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Chapter 14 RISK AND UNCERTAINTY

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Chapter 14 RISK AND UNCERTAINTY Powered By Docstoc
					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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