SK460 Finance Theory, Diderik Lund, 17 January 2001
Risk aversion
(Repeating from \andre avdeling")
For those preference orderings which (i.e., for those individuals who)
satisfy the ve axioms, dene risk aversion.
Compare a gamble W~ = G(a; b: ) with receiving E (W~ ) = a +
(1,)b for sure. Whether the gamble or its expectation is preferred,
depends on the curvature of U :
If U is linear, then U [E (W~ )] = E [U (W~ )], and one is indierent
between gamble and its expectation. One is called risk neutral.
If U is concave, then U [E (W~ )] E [U (W~ )], and one prefers
the expectation. One is called risk averse.
If U is convex, then U [E (W~ )] E [U (W~ )], and one prefers the
gamble. One is called risk attracted.
The inequalities follow from Jensen's inequality (see Sydster,
Strm and Berck). If U is strictly concave or convex, the inequalities are strict, except if W~ is constant with probability one.
Quite possible that many have U functions which are neither everywhere convex, everywhere concave, nor everywhere convex. Then
one does not fall into one of the three categories.
1
SK460 Finance Theory, Diderik Lund, 17 January 2001
Assume risk aversion
(Does not follow from the ve axioms.)
Most common behavior in economic transactions.
Explains the existence of insurance markets.
But what about lotteries, gambling? Expected net result al-
ways negative, so a risk-averse should not participate. Cannot
be explained by theories taught in this course.
Some of our theories will collapse if someone is risk neutral or
risk attracted. Those will take all risk in equlibrium. Does not
happen.
How measure risk aversion?
Natural candidate: ,U 00(w).
Varies with the argument, e.g., high w may give lower ,U 00(w).
Is U () twice dierentiable? Assume yes.
But: The magnitude ,U 00(w) is not preserved if c1U () + c0
replaces U ().
Use instead ,U 00(w)=U 0(w).
This also varies with the argument.
2
SK460 Finance Theory, Diderik Lund, 17 January 2001
Pratt-Arrow measure of absolute risk aversion
For small risks: ,U 00=U 0 meaures how much compensation a
person demands for taking the risk. Called the Pratt-Arrow
measure of absolute risk aversion.
Consider the special case:
{ The wealth W is non-stochastic.
{ A gamble Z~ has expectation E (Z~ ) = 0.
For a person with utility function U () and inital wealth W ,
dene the risk premium associated with the gamble Z~ by
E [U (W + Z~ )] = U (W , ):
Will show the relation between and the risk aversion measure.
3
SK460 Finance Theory, Diderik Lund, 17 January 2001
Risk premium is proportional to risk aversion
(The result holds approximately, for small gambles.)
E [U (W + Z~ )] = U (W
, ):
Take quadratic approximations (second-order Taylor series).
LHS:
1
U (W + z ) U (W ) + zU 0(W ) + z 2U 00(W )
2
which implies
1
E [U (W + Z~ ]) U (W ) + E (Z~ 2 )U 00(W ):
2
RHS:
1
U (W , ) U (W ) , U 0(W ) + 2U 00(W ):
2
Assume last term is very small. Let z2 var(Z~ ). Then:
1 2 00
z U (W ) ,U 0(W )
2
which implies
U 00(W ) 1 2
, 0
:
U (W ) 2 z
4
SK460 Finance Theory, Diderik Lund, 17 January 2001
Stochastic dominance
Two criteria for making decisions without knowing shape of
U ().
May be important for delegation, for research, for prediction.
Work only in a limited number of comparisons. For other comparisons, these decision criteria are inconclusive.
Useful for narrowing down choices by excluding dominated alternatives.
First-order stochastic dominance
A random variable X~ rst-order stochastically dominates another
random variable Y~ if every vN-M expected utility maximizer prefers
X~ to Y~ .
Second-order stochastic dominance
A random variable X~ second-order stochastically dominates another random variable Y~ if every risk-averse vN-M expected utility
maximizer prefers X~ to Y~ .
5
SK460 Finance Theory, Diderik Lund, 17 January 2001
First-order stochastic dominance
(Let the cumulative distribution functions be FX (x) Pr(X~ x)
and GY (y ) Pr(Y~ y ).)
Possible to show that \X~ Y~ by all" is equivalent to the following,
which is one possible denition of rst-order s.d.:
FX (w) GY (w) for all w;
and
FX (wi) < GY (wi) for some wi:
For any level of wealth w, the probability that X~ ends up below
that level is less than the probability that Y~ ends up below it.
6
SK460 Finance Theory, Diderik Lund, 17 January 2001
Second-order stochastic dominance
Possible to show that \X~ Y~ by all risk averters" is equivalent to
the following, which is one possible denition of second-order s.d.:
w
F
(
w
)
dw
X
,1
,1 GY (w)dw for all wi;
Z
and
wi
Z
i
FX (wi) 6= GY (wi ) for some wi:
One distribution is more dispersed (\more uncertain") than the
other.
7
SK460 Finance Theory, Diderik Lund, 17 January 2001
The \mean-variance" approach
Much of elementary nance theory assumes:
Each individual only cares about the expected value (\the
mean") and the variance of wealth.
Very convenient for making more precise theories.
Little doubt that expected value is important.
Variance is one (but only one) measure of uncertainty.
Restrictive: Quite possible that actual people care about other
characteristics of random variables.
Three ways to underpin the assumption, two based on vN-M.
Mean-variance preferences always combined with risk aversion.
Mean-variance versus vN-M expected utility
In general those who maximize E [U (W~ )] care about the whole
distribution of W~ .
Will care about only mean and variance if those two characterize the whole distribution.
Will alternatively care about only mean and variance if U () is
a quadratic function.
The third way to underpin mean-var assumption
Perhaps things are so complicated that people resort to just
considering mean and variance. (Whether they are vN-M people or not.)
8
SK460 Finance Theory, Diderik Lund, 17 January 2001
Mean-var preferences due to distribution
Assume that choices are always between random variables with
one particular type (\class") of probability distribution.
Could be, e.g., choice only between binomially distributed variables. (There are dierent binomial distributions, summarized
in three parameters which uniquely dene each one of them.)
Or, e.g., only between variables with a chi-square distribution.
Or variables with normal distribution. Or variables with a lognormal distribution.
Some of these distributions, such as the normal distribution and
the lognormal distribution, are characterized completely by two
parameters, the mean and the variance.
If all possible choices belong to the same class, then the choice
can be made on the basis of the parameters for each of the
distributions.
Example: Would you prefer a normally distributed wealth with
mean 1000 and variance 40000 or another normally distributed
wealth with mean 500 and variance 10000?
9
SK460 Finance Theory, Diderik Lund, 17 January 2001
If mean and variance characterize each alternative completely,
then all one cares about is mean and variance.
Most convenient: Normal distribution, since sums of normally
distributed variables are also normal. Most opportunity sets
consist of a lot of alternative sums of variables.
Problem: Positive probability for negative outcomes. Share
prices are never negative.
10
SK460 Finance Theory, Diderik Lund, 17 January 2001
Mean-var preferences due to quadratic U
Assume
U (w) ,aw2 + bw + c
where a > 0; b > 0; c are constants. With this U function:
E [U (W~ )] = ,aE (W~ 2) + bE (W~ ) + c
= ,afE (W~ 2) , [E (W~ )]2 g , a[E (W~ )]2 + bE (W~ ) + c
= ,a var(W~ ) , a[E (W~ )]2 + bE (W~ ) + c;
which is a function only of mean and variance of W~ .
Problem: U function is decreasing for large values of W . Must
choose a and b such that those large values have zero probability.
Another problem: Increasing (absolute) risk aversion.
11
SK460 Finance Theory, Diderik Lund, 17 January 2001
Indierence curves in mean-stddev diagrams
If mean andp variance are sucient to determine choices, then
mean and variance are also sucient.
More practical to work with mean () and standard deviation
( ) diagrams.
Common to put standard deviation on horizontal axis.
Will show that indierence curves are increasing and convex in
(; ) diagrams.
Consider normal distribution and quadratic U separately.
Indierence curves are contour curves of E [U (W~ )].
Total dierentiation:
@E [U (W~ )]
@E [U (W~ )]
~
d +
d:
0 = dE [U (W )] =
@
@
12
SK460 Finance Theory, Diderik Lund, 17 January 2001
Indierence curves from quadratic U
Assume W < b=(2a) with certainty in order to have U 0(W ) > 0.
E [U (W~ )] = ,a 2 , a2 + b + c:
First-order derivatives:
@E [U (W~ )]
@E [U (W~ )]
= ,2a < 0;
= ,2a + b > 0;
@
@
Thus the slope of the indierence curves,
~ )]
@E [U (W
d
2a
= , @E [U@(W~ )] =
;
d
,
2
a
+
b
@
is positive, and approaches 0 as ! 0+.
Second-order:
@ 2E [U (W~ )]
@ 2 E [U (W~ )]
@ 2E [U (W~ )]
= ,2a < 0;
= ,2a < 0;
= 0:
@ 2
@2
@@
The function is concave, thus it is also quasi-concave.
13
SK460 Finance Theory, Diderik Lund, 17 January 2001
Indierence pcurves from normally distributed W~
Let f (") (1= 2)e,z2=2 , the std. normal density function. Dene
expected utility as a function
Z 1
~
E [U (W )] = V (; ) = ,1 U ( + ")f (")d":
Slope of indierence curves:
1 U 0( + ")"f (")d"
,
,1
, = 1 U 0( + ")f (")d" :
@V
@
@V
@
R
R
,1
Denominator always positive. Will show that integral in numerator
is negative, so minus sign makes the whole fraction positive.
Integration by parts: Observe f 0(") = ,"f ("). Thus:
Z
Z
0
0
U ( + ")"f (")d" = ,U ( + ")f (") + U 00( + ")f (")d":
First term on RHS vanishes in limit when " ! 1, so that
Z 1
1 0
00
,1 U ( + ")"f (")d" = ,1 U ( + ")f (")d" < 0:
Z
Another important observation:
1 "f (")d"
d ,U 0() R,1
lim
= 0 R1
= 0:
!0+ d
U () ,1 f (")d"
14
SK460 Finance Theory, Diderik Lund, 17 January 2001
To show concavity of V ():
V (1 ; 1) + (1 , )V (2 ; 2)
Z 1
= ,1
[U (1 + 1") + (1 , )U (2 + 2")]f (")d"
Z
1
< ,1
U (1 + 1 " + (1 , )2 + (1 , )2")f (")d"
= V (1 + (1 , )2 ; 1 + (1 , )2 ):
The function is concave, thus it is also quasi-concave.
15