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					            Risk and Risk Measurement (L3)
• Following topics are covered:
   – Risk
   – Defining “More Risk Averse”
       •   Utility functions
       •   Absolute risk aversion
       •   Relative risk aversion
       •   Risk premium
       •   Certainty equivalent
   – Defining “More Risky”
       • Increase in risk
       • Aversion to downside risk
       • Stochastic dominance

   (Materials from Chapters 1&2 of EGS and Chapters 1&2 of HL)

                             L3: Risk and Risk Measurement       1
                                     Risk
• Risk can be generally defined as “uncertainty”.
• Sempronius owns goods at home worth a total of 4000 ducats and in
  addition possesses 8000 decats worth of commodities in foreign countries
  from where they can only be transported by sea, with ½ chance that the
  ship will perish.
• If he puts all the foreign commodity in 1 ship, this wealth, represented by a
  lottery, is x ~ (4000, ½; 12000, ½)
• If he put the foreign commodity in 2 ships, assuming the ships follow
  independent but equally dangerous routes. Sempronius faces a more
  diversified lottery y ~ (4000, ¼; 8000, ½; 12000, ¼)
• In either case, Sempronius faces a risk on his wealth. What are the expected
  values of these two lotteries?
• In reality, most people prefer the latter. Why?




                            L3: Risk and Risk Measurement                     2
       St. Petersburg Paradox
• You pay a fixed fee to enter and then a fair
  coin is tossed repeatedly until a tail appears,
  ending the game. The pot starts at 1 dollar is
  doubled every time a head appears. You win
  whatever is in the pot after the game ends.
  What is the fair price to pay for entering the
  game?
• Are you willing to pay this?
• See page 13, HL.
                  L3: Risk and Risk Measurement     3
         Risk Aversion and Utility Function
• A risk-averse agent is an agent who dislikes zero-mean
  risks.
• Risk aversion: Eu(w  z )  u(w  Ez) .
   – Utility function is the relationship between monetary outcome,
     x, and the degree of satisfaction, u(x).
• Concavity
   – a twice-differentiable function u is concave if and only if its second
     derivative is negative, i.e., if the marginal utility u’(x) is decreasing
     in x.
   – E.g, u(x)=ln(x)
• A decision maker is risk-averse  the agent’s utility
  function u is concave. – proposition 2, page 8, EGS
   – Be able to prove this
• Jensen Inequality
                          L3: Risk and Risk Measurement                      4
Von Neumann_Morgenstern Utility Function

von Neumann-Morgenstern utility describes a utility function
(or perhaps a broader class of preference relations) that has the
expected utility property: the agent is indifferent between
receiving a given bundle or a gamble with the same expected
value.




                     L3: Risk and Risk Measurement              5
       Risk Premium and Certainty Equivalent
• Risk Premium
   – Risk-averse investors may want to purchase risky assets if their
     expected return exceed the risk-free rate.
   – Eu(w+z)=u(w-Π), where z is a zero-mean risk, w is the initial wealth
   – Π is risk premium – cost of risk
• Certainty equivalent
   – The amount of payoff (e.g. money or utility) that an agent would
     have to receive to be indifferent between that payoff and a given
     gamble
   – What is certainty equivalent in the above expression?
• Arrow-Pratt approximation of risk premium
                1 2
   –            A( w) where A(w)=-u’’(w)/u(w)
                2


                          L3: Risk and Risk Measurement                     6
                   Deriving Risk Premium
                (Arrow_Pratt Approximation)
                                         1 2
  Eu( w  z )  E[u ( w)  zu ' ( w)      z u ' ' ( w)]
                                         2
                           1
   u ( w)  u ' ( w) Ez  Ez 2 u ' ' ( w)
                           2
             1
   u ( w)   2 u ' ' ( w)
             2
  where Ez=0 and variance σ2=Ez2.
  Also note that Eu(w+z) = u(w-П), where u(w- П) = u(w)        -   П   u   ’   (   w   )   . Thus proved.


Note: The cost of risk, as measured by risk premium, is approximately
proportional to the variance of its payoffs. This is one reason why
researchers use a mean-variance decision criterion for modeling behavior
under risk. However П=1/2σ2*A(w) only holds for small risk (thus we can
apply for the 2nd-order approximation).




                               L3: Risk and Risk Measurement                                                7
            Further discussions
• Arrow-Pratt expression of risk premium works
  for small risk or special utility functions where
  only mean and variance are applicable to the
  expected utility
  – See page 20, quadratic utility function




                   L3: Risk and Risk Measurement   8
                  Measuring Risk Aversion
• The degree of absolute risk aversion
                              u ' ' ( w)
                 A( w)  
                              u ' ( w)
• For small risks, the risk premium increases with the size of the
  risk proportionately to the square of the size
   – Assuming z=k*ε, where E(ε)=0, σ(ε)=σ
• Accepting a small-mean risk has no effect on the wealth of
  risk-averse agents
                             1 2 2
                             k  A( w)
                             2
• ARA is a measure of the degree of concavity of a utility
  function, i.e., the speed at which marginal utility decreases


                        L3: Risk and Risk Measurement             9
                                                     More Risk Averse Agent

• Consider two risk averse agents u and v. if v is more risk averse than u,
  this is equivalent to that Av>Au
    Proof:

    Suppose v is a concave transformation of u. I.e., v(w) = φ(u(w))                                                                                             ,




    We have v’(w) = φ                 . Hence,       ’       (   u       (   w           )   )   u   ’   (   w       )




                             2
    v’’(w)= φ’   ’   (   u   ( +φw   )   )   (   u     , ’       (   w           )   )                       ’   (   u   (   w   )   )   u   ’   ’   (   w   )




                                  ' ' (u ( w))u ' ( w)
    We have Av ( w)  Au ( w) 
                                      ' (u ( w))


•   Conditions leading to more risk Aversion – page 14-15




                                                                                                             L3: Risk and Risk Measurement                           10
                                   Example

• Two agents’ utility functions are u(w) and v(w).
    u(w)= w , v(w)=ln(w);

       (1) Which agent is more risk averse in small risks and in large risks?
       (2) Suppose they have an initial wealth of 4000 ducats and face a risk
           of (0, ½; 8000, ½). Find their respective risk premiums.




                              L3: Risk and Risk Measurement                     11
                          CARA and DARA
• CARA
• However, ARA typically decreases
    – Assuming for a square root utility function, what would be the risk premium
      of an individual having a wealth of dollar 101 versus a guy whose wealth is
      dollar 100000 with a lottery to gain or lose $100 with equal probability?
• What kind of utility function has a decreasing risk premium?
   Eu(w+z)=u(w-π(w)) Eu’(w+z)=(1- π       ’   (   w   )   )   u   ’   (   w   -   π   (   w   )   )




              u ' ( w   )  Eu ' ( w  z )
    ' ( w) 
                       u' (w   )
   u(w) is an increasing and concave utility function, thus, u’(.)>0
   to have a decreasing risk premium in w, we need u’(w- π                                             )   ≤   E   u   ’   (   w   +   z   )




   Defining v=-u’, we have the following condition for decreasing risk premium
   Ev(w+z) ≤v(w- π )




   The above condition states that ????




                              L3: Risk and Risk Measurement                                                                                    12
                            Prudence
• Thus –u’ is a concave transformation of u.
• Defining risk aversion of (-u’) as –u’’’/u’’
• This is known as prudence, P(w)
• The risk premium associated to any risk z is decreasing in
  wealth if and only if absolute risk aversion is decreasing or
  prudence is uniformly larger than absolute risk aversion
• P(w)≥A(w)




                        L3: Risk and Risk Measurement             13
                     Relative Risk Aversion

• Definition
                     du' ( w) / u ' ( w)    wu ' ' ( w)
           R( w)                                     wA( w)
                          dw / w             u ' ( w)
• Using z for proportion risk, the relation between relative risk premium,
  ПR(z), and absolute risk premium, ПA(wz) is
                           A ( wz ) 1 2
                ( z)  
                 R
                                      R( w)
                             w       2
• This can be used to establish a reasonable range of risk aversion: given that
  (1) investors have a lottery of a gain or loss of 20% with equal probability
  and (2) most people is willing to pay between 2% and 8% of their wealth
  (page 18, EGS).

• CRRA


                            L3: Risk and Risk Measurement                    14
               Some Classical Utility Function

A. Quadratic function: u(w)=aw-1/2w2 – increasing absolute risk aversion
                                     exp(aw)
B. Exponential function: u ( w)              -- constant absolute risk aversion
                                         a
                                      w1
C. Power utility function: u ( w)        -- constant relative risk aversion
                                      1 
D. log utility function: u(w) = ln(w) – constant relative risk aversion


  For more of utility functions commonly used, see HL, pages 25-28
  Also see HL, Page 11 for von Neumann-Morgenstern utility, i.e., utility
  function having expected value
  Appropriate expression of expected utility, see HL, page 6 and 7



                                L3: Risk and Risk Measurement                       15
                         Measuring Risks
• So far, we discuss investors’ attitude on risk when risk is given
   – I.e., investors have different utility functions, how a give risk affects
     investors’ wealth
• Now we move to risk itself – how does a risk change?
• Definition: A wealth distributions w1 is preferred to w2, when
  Eu(w1)≥Eu(w2)
   – Increasing risk in the sense of Rothschild and Stiglitz (1970)
   – An increase in downside risk (Menezes, Geiss and Tressler (1980))
   – First-order stochastic dominance




                            L3: Risk and Risk Measurement                        16
                        Adding Noise

• w1~(4000, ½; 12000, ½)
• w2~(4000, ½; 12000+ε, ½) [adding price risk; or an additional
  noise]
• Then look at the expected utility (page 29)
• General form: w1 takes n possible value w1, w2,w3,…, wn. Let
  ps denotes the probability that w1 takes the value of ws. If w2 =
  w1+ ε  Eu(w2) ≤ Eu(w1).




                        L3: Risk and Risk Measurement            17
      Mean Preserving Spread Transformation
•    Definition:
    – Assuming all possible final wealth levels are in interval
       [a,b] and I is a subset of [a, b]
    – Let fi(w) denote the probability mass of w2 (i=1, 2) at w.
       w2 is a mean-preserving spread (MPS) of w1 if
       1.   Ew2= Ew1
       2.   There exists an interval I such that f2(w)≤ f1(w) for all w in I
•    Example: the figure in the left-handed panel of page 31
•    Increasing noise and mean preserving spread (MPS) are
     equivalent


                            L3: Risk and Risk Measurement                      18
                    Single Crossing Property
• Mean Preserving spread implies that (integration by parts – page 31)
                b
               [ F (s)  F (s)]ds  0
               a
                    2         1


• This implies a “single-crossing” property: F2 must be larger than F1 to the
  left of some threshold w and F2 must be smaller than F1 to its right. I.e.,
                        w
              s( w)   [ F2 ( s)  F1 ( s)]ds  0
                        a




                            L3: Risk and Risk Measurement                       19
                        The Integral Condition
             b                                                  b
Eu ( wi )   u ( w) f i ( w)dw  u ( w) Fi ( w) |    wb
                                                      w a     u ' ( w) Fi ( w)dw
             a                                                 a
                           b
I.e., Eu ( wi )  u (b)   u ' ( w) Fi ( w)dw
                           a
                                              b
It follows that: Eu ( w2 )  Eu ( w1 )   u ' ( w)[ F1 ( w)  F2 ( w)]dw -- page 32 for
                                              a
interpretation

Integrating by parts yields
                                                  b             b
Eu ( w2 )  Eu ( w1 )  u ' ( w) S ( w) |b   u ' ' ( w)(  [ F1 ( w)  F2 ( w)]dw)dw
                                          a
                                                  a             a
                                 b
Thus Eu ( w2 )  Eu ( w1 )   u ' ' ( w) S ( w)dw
                                 a
For a risk averse agent, the above expressive is uniformly negative.




                                     L3: Risk and Risk Measurement                         20
                    MPS Conditions

Consider two random variable w1 and w2 with the same mean,
(1) All risk averse agents prefer w1 to w2 for all concave
    function u
(2) w2 is obtained from w1 by adding zero-mean noise to the
    possible outcome of w1
(3) w2 is obtained w1 by a sequence of mean-preserving spreads
(4) S(w)≥0 holds for all w.




                      L3: Risk and Risk Measurement          21
             Preference for Diversification

• Suppose Sempreonius has an initial wealth of w (in term of
  pounds of spicy). He ships 8000 pounds oversea. Also suppose
  the probability of a ship being sunk is ½. x takes 0 if the ship
  sinks and 1 otherwise. If putting spicy in 1 ship, his final
  wealth is w2=w+8000*x
• If he puts spicy in 2 ships, his final wealth is
  w1=w+8000(x1+x2)/2
• Sempreonius would prefer two ships as long as his utility
  function is concave
• Diversification is a risk-reduction device in the sense of
  Rothschild of Stiglitz (1970).



                       L3: Risk and Risk Measurement            22
                 Variance and Preference

• Two risky assets, w1 and w2, w1 is preferred to w2 iff П2> П1
• For a small risk, w1 is preferred to w2 iff the variance of w2
  exceeds the variance of w1
• But this does not hold for a large risk. The correct statement is
  that all risk-averse agents with a quadratic utility function prefer
  w1 to w2 iff the variance of the second is larger than the variance
  of the first
   – See page 35.




                        L3: Risk and Risk Measurement             23
              Aversion to Downside Risk

• Definition: agents dislike transferring a zero-mean risk from a
  richer to a poor state.
• w2~(4000, ½; 12000+ε, ½)
• w3~(4000+ ε, ½; 12000, ½)
• Which is more risky
• Depends, generally agents dislike risks in lower states more.
  But it depends on agents’ utility function




                       L3: Risk and Risk Measurement            24
        First-Degree Stochastic Dominance
• Definition: w2 is dominated by w1 in the sense of the first-
  degree stochastic dominance order if F2(w)≥F1(w) for all w.
• Consumers would dislike FSD-dominated shifts in the
  distribution final wealth
                              b
      Eu( w2 )  Eu( w1 )   u ( w)[ f 2 ( w)  f1 ( w)]dw
                             a
         b
        u ' ( w)[F1 ( w)  F2 ( w)]dw  0
         a                                     F(w)
                                                              W2
• No longer mean preserving
                                                                   W1



                                                                             w

                            L3: Risk and Risk Measurement               25
            Equivalent Conditions on FSD
• All agents with a nondecreasing utility function prefer w1
  to w2: Eu(w2)<=Eu(w1) for all nondecreasing functions u,
  or stated as, w  w
                  1            2
                      FSD

• W2 is dominated w1 in the sense of FSD: w2 is obtained
  from w1 by a transfer of probability mass from the high
  wealth states to lower wealth states, or F2(w)≥F1(w) for all
  w.
• W1 is obtained from w2 by adding nonnegative noise
                                             d
  terms to the possible outcome of w2:
                                           w1 w2   where є
  is no less than 0 with probability one.


                            L3: Risk and Risk Measurement        26
        Second-degree Stochastic Dominance

• Definition
   – w1 dominates w2 in the sense of the second-degree
     stochastic dominance if and only if

        E ( w1 )  E ( w2 )
                  y
        s( y)   [ F1 ( w)  F2 ( w)]dz  0              y [0,1]
                 0


   – This is same mean preserving spread or adding a
     noise; see page 49, HL.



                          L3: Risk and Risk Measurement               27
                      Allais Paradox
• The Allais paradox is a choice problem designed by Maurice Allais to
  show an inconsistency of actual observed choices with the predictions of
  expected utility theory.
• Two pairs of games




• What is your choice?

                           L3: Risk and Risk Measurement                     28
 Ellsberg Paradox (Ambiguity Aversion)
• The Ellsberg paradox is a paradox in decision theory and experimental
  economics in which people’s choices violate the expected utility
  hypothesis. It is generally taken to be evidence for ambiguity aversion.
• The 1urn paradox: suppose you have an urn containing 30 red balls and 60
  other balls that are either black or yellow. You don't know how many black
  or yellow balls there are, but that the total number of black balls plus the
  total number of yellow equals 60. The balls are well mixed so that each
  individual ball is as likely to be drawn as any other.
• You are now given a choice:
    – Gamble A: You receive $100 if you draw a red ball
    – Gamble B: You receive $100 if you draw a black ball
• Also you are given options between Gamble C and D
    – You receive $100 if you draw a red or yellow ball
    – You receive $100 if you draw a black or yellow ball


                                L3: Risk and Risk Measurement               29
                              Technical Notes

• Taylor Series Expansion: f(x)=


     which in a more compact form can be written




• Integration by parts




                                L3: Risk and Risk Measurement   30
                           Exercises

• EGS, 1.2; 1.4; 2.2




                       L3: Risk and Risk Measurement   31

				
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