Risk Efficiency Criteria

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Risk Efficiency Criteria

• Lecture VIII

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Risk Efficiency Criteria (I)

• Expected Utility Versus Risk Efficiency
• In this course, we started with the precept that
  individuals choose between actions or
  alternatives in a way that maximizes their
  expected utility. Mathematically, this principle
  is based on three axioms (Anderson, Dillon, and
  Hardaker p 66-69)

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Risk Efficiency Criteria (II)

• Ordering and transitivity A person either
  prefers one of two risky prospects a1 and a2 or
  is indifferent between them. Further if the
  individual prefers a1 to a2 and a2 to a3, then he
  prefers a1 to a3.
• Continuity. If a person prefers a1 to a2 to a3,
  then there exists some subjective probability
  level pa1 such that he is indifferent between
  the gamble paying a1 with probability pa1 and
  a3 with probability 1-pa3 which leaves him
  indifferent with a2.

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Risk Efficiency Criteria (III)

• Independence. If a1 is preferred to a2, and a3
  is any other risky prospect, a lottery with a1
  and a3 outcomes will be preferred to a lottery
  with a2 and a3 outcomes when pa1pa2. In
  other words, preference between a1 and a2 is
  independent of a3.

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Risk Efficiency Criteria (IV)

• However, some literature has raised questions
  regarding the adequacy of these assumptions
• Allais (1953) raised questions about the axiom of
  independence.
• May (1954) and Tversky (1969) questioned the
  transitivity of preferences.

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Risk Efficiency Criteria (V)

• These studies question whether preferences under
  uncertainty are adequately described by the
  traditional expected utility framework. One
  alternative is to develop risk efficiency
  criteria rather than expected utility axioms.
• Risk efficiency criteria are an attempt to reduce
  the collection of all possible alternatives to a
  smaller collection of risky alternatives that
  contain the optimum choice.

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Risk Efficiency Criteria (VI)

• One example was the mean-variance derivation of
  optimum portfolios.
• The EV frontier contained the set of possible
  portfolios such that no other portfolio could be
  constructed with a higher return with the same
  risk measured as the variance of the portfolio.
• It was our contention that this efficient set
  contained the utility maximizing portfolio. In
  addition, we derived the conditions which
  demonstrated how the EV framework was consistent
  with expected utility.

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Risk Efficiency Criteria (VII)

• Instead of expected utility justifying risk
  efficiency, we are now interested in the
  derivation of risk efficiency measures under
  their own right.
• An alternative justification of risk efficiency
  measures involves the scenario where the
  individuals risk preferences are difficult to
  elicit.

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Risk Efficiency Criteria (VIII)

• Stochastic Dominance
• One of the most frequently used risk efficiency
  approaches is stochastic dominance. To
  demonstrate the concept of stochastic dominance,
  lets examine the simplest form of stochastic
  dominance (first order stochastic dominance).

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Risk Efficiency Criteria (IX)

• To develop first order stochastic dominance, let
  us assume that the decision maker is faced with
  two alternative investments, a and b.
• Assume that the probability density function for
  alternative a can be characterized by the
  probability density function f(x). Similarly,
  assume that the return on investment b is
  associated with the probability density function
  g(x).

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Risk Efficiency Criteria (X)

• Investment a is said to be first order dominant
  of investment b if and only if

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Risk Efficiency Criteria (XI)
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Risk Efficiency Criteria (XII)

• Thus, investment a is always more likely to yield
  a higher return. Intuitively, one investment is
  going to dominate the other investment if their
  cummulative distribution functions do not cross.
• Economically, the only axiom required for first
  degree stochastic dominance is that the
  individual prefers more to less, or is
  nonsatiated in consumption.

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Risk Efficiency Criteria (XIII)

• This very basic criteria would appear
  noncontroversial, however, it is not very
  discerning. Taking the test data set

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• The Concept of an Efficiency Criteria
• An efficiency criteria is a decision rule for
  dividing alternatives into two mutually exclusive
  groups efficient and inefficient.
• If an alternative is in the efficient group, then
  it is one that an investor may choose.
• An inefficient investment will not be chosen by
  any investor regardless of individual risk
  preferences.

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• From an economic standpoint, the criteria should
  be related to general notions of utility or
  preferences.
• In general, the more global the preference, the
  less discerning the criteria (i.e. the fewer
  alternatives eliminated).
• A smaller efficient set requires more stringent
  requirements on preferences.

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• The most general efficiency criteria relies only
  on the assumption that utility is nondecreasing
  in income, or the decision maker prefers more of
  at least one good to less.
• FSD Rule Given two cummulative distribution
  functions F and G, an option F will be preferred
  to the second option G by FSD independent of
  concavity if F(x) lt G(x) for all return x with
  at least one strict inequality.

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• Intuitively, this rule states that one
  alternative F will dominate G if its cummulative
  distribution function always lies to the left of
  Gs

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• Mathematically, FSD is dependent on the integrals
  of the utility function times each alternative
  distribution function

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• Note that the utility function is the same for
  each investment alternative, but the distribution
  function changes. If investment F dominates
  investment G, then the difference, D, defined as

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• Integrating by parts

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• Second Degree Stochastic Dominance
• Building on FSD, second degree stochastic
  dominance SSD invokes risk aversion by inferring
  that the utility function is concave, implying
  that the second derivative of the utility
  function is negative.
• SSD Rule A necessary and sufficient condition
  for an alternative F to be preferred to a second
  alternative G by all risk averse decision makers
  is that

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• Graphically, another explanation of SSD can be
  determined by Alternative F dominates
  alternative G for all risk averse individuals if
  the cummulative area under F exceeds the area
  under the cummulative distribution function G for
  all values x, or if the cummulative area between
  F and G is non-negative for all x.

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