Financial Intermediation

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Financial Intermediation

• Lecture 7
• Decision-Making Under Uncertainty

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Contents of this chapter

• The difference between certainty and uncertainty
• Decision making under uncertainty
• Theory of expected utility
• Stochastic dominance and risk attitude
• Mean-variance analysis as an example

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Certainty versus uncertainty

• A set of feasible actions a. S set of
  possible states of the world s, C set of
  consequences c c f(s,a)
• If f is constant with respect to the state of the
  world (the state of the world does not influence
  the consequence which arises) decision with
  respect to a is certain. If f is not constant
  uncertainty!

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Example
?
?
?p?(l)-wl
?(l)
?(s2,l)
0
l
l
0
certainty
?(s1,l)
uncertainty
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Decision making under uncertainty

• States of the world are uncertain model this by
  a probability measure on S (see next slide for a
  definition)
• Choice of an action can be seen as (1) choice of
  a state-dependent outcome, or (2) the choice of a
  probability distribution of outcomes

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Probability measure

• E an event which is a subset of the states of
  the world E?S. ? is a set of well-behaved subsets
  (say intervals or unions) of events is called a
  sigma algebra
• A probability measure is a mapping ? ??? with
  the properties (1) ?(E )? 0 for all E ? ?, (2)
  ?(S) 1, (3) if Fi ? ?, i 1,2,3,.. and Fi ? Fj
  ? for all i,j (i?j), then ?(?iFi) ?i?(Fi)
• A triple (S, ?, ?) is a probability space

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Example 1

• Rain s1 and sunshine s2 are the relevant states
• Ss1,s2 set of states
• ??,s1,s2,s1,s2 set of events
  (sigma-algebra)
• ?(?) 0, ?(s1) ½ ?(s2) and ?(S)
  1 is a probability measure

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Example 2

• Suppose we have a distribution function G(x)
  (e.g. the exponential or normal distribution).
  The probability space defined by this
  distribution can be described by
• S ? state space
• ? any interval and its complement sigma
  algebra
• ?( (a,b) ) G(b) - G(a) for any a ? b
  probability measure. Sometimes we see the
  support of a measure the smallest subset of S
  with measure 1

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How do we represent uncertain outcomes? A lottery

• Let S si, i 1,.,T be a finite set of
  outcomes
• Let pi be the nonnegative probability of
  occurrence of si with ?pi 1
• A lottery is a vector (s1,p1s2,p2,.sT,pT)
• We can represent a lottery by (p1,,pT)
• Example for S 3 we have a lottery (p1,p3) in ?2

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Example representation of a lottery
p3
1
p31-p2-p1
p1
0
1
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Alternative representation in terms of
consequences

• In the previous slide we denoted the lottery in
  terms of the interrelation between probabilities
  of various states
• We can also denote the lottery in terms of
  outcomes in the space of consequences
• Take a two by two example c1 and c2 are
  functions of two actions a1 and a2

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Lottery in terms of consequences
c2
c2(a2)
c2(a1)
0
c1
c1(a1)
c1(a2)
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Expected utility

• A decision maker can see a specific choice as a
  state-contingent outcome or as a probability
  distribution
• But how should one order either the
  state-contingent outcomes or the probability
  distributions over outcomes?
• Utility theory gives the answers

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Preferences

• Suppose there is a finite set CSc1,.,cs. If
  preferences on the set CS satisfy completeness,
  transitivity and continuity they can be
  represented by a utility function V(c1,..,cs)

c2
V2
V1
c1
0
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Properties

• Completeness requires the ordering to order any
  pair of probability distributions
• Transitivity if the decision maker prefers p
  over q and q over r she will prefer p over r
• Continuity no abrupt changes of preferences for
  probability distributions

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Probabilities

• Each action determines a vector of probabilities
  from the set ?n(p1,,pn)?pi1 with
  piprob(s?Sf(s,a)ci)
• One can have a preference ordering of such
  probability distributions. If preferences are
  complete, transitive and continuous they can be
  represented by U(p1,,pn)

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Probabilities (2)
p3
1
U1
U1gtU2gtU3
U2
U3
c3 is the most preferred outcome the utility
curves increase in the direction of p3
1
p1
0
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Expected utility

• One additional assumption independence of
  preferences
• Expected utility representation in probabilities
  U(p1,,pn) ?piu(ci), or in state-contingent
  outcomes V(c1,,cn) ?piu(ci). Note that there
  is a linearity. This linearity property changes
  some things

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The consequences of linearity

• Suppose we have a set of lotteries with three
  outcomes x1,x2,x3 and u(x1) gt
  u(x2) gt u(x3). We describe these lotteries by
  (p1, p2, p3) the probabilities of outcomes x1,
  x2, x3 . p2 1 - p1 - p3
• Up1 u(x1) p2 u(x2) p3 u(x3) for various U.
• So p3 U - u(x2)/(u(x3) - u(x2)) -
  u(x1) - u(x2)/(u(x3) - u(x2)) p1

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Indifference curves of the utility function over
probabilities
p3
u(c1) gt u(c2) gt u(c3)
1
1
p1
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Indifference curves for state-contingent outcomes
c2
dc2/dc1 -p1.u(c1)/p2.u(c2) for all 450
tangent slopes
-p1/p2
c1
0
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Continuous time linear adding up changes to
integrals

• g(c) is a density function
• G(c) is a distribution function

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First-order stochastic dominance

• If F(x) ? G(x) for all x ? C the probability
  distribution F dominates G according to
  first-order stochastic dominance

For each probability F has a higher pay-off
Probability
G
F
x
0
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Second-order stochastic dominance

• If ?xF(y)dy ? ?xG(y)dy for all x ? C, F dominates
  G according to second-order stochastic dominance.
• So the surface under the distribution function F
  up to x should be smaller than for G for all x
• A SSD-dominant distribution will have less mass
  in its tails (more concentrated around the mean)
• So SSD can be used to rank distributions with
  equal means according to their riskiness

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Example

• Suppose we have to risk classes H and L
• Both have equal means, so there is no observable
  difference in returns
• But there is SSD of the distribution of returns
  of H over L the riskiness of class L is larger

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Risk attitudes

• u(x) is linear risk neutral, risk premium 0
• u(x) is concave risk averse, risk premium is
  positive
• u(x) is convex risk loving, risk premium is
  negative

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Risk aversion
u(x)
1-2 certainty equivalent
3
2-3 risk premium
2
1
x
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Measures of risk aversion

• Absolute risk aversion Ra(x) -u(x)/u(x)
• Relative risk aversion Rr(x) -x.u(x)/u(x)
• Constant absolute risk aversion (CARA) example
  u(x) exp(?x)
• Certainty equivalent ? - 1/2Ra?2

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Application Mean-variance analysis risk aversion

• Suppose we have a wealth variable W with expected
  value ? and variance ?2. Utility is represented
  by U(W) aW2W for a lt 0 and W lt -1/2a. Ra(W)
  -2a / (2aW1)

u(W)
0
W
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A risk averse portfolio manager

• Reduces the variance for a certain return
• Suppose that a number of risky assets can give a
  certain market return
• Combination with a riskless asset gives an
  income line
• It depends on the shape of the utility function
  which point is optimal

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Mean variance analysis
?W
u(W)
Wealth restriction
?W
0
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Summary

• Distinction between certainty and uncertainty
• Expected utility theory provides a means to order
  preferences
• Properties of distribution functions and utility
  functions
• Risk aversion applies to portfolio analysis