SØK460/ECON460 Finance Theory, Diderik Lund, 28 August 2002
Risk aversion
For those preference orderings which (i.e., for those individuals who)
satisfy the seven axioms, define risk aversion.
Compare a lottery Ỹ = L(a, b, π) (where a, b are fixed monetary
outcomes) with receiving E(Ỹ ) = πa + (1 − π)b for sure. Whether
the lottery or its expectation is preferred, depends on the curvature
of U :
• If U is linear, then U [E(Ỹ )] = E[U (Ỹ )], and one is indifferent
between lottery and its expectation. One is called risk neutral.
• If U is concave, then U [E(Ỹ )] ≥ E[U (Ỹ )], and one prefers the
expectation. One is called risk averse.
• If U is convex, then U [E(Ỹ )] ≤ E[U (Ỹ )], and one prefers the
lottery. One is called risk attracted.
The inequalities follow from Jensen’s inequality (see Sydsæter,
Strøm and Berck, equations 7.15–7.17, or D&D, p.47). If U is
strictly concave or convex, the inequalities are strict, except if Ỹ is
constant with probability one.
Quite possible that many have U functions which are neither everywhere linear, everywhere concave, nor everywhere convex. Then
one does not fall into one of the three categories.
1
SØK460/ECON460 Finance Theory, Diderik Lund, 28 August 2002
Assume risk aversion
(Does not follow from the seven axioms.)
• Most common behavior in economic transactions.
• Explains the existence of insurance markets.
• But what about money games? Expected net result always
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(y).
• Varies with the argument, e.g., high y may give lower −U 00(y).
• Is U () twice differentiable? Assume yes.
• But: The magnitude −U 00(y) is not preserved if c1U () + c0
replaces U ().
• Use instead:
◦ −U 00(y)/U 0(y) measures absolute risk aversion.
◦ −U 00(y)y/U 0(y) measures relative risk aversion.
• In general, these also vary with the argument, y.
2
SØK460/ECON460 Finance Theory, Diderik Lund, 28 August 2002
Arrow-Pratt measures of risk aversion
• For small risks: RA(y) ≡ −U 00(y)/U 0(y) meaures how much
compensation a person demands for taking the risk. Called the
Arrow-Pratt measure of absolute risk aversion.
• RR (y) ≡ −U 00(y) · y/U 0(y) is called the Arrow-Pratt measure
of relative risk aversion.
• Consider the following case (somewhat more general than D&D,
sect. 3.3.1):
– The wealth Y is non-stochastic.
– A lottery Z̃ has expectation E(Z̃) = 0.
• For a person with utility function U () and inital wealth Y ,
define the risk premium Π associated with the lottery Z̃ by
E[U (Y + Z̃)] = U (Y − Π).
• Will show the relation between Π and absolute risk aversion.
3
SØK460/ECON460 Finance Theory, Diderik Lund, 28 August 2002
Risk premium is proportional to risk aversion
(The result holds approximately, for small lotteries.)
E[U (Y + Z̃)] = U (Y − Π).
Take quadratic approximations (second-order Taylor series).
LHS:
1
U (Y + z) ≈ U (Y ) + zU 0(Y ) + z 2U 00(Y )
2
which implies
1
E[U (Y + Z̃]) ≈ U (Y ) + E(Z̃ 2)U 00(Y ).
2
RHS:
1
U (Y − Π) ≈ U (Y ) − ΠU 0(Y ) + Π2U 00(Y ).
2
Assume last term is very small. Let σz2 ≡ var(Z̃). Then:
1 2 00
σz U (Y ) ≈ −ΠU 0(Y )
2
which implies
U 00(Y ) 1 2
Π≈− 0
· σ .
U (Y ) 2 z
4
SØK460/ECON460 Finance Theory, Diderik Lund, 28 August 2002
The U function: Forms which are often used
• Some theoretical results can be derived without specifying form
of U .
• Other results hold for specific classes of U functions.
• Constant absolute risk aversion (CARA) holds for U (y) ≡
−e−ay , with RA(y) = a.
• Constant relative risk aversion (CRRA) holds for U (y) ≡
1 1−g
, with RR (y) = g.
1−g y
• (Exercise: Verify these two claims. (a, g are constants.) Determine what are the permissible ranges for y, a and g, given that
functions should be well defined, increasing, and concave.)
• Essentially, these are the only functions with CARA and CRRA,
respectively, apart from CRRA with RR (y) = 1.
• (Any constant can be added to the functions, and any positive
constant can be multiplied with them.)
• RR (y) ≡ 1 is obtained with U (y) ≡ ln(y).
• Another much used form: U (y) = −ay 2 + by + c, quadratic
utility. Easy for calculations, U 0 linear.
• (What are permissible ranges, given that U should be concave?
Hint: There is a minus sign in front of a.)
• Quadratic U has increasing RA(y) (Verify!), perhaps less reasonable.
• (What happens for this U function when y > b/2a? Is this
reasonable?)
5
SØK460/ECON460 Finance Theory, Diderik Lund, 28 August 2002
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 X̃A first-order stochastically dominates another random variable X̃B if every vN-M expected utility maximizer
prefers X̃A to X̃B .
Second-order stochastic dominance
A random variable X̃A second-order stochastically dominates another random variable X̃B if every risk-averse vN-M expected utility maximizer prefers X̃A to X̃B .
6
SØK460/ECON460 Finance Theory, Diderik Lund, 28 August 2002
First-order stochastic dominance, FSD
(Let the cumulative distribution functions be FA(x) ≡ Pr(X̃A ≤ x)
and FB (x) ≡ Pr(X̃B ≤ x).)
Possible to show that “X̃A X̃B by all” is equivalent to the following, which is one possible definition of first-order s.d.:
FA(w) ≤ FB (w) for all w,
and
FA(wi) < FB (wi) for some wi.
For any level of wealth w, the probability that X̃A ends up below
that level is less than the probability that X̃B ends up below it.
7
SØK460/ECON460 Finance Theory, Diderik Lund, 28 August 2002
Second-order stochastic dominance, SSD
Possible to show that “X̃A X̃B by all risk averters” is equivalent
to the following, which is one possible definition of second-order
s.d.:
Z w
i
−∞
FA(w)dw ≤
Z w
i
−∞
FB (w)dw for all wi,
and
FA(wi) 6= FB (wi) for some wi.
One distribution is more dispersed (“more uncertain”) than the
other. If we restrict attention to variables X̃A and X̃B with the same
expected value, Theorem 3.4 in D&D states that SSD is equivalent
to: X̃A can be written as X̃B + z̃, where the difference z̃ is some
random noise.
8
SØK460/ECON460 Finance Theory, Diderik Lund, 28 August 2002
Risk aversion and simple portfolio problem
(Chapter 4 in Danthine and Donaldson.) Sections 4.1 — 4.4 are
covered in Varian, Microeconomic Analysis, chapter 13. Will not
have time to discuss here. For simple portfolio problem (one risky,
one risk free asset), main result is:
• Absolute sum invested in risky asset is independent of wealth
for CARA, increasing for DARA, decreasing for IARA
• Fraction of wealth invested in risky asset is independent of
wealth for CRRA, increasing for DRRA, decreasing for IRRA
Risk aversion and saving
(Sect. 4.6, D&D.) How does saving depend on riskiness of return?
Rate of return is r̃, (gross) return is R̃ ≡ 1 + r̃. Consider choice of
saving, s, when probability distribution of R̃ is taken as given:
max E[U (Y0 − s) + δU (sR̃)]
s∈R+
where Y0 is a given wealth, δ is (time) discount factor for utility.
9
SØK460/ECON460 Finance Theory, Diderik Lund, 28 August 2002
Risk aversion and saving, contd.
• Savings decision well known topic in microec. without risk
• Typical questions: Depedence of s on Y0 and on E(R̃)
• Focus here: How does saving depend on riskiness of R̃?
• Consider mean-preserving spread: Keep E(R̃) fixed
• Assuming risk aversion, answer is not obvious:
◦ R̃ more risky means saving is less attractive, ⇒ save less
◦ R̃ more risky means probability of low R̃ higher, willing to
give up more of today’s consumption to avoid low consumption levels next period, ⇒ save more
• Need to look carefully at first-order condition
U 0(Y0 − s) = δE[U 0(sR̃)R̃]
• What happens to right-hand side as R̃ becomes more risky?
• Cannot conclude in general, but for some conditions on U
• (Jensen’s inequality:) Depends on concavity of g(R) = U 0(sR)R
• If, e.g., g is concave:
◦ May compare risk with no risk: E[g(R̃)] < g[E(R̃)]
◦ But also some risk with more risk, cf. Theorem 4.7 in D&D.
10
SØK460/ECON460 Finance Theory, Diderik Lund, 28 August 2002
Risk aversion and saving, contd.
max E[U (Y0 − s) + δU (sR̃)]
s∈R+
(assuming, all the time here, U 0 > 0 and risk aversion, U 00 < 0)
• When R̃B = R̃A + ε̃, E(R̃B ) = E(R̃A), will show:
0
◦ If RR
(Y ) ≤ 0 and RR (Y ) > 1, then sA < sB .
0
◦ If RR
(Y ) ≥ 0 and RR (Y ) < 1, then sA > sB .
• First condition on each line concerns IRRA vs. DRRA, but
both contain CRRA.
• Second condition on each line concerns magnitude of RR (also
called RRA): Higher risk aversion implies save more when risk
is high. Lower risk aversion (than RR = 1) implies save less
when risk is high. But none of these claims hold generally; need
0
the respective conditions on sign of RR
.
• Interpretation: When risk aversion is high, it is very important
to avoid the bad outcomes in the future, thus more is saved
when the risk is increased.
• Reminder: This does not mean that a highly risk averse person
puts more money into any asset the more risky the asset is. In
this model, the portfolio choice is assumed away. If there had
been a risk free asset as well, the more risk averse would save
in that asset instead.
11
SØK460/ECON460 Finance Theory, Diderik Lund, 28 August 2002
Risk aversion and saving, contd.
0
Proof for the first case, RR
(Y ) ≤ 0 and RR (Y ) > 1:
Use g 0(R) = U 00(sR)sR + U 0(sR) and g 00(R) = U 000(sR)s2R +
2U 00(sR)s. For g to be convex, need U 000(sR)sR + 2U 00(sR) > 0.
To prove that this holds, use
0
RR
(Y
[−U 000(Y )Y − U 00(Y )]U 0(Y ) − [−U 00(Y )Y ]U 00(Y )
)=
,
[U 0(Y )]2
0
which implies that RR
(Y ) has the same sign as
−U 000(Y )Y − U 00(Y ) − [−U 00(Y )Y ]U 00(Y )/U 0(Y )
= −U 000(Y )Y − U 00(Y )[1 + RR (Y )].
0
When RR
(Y ) < 0, and RR (Y ) > 1, this means that
0 < U 000(Y )Y + U 00(Y )[1 + RR (Y )] < U 000(Y )Y + U 00(Y ) · 2.
(This expression has a typo in D&D, p. 70: U 00 instead of U 000.)
Since this holds for all Y , in particular for Y = sR, we find
U 000(sR)sR + 2U 00(sR) > 0,
and g is thus convex.
12