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) unde

r 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 coul

d 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 densit

y 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) (2) (3) (4) (5 )

(1): Expected return to equity.

(2): the ratio of farm value to equity, Nonnegative

(3): Expected returns from farm capital

(4): the ratio of borrowing or lending to equity

(5): 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 GA3. Returns to S and W are independent4. 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 logn

ormal 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. Identical technology and facing the same risk.

b. The initial value of Genes(Pb ,Ps): Random

c. Maximum debt :bmax= 0.5 ,

d. Ps is nonnegative,Ps=1-Pw ,

f.. 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 inves

tment 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 profit

able 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 model3. 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 αincrease

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 non-uniform 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…’’