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Monotone transformation of utility: Some particular cases
*
Philippe Godfroid
´
Fonds national de la Recherche Scientifique, Catholic Faculties of Mons, 151, Chaussee de Binche, Mons B-7000,
Belgium
Received 20 June 2000; received in revised form 4 January 2001; accepted 10 January 2001
Abstract
In this paper we develop for two particular utility functions a technique that generalizes the fundamental work
th
of Pratt on a ‘more risk averse’ function to the n order of absolute risk aversion. 2001 Published by Elsevier Science B.V.
Keywords: Mixed risk aversion; Transformation of utility
JEL classification: D81
1. Introduction
In many applications of risk theory it is necessary to do some transformations of the utility function in order to define a ‘more risky’ (or a ‘more prudent’) behavior.
Since the fundamental work of Pratt (1964), we know that the composite utility function
V(W )5g U(W ) is more risk averse than U(W ) for g increasing and concave.f g
This paper partly extends (for the CARA and log utility cases) this result to ‘g completely ´
monotone’ (as defined by Caballe and Pomansky (1996) and V(W ) ‘more nth -order risk averse’. For related work see Jullien et al. (1999); Chiu (1997) and Dachraoui et al. (1999). Notice also that Eeckhoudt and Schlesinger (1994) show that under some conditions, g9 convex is sufficient to imply
that V(W ) is also more prudent than U(W ).
The main results presented in this paper are only true for either CARA or logarithmic utility functions. Unfortunately the result does not hold for all risk averse utility functions.
*Tel.:132-65-323-388; fax: 132-65-315-691.
E-mail address: [email protected] (P. Godfroid).
2. The negative exponential case
th
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According to the work of Caballe and Pomansky (1996) we can define the n order index of absolute risk aversion for the function U as:
n11
If the utility function of the final wealth U(W )5 2exp exhibits an nth order absolute index of
1
risk aversion of constant value a , and if we suppose that there exists a completely monotone
V
function g such that the function V(W )5g U(W ) , we can show that A (W ) is always greater thanf g
n U
A (W ).n
Proposition 1. A CARA function transformed by a complete monotone function exhibits always a
th
greater n order absolute index of risk aversion.
Proof. The successive derivatives for V(W ) are given by:
V9(W )5g9fU(W ) Ug 9(W )
(1)
2 V0(W )5g9fU(W ) Ug 0(W )1g0fU(W ) Ugs 9(W )d
In order to simplify the notation we can rewrite (1) into:
( 2 ) s d1 s d2 s d2
s
s d1d
2V 5g U 1g U (2)
while applying this formulation to the CARA case, we obtain for (2):
( 2 ) 2 s d1 2aW 2 s d2 22aW 2 s d1 2aW s d2 22aW
V 5 2a g exp 1a g exp 5 2a
f
g exp 2g expg
th
In a more general way, we can write the n derivative for V(W ):
n
(n) n11 n k11 s dk 2kaW
V 5 2s 1d a
O
s21d f k, n gs d exp (3)k51
where the recurrence equation f k, n is given by:s d
f k, ns 11d 5 kf k, ns d1f ks 21, nd
(4) with f 0, 0s d 5 1 and f k, 0s d5f 0, ns d50
For the sake of convenience, we can illustrate (4) for k and n varying from 0 to 10 in Table 1.
1
Then the n order absolute index of risk aversion is given by
By induction, we can show that (3) is also satisfied for the n11 order of derivation. We have:
This confirms that (3) remains valid for the n11 order of derivation. Moreover we have:
f k, ns 11d$f k, n and f ns d s 11, n11d51
Knowing that s21d g exp is a strictly positive function, we have:
n11 n
We assume now that the individual is characterized by a logarithmic utility function:
U Ws d5ln Ws 1Cd
where C is a positive constant. The successive derivatives are expressed in the following general form:
(n) n11 2n
and the nth order absolute index of risk aversion can be expressed by:
Proposition 2. A logarithmic function transformed by a complete monotone function always exhibits
th
a greater n order absolute index of risk aversion.
th
Proof. The n derivative of the component function is defined by:
n
The recurrence equation is given in this case by:
f k, ns 11d5f ks 21, nd1nf k, ns d
An illustration is given in Table 2 for k and n varying from 0 to 10.
By using an induction mechanism, we must take the derivative of this equation in order to obtain
(n11 )
Table 1
The recurrence Eq. (4) for k and n varying from 0 to 10 for the exponential utility function
n\k 0 1 2 3 4 5 6 7 8 9 10
0 1 0 0 0 0 0 0 0 0 0 0
1 0 1 0 0 0 0 0 0 0 0 0
2 0 1 1 0 0 0 0 0 0 0 0
3 0 1 3 1 0 0 0 0 0 0 0
4 0 1 7 6 1 0 0 0 0 0 0
5 0 1 15 25 10 1 0 0 0 0 0
6 0 1 31 90 65 15 1 0 0 0 0
7 0 1 63 301 350 140 21 1 0 0 0
8 0 1 127 966 1701 1050 266 28 1 0 0
9 0 1 255 3025 7770 6951 2646 462 36 1 0
10 0 1 511 9330 34 105 42 525 22 827 5880 750 45 1
Table 2
For the logarithmic case, we can illustrate (5) for k and n varying from 0 to 10
n\k 0 1 2 3 4 5 6 7 8 9 10
0 1 0 0 0 0 0 0 0 0 0 0
1 0 1 0 0 0 0 0 0 0 0 0
2 0 1 1 0 0 0 0 0 0 0 0
3 0 2 3 1 0 0 0 0 0 0 0
4 0 6 11 6 1 0 0 0 0 0 0
5 0 24 50 35 10 1 0 0 0 0 0
6 0 120 274 225 85 15 1 0 0 0 0
7 0 720 1764 1624 735 175 21 1 0 0 0
8 0 5040 13 068 13 132 6769 1960 322 28 1 0 0
9 0 40 320 109 584 118 124 67 284 22 449 4536 546 36 1 0
10 0 362 880 – – – – 63 273 9450 870 45 1
Acknowledgements
This letter owes much to discussions with my colleague M. Lardinois.
References
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Jullien, B., Salanie, B., Salanie, F., 1999. Should more risk-averse agents exert more effort? The Geneva Papers on Risk and Insurance Theory 24, 19–28.
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Caballe, J., Pomansky, A., 1996. Mixed risk aversion. Journal of Economic Theory 71, 485–513.
Chiu, W.H., 1997. The propensity to self-protect. Working paper, School of Economics Studies, University of Manchester. Dachraoui, K., Dionne, G., Eeckhoudt, L., Godfroid, Ph., 1999. Proper risk behavior. Working paper, Risk Management
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Chair, HEC-Montreal, 99-01.
