2.
Utility functions
How to rank the following 4 investment choices?
Investment A
x
p(x)
4
1
Investment B
x
p(x)
5
1
Investment C
x
p(x)
−5
1/4
0
1/2
40
1/4
Investment D
x
p(x)
−10
1/5
10
1/5
20
2/5
30
1/5
When there is no risk, we choose the investment with the highest
rate of return. — Maximum Return Criterion.
e.g. Investment B dominates Investment A, but this criterion fails
to compare Investment B with Investment C.
Maximum expected return criterion (for risky investments)
Identifies the investment with the highest expected return.
1
1
1
(−5) + (0) + (40) = 8.75
4
2
4
1
2
1
1
ED (x) =
(−10) + (10) + (20) + (30) = 14.
5
5
5
5
Is such procedure well justified?
EC (x) =
St Petersburg paradox (failure of Maximum Expected Return Criterion)
Tossing of a fair coin until the first head shows up. The prize is
2x−1 where x is the number of tosses until the first head shows up
(the game is then ended).
∞
1 x−1
Expected prize of the game =
2
= ∞.
x
2
x=1
Question
What is the certain amount that one would be willing to accept so
that it is indifferent between playing the game for free or receiving
this certain sum?
This certain amount is called the certainty equivalent of the game.
• When people are faced with such a lottery in experimental trials,
they refuse to pay more than a finite price (usually low).
• There is a very small chance to receive large sum of money.
This occurs when x is large.
1
e.g. x = 10, there is a change of 10 to receive 29.
2
• Alternative view In Mark Six, people are willing to pay a few
dollars to bet on winning a reasonably large sum with infinitesimally small chance.
• Certainty equivalent of the game under log utility
∞
∞
1
x−1
x−1
ln
2
=
ln
2
= ln 2
ln c = E[ln R] =
x
x
x=1 2
x=1 2
so that c = 2 is the certainty equivalent.
How much the player is ready to pay to play the game?
We need to solve for
U (w) = E[U (w + y − p)]
where
w = initial wealth
y = price received from the game
p = price that the player is willing to pay to participate in the game.
Pairwise comparison
Consider the set of alternatives B, how to determine which element
in the choice set B that is preferred?
The individual first considers two arbitrary elements: x1, x2 ∈ B. He
then picks the preferred element x1 and discards the other. From the
remaining elements, he picks the third one and compares with the
winner. The process continues and the winner among all alternatives
is identified.
Let the choice set B be a convex subset of the n-dimensional Euclidean space. The component xi of the n-dimensional vector x
may represent xi units of commodity i. By convex, we mean that
if x1, x2 ∈ B, then αx1 + (1 − α)x2 ∈ B for any α ∈ [0, 1].
• Each individual is endowed with a preference relation, .
• Given any elements x1 and x2 ∈ B, x1 x2 means either that x1
is preferred to x2 or that x1 is indifferent to x2.
Three axioms for Reflexivity
For any x1 ∈ B, x1 x1.
Comparability
For any x1, x2 ∈ B, either x1 x2 or x2 x1 .
Transitivity
For x1 , x2, x3 ∈ B, given x1 x2 and x2 x3, then x1 x3.
Remark
1. Without the comparability axiom, an individual could not determine an optimal choice. This is because there would exist at
least two elements of B between which the individual could not
discriminate.
2. The transitivity axiom ensures that the choices are consistent.
Example 1
Let B = {(x, y) : x ∈ [0, ∞) and y ∈ [0, ∞)} represent the set of
alternatives. Let x represent ounces of orange soda and y represent
ounces of grape soda. It is easily seen that B is a convex subset of
R2.
Suppose the individual is concerned only with the total quantity of
soda available, the more the better, then the individual is endowed
with the following preference relation:
For (x1 , y1), (x2, y2 ) ∈ B,
(x1 , y1 ) (x2 , y2) if and only if x1 + y1 ≥ x2 + y2 .
Dictionary order
Let the choice set B = {(x, y) : x ∈ [0, ∞), y ∈ [0, ∞)}, the dictionary
order is defined as follows:
Suppose (x1, y1 ) ∈ B and (x2, y2 ) ∈ B, then
(x1 , y1 ) (x2 , y2) if and only if
[x1 > x2 ] or [x1 = x2 and y1 ≥ y2].
It is easy to check that the dictionary order satisfies the three basic
axioms of a preference relation.
Definition
Given x, y ∈ B and a preference relation satisfying the above three
axioms.
1. x is indifferent to y, written as
x∼y
if and only if
x y and y x.
2. x is strictly preferred to y, written as
xy
if and only if
x y and not x ∼ y.
Axiom 4 – Order Preserving
For any x, y ∈ B where x y and α, β ∈ [0, 1],
[αx + (1 − α)y] [βx + (1 − β)y]
if and only if α > β.
Example 1 revisited
Recall the preference relation defined in Example 1, we take (x1 , y1 ),
(x2 , y2) ∈ B such that (x1 , y1 ) (x2, y2 ) so that x1 + y1 − x2 − y2 > 0.
Take α, β ∈ [0, 1] such that α > β, and observe
α[(x1 + y1 ) − (x2 + y2 )] > β[(x1 + y1) − (x2 + y2)].
Adding x2 + y2 to both sides, we obtain
α(x1 + y1) + (1 − α)(x2 + y2) > β(x1 + y1 ) + (1 − β)(x2 + y2).
Axiom 5 – Intermediate Value
For any x, y, z ∈ B, if x y z, then there exists a unique α ∈ (0, 1)
such that
αx + (1 − α)z ∼ y.
Remark
Given 3 alternatives with rankings of x y z, there exists a
fractional combination of x and z that is indifferent to y. Trade-offs
between the alternatives exist.
Example 1 revisited
Given x1 + y1 > x2 + y2 > x3 + y3 , choose
α=
(x2 + y2 ) − (x3 + y3)
.
(x1 + y1 ) − (x3 + y3)
Rearranging gives
α(x1 + y1 ) + (1 − α)(x3 + y3 ) = x2 + y2
so that
[α(x1 , y1 ) + (1 − α)(x3 , y3)] ∼ (x2, y2 ).
Dictionary order does not satisfy the intermediate value axiom
Suppose (x1 , y1), (x2 , y2 ), (x3 , y3) ∈ B such that (x1, y1 ) (x2 , y2) (x3 , y3) and x1 > x2 = x3 and y2 > y3. For any α ∈ (0, 1), we have
α(x1 , y1) + (1 − α)(x3 , y3 )
= α(x1 , y1) + (1 − α)(x2 , y3 )
= (αx1 + (1 − α)x2 , αy1 + (1 − α)y3 ).
But for α > 0, we have αx1 + (1 − α)x2 > x2 so
α(x1 , y1) + (1 − α)(x3 , y3 ) (x2 , y2)
for all
In other words, there is no α ∈ (0, 1) such that
αx + (1 − α)z ∼ y.
α ∈ (0, 1).
Axiom 6 – Boundedness
There exist x∗, y ∗ ∈ B such that x∗ z y ∗ for all z ∈ B.
• This Axiom ensures the existence of a most preferred element
x∗ ∈ B and a least preferred element y ∗ ∈ B.
Example 1 revisited
Recall B = {(x, y) : x ∈ [0, ∞) and y ∈ [0, ∞)}. Given any (z1, z2) ∈
B, we have
(z1 + 1, z2) (z1, z2) since z1 + z2 + 1 > z1 + z2.
Therefore, a maximum does not exist.
Motivation for defining utility
Knowledge of the preference relation effectively requires a complete listing of preferences over all pairs of elements from the choice
set B. We define a utility function that assigns a numeric value to
each element of the choice set such that larger numeric value implies
higher preference.
Theorem – Existence of Utility Function
Let B denote the set of payoffs from a finite number of securities,
also being a convex subset of Rn. Let denote a preference relation
on B. Suppose satisfies the following axioms
(i)
(ii)
(iii)
(iv)
∀x ∈ B, x x.
∀x, y ∈ B, x y or y x.
For any x, y, z ∈ B, if x y and y z, then x z.
For any x, y ∈ B, x y and α, β ∈ [0, 1],
αx + (1 − α)y βx + (1 − β)y
if and only if
α > β.
(v) For any x, y, z ∈ B, suppose x y z, then there exists a unique
α ∈ (0, 1) such that αx + (1 − α)z ∼ y.
(vi) There exist x∗, y ∗ ∈ B such that ∀z ∈ B, x∗ z y ∗.
Then there exists a utility function U : B → R such that
(a) x y iff U (x) > U (y).
(b) x ∼ y iff U (x) = U (y).
To show the existence of U : B → R, we write down one such
function and show that it satisfies the stated conditions.
Based on Axiom 6, we choose x∗, y ∗ ∈ B such that
x∗ z y ∗
for all
z ∈ B.
Without loss of generality, let x∗ y ∗. [Otherwise, x∗ ∼ z ∼ y ∗ for
all z ∈ B. In this case, U (z) = 0 for all z ∈ B, which is a trivial utility
function that satisfies conditions (a) and (b).]
Consider an arbitrary z ∈ B. There are 3 possibilities:
1.
z ∼ x∗;
2.
x∗ z y ∗;
3.
z ∼ y∗.
We define U by giving its value under all 3 cases:
1. U (z) = 1
2. By Axiom 5, there exists a unique α ∈ (0, 1) such that
[αx∗ + (1 − α)y ∗] ∼ z.
Define U (z) = α.
3. U (z) = 0.
Such U satisfies properties (a) and (b).
Proof of property (a)
Necessity
Suppose z1, z2 ∈ B are such that z1 z2, we need to show
U (z1) > U (z2).
Consider the four possible cases.
1.
2.
3.
4.
z1 ∼ x∗
z1 ∼ x∗
x∗ z1
x∗ z1
Case 1
z2
z2
z2
z2
y∗
∼ y∗
y∗
∼ y∗.
By definition, U (z1) = 1 and U (z2 ) = α, where α ∈ (0, 1)
uniquely satisfies
αx∗ + (1 − α)y ∗ ∼ z2.
Now, U (z1) = 1 > α = U (z2 ).
Case 2
By definition, U (z1) = 1 > 0 = U (z2).
Case 3
By defintion, U (zi) = αi, where αi ∈ (0, 1) uniquely satisfies
αix∗ + (1 − αi)y∗ ∼ zi,
so that
z1 ∼ [α1x∗ + (1 − α)y∗ ] ∼ [α2x∗ + (1 − α2)y∗ ] ∼ z2 .
We claim α1 > α2. Assume not, then α1 ≤ α2 . By Axiom 4,
[α2 x∗ + (1 − α2)y∗ ] [α1x∗ + (1 − α1)y∗ ].
This is a contradiction. Hence, α1 > α2 is true and
U (z1 ) = α1 > U (z2) = α2.
Case 4
By definition, U (z1) = α1, where α1 ∈ (0, 1) uniquely satisfies
α1x∗ + (1 − α1 )y∗ ∼ y1
and
U (z2 ) = 0.
We have
U (z1) = α1 > 0 = U (z2).
Sufficiency
Suppose, given z1, z2 ∈ B, that U (z1) > U (z2), we would like to
show z1 z2. Consider the following 4 cases
1. U (z1) = 1 and U (z2) = α2 , where α2 ∈ (0, 1) uniquely satisfies
[α2x∗ + (1 − α2 )y ∗] ∼ z2.
2. U (z1) = 1, where z1 ∼ x∗ and U (z2 ) = 0, where z2 ∼ y ∗.
3. U (zi) = αi , where αi ∈ (0, 1) uniquely satisfies
[αi x∗ + (1 − αi )y ∗] ∼ zi.
4. U (z1) = α1 and U (z2) = 0, where z2 ∼ y ∗.
Case 1
z1 ∼ x∗ ∼ [1 · x∗ + 0 · y ∗] and z2 ∼ [α2 x∗ + (1 − α2)y ∗].
By Axiom 4, 1 > α2 so that z1 z2.
Case 2
z1 ∼ x∗ y ∗ ∼ z2.
Case 3
z1 ∼ [α1 x∗ + (1 − α1)y ∗]
z2 ∼ [α2x∗ + (1 − α2)y ∗ ]
Since α1 > α2 , by Axiom 4, z1 z2.
Case 4
z1 ∼ [α1 x∗ + (1 − α1)y ∗] and
z2 ∼ y ∗ ∼ [0x∗ + (1 − 0)y ∗].
By Axiom 4 and Axiom 3, since α1 > 0, z1 z2.
Proof of Property (b)
Necessity
Suppose z1 ∼ z2 but U (z1) = U (z2), then
U (z1) > U (z2)
or
U (z2) > U (z1 ).
By property (a), this implies z1 z2 or z2 z1, a contradiction.
Hence,
U (z1) = U (z2).
Sufficiency
Suppose U (z1) = U (z2), but z1 z2 or z1 ≺ z2. By property (a),
this implies U (z1 ) > U (z2) or U (z2) > U (z1), a contradiction. Hence,
z1 ∼ z2 .
Choices among lotteries
How to make a choice between the following two lotteries:
L1 = {p1, A1; p2, A2; · · · ; pn , An}
L2 = {q1, A1; q2, A2; · · · ; qn , An}?
The outcomes are A1, · · · , An ; pi and qi are the probabilities of occurrence of Ai in L1 and L2, respectively. These outcomes are mutually
exclusive and only one outcome can be realized under each investment.
Comparability
When faced by two monetary outcomes Ai and Aj , the investor
must say Ai Aj , Aj Ai or Ai ∼ Aj .
Continuity
If A3 A2 and A2 A1, then there exists U (A2) [0 ≤ U (A2 ) ≤ 1]
such that
L = {[1 − U (A2)], A1; U (A2), A3} ∼ A2.
For a given set of outcomes A1, A2 and A3, these probabilities are
a function of A2, hence the notation U (A2).
Why is it called continuity axiom? When U (A2) = 1, we obtain
L = A3 A2; when U (A2) = 0, we obtain L = A1 ≺ A2. If we
increase U (A2) continuously from 0 to 1, we hit a value U (A2 ) such
that L ∼ A2.
Remark
Though U (A2) is a probability value, we will see that it is also the
investor’s utility function.
Interchangeability
Given L1 = {p1, A1; p2, A2; p3, A3} and A2 ∼ A = {q, A1; (1 − q), A2},
the investor is indifferent between L1 and L2 = {p1, A1; p2 , A; p3, A3}.
Transitivity
Given L1 L2 and L2 L3, then L1 L3.
Also, if L1 ∼ L2 and L2 ∼ L3, then L1 ∼ L3.
Decomposability
A complex lottery has lotteries as prizes. A simple lottery has monetary values A1, A2 etc as prizes.
Consider a complex lottery L∗ = (q, L1; (1 − q), L2), where
L1 = {p1, A1; (1 − p1 ), A2}
and
L2 = {p2 , A1; (1 − p2), A2},
L∗ can be decomposed into a simply lottery L = {p∗, A1; (1−p∗), A2},
with A1 and A2 as prizes where p∗ = qp1 + (1 − q)p2.
Monotonicity
(a) For monetary outcomes, A2 > A1 =⇒ A2 A1.
(b) For lotteries
(i) Let L1 = {p, A1; (1 − p), A2} and L2{p, A1; (1 − p), A3}.
A3 > A2, then A3 A2; and L2 L1.
If
(ii) Let L1 = {p, A1; (1 − p), A2} and L2 = {q, A1; (1 − q), A2}, also
A2 > A1 (hence A2 A1). If p < q, then L1 L2.
Theorem
The optimal criterion for ranking alternative investments is the expected utility of the various investments.
Proof
How to make a choice between L1 and L2
L1 = {p1, A1; p2, A2; · · · ; pn , An}
L2 = {q1, A1; q2, A2; · · · ; qn , An}
A1 < A2 < · · · < An , where Ai are various monetary outcomes?
By comparability axiom, we can compare Ai. Further, by monotonicity axiom, we determine that
A1 < A2 < · · · < An implies A1 ≺ A2 ≺ · · · ≺ An.
Define A∗i = {[1 − U (Ai)], A1; U (Ai), An} where 0 ≤ U (Ai ) ≤ 1.
By continuity axiom, for every Ai, there exists U (Ai) such that
Ai ∼ A∗i .
For A1, U (A1) = 0, hence A∗1 ∼ A1; for An , U (An) = 1. For other
Ai, 0 < U (Ai) < 1. By the monotonicity and transitivity axioms,
U (Ai) increases from zero to one as Ai increases from A1 to An .
Substitute Ai by A∗i in L1 successively and by the interchangeability axiom,
= {p , A∗ ; p , A∗ ; · · · ; pn, A∗ }.
L1 ∼ L
1
1
n
1 2
2
By the decomposability axiom,
∼ L∗ = {Σp [1 − U (A )], A ; Σp U (A ), A }.
L1 ∼ L
n
1
i
i
1
i
i
1
Similarly
L2 ∼ L∗2 = {Σqi[1 − U (Ai)], A1; ΣqiU (Ai), An}.
By the monotonicity axiom, L∗1 L∗2 if
ΣpiU (Ai) > ΣqiU (Ai).
This is precisely the expected utility criterion. The same conclusion
applies to L1 L2, due to transitivity.
Remark
Why does U (Ai) reflect the investor’s utility?
Recall Ai ∼ A∗i = {[1 − U (Ai)], A1; U (Ai), An }, such a function U (Ai)
always exists, though not all investors would agree on the specific
value of U (Ai).
• By the monotonicity axiom, utility is non-decreasing.
• A utility function is determined up to a positive linear transformation, so its value is not limited to the range [0, 1]. “Determined” means that the ranking of the projects by the MEUC
does not change.
• The absolute difference or ratio of the utilities of two investment
choices gives no indication of the degree of preference of one
over the other since utility values can be expanded or suppressed
by a linear transformation.
Characterization of utility functions
1. More is being preferred to less: u(w) > 0
2. Investors’ taste for risk
– averse to risk (reject a fair gamble)
– neutral toward risk (indifferent to a fair gamble)
– seek risk (select a fair gamble)
3. Investors’ preference changes with a change in wealth. Percentage of wealth invested in risky asset changes as wealth changes.
Jensen’s inequality
Suppose u(w) ≤ 0 and X is a random variable, then
u(E[X]) ≥ E[u(X)].
Write E[X] = µ; since u(w) is concave, we have
u(w) ≤ u(µ) + u(µ)(w − µ)
for all values of w.
Risk aversion
Replace w by X and take the expectation on each side
E[u(X)] ≤ u(µ).
Interpretation
E[u(X)] represents the expected utility of the gamble associated
with X. The investor prefers a sure wealth of µ = E[X] rather than
playing the game, if u(w) ≤ 0. This indicates risk aversion.
Insurance premium
Individual’s total initial wealth is w, and the wealth is subject to
random loss Y during the period, 0 ≤ Y < w.
Let π be the insurable premium payable at time 0 that fully reimburses the loss (neglecting the time value of money).
1. If the individual decides not to buy insurance, then the expected
utility is E[u(w − Y )]. The expectation is based on investor’s
own subjective assessment of the loss.
2. If he buys the insurance, the utility at time 1 is u(w − π). Note
that w − π is the sure wealth.
If the individual is risk averse [u(w) ≤ 0], then from Jensen’s inequality (change X to w − Y ), we obtain
u(w − E[Y ]) ≥ E[u(w − Y )].
The fair value of insurance premium π is determined by
u(w − π) = E[u(w − Y )]
so that we can deduce that π ≥ E[Y ]. Suppose the higher moments
of Y are negligible, it can be deduced that the maximum price that
a risk-averse individual with wealth w is willing to pay to avoid a
possible loss of Y is approximately
σY2
π ≈ µY +
RA(w − µ),
2
where RA(w) = −u(w)/u (w), 0 ≤ Y < w and µ = E[Y ] < w. With
higher RA(w), the individual is willing to pay a higher premium to
avoid risk.
Proof
We start from the governing equation for π
u(w − π) = E[u(w − Y )].
Write Y = µ + zV , where V is a random variable with zero mean.
Here, z is a small perturbation parameter. We then have
u(w − π) = E[u(w − µ − zV )].
(1)
We are seeking the perturbation expansion of π in powers of z in
the form
π = a + bz + cz 2 + · · ·
(i)
Setting z = 0, u(w − a) = E[u(w − µ)] = u(w − µ) so that
a = µ.
(ii)
Differentiating (1) with respect to z and setting z = 0,
−π (0)u(w − π) = E[−V u(w − µ)]
since E(V ) = 0 and π (0) = b, so b = 0.
(2)
(iii)
Differentiating (1) twice with respect to z and setting z = 0
−π (0)u(w − π) = E[V 2u(w − µ)]
var(V ) u c = −
.
w−µ
2
u
u(w)
Define the absolute risk aversion coefficient: RA(w) = − , we
u (w)
have
RA(w − µ) 2
z var(V )
2
σY2
RA(w − µ).
= µ+
2
π ≈ µ+
σY2
RA(w − µ) is called the risk premium. For low level of
π−µ =
2
risks, π − µ is proportional to the product of variance of the loss
distribution and individual’s absolute risk aversion.
Relative risk aversion coefficient
2 . The whole
Let X be a fair game with E[X] = 0 and var(X) = σX
wealth w is invested into the game.
Choice A
Choice B
w+X
wC (with certainty)
The investor is indifferent to these two positions iff
E[u(w + X)] = u(wC ).
Note that wC = w − (w − wC ), indicating the payment of w − wC for
Choice B. The difference w − wC represents the maximum amount
the investor would be willing to pay in order to avoid the risk of the
game.
Let q be the fraction of wealth an investor is giving up in order to
w − wC
or wC = w(1 − q). Let Z be the
avoid the gamble; then q =
w
return per dollar invested so that for a fair gamble, E[Z] = 1. Write
2.
var(Z) = σZ
Suppose we invest w dollars, the return would be wZ. Expand u(wZ)
around w:
u(w)
u(wZ) = u(w) + u (w)(wZ − w) +
(wZ − w)2 + · · ·
E[u(wZ)] = u(w) + 0 +
u(w)
2
2
2
w2 σZ
+ ···
On the other hand,
u(wC ) = u(w(1 − q)) = u(w) − qwu (w) + · · · .
Equating u(wC ) with E[u(wZ)], we obtain
u(w) 2 2
w σZ = −u(w)qw
2
so that
2 u (w)
σZ
q=− w .
2 u (w)
u(w)
,
Define RR(w) = coefficient of relative risk aversion = −w u (w)
2
σZ
w − wC
= percentage of risk premium =
RR (w).
then q =
w
2
Types of utility functions
1. Exponential utility
u(x) = 1 − e−ax, x > 0
u(x) = ae−ax
u(x) = −a2e−ax < 0
(risk aversion)
so that RA(x) = a for all wealth level x.
2. Power utility
xα − 1
,
u(x) =
α
u(x) = xα−1
α≤1
u(x) = (α − 1)xα−2
1−α
and RR (x) = 1 − α.
RA(x) =
x
3. Logarithmic utility (corresponds to α → 0+ in power utility)
u(x) = a ln x + b,
u(x) = a/x
u(x) = −a/x2
1
RA(x) = and RR (x) = 1.
x
a>0
Properties of power utility functions: U (x) = xγ /γ, γ ≤ 1
(i) γ > 0, aggressive utility
Consider γ = 1, corresponding to U (x) = x. This is the expected
value criterion.
Recall that the strategy that maximizes the expected value
bets all capital on the most favorable sector – prone to early
bankruptcy.
For γ = 1/2; consider two opportunities:
(a) capital will double with a probability of 0.9 or it will go to
zero with probability 0.10,
(b) capital will increase by 25% with certainty.
√
√
Since 0.9 × 2 > 1.25, so opportunity (a) is preferred to (b).
However, opportunity (a) is certain to go bankrupt.
(ii) γ < 0, conservative utility
For γ = −1/2, consider two opportunities
(a) capital quadruples in value with certainty
(b) with probability 0.5 capital remains constant and with probability 0.5 capital is multiplied by 10 million.
Since −4−1/2 > −0.5 − 0.5(10, 000, 000)−1/2, opportunity (a)
is preferred to (b).
Apparently, the best choice for γ may be negative, but close
to zero. This utility function is close to the logarithm function.
Example
An investor has an initial wealth of w and can allocate funds between
two assets: a risky asset and a riskless asset.
m = expected rate of return on the risky security = piu + (1 − p)id
so that
m − id
iu − m
p=
, 1−p =
.
iu − id
iu − id
• Impose the condition: m > if , otherwise a rational investor will
never invest a positive amount in the risky asset.
• No arbitrage conditions: iu > if > id. Also, iu > m > id for
0 < p < 1.
• Let x be the fraction of initial wealth placed on the risky asset.
wα
, 0 < α < 1.
Choose the power utility function:
α
Investor’s expected utility:
p
{w[(1 + if ) + x(iu − if )]}α
α
+ (1 − p)
{w[1 + if ) + x(id − if )]}α
α
.
Since the utility function is concave, the second order condition
for a maximum is automatically satisfied.
Optimal proportion
where
θ=
x∗ =
(1 + if )(θ − 1)
(iu − if ) + θ(if − id)
[p(iu − if )]1/(1−α)
[(1 − p)(if − id)]1/(1−α)
.
The no-arbitrage condition leads to θ ≥ 1.
(i) x∗ > 0 ⇐⇒ θ > 1.
(ii) As m → if , θ → 1 and x∗ → 0.
A risk-averse investor will prefer the riskless asset if it has the
same return as the expected return on the risky asset.
(iii) x∗ < 1 as long as θ <
1 + iu
.
1 + id
Continuous-time limit of the discrete-time model
Assume stationarity of the return distribution and no transaction
costs, the multi-period case collapses to a series of identical singleperiod problems. The optimal x∗ is the same at each time interval
and it is the same as that of the one-period case.
Assuming discrete random walk of proportional jumps, with h =
time step and σ 2 as the variance rate, we obtain
√
1 + iu = e σ h ,
√
1 + id = e−σ h,
1 + m = eµh,
erh = 1 + rf .
Express iu, id, if and p in terms of h. Taking h → 0,
x∗ =
µ−r
risk premium on risky asset
.
=
2
σ (1− α)
variance × relative risk aversion
Merton’s ratio
• Merton assumes GBM for the asset price process with mean µ
and variance σ 2; and power utility function.
• It is optimal for the investor to maintain a constant proportion
of wealth in the risky asset – continuous rebalancing of portfolio
is required (though at no transaction cost).
Minimum expected excess return required to induce the individual
to invest on the risky asset
r = random rate of return of the single risky asset
x = proportion of wealth invested in the risky asset
w0 = initial wealth
a = wealth invested in the risky asset
= random wealth at the end of the investment period
w
r = riskfree interest rate
= (w0 − a)(1 + r) + a(1 + r) = w0(1 + r) + a(r − r),
w
a = xw0
For the individual to invest at least xw0 amount of his wealth on
d
{E[u(w)]}
≥ 0 or
the risky asset, it must be that
da
E[u(w0 (1 + r) + xw0 (r − r))(r − r)] ≥ 0.
Performing the Taylor expansion of u(w0 (1 + r) + xw0(r − r)) at
w0(1 + r), we obtain
E[u(w0 (1 + r) + xw0 (r − r))(r − r)]
= u(w0(1 + r))E[r − r] + u(w0(1 + r))xw0 E[(r − r)2]
+ o(E[(r − r)2 ]).
Assuming low level of risk of r so that o(E[(r−r)2]) can be neglected.
The minimum risk premium required to induce x portion of wealth
invested on the risky asset is
u(w0(1 + r))
E[r − r] ≥ −xw0 E[(r − r)2 ],
u (w0(1 + r))
for all
0 < x ≤ 1.
Log utility and maximization of the geometric mean return
pj − 1, where r is the
(1
+
r
)
Geometric mean return = rG = n
j
j
j=1
rate of return if outcome j occurs with probability pj .
Taking logarithm on both sides
ln(1 + rG) =
n
pj ln(1 + rj ).
j=1
Consider the log utility function: max E ln w1, where w1 is the endof-period wealth.
Let r be the random rate of return.
Maximization of the expected log utility is given by
w
max E[ln w1 − ln w0] = max E ln 1
w0
= max E[ln(1 + r)]
= max
n
pj ln(1 + rj ).
j=1
Maximization of the geometric mean return is equivalent to the
maximization of the expected utility, where the logarithmic function
is used as the utility function.
Summary on risk aversion coefficients
Absolute risk aversion
U (W )
A(W ) = − U (W )
If A(W ) has the same sign for all values of W , then the investor
has the same risk preference (risk averse, neutral or seeker) for all
values of W (global).
Relative risk aversion
W U (W )
.
R(W ) = −
U (W )
Note that utility functions are only unique up to a strictly positive affine transformation. The second derivative alone cannot be
used to characterize the intensity of risk averse behaviors. The risk
aversion coefficients are invarant to a strictly positive affine transformation of individual utility function.
Changes in Absolute Risk Aversion with Wealth
Condition
Increasing absolute
risk aversion
Constant absolute
risk aversion
Decreasing
absolute risk aversion
Definition
As wealth increases,
hold
fewer dollars in
risky assets
As wealth increases,
hold
some
dollar
amount in risky
assets
As wealth increases,
hold
more dollars in
risky assets
Property
of A(W )
A(W ) >
0
Example
A(W ) =
0
−e−CW
A(W ) <
0
ln W
W −CW
2
Changes in Relative Risk Aversion with Wealth
Condition
Increasing relative
risk aversion
Constant
relative
risk aversion
Decreasing relative
risk aversion
Definition
Percentage
invested
in
risky
assets declines as
wealth increases
Percentage
invested
in
risky
assets is unchanged
as wealth increases
Percentage
invested
in
risky
assets increases as
wealth increases
Property
of
R(W )
R(W ) >
0
Examples
of
Utility
Functions
W − bW 2
R(W ) =
0
ln W
R(W ) <
0
−1/2
2W
−e