2. 2
Topics
- Risk and Uncertainty
- General Risk Categories
- Probability
- Probability Distributions
- Payoff Matrix
- Expected Value
- Variance and Standard Deviation
(Measurement of
Absolute Risk)
- Coefficient of Variation
(Measurement of Relative Risk)
3. 3
Topics (Con’t)
- Risk Attitudes
Risk Aversion
Risk Neutrality
Risk Loving (Taking)
- Utility Theory and Risk Analysis
- Risk Premium
- Decision-Making Under Risk
- Certainty Equivalent
- Game Theory
- Maximin Decision Rule
- Minimax Regret Decision Rule
5. 5
Motivation
Risk – a four-letter word
To make effective investment decisions,
one must understand risk.
Decision makers sometimes know with
certainty the outcomes associated with
each possible course of action.
6. 6
Motivation (Con’t.)
Example:
A firm with BDT 10,00,000 in cash
Decision to make:
(1) Invest in a 91-day Treasury bill yielding 5.66%
interest
(2) Prepay a 15% bank loan
Which course of action to take?
Choose (1) => 14,307 interest income after 91 days
Choose (2) => 37,916 interest expense savings after
91 days
Choose (2) provides 23,609 additional 3-months return
7. 7
Definition of Risk and
Uncertainty
- Risk/Uncertainty: Both concepts deal with
the probability of loss or the chance of
adverse outcomes
- Risk: All possible outcomes of managerial
decisions and their probabilities are
completely known
- Uncertainty: The possible outcomes and
their probabilities are unknown
8. 8
General Risk Categories
Business Risk – the chance of loss associated with a given
managerial decision; typically a by-product of the unpredictable
variation in product demand and cost conditions
Market Risk – the chance that a portfolio of investments can lose
money because of overall swings in financial markets
Inflation Risk – the danger that a general increase in the price level
will undermine the real economic value of corporate agreements
Interest-rate Risk – another type of market risk that can affect the
value of corporate investments and obligations
Credit Risk – the chance that another party will fail to abide by its
contractual obligations
9. 9
General Risk Categories (Con’t.)
Liquidity Risk – the difficulty of selling corporate assets or
investments that have only a few willing buyers or are otherwise
not easily transferable at favorable prices under typical market
conditions
Derivative Risk – the chance that volatile financial derivatives such
as commodity futures and index options could create losses by
increasing rather than decreasing price volatility
Currency Risk – the chance of loss due to changes in the domestic
currency value of foreign profits
10. 10
Probability
- Probability: likelihood of particular outcome occurring, denoted
by p. The number p is always between zero and one.
- Frequency: estimate of probability, p=n/N, where n is number of
times a particular outcome occurred during N trials.
- Subjective probability: If we do not have frequency, we often
resort to informed guesses. Subjective probabilities must follow
the same rules of the probability calculus, if we are dealing with
rational decision-makers.
11. 11
Probability Distribution
Discrete probability distribution:
deals with “events” whose “states of
nature” are discrete. The “event” is
the state of the economy. The “states
of nature” are recession, normal, and
boom.
Continuous probability distribution:
deals with “events” whose “states of
nature” are continuous values. The
“event” is profits, and the “states of
nature” are various profit levels.
Event
State of Economy
P (probabilty)
Recession 0.2
Normal 0.6
Boom 0.2
12. 12
Payoff Matrix
A table that shows outcomes associated with each possible state of nature.
State of Economy Project A Project B Probability of State of Economy
Recession $4,000 $0 0.2
Normal $5,000 $5,000 0.6
Boom $6,000 $12,000 0.2
Project A more desirable in a recession.
Project B more desirable in a boom.
In a normal economy, the projects offer the same profit potential.
Decision to Make:
A firm must choose only one of the two investment projects
(choose Project A or Project B). Each calls for an outlay of
$10,000.
13. 13
Expected Value
The payoffs of all events: x1, x2, …, xN
The probability of each event: p1, p2, …, pN
Expected value of x:
EV(x) is a weighted-average payoff, where the weights are defined by the
probability distribution.
Use the payoff matrix in the previous slide, together with the probability
of each state of the economy.
pxpxpxpxxEV i
N
i
iNN ∑=
=+++=
1
2211 ...)(
15. 15
Variance and Standard
Deviation
Variance and Standard Deviation: measuring risk
The payoffs of all events: x1, x2, …, xN
The probability of each event: p1, p2, …, pN
Expected value of x:
Variance:
Standard deviation: square root of variance
∑
=
−=−++−+−=
N
i
piEVxipNEVxNpEVxpEVx
1
)( 2)( 2...2)2( 2
1)1( 22σ
pxpxpxpxxEV i
N
i
iNN ∑=
=+++=
1
2211 ...)(
16. 16
For project A, what are the variance and standard deviation? EV(A) =
$5,000
Variance (σ2
) = ($4,000-5,000)2
(.2) + ($5,000-$5,000)2
(.6) + ($6,000-
$5,000)2
(.2)
(σ2
) = ($1,000)2
(.2) + ($0)2
(.6) + ($1,000)2
(.2)
(σ2
) = $400,000 (units are in terms of squared dollars)
σA = $632.46
For project B, what are the variance and standard deviation? EV(B) =
$5,400
Variance (σ2
) = ($0-5,400)2
(.2) + ($5,000-$5,400)2
(.6) + ($12,000-$5,400)2
(.2)
(σ2
) = 5,832,000 + 96,000 + 8,712,000 (units are in terms of squared
dollars)
(σ2
) = 14,640,000 (units are in terms of squared dollars)
σB = $3,826.23
Project B has a larger standard deviation; therefore it is the riskier
project
17. 17
Risk Measurement
Absolute Risk:
- Overall dispersion of possible payoffs
- Measurement: variance, standard deviation
- The smaller variance or standard deviation, the
lower the absolute risk.
Relative Risk
- Variation in possible returns compared with the
expected payoff amount
- Measurement: coefficient of Variation (CV),
- The lower the CV, the lower the relative risk.
EV
CV
σ
=
18. 18
Project A
EV(A) = $5,000
σA = $632.46
Project B
EV(B) = $5,400
σ B = $3,826.23
Coefficient of variation
CVA = = 0.1265
CVB = = 0.7086
Coefficient of variation measures the relative risk; the
variation in possible returns compared with the expected
payoff amount.
σ A
E V A( )
C V
E V
=
σ
σ B
E V B( )
19. 19
Risk Aversion
characterizes decision makers who seek to avoid or minimize risk.
Risk Neutrality
characterizes decision makers who focus on expected returns and
disregard the dispersion of returns.
Risk Seeking (Taking)
characterizes decision makers who prefer risk.
Risk Attitudes
20. 20
Scenario: A decision maker has two choices, a sure thing
and a risky option, and both yield the same
expected value.
Risk-averse behavior:
Decision maker takes the sure thing
Risk-neutral behavior:
Decision maker is indifferent between the two choices
Risk-loving (or seeking) behavior:
Decision maker takes the risky option
Risk Attitudes
21. 21
Typically, consumers and investors display risk-averse behavior, especially
when substantial sums of money are involved. Risk aversion is the general
assumption behind decision models in managerial economics.
Examples to the contrary:
State-run lotteries
Casinos (gaming)
Today, U.S. consumers spend more on legal games of chance than on movie
theaters, books, amusement attractions, and recorded music combined!
Source: Wall Street Journal, Ann Davis, September 23, 2004.
Utility Theory and Risk Analysis
23. 23
Examples of utility functions
Let w = income (or profit) or more generally wealth, w > 0
U w w
U w w
U w w
U w w
( )
( ) l n
( )
( )
=
=
=
= +
2
3 1 0
M U w w
w
M U w
w
M U w w
M U w
( )
( )
( )
( )
= =
=
=
=
−1
2
1
2
1
2
3
1
2
Which utility function is consistent with risk-seeking behavior?
Which utility function is consistent with risk neutrality?
Which utility function is consistent with risk aversion?
24. 24
Under risk aversion behavior, the decision rule is to
maximize expected utility.
EV[U(risky option)] = U(w1)p1 + U(w2)p2 + U(wn)pn
Also, it is important to find the level of income, profits, or wealth
that is consistent with the utility level of the expected value of
utility associated with the risky option. Call this level of wealth
w*
.
The difference between the expected value of the risky option
and w*
is the risk premium.
Risk premium = EV(risky option) – w*
25. 25
Example:
Joshua lives in San Francisco, CA, where the probability of an
earthquake is 10%. Suppose that Joshua’s utility function is
given by , where w represents total wealth. If Joshua
chooses not to buy insurance next year, his wealth is $500,000
if no earthquake occurs, and $300,000 if an earthquake occurs.
The reduction in wealth is attributable to the loss of his house
due to the earthquake. The risky option is not buying
insurance.
(a) Is Joshua risk averse, risk loving, or risk neutral?
(b) Find the EV of not buying insurance (the risky option).
(c) Find EV[U(risky option)].
(d) Find the wealth that results in the utility level associated
with the expected value of utility of the risky option.
(e) Find the risk premium.
U w w( ) =
26. 26
(a) Is Joshua risk averse, risk loving, or risk neutral?
MU diminishes with increases in w.
Joshua is risk averse
(b) Find the EV of not buying insurance.
EV(risky option) = ($500,000)(.9) + ($300,000)(.1) =
$480,000
(c) Find EV[U(risky option)].
EV[U(risky option)] =
EV[U(risky option)] = $636.40 + $54.77 = $691.71
(d) What level of wealth is consistent with the utility level of the
EV[U(risky option)]?
(e) Risk premium.
EV(risky option) – w*
= $480,000 - $477,716
U w
M U w
w
=
= =
−1
2
1
2
1
2
1
2
$ 5 0 0 , ( . ) $ 3 0 0 , ( . )0 0 0 9 0 0 0 1+
w
w
=
=
$ 6 9 1 .
$ 4 7 7 ,*
1 7
7 1 6
28. 28
Decision-Making Under Risk
Possible Criteria to consider:
- Maximize expected value
- Minimize variance or standard deviation
- Minimize coefficient of variation
- Incorporate risk attitudes: certainty equivalent
- Maximin criterion
29. 29
Maximizing Expected Value
Event (State of Economy) P Profit
Project A Project B
Recession 0.2 $4,000 $0
Normal 0.6 $5,000 $5,000
Boom 0.2 $6,000 $12,000
EV(A)=$5,000 EV(B)=$5,400
Thinking:
Which project will you choose based on this criterion?
What is ignored using this criterion?
30. 30
Minimizing Variance/Standard
Deviation
Event (State of Economy) P Profit
Project A Project B
Recession 0.2 $4,000 $0
Normal 0.6 $5,000 $5,000
Boom 0.2 $6,000 $12,000
σ A = $632.46 = $3,826.23
Thinking:
Which project will you choose based on this criterion?
What is ignored using this criterion?
σ B
31. 31
Coefficient of Variation: Standard
Deviation Divided by the Expected Value
A B
Expected value $5,000 $5,400
Standard deviation $632.46 $3,826.23
Coefficient of Variation 0.2265 0.7086
Think:
Which project will you choose based on this criterion?
What is ignored?
32. 32
Incorporating Risk Attitudes:
Certainty Equivalent
Suppose that you face the following choices:
(1) Invest $100,000
From a successful project you receive $1,000,000.
If the project fails, you receive $0.
The probability of success is 0.5.
EV(investment) = ($1,000,000)(0.5) + ($0)(.5) = $500,000.
(2) You do not make the investment and keep $100,000.
If you find yourself indifferent between the two alternatives,
$100,000 is your certainty equivalent for the risky expected
return of $500,000.
A certain or riskless amount of $100,000 provides exactly the
same utility as a 50/50 chance to earn $1,000,000 (or $0).
33. 33
In general, any risky investment with a certainty equivalent less than the
expected dollar value indicates risk aversion.
In our case, $100,000 < $500,000 => risk aversion.
Certainly Equivalent Adjustment Factor = = Equivalent Certain Sum
Expected Value of the
Risky Venture
In our case, = $100,000 = .2.
$500,000
The “price” of one dollar in this risky venture is equal to 20¢ in certain dollar
terms.
α
α
34. 34
= Equivalent Certain Sum
Expected Value of the Risky Venture
If Then Implies
Equivalent certain sum < Expected Value of the < 1 Risk aversion
Risky Venture
Equivalent certain sum = Expected Value of the = 1 Risk indifference
Risky Venture (or neutrality)
Equivalent certain sum > Expected Value of the > 1 Risk preference
Risky Venture (or taking)
α
α
α
α
35. 35
Game Theory
- Game Theory dates back to the 1940s by John
von Neuman (Mathematician) and Oskar
Morgenstern (Economist)
- Game Theory is a useful decision framework
employed to make choices in hostile
environments and under extreme uncertainty.
- Use of maximin decision rule
- Use of minimax regret decision rule
(opportunity loss).
36. 36
Maximin Decision Rule
The decision maker should select the alternative
that provides the best of the worst possible
outcomes. Maximize the minimum possible
outcome.
The maximin criterion focuses only on the most
pessimistic outcome for each decision alternative.
The maximin criterion implicitly assumes a very
strong aversion to risk and is quite appropriate for
decisions involving the possibility of catastrophic
outcomes.
37. 37
Minimax Regret Decision Rule
This decision rule focuses on the opportunity loss
associated with a decision rather than on its worst possible
outcome.
The decision maker should minimize the maximum
possible regret (opportunity loss) associated with a wrong
decision after the fact. Minimize the difference between
possible outcomes and the best outcome for each state of
nature.
Opportunity loss => the difference between a given payoff
and the highest possible payoff for the resulting state of
nature. So, find the maximum payoff for a given state of
nature and then subtract from this amount the payoffs that
would result from various decision alternatives.
38. 38
Maximin and
Minimax Regret
Decision Rules
Event (State of
Economy)
Profit
Project A Project B
Recession $4,000 $0
Normal $5,000 $5,000
Boom $6,000 $12,000
Thinking:
Which project will you choose?
- Based on Maximin Decision Rule?
- Based on Minimax Regret Decision Rule?
What is ignored in the respective decisions?
39. 39
Maximin Decision Rule
Example
Minimum possible outcome for project A is $4,000.
Minimum possible outcome for project B is $0.
Therefore by the maximin decision rule, choose project A.
40. 40
Minimax Regret Decision Rule
Calculate the opportunity loss or regret matrix
State of Nature Project A Project B Maximum
Payoff
Recession $4,000-$4,000=$0 $4,000-$0=$4,000 $4,000
Normal $5,000-$5,000=$0 $5,000-$5,000=$0 $5,000
Boom $12,000-$6,000=$6,000 $12,000-$12,000=$0 $12,000
Maximum possible regret $6,000
Project A
$4,000
Project B
Therefore, by the minimax regret decision rule,
choose project B.
