Introduction
Following to the article of “Application
of Pascal’s Triangular in Corporate Financial Strategy” on link: http://emfps.blogspot.com/2012/04/applicationofpascalstriangularin.html and the article of “Monte Carlo Analysis on Case of Nike, Inc.:
Cost of Capital” on link: http://emfps.blogspot.com/2012/01/montecarloanalysisoncaseofnike.html, the purpose of this
article is to use a new simulation model which has been made by the combination
of the Pascal’s Triangular and Monte Carlo method for calculating the risk of
Expected Utility. The most important finding is to track and control the
changes of the risk and Expected Utility when we change our trials. Another
finding is to improve our primary estimation about the probability for Utility
function extracted from some macroeconomic indicators such as inflation rate,
unemployment, CPI, PPI, GDP, interest rate and so on. As the matter of fact, I
have proved that by increasing of the trials, we will have the constant
Expected Utility but the risk of Expected Utility will decrease. Finally, I am
willing to tell you that one of the best ways to solve the complex problems
(Multi Dimension problems) by using of Monte Carlo Simulation Model (due to the
limitation of CPU of our PC) is to breakdown the problem and to utilize the
combination of many methods such as Markov Chain, Fuzzy Logic and so on
accompanied by Monte Carlo Simulation Model. I think that this simulation model
can be used by someone who have been involved in business of Saving Energy, Stock Markets, Forex trading,
Shopping malls such as Carrefour, Giant Hypermarket, IKEA, AEON, Mydin,
Parkson, SOGO, Tesco and so on but the most important usage is to estimate the
risk of deficit financing in
macroeconomic where it was implied in article of “Case Analysis of GAINESBORO
MACHINE TOOLS CORPORATION (CON): A New Financial Simulation Model” on link: http://emfps.blogspot.com/2012/04/caseanalysisofgainesboromachine.html
Expected
Utility Theory
In the reference with Wikipedia on
below link: http://kleene.ss.uci.edu/lpswiki/index.php/Expected_Utility_Theory
The definition of Expected Utility
Theory, Utility and Expected Utility Calculations are as follows:
Expected
utility theory is a tool aimed to help make decisions
amongst various possible choices. It is a way to balance risk versus reward
using a formal, mathematical function.
When faced with a number of different
choices, expected utility theory recommends that you calculate the expected
utility of each choice and then choose the one with highest expected utility.
Utility
Utility is simply a measure of a
person's preferences amongst different things. From these preferences (if they
are rational!) we can deduce a utility
function which represents
preferences by order relations between numbers.
This only works if a person's
preferences are, in a certain sense, rational. If someone prefers the Angels to
the Dodgers, then they shouldn't also prefer the Dodgers to the Angels. And, if
they prefer the Angels to the Dodgers, and the Dodgers to the Giants, then they
shouldn't prefer the Giants to the Angels. We also require that, for any two
things, a person prefers one to the other, or is indifferent between the two.
If a person's preferences are
rational in the above sense, then we can define a utility function as follows:
u(⋅) is
a function that assigns numbers to things (represented by variables x,y,z,…)
u(x)>u(y) if and only if this person prefers x to y
u(x)=u(y) if and only if this person is
indifferent between x and y
The numbers assigned by u(⋅) should
also match how much one thing is preferred to another. If
someone assigns u(Angelswin)=100 and u(Dodgerswin)=1,
then they'd prefer to see an Angels win 100 times more than a Dodgers win.
Example of
Utility Functions
Suppose we want to create a utility
function for a fan of the "Planet of the Apes" movies. There were
five movies in the series: "Planet of the Apes", "Beneath the
Planet of the Apes", "Escape From the Planet of the Apes",
"Conquest of the Planet of the Apes" and "Battle for the Planet
of the Apes".
Our fan likes "Escape"
the best out of the five, prefers "Escape" to "Beneath," is
indifferent between "Beneath" and "Conquest," prefers
"Beneath" to "Planet," and prefers "Planet" to
"Battle." Here is a utility function that could represent our fan's
preferences:
u(Escape)=10,u(Beneath)=u(Conquest)=8,u(Planet)=5,u(Battle)=1
One way to think of utility is in
terms of how much you would pay for each of these things, or how much these
things are worth to you.
Expected Utility
Calculations
How appealing a certain choice is
depends not only on the payoffs of that choice, but how likely those payoffs are. The multimillion dollar payoff of a lottery is
certainly appealing, but it is so unlikely that buying a lottery ticket is
virtually a waste of money. Expected utility calculations are meant to balance
risk versus reward.
We think of an act (like buying a
lottery ticket) as having a number of possible outcomes (in this case, winning
or losing). Given a person's utility function (see above) and their degrees of belief in each of the possible outcomes, we can figure out the expected
utility of any act. This is done as follows:
Let the act in question be labelled A. Let o1,o2,…,on be the various possible outcomes of A (there
needs to be at least one outcome, but there could be many).
To each outcome oi is
an associated probability Pr(oi) which measures how likely that outcome
is, and a utility u(oi) which measures that outcome's spot in
this person's preference relation.
The expected utility of A is:
E(A)=u(o1)⋅Pr(o1)+u(o2)⋅Pr(o2)+…u(on)⋅Pr(on)
Now, when faced with a choice
between multiple acts A1,A2,…An,
expected utility theory says that a person should choose the act with the
highest expected utility. That is, calculate E(A1),E(A2),…E(An) and then choose the act with the
highest associated utility.
Therefore, if we obtain the
probability and Utility function for each outcome, we are able to calculate
Expected Utility for each action.
Methodology
The methodology to make this
simulation model has been explained step by step as follows:
1) To define the acts
2) To determine the numbers of the
total trials or the numbers of the possible outcomes
3) To make the appropriate trials of
Pascal’s Triangular in reference with the numbers of the total trials (the
total possible outcomes) by using of Excel spreadsheet (referred to step 2)
4) To calculate the total amount of outcomes
for each possible outcome
5) To calculate the total sum of
outcomes for the total trials
6) To divide step 4 to step 5 for
each possible outcome to obtain accumulative probability
7) To find out the consequence of all
possible outcomes (which is the same the Binomial Probability Distribution) by
using of accumulative probability (step 6)
8) To find a range for Utility
function by using of marketing research
9) To estimate a primary probability
for each Utility function (primary range) by using of macroeconomic indicators
such as PPI, CPI, GDP, interest rate, unemployment rate, inflation, exchange
currency and so on
10) To calculate Cutoffs or
accumulative primary probability (which is utilized to conduct a Monte Carlo
analysis)
11) Using of the Rand formula = Rand () on our
excel spreadsheet for all Utility functions
12) Using of the VLOOKUP formula =
VLOOKUP (Rand cell, Cutoff, 2) for all Utility functions
13) To multiply each probability of
the possible outcome to each Utility function
14) To sum all results obtained from
step 13
15) The increase iterative
calculations by using of a two –way Table just like Sensitivity analysis in
which row is the numbers of 1 to 10; column is the numbers of 1 to 200 or 400
or 1000 and independent variable is the result of step 14
16) To calculate the standard
deviation (STDEV), coefficient of variation (CV) and the average of Utility
function.
Finding and
discussion
In this case, I started by an
example as follows:
1) I defined the acts as The
Purchasing and Selling
2) Referring to the article of “Application
of Pascal’s Triangular in Corporate Financial Strategy” on link: http://emfps.blogspot.com/2012/04/applicationofpascalstriangularin.html,
I considered the total trials or the total possible outcomes in four levels as
follows:
Ø 999 trials (possible outcomes) where the total number of the people
(participants) were 997 plus Purchasing and Selling
Ø 500 trials (possible outcomes) where the total number of the people
(participants) were 498 plus Purchasing and Selling
Ø 100 trials (possible outcomes) where the total number of the people
(participants) were 98 plus Purchasing and Selling
Ø 10 trials (possible outcomes) where the total number of the people (participants)
were 8 plus Purchasing and Selling
Here is the consequence of the
outcomes for each level in the format of binomial probability distribution curve as
follows:
The level of 999
trials:
 The level of 500
trials:
The level of 100
trials:
3) I considered the primary range for
Utility function and probability for Purchasing and Selling as follows:
Probability of Purchasing

U (Oi)P


0.1

30


0.15

35


0.35

40


0.4

45


Cut offs

U (Oi)P


30


0.1

35


0.25

40


0.6

45

Probability of Selling

U (Oi)S


0.1

45


0.15

40


0.35

35


0.4

30


Cut offs

U (Oi)S


0

45


0.1

40


0.25

35


0.6

30


As you can see, the probability for
Purchasing is exactly vice versa with Selling.
The Results
I calculated the standard deviation
(STDEV), coefficient of variation (CV) and the average of Utility function for
each level of trial as follows:
The level of 999
trials:
Purchasing


STDEV (Ave.)

0.657515


CV (max)

0.017365


CV (average)

0.016342


E (Pur.) Total Ave.

40.23528


Selling


STDEV (Ave.)

0.649111


CV (max)

0.019577


CV (average)

0.018688


E (Pur.) Total Ave.

34.73502


 The level of 500
trials:
Purchasing
Purchasing
STDEV (Ave.)

0.784547


CV (max)

0.02199


CV (average)

0.019497


E (Pur.) Total Ave.

40.23969

Selling
STDEV (Ave.)

0.784866


CV (max)

0.024444


CV (average)

0.022575


E (Sellig) Total Ave.

34.76781


The level of 100
trials:
Purchasing
STDEV (Ave.)

1.175161


CV (max)

0.03183


CV (average)

0.029202


E (Pur.) Total Ave.

40.24228


Selling
STDEV (Ave.)

1.143864


CV (max)

0.034977


CV (average)

0.032905


E (Sellig) Total Ave.

34.76331


 The level of 10
trials:
Purchasing
STDEV (Ave.)

2.066192


CV (max)

0.053684


CV (average)

0.051285


E (Pur.) Total Ave.

40.28859

Selling
STDEV (Ave.)

2.043727


CV (max)

0.063607


CV (average)

0.058879


E (Sellig) Total Ave.

34.7068


I compared the changes of risk for
each level of possible outcomes where this finding has been included in below
diagram:
As you can
see, the risk will decrease if we increase the level of possible outcomes while
the Expected Utility will be the constant for each level of possible outcomes.
Note:
“All spreadsheets and calculation notes are available. The people, who are
interested in having my spreadsheets of this method as a template for further
practice, do not hesitate to ask me by sending an email to: soleimani_gh@hotmail.com or call
me on my cellphone: +989109250225. Please be informed these spreadsheets are
not free of charge.”