Statistical, mathematical and Economic Applications After studying this chapter you will understand how statistics, mathematics, Operation research are used in business decisions. Statistical tools for decision-making: Various techniques we have in statistics up to 2 nd year degree. We see now how they are applied in business decisions. The most powerful techniques are mean and standard deviation. Mean explains the average values of the variable whereas standard deviation explains how far the variable values are deviating from the mean. The coefficient of variation links standard deviation with mean. i.e. S.D x 100/Mean. We know that the lesser the C.V the greater the consistency and vice-versa. Co-efficient of variation measures the spread of a set of data as a proportion of mean. It is used in problems situations where we want to compare the variability, homogeneity, stability and consistency. For example: An investor, Mr. Miky in many portfolios studied 100 companies and obtained the following results for the year 2008 Returns 10-20 20-30 30-40 40-50 Number of companies 19 32 41 8 The average = 18.8%; S.D=8.81; C.V=46.86% Suppose the entire investors C.V. is 30% Mr.Miky’s portfolio is much riskier comparing to market risk. The next question asked by Mr. Miky is that what the probability that will he get assured returns of atleast 15%? In such situation we use normal curve or we use Z distribution to determine the probability of getting at least 15% assured return on his portfolio. Z=(X- )/ where - Average - Standard deviation 15% Z= (15-18.8)/ 8.81 = 0.43 The value for 0.43 in normal distribution table gives probability i.e. 0.6664. This probability explains that 67% chance of assured return of 15%. Suppose he wants to know the probability of return beyond 20%. Z=(X- )/ = (20-18.8)/8.81=0.136 The Z table gives the probability i.e. 0.5-0.0557=0.444 44% chance to earn beyond 20% This helps a person or organization to understand what extent they are taking risk before going for the project.
Transcript
Statistical, mathematical and Economic Applications
After studying this chapter you will understand how statistics, mathematics, Operation
research are used in business decisions.
Statistical tools for decision-making:
Various techniques we have in statistics up to 2nd
year degree. We see now how they are
applied in business decisions.
The most powerful techniques are mean and standard deviation. Mean explains the
average values of the variable whereas standard deviation explains how far the variable
values are deviating from the mean. The coefficient of variation links standard deviation
with mean. i.e. S.D x 100/Mean. We know that the lesser the C.V the greater the
consistency and vice-versa. Co-efficient of variation measures the spread of a set of data
as a proportion of mean. It is used in problems situations where we want to compare the
variability, homogeneity, stability and consistency.
For example: An investor, Mr. Miky in many portfolios studied 100 companies and
obtained the following results for the year 2008
Returns 10-20 20-30 30-40 40-50
Number of
companies
19 32 41 8
The average = 18.8%; S.D=8.81; C.V=46.86%
Suppose the entire investors C.V. is 30% Mr.Miky’s portfolio is much riskier comparing
to market risk.
The next question asked by Mr. Miky is that what the probability that will he get assured
returns of atleast 15%?
In such situation we use normal curve or we use Z distribution to determine the
probability of getting at least 15% assured return on his portfolio.
Z=(X- )/
where - Average
- Standard deviation
15%
Z= (15-18.8)/ 8.81
= 0.43
The value for 0.43 in normal distribution table gives probability i.e. 0.6664. This
probability explains that 67% chance of assured return of 15%.
Suppose he wants to know the probability of return beyond 20%.
Z=(X- )/ = (20-18.8)/8.81=0.136
The Z table gives the probability i.e. 0.5-0.0557=0.444
44% chance to earn beyond 20%
This helps a person or organization to understand what extent they are taking risk before
going for the project.
There are standard probabilities
1 =0.68
2 =0.95
3 =0.99
Exercise:1
Students can try out with their average and standard deviation marks in five semesters to
find probability of scoring beyond 70% or 80% in the last semester.
Exampe-2
Return on market index and a security are given below .
Period Return on Market Index(A) Return on Security(B)
1
2
3
4
5
6
7
8
9
10
10
24
-16
36
32
20
14
30
60
50
12
20
-8
26
28
26
8
36
48
44
1. Calculate regression equation B on A
2. Explain the correlation between return on Market Index and Return on Security
The regression equation B = 4.24 + 0.76A, here b = +0.76 which indicates 1 rupee
change in return of market index. 76 paise change in return on security. b also indicates
there is a positive correlation between these two variables because b is positive.
We can use regression equation to answer certain vital questions under cost volume
analysis. Some of the questions are as follows
1. The firms break-even sales volume
2. How sensitive is the profit in variations in the output.
3. How sensitivity is the profit in variations in selling prices.
4. What should be the sales level in quantitative terms for the firm to earn target
level of earnings.
The regression equation
Y= a + bX, where a = fixed cost
b = variable cost
y = total cost
Example:
The following table gives the repairs and maintenance cost incurred in a cost center for
various levels of annual production. If the budgeted production of the cost center in the
forthcoming year is 8500 units, what would be the estimated repairs and maintenance
cost ignoring possible increase in price levels?
Output (in thousands of
units)
Repairs and Maintenance
Cost Rs.(in thousands
1 15
2 21
2.5 24
3 26
3.5 29
4 32
5 36
6 40
7 44
8 49
Students are expected to calculate by using the formula
Interpretation: b = 4.71 and a = 11.818
Therefore y = 11.818 + 4.71X
The fixed cost is rs.11818 and the variable return and maintenance cost is rs.4.71 per unit
produced.
The total cost when 8500 units are produced (budgeted ) = Fixed component + variable
component = 11818 + (8500 x 4.71) = 51, 853
Standard Error of the Estimate (1 of 3)
Regression line is the line that minimizes the sum of squared deviations of prediction
(also called the sum of squares error).
The standard error of the estimate is a measure of the accuracy of predictions made with a
Regression line Consider the following data.
The second column (Y) is predicted by the first column (X). The slope and Y- Intercept
of the regression line are 3.2716 and 7.1526 respectively. The third column, (Y'),
contains the predictions and is computed according to the formula:
Y' = 3.2716X + 7.1526.
The fourth column (Y-Y') is the error of prediction. It is simply the difference between
what a subject's actual score was (Y) and what the predicted score is (Y').
The sum of the errors of prediction is zero. The last column, (Y-Y')², contains the squared
errors of prediction.
Stats: Coefficient of Determination
Coefficient of Determination
The coefficient of determination is ...
the percent of the variation that can be explained by the regression equation.
the explained variation divided by the total variation
the square of r
What's all this variation stuff?
Every sample has some variation in it (unless all the values are identical, and that's
unlikely to happen). The total variation is made up of two parts, the part that can be
explained by the regression equation and the part that can't be explained by the regression
equation.
Well, the ratio of the explained variation to the total variation is a measure of how good
the regression line is. If the regression line passed through every point on the scatter plot
exactly, it would be able to explain all of the variation. The further the line is from the
points, the less it is able to explain.
Coefficient of Non-Determination
The coefficient of non-determination is ...
The percent of variation which is unexplained by the regression equation
The unexplained variation divided by the total variation
1 - r^2
--------------------------------------------
Decision under uncertainty
Origin of probabilities
Probability is derived from the verb to “probe” meaning to "find out" what is not too
easily accessible or understandable. The word "proof" has the same origin that provides
necessary details to understand what is claimed to be true.
Probabilistic models are viewed as similar to that of a game; actions are based on
expected outcomes.
Probabilistic models
The center of interest moves from the deterministic to probabilistic models using
subjective statistical techniques for estimation, testing, and predictions
In probabilistic modeling, risk means uncertainty for which the probability distribution is
known. Therefore risk assessment means a study to determine the outcomes of decisions
along with their probabilities.
Problem of decision maker?
Problem of decision maker
Decision-makers often face a severe lack of information.
Probability assessment quantifies the information gap between what is known, and what
needs to be known for an optimal decision. The probabilistic models are used for
protection against adverse uncertainty, and exploitation of propitious uncertainty.
Difficulty in probability assessment
arises from information that is scarce, vague, inconsistent, or incomplete
A statement such as "the probability of a power outage is between 0.3 and 0.4" is more
natural and realistic than their "exact" counterpart such as "the probability of a power
outage is 0.36342.“
What are the challenging tasks?
challenging task
compare several courses of action and then select one action to be implemented
The limited information-processing capacity of a decision-maker can be strained when
considering the consequences of only one course of action. Yet, choice requires that the
implications of various courses of action be visualized and compared. In addition,
unknown factors always intrude upon the problem
challenging task
seldom are outcomes known with certainty. Almost always, an outcome depends upon
the reactions of other people who may be undecided themselves
It is no wonder that decision-makers sometimes postpone choices for as long as possible.
Then, when they finally decide, they neglect to consider all the implications of their
decision.
What are the other factors play in decision making?
Emotions and Risky Decision
Most decision makers rely on emotions in making judgments concerning risky decisions.
Many people are afraid of the possible unwanted consequences.
Even though emotions are subjective and irrational (or a-rational), they should be a part
of the decision making process since they show us our preferences. Since emotions and
rationality are not mutually exclusive, because in order to be practically rational, we need
to have emotions. This can lead to an alternative view about the role of emotions in risk
assessment
Fundamental concerns of decision making
The fundamental concerns of decision making are combining information about
probability with information about desires and interests.
For example: how much do you want to meet her, how important is the picnic, how much
is the prize worth?
Does business decision away from all?
Business decision
Always accompanied by conditions of uncertainty
Treating decisions as if they were gambles is the basis of decision theory.
we have to trade off the value of a certain outcome against its probability.
Canons of decision theory
we must compute the value of a certain outcome and its probabilities; hence, determining
the consequences of our choices.
The origin of decision theory is derived from economics by using the utility function of
payoffs. It suggests that decisions be made by computing the utility and probability, the
ranges of options, and also lays down strategies for good decisions.
What are the process of decision making?
Drawback in the Decision Analysis Approach
Criteria always result in selection of only one course of action.
Many decision problems, the decision-maker might wish to consider a combination of
some actions.
For example, in the Investment problem, the investor might wish to distribute the assets
among a mixture of the choices in such a way to optimize the portfolio's return.
Consider the following Investment Decision-Making Example
States of Nature(%)
Actions Growth Medium G No Change Low
G MG NC L
Bonds 12% 8 7 3
Stocks 15% 9 5 -2
Deposit 7 % 7 7 7
States of Nature
The States of Nature are the states of economy during one year. The problem is to decide
what action to take among three possible courses of action with the given rates of return
as shown in the body of the table.
Uncertainties
Make serious business decisions one is to face a future in which ignorance and
uncertainty increasingly overpower knowledge, as ones planning horizon recedes into the
distance.
The deficiencies about our knowledge of the future may be divided into three domains,
each with rather murky boundaries
Risk, uncertainty and black swans
Risk: One might be able to enumerate the outcomes and figure the probabilities.
Uncertainty: One might be able to enumerate the outcomes but the probabilities are
murky. Most of the time, the best one can do is to give a rank order to possible outcomes
and then be careful that one has not omitted one of significance.
Black Swans: The name comes from an Australian genetic anomaly. This is the domain
of events which are either “extremely unlikely” or “inconceivable” but when they
happen, and they do happen, they have serious consequences, usually bad.
Black swan
In fact, all highly man-made systems, such as, large communications networks, nuclear-
powered electric-generating stations and spacecraft are full of hidden “paths to failure”,
so numerous that we cannot think of all of them, or not able to afford the time and money
required to test for and eliminate them. Individually each of these paths is a black swan,
but there are so many of them that the probability of one of them being activated is quite
significant.
Probability is an instrument
Continuum of pure uncertainty and certainty: The domain of decision analysis models
falls between two extreme cases. This depends upon the degree of knowledge we have
about the outcome of our actions, as shown below:
Continuum of pure uncertainty and certainty: The domain of decision analysis models
falls between two extreme cases. This depends upon the degree of knowledge we have
about the outcome of our actions, as shown below:
Ignorance Risky Situation Complete Knowledge
Pure Uncertainty Probabilistic Deterministic
Model Model Model
Probability is an instrument
Probability is an instrument used to measure the likelihood of occurrence for an event.
When you use probability to express your uncertainty, the deterministic side has a
probability of 1 (or zero),
while the other end has a flat (all equally probable) probability. For example, if you are
certain of the occurrence (or non-occurrence) of an event, you use the probability of one
(or zero).
If you are uncertain, and would use the expression "I really don't know," the event may
or may not occur with a probability of 50%.
This is the Bayesian notion that probability assessment is always subjective. That is, the
probability always depends upon how much the decision maker knows.
If someone knows all there is to know, then the probability will diverge either to 1 or 0.
Quantification
The decision situations with flat uncertainty have the largest risk
Probability assessment is nothing more than the quantification of uncertainty.
Different types of decision models
Three most widely used types are:
Decision-making under pure uncertainty
Decision-making under risk
Decision-making by buying information (pushing the problem towards the deterministic
"pole")
Decision-making under pure uncertainty
the decision maker has absolutely no knowledge
not even about the likelihood of occurrence for any state of nature.
In such situations, the decision-maker's behavior is purely based on his/her attitude
toward the unknown.
Some of these behaviors are optimistic, pessimistic, and least regret, among others.
The most optimistic person I ever met was undoubtedly a young artist in Chennai who,
without a franc in his pocket, went into a swanky restaurant and ate dozens of oysters in
hopes of finding a pearl to pay the bill.
Optimist and pessimist
Optimist: The glass is half-full.
Pessimist: The glass is half-empty.
Manager: The glass is twice as large as it needs to be.
Or, as in the following metaphor of a captain in a rough sea:
The pessimist complains about the wind;
the optimist expects it to change;
the realist adjusts the sails.
Optimists are right; so are the pessimists. It is up to you to choose which you will be.
The optimist sees opportunity in every problem; the pessimist sees problem in every
opportunity.
Both optimists and pessimists contribute to our society.
The optimist invents the airplane and the pessimist the parachute.
investment decision-making situation
What will the state of the economy be next year?
Suppose we limit the possibilities to Growth (G), Same (S), or Decline (D). Then, a
typical representation of our uncertainty could be depicted as follows:
Decision Making Under Pure Uncertainty
In decision making under pure uncertainty, the decision-maker has no knowledge
regarding any of the states of nature outcomes, and/or it is costly to obtain the needed
information. In such cases, the decision making depends merely on the decision-maker's
personality type.
Personality Types and Decision Making
1.Pessimism, or Conservative (MaxMin). Worse case scenario. Bad things always happen
to me.
B 3
a) Write min # in each action row S -2) b)Choose max # and do that action. D 7*
Optimism, or Aggressive (MaxMax)
Good things always happen to me.
B 12
a) Write max # in each action row, S 15
b) Choose max # and do that action. D 7
Coefficient of Optimism (Hurwicz's Index)
Middle of the road: I am neither too optimistic nor too pessimistic.
a) Choose an a between 0 & 1, 1 means optimistic and 0 means pessimistic,
b) Choose largest and smallest # for each action,
c) Multiply largest payoff (row-wise) by a and the smallest by (1- a ),
d) Pick action with largest sum.
Example, for a = 0.7,
B(0.7*12)+(0.3*3)=9.3
S(0.7*15)+0.3*(-2)=9.9
*D(0.7*7)+(0.3*7)=7
Minimize Regret: (Savag's Opportunity Loss)
I hate regrets and therefore I have to minimize my regrets. My decision should be made
so that it is worth repeating. I should only do those things that I feel I could happily
repeat. This reduces the chance that the outcome will make me feel regretful, or
disappointed, or that it will be an unpleasant surprise
Regret payoff
Regret is the payoff on what would have been the best decision in the circumstances
minus the payoff for the actual decision in the circumstances. Therefore, the first step is
to setup the regret table:
Regret table
) Take the largest number in each states of nature column (say, L).
b) Subtract all the numbers in that state of nature column from it (i.e. L - Xi,j).
c) Choose maximum number of each action.
d) Choose minimum number from step (d) and take that action.
8(7-7)(7-7)(9-7)(15-7)Deposit
9(7+2)(7-5)(9-9)(15-15)Stocks
4 *(7-3)(7-7)(9-8)(15-12)Bonds
LNCMGG
The Regret Matrix
Limitations of Decision Making under Pure Uncertainty
1.Decision analysis in general assumes that the decision-maker faces a decision problem
where he or she must choose at least and at most one option from a set of options. In
some cases this limitation can be overcome by formulating the decision making under
uncertainty as a zero sum two person game.
2.In decision making under pure uncertainty, the decision-maker has no knowledge
regarding which state of nature is "most likely" to happen. He or she is probabilistically
ignorant concerning the state of nature therefore he or she cannot be optimistic or
pessimistic. In such a case, the decision-maker invokes consideration of security.
Limitations
3.Notice that any technique used in decision making under pure uncertainties, is
appropriate only for the private life decisions. Moreover, the public person (i.e., you, the
manager) has to have some knowledge of the state of nature in order to predict the
probabilities of the various states of nature. Otherwise, the decision-maker is not capable
of making a reasonable and defensible decision.
Decision Making Under Risk
Risk implies a degree of uncertainty and an inability to fully control the outcomes or
consequences of such an action.
Risk or the elimination of risk is an effort that managers employ. However, in some
instances the elimination of one risk may increase some other risks.
Effective handling of a risk requires its assessment and its subsequent impact on the
decision process. The decision process allows the decision-maker to evaluate alternative
strategies prior to making any decision.
process
The problem is defined and all feasible alternatives are considered. The possible
outcomes for each alternative are evaluated.
Outcomes are discussed based on their monetary payoffs or net gain in reference to assets
or time.
Various uncertainties are quantified in terms of probabilities.
The quality of the optimal strategy depends upon the quality of the judgments. The
decision-maker should identify and examine the sensitivity of the optimal strategy with
respect to the crucial factors.
Assign subjective probability
Whenever the decision maker has some knowledge regarding the states of nature, he/she
may be able to assign subjective probability estimates for the occurrence of each state. In
such cases, the problem is classified as decision making under risk. The decision-maker
is able to assign probabilities based on the occurrence of the states of nature
Risk process
a) Use the information you have to assign your beliefs (called subjective probabilities)
regarding each state of the nature, p(s),
b) Each action has a payoff associated with each of the states of nature X(a,s),
c) We compute the expected payoff, also called the return (R), for each action R(a) =
Sums of [X(a,s) p(s)],
d) We accept the principle that we should minimize (or maximize) the expected payoff,
e) Execute the action which minimizes (or maximize) R(a).
Expected Payoff
The actual outcome will not equal the expected value. What you get is not what you
expect, i.e. the "Great Expectations!"
a) For each action, multiply the probability and payoff and then,
b) Add up the results by row,
c) Choose largest number and take that action.
The Most Probable States of Nature
good for non-repetitive decisions
a) Take the state of nature with the highest probability (subjectively break any ties),
b) In that column, choose action with greatest payoff.
In our numerical example, there is a 40% chance of growth so we must buy stocks.
Expected Opportunity Loss (EOL):
a) Setup a loss payoff matrix by taking largest number in each state of nature column(say
L), and subtract all numbers in that column from it, L - Xij,
b) For each action, multiply the probability and loss then add up for each action,
c) Choose the action with smallest EOL.
Expected Value of Perfect Information (EVPI )
EVPI helps to determine the worth of an insider who possesses perfect information.
Recall that EVPI = EOL.
a) Take the maximum payoff for each state of nature,
b) Multiply each case by the probability for that state of nature and then add them up,
c) Subtract the expected payoff from the number obtained in step (b)
I Know Nothing
The Laplace equal likelihood principle
Every state of nature has an equal likelihood. Since I don't know anything about the
nature, every state of nature is equally likely to occur:
a) For each state of nature, use an equal probability (i.e., a Flat Probability),
b) Multiply each number by the probability,
c) Add action rows and put the sum in the Expected Payoff column,
d) Choose largest number in step (c) and perform that action.
I Know Nothing
The Laplace equal likelihood principle
Every state of nature has an equal likelihood. Since I don't know anything about the
nature, every state of nature is equally likely to occur:
a) For each state of nature, use an equal probability (i.e., a Flat Probability),
b) Multiply each number by the probability,
c) Add action rows and put the sum in the Expected Payoff column,
d) Choose largest number in step (c) and perform that action.
Expected Opportunity Loss (Expected Regret)
Comparing a decision outcome to its alternatives appears to be an important component
of decision-making.
One important factor is the emotion of regret. This occurs when a decision outcome is
compared to the outcome that would have taken place had a different decision been
made.
This is in contrast to disappointment, which results from comparing one outcome to
another as a result of the same decision. Accordingly, large contrasts with counterfactual
results have a disproportionate influence on decision making.
regret
regret may be related to the distinction between acts and omissions
Some studies have found that regret is more intense following an action, than an
omission. For example, in one study, participants concluded that a decision maker who
switched stock funds from one company to another and lost money, would feel more
regret than another decision maker who decided against switching the stock funds but
also lost money. People usually assigned a higher value to an inferior outcome when it
resulted from an act rather than from an omission. Presumably, this is as a way of
counteracting the regret that could have resulted from the act.
Bayesian Approach
Making a Better Decision by Buying Reliable Information
In many cases, the decision-maker may need an expert's judgment to sharpen his/her
uncertainties with respect to the probable likelihood of each state of nature. For example,
consider the following decision problem a company is facing concerning the
Sensitivity analysis is a technique for determining how much an expected payoff will
change in response to a given change in an input variable (all other things remaining
unchanged).
Steps in Sensitivity Analysis
Begin with consideration of a nominal base-case situation, using the expected values for
each input.
Calculate the base-case output.
Consider a series of "what-if" questions, to determine by how much the output would
deviate from this nominal level if input values deviated from their expected values.
Each input is changed by several percentage points above and below its expected value,
and the expected payoff is recalculated.
The set of expected payoff is plotted against the variable that was changed.
The steeper the slope (i.e., derivative) of the resulting line, the more sensitive the
expected payoff is to a change in the variable.
Scenario Analysis
Scenario analysis is a risk analysis technique that considers both the sensitivity of
expected payoff to changes in key variables and the likely range of variable values. The
worst and best "reasonable" sets of circumstances are considered and the expected payoff
for each is calculated, and compared to the expected, or base-case output.
Scenario analysis also includes the chance events, which could be rare or novel events
with potentially significant consequences for decision-making in some domain.
Integer Linear optimization Application:
Suppose you invest in project (i) by buying an integral number of shares in that project,
with each share costing Ci and returning Ri. If we let Xi denotes the number of shares of
project (i) that are purchased, then the decision problem is to find nonnegative integer
decision variables X1, X2,…, Xn --- when one can invest at most M in the n project --- is
to:
Integer Linear optimization Application
Maximize S Ri Xi
Subject to:
SXi Ci Rs. M
Application: Suppose you have 25 to invest among three projects whose estimated cost
per share and estimated return per share values are as follows:
Project Cost Return
1 5 7
2 9 12
3 15 22
Maximize 7X1 + 12X2 + 22X3
Subject to:
5X1 + 9X2 + 15X3 Rs. 25
7X1 +12X2 + 22X3
Using any linear integer programming software package, the optimal strategy is X1 = 2,
X2 = 0, and X3 = 1 with $36 as its optimal return.
Exercise-2
New Horizons Ltd wants to go in for the public share issue of Rs. 10 lakhs(1 lakhs
shares of Rs. 10 each) as a part of its effort to raise capital needed for its expansion
programme. The company is optimistic that if the issue were made now, it would be fully
taken up at a price or Rs. 30 per share.
However the company is facing situations both of which may influence the share prices
in the near future namely
a) An impending wage dispute with assembly workers which assembly workers which
could lead to strike in the whole factory could have an adverse effect on the share.
b) The possibility of a substantial business in the export market, which would increase
the share price.
The four possible events and their expected effect on the Company’s share prices are
envisaged as:
E1: No strike and export business obtained-share price rises to Rs. 34
E2:strike and export business obtained-share price rises to Rs. 30
E3:No strike and export business lost-share price rises to Rs. 32
E4: strike and export business lost-share price drops to Rs. 16
And the management has identified three possible strategies that the company could
adopt:
S1-Issue 1,00,000 shares now.
S2- issue 1,00,000 shares only after the outcome of (a) & (b) are known.
S3- Issue 50,000 shares now and 50,000 shares after the outcome (a) & (b) are known.
Calculate 1. MINIMAX Regret 2. Maxmax 3. Expected value if probability of strike is
55% and chance of getting export business is 65% 4. Expected value of perfect
information.
Answer
Pay off table (Rs. in lakhs)
30
16
23
30
32
31
30
32
30
30
34
32
S1
S2
S3
E4E3E2E1Event strategies
Calculations: see next page
Calculations
E1- 50,000*30+50,000*34 =32,00,000
E2- 1,00,000*30 = 30,00,000
E3- 50,000*30+ 50,000*32=31,00,000
E4- 50,000*30+ 50,000*16= 23,00,000
Regret table
0 4
14 14
7 7
2
0
1
0
0
0
4
0
2
S1
S2
S3
E4 Max.reg
E3 E2E1Event strategies
Minimax regret solution is S1 ie 4. alternatively include Maxmin ie. Strategy
With highest minimum pay off to which is S1 i.e. 34
Joint probability
Probability of outcome are not given directly but can be easily calculated:
E1 0.45*0.65=0.2925
E2 0.55*0.65=0.3575
E3 0.45*0.35=0.1575
E4 0.55*0.35=0.1925
Maximising expected pay off
30
28.79
29.40
30*.1975
16*.1975
23*.1975
30*.1575
32*.1575
31*.1575
30*.3575
30*.3575
30*.3575
30*.2925
34*.2925
32*.2925
S1
S2
S3
EXP.PAY OFF
E4E3E2E1Event strategies
S1 HAS THE HIGHEST EXPECTED PAYOFF I.E. 30(Rs. 30 lakhs)
Expected value of perfect information
9.94
10.72
5.04
5.78
31.48
0.2925
0.3575
0.1575
0.1925
34
30
32
30
E1
E2
E3
E4
TOTAL
EVPIJoint probability
Max.payoff
EXPECTED VALUE OF PERFECT
INFORMATION=31.48-30=1.48 OR RS.1,48,000
Example Engineering Ltd. Manufacture engines.They have been asked to bid on prospective contract for 90 engines for cars. They have completed an initial run of 30 of these mounting at the following costs:
Direct material Rs. 20,000; Direct labour(6000 hours at Rs.4 per hour)-24000; tooling cost (re-usable)- Rs.3000; variable overhead(Rs.0.50 per labour hour)-Rs.3000;Fixed overhead(Rs.0.50 per labour hour)-Rs.6000.
If 80% learning curve is thought to be pertinent in this case. The Marketing Director believes that the quotation is unlikely to be accepted if it exceeds Rs. 1,10,000 and as the Company are short of work, he believes the contract to be vital.
You are required to comment whether it is worth accepting at Rs. 1,10,000.
No cumulative cumulative cumulative average
Qty.manufa. Hours hours per unit
1. 30 3000 200 ie 6000/30
2. 60 9600(160*60) 160 ie 80% of 200
3. 120 15,369(120*128) 128 ie.80%of 160
Additional hours for 90 additional engines= 15369-6000=9360 hours
Incremental costs for 90 engines:
direct material Rs.60,000
direct labour(9360*4) 37,440
Tooling cost nil
variable overheads(9360*0.5) 4,680
fixed overhead nil
total 1,02,120
Net saving=1,10,000-1,02,120=Rs.7880
3. Indian would like to have travelers cheques: GBP-
STERLING 72.70-73.25
A) explain the quote
B) compute the spread
C) How much would you pay for purchasing 250 pounds in
TCS?
D) If you have a balance of pounds 23 in travellers cheques ,
how many rupees would you receive if the bank in india
quotes 73.65-73.92?
4.Explain the sections 2(1B) and 72A of Income tax Act with
respect to amalgamation/absorption
1. Distinguish between Forwards and Futures
2. Distinguish between spread and swap points.
. Explain the following terms: a) Strike price b) forward price c)in
the money d)Bid and Ask e)holder and writer
4. The current market price is Rs. 50 has the following exercise
price and cal option premium. Compute intrinsic value and time
value
Exercise
price
premium
45
48
50
52
55
5
6
4
5
7
1. consider the following Euro/USD direct quote 0.9345-0.9375
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