LECTURE NOTES 2 EC 403
RISK ANALYSIS
Introduction
Managers are expected to come up with good decisions. A good decision is one that is based on logic,
considers all available data and possible alternatives and applies the quantitative approach.
Occasionally, a good decision results in an unexpected or unfavourable outcome. If it is made properly
it still is a good one.
A bad decision is one that is not based on logic, does not consider all alternatives and does not employ
appropriate quantitative techniques. If a manager makes a bad decision, but is luck and a favourable
outcome occurs, he still has made a bad decisions.
Steps in Decision making
There are ten steps
(i)
Realize that there is a problem and clearly define the problem at hand.
(ii)
Obtain suitable information on all aspects of the problem.
(iii)
Judge the relevance and validity of all information obtained.
(iv)
List the possible alternatives and Evaluate these alternatives.
(v)
Identify the possible outcomes.
(vi)
List the pay-off or profit of each combination of alternatives and outcomes in a decision table.
(vii)
Select one of the mathematical decision theory models. Selection of the model depends the
environment the firm is operating in and the amount of risk and uncertainty involved.
(viii)
Mobilize the resources required.
(ix)
Apply the model and make your decision.
(x)
Check to determine if the problem has been solved.
Types of Decision making Environments
There are alternative states of information or decision making environments.
1. Certainty- Decision making under this environment, the decision-maker knows with certainty the
consequences of every alternative or decision choice. The decision-maker is perfectly informed in
advance about the outcome of his decisions. For each decision there is only one possible outcome
which is known to the decision-maker. Under conditions of certainty there is accurate, measurable
reliable information available on which to base a decision. For example a man who has $100 to
invest for one year may open a savings account paying 5% interest and the second is to invest
same amount in treasury bills paying 10% interest, there is certainty the treasury bill will be a
better investment and will be chosen.
2. Risk- Risk is probability that an outcome will not be as expected. Decision making under risk is
where predictability is lower. Complete information is unavailable. A situation of risk is when
either of two or more events (outcomes) may follow an act (decision) and where all of these events
and the probability of each occurring, are known to the decision-maker. In other words in this
situation of risk a decision may have more than one possible outcome, so that certainty no longer
exists and the decision maker is aware of all possible outcomes and knows the probability of each
one occurring. In this environment, the decision-maker will attempt to maximize his/her expected
well being. In this environment decision theory models for business problems typically employ
two equivalent criteria- maximization of expected monetary value and minimization of expected
loss.
3. Uncertainty- Decision environment here, very little is known. In this situation a decision may have
more than one outcome and the decision-maker does not know the precise nature of these
outcomes, nor can he objectively assign a probability to the outcomes. In other words not all
outcomes may be accurately foreseen and the probabilities cannot be deduced or based on
empirical data. The decision maker has to use intuition, judgement and experience to assign the
probabilities to the outcomes considered possible in such a situation. Assignment of probabilities is
on a subjective basis. Instead of immediately identifying the appropriate course of action, or
solution to a problem,(because of limited knowledge and experience), actions and solutions are
arrived at through a process of searching through sequences of possible alternatives, using past
experience and rules of thumb as guidelines.
1
Techniques for decision making under risk.
Decision making under risk is a probabilistic situation. Several states of nature may occur, each with a
given probability. The probabilities are known.
Measures of risk
(i)
Standard deviation- the lower the standard deviation the lower the risk.
 
 p (x
i
i
 x) 2 where pi is probability of outcome i; xi is value of outcome i; x is
the mean value or expected value of all outcomes. When considering choices between
alternative courses of action, decision markets may be thought of as deciding on different
combinations of return on the one hand, measured by the expected value of returns and risk on
the other, measured by the standard deviation of those returns. The smaller the standard
deviation, the lower the riskiness of the alternative.
Suppose there are two projects A and B whose returns are as stated below and probabilities of
states of nature also given.
Project A
State of economy
Returns if state occurs
Probability p
i
Boom
Normal
Recession
xi
0.2
0.6
0.2
Project B
State of economy
Boom
Normal
Recession
$600
$500
$400
Probability
Returns if state occurs
pi
xi
0.2
0.6
0.2
$1000
$500
$0
The standard deviations are calculated as shown below for the two projects
Project A
State
of
economy
Boom
Normal
Recession
Expected value
Probability
pi
0.2
0.6
0.2
Standard deviation
Project B
State
of
economy
Boom
Normal
Recession
Expected value
pi
if
$600
$500
$400
 x )2
Expected
value
xi  x
$120
$300
$80
$500
100
10 000
0
0
(100)
10 000
Variance
Expected
value
xi  x
$200
$300
$0
$500
500
250 000
0
0
(500)
250 000
Variance
( xi
pi ( xi  x )2
2000
0
2000
4000
 = 4000  $63.00
Probability
0.2
0.6
0.2
Outcome
state occurs
Outcome
state occurs
$1000
$500
$0
if
( xi
 x )2
pi ( xi  x )2
50 000
0
50 000
$100 000
Standard deviation  = 100000  $316.23
The standard deviation of Project A is $63.25; that of B is $316.23. By the standard deviation
criterion, project B is riskier since its standard deviation is much larger than the standard
deviation for project A. Since the expected values for the returns from the two projects are
equal at $500, project A would be preferred. The standard deviation as a measure of risk has
problems where projects have the same standard deviation but different expected returns, the
standard deviation alone cannot bee used hence the risk per dollar is a much better measure.
The project with a lower risk per dollar will be selected.
2
(ii)
Coefficient of Variation- this can be used as an alternative measure to the standard deviation.
It handles the problem mentioned above. It divides the standard deviation by the mean
deviation by the mean or expected value of the net cash flows, to obtain the coefficient of
variation (CV).
In formula terms, CV=
Project C
Project D
(iii)

x
where
x is the mean or expected value of net cash flows.
Standard deviation
300
Standard deviation
300
Expected value
$1000
Expected value
$4000
C.V.
0.3
C.V.
0.075
Since D has a lower coefficient of variation, of variation, it has less risk per unit of return than
investment C.
EXPECTED MONETARY VALUE (EMV)
The expected monetary value (EMV) of an event is the payoff, should that event occur,
multiplied by the probability (the probability is known) that the event will occur. In other
words, the EMV for an alternative is just the sum of possible pay-offs of the alternative, each
weighted by the probability of that pay-off occurring. It is the mean of the probability
distribution in question.
Formula:
EMV   Ri Pi where Ri is return or pay-off of the ith event, and Pi is the
probability of the ith event.
Case study: Mr. Thambolenyoka is a director of a profitable firm. He has identified the
problem whether to expand his product line by manufacturing and marketing a new product,
backyard storage sheds. He has three options; to construct:
(1) A large new plant to manufacture the storage sheds.
(2) A small plant or
(3) No plant at all.
Thambo has determined that there are only two possible outcomes: the market for the sheds
could be favourable (high demand for the sheds) or unfavourable (low demand for the sheds)
He has already evaluated the potential profits associated with the various outcomes. With a
favourable demand he thinks a large facility would result in a net profit of $200 000, to his
firm. This profit is conditional upon building a large factory and there is a good (favourable)
market. If the market is unfavourable there would be a net loss of $180 000. A small plant
would result in a net profit of $100 000 in a favourable market but a net loss of $20 000 would
occur if the market was unfavourable. Finally doing nothing would result in a $0 profit in
either market. Using the EMV what should Thambo do?
To solve the problem, construct a decision table, also called a pay-off table or pay-off matrix.
A pay-off matrix is a table that shows the possible outcomes or results of each strategy under
each state of nature. States of nature refers to conditions in the future that will have a
significant effect on the degree of success or failure of any strategy.
Decision table with conditional values
ALTERNATIVES
STATES OF NATURE
(or strategies)
Favourable market ($)
Unfavourable market ($)
Construct a large plant
$200 000
-180 000
Construct a small plant
$100 000
- 20 000
Do nothing
0
0
The alternatives listed in the first column can also be referred to as strategies. A strategy is
one of several alternative courses of action that a decision maker can take to achieve a goal.
The alternative that gives the highest EMV is the one to go for.
Alternative 1 :Large facility
States of nature
Pay-off (Ri)
Favourable market
$200 000
Unfavourable market
($180 000)
EMV
Probability (Pi)
0.5
0.5
3
RiPi
$100 000
($90 000)
$ 10 000
Alternative 2: Small facility
States of nature
Pay-off (Ri)
Favourable market
$100 000
Unfavourable market
($ 20 000)
EMV
Probability (Pi)
0.5
0.5
RiPi
$50 000
($10 000)
$ 40 000
Alternative 3 :Do nothing
States of nature
Pay-off (Ri)
Probability (Pi)
RiPi
Favourable market
$0
0.5
$0
Unfavourable market
$0
0.5
$0
EMV
$0
Since the largest EMV ($40 000) results from the second alternative, building a small facility,
Thambo should to put up a small plant to manufacture the sheds.
Limitations of expected values
1. Expected value is an average and therefore only really applicable when there are repeated
trials or decisions. Many business decisions are one-offs however.
2. The probabilities employed will often be subjective estimates by managers and therefore
influenced by their personalities (optimistic, pessimistic) and objectives of the individuals
making the estimates.
3. Although the EMV provides an average of the future values, it does not measure the
degree of possible spread around the average (i.e. risk of the decision).
4. It does not reflect the personal attitudes to risk of either individual managers or a group of
managers.
5. Individuals will not accept fair bets involving large amounts of money because they ‘care’
more about the possibility of loss than they do about the possibility of an equal gain.
(Davies and Lam(2001:242).
EXPECTED VALUE OF PERFECT INFORMATION (EVPI)
Suppose a KMO Marketing Company claims that its technical analysis will tell Mr. Thambo
with certainty whether or not the market is favourable for his proposed product. In other
words KMO claims it can change his environment from one of decision making under risk to
one of decision making under certainty. This information could prevent Thambo from making
a very expensive mistake. For this service (information), KMO will charge $65 000. The
question now is whether Thambo should hire the firm or not and whether even if the
information is accurate, it is worth the amount. The value of such perfect information can be
useful as it places an upper bound on what Thambo would be willing to spend on information
such as being sold by KMO.
The Expected Value of Perfect Information (EVPI) and the Expected Value With Perfect
Information (EVWPI) can help Thambo make his decision about the marketing consultant.
Expected Value With Perfect Information (EVWPI)
Is the average or expected value of the decision if you know what would happen ahead of
time. You have perfect knowledge before a decision has to be made. In order to calculate this
value choose the best alternative for each state of nature and multiply its pay-off times the
probability of occurrence of that state of nature.
Expected Value With Perfect Information = (Best outcome for 1st state of nature) x (Prob. of
1st state of nature) + (best outcome for 2nd state of nature) x (Prob. of 2nd state of nature)-------------- + (Best outcome for last state of nature ) x (Prob. of last state of nature)
The Expected Value of Perfect Information, EVPI is the expected outcome with perfect
information minus the expected outcome without perfect information, the maximum, EMV.
EVPI= Expected value with perfect information – maximum EMV or
EVPI=EVWPI-EMVmax
ALTERNATIVES
Construct a large plant
Construct a small plant
Do nothing
STATES OF NATURE
Favourable market ($)
Unfavourable market ($)
$200 000
-180 000
$100 000
- 20 000
0
0
4
Under favourable market the best outcome is a pay-off of $200 000 (when he constructs a
large facility) while under the unfavourable market condition the best is a pay-off of $0 when
he does nothing.
EVWPI= $200 000(0.5) + $0 (0.5) = $100 000. Thus if there is perfect information, we would
expect on the average $100 000 if the decision could be repeated many times.
The maximum EMV is $40 000, which is the expected outcome without perfect information.
EVPI=EVWPI –EMVmax = $100 000 - $40 000= $60 000. Thus, basing on the assumption that
the probability of each state of nature is 0.5, the most Thambo would be willing to pay for
perfect information is $60 000. He would thus not hire the firm to pay $65 000.
EXPECTED OPPORTUNITY LOSS (EOL)
This is an alternative approach to maximizing the EMV. In this approach the aim is to minimize the
expected opportunity loss (EOL). Opportunity loss, sometimes called regret, refers to the difference
between the optimal profit or pay-off and the actual pay-off received, i.e. it is the amount lost by not
picking the best alternative. The minimum expected opportunity loss is found by constructing an
opportunity loss table and computing EOL for each alternative. Using the case study above Thambo’s,
the steps involved are:
1. Create the opportunity loss table. This is done by determining the opportunity loss for not
choosing the best alternative for each state of nature. Opportunity loss for any state of nature
or any column is calculated by subtracting each outcome in the column from the best
outcome in the same column. For a favourable market the best outcome is $200 000, as a
result of the first alternative, building a large facility. For an unfavourable market, the best
outcome is $0, as a result of the third alternative, doing nothing.
Favourable market
$200 000 – 200 000 = $0
$200 000 – 100 000 = $100 000
$200 000 – 0 = $200 000
States of Nature
Unfavourable Market
0- (-180 000) = $180 000
0- (-20 000) = $ 20 000
0- 0 = $0
The values in this table represent the opportunity loss for each state of nature for not choosing
the best alternative.
Alternatives
Large facility
Small facility
Do nothing
Probabilities
2.
Favourable market $
$0
$100 000
$200 000
0.5
States of Nature
Unfavourable market $
$180 000
$ 20 000
$0
0.5
Compute the EOL by multiplying the probability of each state of nature times the appropriate
opportunity loss value.
EOL (Building large facility) = 0.5(0) + 0.5($180 000)= $90 000.
EOL (Building a small facility) = 0.5 ($100 000) + 0.5 ($20 000) = $60 000
EOL (Doing Nothing) = 0.5($200 000) + 0.5($0) = $100 000
Using minimum EOL as the decision criterion, the best decision would be the second
alternative, build a small facility. N.B. Minimum EOL will always result in the same decision
as maximum EMV, and that the following relationship always holds: EVPI = minimum EOL.
DECISION MAKING UNDER UNCERTAINTY
When the probability of occurrence of each state of nature can be assessed, the EMV or EOL decision
criteria are usually appropriate. When a manager cannot assess the outcome p or when virtually no
probability data are available, other decision criteria are required. This type of problem is referred to as
decision making under uncertainty.
The decision criteria that can be used are:
(i)
Maximax.
(ii)
Maximin.
(iii)
Equally likely.
(iv)
Criterion of realism
(v)
Minimax
5
The first four can be computed directly from the decision table, while the minimax criterion normally
requires the use of the opportunity loss table. (NB. No probability about the outcomes is available to
the manager.)
MAXIMAX
This is an optimistic decision making criterion. It presumes that the best will always happen. It is an
optimistic criterion that maximizes the maximum pay-off. The criterion finds the alternative that
maximizes the maximum outcome or consequence for every alternative. Procedure:First locate the
maximum outcome within every alternative and then pick that alternative with the maximum number.
Since this decision criterion locates the alternative with the highest possible gain, it has been called an
optimistic decision criterion.
In Thambo’s case, the maximax choice is to build a large facility. This is the maximum of the
maximum of the maximum number within each row or alternative.
Thambo’s maximax decision
STATES OF NATURE
ALTERNATIVES
Favourable market ($)
Unfavourable
market Maximum in row
($)
Construct a large plant $200 000
-180 000
$200 000
Construct a small plant
Do nothing
$100 000
0
- 20 000
0
$100 000
0
Maximum
MAXIMIN
This is a pessimistic decision making criterion which maximizes the minimum pay-off. It involves
choosing the strategy that promises the best of the worst possible outcomes. The criterion finds the
alternative that maximizes the minimum outcome or consequence for every alternative. To use the
criterion, first locate the minimum outcome within every alternative and then pick that alternative with
the maximum number. Since this decision criterion locates the alternative that has the least possible
loss, it has been called a pessimistic decision criterion. Thambo’s Maximin choice is to do nothing and
the decision is shown in the table below.
MAXIMIN DECISION
ALTERNATIVES
Construct a large plant
Construct a small
plant
Do nothing
STATES OF NATURE
Favourable market ($)
Unfavourable
($)
$200 000
-180 000
$100 000
- 20 000
0
market
Minimum in row
- 180 000
- 20 000
0
0
Maximin
EQUALLY LIKELY
This is also called Baye’s criterion or insufficient reason. This is a decision criterion that places an
equal weight on all states of nature. The equally likely decision criterion finds that alternative with the
highest average outcome. To use the criterion, first calculate the average outcome for every alternative,
which is the sum of all outcomes divided by the number of outcomes. Pick that alternative with the
maximum number. The equally likely choice for Thambo is the second alternative, to build a small
plant. This strategy shown in the table below is the maximum of the average outcome of each
alternative.
Thambo’s Equally Likely Decision
STATES OF NATURE
Row Average
ALTERNATIVES
Favourable market ($)
Unfavourable market ($)
Construct a large plant $200 000
-180 000
$10 000
Construct a small plant $100 000
- 20 000
$40 000
Do nothing
0
0
$0
Equally Likely
6
CRITERION OF REALISM (HURWICZ CRITERION)
This is often called the weighted average. It is a middle ground criterion between Maximax and
Maximin. In other words it is a compromise between an optimistic and a pessimistic decision criterion.
It requires the decision maker to select or specify a coefficient or index of optimism (also called
coefficient of realism) alpha,  ( 0    1 ). In other words, the coefficient is between 0 and 1.
When  is close to 1, the decision maker is optimistic about the future. When  is close to 0, the
decision maker is pessimistic about the future. The advantage of this approach is that it allows the
decision maker to build in, personal feelings about relative optimism and pessimism.
To apply this criterion to decision making, the decision maker must determine first both the maximum
and minimum pay-off for each decision alternative. The maximum pay-off for each decision alternative
is indicated in colour and the minimum pay-off in black.
STATES OF NATURE
Favourable market ($)
Unfavourable market ($)
$200 000
-180 000
$100 000
- 20 000
0
0
ALTERNATIVES
Construct a large plant
Construct a small plant
Do nothing
For each decision alternative the realism value is a measure of realism or
Criterion of Realism =  (maximum pay-off in row) + (1-  )(minimum pay-off in a row).
Going back to Thambo’s case study, if he sets his coefficient of realism,  to be 0.8 the outcomes
would be:
Construct a large plant
Criterion of Realism = 0.8(200 000) +0.2(-180 000)= $124 000
Construct small plant
Criterion of Realism = 0.8(100 000) +0.2(-20 000) = $ 76 000
Do nothing
Criterion of Realism = 0.8(0) + 0.2(0) = $0
The best decision would thus be to build a large plant (it has the highest weighted average of $124
000).
SUMMARY OF CRITERION OF REALISM DECISION
Criterion of
STATES OF NATURE
ALTERNATIVES Favourable market Unfavourable market Realism or
weighted average
($)
($)
 = 0.8
Construct a large $200 000
-180 000
$124 000
plant
Construct a small $100 000
- 20 000
$ 76 000
plant
Do nothing
0
0
$0
Realism
Suppose another decision maker has the following alternatives for his firm in the
decision table below:
Decision Alternative
STATES OF NATURE (DEMAND)
High
Moderate
Low
Failure
Expand
$500 000
$250 000
-$250 000
-$450 000
Build
$700 000
$300 000
-$400 000
-$800 000
Subcontract
$300 000
$150 000
-$ 10 000
-$100 000
If in this case the decision maker feels fairly optimistic and assigns an  = 0.7 for the three decision
alternatives, the realism values are:
Expand 0.7(500 000) + 0.3 (-450 000) = $215 000
7
Build
0.7(700 000) + 0.3 (-800 000) = $250 000
Subcontract
0.7(300 000) + 0.3 (-100 000) = $180 000
In this situation the decision is to build. The advantage of using the criterion of realism is the decision
maker is able to introduce his own personal feelings of relative optimism or pessimism into the
decision process.
MINIMAX or Minimax Regret Decision Rule
This criterion minimizes the maximum opportunity loss. It finds that alternative that minimizes the
maximum opportunity loss within each alternative. To apply the criterion first find the maximum
opportunity loss within each alternative, then pick that alternative with the minimum number.
Opportunity Loss is the amount lost by not picking the best alternative and is equal to the difference
between optimal profits or pay-off and the actual pay-off received. The Opportunity loss for any state
of nature or any column is calculated by subtracting each outcome in the column from the best outcome
in the same column e.g. for favourable market best outcome is $200 000 as a result of first alternative,
build large facility.
Thambo’s Minimax Decision
ALTERNATIVES
Construct a large plant
Construct a small plant
Do nothing
Maximum in column
STATES OF NATURE
Favourable market ($)
Unfavourable market ($)
$200 000
-180 000
$100 000
- 20 000
0
0
$200 000
0
Opportunity Loss Table
ALTERNATIVES
Construct a large plant
Construct a small plant
Do nothing
ALTERNATIVES
Construct a large plant
Construct a small plant
STATES OF NATURE
Favourable market ($)
Unfavourable market ($)
$200 000-$200 000
0 - (-180 000)
=0
=180 000
$200 000-$100 000
0 - (- 20 000 )
=$100 000
= $20 000
$200 000 -0
0 -0
=$200 000
=0
STATES OF NATURE
Favourable market ($)
Unfavourable
($)
0
180 000
$100 000
20 000
market
Maximum in row
$
180 000
100 000
Do nothing
$200 000
0
200 000
Minimax
DECISION TREES
Any problem that can be presented in a decision table can also be graphically illustrated in a decision
tree.
When trees contain both probabilities of outcomes and conditional monetary values of those outcomes,
such that expected values can be computed, the common practice is to refer to them as decision trees.
Or A decision tree is a graphic display of the decision alternatives, states of nature, probabilities,
probabilities attached to the states of nature and conditional benefits and losses.
All decision trees contain decision nodes and state of nature nodes. A decision node or a decision fork
is a node from which one of several alternatives may be chosen. The decision maker is in control.
A state of nature node is a node out of which one state of nature will occur. This is also called a chance
fork.
8
Decision node
State of nature node
Steps in Solving Problems with Decision Trees
1. Define the problem.
2. Structure or draw the decision tree.
3. Assign probabilities to the state of nature.
4. Estimate pay-offs for each possible combination of alternatives and states of nature.
5. Solve the problem by computing expected monetary values (EMV) for each state of nature
node. (This is done by working backward that is starting at the right of the tree and working
back to the decision nodes on the left. This is called the roll back approach.)
Solved decision tree diagram for Tambolenyoka
Pay-off
Favourable market (0.5) $200 000
Pr x Pay-off
$100 000
Node 1
Large plant
Unfavourable Market(0.5) -180 000
Total EMV
Favourable market (0.5) $100 000
-$90 000
$ 10 000
$ 50 000
Unfavourable market(0.5) -$20 000
Total EMV
Favourable market (0.5) $0
-$10 000
$ 40 000
Node 2
Small plant
Do nothing
2
Total EMV
$0
Unfavourable market (0.5) $0
NB. If cost of project is subtracted from the Expected Values, the result is Net Expected Monetary
value.
The branch leaving the decision node leading to the state of nature node with the highest EMV will be
chosen, a small plant.
Case Study
Phillips Electrical must decide to build a large or small plant to produce a new Vacuum cleaner which
is expected to have a market life of 10 years. A large plant will cost $28m to build and put into
operation, while a small plant will cost only $14m to build and put into operation. The company’s best
estimate of a discrete distribution of sales over the 10 year period is:
High Demand
Probability
= 0.5
Moderate Demand
Probability
= 0.3
Low Demand
Probability
= 0.2
Cost volume profit analysis done by the management at Phillips Electrical, indicates these conditional
outcomes under the various combinations of plant size and market size:
1. A large plant with high demand would yield $10m annually in profits.
9
2.
3.
4.
A large plant with moderate demand would yield $6m annually in profits.
A large plant with low demand would lose $2m annually because of production inefficiencies.
A small plant with high demand would yield only $2.5m annually in profits, considering the
cost of the lost sales because of inability to supply customers.
5. A small plant with moderate demand would yield $4.5m annually in profits because the cost
of lost sales would be somewhat lower.
6. A small plant with low demand would yield $5.5m annually because the plant size would be
matched fairly optimally.
Which project should Phillips Electrical embark on? (NB. There is no discounting in this case, but
under normal circumstances there should be. Also subtract the Plant cost from the expected value to get
a net expected value.)
10