A. Penati - G. Pennacchi
Choice Under Uncertainty
Let us consider how individuals might value assets that have random payoffs.
One possible measure of the attractiveness of a risky asset is the expected value of the
asset’s random payoff. Suppose an asset (or gamble) offers a single random payoff that has
a discrete distribution with outcomes (x1 , ..., xn ) and corresponding probabilities (p1 , ..., pn ),
where
n pi = 1. Then the expected value of the payoff is x̄ = n xipi.
i=1
i=1
How reasonable is it to think that individuals value risky assets based on the assets’ expected
values? In 1728, Nicholas Bernoulli clearly demonstrated a weakness of using expected value as
the sole measure of preferences with his “St. Petersberg Paradox”: Suppose you are offered the
following gamble (or asset). The gamble’s payoff is determined by a sequence of coin flips. If
on the first flip, the coin comes up “tails,” you receive nothing and the game is over. However,
if it comes up “heads,” you receive $1 and also obtain the right to flip again. If on the second
flip, “tails” occurs, the game is over, but if “heads” occurs you receive another $2 and the right
to flip once again. If “heads” comes up on the third flip, you receive $22 = $4 and so on. Thus,
given the coin came up “heads” on the first
n + 1th
n flips, your payoff if a “heads” is obtained on the
flip is $2n .
How much would an individual pay to participate in this game? Or, in other words, what
would be an individual’s valuation placed on this gamble (asset)? If the individual values the
gamble’s payoff by its expected value, it will be
x̄ =
1
1
1
2 $1 + 4 $2 + 8 $4 + ...
(1)
= 12 ($1 + 12 $2 + 14 $4 + ...
∞
= 12 ($1 + $1 + $1 + ... = $
Thus, the expected value of this gamble is infinite. Clearly, most individuals would pay
1
only a moderate, not infinite, amount for this gamble. In 1738, Daniel Bernoulli, a cousin of
Nicholas, provided an explanation for the St. Petersberg Paradox by stating that people cared
about the expected “utility” of a gamble’s payoff, not the expected value of its payoff. As
an individual’s wealth increases, the “utility” that one receives from the additional increase in
wealth grows less than proportionally. In the St. Petersberg Paradox, prizes go up at the same
rate that the probabilities decline. In order to obtain a finite valuation, the trick would be to
allow the “value” or “utility” of prizes to increase slower than the rate probabilities decline.
The first complete axiomatic development of expected utility is due to von Neumann and
Morgenstern (1944), which we now illustrate.
Define a lottery to be risky payoff (gamble
or asset) and consider an individual’s optimal choice of a lottery from a given set of different
lotteries. All lotteries have possible payoffs that are contained in the set
{x , ..., xn }. In general,
1
the elements of this set can be viewed as different, uncertain outcomes. For example, they
could be interpreted as particular consumption levels (bundles of consumption goods) that the
individual obtains in different states of nature or, more simply, different monetary payments
received in different states of the world. A given lottery can be characterized as an ordered set
n
pi = 1 and pi ≥ 0. A different lottery
of probabilities P = {p1, ..., pn }, where, of course,
i=1
is characterized by another set of probabilities, for example, P ∗ = {p∗1 , ..., p∗n }.
Let , ≺, and
∼ denote preference and indifference between gambles. We will show that if an individual’s
preferences satisfy the following conditions (axioms), then these preferences can be represented
by a real-valued utility function defined over a given lottery’s probabilities, that is, a function
V (p1 , ..., pn ).
Axioms:
1) Completeness
For any two lotteries P ∗ and P , either P ∗ P , or P ∗ ≺ P , or P ∗ ∼ P .
2) Transitivity
If
P ∗∗ P ∗and P ∗ P , then P ∗∗ P .
3) Continuity
P ∗∗ P ∗ P , there exists some λ ∈ [0, 1] such that P ∗ ∼ λP ∗∗ + (1 − λ)P , where
λP ∗∗ + (1 − λ)P denotes a “compound lottery or gamble,” namely with probability λ one
receives the lottery P ∗∗ and with probability (1 − λ) one receives the lottery P .
If
2
These three axioms are analogous to those in standard consumer theory and are needed to
establish the existence of a real-valued utility function. The fourth axiom is crucial to expected
utility theory.
4) Independence
For any two lotteries
P and P ∗ , P ∗ P
if and only if for all
λ ∈(0,1] and all P ∗∗ :
λP ∗ + (1 − λ)P ∗∗ λP + (1 − λ)P ∗∗
Further, for any two lotteries
P
and
P †, P ∼ P † if and only if for all λ ∈(0,1] and all P ∗∗ :
λP + (1 − λ)P ∗∗ ∼ λP † + (1 − λ)P ∗∗
To better understand the meaning of the independence axiom, note that
P∗
is preferred
λP ∗ + (1 − λ)P ∗∗ and λP + (1 − λ)P ∗∗ is
equivalent to a toss of a coin that has a probability (1 − λ) of landing “tails”, in which case
both compound lotteries are equivalent to P ∗∗ , and a probability λ of landing “heads,” in which
case the first compound lottery is equivalent to the single lottery P ∗ and the second compound
lottery is equivalent to the single lottery P . Thus, the choice between λP ∗ + (1 − λ)P ∗∗ and
λP + (1 − λ)P ∗∗ is equivalent to being asked, prior to the coin toss, if one would prefer P ∗ to
P in the event the coin lands “heads.”
to
P
by assumption.
Now the choice between
It would seem reasonable that should the coin land “heads,” we would go ahead with our
original preference in choosing
P∗
over
P.
The independence axiom assumes that preferences
over the two lotteries are independent of the way in which we obtain them.1 For this reason,
it is also known as the “no regret” axiom. However, experimental evidence finds some systematic violations of this independence axiom. See the Machina (1987) Journal of Economic
Perspectives article and, in particular, the Allais Paradox regarding this point.
1
In the context of standard consumer choice theory,
λ
would be interpreted as the amount (rather than
C ) and (1 − λ) as the amount of another
C ∗∗ ). In this case, it would not be reasonable to assume that the choice
probability) of a particular good or bundle of goods consumed (say
good or bundle of goods consumed (say
of these different bundles is independent. This is due to some goods being substitutes or complements with other
goods. Hence, the validity of the independence axiom is linked to outcomes being uncertain (risky), that is, the
interpretation of
λ as a probability rather than a deterministic amount.
3
The final axiom is similar to the independence and completeness axioms.
5) Dominance
Let P 1 be the compound lottery λ1 P ‡ + (1 − λ1)P †
λ2P ‡ + (1 − λ2)P † .
If
P ‡ P † , then P 1
P2
and
be the compound lottery
P 2 if and only if λ1 > λ2.
Given preferences characterized by the above axioms, we will now show that the choice
between any two (or more) arbitrary lotteries will be that which has the higher (highest)
expected utility.
The completeness axiom’s ordering on lotteries naturally induces an ordering on the set
of outcomes. Without loss of generality, suppose that the outcomes are ordered such that
xn
xn−1 ... x1.
This follows from the completeness axiom for the case of
n
degenerate
or “primitive” lotteries where the ith primitive lottery is defined to return outcome
xi
with
probability 1 and all of the other outcomes have probability zero. Note that this ordering may
not necessarily coincide with ranking the elements of
payoffs, as the state of nature for which
for which
xj
xi
x
strictly by the size of their monetary
is the outcome may differ from the state of nature
is the outcome, and these states of nature may have different effects on how an
individual values the same monetary outcome. For example,
xi
may be received in a state of
nature when the economy is depressed, and monetary payoffs may be highly valued in this state
of nature. In contrast,
xj
may be received in a state of nature characterized by high economic
expansion, and monetary payments may not be as highly valued.
From the continuity axiom, we know that for each
xi , there
exists a
Ui ∈ [0, 1] such that
xi ∼ Ui xn + (1 − Ui )x1
and for
Ui
i
= 1, this implies
U1
= 0 and for
i
=
n,
this implies
(2)
Un
= 1. The values of the
weight the most and least preferred outcomes such that the individual is just indifferent
between a combination of these polar payoffs and the payoff of
xi .
The
Ui
can adjust for
both differences in monetary payoffs and differences in the states of nature during which the
outcomes are received.
Now consider a given arbitrary lottery,
lottery over the
P
= {p1 , ..., pn }. This can be considered a compound
n primitive lotteries, where the ith primitive lottery which pays xi with certainty
is obtained with probability
pi .
By the independence axiom, and using equation (2), the
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individual is indifferent between the compound lottery,
P,
and the following lottery given on
the right-hand-side of the equation below:
p1 x1 + ... + pn xn ∼ p1 x1 + ... + pi−1 xi−1 + pi [Ui xn + (1 − Ui )x1 ] + pi+1xi+1 + ... + pn xn
where we have used the indifference relation in equation (2) to substitute for
hand side of (3). By repeating this substitution for all i,
will be indifferent between
P , given by
p1 x1 + ... + pn xn ∼
≡
on the right
i = 1, ..., n, we see that the individual
the left hand side of (3), and
n
i=1
− pU x
pi Ui xn + 1
n
pi Ui . Thus, we see that lottery
i=1
consisting of a Λ probability of obtaining xn and a (1
Now define Λ
xi
(3)
P
n
i=1
i i
1
(4)
is equivalent to a compound lottery
− Λ) probability of obtaining x1. In a
similar manner, we can show that any other arbitrary lottery P ∗ = {p∗1, ..., p∗n } is equivalent to
a compound lottery consisting of a Λ∗ probability of obtaining
n
of obtaining x1, where Λ∗
p∗i Ui .
≡
i =1
xn
and a (1 − Λ∗ ) probability
P if and only if Λ∗
Thus, we know from the dominance axiom that P ∗
n
n
p∗i Ui > pi Ui . So defining an expected utility function as
i=1
i=1
V (p1, ..., pn ) =
will imply that P ∗
n
i=1
pi Ui
> Λ , which implies
(5)
P if and only if V (p∗1, ..., p∗n ) > V (p1, ..., pn ).
The utility function given in equation (5) is known as a von Neumann - Morgenstern utility
function. Note that it is linear in the probabilities and is unique up to a linear transformation, that is, the utility function is “cardinal,” unlike the ordinal utility functions of standard
consumer theory. For example, if
same under the transformation
the “shape” of
Ui
=
U (xi ),
aU (xi ) + b,
an individual’s choice over gambles will be the
but not a non-linear transformation that changes
U (xi ).
The von Neumann-Morgenstern expected utility framework may only partially explain the
phenomenon illustrated by the St. Petersberg Paradox. Suppose utility is given by the square
5
Ui
root of a monetary payoff, that is,
concave function of
=
U (xi )
√xi.
=
This is a monotonically increasing,
x, which we here assume is simply a monetary amount.
Then the expected
utility (value) of the payoff of the St. Petersberg gamble is
V
=
n
pi Ui
i=1
∞
=
i=1
1
2i
√
∞ −
2 −1 =
i
i
=2
2
i
2
(6)
−
−
= 2 2 + 2 2 + ...
2
3
∞ √1 − 1 − √1 =
1√ ∼= 1.7071
2 2− 2
i
=
i
2
=0
which is finite. The gamble would be worth $1.71 to a square-root utility maximizer.
However, the reason that this is not a complete resolution of the paradox is that one can
always construct a “super St. Petersberg paradox” where even expected utility is infinite. Note
that in the regular St. Petersberg paradox, the probability of winning declines at rate 2n while
the winning payoff increases at rate 2n . In a super St. Petersberg paradox, we can make the
xn = U −1(2n ) and expected utility would no longer be finite.
square-root utility, let the winning payoff be xn = 22n−2 , that is,
winning payoff increase at a rate
If we take the example of
x1
= 1,
x2
= 4,
x3
= 16, etc. In this case, the value placed on the gamble by a square-root
expected utility maximizer is
V
n
=
i=1
pi Ui
∞
=
i=1
√
1
22i−2 =
2i
∞
(7)
Should we be concerned by the fact that if we let the prizes grow quickly enough, we can
get infinite valuations for any chosen form of expected utility function? Maybe not. One
could argue that St. Petersberg gambles are unrealistic, particularly ones where the payoffs are
assumed to grow rapidly. The reason is that any person offering this gamble has finite wealth
(even Bill Gates). This would set an upper bound on the amount of prizes that could feasibly
be paid, making expected utility, and even the expected value of the payoff, finite.
The von Neumann-Morgenstern expected utility approach can be generalized to the case of
a continuum of outcomes and lotteries having possibly continuous probability distributions. For
6
example, if outcomes are a possibly infinite number of purely monetary payoffs or consumption
levels denoted by the variable
x,
a subset of the real numbers, then a generalized version of
equation (5) is
V (F ) =
where
F (x)
U (x) dF (x)
(8)
is a given lottery’s cumulative distribution function over the payoffs,
the generalized lottery represented by the distribution function
lottery represented by the discrete probabilities
P
F
x.2
Hence,
is analogous to our previous
= {p1 , ..., pn }. For a given gamble or lottery,
can also be represented as
expected utility defined over the random payoff x
E [U (x)] =
2
U (x) dF (x)
(9)
When the random payoff, x, is absolutely continuous, then expected utility can be written in terms of the
probability density function, f (x), as V (f) = U (x) f (x) dx.
7
A. Penati - G. Pennacchi
Risk Aversion and Risk Premia
An individual is said to be risk averse if and only if the individual’s utility function is
concave. This aversion to risk implies that the individual would not accept a “fair” gamble. To
see why this is the case, consider the following “fair” lottery (gamble). The lottery consists of
a random payoff, ε, where


 h1 with probability p
ε = 
 h with probability 1 − p
2
(1)
The requirement that the lottery be a fair (pure risk) lottery means that its expected value
equals zero, that is,
E [ε] = ph1 + (1 − p)h2 = 0
(2)
which implies h1/h2 = − (1 − p) /p, or, solving for p, p = −h2/ (h1 − h2).
Now suppose a von Neumann-Morgenstern expected utility maximizer whose current wealth
equals W is offered the above lottery. Will this individual accept it? In other words, would he
place a positive value on this lottery?
If the gamble is accepted, expected utility is given by V = E [U (W + ε)]. If the gamble
is not accepted, expected utility is given by V = E [U (W )] = U (W ). Thus, if the individual
refuses to accept a fair gamble, it implies
U (W ) > E [U (W + ε)] = pU (W
+ h1) + (1 − p)U (W + h2)
(3)
To show that this is equivalent to having a concave utility function, note that U (W ) can be
re-written as
U (W ) = U (W + ph1 + (1 − p)h2 )
(4)
since ph1 + (1 − p)h2 = 0 by assumption that the gamble is fair. Re-writing inequality (3), we
1
have
U (W + ph1 + (1 − p)h2 ) > pU (W + h1 ) + (1 − p)U (W + h2)
(5)
which is the definition of U being a concave function. A function is concave if a line joining any
two points of the function lies entirely below the curve. When U (W ) is concave, a line connecting
the points U (W + h2) to U (W + h1) lies below U (W ) for all W such that W +h2 < W < W +h1.
pU (W + h1) + (1 − p)U (W + h2) is exactly the point on this line directly below U (W ). This
is clear by substituting p = −h2/(h1 − h2). Note that when U (W ) is a continuous, second
differentiable function, concavity implies that its second derivative, U (W ), is less than zero.
To show the reverse, that concavity of utility implies the unwillingness to accept a fair
gamble, we can use a result from statistics known as Jensen’s inequality. If
concave function, and x is a random variable, then Jensen’s inequality says that
E [U (x̃)] < U (E [x̃])
U( )
·
is some
(6)
Therefore, substituting x̃ = W + ε, with E [ε] = 0, we have
E [U (W + ε)] < U (E [W + ε]) = U (W )
(7)
which is the desired result.
We have defined risk aversion in terms of the individual’s utility function. We now consider
how this aversion to risk can be quantified. This is done by defining a risk premium, the amount
an individual is willing to pay to avoid a risk.
Let
π
denote the individual’s risk premium for a particular gamble, ε. It can be likened
to the maximum insurance payment an individual would pay to avoid a particular risk. Pratt
(1964) defined the risk premium for gamble (risk) ε as
U (W − π) = E [U (W + ε)]
(8)
This is the definition of a risk premium that is commonly used in the insurance literature. In
financial economics, a somewhat different concept is often used, namely, that an asset’s risk
2
utility
U(WH)
y
i ))
U(E( w
i ))
E(U( w
π2
π
U(WL)
WL
CE
W
WH
wealth
Certainty equivalent
i < U[E(w)]
i
Jensen's inequality: E[U(w)]
y = the expected utility cost of the randomness (risk)
π2 = wealth you need to receive to voluntarily choose the gamble (“extra return”)
π = insurance premium you would pay to avoid the gamble
U(w-π)=E[U(w+ε)]
Taylor Series:
2
3


U "(w)π 2
U "(w)ε
U "'( w)ε

+ ....
+ .... = E U ( w) + U '( w)ε +
+
U ( w) − U '( w)π +
2
2
6




2
U "( w)σ ε
U ( w) − U '( w)π ≈ U ( w) +
2
2
−U "( w) σ ε
π=
U '( w) 2
−U "( w)
= R( w) ← Arrow-Pratt measure
So π is proportional to
U '(w)
Absolute risk aversion = R(w) ← dealing with amounts of wealth
Relative risk aversion = w R(w) ← dealing with rates of return
Risk tolerance = T(w) =
1
R(w)
premium is its expected rate of return in excess of the risk-free rate of return. This alternative
concept will be considered later.
To analyze this Pratt (1964) risk premium, we continue to assume the individual is an
expected utility maximizer and that ε is a fair gamble, that is, its expected value equals zero.
Further, let us consider the case of ε being “small,” so that we can study its effects by taking
a Taylor series approximation of equation (8) around the point ε = 0 and π = 0.1 Expanding
the left hand side of (8) around
π = 0 gives
U (W − π) ∼= U (W ) − πU (W )
(9)
and expanding the right hand side of (8) around ε = 0 (and taking a three term expansion since
E [ε] = 0 implies that a third term is necessary for a limiting approximation) gives
E [U (W + ε)] ∼
= E U (W ) + εU (W ) + 12 ε2 U
=
where
W)
(
(10)
U (W ) + 12 σ2U (W )
σ2 ≡ E ε2 is the gamble’s variance. Equating the results in (9) and (10), we have
W ) ≡ 1 σ2R(W )
π = − 12 σ2 UU ((W
) 2
(11)
where R(W ) ≡ −U (W )/U (W ) is the Pratt (1964) - Arrow (1970) measure of absolute risk
aversion. Note that the risk premium, π, depends on the uncertainty of the risky asset, σ2, and
on the individual’s coefficient of absolute risk aversion. Since σ2 and U (W ) are both greater
than zero, concavity of the utility function ensures that π must be positive.
From (11) we see that the concavity of the utility function, U (W ), is not sufficient to
quantify the risk premium an individual is willing to pay, even though it is necessary and
sufficient to indicate if an individual is risk-averse or not. We also need the first derivative,
U (W ), which tells us the marginal utility of wealth. An individual may be very risk averse
By describing the random variable ε as “small” we mean that its probability density is concentrated around
its mean 0.
1
3
−U (W ) is large), but he may be unwilling to pay a large risk premium if he is poor since his
(
U W
marginal utility will be high ( ( ) will be large).
To illustrate this point, consider the following utility function:
(12)
U (W ) = −e ,b > 0
> 0 and U (W ) = −b2e < 0. Consider the behavior of a very
−
Note that
U (W ) = be
bW
−
bW
bW
−
wealthy individual, that is, one whose wealth approaches infinity:
lim
W →∞
U (W ) = lim U (W ) = 0
W →∞
As W → ∞, the utility function is a flat line.
(13)
Concavity disappears, which might imply that
this very rich individual would be willing to pay very little for insurance against a random
ε
event, , certainly less than a poor person with the same utility function. However, this is not
true because the marginal utility of wealth is also very small. This neutralizes the effect of
smaller concavity. Indeed:
R(W ) = bbee
2
−
−
bW
bW
=
b
(14)
which is a constant. Hence, this utility function is known as a constant absolute risk aversion
utility function.
If we want to assume that absolute risk aversion is declining in wealth, a necessary, though
not sufficient, condition for this is that the utility function have a positive third derivative,
since
∂R(W ) = − U (W )U (W ) − [U (W )]2
∂W
[U (W )]2
(15)
One can show that the coefficient of risk aversion contains all relevant information about
the individual’s risk preferences. Note that
U (W )])
R(W ) = − UU ((WW)) = − ∂ (ln [∂W
Integrating both sides of (16), we have
4
(16)
− R(W )dW = ln[U (W )] + c
(17)
Taking the exponential function of (17)
e
−
R(W )dW =
U (W )ec
(18)
Integrating once again gives
e−
R(W )dW dW = ec U (W ) + d
˜
U (W )
Because expected utility functions are unique only up to a linear transformation,
UW
(19)
e U (W )+d
c
reflects the same risk preferences as ( ).
Relative
risk aversion is another frequently used measure of risk aversion and is defined
simply as
R (W ) = WR(W )
r
(20)
Some useful utility functions:
a. Negative exponential (constant absolute risk aversion)
U (W ) = −e− ,b > 0
bW
As we saw earlier, R(W ) = b, so that Rr (W ) = bW .
b. Power (constant relative risk aversion)
U (W ) = γ1 W γ ,γ < 1
−1)W
implying that ( ) = − γ(γγW
γ −1
RW
c. Logarithmic
γ −2
= (1W−γ) and, therefore, Rr (W ) = 1 − γ.
This is a limiting case of power utility. To see this, write the power utility function as
γ − γ1 = W γγ−1 . (Recall that we can do this because utility functions are unique up to a
γW
1
5
linear transformation.) Now consider taking the limit of this utility function as γ
→ 0. Note
that the numerator and denominator both go to zero, so that the limit is not obvious. However,
we can re-write the numerator in terms of an exponential and natural log function and apply
L’Hospital’s rule to obtain:
Wγ −
γ →0
γ
lim
1
→
= lim
γ
0
eγ ln(W )
γ
−
1
γ →0
= lim
W W γ = ln(W )
ln( )
1
Thus, logarithmic utility is equivalent to power utility with
γ = 0, or a coefficient of relative
risk aversion of unity:
R(W ) = − WW −− = W1 and Rr (W ) = 1.
2
1
d. Quadratic
U (W ) = W − 2 W 2,b > 0
b
U (W ) = 1 − bW , which is > 0 if and only if b < 1 . Thus, this utility function only
makes sense when W < 1 , which is know as the “bliss point.” We have R(W ) = 1− and
R (W ) = 1− .
Note that
W
b
bW
b
bW
bW
r
e.
HARA (hyperbolic absolute risk aversion)
γ
+β
U (W ) = 1 −γ γ 1αW
−γ
subject to the restrictions γ = 1, α > 0, 1αW
−γ + β > 0, and β = 1 if γ = −∞. Thus,
−1
R(W ) = 1W−γ + βα . Since R(W ) must be > 0, it implies β > 0 when γ > 1. Rr (W ) =
W 1W−γ + αβ −1. Note that HARA utility nests constant absolute risk aversion (γ = −∞,
β = 1), constant relative risk aversion (γ < 1, β = 0), and quadratic (γ = 2) utility functions.
Thus, depending on the parameters, it is able to display constant absolute risk aversion or
relative risk aversion that is increasing, decreasing, or constant.
Kenneth Arrow (1970) independently derived a coefficient of risk aversion that is identical
to Pratt’s measure, but starts from a definition of a risk premium that is close to what is
typically utilized in financial markets. Suppose that the gamble, ε, has the following payoffs
and probabilities (this could be generalized to other fair payoffs):
6



+h with probability 12
 −h with probability 1
2
ε = 
(21)
Note that, as before, E [ε] = 0. Now consider the following question. By how much should
we change the expected value (return) of the gamble, by changing the probability of winning,
in order to make the individual indifferent between taking and not taking the risk? If p is the
probability of winning, we can define the risk premium as
θ = prob (ε = +h) − prob (ε = −h) = p − (1 − p) = 2p − 1
(22)
Therefore, from (22) we have
prob (ε = +h) ≡ p = 12 (1 + θ)
prob (ε = −h) ≡ 1 − p = 12 (1 − θ)
(23)
These new probabilities of winning and losing are equal to the old probabilities, 12 , plus half
of the increment, θ. Thus, the premium, θ, that makes the individual indifferent between
accepting and refusing the gamble is
U (W ) =
1 (1 + θ)U (W + h) + 1 (1 − θ)U (W − h)
2
2
(24)
Taking a Taylor series approximation around h = 0, gives
1
U (W ) = (1 + θ) U (W ) + hU (W ) + 12 h2U (W )
2
1
+ (1 − θ) U (W ) − hU (W ) + 12 h2U (W )
2
= U (W ) + hθU (W ) + 12 h2U (W )
Re-arranging (25) implies
7
(25)
θ = 12 hR(W )
(26)
which, as before, is a function of the coefficient of absolute risk aversion. Note that the Arrow
premium, θ, is in terms of a probability, while the Pratt measure, π, is in units of a monetary
payment. If we multiply θ by the monetary payment received, h, then (26) becomes
hθ = 12 h2R(W )
(27)
Noting that h2 is the variance of the random payoff, ε, shows that the Pratt and Arrow measures
of risk premia are equivalent. Both were obtained as a linearization of the true function around
ε = 0.
8
A. Penati - G. Pennacchi
Risk Aversion and Portfolio Choice
Having developed a concept of risk aversion, we will now consider the relation between risk
aversion and portfolio choice in a single period context. For simplicity, we assume there is a
riskless security, which pays a rate of return equal to rf , and a single risky security, which pays
a stochastic rate of return equal to r.
Let W0 be the individual’s initial wealth, and let A be the dollar amount that the individual
initially invests in the risky asset. Thus, W0 − A is the amount invested in the riskless security.
Denoting the individual’s end of period wealth as W̃ , it satisfies:
W̃ = (W0
− A)(1 + rf ) + A(1 + r̃)
(1)
= W0 (1 + rf ) + A(r̃ − rf )
We assume that the individual cares only about consumption at the end of this single
period. Therefore, maximizing end-of-period consumption is equivalent to maximizing end-ofperiod wealth. Assuming that the individual is a von Neumann-Morgenstern expected utility
maximizer, she chooses her portfolio by maximizing the expected utility of end-of-period wealth:
max E [U (W̃ )] = max E [U (W0 (1 + rf ) + A(r̃
A
A
− rf ))]
(2)
Taking the first order condition with respect to A, we obtain
E U W̃ (r̃
− rf )
=0
(3)
This condition determines the amount, A, that the individual invests in the risky asset. Note
from our earlier results that a risk averse individual would choose to hold positive amounts
of a risky asset (gamble) only if E [r̃
− rf ]
> 0. Now, we can go further and explore the
1
relationship between A and the individual’s initial wealth, W0 . Using the envelope theorem,
we can differentiate the first order condition to obtain1
E U (W̃ )(r̃
−
rf )(1 + rf ) dW0 + E U (W̃ )(r̃
−
rf )2 dA
=0
(4)
or
(1
+
r
)
E
U
(
W̃
)(r̃
−
r
)
f
f
dA
=
dW0
)(r̃ − rf )2
−E U ( W
(5)
The denominator of (5) is positive because concavity of the utility function ensures that
U
(W̃ ) is negative. Therefore, the sign of the expression depends on the numerator, which can
be of either sign because realizations of (r̃ − rf ) can turn out to be both positive and negative.
To characterize situations in which the sign of (5) can be determined, let us first consider
the case where the individual has absolute risk aversion that is decreasing in wealth. Let rh
denote a realization of r̃ such that it exceeds rf , and let W h be the corresponding level of W̃ .
Then for A
0, we have W h W0(1 + rf ).
If absolute risk aversion is decreasing in wealth,
this implies
R
where, as before, R(W ) =
−U
W h R (W0(1 + r ))
(6)
f
(W )/U (W ). Multiplying both terms of (6) by
−U (W h)(rh −
rf ), which is a negative quantity, the inequality sign changes:
U (W h )(rh
− rf ) −U (W h )(rh − rf )R (W0(1 + rf ))
(7)
Next, let rl denote a realization of r̃ such that it is lower than rf , and let W l be the corresponding
The envelope theorem applies to the problem of examining how the maximized value of the objective function
and the
variable change when one of the model’s parameters changes. In our context, define f (A, W0 ) ≡
control
f (A, W0 ) is the maximized value of the objective function when the control
E U W
so that v (W0 ) =max
A
(W 0 )
0 ) dA(W0 ) + ∂f (A,W0 ) .
variable, A, is optimally chosen. Then applying the chain rule, we have dvdW
= ∂f (A,W
∂A
dW0
∂W0
0
0 ) = 0, from the first order condition, this simplifies to just dv(W0 ) = ∂f (A,W0 ) Again applying
But since ∂f (A,W
∂A
dW0 2
∂W0
2
0 )/∂A) = 0 = ∂ f (A,W0 ) dA(W0 ) + ∂ f (A,W0 ) .
the chain rule to the first order condition, one obtains ∂ (∂f (A,W
2
∂W0
dW0
∂A∂W0
∂A
(W 0 )
∂ 2 f (A,W0 ) / ∂ 2 f (A,W0 ) , which is equation (5).
=
−
Re-arranging gives us dAdW
∂A∂W0
∂A2
0
1
.
2
level of
W̃ . Then for A 0, we have W W0(1 + r ). If absolute risk aversion is decreasing
l
f
in wealth, this implies
R(W l )
Multiplying (8) by
R (W0(1 + rf ))
(8)
−U (W l )(rl − rf ), which is positive, so that the sign of (8) remains the
same, we obtain
U (W l )(rl
− rf ) −U (W l )(rl − rf )R (W0(1 + rf ))
(9)
Notice that inequalities (7) and (9) are of the same form. The inequality holds whether the
realization is r̃ = rh or r̃ = rl . Therefore, if we take expectations over all realizations, where r̃
can be either higher than or lower than rf , we obtain
E U (W̃ )(r̃
− rf ) −E
U (W̃ )(r̃
− rf )
R (W0 (1 + rf ))
(10)
Since the first term on the right-hand-side is just the first order condition, inequality (10)
reduces to
E U (W̃ )(r̃
− rf ) 0
(11)
Thus, the first conclusion that can be drawn is that declining absolute risk aversion implies
dA/dW0 > 0, that is, the individual will invest an increasing amount of wealth in the risky
asset for larger amounts of initial wealth. For two individuals with the same utility function but
different initial wealths, the more wealthy one will invest a greater dollar amount in the risky
asset if utility is characterized by decreasing absolute risk aversion. While not shown here, the
opposite is true, namely, that the more wealthy individual will invest a smaller dollar amount
in the risky asset if utility is characterized by increasing absolute risk aversion.
Thus far, we have not said anything about the proportion of initial wealth invested in the
risky asset. To analyze this issue, we need the concept of relative risk aversion. Define
dA W0
η ≡ dW
0 A
3
(12)
which is the elasticity measuring the proportional increase in the risky asset for an increase in
initial wealth. Adding 1 −
A
A
to the right hand side of (12) gives
η = 1+
(dA/dW0)W0 − A
(13)
A
Substituting the expression dA/dW0 from equation (5), we have
W0 (1 + rf )E U (W̃ )(r̃ rf ) + AE U (W̃ )(r̃
η =1+
AE U (W̃ )(r̃ rf )2
−
−
Collecting terms in
U
− rf )2
−
(14)
(W̃ )(r̃ − rf ), this can be re-written as
η =1+
E U (W̃ )(r̃
− rf ){W0(1 + rf ) + A(r̃ − rf )}
−AE U (W̃ )(r̃ − rf )2
= 1+
E U
−AE
(W̃ )(r̃ − rf )W̃
U (W̃ )(r̃
(15)
− rf )2
The denominator is always positive. Thus, we can see that the elasticity, η, will be greater than
one, so that the individual will invest proportionally more in the risky asset with an increase in
wealth, if E U (W̃ )(r̃ − rf )W̃ 0. Can we relate this to the individual’s risk aversion? The
answer is yes and the derivation is almost exactly the same as that just given.
Consider the case where the individual has relative risk aversion that is decreasing in wealth.
Let rh denote a realization of r̃ such that it exceeds rf , and let W h be the corresponding level
of W̃ . Then for A 0, we have W h
W0(1 + rf ). If relative risk aversion, Rr (W ) ≡ WR(W ),
is decreasing in wealth, this implies
W h R(W h ) W0
Multiplying both terms of (16) by
(1 +
rf )R (W0(1 + rf ))
(16)
−U (W h)(rh −rf ), which is a negative quantity, the inequality
4
sign changes:
W h U (W h )(rh
− rf ) −U (W h )(rh − rf )W0(1 + rf )R (W0(1 + rf ))
(17)
Next, let rl denote a realization of r̃ such that it is lower than rf , and let W l be the corresponding
W0(1 + rf ). If relative risk aversion is decreasing in
level of W̃ . Then for A 0, we have W l
wealth, this implies
W l R(W l )
Multiplying (18) by
W0(1 + rf )R (W0(1 + rf ))
(18)
−U (W l )(rl − rf ), which is positive, so that the sign of (18) remains the
same, we obtain
W l U (W l )(rl
− rf ) −U (W l )(rl − rf )W0(1 + rf )R (W0(1 + rf ))
(19)
Notice that inequalities (17) and (19) are of the same form. The inequality holds whether the
realization is r̃ = rh or r̃ = rl . Therefore, if we take expectations over all realizations, where r̃
can be either higher than or lower than rf , we obtain
E W̃ U (W̃ )(r̃
− rf ) −E
U (W̃ )(r̃
− rf )
W0 (1 + rf )R(W0 (1 + rf ))
(20)
Since the first term on the right-hand-side is just the first order condition, inequality (20)
reduces to
E W̃ U (W̃ )(r̃
− rf ) 0
Thus, we see that an individual with decreasing relative risk aversion will have
(21)
η > 1 and
will invest proportionally more in the risky asset as wealth increases. The opposite is true for
increasing relative risk aversion:
η < 1 so that this individual will invest proportionally less in
the risky asset as wealth increases.
5