Download Lecture 04: Risk Preferences and Risk Preferences and Expected

Fin 501: Asset Pricing
Lecture 04: Risk Preferences and
Expected Utility Theory
• Prof. Markus K. Brunnermeier
Slide 04
04--1
Fin 501: Asset Pricing
O
Overview:
i
Ri k Preferences
Risk
P f
1. State
State--byby-state dominance
2. Stochastic dominance
3. vNM expected utility theory
a) Intuition
b) Axiomatic
A i
ti foundations
f d ti
[DD4]
[L4]
[DD3]
4. Risk aversion coefficients and portfolio choice [DD5,L4]
5 Prudence coefficient and precautionary savings [DD5]
5.
6. Mean
Mean--variance preferences
[L4.6]
Slide 04
04--2
Fin 501: Asset Pricing
St t -by
StateState
b -state
byt t Dominance
D i
- State-by-state
y
dominance Ä incomplete
p
rankingg
- « riskier »
Table 2.1 Asset Payoffs ($)
t=0
Cost at t=0
investment 1
i
investment
t
t2
investment 3
- 1000
- 1000
- 1000
t=1
Value at t=1
π1 = π2 = ½
s=1
s=2
1050
1200
500
1600
1600
1050
- investment 3 state by state dominates 1.
Slide 04
04--3
Fin 501: Asset Pricing
St t -by
StateState
b -state
byt t Dominance
D i
((ctd.)
td )
Table 2.2 State Contingent ROR (r )
Investment 1
Investment 2
Investment 3
State
St
t C
Contingent
ti
t ROR (r
( )
s=1
s=2
Er
σ
5%
20%
12.5%
7.5%
-50%
60%
5%
55%
5%
60%
32.5%
27.5%
- Investment 1 mean-variance dominates 2
- BUT investment 3 does not m-v dominate 1!
Slide 04
04--4
Fin 501: Asset Pricing
St t -by
StateState
b -state
byt t Dominance
D i
((ctd.)
td )
Table 2.3 State Contingent Rates of Return
investment 4
investment 5
State Contingent Rates of Return
s=1
s=2
3%
5%
3%
8%
π1 = π2 = ½
E[r4] = 4%; σ4 = 1%
E[r5] = 5.5%;
5 5%; σ5 = 2.5%
2 5%
- What is the trade-off between risk and expected return?
- Investment
I
4 hhas a hi
higher
h Sharpe
Sh
ratio
i (E[r]-r
(E[ ] f)/σ
)/ than
h iinvestment 5
for rf = 0.
Slide 04
04--5
Fin 501: Asset Pricing
O
Overview:
i
Ri k Preferences
Risk
P f
1. State
State--byby-state dominance
2. Stochastic dominance
3. vNM expected utility theory
a) Intuition
b) Axiomatic
A i
ti foundations
f d ti
c) Risk aversion coefficients
[DD4]
[L4]
[DD3]
[DD4,L4]
4 Risk aversion coefficients and portfolio choice [DD5,L4]
4.
[DD5 L4]
5. Prudence coefficient and precautionary savings [DD5]
6. Mean
Mean--variance preferences
[L4.6]
Slide 04
04--6
Fin 501: Asset Pricing
St h ti Dominance
Stochastic
D i
ƒ Still incomplete
p
ordering
g
ƒ “More complete” than state-by-state ordering
ƒ State-by-state dominance ⇒ stochastic dominance
ƒ Risk preference not needed for ranking!
ƒ independently of the specific trade-offs (between return, risk and other
characteristics of probability distributions) represented by an agent
agent'ss
utility function. (“risk-preference-free”)
ƒ Next Section:
ƒ Complete preference ordering and utility
representations
H
Homework:
k Provide
P id an example
l which
hi h can be
b ranked
k d
according to FSD , but not according to state dominance.
Slide 04
04--7
Table 3-1 Sample Investment Alternatives
States of nature
Payoffs
Proba Z 1
Proba Z 2
1
2
10
100
.4
.6
.4
.4
EZ 1 = 64, σ z 1 = 44
Fin 501: Asset Pricing
3
2000
0
.2
EZ 2 = 444, σ z 2 = 779
Pr obability
F1
10
1.0
0.9
F2
0.8
07
0.7
0.6
0.5
F 1 and F 2
0.4
0.3
0.2
01
0.1
Payoff
0
10
100
2000
Slide 04
04--8
Fin 501: Asset Pricing
Fi t Order
First
O d Stochastic
St h ti Dominance
D i
ƒ Definition 33.1
1 : Let FA(x) and FB(x) , respectively,
respectively
represent the cumulative distribution functions of two
random variables ((cash payoffs)
p y ff ) that,, without loss off
generality assume values in the interval [a,b]. We say
that FA(x) first order stochastically dominates (FSD)
FB(x) if and only if for all x ∈ [a,b]
FA(x) ≤ FB(x)
H
Homework:
k Provide
P id an example
l which
hi h can be
b ranked
k d
according to FSD , but not according to state dominance.
Slide 04
04--9
Fin 501: Asset Pricing
Fi t Order
First
O d Stochastic
St h ti Dominance
D i
1
0.9
0.8
FB
0.7
FA
0.6
0.5
0.4
0.3
0.2
0.1
0
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
X
Slide 04
04--10
Fin 501: Asset Pricing
Table 3-2 Two Independent Investments
Investment 3
Payoff
4
5
12
Investment 4
Prob.
0 25
0.25
0.50
0.25
Payoff
1
6
8
Prob.
0 33
0.33
0.33
0.33
1
0.9
0.8
0.7
0.6
investment 4
0.5
0.4
0.3
0.2
investment 3
0.1
0
0
1
2
3
4
5
6
7
8
9
10
11
12
13
Figure 3-6 Second Order Stochastic Dominance Illustrated
Slide 04
04--11
Fin 501: Asset Pricing
S
Second
dO
Order
d St
Stochastic
h ti Dominance
D i
ƒ Definition 3.2:
3 2: Let FA ( ~x ) , FB ( ~x ) , be two
cumulative probability distribution for
random payoffs in [a, b]. We say that FA ( ~x )
second order stochastically dominates
(SSD) FB ( ~x ) if and only if for any x :
x
∫
-∞
[ FB (t) - FA (t) ] dt ≥ 0
(with strict inequality for some meaningful
interval of values of t).
Slide 04
04--12
Fin 501: Asset Pricing
Mean Preserving Spread
xB = xA + z
((3.8))
where z is independent of xA and has zero mean
for normal distributions
fA (x)
f B (x)
μ = ∫ x fA (x)dx
d = ∫ x f B (x)dx
d
~
x , Payoff
Figure 3-7 Mean Preserving Spread
Slide 04
04--13
Fin 501: Asset Pricing
Mean Preserving Spread & SSD
ƒ Theorem 3.4 : Let FA(•) and FB(•) be two distribution
functions defined on the same state space with identical
means. Then the follow statements are equivalent :
x ) SSD FB ( ~
x)
ƒ FA ( ~
x ) is a mean p
preserving
ese ving spread
sp ead of FA ( ~x )
ƒ FB ( ~
in the sense of Equation (3.8) above.
Slide 04
04--14
Fin 501: Asset Pricing
O
Overview:
i
Ri k Preferences
Risk
P f
1. State
State--byby-state dominance
2. Stochastic dominance
3. vNM expected utility theory
a) Intuition
b) Axiomatic
A i
ti foundations
f d ti
[DD4]
[L4]
[DD3]
4. Risk aversion coefficients and portfolio choice [DD4,5,L4]
5 Prudence coefficient and precautionary savings [DD5]
5.
6. Mean
Mean--variance preferences
[L4.6]
Slide 04
04--15
Fin 501: Asset Pricing
AH
Hypothetical
th ti l Gamble
G bl
ƒ Suppose someone offers you this gamble:
ƒ "I have a fair coin here. I'll flip it, and if it's tail I pay
you $1 and the gamble is over. If it's head, I'll flip
again. If it's tail then, I pay you $2, if not I'll flip again.
With every round, I double the amount I will pay to
you if it's
it s tail
tail."
ƒ Sounds like a good deal. After all, you can't loose.
So here
here'ss the question:
ƒ How much are you willing to pay to take this
gamble?
g
Slide 04
04--16
Fin 501: Asset Pricing
P
Proposal
l 1:
1 Expected
E
t d Value
V l
ƒ
ƒ
ƒ
ƒ
With probability 1/2 you get $1.
With probability 1/4 you get $2.
$2
With probability 1/8 you get $4.
etc.
Ê
(12 )1 times
(12 )2 times
(12 )3 times
20
21
22
The expected payoff is the sum of these
payoffs weighted with their probabilities,
payoffs,
probabilities
∞
t
so
∞
1
⎛1⎞
∑
t =1
t −1
=
⎜ ⎟ ⋅ 2
2⎠
⎝{
123
probability
∑2
t =1
1
=∞
payoff
Slide 04
04--17
Fin 501: Asset Pricing
An Infinitely Valuable Gamble?
ƒ You should pay
everything you own and
more to purchase the right
to take this gamble!
ƒ Yet, in practice, no one is
prepared to pay such a
high price. Why?
ƒ E
Even though
th
h the
th expected
t d
payoff is infinite, the
distribution of payoffs is
not attractive…
probability
0.5
0.4
0.3
0.2
0.1
0
0
20
40
60 $
ƒ With 93% probability
we get $8 or less, with
99% probability
b bili we get
$64 or less.
Slide 04
04--18
Fin 501: Asset Pricing
What Sho
Should
ld We Do?
ƒ How can we decide in a rational fashion about such
gambles (or investments)?
ƒ Proposal 2: Bernoulli suggests that large gains should be
weighted less. He suggests to use the natural logarithm.
[Cremer - another great mathematician of the time - suggests the
square root
root.]]
∞
∑
t =1
t
expected utility
⎛1⎞
t −1
<∞
⎜ ⎟ ⋅ ln(2 ) = ln(2) =
of gamble
2⎠
⎝{
14243
probability
Ê
utility of payoff
Bernoulli would have paid at most eln(2) = $2 to
participate in this gamble.
Slide 04
04--19
Fin 501: Asset Pricing
Ri
Risk
Riskk-Aversion
A
i andd C
Concavity
it
u(x)
( )
ƒ The shape of the von
Neumann Morgenstern (NM)
utility function contains a lot
of information.
ƒ Consider a fifty-fifty lottery
with final wealth of x0 or x1
u(x1)
E{u(x)}
u(x0)
x0
E[x]
x1 x
Slide 04
04--20
Fin 501: Asset Pricing
Ri
Risk
Riskk-aversion
i andd concavity
it
u(x)
( )
ƒ Risk-aversion means that the
certainty equivalent is smaller
than the expected prize.
prize
Ê
We conclude that a
risk-averse vNM
utility function
must be concave.
u(x1)
u(E[x])
E[u(x)]
u(x0)
x0
u-1(E[u(x)])
E[x]
x1 x
Slide 04
04--21
Fin 501: Asset Pricing
J
Jensen’s
’ IInequality
lit
Theorem 3.1
3 1 (Jensen
(Jensen’ss Inequality):
ƒ Let g( ) be a concave function on the interval
[[a,b],
] and be a random variable such that
Prob (x ∈[a,b]) =1
ƒ Suppose the expectations E(x) and E[g(x)] exist;
then
E [g ( ~
x )] ≤ g [E ( ~
x )]
Furthermore, if g(•) is strictly concave, then the
inequality is strict.
Slide 04
04--22
Fin 501: Asset Pricing
R
Representation
t ti off Preferences
P f
A preference ordering is (i) complete, (ii) transitive, (iii)
continuous and [(iv) relatively stable] can be represented
by a utility function, i.e.
(c0,cc1,…,ccS) Â (c
(c’0,cc’1,…,cc’S)
⇔ U(c0,c1,…,cS) > U(c’0,c’1,…,c’S)
(preference ordering over lotteries –
(S+1)-dimensional
(S
)
space)
p )
Slide 04
04--23
Fin 501: Asset Pricing
P f
Preferences
over P
Prob.
b Di
Distributions
t ib ti
ƒ Consider c0 fixed, c1 is a random variable
ƒ Preference ordering over probability distributions
ƒ Let
ƒ P bbe a sett off probability
b bilit distributions
di t ib ti
with
ith a finite
fi it
support over a set X,
ƒ  a (strict) preference ordering over P
P, and
ƒ Define % by p % q if q ¨ p
Slide 04
04--24
Fin 501: Asset Pricing
ƒ S states of the world
ƒ Set of all possible lotteries
ƒ Space with S dimensions
ƒC
Can we simplify
i lif the
th utility
tilit representation
t ti off
preferences over lotteries?
ƒ Space
S
with
ith one dimension
di
i – income
i
ƒ We need to assume further axioms
Slide 04
04--25
Fin 501: Asset Pricing
E
Expected
t d Utility
Utilit Th
Theory
ƒ A binary relation that satisfies the following
three axioms if and only
y if there exists a function
u(•) such that
p  q ⇔ ∑ u(c) p(c) > ∑ u(c) q(c)
i.e. preferences correspond to expected utility.
Slide 04
04--26
Fin 501: Asset Pricing
vNM
NM Expected
E
t d Utility
Utilit Theory
Th
ƒ Axiom 1 ((Completeness
p
and Transitivity):
y)
ƒ Agents have preference relation over P (repeated)
ƒ Axiom 2 (Substitution/Independence)
ƒ For all lotteries p,q,r ∈ P and α ∈ (0,1],
p < q iff α p + ((1-α)) r < α q + ((1-α)) r ((see next slide))
ƒ Axiom 3 (Archimedian/Continuity)
ƒ For all lotteries p,q,r
p q r ∈ P,
P if p  q  r,
r then there
exists a α , β ∈ (0,1) such that
α p + (1( α) r  q  β p + ((1 - β) r..
Problem: p you get $100 for sure, q you get $ 10 for sure, r you are killed
Slide 04
04--27
Fin 501: Asset Pricing
I d
Independence
d
Axiom
A i
ƒ Independence of irrelevant alternatives:
π
p<q
p
π
q
<
⇔
r
r
Slide 04
04--28
Fin 501: Asset Pricing
Allais Paradox –
Violation of Independence Axiom
10%
10’
≺
0
9%
15’
0
Slide 04
04--29
Fin 501: Asset Pricing
Allais Paradox –
Violation of Independence Axiom
10%
10’
≺
9%
0
100%
15’
0
10’
90%
15’
Â
0
0
Slide 04
04--30
Fin 501: Asset Pricing
Allais Paradox –
Violation of Independence Axiom
10%
10’
9%
≺
0
100%
0
10’
90%
Â
10%
15’
15’
10%
0
0
0
0
Slide 04
04--31
Fin 501: Asset Pricing
vNM
NM EU Th
Theorem
ƒ A binary relation that satisfies the axioms 1-3 if
and only
y if there exists a function u(•)
( ) such that
p  q ⇔ ∑ u(c)
( ) p(
p(c)) > ∑ u(c)
( ) q(
q(c))
i.e. p
preferences correspond
p
to expected
p
utility.
y
Slide 04
04--32
Fin 501: Asset Pricing
E
Expected
t d Utility
Utilit Th
Theory
U(Y)
U( Y0 + Z 2 )
~
U( Y0 + E( Z))
~
EU( Y0 + Z)
U( Y0 + Z1 )
~
CE( Z)
Y0
Y0 + Z1
Π
~
~
CE ( Y0 + Z) Y0 + E(Z)
Y0 + Z 2
Y
Slide 04
04--33
Fin 501: Asset Pricing
Expected Utility & Stochastic Dominance
ƒ Theorem 3. 2 : Let FA ( ~x ) , FB ( ~x ) , be two cumulative
x ∈ [a , b].
probability
b bilit distribution
di t ib ti for
f random
d payoffs
ff ~
Then FA ( ~x ) FSD FB ( ~x ) if and only if
for all non decreasing utility functions U(
U(•)).
E A U(~
x) ≥ E B U(~
x)
Slide 04
04--34
Fin 501: Asset Pricing
Expected Utility & Stochastic Dominance
ƒ Theorem 3. 3 : Let FA ( ~x ) , FB ( ~x ) , be two cumulative
probabilityy distribution
p
x defined on [a , b] .
for random payoffs ~
Then, FA ( ~x ) SSD FB ( ~x ) if and only if E A U(~x) ≥ E B U(~x)
f all
for
ll non decreasing
d
andd concave U.
Slide 04
04--35
Fin 501: Asset Pricing
Digression:
S bj ti EU Theory
Subjective
Th
ƒ Derive perceived probability from preferences!
ƒ Set S of prizes/consequences
ƒ Set Z of states
ƒ Set of functions f(s) ∈ Z
Z, called acts (consumption plans)
ƒ Seven SAVAGE Axioms
ƒ Goes
G
bbeyond
d scope off thi
this course.
Slide 04
04--36
Fin 501: Asset Pricing
Digression:
Ell b
Ellsberg
Paradox
P d
ƒ 10 balls in an urn
Lottery 1: win $100 if you draw a red ball
L tt
Lottery
2:
2 win
i $100 if you ddraw a bl
blue bball
ll
ƒ Uncertainty: Probability distribution is not known
ƒ Risk:
Probability distribution is known
(5 balls are red, 5 balls are blue)
ƒ Individuals are “uncertainty/ambiguity averse”
(non-additive probability approach)
Slide 04
04--37
Fin 501: Asset Pricing
Digression: Prospect
P
t Th
Theory
ƒ Value function ((over ggains and losses))
ƒ Overweight low probability events
ƒ Experimental evidence
Slide 04
04--38
Fin 501: Asset Pricing
I diff
Indifference
curves
x2
Any point in
this plane is
a particular
lottery.
Where is the
set of riskffree
lotteries?
45°
x1
If x1=x
x2,
then the
lottery
contains no
risk.Slide 04
04--39
Fin 501: Asset Pricing
I diff
Indifference
curves
x2
π
z
45°
z
x1
Where is the
set of lotteries
with expected
prize E[L]=z?
It's a
straight line,
and the
slope is
given by the
relative
probabilities
of the two
states.
Slide 04
04--40
Fin 501: Asset Pricing
Suppose the
agent is risk
averse. Where
is the set of
lotteries which
are indifferent
to (z,z)?
That's not
right! Note
th t there
that
th
are risky
lotteries
with smaller
expected
x1 prize and
which are
Slide 04
04--41
preferred.
I diff
Indifference
curves
???
x2
π
z
45°
z
Fin 501: Asset Pricing
I diff
Indifference
curves
x2
π
z
So the
indifference
curve must be
tangent to the
iso-expectedprize line.
This is a direct
p
of
implication
risk-aversion
alone.
45°
z
x1
Slide 04
04--42
Fin 501: Asset Pricing
I diff
Indifference
curves
x2
π
But riskrisk
aversion does
not imply
convexity.
This
indifference
d ff
curve is also
compatible
p
with riskaversion.
z
45°
z
x1
Slide 04
04--43
Fin 501: Asset Pricing
I diff
Indifference
curves
x2
∇ V(z,z)
π
z
The tangency
implies that
the gradient
of V at the
point (z,z) is
collinear to π.
Formally,
∇ V(z,z) = λπ,
for some λ>0.
45°
z
x1
Slide 04
04--44
Fin 501: Asset Pricing
C i
Certainty
Equivalent
E i l andd Risk
Ri k Premium
P
i
(3.6)
(3.7)
~
~
EU(Y + Z ) = U(Y + CE(Y, Z ))
~
~
= U(Y +E Z - Π(Y, Z ))
Slide 04
04--45
Fin 501: Asset Pricing
C i
Certainty
Equivalent
E i l andd Risk
Ri k Premium
P
i
U(Y)
U( Y0 + Z 2 )
~
U( Y0 + E( Z))
~
EU( Y0 + Z)
U( Y0 + Z1 )
~
CE( Z)
Y0
Y0 + Z1
Π
~
~
CE ( Y0 + Z) Y0 + E(Z)
Y0 + Z 2
Y
Figure
g
3-3 Certainty
y Equivalent
q
and Risk Premium: An Illustration
Slide 04
04--46
Fin 501: Asset Pricing
O
Overview:
i
Ri k Preferences
Risk
P f
1. State
State--byby-state dominance
2. Stochastic dominance
3. vNM expected utility theory
a) Intuition
b) Axiomatic
A i
ti foundations
f d ti
[DD4]
[L4]
[DD3]
4. Risk aversion coefficients and portfolio choice [DD4,5,L4]
5 Prudence coefficient and precautionary savings [DD5]
5.
6. Mean
Mean--variance preferences
[L4.6]
Slide 04
04--47
Fin 501: Asset Pricing
M
Measuring
i Ri
Riskk aversion
i
U(W)
tangent lines
li
U(Y+h)
U[0.5(Y+h)+0.5(Y-h)]
0.5U(Y+h)+0.5U(Y-h)
U(Y-h)
Y-h
Y
Y+h
W
Figure 3-1 A Strictly Concave Utility Function
Slide 04
04--48
Fin 501: Asset Pricing
A
ArrowArrow
-Pratt
P tt measures off risk
i k aversion
i
and their interpretations
ƒ absolute risk aversion = - U" ((Y)) ≡ R A (Y)
U' (Y)
ƒ relative risk aversion
Y U" (Y)
=≡ R R (Y)
U' ((Y))
ƒ risk tolerance
=
Slide 04
04--49
Fin 501: Asset Pricing
Ab l t risk
Absolute
i k aversion
i coefficient
ffi i t
π
Y+h
1−π
Y-h
Y
Slide 04
04--50
Fin 501: Asset Pricing
R l ti risk
Relative
i k aversion
i coefficient
ffi i t
π
Y(1+θ)
1−π
Y(1-θ)
Y
Homework: Derive this result.
Slide 04
04--51
Fin 501: Asset Pricing
CARA and
d CRRA
CRRA--utility
tilit functions
f ti
ƒ Constant Absolute RA utility function
ƒ Constant
C
R
Relative
l i RA utility
ili function
f
i
Slide 04
04--52
Fin 501: Asset Pricing
Investor ’s
’ Levell off Relative
l i Risk
i k Aversion
A
i
1− γ
( Y + CE)
1- γ
Y=0
Y=100,000
( Y + 50,000 )1 − γ 12 ( Y + 100,000 )1 − γ
=
+
1- γ
1- γ
1
2
γ=0
γ=1
γ=2
γ=5
γ = 10
γ = 20
γ = 30
CE = 75,000 (risk neutrality)
CE = 70,711
CE = 66,246
66 246
CE = 58,566
CE = 53,991
CE = 51,858
CE = 51,209
γ=5
CE = 66,530
Slide 04
04--53
Fin 501: Asset Pricing
Ri k aversion
Risk
i andd Portfolio
P tf li Allocation
All ti
ƒ No
N savings
i
ddecision
i i (consumption
(
i occurs only
l at t=1)
1)
ƒ Asset structure
ƒ One risk free bond with net return rf
ƒ One risky asset with random net return r (a =quantity of risky assets)
Slide 04
04--54
Fin 501: Asset Pricing
• Theorem 4.1: Assume U'( ) > 0,, and U"( ) < 0 and let â
denote the solution to above problem. Then
â > 0 if and only if E~r > rf
â = 0 if and only if E~r = r
f
â < 0 if and only if E~r < rf .
• Define W(a ) = E{U (Y0 (1 + rf ) + a (~r − rf ))}. The FOC can
then be written W′(a ) = E[U′(Y0 (1 + rf ) + a (~r − rf ))(~r − rf )] = 0 .
By risk aversion (U''<0)
(U''<0), W′′(a ) = E[U′′(Y0 (1 + rf ) + a (~r − rf ))(~r − rf )2 ]
< 0, that is, W'(a) is everywhere decreasing. It follows that
â will be positive if and only if W′(0) = U′(Y0 (1 + rf ))E(~r − rf ) > 0
(since then a will have to be increased from its value of 0 to
achieve equality in the FOC). Since U' is always strictly
positive this implies aâ > 0 if and only if E(~r − rf ) > 0 .
positive,
W’(a)
W
(a)
The other assertion follows similarly.
a
0
Slide 04
04--55
Fin 501: Asset Pricing
P tf li as wealth
Portfolio
lth changes
h
• Theorem 4.4 (Arrow, 1971): Let
solution to max-problem above; then:
be the
(i)
(ii)
(iii) .
Slide 04
04--56
Fin 501: Asset Pricing
P tf li as wealth
Portfolio
lth changes
h
• Theorem 4.5 (Arrow 1971): If, for all wealth levels Y,
(i)
(ii)
( )
(iii)
where
η=
da/a (elasticity)
dY /Y
Slide 04
04--57
Fin 501: Asset Pricing
L utility
Log
tilit & Portfolio
P tf li All
Allocation
ti
U(Y) = ln Y.
2 states
states, where r2 > rf > r1
Constant fraction of wealth is invested in risky asset!
Slide 04
04--58
Fin 501: Asset Pricing
Portfolio of risky assets as wealth changes
Now -- many risky assets
ƒ Theorem 4.6 (Cass and Stiglitz,1970). Let the vector
⎡ â1 ( Y0 ) ⎤
p
y invested in the J risky
y assets if
⎢ . ⎥ denote the amount optimally
⎢
⎥
⎢ . ⎥
⎡ â1 ( Y0 ) ⎤ ⎡ a1 ⎤
⎢
⎥
â
(
Y
)
⎣ J 0 ⎦
⎢ . ⎥ ⎢.⎥
⎥ = ⎢ ⎥f ( Y0 )
the wealth level is Y0.. Then ⎢
⎢ . ⎥ ⎢.⎥
⎢
⎥ ⎢ ⎥
â
(
Y
)
J
0
⎣
⎦ ⎣a J ⎦
if andd only
l if either
ith
Δ
(i) U ' ( Y0 ) = ( θY0 + κ )
(ii) U ' ( Y ) = ξe − νY0 .
or
0
ƒ In words, it is sufficient to offer a mutual fund.
Slide 04
04--59
Fin 501: Asset Pricing
LRT/HARA--utility
LRT/HARA
tilit functions
f ti
ƒ Linear Risk Tolerance/hyperbolic absolute risk aversion
ƒ Special Cases
ƒ B=0, A>0 CARA
ƒ B ≠ 0, ≠1 Generalized Power
ƒ B=1 Log utility
u(c) =ln (A+Bc)
ƒ B=-1 Quadratic Utility
u(c)=-(A-c)2
ƒ B ≠ 1 A=0
A 0 CRRA Utilit
Utility function
f ti
Slide 04
04--60
Fin 501: Asset Pricing
O
Overview:
i
Ri k Preferences
Risk
P f
1. State
State--byby-state dominance
2. Stochastic dominance
3. vNM expected utility theory
a) Intuition
b) Axiomatic
A i
ti foundations
f d ti
[DD4]
[L4]
[DD3]
4. Risk aversion coefficients and portfolio choice [DD4,5,L4]
5 Prudence coefficient and precautionary savings [DD5]
5.
6. Mean
Mean--variance preferences
[L4.6]
Slide 04
04--61
Fin 501: Asset Pricing
I t d i Savings
Introducing
S i
• Introduce savings decision: Consumption at tt=00 and tt=11
• Asset structure 1:
– risk free bond Rf
– NO risky asset with random return
– Increase Rf:
– Substitution effect: shift consumption from t=0 to t=1
⇒ save more
– Income
I
effect:
ff t agentt is
i “effectively
“ ff ti l richer”
i h ” andd wants
t to
t
consume some of the additional richness at t=0
⇒ save less
– For log-utility (γ=1) both effects cancel each other
Slide 04
04--62
Fin 501: Asset Pricing
P d
Prudence
and
dP
Pre-cautionary
Preti
Savings
S i
• IIntroduce
t d
savings
i
decision
d ii
Consumption at t=0 and t=1
• Asset structure 2:
– NO risk free bond
– One risky asset with random gross return R
Slide 04
04--63
Fin 501: Asset Pricing
P d
Prudence
and
dS
Savings
i
B
Behavior
h i
ƒ Risk aversion is about the willingness to insure …
ƒ … but not about its comparative statics.
ƒ How
H ddoes th
the behavior
b h i off an agentt change
h
when
h
we marginally increase his exposure to risk?
ƒ An old hypothesis (going back at least to
J.M.Keynes) is that people should save more now
when they face greater uncertainty in the future.
ƒ The idea is called precautionary saving and has
intuitive appeal.
Slide 04
04--64
Fin 501: Asset Pricing
P d
Prudence
and
dP
Pre-cautionary
Prei
Savings
S i
ƒ Does not directlyy follow from risk aversion alone.
ƒ Involves the third derivative of the utility function.
ba ((1990)
990) de
defines
es abso
absolute
ute p
prudence
ude ce as
ƒ Kimball
P(w) := –u'''(w)/u''(w).
p
ƒ Precautionaryy savingg if anyy onlyy if theyy are prudent.
ƒ This finding is important when one does comparative
statics of interest rates.
ƒ Prudence seems uncontroversial, because it is weaker
than DARA.
Slide 04
04--65
Fin 501: Asset Pricing
Pre--cautionary Saving (extra material)
Pre
(+)
((-))
in s
Is saving s increasing/decreasing in risk of R?
Is RHS increasing/decreasing is riskiness of R?
Is U’() convex/concave?
Depends on third derivative of U()!
N.B: For U(c)=ln c, U’(sR)R=1/s does not depend on R.
Slide 04
04--66
Fin 501: Asset Pricing
P -cautionary
PrePre
ti
Saving
S i (extra material)
2 effects: Tomorrow consumption is more volatile
• consume more today, since it’s not risky
• save more for precautionary reasons
~
~
R
R
Theorem 4.7 (Rothschild and Stiglitz,1971) : Let A , B
be
~ two
~ return distributions with identical means such that
RB = RA + e, (where e is white noise) and let s and s be,
A
B
respectively, the savings out of Y0 corresponding to the
~
~
return distributions R
and
.B
R
A
If R ' R ( Y ) ≤ 0 and RR(Y) > 1, then sA < sB ;
If R ' R ( Y ) ≥ 0 and RR(Y) < 1, then sA > sB
Slide 04
04--67
Fin 501: Asset Pricing
Prudence & PrePre-cautionary Saving
P(c) =
− U ' ' ' (c)
U ' ' (c)
− cU ' ' ' (c)
P(c)c =
U ' ' (c)
~ ~
ƒ Th
Theorem 4.8
4 8 : Let
L tR
, be
b two
t return
t
distributions
di t ib ti
suchh
A RB
~ SSD ~ , and let s and s be, respectively, the
that R
RB
A
B
A
savings out of Y0 corresponding to the return distributions
~
~
R Aand R B. Then,
s A ≥ s B iff cP(c) ≤ 2, and conversely,
iff cP(c) > 2
sA < sB
Slide 04
04--68
Fin 501: Asset Pricing
O
Overview:
i
Ri k Preferences
Risk
P f
1. State
State--byby-state dominance
2. Stochastic dominance
3. vNM expected utility theory
a) Intuition
b) Axiomatic
A i
ti foundations
f d ti
[DD4]
[L4]
[DD3]
4. Risk aversion coefficients and portfolio choice [DD4,5,L4]
5 Prudence coefficient and precautionary savings [DD5]
5.
6. Mean
Mean--variance preferences
[L4.6]
Slide 04
04--69
Fin 501: Asset Pricing
M -variance
MeanMean
i
Preferences
P f
ƒ Early researchers in finance, such as Markowitz
and Sharpe, used just the mean and the variance
of the return rate of an asset to describe it.
ƒ Mean-variance characterization is often easier
than using an vNM utility function
ƒ But is it compatible with vNM theory?
ƒ The answer is yes … approximately … under
some conditions.
Slide 04
04--70
Fin 501: Asset Pricing
M -Variance:
MeanMean
V i
quadratic
d ti utility
tilit
Suppose utility is quadratic, u(y) = ay–by2.
Expected utility is then
2
E [u ( y )] = aE [ y ] − bE [ y ]
= aE
E [ y ] − b ( E [ y ] 2 + var[[ y ]).
])
Thus, expected utility is a function of the
Thus
mean, E[y], and the variance, var[y], only.
Slide 04
04--71
Fin 501: Asset Pricing
M -Variance:
MeanMean
V i
joint
j i t normals
l
ƒ Suppose all lotteries in the domain have normally
distributed prized. (independence is not needed).
ƒ This
Thi requires
i an infinite
i fi i state space.
ƒ Any linear combination of normals is also normal.
ƒ The normal distribution is completely described by its
first two moments.
ƒ Hence, expected utility can be expressed as a function of
just these two numbers as well.
Slide 04
04--72
Fin 501: Asset Pricing
MeanMean
e -V
Variance:
ce:
linear distribution classes
ƒ Generalization of joint nomarls.
ƒ Consider a class of distributions F1, …, Fn with the
f ll i property:
following
ƒ for all i there exists (m,s) such that Fi(x) = F1(a+bx) for all x.
ƒ Thi
This iis called
ll d a li
linear di
distribution
t ib ti class.
l
ƒ It means that any Fi can be transformed into an Fj by an
appropriate shift (a) and stretch (b).
(b)
ƒ Let yi be a random variable drawn from Fi. Let μi = E{yi}
and σi2 = E{(yi–μi)2} denote the mean and the var of yi.
Slide 04
04--73
Fin 501: Asset Pricing
MeanMean
e -V
Variance:
ce:
linear distribution classes
ƒ Define then the random variable x = (yi–μi)/σi. We
denote the distribution of x with F.
ƒ Note that the mean of x is 0 and the variance is 1, and F
is p
part of the same linear distribution class.
ƒ Moreover, the distribution of x is independent of which
i we start with.
Ê
We want to evaluate the expected utility of yi ,
+∞
∫− ∞ v(z)dFi (z).
Slide 04
04--74
Fin 501: Asset Pricing
MeanMean
e -V
Variance:
ce:
linear distribution classes
But yi = μi + σi x, thus
+∞
+∞
∫− ∞ v(z)dFi (z) = ∫− ∞ v(μi + σi z)dF (z)
=: u(μ i , σi ).
The expected utility of all random variables
d
drawn
ffrom the
h same linear
l
distribution
d
b
class can be expressed as functions of the
mean and the standard deviation only.
only
Slide 04
04--75
Fin 501: Asset Pricing
M -Variance:
MeanMean
V i
small
ll risks
ik
ƒ Justification for mean-variance for the case of small risks.
ƒ use a second order (local) Taylor approximation of vNM
U(c).
( ) is concave,, second order Taylor
y approximation
pp
is a
ƒ If U(c)
quadratic function with a negative coefficient on the
quadratic term.
ƒ Expectation
E
i off a quadratic
d i utility
ili function
f
i can be
b
evaluated with the mean and variance.
Slide 04
04--76
Fin 501: Asset Pricing
Mean--Variance: small risks
Mean
ƒ Let f : R Æ R be a smooth function.
function The Taylor
approximation is
( x − x0 )1
( x − x0 )2
+ f ' ' ( x0 )
+
f ( x ) ≈ f ( x0 ) + f ' ( x0 )
1!
2!
( x − x0 )3
+L
f ' ' ' ( x0 )
3!
ƒ Use Taylor approximation for E[u(x)].
Slide 04
04--77
Fin 501: Asset Pricing
Mean--Variance: small risks
Mean
ƒ Since E[u(w+x)] = u(cCE), this simplifies to
var( x)
w − cCE ≈ RA( w)
.
2
Ê
w – cCE is the risk premium.
Ê
We see here
W
h
that
th t the
th risk
i k premium
i
is
i
approximately a linear function of the
variance of the additive risk
risk, with the
slope of the effect equal to half the
coefficient of absolute risk.
Slide 04
04--78
Fin 501: Asset Pricing
Mean--Variance: small risks
Mean
ƒ The same exercise can be done with a multiplicative risk.
ƒ Let y = gw, where g is a positive random variable with
unit mean.
ƒ Doing
D i the
th same steps
t
as before
b f
leads
l d to
t
var( g )
1 − κ ≈ RR ( w)
,
2
where κ is the certainty equivalent growth rate,
u( w) = E[u(gw)].
u(κw)
E[u(gw)]
The coefficient of relative risk aversion is
relevant for multiplicative risk
risk, absolute risk
aversion for additive risk.
Slide 04
04--79
12:30 Lecture
Risk Aversion
Ê
Fin 501: Asset Pricing
E t material
Extra
t i l follows!
f ll
!
Slide 04
04--80
Fin 501: Asset Pricing
Joint savingsaving-portfolio problem
• Consumption at t=0 and t=1. (savings decision)
• Asset structure
– One risk free bond with net return rf
– One riskyy asset ((a = quantity
q
y of riskyy assets))
max U ( Y0 − s ) + δEU ( s(1 + rf ) + a ( ~r − rf ))
{ a ,s }
FOC:
s:
a:
((4.7))
U’(ct)
= δ E[U’(ct+1)(1+rf)]
E[U’(c
E[U
(ct+1)(r-r
)(r rf)] = 0
Slide 04
04--81
Fin 501: Asset Pricing
for CRRA utility functions
s:
a:
( Y0 − s) − γ ( −1) + δE ([s(1 + rf ) + a ( ~r − rf )]− γ (1 + rf ) ) = 0
[
]
E (s(1 + rf ) + a ( ~r − rf )) − γ ( ~r − rf ) = 0
Where s is total saving and a is amount invested in risky asset.
asset
Slide 04
04--82
Fin 501: Asset Pricing
MultiM lti-period
Multi
i d Setting
S tti
ƒ Canonical framework (exponential discounting)
U(c) = E[ ∑ βt u(ct)]
ƒ prefers earlier uncertainty resolution if it affect action
ƒ indifferent, if it does not affect action
ƒ Time-inconsistent (hyperbolic discounting)
Special case: β−δ formulation
U(c) = E[u(c0) + β ∑ δt u(ct)]
ƒ Preference for the timing of uncertainty resolution
recursive utility formulation (Kreps-Porteus
(Kreps Porteus 1978)
Slide 04
04--83
Fin 501: Asset Pricing
MultiM lti-period
Multi
i d Portfolio
P tf li Choice
Ch i
Theorem 4.10 ((Merton, 1971):
) Consider the above canonical
multi-period consumption-saving-portfolio allocation problem.
Suppose U() displays CRRA, rf is constant and {r} is i.i.d.
Then a/st is time invariant.
invariant
Slide 04
04--84
Fin 501: Asset Pricing
Preference
e e e ce for
o thee
timing of uncertainty resolution
Digression:
g ess o :
π
0
$100
$150
$100
$ 25
$150
$100
Kreps-Porteus
π
$ 25
Early (late) resolution if W(P1,…) is convex (concave)
Marginal rate of temporal substitution Ÿ risk aversion
Slide 04
04--85