# Chapter 14 RISK AND UNCERTAINTY

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```					Hirschey Chapter 14

RISK ANALYSIS
   In the real world, most future events
are not known with any degree of
certainty.
   Managers must make decisions relying
on estimates that involve some
uncertainty.
   Risk and uncertainty will be a part of all
analyses we will carry out throughout
the semester.
Risk vs Uncertainty
   Risk and uncertainty are used
interchangeably in economics and
finance.
   Most future events are not known with
certainty and some of these events can
be assigned probabilities.
   We are talking about risk when future
events can be defined and probabilities
can be assigned.
The Measures of Risk
   When there is a number of alternative
outcomes, each outcome will have a
probability attached to it.
   A probability distribution describes
the chances of all possible occurrences
in percentage terms.
Probability Distributions
Cash Inflow    Probability
3,000           0.10
4,000           0.20
5,000           0.40
6,000           0.20
7,000           0.10
Probability Distributions
   Expected Value: Average of all
possible outcomes weighted by their
respective probabilities.
   E(Cash Flow) = (3,000 x 0.1) + (4,000
x 0.2) + (5,000 x 0.4) + (6,000 x 0.2)
+ (7,000 x 0.1) = 5,000
Expected Value

n
E(X)   X ipi
i1
Probability Distribution
   Risk is the dispersion of possible
outcomes around the expected value.
   Risk is higher when the potential
differences from the average are high.
   Standard Deviation: Square root of
the weighted average of the squared
deviations of all possible outcomes from
the expected value.
Standard Deviation
   (Cash Flow) = [(3,000 - 5,000)2(0.1)
+ (4,000 - 5,000)2(0.2) + (5,000 -
5,000)2(0.4) + (6,000 - 5,000)2(0.2) +
(7,000 - 5,000)2(0.1)]1/2 = 1,095
Standard Deviation

n
σ     (Xi  X) pi
i1
2
Standard Deviation as a
Measure of Risk
   When the expected values of two
alternatives are equal or close to one
another, the standard deviation is a
proper measure of risk.
   In such a case, the standard deviation
shows the amount of dispersion (risk)
around the expected value.
Coefficient of Variation
 If two alternatives have divergent
expected values, standard deviation will
not be an appropriate measure of risk.
 E.g.

Exp Value      Std Dev
Project A           100           30
Project B             50          20
Coefficient of Variation
   Project A has both the larger expected
value and the larger standard deviation.
   We need a “relative” measure of risk to
compare these two projects.
   Coefficient of Variation measures
risk relative to expected value.
   It shows the amount of risk per unit of
return.
Coefficient of Variation

CV  σ
X
Coefficient of Variation
   From the example:
CVA = 30/100 = 0.30
CVB = 20/50 = 0.40
   The CV is greater for Project B.
   Project A has a larger absolute risk ()
but its relative risk (CV) is lower.
   A is preferred over B since its expected
value is higher and its relative risk is
lower.
Standard Normal Distribution
   When the probability distribution of
outcomes is known,
   the risk of a given course of action can be
measured by using the expected value and
the standard deviation of the distribution.
   Normal Distribution:
   A special case probability distribution
where the dispersion about the expected
value is symmetrical.
   Since it is symmetrical, it is possible to
measure the probability of a certain
outcome in a standardized manner:

   The actual outcome of the decision will lie
   Within ± 1 standard deviation of the mean
about 68% of the time
   Within ± 2 standard deviations of the mean
about 95% of the time
   Within ± 3 standard deviations of the mean
about 99% of the time
Probability Ranges for a Normal Distribution

68.26%

95.46%

99.74%

-3σ      -2σ   -1σ             +1σ +2σ   +3σ

Mean or expected value
The Standardized Variable
   A standardized variable has
   a mean of 0,
   a standard deviation of 1.
   Any distribution can be standardized in the following
way:
x μ
z
σ
z : standardized variable
x : outcome of interest
μ : mean of the distribution
σ : standard deviation of the distribution
Interpretation of the
Standardized Variable

   When the outcome is 1 σ away from
the mean, x – μ = σ. This means z = 1.
   So, when z = 1, the point of interest is
1 σ away from the mean.
   Or, when z = 2, the point of interest is
2 σ away from the mean.
Utility Theory
   Assumptions about risk is central to many
decisions in managerial economics.
   When analyzing human behavior, we make
assumptions about attitudes towards risk.
   Theoretically, there are three different
attitudes:
 Risk Aversion

 Risk Neutrality

 Risk Seeking
Risk Attitudes
   Risk Aversion: Individuals seek to
avoid or minimize risk.
   Risk Neutrality: Individuals focus on
expected returns and disregard the
dispersion of returns (risk).
   Risk Seeking: Individuals prefer risk.
Risk Attitudes
   Choice between more risky and less
risky investments with identical returns:
   Risk averter chooses the one with less risk
   Risk seeker chooses the one with high risk
   Risk neutral is indifferent between the
choices
Risk Attitudes

   Business managers and investors are
mostly risk averters.
   Some individuals prefer risk when small
amounts of money are involved
(entrepreneurs, innovators, inventors,
speculators, lottery ticket buyers).
   However, the assumption of risk aversion
generally holds.
   Why should risk aversion generally hold?
 Utility theory provides a satisfying
answer.
Utility Theory
   In order to understand risk aversion, we
have to look at the relation between
money and its utility.
   Diminishing Marginal Utility
When additional increments of money
brings ever smaller increments of added
benefit, diminishing marginal utility is
observed.
   For risk averters, money has diminishing
marginal utility.
   For such an individual, a less than proportional
relation holds between total utility and money.
 The utility of a doubled quantity of money is
less than twice the utility of the original level.
   For a risk neutral individual, there is a strictly
proportional relationship between total utility and
money.
   For a risk seeker, there is a more than
proportional relationship between total utility and
money.
Total Utility
Risk Averter
Income or Wealth
Total Utility

Indifferent to Risk
Income or Wealth
Total Utility

Risk Seeker
Income or Wealth
Risk Aversion Example
Alternative 1: Riskless government securities offer
a 9% return on an annual basis. If you invest
\$10,000 in government securities, you will earn \$900
at year-end and your total wealth will become
\$10,900.
Alternative 2: If you invest \$10,000 in a risky oil-
drilling venture, with 60% chance, oil will be
discovered and you will earn \$10,000 and your total
wealth will become \$20,000. With 40% chance,
there will be a dry hole and you will recover only
\$5,000 of your initial investment and your total
wealth will be \$5,000.
Which alternative should you choose?

Expected Value from Alternative 1:
E(R) = \$10,900 (no calculation necessary
since there is no risk)

Expected Value from Alternative 2:
E(R) = 0.6(\$20,000) + 0.4(\$5,000) = \$14,000

E(R) suggests that we should choose
Alternative 2. How about the risk?
Assume the following schedule for utilities:

40
35
Total Utility

30
25
20
15
10
5
0
0            5000         10000         15000       20000
Income or Wealth

Diminishing MU     Constant MU      Increasing MU
Reading from the schedule, for a diminishing
marginal utility (risk averter):
TU (government bond) = 10.7
TU (the oil drill) = 0.6(13) + 0.4(6) = 10.2.

The risk averse person will choose the
government bond since it has the larger
expected total utility.
Firm Value Under Risk
   The valuation model
N
CFt
V            t
t 1 (1  i)

needs to be adjusted for risk.
The valuation model can be adjusted by
reflecting risk either in CFs or in the
discount rate (i).
Adjustment to CFs
   Certainty Equivalent Adjustment
Specify the certain sum that is
comparable to the expected value of
a risky investment alternative.
   The certainty equivalent of an expected
value has the same total utility but it
differs in dollar/lira terms.
Certainty Equivalent
Adjustment
   E.g. Invest \$100,000 in a risky project.
If successful, earn \$1,000,000 (50% chance)
If unsuccessful, earn \$0 (50% chance)
Expected Value = 0.5(\$1,000,000) + 0.5(\$0)
= \$500,000

If you do not invest, you keep the \$100,000.
Certainty Equivalent
Adjustment
   If you are indifferent between investing
and not investing, then
Risky expected return = \$500,000
Certainty equivalent = \$100,000
and these two sums have the same total
utility.
Note that a much smaller certain sum
(\$100,000) has the same total utility as
the much larger risky sum (\$500,000).
Certainty Equivalent
Adjustment
Equivalent certain sum < Expected risky sum
 Risk aversion

Equivalent certain sum = Expected risky sum
 Risk neutrality (indifference)

Equivalent certain sum > Expected risky sum
 Risk preference
Certainty Equivalent
Adjustment
Certainty Equivalent Adjustment Factor = 

 = equivalent certain sum / expected risky sum

From the example,

 = \$100,000 / \$500,000 = 0.20.
Risk-Adjusted Valuation Model
   The valuation model becomes
N
E(CFt )
V                   t
t 1   (1  i)
where we convert the expected value of
cash flows into their certainty
equivalent and use the riskless discount
factor i in the denominator.
Adjusting the Discount Rate
for Risk
   The valuation model can be adjusted
for risk by reflecting the risk in the
discount rate while using the expected
cash flows in the numerator:
N
E(CFt )
V            t
t 1 (1  k)

where k = risk-free rate (i) + risk premium.

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 views: 6 posted: 2/29/2012 language: pages: 39