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The Evolution of Portfolio Rules and the Capital Asset Pricing Model

The Evolution of Portfolio Rules and the Capital Asset Pricing Model. Emanuela Sciubba. 0. Abstract 1. Introduction 2. The Model 2.1 The Dynamics of Wealth Shares 2.2 Types of Traders 3. Dynamics with Traders who Believe in CAPM 3.1 Trivial Cases 3.1.1 No Aggregate

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The Evolution of Portfolio Rules and the Capital Asset Pricing Model

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  1. The Evolution of Portfolio Rules and the Capital Asset Pricing Model Emanuela Sciubba

  2. 0. Abstract 1. Introduction 2. The Model 2.1 The Dynamics of Wealth Shares 2.2 Types of Traders 3. Dynamics with Traders who Believe in CAPM 3.1 Trivial Cases 3.1.1 No Aggregate 3.1.2 Constant Absolute Risk Aversion 3.2 Existence of Equilibrium 3.3 The main Result 3.4 Extensions 4. Genuine Mean-Variance Behavior 5. Concluding Remarks

  3. Abstract • The aim : test the performance of the standard • version of CAPMin an evolution framework . • Prove : traders who either “believe”in CAPM and use it as a rule of thumb ,or are endowed with genuine mean-variance preferences ,under some very weak condition ,vanish in the long run . • A sufficient condition to drive CAPM or mean variance traders’ wealth shares to zero is that an investor endowed with a logarithmic utility function enters the market .

  4. 1. Introduction • 1.1 Motivation • Imagine a heterogeneous population of long-lived agents • who invest according to different portfolio rules and ask • what is the asymptotic market share of those who happen • to behave as prescribed by CAPM . • The result proves : • 1.CAPM is not robust in an evolution sense • 2.it triggers once again the debate on the normative appeal • and descriptive appeal of logarithmic utility approach as • opposed to mean-variance approach in finance .

  5. The debate originates from the dissatisfaction with the mean- • variance approach which fails to single out a unique optimal • portfolio . • Kelly criterion :That a rational long run investor should • maximise the expected growth rate of his wealth share and • should behave as if he were endowed with a logarithmic • utility function . • The evolutionary framework adapted in this paper suggests • that maximising a logarithmic utility function might not make • you happy ,but will definitely keep you alive

  6. 1.2 Related Literature • Debate on bounded rationality in economics and find • motivation in the simple idea that individuals “may be • irrational and yet markets quite rational “ • Becker (1962) and numerous studies • Evolutionary model of an industry • Luo (1995) • Noise trading • Shefrin and Statman (1994) • De long et al. (1990,1991) • Biais and Shadur(1994)

  7. Blume and Easley (1992,1993) • :in the long run ,traders who are endowed with a logarithmic • utility function will survive ,as well as successful imitators . • Cannot directly apply Blume and Easley results: • Two major reasons : • 1 .Blume and Easley’s result on logarithmic traders’dominance • do not necessarily imply that CAPM traders would vanish . • 2 .both CAPM and mean-variance trading rules do not satisfy • a crucial boundedness assumption which Blume and Easley • impose .

  8. 2. The Model • Time is discrete : t • There are S states of the world : s • States follow an i.i.d process with distribution • Let denote the product σ-field on Ω • denote the sub-σ-field σ(ωt) of .

  9. wst :total wealth in the economy at time t if state s occurs . • :the price of asset s at date t . • :denotes his demand of asset s at time t . • αsti :the fraction of trader i’swealth at the beginning of t , • that he invests in asset s . (1) (2)

  10. and (1) (3) (4)

  11. In equilibrium ,prices must be such that markets clear , • i.e. total demand equals total supply (5) (6) (4) • Market prices are related to wealth shares .

  12. 2.1 The Dynamics of Wealth Shares • Trader i’s wealth share (8) • Market saving rate (9) (10)

  13. (11) • Using our price normalisation : (12) • Trader i’s wealth share will increase if he scores a payoff which is high than the average population payoff . • The fittest behaviour is that which maximises the expected • growth rate of wealth share accumulation . • is a weighted average across traders of , where weight are given by wealth shares at the beginning of period t . (15)

  14. Define a formal notion of “dominance”

  15. Blume and Easley justify the word “dominates” as follows: • “ When saving rates are identical a trader who dominates • actually determines the price asymptotically . • His wealth share need not converge to one because • there may be other traders who asymptotically have • the same portfolio rule ,but prices adjust • so that his conditional expected gains converge to zero ” • Assumption 1 For all t and all i , • and • Assumption 2 There exists a real number • such that ,for all ifor all s .

  16. (12) :the indicator function that is equal to 1 if state s occurs at date t and equal 0 to otherwise . • The expected values of conditional on the information • available at time t-1 : (13) (14)

  17. Intuitions : • 1 .the dominating traders are those who are better than the others • in maxinising the expected growth rate of their wealth shares . • 2 .condition (c) implies that conditions (b) and a fortiori(a) fail . • condition (c) puts a restriction on the rate at which • diverge . • 3 .if all traders have the same rate ,the dominating trader • determines market prices asymptotically and his wealth share • need not converge to 1 because there might be other suriving • traders .

  18. Proof :Under simplifying assumption :all traders have • identical savings rates .

  19. 2.2 Types of Traders • Three different types of traders :Type CAPM,Type L,Type MV • First type : Agents who believe in CAPM(Type CAPM)

  20. Second type: Agents who are endowed with a logarithmic utility function(Type L) and who maximise the growth rate of their wealth share and invest according to a “simple” portfolio rule : (22) • More generally ,a rational trader i will choose • so as to maximise : • subject to the constraint that investment expenditure at each date is less than or equal to the amount of wealth saved in the pervious period . • If is logarithmic ,it follows that • and that (23) (1)

  21. Third type : Agents who display a genuine mean-variance behavior (Type MV) and are endowed with a quadratic utility function : where (24) • Substituting (24)into (23) and solving for using the first • order conditions ,we obtain : • where: (25) (26) is the wealth share of mean-variance traders at date t

  22. According to (1),(4),(8)and (25) : (27) • If for some s ,then both and • so that theorem 1 in section 2.1 does not apply . (19)

  23. 3. Dynamics with Traders who Believe in CAPM • Assumption : • Only two types of traders in the economy : 1.believe inCAPM 2. Logarithmic utility function(MEL traders) • is the quantity (share) of each asset s that trader i demands at time t . • is the share of aggregate wealth which belong to type L • is the share of aggregate wealth which belong to type CAPM at the beginning of period t . • The degree of risk aversion is homogeneous in the population of traders who believe in CAPM ,so that and

  24. 3.1 Trivial Cases3.1.1 No Aggregate Risk • Remark 1 With no aggregate risk ,in a population of traders • who believe in CAPM and traders with logarithmic utility • function ,the behavior of traders who believe in CAPM and • traders with a logarithmic utility function coincide . • Formally ,if then • Intuition :Because market and risk-free portfolio coincide , • traders who believe in CAPM invest only according to the • market portfolio ,so that their behaviour is purely imitative . • As a result ,when a logarithmic utility maximiser enters the • economy ,everyone invests according to his portfolio rule .

  25. 3.1.2 Constant Absolute Risk Aversion • All investors are risk averse and that the degree of risk aversion • does not change with wealth i.e.constant absolute risk aversion . • Remark 2 under the CARA assumption ,in a population of • traders who believe in CAPM and traders with logarithmic utility • function .if the behaviour of traders who believe • in CAPM and traders with a logarithmic utility function • coincides .i.e.

  26. 3.2 Existence of Equilibrium • Two types of traders : 1. believe in CAPM and • 2. endowed with a logarithmic utility function • Traders’ demands are : (28) (29) • There is only unit available of each asset : (30) (31)

  27. Definition 3 Market clearing equilibrium at date t for for this economy is an array of portfolios and assets’ prices such that ,

  28. Proposition 2 Provided that ,at each date • there exists a unique market clearing equilibrium . (31) • A corollary of equation 31 :if all traders behave according to • CAPM rule that there is no market clearing equilibrium . • Intuition : in such an economy (CAPM) every trader would • like to invest his whole wealth in the risk-free portfolio . • However ,as long as there is aggregate uncertainty ,for an • equilibrium to exist some traders must bear the risk . • A unique equilibrium exists in an economy populated only by • traders who are endowed with a logarithmic utility function . • Equilibrium prices are equal to probabilities : • (Substituting into(31) )

  29. Characterise the limiting behavior of prices as • equilibrium prices move towards a vertex of of the price • simplex .Only the market of asset 1(the asset with the lowest • payout) clears with a strictly positive price . • Proposition 3 When while • In compact notation:

  30. (pf)In the limit ,non-negativity of prices requires while market clearing requires The unique limiting value for that satisfies both is : (32) Implies • Consequence of proposition 3 : that portfolio weights of traders • who believe in CAPM are not bounded away from zero on those • sample paths where So theorem 1 does not apply .In particular,we can not use it to show that log traders dominate, since we would need to assume their dominance( ) in order to apply the theorem.

  31. Corollary 4 according to (28)(29)(31)(32) • Notice that ,so that there is • market clearing • Both types of traders invests ; • only CAPM traders invest in asset 1 .

  32. 3.3 The Main Result (1) • In this section we prove our results under a simplifying assumption: Assumption 3: • We present our first two main results as separate propositions which accords with Blume and Easley (1992): -Proposition 5:Under assumption 1 and 3, in a population of traders who believe in CAPM and traders who are endowed with a logarithmic utility function, the latter dominate almost surely. Formally: (pf steps) converge almost surely to

  33. The Main Result (2) -Proposition 6:Under assumption 1 and 3, in a population of traders who believe in CAPM and traders who are endowed with a logarithmic utility function, the latter dominate almost surely,so that, (Note)MEL dominate Because it is possible that and yet • Extinction of traders who believe in CAPM is the last main result, and one could not directly anticipate that through Blume and Easley’s theorem 1.In fact, We have examined two trivial cases as examples that traders who believe in CAPM survive because they behave as MEL. To prove this result,we need to make a further assumption on traders’ behavior towards risk.

  34. The Main Result (3) Assumption 4:The portion of wealth that traders who believe in CAPM decide to invest in the risk free portfolio, ,is a monotonic function of their level of wealth, -Proposition 7:Under assumption 1, 3 and 4 and in presence of aggregate uncertainty, in a population of traders who believe in CAPM and traders who are endowed with a logarithmic utility function, the former vanish almost surely. (Intuitive Proof)Dominance of MEL requires that in the long run all surviving traders invest according to the Kelly criterion.We prove that the CAPM rule does not succeed in fully imitating the behavior of MEL traders.We find that the market portfolio weights converge to probabilities,but risk-free portfolio do not if there is aggregate uncertainty.And under assumption 4, there is no sample path for such that CAPM traders asymptotically invest only according to the market portfolio.

  35. 3.4 Extensions In this section, our aim is to check the robustness of our main results in three more general settings: • A Multipopulation Model • Heterogeneous Risk Attitudes • Traders with Different Savings Rates

  36. A Multipopulation Model (1) • Consider a population of traders who believe in CAPM, and suppose a MEL trader enters the market with N other types of traders with portfolio rules and n=1,…N. • For simplicity we also assume that:

  37. A Multipopulation Model (2) • Assumption 5 allows us to apply corollary 4.1 in Blume and Easley (1992). • Assumption 6 is without loss of generality: even if all the results in this section would still apply by proposition 5, 6 and 7. • It is possible to show that, provided that , then a market clearing equilibrium exists at each date.In particular,as ,equilibrium prices for some s and therefore for some s, so that, despite ass.5, theorem 1 is not applicable.

  38. A Multipopulation Model (3) • Proposition 8:Under assumptions 1,3 and 5,given a population of traders who believe in CAPM, suppose that a trader with log utility function and N other traders with portfolio rules andn=1,…N, enter the market.Traders endowed with a log utility function will dominate almost surely and determine asset prices asymptotically. (Pf Steps)We first show that log utility maximizers outperform each of the N new types of traders.We then prove that LOG traders dominate by similar arguments to those used for proposition 5.

  39. A Multipopulation Model (4) • Let be the limiting values of respectively, as t→∞. • Proposition 9:Under assumptions 1,3,4,5,and 6, given a population of traders who believe in CAPM, suppose that a trader with log utility function and N other traders enter the market.Unless the evolution of the system is such that, : (36) Traders who believe in CAPM vanish.( a.s.)

  40. A Multipopulation Model (5) • Condition (36)can also be express as follows: What (36) requires is that the N new rules should complement CAPM behavior so that we could think of them as of a single trader whose portfolio rules are asymptotically equal to probabilities.As a result, even no traders asymptotically behaves as a log utility maximizer, all traders survive. • This condition is severe,so we claim that extinction of CAPM believer is “generic”.Survival of CAPM traders is not robust to small change to the set of the new N types of traders introduced in the market.

  41. Heterogeneous Risk Attitudes (1) • In this section, we show that our results are robust when allowing for heterogeneity in the degree of risk aversion among CAPM traders. • In fact, we can deal with heterogeneity thinking of a population of traders endowed with different degrees of risk aversion as of a single “average” trader whose portfolio rules are given by an appropriate weighted average of each trader’s portfolio rules.

  42. Heterogeneous Risk Attitudes (2) • Consider a population of CAPM traders, indexed by ;trader j’s portfolio rules at t will be , and assumption 4 holds for each j. • Denote by and the wealth shares of MEL traders and of CAPM trader j, respectively. • Proposition 10:Under assumption 1,3 and 4, log utility maximizers dominate and drive to extinction a population of heterogeneous traders who believe in CAPM.Formally,

  43. Heterogeneous Risk Attitudes (3) (pf steps) We first show that log utility maximizers dominate in a world of aggregate uncertainty. Again, an immediate corollary of this result is that price converge to probabilities. Finally, assuming that is a monotonic function of wealth is a sufficient condition for all CAPM traders to vanish.

  44. Traders with Different Saving Rates (1) • If saving rates are different across traders, by theorem 1, trader i dominates on those sample paths where: So, the market selects for most patient investors, i.e., those whose savings rate is larger w.r.t. the average . • Obviously, if , the MEL traders will dominate and drive CAPM traders to extinction.

  45. Traders with Different Saving Rates (2) • Proposition 11:Under assumptions 1&4, in a population of traders who believe in CAPM and of log utility maximizers, the latter dominate, provided that their savings rate is at least as large as the average savings rate, and drive to extinction the population of traders who believe in CAPM.Formally,if then, • However,by assuming that,we ignore the fact that MEL traders have a “comparative advantage”, so we will prove their dominance under a weaker assumption.

  46. Traders with Different Saving Rates (3) • Proposition 12:Under assumptions 1 and 4, in a population of traders who believe in CAPM and traders with a log utility function, the latter dominate and drive CAPM traders to extinction if a.s. • This condition is weaker than Namely: , while the converse is not true. It is not the weakest one could impose; however, it shows that in Blume and Easley (1992) can be relaxed.

  47. 4. Genuine Mean-Variance Behavior • Traders who believe in CAPM do not display a genuine mean-variance behavior: they know what the two-fund separation theorem prescribes, believe it works in reality and only partially optimize between the risk-free and market portfolios. • In this section, we show that, in an evolutionary framework, traders with mean-variance preferences will not do any better than traders who believe in CAPM. 4.1 Existence of Equilibrium 4.2 The Evolution of Wealth Shares

  48. 4.1 Existence of Equilibrium (1) • Suppose that there are two types of rational traders in the market:traders who are endowed with a quadratic utility function(and display a genuine mean-variance behavior)and traders who are endowed with a log utility function. • From an analytical point of view, the equilibrium existence problem in this setting is equivalent to the general equilibrium problem in a pure exchange economy.

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