Contents

(A) First and Second Order Stochastic Dominance

(B) The Characterization of Increasing Risk

(C) Application: Portfolio Allocation

(D) Alternative Measures of Increasing Risk

**(A) First and Second Order Stochastic Dominance**

When is a particular random prospect "better" than another random prospect? As von Neumann-Morgenstern argue, the answer is readily available for a single person: the expected utility of the first prospect is greater than the second. However, for most economic applications this does not really suffice as we cannot observe "utility". Might there be a way of comparing the random prospects by the observable properties of the different random prospects? Perhaps.

For uncertainty with univariate payoffs (i.e. "wealth" or "money" payoffs), a random variable can be described via a cumulative distribution function. This will contain effectively all of the relevant information about the "objective" properties of the random variable. Thus, comparison between different random prospects can be approached by examining their associated cumulative distribution functions.

Suppose over a compact support [a, b], we have two random variables, one
of which is associated with cumulative distribution F and another with cumulative
distribution G. When can we say that F is "better" than G for a particular
agent? Expected utility be constructed as an integral over this support via the cumulative
distribution function. Specifically, expected utility from the random variable with
cumulative distribution F can be written U(F) = ・/font> _{a}^{b}
u(x) dF(x) and expected utility of the random variable with cumulative distribution G is
U(G) = ・/font> _{a}^{b} u(x) dG(x). Note that U(ｷ)
is the von Neumann-Morgenstern utility function whereas u(ｷ)
is the associated elementary utility function. If U(F) ｳ U(G),
then F is preferred to G by the relevant agent.

If every reasonable agent prefers prospect F to prospect G, then we can
say that, on the whole, F *dominates* G. We ought to make this more precise. Letting
U^{0} be the set of all elementary utility functions u: X ｮ
R that are monotonically increasing, then let us define the following:

Dominance: Let ｳ_{1}be a transitive and incomplete binary relation on the set of cumulative distribution functions on X. We say FdominatesG if for every u ﾎ U^{0}, F ｳ_{1}G.

Thus, a random variable with distribution F "dominates" another
random variable with distribution G if *everybody* who has a utility function which
is monotonically increasing thinks F is better than G.

However, returning to the empirical problem, where utility is not
observable, when can we say that prospect F dominates G? In other words, by *only*
observing the properties of the cumulative distribution functions, can we say whether F
dominates G or not? In general, no; in certain cases, perhaps. One ought to expect, for
instance, that just about everyone will consider a prospect which has a lot of probability
mass skewed towards higher returns to be better than a prospect which has a lot of
probability mass skewed towards low returns. If such is the case, then we have what is
called "first order stochastic dominance". This is defined as follows:

First Order Stochastic Dominance: F is said to dominate G according tofirst order stochastic dominanceif F ｳ_{1}G ﾞ F(x) ｣ G(x) for all x ﾎ [a, b].

We can see this more or less in Figure 1 below when comparing the
cumulative distribution function H with either F or G. Specifically, note that for any x ﾎ [a, b], F(x) ｣ H(x) and G(x) ｣ H(x) as H lies uniformly above F or G. Considering any x ﾎ [a, b], then we see immediately that there is more
"area" under the curve between a and x then there is under the F or G curve
between a and x. Thus, there is a greater probability mass under H(x) than F(x) or G(x)
for any x ﾎ [a, b], i.e. the probability that any t is less
than x under H is greater than the probability that any t is less than x under F or G.
Thus both F and G *dominate* H.

Is this consistent with the von
Neumann-Morgenstern theory? In other words, can we say that if F is preferred to G
according to the first order stochastic dominance criteria, then necessarily that implies
that expected utility of F is greater than expected utility of G for *any* agent with
a monotonically increasing utility function? Indeed, this is so:

Proposition: F ｳ_{1}G if and only if ・/font>_{a}^{b}u(x)dF(x) ｳ ・/font>_{a}^{b}u(x)dG(x) for all u ﾎ U^{0}.

Proof: This is an if and only if statement, thus we must prove it both
ways. For the "if" part, suppose F ｳ _{1} G
holds, i.e. F(x) ｣ G(x) for all x ﾎ
[a, b] or [F(x) - G(x)] ｣ 0 for all x. Now, for any u ﾎ U^{0}, consider the integral ・/font>
_{a}^{b} u(x)[dF(x) - dG(x)]. Then integrating by parts:

・/font>

_{a}^{b}u(x)[dF(x) - dG(x)] = u(x)[F(x)-G(x)]_{a}^{b}- ・/font>_{a}^{b}u｢ (x)[F(x)-G(x)]dx ｳ 0

as [F(x) - G(x)]_{a}^{b} = 0 and u｢
(x) > 0 thus, then the inequality follows from [F(x) - G(x)] ｣
0 for all x. As this holds true for any u ﾎ U^{0},
then ・/font> _{a}^{b} u(x)dF(x) ｳ
・/font> _{a}^{b} u(x)dG(x) for all u ﾎ U^{0}. Conversely, for the "only if" part,
suppose it is *not *the case that F(x) ｣ G(x) for all x ﾎ [a, b]. Then, there is an x_{0} such that F(x_{0})
> G(x_{0}). By right continuity, there is a neighborhood Ne
(x_{0}) such that G(x) < F(x) for all x ﾎ Ne (x). Let u ﾎ U^{0} be such that
it takes on constant values outside Ne (x_{0}) but
increasing within Ne (x_{0}), i.e. u｢
(x) > 0 for x ﾎ Ne (x_{0})
and u｢ (x) = 0 for x ﾏ Ne (x_{0}). Thus, ・/font> _{a}^{b}
u｢ (x)[F(x) - G(x)dx = ・/font> _{N(x)}
u｢ (x)[F(x) - G(x)]dx > 0 or -・/font>
_{N(x)} u｢ (x)[F(x) - G(x)]dx < 0. But, we know
from integrating by parts that:

・/font>

_{a}^{b}u(x)[dF(x) - dG(x)] = -・/font>_{N(x)}u｢ (x)[F(x) - G(x)]dx

But as the right hand side is negative, then ・/font>
_{a}^{b} u(x)dF(x) < ・/font> _{a}^{b}
u(x)dG(x), thus it is *not* true that F ｳ _{1} G.
Thus, Fｳ _{1}G ﾞ F(x) ｣ G(x) for all x ﾎ [a, b].ｧ

Figure 1- First and Second Order Stochastic Dominance

The first order stochastic dominance criteria seems apt in comparing H
with either F or G in Figure 1. However, note that this criteria fails when comparing the
cumulative distribution functions F and G with each other. Obviously, neither F nor G are
uniformly above each other thus it neither F dominates G nor G dominates F. In particular,
note that *below* point z, F(x) ｣ G(x), whereas *above*
point z, F(x) ｳ G(x), or [1-G(x)] ｣
[1-F(x)]. As we cannot compare F and G according to the first order stochastic dominance
criteria, then ｳ _{1} is only a partial ordering on
the space of cumulative distribution functions.

The purpose of the first order stochastic dominance is to enable us to
order (if only partially) distributions according to their return: in other words, F >_{1}
H implies that F is unambiguously "better" than H because it deposits the bulk
of its probability mass at a higher return, and thus will have a higher expected return
(and utility). But, as we see in Figure 1, we cannot compare F and G by this criteria. If
z is the mean, then F and G have the same expected return.

However, in Figure 1, it also seems as if the probability mass of F is
"less dispersed" than G, and thus, in a sense, ought to be considered "less
risky". An alternative criteria, then, would be argue that a particular distribution
is better than another (at least for risk-averse people) if it has unambiguously
"lower" risk. This is the purpose of the *second order stochastic dominance*
criteria.

Second order stochastic dominance criteria helps us rank distributions
according to relative riskiness in terms of the spread of the probability mass of the
cumulative density functions. In order to move in that direction, let us define T(x) as
the area *between* the two curves F and G in [a, x], specifically T(x) = ・/font> _{a}^{x}[G(t) - F(t)]dt. In Figure 1, we can
notice that everywhere below z, G is above F, thus the area in between the curves below z
is T_{1} = ・/font> _{a}^{z}[G(t) - F(t)]dt
> 0. However, above z, G is below F, thus (negative of) the area between the curves
above z is ・/font> _{z}^{b}[G(t) - F(t)]dt < 0.
But notice that T(x) is *cumulative*, i.e. T(x) = ・/font> _{a}^{z}[G(t)
- F(t)]dt + ・/font> _{z}^{x}[G(t) - F(t)]dt for all
x ﾎ [a, b], thus we add up the areas where G is above F and
subtract from it the areas where G is below F (note: we can allow G and F to rise above
and dip below each other several times over the range). Thus, in Figure 1, T(y) = ・/font> _{a}^{y}[G(t) - F(t)]dt is the sum of the areas T_{1}
and T_{2} where, notice, T_{1} > 0 and T_{2} < 0.

If T(x) ｳ 0 for all x ﾎ [a, b], then the cumulative area where G is above F is greater than the area where F is above G, i.e. the probability mass of G is more spread out than the probability mass of F - which we may say implies that G is "riskier" than F - which is what we see in Figure 1. Conversely, if T(x) ｣ 0 for all x ﾎ [a, b], then the mass of F is more spread out than G.

[However, one must be careful here. Spread is *not* the same as
"variance". There are many situations where for two distributions, F and G, the
distribution F has the *same* mean and *less* variance than G and yet *also *yields
lower expected utility than G. To visualize such a situation, imagine a random prospect
with three returns (x_{1}, x_{2}, x_{3}), where x_{1} <
x_{2} < x_{3}, where x_{2} happens to be the mean. We can
decrease x_{1} a little and increase x_{3} a little so that we retain the
same mean, and we can even reduce the probability of either x_{1} or x_{3}
occurring (and thereby increase the probability of x_{2}) so that variance is
reduced. Yet with a concave utility function, it is still entirely likely that expected
utility is *lowered* as a consequence. Alternatively, imagine two skewed
distributions with the same mean, albeit with the one with higher variance being skewed
upwards and thus potentially preferable. Reduction in variance, keeping mean constant,
does *not* necessarily mean a rise in expected utility (there are cases when a
reduction in variance *does* increase expected utility unambiguously, but we must
then restrict the kinds of utility functions we can accept, e.g. quadratic utility, or the
distributions of returns must be completely describable by mean and variance).]

We want to "rank" distributions F and G according to whether T(x) is positive or negative over the entire range. Thus, we now define the following:

Second Order Stochastic Dominance: we say F dominates G according to second order stochastic dominance, or F ｳ_{2}G, if T(x) = ・/font>_{a}^{x}[G(t)-F(t)]dt ｳ 0 for all x ﾎ [a, b].

Notice in Figure 1 that it is quite probable that F ｳ
_{2} G by this criterion, thus F dominates G according to second order stochastic
dominance. Notice also that if T(x) compares H with F or G, then it is indeed true that F ｳ _{2} H and G ｳ _{2} H.
Thus, first order stochastic dominance implies second order stochastic dominance, but not
vice-versa.

Proposition: Letting U^{1}be the set of all increasing,concaveutility functions on [a, b], then F ｳ_{2}G if and only if ・/font>_{a}^{b}u(x)dF(x) ｳ ・/font>_{a}^{b}u(x)dG(x) for all u ﾎ U^{1}.

Proof: Suppose F ｳ _{2 }G and consider
・/font> _{a}^{b} u(x)[G(x) - F(x)]dx. Integrating by
parts:

・/font>

_{a}^{b}u(x)[G(x) - F(x)]dx = u(x)[G(x) - F(x)]_{a}^{b}-・/font>_{a}^{b}u｢ (x)[G(x) - F(x)]dx.

as [G(x) - F(x)]_{a}^{b} = 0, then:

・/font>

_{a}^{b}u(x)[G(x) - F(x)]dx = -・/font>_{a}^{b}u｢ (x)[G(x) - F(x)]dx.

integrating the right-hand side by parts again:

・/font>

_{a}^{b}u(x)[G(x) - F(x)]dx = -u｢ (x)・/font>_{a}^{x}[G(t) - F(t)]dt|_{a}^{b}+ ・/font>_{a}^{b}u｢ ｢ (x)(・/font>_{a}^{x}[G(t) - F(t)]dt)dx

or, noticing that T(x) = ・/font> _{a}^{x}
[G(t) - F(t)]dt, then:

・/font>

_{a}^{b}u(x)[G(x) - F(x)]dx = -u｢ (x)[T(x)]_{a}^{b}+ ・/font>_{a}^{b}u｢ ｢ (x)T(x)dx

Now, as [T(x)]_{a}^{b} = 0, then this reduces to:

・/font>

_{a}^{b}u(x)[G(x) - F(x)]dx = ・/font>_{a}^{b}u｢ ｢ (x)T(x)dx ｣ 0

where the inequality follows from the fact that u｢
｢ (x) ｣ 0 by concavity of u ﾎ U^{1} and T(x) ｳ 0 by the
assumption of second order stochastic dominance. Thus:

・/font>

_{a}^{b}u(x)dF(x) ｳ ・/font>_{a}^{b}u(x)dG(x).

Conversely, for the "only if" part, suppose it is *not* the
case that F ｳ _{2} G. Then there is an x_{0 ﾎ }[a, b] such that T(x_{0}) < 0. By continuity,
define Ne (x_{0}) as the neighborhood around x_{0}
where T(x) < 0 for all x ﾎ Ne
(x). Now, consider a u which is linear on [a, x_{0}-e ]
and on [x_{0}+e ] and concave on Ne
(x_{0}). Recall from the earlier proof that:

・/font>

_{a}^{b}u(x)[G(x) - F(x)]dx = -u｢ (x)[T(x)]_{a}^{b}+ ・/font>_{a}^{b}u｢ ｢ (x)T(x)dx

thus as u｢ ｢ (x) =
0 for all x ﾏ Ne (x_{0})
and [T(x)]_{a}^{b} = 0, then:

・/font>

_{a}^{b}u(x)[G(x) - F(x)]dx = ・/font>_{N(x)}u｢ ｢ (x)T(x)dx

As u｢ ｢ (x) ｣ 0 for x ﾎ Ne
(x) and T(x) < 0 by hypothesis, then ・/font> _{a}^{b}
u(x)[G(x) - F(x)]dx > 0 as a result, or *not* ・/font> _{a}^{b}
u(x)F(x)dx > ・/font> _{a}^{b} u(x)G(x)dx. Thus, ・/font> _{a}^{b} u(x)dF(x) > ・/font>
_{a}^{b}u(x)dG(x) implies F ｳ _{2} G.ｧ

**(B) The Characterization of Increasing Risk**

Let us now turn to the definition of "riskiness". A series of
similar definitions was pursued independently by J. Hadar and W. Russell (1969), G. Hanoch
and H. Levy (1969) and, perhaps most famously, Michael Rothschild and Joseph E. Stiglitz (1970, 1971). The first definition we
would like apply is almost behavioral in its implications. Namely, we can say that a
distribution G is "riskier" than a distribution F if risk-averters prefer F to
G. A risk-averter, recall, has a concave utility function, u ﾎ
U^{1}. Thus, we would like to use the following definition (m
_{F} is the mean of distribution F):

Riskiness(R.1): F isless riskythan G if m_{F}= m_{G}but for every u ﾎ U^{1}, ・/font> u(x)dF(x) ｳ ・/font> u(x)dG(x).

In other words, a distribution G is riskier than F if, controlling for the
same expected value (m _{F} = m
_{G}), the expected utility of F is greater than the expected utility of G of
risk-averse agents (u ﾎ U^{1} implies u concave).

Another notion of higher risk is the Rothschild and Stiglitz (1970) idea of a *mean-preserving
increase in spread* (MPIS). This means that given a distribution F, we construct
another distribution G by "increasing" the spread, i.e. moving mass away from
the center of the distribution to its tails in a manner such that the mean remains the
same. This is depicted in Figure 2. Moving from F to G, we have "fattened" the
tails but maintained the same mean m _{F} and m _{G}, thus a move from F to G is a MPIS. Of course, in
Figure 2, H is *not* a mean-preserving increase in spread on either F nor G as the
mean of H (m _{H}) is below that of G and F. Notice
that for distribution J, a move from F to J *is* a MPIS as spread increases and m _{J} = m _{F}, but a
move from G to J is *not* MPIS because even though the mean is preserved (m _{J} = m _{G}), the
spread has not increased for all x. Thus, a MPIS move from any distribution to another
implies there is a *single-crossing* property between the distributions, i.e. they
cross only once - and that is at the mean. In Figure 2, J crosses F only once, but it
crosses G at several points.

Figure 2- Mean-Preserving Increase in Spread

The MPIS of Rothschild-Stiglitz (1970) implies another definition of riskiness, namely F is less risky that G if G is generated by an mean-preserving increase in spread on F, i.e.

Riskiness(R.2): F isless riskythan G if the following two conditions hold:

(i) T(b) = ・/font>

_{a}^{b}[G(x)-F(x)]dx = 0 (mean-preservation)(ii) T(y) = ・/font>

_{a}^{y}[G(x)-F(x)]dx ｳ 0 (increase in spread)

We actually ought to verify that T(b) = 0 implies equality of mean. Specifically:

Proposition:T(b) = ・/font>_{a}^{b}[G(x) - F(x)]dx = 0 ﾛ m_{F}= m_{G}

Proof: Note that if T(b) = 0, then ・/font> _{a}^{b}[G(x)
- F(x)dx = 0. Or, integrating by parts:

・/font>

_{a}^{b}[G(x) - F(x)]dx = x[G(x)-F(x)]|_{a}^{b}- ・/font>_{a}^{b}x[G｢ (x) - F｢ (x)]dx= - ・/font>

_{a}^{b}x[G｢ (x) - F｢ (x)]dx= - ・/font>

_{a}^{b}x[g(x) - ｦ (x)]dx= -(m

_{G}- m_{F}) = 0

as m _{F} = ・/font>
_{a}^{b}xｦ (x)dx and m
_{G} = ・/font> _{a}^{b}xg(x)dx and ｦ = F｢ and g = G｢
are probability distributions. The converse applies in reverse.ｧ

Thus, the first part of the (R.2) definition of riskiness, T(b) = 0, implies the preservation of mean while the second part, T(y) ｳ 0, refers to the increase in spread. Notice that we do not make use of "utility" in this definition thus it may seem "less" behavioral than the previous (R.1) definition. However, they lead to the same result, namely:

Theorem: F is less risky than G by the first definition (R.1) if and only if F is less risky than G by the second definition (R.2).

Proof: The proof is the same as the proof of second-order stochastic dominance in the previous section, only now we add T(b) = 0 in addendum.ｧ

Notice the implication that the MPIS criteria effectively implies second
order stochastic dominance and, if we confine our attention to distributions with the same
mean, second order stochastic dominance implies MPIS. This is evident from Figure 2: by
the MPIS criteria, we cannot compare distributions G and J, but we can by the second-order
stochastic dominance criteria. Specifically, note that T(b) = ・/font>
_{a}^{b}[J(x) - G(x)]dx is the *difference* between the
lightly-shaded areas (where J lies above G) and the darkly-shaded areas (where G lies
above J). Thus, if the sum of the lightly-shaded areas are greater than the sum of the
darkly shaded areas, then T(b) > 0 and J is said to dominate G by second-order
stochastic dominance. Thus, by *imposing* T(b) = 0, we are omitting comparisons along
these lines - and thus we obtain the single-crossing property and our definition of MPIS.

Finally, we provide the third definition of riskiness, also due to Rothschild-Stiglitz

Riskiness(R.3): The random variable y is riskier than random variable x if there is a random variable e such that y = x + e and E[e |x] = 0 for all x.

Thus, we construct random variable y from x by adding an extra degree of riskiness or "noise", e , to the outcomes of x. In this case, y is unambiguously riskier than e . One can visualize that if F is the distribution associated with random variable x and G is the distribution of random variable y, then G is a mean-preserving increase in spread on F. Thus, as Rothschild and Stiglitz (1970) prove (although the proof stretches back to Hardy, Littlewood and Polya in the 1930s), the three definitions of riskiness are equivalent. In short, increasing risk is what risk-averters dislike (R.1), the movement of probability mass from center of a distribution to its tails (definition R.2) and the adding of noise to a random variable (definition R.3).

**(C) Application**

With the definition of riskiness in hand, let us consider the old portfolio allocation problem where agents choose their optimal portfolio allocation a * between a riskless asset with no return and a risky asset with return x and E(x) > 0. Recall that we showed that in such a case, more risk-averse agents will choose a smaller a *. Consequently, we would like to hypothesize that if the risky asset became riskier by a mean-preserving increase in spread, then a risk-averse agent would move away from it, i.e. the optimal portfolio allocation a * would decline. Unfortunately, as we shall see, that is not necessarily true. Increases in risk may decrease expected utility, but that does not mean that a * declines.

To see this, consider the following. Normalizing initial wealth w = 1, then the agent's consumer's optimal portfolio decision a * is made by maximizing expected utility E[u(1+a x)] where x is the return on the risky asset. The first order conditions for a maximum imply:

ｶ E[ｷ]/ｶ a = E[u｢ (1-a x)ｷx] = 0

or assuming the risky asset can take a continuum of values in a closed interval [a, b] ﾌ R and is characterized by the cumulative distribution F, then this becomes

・/font>

_{a}^{b}[u｢ (1-a x)ｷx]dF(x) = 0

Now, let us define:

H(ｷ, r) = (1-r)F(ｷ) + rG(ｷ)

where r ﾎ [0, 1]. Thus, H(ｷ, r) generates a family of distributions where, if r = 0, then H(ｷ, r) = F(ｷ) and if r = 1, then H(ｷ, r) = G(ｷ). Let G be "more risky" than F, in other words, G is a mean-preserving increase in spread over F. Then the family generated by H(ｷ, r) as r goes from 0 to 1 is a linear path or sequence of riskier (MPIS) distributions from F to G. Consider now the following:

Theorem: If marginal utility u_{a}(ｷ) is concave in x (i.e. u_{axx}< 0), then an increase in riskiness (i.e. a rise in r), will lead to a reduction in the proportion of a portfolio allocated to the risky asset, i.e. da */dr < 0.

Proof: For any distribution H(x, r) we have the first order condition:

・/font>

_{a}^{b}[u｢ (1-a x)ｷx]dH(x, r) = 0

where, again, if r = 0, then H(x, r) = F(x) and if r = 1, then H(x, r) =
G(x). As different a are generated for different distributions,
we see, then, that a will be some function of r. To make our
notation compact, let u(1-a x) = u(a
, x) and thus u_{a}(a , x) =
u｢ (1-a x)ｷx and u_{aa} (a , x) = u｢
｢ (1-a x)ｷx^{2}. Now,
totally differentiating the first order condition at a *:

[・/font>

_{a}^{b}u_{aa}(a *, x)dH(x, r)]da + [・/font>_{a}^{b}u_{a}(a *, x)dH_{r}(x, r)]dr = 0

or:

da */dr = - [・/font>

_{a}^{b}u_{a}(a *, x)dH_{r}(x, r)]/[・/font>_{a}^{b}u_{aa}(a *, x)dH(x, r)]

Now, H_{r}(x, r) = dH(x, r)/dr = -dF(x) + dG(x) or simply:

H

_{r}(x, r) = d[G(x) - F(x)]

As a result:

da */dr = - [・/font>

_{a}^{b}u_{a}(a *, x)d[G(x) - F(x)]]/[・/font>_{a}^{b}u_{aa}(a *, x)dH(x, r)]

Now the denominator is clear enough: as u_{aa}
< 0 by the concavity of u, then the denominator is negative. The numerator is
more complicated. We know also that u_{a} > 0,
but we do not know the implications of d[G(x) - F(x)] on the sign. So, integrating the
(negative of the) numerator by parts, we obtain:

・/font>

_{a}^{b}u_{a}(a *, x)d[G(x) - F(x)] = u_{a}(a *, x)[G(x) - F(x)]|_{a}^{b}- ・/font>_{a}^{b}u_{ax}(a *, x)[G(x) - F(x)] dx

where as the first term [G(x)-F(x)]|_{a}^{b} = 0, then:

・/font>

_{a}^{b}u_{a}(a *, x)d[G(x) - F(x)] = - ・/font>_{a}^{b}u_{ax}(a *, x)[G(x) - F(x)]dx

The term ua _{x}(a
*, x) is the derivative of *marginal* utility with respect to x. Now, integrating the
right-hand side by parts:

・/font>

_{a}^{b}u_{a}(a *, x)d[G(x) - F(x)] = -u_{ax}(a *, x)(・/font>_{a}^{x}[G(t) - F(t)]dt)|_{a}^{b}+ ・/font>_{a}^{b}[u_{axx}(a *, x) ・/font>_{a}^{x}[G(t) - F(t)]dt]dx.

or, as (・/font> _{a}^{x}[G(t) -
F(t)]dt)|_{a}^{b} = T(b) = 0 by the assumption of MPIS and ・/font> _{a}^{x}[G(t) - F(t)]dt] = T(x), then this
becomes:

・/font>

_{a}^{b}u_{a}(a *, x)d[G(x) - F(x)] = ・/font>_{a}^{b}[u_{axx}(a *, x)T(x)]dx.

As G(x) is an MPIS on F(x), then we know that T(x) > 0. But what about
u_{axx}? If we assume that *marginal* utility is
concave in x, then u_{axx} < 0. If this is true,
then ・/font> _{a}^{b} u_{a}(a *, x)d[G(x) - F(x)] < 0 and so the numerator is positive, so, in
conclusion, term da */dr > 0, i.e. an *increase* in
risk defined via MPIS will lead to a *reduction* in investment in the risky asset.ｧ

The question that imposes itself here, of course, is what
economically-intuitive conditions on behavior guarantee that marginal utility is concave
in x? None, really. Neither risk aversion, nor increasing/decreasing absolute risk
aversion, nor much of anything else yields us u_{axx}
< 0. Taking our explicit form, we can see that:

u = u(w + a wx) ｳ 0

u

_{a}= u｢ (w+a wx)ｷwx > 0u

_{ax }= u｢ ｢ (w+a wx)a w^{2}x + u｢ (w+a wx)wu

_{axx}= u｢ ｢ ｢ (w+a wx)a^{2}w^{3}x + 2u｢ ｢ (w+a wx)a x^{2}

we are certain that the last part of the last term 2u｢
｢ (w+a wx)a
x^{2 }is negative, but we have no assumptions on the *third* derivative of
the utility function (u｢ ｢ ｢ ) we need to derive the concavity of marginal utility.

To see that assuming decreasing absolute risk aversion (DARA) does not help us, recall that DARA says d[-u｢ ｢ (w)/u｢ (w)]/dw < 0 or:

d[-u｢ ｢ (w)/u｢ (w)]/dw = [-u｢ (w)u｢ ｢ ｢ (w) + u｢ ｢ (w)

^{2}]/u｢ (w)^{2}< 0

where we see the third derivative u｢ ｢ ｢ (w). Now, the denominator is
positive, thus the numerator must be negative, which implies that u｢
｢ ｢ (w) > 0. This is great, but
that does not help us to find the sign of u_{axx}. Why?
Because u_{axx} = u｢ ｢ ｢ (w+a wx)a ^{2}w^{3}x + 2u｢ ｢ (w+a wx)a x^{2},
but the rightmost term is negative; thus having found that u｢ ｢ ｢ (w) > 0 via DARA only tells us we
are adding a positive to a negative. That is not enough to tell whether u_{axx} is positive or negative. Might *increasing* absolute
risk-aversion help? Not quite. All that IARA implies is that u｢
｢ (w)^{2} > u｢ (w)u｢ ｢ ｢ (w).
As u｢ ｢ (w)^{2} > 0 and
u｢ (w) > 0, then u｢ ｢ ｢ (w) can now be *either*
positive or negative to fulfill IARA. Thus we now can't get u｢
｢ ｢ and thus have even less hope
of establishing the sign of u_{axx}. In sum, economic
reasoning is of little assistance. We simply have to make an exogenous assumption that u_{a} is concave in x.

**(D) Alternative Measures of Increasing Risk**

An alternative characterization of increasing risk was provided by Peter Diamond and Joseph E. Stiglitz (1974) in the form of *mean
utility-preserving increase in spread* (MUPIS). This can be visualized by appealing to
the two-state diagram in Figure 3. Suppose we begin at certain allocation W. Notice that
we have two indifference curves passing through W: u_{0} and v where, notice, v is
more convex than u, thus v is more risk-averse. Now, if we attempted a mean-preserving
increase in risk (MPIS) that would involve a straight-line movement from W to a point such
as W+e and this movement would have a slope of -p/(1-p).
However, instead of doing this, let us instead move to a point such as A. The movement
from W to A is an increase in risk as we are moving away from the certainty line, but it
is *not* a "mean-preserving" increase because the movement from W to A is
not at angle -p/(1-p). However, notice that at A, the utility of the more risk-averse
agent is lower (at v｢ ) but that of the less risk-averse agent
remains the same at A as it was at W, i.e. it stays at u_{0}.

Figure 3- Mean Utility-Preserving Increase in Spread

However, from the point of view of agent u_{0}, the movement from
W to A is a *mean utility preserving increase in spread*, or a MUPIS. Why? It is
obvious that expected *utility* is the same, but expected *return* is higher, as
we can see by the intersection of the -p/(1-p) curve at point A. Thus, it is as if we
using a higher expected return to compensate agent u_{0} for the loss in utility
that arises from the increase in risk - so his expected utility remains the same. Of
course, from the perspective of agent v, the movement from W to A is a mean-utility *decreasing*
increase in spread.

The basic idea behind Diamond and Stiglitz (1974) is to exploit this notion. Now if w is a random variable, we know that as a consequence, u(w) is a random variable as well. If F(x) is the distribution of random variable w, then, correspondingly, we can posit F (u) as the related distribution of random variable u(w). Consequently, we can define a MUPIS by using the definition of MPIS but only replacing distributions F(x) and G(x) with analogous F (u) and G (u). It is easy to note that F(x) = F (u(x)), where, note, F = F ｰ u, thus F is some transformation of u which makes it equal to the distribution of the random variable, F.

As x ﾎ [0, 1], then we can normalize u via some projection of von Neumann-Morgenstern utility so that u ﾎ (0, 1), thus x and u are defined over the same interval [a, b]. Thus we can now define a mean utility preserving increase in spread (MUPIS) as follows:

(i) ・/font>

_{a}^{b}[G (u) - F (u)]du = 0 (preservation of mean utility)(ii) ・/font>

_{a}^{y}[G (u) - F (u)]du ｳ 0 (increase in spread)

where we can immediately note the analogue with MPIS, thus G is *riskier* than F in the MUPIS
sense. However, F = F ｰ u, then, by
an appropriate change of variable technique, we can rewrite the MUPIS conditions as:

(i) ・/font>

_{a}^{b }u｢ (x)[G(x) - F(x)]dx = 0(ii) ・/font>

_{a}^{y}u｢ (x)[G(x) - F(x)]dx ｳ 0

where u｢ (x) is marginal utility.

Let us now turn to the mean utility *decreasing* increase in spread
for the more risk-averse agent v. Letting v = T(u(x)) where T is concave because v is more
risk-averse than u, then we know that agent v's change in expected utility can be written:

・/font>

_{a}^{b }v(x)[G(x) - F(x)]dx = ・/font>_{a}^{b }T(u(x))[G(x) - F(x)]dx

or, integrating the right-hand side by parts:

・/font>

_{a}^{b }v(x)[G(x) - F(x)]dx = T(u(x))[G(x)-F(x)]|_{a}^{b}- ・/font>_{a}^{b}T｢ (u(x))u｢ (x)[G(x)-F(x)]dx

Or, as [G(x)-F(x)]|_{a}^{b} = 0, then this reduces to:

・/font>

_{a}^{b }v(x)[G(x) - F(x)]dx = - ・/font>_{a}^{b}T｢ (u(x))u｢ (x)[G(x)-F(x)]dx

By now we should have learnt that whenever we are in a fix, we should integrate by parts again - and thus we do so:

・/font>

_{a}^{b }v(x)[G(x) - F(x)]dx = - T｢ (u(x))[・/font>_{a}^{y}u｢ (t)[G(t)-F(t)]dt]|_{a}^{b}+ ・/font>_{a}^{b}T｢ ｢ (u(x))u｢ (x)[・/font>_{a}^{x}u｢ (t)(G(t)-F(t))dt]dx

Now, we know by condition (i) of MUPIS that [・/font>
_{a}^{y} u｢ (t)[G(t)-F(t)]dt]|_{a}^{b}
= 0, thus we are left with:

・/font>

_{a}^{b }v(x)[G(x) - F(x)]dx = ・/font>_{a}^{b}T｢ ｢ (u(x))u｢ (x)[・/font>_{a}^{x}u｢ (t)(G(t)-F(t))dt]dx

Now, T｢ ｢ (ｷ)
< 0 by concavity (from the greater risk-aversion of v), u｢
(x) by assumption so, finally, by condition (ii) of MUPIS, we know that [・/font> _{a}^{x} u｢
(t)(G(t)-F(t))dt] ｳ 0. Thus the whole term is negative, i.e.

・/font>

_{a}^{b }v(x)dG(x) < ・/font>_{a}^{b }v(x)dF(x)

so a mean utility-preserving increase in spread for u leads to a mean
utility-decreasing increase in spread for v. Diamond
and Stiglitz (1974) use this relationship,
in fact, to provide *another* definition of risk-aversion, i.e. v is more risk-averse
than u if an increase in spread that maintains expected utility of u implies that the
expected utility of v is lower.

** **

P. Diamond and J.E. Stiglitz (1974) "Increases in Risk and in Risk
Aversion", *Journal of Economic Theory*, Vol. 8, p.337-60.

J. Hadar and W. Russell (1969) "Rules for Ordering Uncertain
Prospects", *American Economic Review*, Vol. 59, p.25-34.

G. Hanoch and H. Levy (1969) "The Efficiency Analysis of Choices
involving Risk", *Review of Economic Studies*, Vol. 36, p.335-46.

M. Rothschild and J.E. Stiglitz (1970) "Increasing Risk I: a
definition", *Journal of Economic Theory*, Vol. 2 (3), p.225-43.

M. Rothschild and J.E. Stiglitz (1971) "Increasing Risk II: its
economic consequences", *Journal of Economic Theory*, Vol. 3 (1), p.66-84.