# Example of second-order probabilities and utilities

Document Sample

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 =

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
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

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

0.199600533      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
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

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
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

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
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

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

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

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
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

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

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

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
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

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

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.
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

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
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

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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