Agent-Based Modeling of Portfolio Theory (I)


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Agent-Based Modeling of Portfolio Theory (I)

  1. 1. 1 Explaining the facts with adaptive agents: The case of mutual fund flows Introduction A financial market with adaptive agents Mutual fund flows and GA learning Conclusion 報告人 : 范志萍
  2. 2. 2 Introduction (1) The paper studies portfolio decisions of boundedly rational agents in a financial market. The data set set uses aggregate flows instead of flows of individual mutual funds. Agents have to decide how much to invest in a single risky asset. Their investment horizon is one-period so that learning takes place as repeated one-short investment decisions.
  3. 3. 3 Introduction (2) Two version of the model are studied. The first models a population of agents whose investment portfolio converges to a common value over the lifetime of the agents. The second model consists of a population of agents with new agents coming into the market and some of the existing agents leaving the market. The models with adaptive agents provide a viable alternative to conventional rational models in explaining observed behavior of participants in financial markets.
  4. 4. 4 Introduction (3) In Markowitz (1952, 1959) and Sharpe (1964) the optimal portfolio strategy is a passive one, i.e.,investment flows should not follow any pattern and can only be due to liquidity needs. Wang (1993) presents a model which is partially consistent with the mutual fund flow patterns.In his dynamic model with asymmetric information and noise trading, the uninformed speculator behave under certain parameter value like price chaser.
  5. 5. 5 A financial market with adaptive agents The model The genetic algorithm Decision rules The GA implementation Simulation results
  6. 6. 6 The model(1) Single asset. Agent’s utility is U(w)=- exp(-r w) p0 be the price pf asset and exogenously determined. How many units s of the risky asset she should buy? net payoff w=s(v-p0 ) Constant absolute risk aversion with coefficient
  7. 7. 7 The model(2)  The utility of the payoff is evaluated every period so that there is no dynamic link between periods.  Under full rationality the optimal portfolio is linear function of mean value of the asset and the current price p0, with optimal and  To calculate her optimal portfolio the agent must know all the relevant parameters: the distribution of the asset and her own utility function. v 0 * 2 * 1 * pvs αα −= * 1α * 2α
  8. 8. 8 The model(3) Agents observe the current market outcome and revise their next periods portfolio using observation. The decision of mutual fund investors to consciously put their money into the hands of fund managers can be viewed as a decision in terms of an optimal allocation of time. This paper assume a weaker notion of noise trading in that investors use a simple learning method to improve decisions. Investors use GA to update their portfolio decisions.
  9. 9. 9 The genetic algorithm(1) The initial population is generated randomly. GA first randomly selects copies of strings in the current population. [Proportionate Selection]The probability that a given string is copied to the new population is based on its performance or fitness: a string that did well according to the fitness measure will be more likely to be copied than a string with a lower fitness. The GA then introduces new strings through crossover and mutation that alter some strings of the population. Did not use the election operator? Why?
  10. 10. 10 Crossover It supports the development of compact building blocks in the population. A compact building blocks is a block of genes that are located next to each other. These building blocks tend to survive crossover and in the long run only successful blocks will prevail. Crossover guarantees that all building blocks of the search space are sampled at a rate proportional to their fitness.
  11. 11. 11 Mutation Mutation simply flips a 0 to a 1 and vice versa. Each chromosome undergoes mutation with a given (small) probability. The purpose of mutation is to introduce new genetic material and to avoid the development of a uniform population which will be incapable of further evolution over time.
  12. 12. 12 Decision rules  A strategy or decision rule maps the observable parameters to the demand for the asset. In this model, the observable parameters are p0 and and the asset demand decision can be written as s(p0, ).  This model choose a parsimonious encoding of the demand function. I assume that p0 and are constant and known to the agents. Hence, the agent’s problem is reduced to finding a scalar in the linear function v v v tα )0( pvs tt −=α
  13. 13. 13 The GA implementation(1) Let T represent the lifespan of each agent. Let J be the (constant) number of agents in each period. Each string of length L represents a value for a parameter in the demand function of one agent. The range of values for the parameter is the interval [MIN,MAX]. Let denote the ith bit of a string in period t. then the value of parameter in the demand function decoded by the string is given by }1,0{, ∈tiw tα )....( ,,1 tLtt www = 12 2 )( 1 1 , − −+= ∑ = − L L j j ti t w MINMAXMINα
  14. 14. 14 The GA implementation(2) In each period t the agents face S portfolio decisions, so that each agent makes a total number of T*S decisions in her lifespan. The population of decision rules is constant for all S drawings in a period t. After S realizations the population is updated using crossover and mutation. Each decision rule is assigned a probability of being copied to the next generation based on its performance. The probability is ∑ = = J j j i i V V p 11 1 be the cumulative utility after S drawings.  ∑ == S j jii wUV 1 )(
  15. 15. 15 Simulation Let the number of strings in the population is J=30,and each string is of length L=20, the probability of crossover is 0.4, The initial mutation rate is set to MUT=0.08. The strings decode possible parameter values between MIN=-4 and MAX=4. Each simulation is repeated 25 times for a given set of parameter value. The variance of the asset value and the coefficient of absolute risk aversion are set to unity.
  16. 16. 16 γbias bias2 vσ S 相同, T 較 小 bias 較大 當 s ,與 optimal portfolio 差距 愈大,表示 GA anent 持有更多 的風險資產 Simulation results (1)
  17. 17. 17 S 較小 時, bias 為 正 隨著 sbias0 Simulation results (2)
  18. 18. 18  bias v S 大 bias 小 Sbias0 Simulation results (3)
  19. 19. 19 Simulation results (4) The average utility for very low S is very small. When S is high, the demand parameter is updated rarely resulting again in lower utility.
  20. 20. 20 Simulation results (5) 300 期以 後 mean1 ∵initial 為 randomly ∴ 第 1 期 mean 0 300 期以後 variance=0 mutation 還在運作
  21. 21. 21 Mutual fund flows and GA learning Mutual fund flows: Some empirical evidence Financial flows in models with rational agents A learning model with entry and exit
  22. 22. 22 Mutual fund flows An empirical investigation of flows into and out of commonly held mutual funds. The data set consists of monthly data starting in February 1985 and ending in December 1992. The four fund groups are aggressive growth funds(AG), growth funds(GR), growth
  23. 23. 23 AG, GR, GI, BP AG seek to maximize capital gains thus focusing on risky stocks. GR invest in common stocks of well- established companies. GI invest in stocks of companies with a solid record of paying high dividends. BP have a portfolio mix of bonds, preferred stocks, and common stocks.
  24. 24. 24 Statistics of return and fund flows 1.highest average asset value 2.highest average flows 3.lowest average return 4.lowest standard deviation of asset flows 1. Lowest risk 2.lowest average asset value 3.highest average flow in % of asset value 1. highest risk 2. highest average return 3. highest standard deviation of return 4. highest standard deviation of asset flows
  25. 25. 25 AG’s returns and flows The correlation coefficient of two series is 0.73 . Whenever the is positive, flows into aggressive growth funds tend to be positive. At 1987/10 , AG funds a return of about –25%, AG funds lost about 10% of its assets net the capital loss.
  26. 26. 26 GR ’s returns and flows The positive tend in flows of growth funds especially between 1988~1992 年 . The correlation coefficient of two series is 0.54
  27. 27. 27 GI ’s returns and flows The correlation coefficient of two series is 0.24
  28. 28. 28 BP’s returns and flows The correlation coefficient of two series is 0.10 . 1985~1986 年, BP funds experienced very high inflow. 若不看 1985~1986 年的資料,則 BP 落在 GI的範圍 .
  29. 29. 29 Summarize the behavior of mutual funds Flows into mutual funds are positively correlated with returns. Flows are more sensitive to negative returns than to positive ones. Evidence is stronger for riskier mutual funds.
  30. 30. 30 Financial flows in models with rational agents Markowitz (1952, 1959) and Sharpe (1964) is also not able to explain why mutual investors are changing the portfolio composition after observing the return of their investment portfolio. Wang (1993) shows the uninformed agents behave like trend chasers. This leads to a positive correlation between changes in the holdings of the risky asset in the portfolio of the uninformed investors and price changes.
  31. 31. 31 A learning model with entry and exit A population of adaptive agents will not be able to reproduce the behavior of the mutual fund investors. The simplest way to achieve a heterogeneous population even in the long run is to insert new random strings over time and to delete some of the existing strings from the population.
  32. 32. 32 Simulations(1) The population consists of 60 agents. In each period t=1,…,T three agents, chosen randomly, exit the market and are replaced with new agents who start with a random strategy. Agents update their decision rule after each market outcome, i.e. S=1. The coefficient of absolute risk aversion is set to unity. To let the population settle down, it evolve for 5000 periods before starting with the analysis.
  33. 33. 33 Simulations(2) > < < The model with GA agents exhibit the same asymmetry after positive and negative returns realizations while the coefficients β+and β- are both bigger than for the AG funds (Table 3).
  34. 34. 34 Simulations(3) As investors in mutual fund, the GA population of adaptive agents adjusts its portfolio after each market outcome. The GA population exhibits the same asymmetry after positive and negative returns realizations.
  35. 35. 35 Simulations(4) Plots the change of the portfolio of the GA population while the Table 4 presents the corresponding correlation and OLS regression results.The results are very similar to the case with normal distributions. The portfolio adjustment is more extreme for high risk mutual returns.
  36. 36. 36 Simulations(4) Fig. 5B demonstrates this overreaction of the GA agents when compared to investors into aggressive growth funds (Fig. 5A ) very clearly.
  37. 37. 37 Conclusion (1) Mutual fund investors change their portfolio composition after observing marker outcomes. The correlation between flows into mutual funds and returns is positive and large, at least for riskier funds. Mutual fund investors exhibit an asymmetric response after positive and negative returns. They tend to react more heavily after negative market outcomes than after positive ones. In the first version of the model the adaptive algorithm is implemented so that the agent’s investment strategy converges to a fixed portfolio as time grows.
  38. 38. 38 Conclusion (2) They tend to hold a portfolio which is too risky compared to the portfolio of a rational agent who maximizes expected utility. A second model with a varying set of agents is introduced. The behavior of the population of adaptive agents is consistent with the mutual fund flow data. They adjust their portfolio after observing the market outcome just like mutual fund investors.
  39. 39. 39 Conclusion (3) The GA population also reacts asymmetrically after positive and negative returns just as the mutual fund investors The learning approach as used in this paper produces behavioral patterns which are consistent with most aspects of mutual investor’s strategies in real financial markets and thus provide a simpler and better fitting theory of observed behavior than standard theories with rational agents.
  40. 40. 40 Discussion Not using the election operator. Why? Role of S: Why is the performance so much dependent on S? When S is large: When S is small: Trade-off between S and T: GA Learner and Bayesian: Population Allowing for Entry and Exit:
  41. 41. 41 Election Operator Read the last line of p. 1126 to the end of first paragraph of p.1127. Read the first paragraph of p. 1132 with Figures 2.A and 2.B very carefully. This pattern of behavior is a very typical one for the GA agent. In what sense is it typical?
  42. 42. 42 When S is large.. Lettaul (1997) provides an analysis of the relation between and Using Monte-Carol Simulation, Lettaul (1997) showed that
  43. 43. 43 When S is small.. Since is chosen to be positive, this tendency is more pronounced for weights close to MAX. In other words, since there are more `good’ states of the world with a positive payoff than `bad’ states with negative outcomes, the riskier strategies are more successful than safer ones in more than 50% of market realizations.
  44. 44. 44 Behavior Implication This explains why the GA selects portfolio weights that are too high as long as S is low. If she observes only a few realization of the uncertain asset, she selects portfolios with too much risk since she does not take rare negative events correctly into account.
  45. 45. 45 Under survival pressure, agents tend to behave as if they are less risk-averse than otherwise. This result should be compared to Szpiro (1997).
  46. 46. 46 S, T and ST The second contribution of Lettau is that he actually made an economic analysis of the optimal combination of T and S given a fixed amount of resource to be spent in search. But, usually S is set exogenously by the institution, and T has the sky as the limit.
  47. 47. 47 Modeling Survival Pressure with Institutional Arrangements This is a research topic open for students who are interested in doing research on the relation among S, T and ST.
  48. 48. 48 Fundamental Failure of Conventional Financial Modeling based on Bayesian Learning In general, it seems unlikely that models with agents who are using Bayes’ rule to update their beliefs are able to generate substantial trading volume in steady state without relying on unreasonable large amount of noise trading. Intuitively, the reason is that in dynamic setting, Bayes’ rule puts little weight on recent information and relies more on the past.
  49. 49. 49 Cognitive Features of Bayesian Thus, any new information does not change beliefs too much and thus does not cause substantial portfolio updates. In other words, Bayesians tend to have a long window of memory, and normally the window size is fixed during the adaptation process is fixed.
  50. 50. 50 Exit and Entry While Lettau (1997) considered the population which allowed for exit and entry, it is not clear how this added mechanism can help match the empirical features of the mutual fund markets. The most interesting part of the empirical behavior is the asymmetric response to positive returns and negative returns.
  51. 51. 51 But, to replicate the pattern of asymmetric response, it seems enough to have a small S, as the author also well noticed: ``The reason for this asymmetry is the risk-taking bias as discussed in Section 4.2.’’ (p. 1142) Therefore, it has nothing to do with the added entry and exit mechanism.
  52. 52. 52 While S plays the major role in Section 3, the author did not carefully make this distinction by only saying ``Agents update their decision rule after each market outcome, i.e, S=1.’’ (p. 1141) My conjecture is that if the S is enlarged to 1000, there should be no discernible pattern between the adjustment of portfolio and the returns.
  53. 53. 53 Therefore, Lettau’s experiment suggested an empirical size of S, which seems to be small, while not necessary one. This study provides probably the only empirical evidence on the length of evaluation cycle. Since the evidence is in favor of a small S, there is no ground for the choice of a large S in the GA simulation as many existing studies did.
  54. 54. 54 Also see LeBaron (1999) for a different and related study on the choice of time horizon. There is no natural economic interpretation of entry and exit used in this paper. Dawid and Kopel (1998) is probably the only paper considering entry and exit as a part of decision variable to determined by GA.