Download Agent-Based Modeling of Portfolio Theory (III)

A genetic algorithm approach to
farm investment
Oscar Cacho and Phil Simmons(1999)
經碩二 88258023 金甫均
Genetic Algorithm(GA)
 The purpose of using GA model
1.Methodological instrument to maximize functions
- highly non-linear functions
- a large number of control variables’ functions
2. Test to the specific behavioral hypotheses
- monetary theory, index design, dynamic cobweb,
stock behavior, optimization under imperfect competition
learning model and so on
 The elements of GA : Goldberg(1989)
- Evaluation (fitness function) , selection, crossover
mutation.
The investment behavior in a GA model
 Szpiro(1997): “The emergence of risk aversion”
1.Using GA to study the investment behavior in the stock
market
2. Concluded that when firm was selected against on the
basis of poor profit performance, they developed caution
in their investment strategies.
 firms reflect the an alternative approach to risk behavior
based on competitive adaptation rather than maximizing
certainty equivalent value
Farm Investment behavior in GA Model
 Oscar Cacho and Phil Simmons(1999)
 Applying GA model to test farm investment (portfolio)
under the various environments
 Show that the assumption of competitive adaptation lead to
a violation normative efficiency.
 Those who survive are not the most efficient in a
normative sense.
 Traditional Economics :
prefect competition = most efficiency (guarantee survival)
The Analytical Methodology
 Applying Szpiro’model(1997) to farm investment
 Augment model to test whether normative efficiency
could be undertaken or not under the various environment
in which the farmer take the different risk attitudes.
 Traditional Economic Model is transferred to GA model
 The various environment is defined by the distinct density
of risk (2 methods)
- Adjusting investment horizon & the probability of
survival
Economic model Description
 Portfolio investment model [CTRS]
 two risk enterprise and a risk-less asset
 risk neural at the initial point
 Farmer’ Choice: Crop mix (S, W), Borrowing or Lending
 Farmer income = Return to equity [one unit of equity].
 S and W market is separated, Returns to S and W is
independent
The expected return from farm capital (1):
(1)
(2)
(3)
(4)
(5)
(1): Expected return from farm capital.
(2): Proportions of farm capital allocated to S, Nonnegative
(3): Expected returns from S
(4): Proportions of farm capital allocated to W, Nonnegative
(5): Expected returns from W
 Cov(rw,rs )=0 and Pw = 1 – Ps
The expected return to equity (2):
(1)
(1):
(2):
(3):
(4):
(5):




(2)
(3)
(4)
(5 )
Expected return to equity.
the ratio of farm value to equity, Nonnegative
Expected returns from farm capital
the ratio of borrowing or lending to equity
Risk free rate : at this rate, farmers lend or borrow
if the farmer is a lender
if the farmer is a borrower
the borrowing limit
additive condition
The expected return as the farm income (3)
From Eq(1)and (2) and two additive restrictions
1. The choice of farmer (at the specific risk free rate)
 Corps mix and the amount of borrowing or lending.
2. Design variables Pa and Pb
 “Genes” in the GA
3. Returns to S and W are independent
4. S and W market separate.
:
Production shock (4) :
(1)
(2)
(3)
(4) (5)
(1): the expected returns from S and W
(2) and (3): production shocks (j=S,W)
(4) Cj>0 is constant
 The expected return from S and W have bivariable
lognormal distributions bounded by Cj and unbounded
above.
 Production Risk are Price makers Corr(U1j,U2j)=p12j<0
Andelerson, Dillon and Hardaker (1977 pp. 171-2)
The genetic algorithm
 Elements in GA: evaluation, selection,crossover,mutation
 Population :100 farmers
a.
b.
c.
d.
f..
Identical technology and facing the same risk.
The initial value of Genes(Pb ,Ps): Random
Maximum debt :bmax= 0.5 ,
Ps is nonnegative,Ps=1-Pw ,
Each gene is a ten-character binary string
Evaluation
 The investment performance at the end of k- the period.
 Using value function to sum return to equity over the
investment horizon
where reik for agent i in period k is
estimated Eq3
Ri: Raw fitness
 Fitness function(Goldberg1989. pp76-9)
 Calculated by subjecting the value function to linear
scaling
 The highest “raw fitness(R)”produce an expected two
springs
 The average produce one expected offspring per
generation
Selection
 Mitchell(1997,pp166-7)
 Using the classical roulette wheel selection
1. The probability of selection is proportional to the fitness
of the individual relative to the rest of population
2. Genes belong to agents who are not selected for
reproduction disappear from the population
Crossover & Pairing (1)
 Selection Crossover with Paring.
 Each pair reproduce two offspring and disappear
1.Leaving population size in the second generation the
same as in the first
2. Transmission of genes between generations
 Adapt to the stochastic environment easily
3. “Bit string swapping”(Goldberg 1989)
The copies from genes of the two parents
 The Probability of single-point crossover: 0.6 in this test
 the 0.4 probability of offspring is identified to their
parents
Crossover & Pairing (2)
 The meaning of the Crossover and Pairing in the GA
1. Based on the value function, poor performers disappear
in the population
only the fittest survive
2. Pairing allow that inferior genes can survive, but in
proportions deceasing with each generation
 GA model have “ genetic memory”
3. The inferior genes can be dominant within a few
generation
4. But, a gene eventually disappear if it don not contribute to
overall population fitness
Mutation
 It consists of flipping a random bit(zero changes to one) in
the binary representation of the parameter
 In this paper: the probability of this is 0.01.
 The reconsideration of Mutation
1.Mutation in the GA is designed for the maximization,
since it prevent the population from converging to
a local maximum.
The fitness function lead the wealth maximization
2.In the stochastic environment, mutation play the role of
allowing agents to explore strategies which might
otherwise have been left untested.
Environment design for the experiments
 The Method of the risk adjustment in GA model
1.The increase in the period of evaluation
(investment horizon)
which reduce the variance of the value function.
2.Varies the severity of the selection process by adjusting
the probability of survival.
3.Experiments for the test (3 cases)
- Standard environment [S]
- Harsh environment [H]
- Mild environment [M]
Parameter values used in Experiment
 Population : 100 agents
 Run :200 generation
Standard environment [S]
 Risk-free rate(rb) : 8 to 10
 Critical value: 9.227 (the expected return of the most
profitable crop)
the risk neutral agent switch from borrowing to lending
Crop
Expected value
Standard deviation
rs
9.227
8.684
rw
7.486
4.722
Harsh environment [H]
1.Increases the severity of selection by reducing the raw
fitness value of the agents if they have negative wealth .
2.The worse performer can not reproduce.
3.Debt Penalty
(.) assume a value if true, zero if false.
4. Scaling factor (w)
 Prevent premature convergence early in a run and slow
convergence in later generation
.
5. Fitness function
: linear scaling parameters to get the desired number of
offspring from the fittest individual.
Mild environment [M]
 Increases the investment horizon: from 3 to15
 Reducing hostility in the experiment
 Making farmers less cautious
 Agent offsets losses in poor periods against gains from
favorable period prior to evaluation and selection
Model behavior in the farm investment

Deterministic environment
1.Risk free rate : rb=8 : rb=9.3 , rb=10
2.Simulation: all
to zero
3. In figure 1,
 For three rb , Convergence was accompanied by a
rapid drop in the genetic variability denoted by SD
of Pb and Ps
 the gene of the fittest agents took over an increasing
proportion of the population.
 at the16 to 30 generation, convergence is slower
rb was close to the critical point
Results from the deterministic runs ate the three
values of the risk free rate ( Fig 1)
Stochastic environment
IN figure 2,
 Convergence have evolved over 200 generations exhibit
risk aversion behavior, since Borrowing less than bmax
and invest partially on the high risk, high return crop.
In the H, the decrease in the borrow can be viewed as more
risk aversion.
 Ps have the similar values as the environment harsher,
since agents adjust borrowing rather than the crop mix.
(See Separation theorem)
Behavior of GA under stochastic condition(rb=8)
Convergence and the role of mutation
 The definition of convergence in GA
1. when entire population has evolved to the same genetic
make-up within the desired tolerance.
2. In strictly viewpoint, convergence never occurs in GA
since mutation.
 The role of Mutation: (see Mutation)
 the probability of mutation means new gene occur any
time.
 In figure 3, mutation prevent early convergence in local
sense.
 In 0 to 70th generation, converge to –2.0
 In 120th generation, large mutation occurred and
redirected the algorithm to the true optimum
Result in Standard environment (rb=9.23)
Spread of surviving agents
 Spread of surviving agents shown in Figure 4,
 The effect of drawing the initial population from a
uniform distribution
1. Initial average parameter values occur close to the
central of feasible zone.
2. Initial variability among individual agents is high
(see Figure1)
 After 200th generations, the parameter values of surviving
agents are concentrated within a small area in the feasible
space
 Moving from a to c and d , Risk neutrality is far away.
Spread of surviving agents in parameter space
under alternative assumption
Monte Carlo Experiments
 GA in the 4*3 factorial design: rb (8,9,9.3,10) and (M,S,H)
 12 different treatment
 per treatment yield 10,000 surviving agents
(100 individual per experiment * 100 experiment)
 In figure 5, Table2
1. Risk aversion increased as the environment changed
from M to S to H
2.at the below critical risk free value,
 In H, average agent lending 41%, risk neutral agent
still borrows as much as possible
 In S and M, agent borrow less than 4% of maximum
allowed.
3. Mild gambling behavior in all three M, S, H at the above.
critical risk free value(=9.3)
a. 0.19 –0.49 of capital available was lent at the higher
risk free rate.
b. the remainder was invested in farm production which
yields a lower expected return
 the sufficient compensation provided by high prices
and yields
 it makes the gamble more attractive.
4. Risk aversion index (Szpiro 1997)
the distance the between the maximum borrowing rate and
the actual borrowing rate in the surviving population.
Effect of the risk free rate on the average borrowing
behavior of surviving agents under M,S,H
K=3
K=15
Average results of Monte Carlo experiments
 In figure 6,
1.Using Cumulative density functions
2.Exit the considerable variation among survival agents in
each population
 But Clear Patterns still emerge which can support the
above discussion
3. The proportion of surviving agents, borrowed more than 4
times the available equity, decreased as changing from M
to S to H. (M,S,H)=(1, 0.75,0.2)
4. The proportion of surviving agents, invest more than 0.8
capital in the high return,high risk crop, decreased.
(M,S,H)=(0.55,0.2,0.05)
Cumulative functions of parameter values (rb=0.8)
Figure 6
The Separation Theorem
 Tobin state “ if capital markets are efficient and there exists
a risk free asset, then the crop mix is not influenced by
risk preference”
 Efficient farmer is restricted to rb z as indicated by the
indifference curves
 risk attitudes are separated from the crop mix decision
SW : the farmer income risk frontier
rb z : the risk efficient frontier which comprise different combinations
of risky farm capital and risk free asset
Test for the Separation theorem
 Manipulation for the test
1. allow only the level of risk aversion to vary at the given
risk efficient frontier.
2. Redefinition of fitness function to account for risk aversion
 function form as the numerical model
3. Utility function(AndersonDillon and Hardaker 1997,pp99)
α: a coefficient of risk aversion
 The spread in αvalues in these experiments is
proportional to the spread in risk aversion indexes
between S and H
IN figure 8,
1.Higher risk aversion result in lower borrowings (Fig.8A)
2. As the increase inα,the proportion of the surviving
population that borrowed within one unit of the limit
decreased (0.67 to 0.46)
3.Results on crop mix are not as clear cut as those on
borrowing (Fig 8B)
4.the mean of two experiments are similar (0.507 and 0.515)
 but the hypothesis was rejected (p<0.05)by an F test
(F=7.26)
5. The proportion of surviving agents, invested less than 0.8
of their capital in the high risk crop, increase as the α
increase
Cumulative density function of parameter values
under the utility maximization (Figure 8)
Conclusion.
1.The main factor of the cautious behavior come directly
from selection.
2.The fittest agent make a balance between too much and too
little caution
 similar to the agent who maximize certainty equivalent
income with risk aversion
3.The test for the normative efficiency based on the
Separation Theorem
 the fittest agents may not be most efficient in
normative sense, the most efficient may not be the
most likely to survive.
Discussion
 Alternative approach to investment under
uncertainty:
– Competitive Adaptation
 Capital Structure of Firms
 Survival and Efficiency
 Not Using the Election operator
 Role of K (Evaluation Horizon)
 Linear Scaling (Goldberg, 1989)
 Roulette-wheel selection vs. Tournament
Selection
Election Operator
 On discussing the role of mutation, the
authors neglected the election operator (See
conclusion).
 They, however, mentioned the nonuniform mutation operator introduced by
Michalewicz (1994).
Role of K
 Related to Lettau (1997)
 Lettau had only one way to test the role of
K on the potential conflict between
efficiency and survivability.
 However, here, you see a different measure
to be taken, i.e., fitness re-scaling.
 Lettau (1997) also did not endow K with a
intuitive interpretation, while his K ranged
from 1 to 1000.
Role of K
 Here, Cacho and Simmons interpreted K in
a sense of adversity.
 I highly recommend students to read these
four papers together: Lettau (1997),
LeBaron (1999), Szpiro (1997) and this
paper.
 For those students who are interesting in
investment under uncertainty, you are also
referred to Lensberg (1999).
Scaling
 Scaling may prevent two problems:
– Premature convergence early in a run
– Slow convergence in later generations
Selection
 It was evidenced that roulette-wheel
selection tended to have a weak selection
when fitness of chromosomes are very close.
Therefore, tournament selection is preferred
to roulette-wheel selection in this regard.
 See Arifovic (1995) for an example, and
Bullard and Duffy’s paper for the discussion
of this issue.
 In this paper, read p.311, line 5 carefully:
``Convergence was slower…’’