A Numerical Study on Portfolio Optimization:

Is CAPM Fit for Nasdaq?

Guido Caldarelli

Marina Piccioni

Emanuela Sciubba

Presented by Chung-Ching Tai

Contents

MotivationA Grab for CAPMThe ModelComputer SimulationNumerical ResultsConclusionsDiscussions

Motivation

The DebateThe IssueThe Main Goals of this Research

The Debate

The mean variance approach sets a standard in financial economics, and its main corollary in asset pricing. (CAPM)Some argued that a rational investor with a long time horizon should maximize the expected rate of growth of the wealth. This type of behavior is equivalent to that of maximizing a logarithmic utility function. (the Kelly criterion)

The Debate

Central to the debate is whether maximizing a logarithmic utility function is a “more rational ” objective to follow for a trader with a long time horizon.

The Issue

Blume and Easley (1992) show that if all traders save at the same rate and under some uniform boundedness conditions on portfolio, then there exists one globally fittest portfolio rule which is prescribed by logarithmic utility maximization.Namely, if there is a logarithmic utility maximizer in the economy, he will dominate, determine asset prices asymptotically and drive to extinction any other trader that does not behave as a logarithmic utility maximizer.

The Issue

Sandroni (1999) show that provided that agents’ utilities satisfy Inada conditions, then all traders survive. A rational trader can avoid extinction by suitably modifying his investment intensity.Mean-variance preferences do not satisfy Inada conditions, and do not necessarily uniform boundedness properties !

Sciubba (1999) has shown that under the same saving rate, logarithmic traders will dominate and drive to extinction those with mean-variance preferences (or use their theoretical predictions ‘CAPM’ as a rule of thumb).How about heterogeneous saving rates?

The Main Goals of this Research

We believe that a useful contribution to this debate comes from the adoption of an evolutionary technique.We aims at studying long run financial market outcomes as the result of a process akin to natural selection.Two points: Dominance ? the threshold in the saving rate

A Grab for CAPM

Mean-variance preference:

Asset Portfolio & Efficient Frontier:

N ∞ :

Var(M) = Cov

Riskless and Risky Assets:

Market portfolio

Beta:

CAPM:

The Model

Basic AssumptionsThe DynamicsTypes of Traders

Basic Assumptions

Discrete time: t =1, 2, … States of the world: s =1, 2, … SStates follow an i.i.d. process with distribution p =(p1, p2, … pS) where ps >0 sA finite set of assets, asset s{1, 2, … S } pays a positive dividend ds>0 when state s{1, 2, … ,S } occurs and 0 otherwiseAt each date there is only 1 unit of each asset available.

Wealth will be distributed among traders proportionately according to the share of asset s that each of them ownsst is the market price of asset s at date t

A heterogeneous population of long-lived agents, indexed by i {1, 2, … I }

Agents

Agent i can be described as a triple:

The total amount invested by agent i at date t:

Asset Prices

Prices must be such that markets clear:

Price normalization:

investment share of agent i at date t

The Dynamics

Agents’ investment shares changes period to period dynamically.If state s occurs at date t, total wealth in the economy, ds, gets distributed to traders according to the share of asset s that each of them owns.

Agent’s endowment at date t+1:

( )Market average investment rate:

Agent’s investment in period t+1:

The new investment share:

the period to period dynamics of the investment share of trader i

To see whether a specific trader survives or vanishes, just consider the asymptotic value of his investment share to check if it is bounded away from zero.

Types of Traders

CAPM believer:At the beginning of each period, they observe payoffs and market prices and work out the composition of the market and the risk-free portfolios. Finally, according to their degree of risk aversion they choose their preferred combination between the two.

They choose such that

Log-utility trader:maximize the growth of their wealth share

max

Computer Simulation

Price determination:

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ solved through iteration by a numerical

technique called relaxation

relaxation

Start from a trial value for πst, compute a new value through

and then iterate this procedure until a fixed point is reached, i.e. when the difference of π between two successive periods is negligible (tolerance parameter = 10-5).

Two sets of simulations

To detect the time of convergence of the stochastic process given the wealth shares.To check the robustness of the results to heterogeneity in saving rates.

The time of convergence

Record the “time” if CAPM traders are extinct.Change γ (the risk-aversion of CAPM traders) to see whether the situation will differ.15000 realizations: 100 assets 100 states The prob. of the states distributed uniformly ds ~ N (μ,σ) Equal initial investment shares Equal saving rates

The robustness under heterogeneous saving rates

To check whether Log dominance results are robust when CAPM traders save at a higher rate than logarithmic utility maximizers. (normalize the saving rate of CAPM traders to be 1)1000 simulations under the same settings of the previous set of experiments.

Numerical Results

In all the runs CAPM traders see their investment shares reduced until extinction takes place.The density function describing the probability that CAPM traders survive up to a time t when interacting with LOG traders:

The more risk-averse the CAPM traders are, the faster their wealth share converges to zero.

The simulations show that exponential decay is robust with respect to the values of γ used: (hypothetical)

In this setting, a CAPM trader with an extremely low risk-aversion (γ 0 ) would indeed survive.

collapse plot

The higher the variance of the dividend stream, the higher is the advantage of logarithmic utility maximizers over CAPM traders.Intuition: with a large variance of dividends, the behavior prescribed by a logarithmic utility differs greatly from that prescribed by CAPM.

Conclusions

The wealth share of CAPMers converges almost surely to zero when LOGers with saving rates at least as large as that of CAPMers enter the market.When saving rates are identical across two types of traders, the wealth share of CAPMers decreases exponentially fast toward 0.

The degree of risk aversion of CAPMers has a role in determine the speed of convergence: the more risk-averse the CAPMers, the faster their shares converge to 0.LOGers dominate even when their saving rates are lower (but not too much) than those of CAPMers.

The difference between saving rates might serve as a measure of the fitness of LOGers with respect to CAPMers it’s increasing in the variance of the dividend stream.That seems to suggest that, from an evolutionary perspective, if it’s true that CAPM could perform almost satisfactorily as log utility maximization in markets with low volatility, it proves particularly unfit for highly risky environments.

Discussions

An environment with natural selection but without adpatationAre market prices determined mainly by certain kind of traders?Questions: Do LOG traders have additional information

compared with CAPM traders? What if the states of nature is not uniformly

distributed? Do LOG traders not care about the risk at all? Actual prices should have something to do with the

risks.

Guido Caldarelli

the Physics Department of the University of Rome La Sapienza

Marina Piccioni

PhD, Università di Napoli “Federico II”

Emanuela Sciubba

University Lecturer in Economics at the University of Cambridge and Fellow of Newnham College