 The dividend depends on the possible states of nature that are determined in each period in time by the realization of a stationary stochastic process Histories ω t = (..., ω 1, ω 0, ω 1,...) where ω t S is the state revealed at beginning of period t and S is a finite set Every state is drawn with some strictly positive probability The number of stocks that one holds is θt,k i = λi k w t i pt k Demand is equal to supply so I i=1 θi t,k = sk t, k = 0, 1,..., K

27 The Model Volatility and Growth Early Research The Basic Assumptions of the Model The Model Supply is exogenous in this model and is normalized to one. Therefore, I i=1 θi t,k = 1 or I i=1 = 1 so λ i k w i t p k t p k t = I i=1 λi k w i t Lucas Framework: w i t+1 = K k=1 ( D k t+1 ( ω t+1 ) + p k t+1) θ i t,k Total dividend of the economy is D t K k=1 Dk t and the total wealth of the economy is W t I i=1 w i t Total consumption of the economy equals to the total dividend in order so λ 0 W t = D t or W t = Dt λ 0 Define investor s individual return as r i t = w i t W t relative dividend as d k t+1 = Dk t+1 D t+1 and asset k s

28 The Model Volatility and Growth Early Research The Basic Assumptions of the Model The Model Market Selection [ ( Mechanism: λ 0 dt+1 k r i t+1 = K k=1 ( ω t+1 ) + I i=1 λi k r i t+1 Vector Notation: λ k = ( λ 1 k, λ2 k,..., k) λi, rt+1 T = ( rt+1 1, r t+1 2,..., r t+1) I and Λ T = ( λ T 1, λt 2,..., K) λt R I K Solution: r t+1 = λ 0 [I [ λ i k r i t λ k r t ] k i Assumption ) λ i k r i t I i=1 λi k r i t ] 1 [ K Λ k=1 d t+1 k ( ω t+1 ) λ i k r t i λ k r t ]i I.i.d. dividend payments d k t (ω t ) = d k (ω t ), for all k = 1, 2,....K and the state of nature ω t follows an i.i.d. process. ]

29 The Model Volatility and Growth Early Research The Basic Assumptions of the Model The Model We consider the linearization of the local dynamics at the fixed points for only two rules h(ω t+1,rt 1 ) = 1 λ rt λ K 0 k=1 d t+1 k (ω t+1) λ2 k λ 1 k r 1 t =1 The exponential growth rate is ( g λ 1 λ 2 ) ˆ [ K ] = ln 1 λ 0 + λ 0 dt+1 k (s) λ2 k S λ 1 dp k k=1 In more general form ( g λ j λ i ) [ ( )] = E ln 1 λ 0 + λ K 0 k=1 d t+1 k (s) λi k λ j k

30 The Model Volatility and Growth Early Research The Basic Assumptions of the Model The Model ( If the growth rate is negative, g λ j λ i ) < 0, portfolio rule λ i loses market share while λ j s market share tends to one. Therefore, portfolio rule λ j is considered stable ( against λ i. On the contrary, if the growth rate is positive g λ j λ i ) > 0, portfolio rule λ i gains market share while λ j s market share falls. Thus, portfolio rule λ j is considered unstable against λ i.

31 The Main Result Volatility and Growth Early Research The Basic Assumptions of the Model The Model Theorem The portfolio rule λ, defined by λ k = (1 λ 0 ) Ed k (s), k = 1, 2,..., K is the only investment strategy that is evolutionary stable against any other portfolio rule. That is, g λ (λ) < 0 and g λ (λ ) > 0 for all λ λ.

32 An Important Corollary and Remark Early Research The Basic Assumptions of the Model The Model The following Corollary is important for the choice in the participating set of strategies: Corollary Suppose some incumbent rule λ j with λ j k = 0 for some asset k has conquered the market. Then any portfolio rule λ i with λ i k > 0 for all k ( grows against λ j, i.e. g ) λ j λ i > 0. In fact the growth rate is arbitrarily large. The following remark shows the relation between the evolutionary portfolio rule and the betting your beliefs strategy: Remark The evolutionary portfolio rule λ k = (1 λ 0) Ed k (s) reduces to the betting your beliefs strategy, i.e. λ s = p s when assets are arrow securities.

33 Participating Set of Strategies Preliminary Results New Results Time Varying Portfolio Strategies Evolutionary Portfolio Rule: λ k = (1 λ 0) Ed k (s) Illusionary Rule: λ illu k = (1 λ 0) K Risk Lover: λ RL k = (1 λ 0 ) σd k Risk Averter: λ RA k = (1 λ 0 ) ( 1 σ dk ) Notice: All Strategies above are well diversified, i.e. No zero weight for any stock Also in some simulations we use some Noise Strategies, i.e. strategies that they are not based in any specific rule

34 Preliminary Results New Results Time Varying Portfolio Strategies with Simple Random Generated Dividends ,000 Evolutionary illusionary Noise1 Noise2 Noise3 These results are typical of the existing results in the literature However, here dividends are stationary Relative prices are smooth-no Volatility

35 with New Strategies Preliminary Results New Results Time Varying Portfolio Strategies ,000 Evolutionary illusionary Noise1 RiskLover RiskAverter Noise2 Noise ,000 StockPrices Although λ is doing relatively well from the first year, λ RL is doing even better for the first 1,000 time periods! Relative prices are not very smooth

36 Preliminary Results New Results Time Varying Portfolio Strategies Introduction of Nonconstant Strategies λ Smean t τ=1 βt τ d k (s τ ) with β = 0.95 can challenge λ when dividends are specifically calibrated Notice the performance of the nonconstant Momentum strategy which follows a relatively steady path Volatility is dramatically increased by these adaptive strategies

37 Future Directions Volatility and Growth Extend Current Research Conclusion A new evolution of wealth function could be developed in order to theoretically justify a large variety of fixed-mix rules Introduce external growth and study its effect on the wealth dynamics Develop an evolutionary framework where wealth dynamics would be explored in high trading frequencies

38 Conclusions Volatility and Growth Extend Current Research Conclusion Conclusions about multiperiod investment situations are not mere variations of single-period conclusions rather they often reverse those earlier conclusions Conclusions such as volatility is bad or diversification can be achieved with a portfolio of 30 stocks are no universal truths. The story is much more interesting Volatility is not the same as risk. Volatility is an opportunity Evolutionary Portfolio Rules provides a good strategies for Pension/Insurance Funds We have discovered some new fixed-mix strategies that could be profitable for some more risk aggressive investors (i.e. Hedge Funds)

39 Extend Current Research Conclusion P.H. Algoet and T.M. Cover. Asymptotic Optimality and Asymptotic Equipartition Properties of Log-Optimum Investment, Annals of Probability, 16: , L. Blume and D. Easley. Evolution and Market Behavior, Journal of Economic Theory, 58:9-40, L. Blume and D. Easley. If you are smart why aren t you rich? Belief Selection in Complete and Incomplete Markets, Econometrica, 74: , L. Breiman. Optimal Gambling Systems for Favorable Games. Fourth Berkeley Symposium on Mathematical Statistics and Probability, 1:65-78, M. A. H. Dempster, I. V. Evstigneev and K. R. Schenk-Hoppé. Volatility-induced financial growth, Quantitative Finance 7(2): , 2007.

40 Extend Current Research Conclusion I.V. Evstigneev, T. Hens, and K.R. Schenk-Hoppé. Evolutionary finance, in Handbook of Financial Markets: Dynamics and Evolution, T. Hens and K. R. Schenk-Hoppé, eds. North-Holland, I.V. Evstigneev, T. Hens, and K.R. Schenk-Hoppé. Evolutionary Stable Stock Markets, Economic Theory, 27: , R.Fernholz. Stochastic Portfolio Theory. Springer-Verlag, 2002 R. Fernholz and C. Maguire. The statistics of statistical arbitrage in stock markets, Financial Analysts Journal 63(5):46 52, 2007.

41 Extend Current Research Conclusion N. Hakansson. Optimal Investment and Consumption Strategies Under Risk for a Class of Utility Functions, Econometrica, 38: , T. Hens, K.R. Schenk-Hoppé and M. Stalder. An Application of to Firms Listed in the Swiss Market Index. Swiss Journal of Economics and Statistics, 138: , T. Hens, K.R. Schenk-Hoppé. Survival of the Fittest on Wall Street, In: Proceedings of the VI. Buchenbach Workshop, Institutioneller Wandel, Marktprozesse und dynamische Wirtschaftspolitik, M. Lehmann-Waffenschmidt et al., (eds) Metropolis- Verlag, Marburg, , I. Karatzas and S. Shreve. Methods of Mathematical Finance. Springer-Verlag, New York, 1998.

42 Extend Current Research Conclusion J.L. Kelly. A New Interpretation of Information Rate, Bell System Technical Journal, 35: , D. G. Luenberger. Investment Science, Oxford University Press, R. Lucas. Asset Prices in an Exchange Economy, Econometrica, 46: , K. Mavroudis and Craig A. Nolder. Fixed-Mix Rules in an Evolutionary Market Using a Factor Model for Dividends, International Journal of Theoretical and Applied Finance,14(8): , A. Sandroni. Do Markets Favor Agents Able to make Accurate Predictions, Econometrica, 68: , 2000.

43 Extend Current Research Conclusion R. Shiller. Do Stock Prices Move Too Much to be Justified by Subsequent Changes in Dividends? The American Economic Review, 71(3): , E.O. Thorp, Portfolio Choice and the Kelly Criterion, In: Ziemba, W.T., Vickson, R.G., (eds) Stochastic Models in Finance, , New York: Academic Press, 1971.

44 Extend Current Research Conclusion THANK YOU!