Example of second-order probabilities and utilities

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Example of second-order probabilities and utilities Powered By Docstoc
					Worksheet to illustrate calculation of risk premia in 2x2 model of second-order utilities and probabilities (SOUP)
Status quo wealth = zero, utility functions = exponential with unit slope at zero, risk neutral probabilities = true joint pr
Orange cells = changeable inputs, yellow cells = key outputs, run Solver to re-compute exact risk premia after change

                             States                   Risk neutral probabilities                                Payoffs
                             A_1 & B_1                 pi_11 =            0.25                                  z_11 =
                             A_1 & B_2                 pi_12 =            0.25                                  z_12 =
                             A_2 & B_1                 pi_21 =            0.25                                  z_21 =
                             A_2 & B_2                 pi_22 =            0.25                                  z_22 =

                                                                             Risk-neutral expected value of spread
                                                                                Buying price for spread (via Solver)
                     Delta                0.1                                                  Exact risk premium

    1st-order parameters         Probability     Risk tolerance                     First-order utilities
                 B_1|A_1                0.5                 100                                              v_11(z_11)
                 B_2|A_1                0.5                 100                                              v_12(z_12)
                                                                                                                 v_1(z)

                 B_1|A_2                  0.5               100                                              v_21(z_12)
                 B_2|A_2                  0.5               100                                              v_22(z_22)
                                                                                                                 v_2(z)
   2nd-order parameters                                                             Second-order utilities
                   A_1                    0.5                10                                              u_1(v_1(z))
                   A_2                    0.5                10                                              u_2(v_2(z))
                                                                    Total utility                                  U(z)
 This is a "normal" example of uncertainty aversion in the
 two-color Ellsberg paradox. Events A1 and A2 correspond
 to the draw of a red ball or a black ball from urn I (with
 unknown proportions of red and black), while B1 and B2                   Conditional risk-neutral expected values
 correspond to the draw of a red ball or black ball from urn II                                            z-bar_1
 (with known 50-50- proportions). The second-order                                                         z-bar_2
 probability distribution and conditional first-order probability
 distributions are all uniform, and the 2nd-order utility
 functions are highly concave (risk averse) while the 1st-                     s
 order utility functions are only slightly concave (nearly                   0.1                   pi_1*s_1*(z-bar_1)^2
 linear). Hence the individual is significantly risk averse with             0.1                   pi_2*s_2*(z-bar_2)^2
 respect to bets on the A-events and nearly risk-neutral with
 respect to bets on the B-events. The risk premium for the                     t
 A-spread is more than 10 times larger than the risk
                                                                           0.01                      pi_11*t_11*z_11^2
 premium for the B-spread. Details of approximation
                                                                           0.01                      pi_12*t_12*z_12^2
 formulas for the risk premia are shown at the bottom of the
 worksheet (scroll down).                                                  0.01                      pi_21_t_21*z_21^2
                                                                           0.01                      pi_22*t_22*z_22^2

                                                Approximate risk premium (via formula in Theorem 1 part (d))




                                                                                                                           Diagonal part
                                                                           0.0




                                                                Total risk ave
                                                                            0.0
                                                                            0.0




                                                                Pi * Risk avers




Approximate risk premium (via matrix formula from 2003 paper)
probabilities (SOUP)
probabilities = true joint probabilities at status quo wealth
t risk premia after changes

                     A-spread        B-spread      Net A-spread      Net B-spread
                           0.4             0.4     0.408796821       0.400800048
                           0.4            -0.4     0.408796821      -0.399199952
                          -0.4             0.4    -0.391203179       0.400800048
                          -0.4            -0.4    -0.391203179      -0.399199952

                            0               0
                 -0.008796821    -0.000800048
                  0.008796821     0.000800048 <--Individual is uncertainty averse if A-spread risk premium is larger than that of B-spread


                 0.199600533      0.199600533      0.203981192       0.199998958
                 0.199600533     -0.200400534      0.203981192      -0.199998908
                 0.399201066     -0.000800001      0.407962384       5.00131E-08

                 -0.200400534     0.199600533     -0.195984689       0.199998958
                 -0.200400534    -0.200400534     -0.195984689      -0.199998908
                 -0.400801068    -0.000800001     -0.391969377       5.00131E-08

                  0.195668985    -0.000400017      0.199876369      2.50066E-08
                 -0.204470768    -0.000400017     -0.199876369      2.50066E-08
                 -0.008801783    -0.000800033      5.27356E-13      5.00131E-08

                                                   2.50131E-15 Sum of squared deviations from status quo utility of zero
ected values                                                   (minimize with Solver to determine break-even buying prices)
                           0.4               0
                          -0.4               0



                         0.008               0
                         0.008               0
                         0.016               0

                       0.0004           0.0004
                       0.0004           0.0004
                       0.0004           0.0004
                       0.0004           0.0004
                       0.0016           0.0016
                       0.0088           0.0008

               Block-diagonal part of risk aversion matrix
                         0.05              0.05
                         0.05              0.05
                                                             0.05            0.05
                                                             0.05            0.05

               Diagonal part of risk aversion matrix
           0.01
                           0.01
                                     0.01
                                              0.01

Total risk aversion matrix
            0.06           0.05
            0.05           0.06
                                     0.06     0.05
                                     0.05     0.06

Pi * Risk aversion matrix
           0.015          0.0125        0        0
          0.0125           0.015        0        0
               0               0    0.015   0.0125
               0               0   0.0125    0.015

        0.0088           0.0008
um is larger than that of B-spread




 quo utility of zero
ak-even buying prices)
Worksheet to illustrate calculation of risk premia in 2x2 model of second-order utilities and probabilities (SOUP)
Status quo wealth = zero, utility functions = exponential with unit slope at zero, risk neutral probabilities = true joint pr
Orange cells = changeable inputs, yellow cells = key outputs, run Solver to re-compute exact risk premia after change

                             States                   Risk neutral probabilities
                             A_1 & B_1                 pi_11 =            0.12
                             A_1 & B_2                 pi_12 =            0.28
                             A_2 & B_1                 pi_21 =            0.48
                             A_2 & B_2                 pi_22 =            0.12

                                                                                Risk-neutral expected value of spread
                                                                                   Buying price for spread (via Solver)
                     Delta                0.1

    1st-order parameters          Probability    Risk tolerance
                 B_1|A_1                 0.3                100
                 B_2|A_1                 0.7                 50


                  B_1|A_2                 0.8                   75
                  B_2|A_2                 0.2                  100

   2nd-order parameters
                   A_1                    0.4                  20
                   A_2                    0.6                  10
                                                                       Total utility
  Here is a variation in which the probabilities are non-
  uniform and the risk tolerances of the utility functions
  are also state-dependent. The qualitative result is the                    Conditional risk-neutral expected values
  same: the risk premium of the A-spread is much
  larger than that of the B-spread, hence the individual
  is still locally uncertainty averse. This illustrates that
  uncertainty aversion can be measured even when
  events are asymmetric and utilities are state-
  dependent. (The utility functions could also have                              s
  state-dependent scale factors, and hence different                          0.05
  slopes at the origin, and this wouldn't matter either: it                    0.1
  would merely be equivalent to an additional distortion
  of the probabilities.)                                                         t
                                                                             0.01
                                                                             0.02
                                                                     0.013333333
                                                                             0.01

                                                Approximate risk premium (via formula in Theorem 1 part (d))
                                                                Diagonal part
                                                                          0.0




                                                                Total risk ave
                                                                           0.02
                                                                           0.01




                                                                Pi * Risk avers




Approximate risk premium (via matrix formula from 2003 paper)
del of second-order utilities and probabilities (SOUP)
h unit slope at zero, risk neutral probabilities = true joint probabilities at status quo wealth
s, run Solver to re-compute exact risk premia after changes

                                         Payoffs          A-spread        B-spread      Net A-spread      Net B-spread
                                         z_11 =        0.833333333     0.833333333      0.840442991       0.834662472
                                         z_12 =        0.357142857    -0.357142857      0.364252515      -0.355813719
                                         z_21 =       -0.208333333     0.208333333     -0.201223676       0.209662472
                                         z_22 =       -0.833333333    -0.833333333     -0.826223676      -0.832004195

         Risk-neutral expected value of spread                   0               0
            Buying price for spread (via Solver)      -0.007109658    -0.001329138
                           Exact risk premium          0.007109658     0.001329138 <--Individual is uncertainty averse if A-spread risk premiu

             First-order utilities
                                      v_11(z_11)      0.248961221      0.248961221      0.251076343       0.249356651
                                      v_12(z_12)      0.249109265     -0.250894987      0.254050252      -0.249957933
                                          v_1(z)      0.498070486     -0.001933766      0.505126595      -0.000601282

                                      v_21(z_12)      -0.166898363     0.166435399     -0.161195086       0.167495751
                                      v_22(z_22)      -0.167363044    -0.167363044     -0.165929265      -0.167094994
                                          v_2(z)      -0.334261407    -0.000927645      -0.32712435       0.000400757
             Second-order utilities
                                      u_1(v_1(z))      0.196767918    -0.000773544      0.199520455      -0.000240517
                                      u_2(v_2(z))     -0.203946426    -0.000556613     -0.199520214       0.000240449
                                            U(z)      -0.007178508    -0.001330156      2.40717E-07      -6.70999E-08

                                                                                        6.24469E-14 Sum of squared deviations from status q
       Conditional risk-neutral expected values                                                     (minimize with Solver to determine brea
                                        z-bar_1                 0.5   -5.55112E-17
                                        z-bar_2       -0.333333333               0



                            pi_1*s_1*(z-bar_1)^2             0.005     6.16298E-35
                            pi_2*s_2*(z-bar_2)^2      0.006666667                0
                                                      0.011666667      6.16298E-35

                              pi_11*t_11*z_11^2       0.000833333      0.000833333
                              pi_12*t_12*z_12^2       0.000714286      0.000714286
                              pi_21_t_21*z_21^2       0.000277778      0.000277778
                              pi_22*t_22*z_22^2       0.000833333      0.000833333
                                                       0.00265873       0.00265873
premium (via formula in Theorem 1 part (d))           0.007162698      0.001329365

                                                    Block-diagonal part of risk aversion matrix
                                                             0.015            0.035
                                                             0.015            0.035
                                                                                                  0.08            0.02
                                                                                                  0.08            0.02
                                             Diagonal part of risk aversion matrix
                                                       0.01
                                                                         0.02
                                                                                0.013333333
                                                                                                 0.01

                                             Total risk aversion matrix
                                                        0.025           0.035
                                                        0.015           0.055
                                                                                0.093333333      0.02
                                                                                        0.08     0.03

                                             Pi * Risk aversion matrix
                                                        0.003          0.0042             0         0
                                                       0.0042          0.0154             0         0
                                                            0               0        0.0448    0.0096
                                                            0               0        0.0096    0.0036

emium (via matrix formula from 2003 paper)      0.007162698     0.001329365
 ainty averse if A-spread risk premium is larger than that of B-spread




m of squared deviations from status quo utility of zero
nimize with Solver to determine break-even buying prices)
Worksheet to illustrate calculation of risk premia in 2x2 model of second-order utilities and probabilities (SOUP)
Status quo wealth = zero, utility functions = exponential with unit slope at zero, risk neutral probabilities = true joint pr
Orange cells = changeable inputs, yellow cells = key outputs, run Solver to re-compute exact risk premia after change

                            States                  Risk neutral probabilities
                            A_1 & B_1                pi_11 =            0.25
                            A_1 & B_2                pi_12 =            0.25
                            A_2 & B_1                pi_21 =            0.25
                            A_2 & B_2                pi_22 =            0.25

                                                                           Risk-neutral expected value of spread
                                                                              Buying price for spread (via Solver)
                    Delta               0.1

    1st-order parameters        Probability    Risk tolerance
                 B_1|A_1               0.5                100
                 B_2|A_1               0.5                100


                 B_1|A_2                0.5              100
                 B_2|A_2                0.5              100

   2nd-order parameters
                   A_1                  0.5                10
                   A_2                  0.5              -100
                                                                  Total utility

    In this "abnormal" example, the 2nd-order risk
    tolerance is positive and relatively small in magnitude             Conditional risk-neutral expected values
    in event A_1, while it is negative and relatively large
    in magnitude in event A_2. The individual is still
    locally uncertainty averse, i.e., the risk premium of
    the A-spread is still much larger than that of the B-
    spread, despite the fact that 2nd-order utility is
    convex (risk seeking) in event A_2, because 2nd-                        s
    order utility is much more strongly concave (risk                     0.1
    averse) in event A_1.                                               -0.01

                                                                             t
                                                                         0.01
                                                                         0.01
                                                                         0.01
                                                                         0.01

                                              Approximate risk premium (via formula in Theorem 1 part (d))
                                                                Diagonal part
                                                                          0.0




                                                                Total risk ave
                                                                            0.0
                                                                            0.0




                                                                Pi * Risk avers




Approximate risk premium (via matrix formula from 2003 paper)
del of second-order utilities and probabilities (SOUP)
h unit slope at zero, risk neutral probabilities = true joint probabilities at status quo wealth
s, run Solver to re-compute exact risk premia after changes

                                         Payoffs          A-spread        B-spread      Net A-spread     Net B-spread
                                         z_11 =                 0.4             0.4     0.404426165      0.400800033
                                         z_12 =                 0.4            -0.4     0.404426165     -0.399199967
                                         z_21 =                -0.4             0.4    -0.395573835      0.400800033
                                         z_22 =                -0.4            -0.4    -0.395573835     -0.399199967

         Risk-neutral expected value of spread                   0               0
            Buying price for spread (via Solver)      -0.004426165    -0.000800033
                           Exact risk premium          0.004426165     0.000800033 <--Individual is uncertainty averse if A-spread risk premiu

             First-order utilities
                                      v_11(z_11)      0.199600533      0.199600533      0.201804732      0.199998951
                                      v_12(z_12)      0.199600533     -0.200400534      0.201804732     -0.199998916
                                          v_1(z)      0.399201066     -0.000800001      0.403609464      3.53725E-08

                                      v_21(z_12)      -0.200400534     0.199600533      -0.19817863      0.199998951
                                      v_22(z_22)      -0.200400534    -0.200400534      -0.19817863     -0.199998916
                                          v_2(z)      -0.400801068    -0.000800001     -0.396357261      3.53725E-08
             Second-order utilities
                                      u_1(v_1(z))      0.195668985    -0.000400017      0.197786459      1.76862E-08
                                      u_2(v_2(z))     -0.199999466    -0.000399999     -0.197786401      1.76862E-08
                                            U(z)      -0.004330481    -0.000800015      5.78818E-08      3.53725E-08

                                                                                        4.60152E-15 Sum of squared deviations from status q
       Conditional risk-neutral expected values                                                     (minimize with Solver to determine brea
                                        z-bar_1                 0.4               0
                                        z-bar_2                -0.4               0



                            pi_1*s_1*(z-bar_1)^2              0.008               0
                            pi_2*s_2*(z-bar_2)^2            -0.0008               0
                                                             0.0072               0

                              pi_11*t_11*z_11^2             0.0004           0.0004
                              pi_12*t_12*z_12^2             0.0004           0.0004
                              pi_21_t_21*z_21^2             0.0004           0.0004
                              pi_22*t_22*z_22^2             0.0004           0.0004
                                                            0.0016           0.0016
premium (via formula in Theorem 1 part (d))                 0.0044           0.0008

                                                    Block-diagonal part of risk aversion matrix
                                                              0.05              0.05
                                                              0.05              0.05
                                                                                             -0.005             -0.005
                                                                                             -0.005             -0.005
                                             Diagonal part of risk aversion matrix
                                                       0.01
                                                                         0.01
                                                                                         0.01
                                                                                                    0.01

                                             Total risk aversion matrix
                                                         0.06           0.05
                                                         0.05           0.06
                                                                                        0.005     -0.005
                                                                                       -0.005      0.005

                                             Pi * Risk aversion matrix
                                                        0.015          0.0125               0          0
                                                       0.0125           0.015               0          0
                                                            0               0         0.00125   -0.00125
                                                            0               0        -0.00125    0.00125

emium (via matrix formula from 2003 paper)           0.0044           0.0008
 ainty averse if A-spread risk premium is larger than that of B-spread




m of squared deviations from status quo utility of zero
nimize with Solver to determine break-even buying prices)
Worksheet to illustrate calculation of risk premia in 2x2 model of second-order utilities and probabilities (SOUP)
Status quo wealth = zero, utility functions = exponential with unit slope at zero, risk neutral probabilities = true joint pr
Orange cells = changeable inputs, yellow cells = key outputs, run Solver to re-compute exact risk premia after change

                            States                  Risk neutral probabilities
                            A_1 & B_1                pi_11 =            0.12
                            A_1 & B_2                pi_12 =            0.28
                            A_2 & B_1                pi_21 =            0.48
                            A_2 & B_2                pi_22 =            0.12

                                                                           Risk-neutral expected value of spread
                                                                              Buying price for spread (via Solver)
                    Delta               0.1

    1st-order parameters        Probability    Risk tolerance
                 B_1|A_1               0.3                100
                 B_2|A_1               0.7                 50


                 B_1|A_2                0.8               75
                 B_2|A_2                0.2              100

   2nd-order parameters
                   A_1                  0.4               10
                   A_2                  0.6             -100
                                                                  Total utility


                                                                        Conditional risk-neutral expected values
   Here is a variation on the preceding scenario in
   which the probabilities and first-order risk
   tolerances are non-constant. Again, the individual
   is locally uncertainty-averse because her
   2nd-order utility is much more concave in event
   A_1 and than it is convex in event A_2.                                  s
                                                                          0.1
                                                                        -0.01

                                                                            t
                                                                        0.01
                                                                        0.02
                                                                0.013333333
                                                                        0.01

                                              Approximate risk premium (via formula in Theorem 1 part (d))
Approximate risk premium (via matrix formula from 2003 paper)
del of second-order utilities and probabilities (SOUP)
h unit slope at zero, risk neutral probabilities = true joint probabilities at status quo wealth
s, run Solver to re-compute exact risk premia after changes

                                         Payoffs          A-spread        B-spread      Net A-spread     Net B-spread
                                         z_11 =        0.833333333     0.833333333      0.839406734      0.834732979
                                         z_12 =        0.357142857    -0.357142857      0.363216258     -0.355743211
                                         z_21 =       -0.208333333     0.208333333     -0.202259933      0.209732979
                                         z_22 =       -0.833333333    -0.833333333     -0.827259933     -0.831933687

         Risk-neutral expected value of spread                   0               0
            Buying price for spread (via Solver)      -0.006073401    -0.001399646
                           Exact risk premium          0.006073401     0.001399646 <--Individual is uncertainty averse if A-spread risk premiu

             First-order utilities
                                      v_11(z_11)      0.248961221      0.248961221      0.250768066      0.249377627
                                      v_12(z_12)      0.249109265     -0.250894987       0.25333013     -0.249908225
                                          v_1(z)      0.498070486     -0.001933766      0.504098196     -0.000530598

                                      v_21(z_12)      -0.166898363     0.166435399     -0.162026324          0.167552
                                      v_22(z_22)      -0.167363044    -0.167363044     -0.166138237     -0.167080774
                                          v_2(z)      -0.334261407    -0.000927645     -0.328164561      0.000471225
             Second-order utilities
                                      u_1(v_1(z))      0.194348067    -0.000773581      0.196641312     -0.000212245
                                      u_2(v_2(z))     -0.200222025    -0.000556584     -0.196576014      0.000282736
                                            U(z)      -0.005873958    -0.001330165      6.52989E-05      7.04911E-05

                                                                                        9.23294E-09 Sum of squared deviations from status q
       Conditional risk-neutral expected values                                                     (minimize with Solver to determine brea
                                        z-bar_1                 0.5   -5.55112E-17
                                        z-bar_2       -0.333333333               0



                            pi_1*s_1*(z-bar_1)^2               0.01     1.2326E-34
                            pi_2*s_2*(z-bar_2)^2      -0.000666667               0
                                                       0.009333333      1.2326E-34

                              pi_11*t_11*z_11^2       0.000833333     0.000833333
                              pi_12*t_12*z_12^2       0.000714286     0.000714286
                              pi_21_t_21*z_21^2       0.000277778     0.000277778
                              pi_22*t_22*z_22^2       0.000833333     0.000833333
                                                       0.00265873      0.00265873
premium (via formula in Theorem 1 part (d))           0.005996032     0.001329365

                                                    Block-diagonal part of risk aversion matrix
                                                              0.03              0.07              0                  0
                                                              0.03              0.07              0                  0
                                                                 0                 0         -0.008             -0.002
                                                                 0                 0         -0.008             -0.002
                                             Diagonal part of risk aversion matrix
                                                       0.01                 0             0           0
                                                           0             0.02             0           0
                                                           0                0   0.013333333           0
                                                           0                0             0        0.01

                                             Total risk aversion matrix
                                                         0.04           0.07
                                                         0.03           0.09
                                                                                0.005333333      -0.002
                                                                                      -0.008      0.008

                                             Pi * Risk aversion matrix
                                                       0.0048          0.0084             0           0
                                                       0.0084          0.0252             0           0
                                                            0               0       0.00256    -0.00096
                                                            0               0      -0.00096     0.00096

emium (via matrix formula from 2003 paper)      0.005996032     0.001329365
 ainty averse if A-spread risk premium is larger than that of B-spread




m of squared deviations from status quo utility of zero
nimize with Solver to determine break-even buying prices)
Worksheet to illustrate calculation of risk premia in 2x2 model of second-order utilities and probabilities (SOUP)
Status quo wealth = zero, utility functions = exponential with unit slope at zero, risk neutral probabilities = true joint pr
Orange cells = changeable inputs, yellow cells = key outputs, run Solver to re-compute exact risk premia after change

                             States                   Risk neutral probabilities
                             A_1 & B_1                 pi_11 =            0.12
                             A_1 & B_2                 pi_12 =            0.28
                             A_2 & B_1                 pi_21 =            0.48
                             A_2 & B_2                 pi_22 =            0.12

                                                                             Risk-neutral expected value of spread
                                                                                Buying price for spread (via Solver)
                     Delta                0.1

    1st-order parameters         Probability     Risk tolerance
                 B_1|A_1                0.3                 100
                 B_2|A_1                0.7                  50


                  B_1|A_2                 0.8               50
                  B_2|A_2                 0.2              100

   2nd-order parameters
                   A_1                    0.4              100
                   A_2                    0.6              -10
                                                                    Total utility


                                                                          Conditional risk-neutral expected values
   Here the 2nd-order risk tolerances have been
   flipped so that 2nd-order utility is highly convex in
   event A_2 and only slightly concave in event A_1.
   Now the risk premium for the A-spread is smaller
   than that of the B-spread--in fact, it is negative.
   Hence, the individual is not only                                          s
   uncertainty-seeking, but she is also risk-seeking                       0.01
   with respect to bets on the A-events.                                   -0.1

                                                                               t
                                                                           0.01
                                                                           0.02
                                                                           0.02
                                                                           0.01

                                                Approximate risk premium (via formula in Theorem 1 part (d))
Approximate risk premium (via matrix formula from 2003 paper)
del of second-order utilities and probabilities (SOUP)
h unit slope at zero, risk neutral probabilities = true joint probabilities at status quo wealth
s, run Solver to re-compute exact risk premia after changes

                                         Payoffs          A-spread        B-spread      Net A-spread     Net B-spread
                                         z_11 =        0.833333333     0.833333333      0.831878001      0.834732757
                                         z_12 =        0.357142857    -0.357142857      0.355687525     -0.355743433
                                         z_21 =       -0.208333333     0.208333333     -0.209788666      0.209732757
                                         z_22 =       -0.833333333    -0.833333333     -0.834788666      -0.83193391

         Risk-neutral expected value of spread                   0               0
            Buying price for spread (via Solver)       0.001455332    -0.001399424
                           Exact risk premium         -0.001455332     0.001399424 <--Individual is uncertainty averse if A-spread risk premiu

             First-order utilities
                                      v_11(z_11)      0.248961221      0.248961221      0.248528241      0.249377561
                                      v_12(z_12)      0.249109265     -0.250894987      0.248097768     -0.249908382
                                          v_1(z)      0.498070486     -0.001933766       0.49662601     -0.000530821

                                      v_21(z_12)      -0.167014372     0.166319926     -0.168183516      0.167434795
                                      v_22(z_22)      -0.167363044    -0.167363044     -0.167656548     -0.167080819
                                          v_2(z)      -0.334377416    -0.001043118     -0.335840064      0.000353975
             Second-order utilities
                                      u_1(v_1(z))      0.198732869    -0.000773514      0.198157945     -0.000212329
                                      u_2(v_2(z))     -0.197309277    -0.000625838     -0.198157945      0.000212389
                                            U(z)       0.001423591    -0.001399352     -4.55922E-10      6.00328E-08

                                                                                        3.60414E-15 Sum of squared deviations from status q
       Conditional risk-neutral expected values                                                     (minimize with Solver to determine brea
                                        z-bar_1                 0.5   -5.55112E-17
                                        z-bar_2       -0.333333333               0



                            pi_1*s_1*(z-bar_1)^2              0.001     1.2326E-35
                            pi_2*s_2*(z-bar_2)^2      -0.006666667               0
                                                      -0.005666667      1.2326E-35

                              pi_11*t_11*z_11^2        0.000833333     0.000833333
                              pi_12*t_12*z_12^2        0.000714286     0.000714286
                              pi_21_t_21*z_21^2        0.000416667     0.000416667
                              pi_22*t_22*z_22^2        0.000833333     0.000833333
                                                       0.002797619     0.002797619
premium (via formula in Theorem 1 part (d))           -0.001434524      0.00139881

                                                    Block-diagonal part of risk aversion matrix
                                                             0.003            0.007                0                 0
                                                             0.003            0.007                0                 0
                                                                 0                0            -0.08             -0.02
                                                                 0                0            -0.08             -0.02
                                             Diagonal part of risk aversion matrix
                                                       0.01                 0             0         0
                                                           0             0.02             0         0
                                                           0                0          0.02         0
                                                           0                0             0      0.01

                                             Total risk aversion matrix
                                                        0.013           0.007
                                                        0.003           0.027
                                                                                       -0.06     -0.02
                                                                                       -0.08     -0.01

                                             Pi * Risk aversion matrix
                                                     0.00156         0.00084               0         0
                                                     0.00084         0.00756               0         0
                                                            0              0         -0.0288   -0.0096
                                                            0              0         -0.0096   -0.0012

emium (via matrix formula from 2003 paper)     -0.001434524      0.00139881
 ainty averse if A-spread risk premium is larger than that of B-spread




m of squared deviations from status quo utility of zero
nimize with Solver to determine break-even buying prices)
Plot of normalized exponential utility as used in examples: marginal utility (slope) is normalized to 1 at z=0
Utility function is concave for R>0, convex for R<0, approaches linear utility as R goes to plus or minus infinity

Risk tolerance (R)                    1

                 z      R*(1-exp(-z/R))
                -1       -1.718281828
             -0.95       -1.585709659
              -0.9       -1.459603111
             -0.85       -1.339646852
              -0.8       -1.225540928
             -0.75       -1.117000017
              -0.7       -1.013752707
             -0.65       -0.915540829
              -0.6          -0.8221188
             -0.55       -0.733253018
              -0.5       -0.648721271
             -0.45       -0.568312185
              -0.4       -0.491824698
                                                    -1         -0.8       -0.6       -0.4
             -0.35       -0.419067549
              -0.3       -0.349858808
             -0.25       -0.284025417
              -0.2       -0.221402758
             -0.15       -0.161834243
              -0.1       -0.105170918
             -0.05       -0.051271096
                 0                   0
              0.05        0.048770575
               0.1        0.095162582
              0.15        0.139292024
               0.2        0.181269247
              0.25        0.221199217
               0.3        0.259181779
              0.35         0.29531191
               0.4        0.329679954
              0.45        0.362371848
               0.5         0.39346934
              0.55         0.42305019
               0.6        0.451188364
              0.65        0.477954223
               0.7        0.503414696
              0.75        0.527633447
               0.8        0.550671036
              0.85        0.572585068
               0.9         0.59343034
              0.95        0.613258977
                 1        0.632120559
 e) is normalized to 1 at z=0
R goes to plus or minus infinity




                         1




                       0.5




                         0
               -0.2          0     0.2   0.4   0.6   0.8   1



                      -0.5




                        -1




                      -1.5




                        -2

				
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