This paper characterizes downside risk aversion in a simple and intuitive manner. It is shown that using this characterization one can simplify considerably a theorem by Jindapon (2010) relating to greater downside risk aversion as measured by the prudence probability premium. The comparative statics of downside risk aversion in risk-free wealth are also considered.downside risk aversion, prudence

Richard Watt

English

Research Papers in Economics

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R. Watt 
 
 
 
A NOTE ON GREATER DOWNSIDE RISK AVERSION 
 
 
 
Working Paper No.17/2011 
 
 
 
 
 
 
 
. A NOTE ON GREATER DOWNSIDE RISK AVERSION
RICHARD WATT
Abstract. This paper characterizes downside risk aversion in a simple and
intuitive manner. It is shown that using this characterization one can simplify
considerably a theorem by Jindapon (2010) relating to greater downside risk
aversion as measured by the prudence probability premium. The comparative
statics of downside risk aversion in risk-free wealth are also considered.
Keywords: downside risk aversion, prudence
JEL code: D8
1. Introduction
Traditionally, downside risk aversion is determined as the preference for locating
a zero-mean risk on the “upside” rather than on the “downside” of a primary lottery
with two equally probable states. The upside (downside) of the primary lottery is
simply the state of nature with the largest (smallest) payoﬀ. Thus, say the primary
gives the individual wealth of w1 in state 1 and w2 in state 2, where w1 >w 2,a n d
where both state 1 and state 2 occur with probability 1
2, then utility function u is
downside risk averse if and only if
1
2
Eu(w1 + e x)+
1
2
u(w2) >
1
2
u(w1)+
1
2
Eu(w2 + e x) (1)
where Ee x =0 , and random variable e x has positive variance. It is well known that
as u ﬃcient and necessary condition for (1) is that marginal utility is convex, that
is, u000, which together with the assumptions of positive marginal utility and risk
aversion (concavity of utility), implies that the decision maker is “prudent”.
In order to measure an intensity of downside risk aversion, the literature has often
considered compensation amounts under which the decision maker would accept
to have the risk on the downside rather than the upside. More particularly, the
literature has considered the minimum level of compensation such that the decision
maker is indiﬀerent between having the risk on the upside (without compensation)
or on the downside (with compensation). For example, Crainich and Eeckhoudt
Department of Economics and Finance, University of Canterbury, Private Bag 4800, Christchurch
8140, New Zealand. E-mail: richard.watt@canterbury.ac.nz. I am grateful to the hospitality of
ICER in Turin (Italy), where I was visiting while this paper was written.
12
(2009) consider the wealth compensation m such that
1
2
Eu(w1 + e x)+
1
2
u(w2)=
1
2
u(w1 + m)+
1
2
Eu(w2 + e x)
Taking a diﬀerent perspective on the problem, Jindapon (2010) considers the prob-
ability premium, q, such that
1
2
Eu(w1 + e x)+
1
2
u(w2)=
µ
1
2
+ q
¶
u(w1)+
µ
1
2
− q
¶
Eu(w2 + e x)
2. Greater Downside Risk Aversion
Looking at compensation amounts with the objective of characterizing greater
downside risk aversion normally requires an assumption of small risks. Instead of
looking at a compensation amount for carrying the risk on the downside rather
t h a nt h eu p s i d e ,Ic h a r a c t e r i z ed o w n s i d er i s ka v e r s i o ni nt e r m so ft h es i z eo ft h e
set of lotteries for which the risk e x w o u l db el o c a t e do nt h ed o w n s i d e( t h es m a l l e r
is this set, the more downside risk averse is the decision maker). In so doing, the
analysis is valid for risks e x of any size (small and large).
Notice that if the probability of state 1 was equal to 1, then the zero-mean
risk would always be optimally located in state 2, as that way it can be avoided
completely which under risk aversion would of course be optimal. Thus, if a utility
function is downside risk averse, and risk averse, by continuity there must exist a
probability p∗,d e ﬁned on 1
2 <p ∗ < 1,s u c ht h a t
p∗Eu(w1 + e x)+( 1− p∗)u(w2)=p∗u(w1)+( 1− p∗)Eu(w2 + e x) (2)
The number p∗ is in and of itself a natural measure of the concept of downside
risk aversion. Notice that p∗ splits the set p =[ 0 ,1] into two mutually independent
parts. For lotteries with p<p ∗, the decision maker prefers to locate the risk e x on
the upside of the primary lottery, and for lotteries with p>p ∗ locating the risk e x
on the downside is preferred. Therefore, the greater is p∗ the smaller is the set of
lotteries for which locating e x on the downside is preferred. It is thus very natural
to state that the greater is p∗, the more downside risk averse is our decision maker.
Such a deﬁnition of greater downside risk aversion mimics to a certain degree the
traditional sorting of utility functions according to their absolute risk aversion — the
more risk averse is a utility function, the smaller is the set of lotteries that would
be voluntarily exchanged for a given risk-free wealth allocation.
Note from (2) that p∗ can be written as
p∗ =
u(w2) − Eu(w2 + e x)
[u(w2) − Eu(w2 + e x)] + [u(w1) − Eu(w1 + e x)]3
or, deﬁning f(wi,e x) ≡ u(wi) − Eu(wi + e x),w eh a v e
p∗ =
f(w2,e x)
f(w2, e x)+f(w1, e x)
(3)
The function f(wi, e x) is habitually referred to as the “utility premium” associated
with random variable e x at wealth level wi. It is well known that the utility premium
is strictly positive under risk aversion, and strictly decreasing in wi if u000 is positive,
that is, if the utility function displays positive prudence (see Hanson and Menezes,
1971, and Eeckhoudt and Schlesinger, 2006).1 In all that follows, the secondary
risk e x will not be altered, and so we shall simplify the notation by eliminating e x
from the expression for the utility premium. Thus, we write (3) as
p∗ =
f(w2)
f(w2)+f(w1)
(4)
Using (4) we can easily characterize changes that will increase (resp. decrease)
the value of p∗:
Proposition 1. Any alteration in u(w) that causes a relative change in f(w2) that
is greater than (resp. less than, equal to) the relative change in f(w1) will serve to
increase (resp. decrease, not alter) the value of p∗.
Consider a transformation of u(w),s a yv(w)=G(u(w)). Then utility function
v(w) will be more downside risk averse than utility function u(w), in the sense that
p∗
v >p ∗
u,i f 2
fv(w2) − fu(w2)
fu(w2)
>
fv(w1) − fu(w1)
fu(w1)
fv(w2)
fu(w2)
− 1 >
fv(w1)
fu(w1)
− 1
fv(w2)
fu(w2)
>
fv(w1)
fu(w1)
fv(w2)
fv(w1)
>
fu(w2)
fu(w1)
(5)
This can be summed up as:
Proposition 2. The intensity of downside risk aversion can be measured by the
ratio of the utility premium in state of nature 2 to the utility premium in state of
nature 1,
f(w2)
f(w1).
1Notice that f(wi, h x) is unchanged by an additive alteration to utility, but is altered by a multi-
plicative change to utility. Thus the utility premium is not comparable over diﬀerent individuals.
However, p∗ is unaﬀected by multiplicative changes to utility, and thus p∗ is immune to any linear
transform of the underlying utility function.
2The following result can also be proved by simply requiring that 1
p∗
v
< 1
p∗
u
, and using (4).4
We can also get a relatively simple suﬃcient condition for the inequality (5).
Notice that in the line immediately preceding (5), the condition is equivalent to
fv(w2)
fu(w2) >
fv(w1)
fu(w1).S i n c ew1 >w 2, this would hold for sure if ∂
∂w
fv(w)
fu(w) < 0. Carrying
out the derivative and simplifying, this reduces to:
Proposition 3. As u ﬃcient condition for utility function v to be more downside
risk averse than utility function u is, for all w
−
f0
v(w)
fv(w)
> −
f0
u(w)
fu(w)
This result relates to Proposition 2 in Jindapon (2010), in which −
f0(w)
f(w) is re-
ferred to as the measure of “pain elasticity”.3 In his Proposition 2 (but using the
notation in the present paper), Jindapon ﬁnds that, for the case of large risks, utility
function v is more downside risk averse (in the sense of having a larger probability
premium) than is utility function u if v is both more risk averse and has more pain
elasticity than does u. Here, using the characterization of downside risk aversion
given by the size of the set of lotteries for which locating the secondary risk on the
downside is preferred, we only need the condition on pain elasticity, and not the
condition on risk aversion.
3. Changes in risk-free wealth
We can also easily enquire how p∗ changes with wealth, that is, we can con-
sider the question of whether downside risk aversion is increasing or decreasing in
wealth. In order to approach this question, we retain all the assumptions on utility
from above, namely marginal utility is positive, decreasing and convex (u0(w) > 0,
u00(w) < 0, u000(w) > 0). Recall that the assumption that u000(w) > 0 implies that
our decision maker also displays strictly positive prudence, −
u000(w)
u00(w) > 0,a n di s
downside risk averse in the sense that (1) holds.
Also, from now on we shall assume that there exists some non-random initial
wealth, that is wealth in state i is given by wi = w0 + yi,w h e r eyi is the outcome
of a primary lottery e y in state i,f o ri =1 ,2. We continue assuming that y1 >y 2.
Under these assumptions, an increase in w0 must decrease the utility premium
(since u000 > 0).
Derivin p∗ with respect to w0 and simplifying yields
∂p∗
∂w0
=
f0(w0 + y2)f(w0 + y1) − f(w0 + y2)f0(w0 + y1)
[f(w0 + y2)+f(w0 + y1)]
2
3Jindapon works with the function −f(w) rather than with f(w), which is why he interprets his
measures in terms of pain whereas here we are using the standard utility premium. Of course,
when the elasticity measure is considered, it makes no diﬀerence if the underlying concept is the
positive or the negative of the utility premium.5
This is negative, i.e. downside risk aversion as deﬁned by p∗ decreases with w0 if
f0(w0 + y2)f(w0 + y1) <f(w0 + y2)f0(w0 + y1)
that is, if
f0(w0 + y2)
f(w0 + y2)
<
f0(w0 + y1)
f(w0 + y1)
Since our assumption of positive prudence implies that f0(w0 + yi) < 0 for both
i =1 ,2,i ti sc o n v e n i e n tt om u l t i p l yb o t hs i d e sb y−1 to get
−
f0(w0 + y2)
f(w0 + y2)
> −
f0(w0 + y1)
f(w0 + y1)
(6)
We can see that this says that, for an increase in wealth to decrease the value of
p∗,i ti ss u ﬃcient that the (absolute value of the) ratio of the ﬁrst derivative of the
utility premium to the utility premium itself measured in state 2 of the primary
lottery be greater (in absolute value) than the same ratio measured in state 1.
Note the similarity with the result in Proposition 3. In fact the two results can
be seen to be telling a very similar story. Indeed, we can easily show that (6) is
nothing more than a special case of Proposition 3. Concretely, deﬁne the particular
transform u(w) ≡ v(w + z), and then use w = w0 + y2 and z = y1 − y2 > 0.T h e n
(6) says that utility function u is less downside risk averse than is utility function
v if
−
f0
v(w)
fv(w)
> −
f0
u(w)
fu(w)
which is exactly the same result as in Proposition 3.
Finally, since we have y1 >y 2,as u ﬃcient condition for (6) is that for all w0 and
y, it holds that
∂
∂y
µ
−
f0(w0 + y)
f(w0 + y)
¶
< 0
Carrying out the derivative gives the following condition:
−
f00(w0 + y)f(w0 + y) − f0(w0 + y)2
f(w0 + y)2 < 0
Simplifying this (and recalling that f0(w0 + y) < 0)w eg e tt h ec o n d i t i o n
−
f00(w0 + y)
f0(w0 + y)
> −
f0(w0 + y)
f(w0 + y)
(7)
Of course, (7) is a stronger condition than is (6), since it must hold for all y, but
it gives us some further intuition about what type of utility function is characterized
by decreasing downside risk aversion.
Once again, the condition (7) is a special case of Proposition 3, but where now
the transformation in question is just v(w) ≡− u0(w). T os e et h i s ,g ob a c kt o
the deﬁnition of f(w0 + y).W e h a v e f(w0 + y)=u(w0 + y) − Eu(w0 + y + e x).
Deriving with respect to w0,w eh a v ef0(w0 + y)=u0(w0 + y) − Eu0(w0 + y + e x),6
and f00(w0 + y)=u00(w0 + y) − Eu00(w0 + y + e x). In that way, we can easily
see that −f0(w0 + y)=−[u0(w0 + y) − Eu0(w0 + y + e x)] is the utility premium
corresponding to the utility function v(w)=−u0(w), and of course −f00(w0 +y)=
−[u00(w0 + y) − Eu00(w0 + y + e x)] is the slope of the utility premium of −u0(w).
Thus, we can see that (7) simply states that, in order for u(w) to display decreasing
downside risk aversion, it is suﬃcient that −u0(w) be more downside risk averse
than u(w).4 This result can be easily seen to mirror the well known result that for
u(w) to display decreasing absolute risk aversion, the function −u0(w) should be
more risk averse than the function u(w) itself.5
References
[1] Crainich, D. and Eeckhoudt, L., 2008, On the intensity of downside risk aversion. Journal of
Risk and Uncertainty 36, 267-276.
[2] Eeckhoudt, L. and Schlesinger, H., 2009, On the utility premium of Friedman and Savage.
Economics Letters 105, 46-48.
[3] Hanson, D. and Menezes, C., 1971, On a neglected aspect of the theory of risk aversion.
Western Economic Journal 9, 211-217.
[4] Jindapon, P., 2010, Prudence probability premium. Economics Letters 109, 34-37.
4Of course, multiplying both sides of (7) by w0 + y indicates that, in order for downside risk
aversion (as deﬁned herein) be decreasing in wealth, it is suﬃcient that the (absolute value of
the) elasticity of the utility premium (or, alternatively, the elasticity of pain) be greater for utility
function −u0(w) than for utility function u(w).
5There is a slight diﬀerence; the condition on negative marginal utility is both suﬃcient and
necessary for the case of DARA, but here it has only been shown to be suﬃcient.

A note on greater downside risk aversion

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