Research Methods/ Statistical Methods, Risk and Uncertainty,

McCarl, Bruce A.

English

Research Papers in Economics

Interpretations and Transformations of
Scale  for the Pratt-Arrow Absolute  Risk
Aversion  Coefficient:  Implications for
Generalized  Stochastic  Dominance:
Comment
Bruce A. McCarl
Raskin and Cochran (RC) recently presented a use-
ful paper  dealing with several  items relating to the
Pratt-Arrow  risk  aversion  coefficient  (RAC).  One
aspect of that paper deals with the way that the RAC
changes with respect to arithmetic manipulations of
the outcome variable. In particular, the authors show
the effects of  multiplying and adding constants. This
note takes issue with RC's statements regarding the
addition of constants, i.e., their theorem 2. In turn,
the implications of the criticisms in terms of RC's
conclusions are  explored.
Theorem  2 in the RC article  states that, given a
utility function u(w) and its associated risk aversion
function r(w),  "If v = x  +c where  c  is a constant,
then r(v)  = r(x). Therefore,  the magnitude  of the
risk aversion coefficient  is unaffected by the  use of
incremental rather than absolute returns..  ." (p. 207).
The first  part  of the statement  of the  theorem  is
Bruce A. McCarl is a professor in the Department of Agricultural
Economics, Texas  A&M University.
Technical paper no. 22806 of the Texas Agricultural Experiment
Station. The author thanks Francis McCamley for aid in improving
the presentation  and for catching errors.
accurate and properly proven in the appendix; how-
ever, the basic contention of the second half of the
theorem is that the RAC at income level x is equal
to the RAC at wealth  level v = x  + c. This is not
equivalent  to saying  r(v) = r(x) as RC state in  the
first part of the theorem but rather one  must show
that
d
2 u(x  + c)
dx2
du(x + c)
dx
d
2u(x)
dx2
du(x)
dx
However, this is not generally true,  as can be dem-
onstrated  with  a  counter-example.  Consider  the
commonly  posited polynomial utility form, specif-
ically (and purely for expository purposes), the qua-
dratic
u(x) = a + bx + dx2;
then if =  x  +  c,
u(v)  = u(x  +  c)
= a + b(x  + c) + d(x  + c)2,
Table  1.  Utility Functions Used
Number  Source  Utility Function
1  Lin, Dean, and Moore subject  la  U = 44.52 +  1.96 W - .0099 W
2
2  Lin, Dean, and Moore subject  3  U = 55.74  +  1.27W - .0031 W2
3  Lin, Dean,  and Moore subject  5  U = -54.01  + 9.67 W-  .19W
2 + .0012W
3
4  Lin, Dean, and Moore subject  6  U = 70.01  +  1.30W - .0064W2
5  Kaufmanb  U= -263.61  + 22.093  ln(W +  150,000)
a The  Lin, Dean,  and Moore  functions are drawn from their table  2 on  page 504.  W was elicited in thousands of dollars  of net farm
income. Linear  functions were omitted as they exhibit zero  RAC's.
b The Kaufman  function exhibits a decreasing  Pratt-Arrow coefficient over  its whole domain.
Western Journal  of Agricultural  Economics, 12(2):  228-230
Copyright  1987 Western Agricultural Economics  AssociationRisk Aversion Coefficient: Comment  229
Table 2.  Risk Aversion  Coefficients  at Various Wealth Levels
Size of Snize  ofm  Utility Function
Incrementa
(%)  1  2  3  4  5
-50  1.352E-02  5.561E-03  8.264E-02  1.306E-02  5.714E-06
-10  1.852E-02  6.256E-03  -4.000E-01  1.768E-02  5.128E-06
-1  2.021E-02  6.438E-03  -7.396E-02  1.921E-02  5.013E-06
-. 1  2.039E-02  6.456E-03  -6.189E-02  1.938E-02  5.001E-06
0  2.041E-02  6.458E-03  -6.061E-02  1.939E-02  5.000E-06
+.1  2.043E-02  6.460E-03  -5.934E-02  1.941E-02  4.999E-06
+ 1  2.062E-02  6.479E-03  -4.836E-02  1.958E-02  4.988E-06
+10  2.273E-02  6.674E-03  4.706E-02  2.148E-02  4.878E-06
+50  4.168E-02  7.702E-03  -1.127E-01  3.765E-02  4.444E-06
a Relative to initial wealth.
the risk aversion coefficient of the function  u(x) is
-2d
b  + 2dx '
while  the risk aversion coefficient  of u(x + c) is
-2d
b + 2d(x  + c)'
Here  these  coefficients  are  equal  only if c  equals
zero. This is clearly a counter-example to the second
part of RC's theorem  2, rendering it invalid.
Consequently,  RC cannot claim  "the magnitude
of the risk aversion coefficient  is unaffected  by the
use of incremental rather than absolute returns"  (p.
207).  This would be the case with certain  very re-
strictive functional forms,  i.e.,  linear and constant
risk aversion negative exponential functions, or zero
wealth.  Furthermore,  the Pratt-Arrow  RAC  [r(w)]
has  been  proposed  as  a risk  measure  in  terms  of
wealth  (w).  If RC's theorem were valid,  one  could
abandon the wealth concept and look only at wealth
increments  (i.e.,  income).  The  invalidity  of this
theorem leaves wealth as the item of focus.
The  above  findings  have  implications  for  RC's
examples  2 and 3. RC contend in examples  2 and
3 that when the size of the risky prospect is changed,
the RAC should be changed in a reciprocal fashion;
i.e.,  in example  3 (p.  207) RC state that when  de-
ciding on annual income vis-a-vis ten-year net pres-
ent value, "The r over the new ten-year [period]  ..
would be obtained by dividing the old r by the ten-
year NPV." In example 2, RC indicate when going
from a whole farm to  a single-acre  basis, r should
be divided by the reciprocal of the number of acres.
In both cases,  the rules  are  strictly correct  only if
wealth is zero or is divided by the same  amount.
This may  again  be illustrated through  example.
Table 1 shows utility functions reported in Lin, Dean,
and Moore;  and Kaufman (p. 178 or as reported in
Keeney  and Raiffa,  p. 205).  Evaluating  the associ-
ated  RAC's  at  an  initial  wealth  level  of $50  for
utility functions  1-4  and $50,000 for number 5 as
well as at wealth plus and minus an increment equal-
ing  .1%,  1%,  10%,  and 50% of initial wealth yields
the data in table  2.  Dividing the resultant RAC by
the RAC  at  initial wealth  leads to  the data in  ta-
ble  3.
Table 3.  Comparison of Proportional Change  in the Risk Aversion  Coefficient  With Those
Predicted by the Raskin and Cochran Formula
Size of  Change Predicted Size o f enProportional  Change by Utility Functionb 
h ange redicted
Increment
a by Raskin and
(%)  1  2  3  4  5  Cochran Formula
-50  .66  .86  -1.36  .67  1.14  2.00
-10  .91  .97  6.60  .91  1.03  1.11
-1  .99  1.00  1.22  .99  1.00  1.01
-. 1  1.00  1.00  .98  1.00  1.00  1.00
+.1  1.00  1.00  .98  1.00  1.00  1.00
+1  1.01  1.00  .80  1.01  1.00  .99
+10  1.11  1.03  -. 78  1.11  .98  .91
+50  2.04  1.19  1.86  1.94  .89  .67
a Relative  to initial wealth.
b Formed by dividing the RAC at terminal wealth by the RAC at initial wealth.
McCarlWestern Journal  of  Agricultural Economics
The last  column  of table  3 gives  RC's formula
evaluated at the change in the risk aversion coeffi-
cient when wealth is not divided by the increase in
the bet size; i.e., one  over the proportional change.
Thus, if the income level is raised by  1.5, then RC
would forecast 1/1.5 or .67 of that before. The data
show  that r(w)  potentially  does  change  as  the in-
cremental income gets large relative to initial wealth.
However, as can be seen from table 3, RC's forecasts
are  not  very  accurate.  Under  the  increasing  risk
aversion functions  (1,  2,  and  3), the forecast  is in
the wrong direction, while, with the decreasing RAC
function (5) RC forecast,  it is too large a change.
In summary,  Raskin and Cochran properly con-
clude that the units of r and x are inversely related
but improperly  conclude  that risk  aversion coeffi-
cients are unaffected  by  the addition  of constants.
Furthermore,  the RAC cannot be considered solely
with respect to the size of the risky prospect at RC's
discussion implies, but rather wealth  must also be
considered.  Consequently,  when  there  is  nonzero
wealth,  the  magnitude  of the risk aversion  coeffi-
cient does not vary in a reciprocal relationship with
the size of the risky prospect unless wealth  is also
scaled accordingly.
[Received February 1987;  final revision
received June 1987.]
References
Kaufman,  G.  M.  Statistical  Decision and  Related Tech-
niques in Oil and Gas Exploration. Englewood Cliffs
NJ: Prentice Hall,  1963.
Keeney,  R.  L.,  and  H.  Raiffa,  Decisions with Multiple
Objectives:  Preferences  and Value Tradeoffs. New York:
John Wiley & Sons,  1976.
Lin,  W., G.  W. Dean, and C. V. Moore.  "An Empirical
Test of Utility vs. Profit Maximization in Agricultural
Production." Amer. J. Agr. Econ. 56(1974):498-508.
Raskin,  R.,  and  M.  J.  Cochran.  "Interpretations  and
Transformations of Scale  for the Pratt-Arrow  Abso-
lute Risk Aversion Coefficient:  Implications for Gen-
eralized Stochastic Dominance."  West. J. Agr. Econ.
11(1986):204-10.
230  December 1987

An Empirical Test of Utility vs. Profit Maximization in Agricultural Production.&quot;

Decisions with Multiple Objectives: Preferences and Value Tradeoffs.

Interpretations and Transformations of Scale for the Pratt-Arrow Absolute Risk Aversion Coefficient: Implications for Generalized Stochastic Dominance.&quot;

Statistical Decision and Related Techniques in Oil and Gas Exploration. Englewood Cliffs NJ:

INTERPRETATIONS AND TRANSFORMATIONS OF SCALE FOR THE PRATT-ARROW ABSOLUTE RISK AVERSION COEFFICIENT: IMPLICATIONS FOR GENERALIZED STOCHASTIC DOMINANCE: COMMENT

INTERPRETATIONS AND TRANSFORMATIONS OF SCALE FOR THE PRATT-ARROW ABSOLUTE RISK AVERSION COEFFICIENT: IMPLICATIONS FOR GENERALIZED STOCHASTIC DOMINANCE: COMMENT

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