markus k. brunnermeier lecture 4: risk preferences
TRANSCRIPT
FIN501 Asset PricingLecture 04 Risk Prefs & EU (1)
LECTURE 4: RISK PREFERENCES & EXPECTED UTILITY THEORY
Markus K. Brunnermeier
FIN501 Asset PricingLecture 04 Risk Prefs & EU (2)
specifyPreferences &
Technology
observe/specifyexisting
Asset Prices
State Prices q(or stochastic discount
factor/Martingale measure)
derivePrice for (new) asset
โข evolution of statesโข risk preferencesโข aggregation
absolute asset pricing
relativeasset pricing
NAC/LOOP
LOOP
NAC/LOOP
Only works as long as market completeness doesnโt change
deriveAsset Prices
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Overview: Risk Preferences
1. State-by-state dominance
2. Stochastic dominance [DD4]
3. vNM expected utility theorya) Intuition [L4]
b) Axiomatic foundations [DD3]
4. Risk aversion coefficients and portfolio choice [DD5,L4]
5. Uncertainty/ambiguity aversion
6. Prudence coefficient and precautionary savings [DD5]
7. Mean-variance preferences [L4.6]
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State-by-state Dominance
- State-by-state dominance is an incomplete ranking
- Investment 3 state-by-state dominates investment 1
- Recall: ๐ฆ, ๐ฅ โ โ๐
๐ฆ โฅ ๐ฅ โ ๐ฆ๐ โฅ ๐ฅ๐ for each ๐ = 1,โฆ , ๐
๐ฆ > ๐ฅ โ ๐ฆ โฅ ๐ฅ, ๐ฆ โ ๐ฅ
๐ฆ โซ ๐ฅ โ ๐ฆ๐ > ๐ฅ๐ for each ๐ = 1,โฆ , ๐
๐ = ๐ ๐ = ๐
Cost Payoff ๐1 = ๐2 = 1/2
๐ = 1 ๐ = 2
Investment 1 -1000 1050 1200
Investment 2 -1000 500 1600
Investment 3 -1000 1050 1600
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State-by-state Dominance (ctd.)
โข Investment 1 mean-variance dominates 2
โข But, investment 3 does not mean-variance dominate 1
๐ = ๐ ๐ = ๐
Cost Return ๐1 = ๐2 = 1/2 E[Return] ๐
๐ = 1 ๐ = 2
Investment 1 -1000 + 5% + 20% 12.5% 7.5%
Investment 2 -1000 - 50% + 60% 5% 55%
Investment 3 -1000 + 5% + 60% 32.5% 27.5%
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State-by-state Dominance (ctd.)
โข What is the trade-off between risk and expected return?
โข Investment 4 has a higher Sharpe ratio ๐ธ ๐ โ๐๐
๐than investment 5 for ๐๐ = 0
๐ = ๐ ๐ = ๐
Cost Return ๐1 = ๐2 = 1/2 E[Return] ๐
๐ = 1 ๐ = 2
Investment 4 -1000 + 3% + 5% 4.0% 1.0%
Investment 5 -1000 + 3% + 8% 5.5% 2.5%
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Overview: Risk Preferences
1. State-by-state dominance
2. Stochastic dominance [DD4]
3. vNM expected utility theorya) Intuition [L4]
b) Axiomatic foundations [DD3]
4. Risk aversion coefficients and portfolio choice [DD5,L4]
5. Prudence coefficient and precautionary savings [DD5]
6. Mean-variance preferences [L4.6]
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Stochastic Dominance
โข No state-space โ probabilities are not assigned specific states
Only applicable for final payoff gamble โข Not for stocks/lotteries that form a portfolio (whose payoff is final)
Random variables before introduction of (ฮฉ, โฑ, ๐)โข Still incomplete ordering
โMore completeโ than state-by-state ordering State-by-state dominance โ stochastic dominance Risk preference not needed for ranking!
โข independently of the specific trade-offs (between return, risk and other characteristics of probability distributions) represented by an agent's utility function. (โrisk-preference-freeโ)
โข Next Section: Complete preference ordering and utility representations
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From payoffs per state to probability
state ๐๐ ๐๐ ๐๐ ๐๐ ๐๐
probability ๐1 ๐2 ๐3 ๐4 ๐5
Payoff x 10 10 20 20 20
Payoff y 10 20 20 20 30
payoff 10 20 30
Prob x ๐10 = ๐1 + ๐2 ๐20 = ๐3 + ๐4 + ๐5 ๐30 = 0
Prob y ๐10 = ๐1 ๐20 = ๐2 + ๐3 + ๐4 ๐30 = ๐5
Expressed in โprobability lotteriesโ โ only useful for final payoffs (since some cross correlation information ins lost)
Preference ๐ฅ โป ๐ฆ โ โ๐ expressed in probabilities ๐๐ฅ โป ๐๐ฆ
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0 10 100 2000
0.1
0.4
0.2
0.3
0.5
0.6
0.7
0.8
0.9
1.0
F1
and F2
F1
F2
Payoff
obabilityPr
ExpectedPayoff
Standard Deviation
Payoffs 10 100 2000
Probability 1 .4 .6 0 64 44
Probability 2 .4 .4 .2 444 779
Note: Payoff of $10 is equally likely, but can bein different states of the world
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โข Definition: Let ๐น๐ด ๐ฅ , ๐น๐ต ๐ฅ , respectively, represent the cumulative distribution functions of two random variables (cash payoffs) that, without loss of generality assume values in the interval ๐, ๐ . We say that ๐น๐ด ๐ฅ first order stochastically dominates (FSD) ๐น๐ต ๐ฅ
if and only if for all ๐ฅ โ ๐, ๐๐น๐ด ๐ฅ โค ๐น๐ต ๐ฅ
Homework: Provide an example which can be ranked according to FSD, but not according to state dominance.
First Order Stochastic Dominance
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X
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
FA
FB
First Order Stochastic Dominance
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0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 1 2 3 4 5 6 7 8 9 10 11 12 13
investment 3
investment 4
CDFs of investment 3 and 4
Payoff 1 4 5 6 8 12
Probability 3 0 .25 0.50 0 0 .25
Probability 4 .33 0 0 .33 .33 0
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โข Definition: Let ๐น๐ด ๐ฅ , ๐น๐ต ๐ฅbe two cumulative probability distribution for random payoffs in ๐, ๐ . We say that ๐น๐ด ๐ฅ second order stochastically dominates (SSD) ๐น๐ต ๐ฅif and only if for any ๐ฅ โ ๐, ๐
โโ
๐ฅ
๐น๐ต ๐ก โ ๐น๐ด ๐ก ๐๐ก โฅ 0
(with strict inequality for some meaningful interval of values of t).
Second Order Stochastic Dominance
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f xA
f xB
~,x Payoff x f x dx x f x dxA B
Mean Preserving Spread
๐ฅ๐ต = ๐ฅ๐ด + ๐ง
(where ๐ง is independent and has zero mean)
Mean Preserving Spread:
(for normal distributions)
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โข Theorem: Let ๐น๐ด ๐ฅ and ๐น๐ต ๐ฅ be two distribution functions defined on the same state space with identical means. Then the following statements are equivalent :
๐น๐ด ๐ฅ SSD ๐น๐ต ๐ฅ
๐น๐ต ๐ฅ is a mean-preserving spread of ๐น๐ด ๐ฅ
Mean Preserving Spread & SSD
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Overview: Risk Preferences
1. State-by-state dominance
2. Stochastic dominance [DD4]
3. vNM expected utility theorya) Intuition [L4]
b) Axiomatic foundations [DD3]
4. Risk aversion coefficients and portfolio choice [DD5,L4]
5. Prudence coefficient and precautionary savings [DD5]
6. Mean-variance preferences [L4.6]
FIN501 Asset PricingLecture 04 Risk Prefs & EU (18)
A Hypothetical Gamble
โข Suppose someone offers you this gamble: "I have a fair coin here. I'll flip it, and if it's tails I
pay you $1 and the gamble is over. If it's heads, I'll flip again. If it's tails 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 turns up tails."
โข Sounds like a good deal. After all, you can't lose. So here's the question: How much are you willing to pay to take this
gamble?
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Proposal 1: Expected Value
โข With probability 1
2you get $1,
1
2
1ร 20
โข With probability 1
4you get $2,
1
2
2ร 21
โข With probability 1
8you get $4,
1
2
3ร 22
โข โฆ
The expected payoff is given by the sum of all these terms, i.e.
๐ก=1
โ1
2
๐ก
ร 2๐กโ1 =
๐ก=1
โ1
2= โ
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St. Petersburg Paradox
โข You should pay everything you own and more to purchase the right to take this gamble!
โข Yet, in practice, no one is prepared to pay such a high price. Why?
โข Even though the expected payoff is infinite, the distribution of payoffs is not attractiveโฆ
with 93% probability we get $8 or less;
with 99% probability we get $64 or less
0
0.1
0.2
0.3
0.4
0.5
0 20 40 60
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Proposal 2
โข 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.]
๐ก=1
โ1
2
๐ก
ร ln 2๐กโ1 = ln 2 < โ
According to this Bernoulli would pay at most
๐ln 2 = 2 to participate in this gamble
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Representation of Preferences
A preference ordering is (i) complete, (ii) transitive, (iii) continuous and [(iv) relatively stable] can be represented by a utility function, i.e.
๐0, ๐1, โฆ , ๐๐ โป ๐0โฒ , ๐1
โฒ , โฆ , ๐๐โฒ
โ ๐ ๐0, ๐1, โฆ , ๐๐ > ๐ ๐0โฒ , ๐1
โฒ , โฆ , ๐๐โฒ
(preference ordering over lotteries โ(๐ + 1)-dimensional space)
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Indifference curvesin โ2 (for ๐ = 2)
45ยฐ
๐1
๐2
๐
๐ง
๐ง
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Preferences over Prob. Distributions
โข Consider ๐0 fixed, ๐1 is a random variable
โข Preference ordering over probability distributions
โข Let
๐ be a set of probability distributions with a finite support over a set ๐,
โฝ preference ordering over ๐ (that is, a subset of ๐ ร ๐)
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Prob. Distributions
โข S states of the worldโข Set of all possible lotteries
๐ = ๐ โ โ๐ ๐ ๐ โฅ 0, ๐ ๐ = 1
โข Space with S dimensions
โข Can we simplify the utility representation of preferences over lotteries?
โข Space with one dimension โ incomeโข We need to assume further axioms
FIN501 Asset PricingLecture 04 Risk Prefs & EU (26)
Expected Utility Theory
โข A binary relation that satisfies the following three axioms if and only if there exists a function ๐ข โ such that
๐ โป ๐ โ ๐ ๐ ๐ข ๐ > ๐ ๐ ๐ข ๐
i.e. preferences correspond to expected utility.
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vNM Expected Utility Theory
โข Axiom 1 (Completeness and Transitivity): Agents have preference relation over ๐ (repeated)
โข Axiom 2 (Substitution/Independence) For all lotteries ๐, ๐, ๐ โ ๐ and ๐ผ โ 0,1 ,
๐ โฝ ๐ โ ๐ผ๐ + 1 โ ๐ผ ๐ โฝ ๐ผ๐ + 1 โ ๐ผ ๐
โข Axiom 3 (Archimedian/Continuity) For all lotteries ๐, ๐, ๐ โ ๐ if ๐ โป ๐ โป ๐ then there
exists ๐ผ, ๐ฝ โ 0,1 such that,๐ผ๐ + 1 โ ๐ผ ๐ โป ๐ โป ๐ฝ๐ + 1 โ ๐ฝ ๐
Problem: ๐ you get $100 for sure, ๐ you get $ 10 for sure, ๐ you are killed
FIN501 Asset PricingLecture 04 Risk Prefs & EU (28)
Independence Axiom
โข Independence of irrelevant alternatives:
๐ โฝ ๐ โ
๐ ๐๐
๐ ๐
๐
โฝ
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Allais Paradox โViolation of Independence Axiom
10โ
0
15โ
0
9%10%โบ
FIN501 Asset PricingLecture 04 Risk Prefs & EU (30)
Allais Paradox โViolation of Independence Axiom
10โ
0
15โ
0
9%10%โบ
10โ
0
15โ
0
90%100%
โป
FIN501 Asset PricingLecture 04 Risk Prefs & EU (31)
10โ
0
15โ
0
9%10%โบ
10โ
0
15โ
0
90%100%
โป10%
0
10%
0
Allais Paradox โViolation of Independence Axiom
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vNM EU Theorem
โข A binary relation that satisfies the axioms 1-3 if and only if there exists a function ๐ข โ such that
๐ โป ๐ โ ๐ ๐ ๐ข ๐ > ๐ ๐ ๐ข ๐
i.e. preferences correspond to expected utility.
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โข The shape of the von Neumann Morgenstern (NM) utility function reflects risk preference
โข Consider lottery with final wealth of ๐1 or ๐2
๐ธ ๐ข ๐
Risk-Aversion and Concavity
๐
๐ข ๐
๐1 ๐2
๐ข ๐1
๐ข ๐2
๐ธ ๐
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โข Risk-aversion means that the certainty equivalent is smaller than the expected prize. We conclude that a risk-
averse vNM utility function must be concave. ๐ธ ๐ข ๐
Risk-aversion and concavity
๐ฅ
๐ข ๐ฅ
๐1 ๐2
๐ข ๐1
๐ข ๐2
๐ธ ๐
๐ขโ1 ๐ธ ๐ข ๐
๐ข ๐ธ ๐
FIN501 Asset PricingLecture 04 Risk Prefs & EU (35)
Theorem:
โข Let ๐ โ be a concave function on the interval ๐, ๐ , and ๐ฅ be a random variable such that
๐ ๐ฅ โ ๐, ๐ = 1
โข Suppose the expectations ๐ธ ๐ฅ and ๐ธ ๐ ๐ฅ exist; then
๐ธ ๐ ๐ฅ โค ๐ ๐ธ ๐ฅ
Furthermore, if ๐ โ is strictly concave, then the inequality is strict.
Jensenโs Inequality
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โข Theorem: Let ๐น๐ด ๐ฅ , ๐น๐ต ๐ฅ be two cumulative probability distribution for random payoffs ๐ฅ โ ๐, ๐ . Then ๐น๐ด ๐ฅFSD ๐น๐ต ๐ฅ if and only if ๐ธ๐ด[๐ข ๐ฅ ] โฅ ๐ธ๐ต[๐ข ๐ฅ ] for all non decreasing utility functions ๐ โ .
โข Theorem: Let ๐น๐ด ๐ฅ , ๐น๐ต ๐ฅ be two cumulative probability distribution for random payoffs ๐ฅ โ ๐, ๐ . Then ๐น๐ด ๐ฅSSD ๐น๐ต ๐ฅ if and only if ๐ธ๐ด[๐ข ๐ฅ ] โฅ ๐ธ๐ต[๐ข ๐ฅ ] for all non decreasing concave utility functions ๐ โ .
Expected Utility & Stochastic Dominance
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Certainty Equivalent and Risk Premium
๐ธ ๐ข ๐ + ๐ = ๐ข ๐ + ๐ถ๐ธ ๐, ๐
๐ธ ๐ข ๐ + ๐ = ๐ข ๐ + ๐ธ ๐ โ ฮ ๐, ๐
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Certainty Equivalent and Risk Premium
Certainty Equivalent and Risk Premium
U(Y)
Y
~
~ ~
P
)ZY(U 20 +
))Z~
(EY(U 0 +
)ZY(EU 0 +
)ZY(U 10 +
)Z~
(CE
0Y 10 ZY + )ZY(CE 0 + )Z(EY0 + 20 ZY +
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Utility Transformations
โข General utility function: Suppose ๐ ๐0, ๐1, โฆ , ๐๐ > ๐ ๐0
โฒ , ๐1โฒ , โฆ , ๐๐
โฒ represents complete, transitive,โฆ preference ordering,
then ๐ โ = ๐ ๐ โ , where ๐ โ is strictly increasingrepresents the same preference ordering
โข vNM utility function Suppose ๐ธ ๐ข ๐ represents preference ordering
satisfying vNM axioms,
then ๐ฃ ๐ = ๐ + ๐๐ข ๐ represents the same.โaffine transformationโ
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Overview: Risk Preferences
1. State-by-state dominance
2. Stochastic dominance [DD4]
3. vNM expected utility theorya) Intuition [L4]
b) Axiomatic foundations [DD3]
4. Risk aversion coefficients and portfolio choice [DD5,L4]
5. Uncertainty/ambiguity aversion
6. Prudence coefficient and precautionary savings [DD5]
7. Mean-variance preferences [L4.6]
FIN501 Asset PricingLecture 04 Risk Prefs & EU (41)
Measuring Risk aversionU(W)
WY-h Y Y+h
U(Y-h)
U[0.5(Y+h)+0.5(Y-h)]
0.5U(Y+h)+0.5U(Y-h)
tangent lines
U(Y+h)
A Strictly Concave Utility Function
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Arrow-Pratt Measures of Risk aversion
โข absolute risk aversion = โ๐ขโฒโฒ ๐
๐ขโฒ ๐โก ๐ ๐ด ๐
โข relative risk aversion = โ๐๐ขโฒโฒ ๐
๐ขโฒ ๐โก ๐ ๐ ๐
โข risk tolerance =1
๐ ๐ด
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Absolute risk aversion coefficient
๐ ๐ด = โ๐ขโฒโฒ ๐
๐ขโฒ ๐
๐ ๐, ฮ =1
2+
1
4ฮ๐ ๐ด ๐ + ๐ป๐๐
๐
๐ + ฮ
๐ โ ฮ
๐
1 โ ๐
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Relative risk aversion coefficient
๐ ๐ = โ๐ขโฒโฒ ๐
๐ขโฒ ๐๐
๐ ๐, ๐ =1
2+
1
4๐ฟ๐ ๐ ๐ + ๐ป๐๐
Homework: Derive this result.
๐
๐(1 + ๐ฟ)
๐(1 โ ๐ฟ)
๐
1 โ ๐
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CARA and CRRA-utility functions
โข Constant Absolute RA utility function๐ข ๐ = โ๐โ๐๐
โข Constant Relative RA utility function
๐ข ๐ =๐1โ๐พ
1 โ ๐พ, ๐พ โ 1
๐ข ๐ = ln ๐ , ๐พ = 1
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Level of Relative Risk Aversion
๐ + ๐ถ๐ธ 1โ๐พ
1 โ ๐พ=
12
๐ + 50000 1โ๐พ
1 โ ๐พ+
12
๐ + 100000 1โ๐พ
1 โ ๐พ
๐ = 0 = 0 CE = 75,000 (risk neutrality)
= 1 CE = 70,711
= 2 CE = 66,246
= 5 CE = 58,566
= 10 CE = 53,991
= 20 CE = 51,858
= 30 CE = 51,209
๐ = 100000 = 5 CE = 66,530
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Risk aversion and Portfolio Allocation
โข No savings decision (consumption occurs only at t=1)
โข Asset structure
One risk free bond with net return ๐๐
One risky asset with random net return ๐(a =quantity of risky assets)
max๐
๐ธ ๐ข ๐0 1 + ๐๐ + ๐ ๐ โ ๐๐
FOC โ ๐ธ ๐ขโฒ ๐0 1 + ๐๐ + ๐ ๐ โ ๐๐ ๐ โ ๐๐ = 0
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a
Wโ(a)
0
โข Theorem 4.1: Assume ๐โฒ > 0,๐โฒโฒ < 0 and let ๐ denote the solution to above problem. Then
๐ > 0 โ ๐ธ ๐ > ๐๐ ๐ = 0 โ ๐ธ ๐ = ๐๐ ๐ < 0 โ ๐ธ ๐ < ๐๐
โข Define ๐ ๐ = ๐ธ ๐ข ๐0 1 + ๐๐ + ๐ ๐ โ ๐๐ . The FOC can then be
written ๐โฒ ๐ = ๐ธ ๐ขโฒ ๐0 1 + ๐๐ + ๐ ๐ โ ๐๐ ๐ โ ๐๐ = 0.
โข By risk aversion ๐โฒโฒ ๐ = ๐ธ ๐ขโฒโฒ ๐0 1 + ๐๐ + ๐ ๐ โ ๐๐ ๐ โ ๐๐2
<
0, that is, ๐โฒ ๐ is everywhere decreasing It follows that ๐ will be positive โ ๐โฒ 0 > 0
โข Since ๐ขโฒ > 0 this implies that ๐ > 0 โ ๐ธ ๐ โ ๐๐ > 0 The other assertion follows similarly
Risk aversion and Portfolio Allocation
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Portfolio as wealth changes
โข Theorem (Arrow, 1971):Let ๐ = ๐ ๐0 be the solution to max-problem above; then:
i.๐๐ ๐ด
๐๐< 0 (DARA) โ
๐ ๐
๐๐0> 0
ii.๐๐ ๐ด
๐๐= 0 (CARA) โ
๐ ๐
๐๐0= 0
iii.๐๐ ๐ด
๐๐> 0 (IARA) โ
๐ ๐
๐๐0< 0
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Portfolio as wealth changes
โข Theorem (Arrow 1971): If, for all wealth levels Y,
i.๐๐ ๐
๐๐= 0 (CRRA) โ ๐ = 1
ii.๐๐ ๐
๐๐< 0 (DRRA) โ ๐ > 1
iii.๐๐ ๐
๐๐> 0 (IRRA) โ ๐ < 1
where ๐ = ๐๐ ๐
๐๐ ๐
FIN501 Asset PricingLecture 04 Risk Prefs & EU (51)
Log utility & Portfolio Allocation
๐ข ๐ = ln๐
๐ธ ๐ โ ๐๐
๐0 1 + ๐๐ + ๐ ๐ โ ๐๐= 0
๐
๐0=
1 + ๐๐ ๐ธ ๐ โ ๐๐
โ ๐1 โ ๐๐ ๐2 โ ๐๐> 0
2 states, where ๐2 > ๐๐ > ๐1
Constant fraction of wealth is invested in risky asset!Homework: show that this result holds for
โข any CRRA utility functionโข any distribution of r
FIN501 Asset PricingLecture 04 Risk Prefs & EU (52)
โข Theorem (Cass and Stiglitz,1970): Let the vector ๐1 ๐0
โฎ ๐๐ฝ ๐0
denote the
amount optimally invested in the ๐ฝ risky assets if the wealth level is ๐0.
Then ๐1 ๐0
โฎ ๐๐ฝ ๐0
=
๐1
โฎ๐๐ฝ
๐ ๐0 if and only if either
i. ๐ขโฒ(๐0) = ๐ต๐0 + ๐ถ ฮ orii. ๐ขโฒ ๐0 = ๐๐โ๐๐0
โข In words, it is sufficient to offer a mutual fund.
Risk aversion and Portfolio Allocation
FIN501 Asset PricingLecture 04 Risk Prefs & EU (53)
LRT/HARA-utility functions
โข Linear Risk Tolerance/hyperbolic absolute risk aversion
โ๐ขโฒโฒ ๐
๐ขโฒ ๐=
1
๐ด + ๐ต๐
โข Special Cases
๐ต = 0, ๐ด > 0 CARA ๐ข ๐ =1
๐ตโ1๐ด + ๐ต๐
๐ตโ1
๐ต
๐ต โ 0,โ 1 Generalized Power
โข ๐ต = 1 Log utility ๐ข ๐ = ln ๐ด + ๐ต๐
โข ๐ต = โ1 Quadratic Utility ๐ข ๐ = โ ๐ด โ ๐ 2
โข ๐ต โ 1, ๐ด = 0 CRRA Utility function ๐ข ๐ =1
๐ตโ1๐ต๐
๐ตโ1
๐ต
FIN501 Asset PricingLecture 04 Risk Prefs & EU (54)
Overview: Risk Preferences
1. State-by-state dominance
2. Stochastic dominance [DD4]
3. vNM expected utility theorya) Intuition [L4]
b) Axiomatic foundations [DD3]
4. Risk aversion coefficients and portfolio choice [DD5,L4]
5. Uncertainty/ambiguity aversion
6. Prudence coefficient and precautionary savings [DD5]
7. Mean-variance preferences [L4.6]
FIN501 Asset PricingLecture 04 Risk Prefs & EU (55)
Digression: Subjective EU Theory
โข Derive perceived probability from preferences!
Set ๐ of prizes/consequences
Set ๐ of states
Set of functions ๐ ๐ โ ๐, called acts (consumption plans)
โข Seven SAVAGE Axioms
Goes beyond scope of this course.
FIN501 Asset PricingLecture 04 Risk Prefs & EU (56)
Digression: Ellsberg Paradox
โข 10 balls in an urn Lottery 1: win $100 if you draw a red ballLottery 2: win $100 if you draw a blue ball
โข Uncertainty: Probability distribution is not known
โข Risk: Probability distribution is known(5 balls are red, 5 balls are blue)
โข Individuals are โuncertainty/ambiguity averseโ(non-additive probability approach)
FIN501 Asset PricingLecture 04 Risk Prefs & EU (57)
Digression: Prospect Theoryโข Value function (over gains and losses)
โข Overweight low probability events
โข Experimental evidence
FIN501 Asset PricingLecture 04 Risk Prefs & EU (58)
Overview: Risk Preferences
1. State-by-state dominance
2. Stochastic dominance [DD4]
3. vNM expected utility theorya) Intuition [L4]
b) Axiomatic foundations [DD3]
4. Risk aversion coefficients and portfolio choice [DD5,L4]
5. Uncertainty/ambiguity aversion
6. Prudence coefficient and precautionary savings [DD5]
7. Mean-variance preferences [L4.6]
FIN501 Asset PricingLecture 04 Risk Prefs & EU (59)
Introducing Savings
โข Introduce savings decision: Consumption at ๐ก = 0 and ๐ก = 1
โข Asset structure 1:
โ risk free bond ๐ ๐
โ NO risky asset with random return
โ Increase ๐ ๐:โ Substitution effect: shift consumption from ๐ก = 0 to ๐ก = 1
โ save more
โ Income effect: agent is โeffectively richerโ and wants to consume some of the additional richness at ๐ก = 0โ save less
โ For log-utility (๐พ = 1) both effects cancel each other
FIN501 Asset PricingLecture 04 Risk Prefs & EU (60)
Savings: Euler Equationfor CRRA: ๐ข ๐ = ๐1โ๐พ
1โ๐พ
โข max๐0,๐1
๐ข ๐0 + ๐ฟ๐ข(๐1)
s.t. ๐1 = ๐ ๐(๐0โ๐0) + ๐1
โข max๐0
๐ข ๐0 + ๐ฟ๐ข(๐ ๐๐0 + ๐1)
โข FOC: 1 = ๐ฟ๐ขโฒ ๐1๐ขโฒ ๐0
๐ ๐ 1 = ๐ฟ ๐1๐0
โ๐พ๐ ๐
๐ฃ๐ โ ln๐ ๐ = โ ln ๐ขโฒ ๐1๐ขโฒ ๐0
โ ๐๐๐ฟ
for log: ๐ข ๐ = ln ๐ & ๐1 = 0
๐0 = 1
๐ฟ(๐ฟ+1)[๐0 + 1
๐ ๐1]
๐1 = 1 โ 1
๐ฟ ๐ฟ+1[๐ ๐0 + ๐_1]
for e1 = 0 saving does not depend on (risk of) ๐ ๐: 1 = ๐ฟ ๐0
๐ ๐(๐0โ๐0)๐ ๐
FIN501 Asset PricingLecture 04 Risk Prefs & EU (61)
Intertemporal Elasticity of Substitution
โข ๐ผ๐ธ๐ โ๐ ln
๐1๐0
๐๐= โ
๐ ln๐1๐0
๐ ln๐ขโฒ(๐1)
๐ขโฒ(๐0)
โข For CRRA ๐ข ๐ = ๐1โ๐พ
1โ๐พ๐ผ๐ธ๐ = 1
๐พ
FIN501 Asset PricingLecture 04 Risk Prefs & EU (62)
Investment Risk
โข Savings decision: Consumption at ๐ก = 0 and ๐ก = 1โข No endowment risk at ๐ก = 1โข Asset structure 2: (no portfolio choice yet)
Single risky asset only No risk-free asset
โข Theorem (Rothschild and Stiglitz, 1971): For ๐ ๐ต = ๐ ๐ด + ฮต, where E ํ = 0 and ํ โฅ ๐ ๐ด, then respective savings ๐ ๐ด , ๐ ๐ต out of initial wealth level ๐0 are If ๐๐ ๐
๐๐0โค 0 and ๐ ๐ > 1, then ๐ ๐ด < ๐ ๐ต.
If ๐๐ ๐ ๐๐0
โฅ 0 and ๐ ๐ < 1, then ๐ ๐ด > ๐ ๐ต.
FIN501 Asset PricingLecture 04 Risk Prefs & EU (63)
Investment Risk with Portfolio and Savings Decision
โข Savings decision: Consumption at ๐ก = 0 and ๐ก = 1
โข No endowment risk at ๐ก = 1, ๐1 = 0
โข Asset structure 3: portfolio shares ๐ผ๐
โข max๐0,๐1,๐ถ๐๐ข ๐0 + ๐ฟ๐ธ0[๐ข(๐1)]
s.t. ๐1 = ๐ผ0๐๐ 1
๐(๐0 โ ๐0)
๐ขโฒ ๐0 = ๐ธ0 ๐ฟ๐ขโฒ ๐1 ๐ ๐ โ๐
FIN501 Asset PricingLecture 04 Risk Prefs & EU (64)
Investment Risk: Excess Return
1 = ๐ธ0 ๐ฟ๐ข(๐1)๐ข(๐0)
๐ ๐ โ๐
โข For CRRA 1 = ๐ฟ๐ธ0๐1๐0
โ๐พ๐ ๐
โข In โlog-notationโ: ๐๐ก โก log ๐๐ก, ๐ฃ๐ก๐โก log๐ ๐ก
๐
1 = ๐ฟ๐ธ0 ๐โ๐พ ๐1โ๐0 +๐ฃ๐
โข Assume ๐๐ก, ๐ฃ๐ก๐~๐ฉ
1 = ๐ฟ[๐โ๐พ๐ธ0 ฮ๐1 +๐ธ0 ๐ฃ๐ +12๐๐๐0 โ๐พฮ๐1+๐ฃ๐ ]
0 = ln ๐ฟ โ ๐พ๐ธ0 ฮ๐1 + ๐ธ0 ๐ฃ๐ +๐พ2
2๐๐๐0 ฮ๐1 +
1
2๐๐๐0 ๐ฃ๐ โ ๐พ๐ถ๐๐ฃ0[ฮ๐1, ๐ฃ
๐]
โข For risk free asset:
๐ฃ๐ = โ ln ๐ฟ + ๐พ๐ธ0 ฮ๐1 โ๐พ2
2๐๐๐0 ฮ๐1
โข Excess return of any asset:
๐ธ0 ๐ฃ๐ +1
2๐๐๐0 ๐ฃ๐ โ ๐ฃ๐ = ๐พ๐ถ๐๐ฃ0[ฮ๐1, ๐ฃ
๐]
FIN501 Asset PricingLecture 04 Risk Prefs & EU (65)
Investment Risk: Portfolio Shares
โข Excess return
๐ธ0 ๐ฃ๐ +1
2๐๐๐0 ๐ฃ๐ โ ๐ฃ๐ = ๐พ๐ถ๐๐ฃ0[ฮ๐1, ๐ฃ
๐]
โข If consumption growth ฮ๐1 = ฮ๐จ1 wealth growth
โข ๐ถ๐๐ฃ0 ฮ๐จ1, ๐ฃ๐ = ๐ถ๐๐ฃ0 ๐ผ0
๐๐ฃ๐, ๐ฃ๐ = ๐ผ0
๐๐๐๐0[๐ฃ
๐]
โข Hence, optimal portfolio share
๐ผ0๐=
๐ธ0 ๐ฃ๐ +12๐๐๐0 ๐ฃ๐ โ๐ฃ๐
๐พ๐๐๐0[๐ฃ๐]
FIN501 Asset PricingLecture 04 Risk Prefs & EU (66)
Making ฮ๐จ1 Linear in ๐0 โ ๐จ0
โข ๐1 = ๐ผ0๐๐ 1
๐(๐0 โ ๐0) recall ๐1 = 0
โข ๐1๐0
= ๐ผ0๐๐ 1
๐(1 โ ๐0
๐0) let ๐ 1
๐= ๐ผ0
๐๐ 1
๐
โข In โlog-notationโ: ฮ๐จ1 = ๐ฃ1๐+ log(1 โ ๐๐0โ๐จ0)
โข Linearize using Taylor expansion around ๐ โ ๐จ
โข ฮ๐จ1 = ๐ฃ1๐+ ๐ + 1 โ 1
๐๐0 โ ๐จ0
Where ๐ โก log ๐ + 1 โ ๐ log 1โ๐
๐, ๐ = 1 โ ๐๐โ๐จ
Hint: in continuous time this approximation is precise
nonlinear
FIN501 Asset PricingLecture 04 Risk Prefs & EU (67)
Endowment Risk: Prudence and Pre-cautionary Savings
โข Savings decisionConsumption at ๐ก = 0 and ๐ก = 1
โข Asset structure 2: No investment risk: riskfree bond
Endowment at ๐ก = 1 is random (background risk)
โข 2 effects: Tomorrow consumption is more volatile consume more today, since itโs not risky
save more for precautionary reasons
FIN501 Asset PricingLecture 04 Risk Prefs & EU (68)
โข Risk aversion is about the willingness to insure โฆ
โข โฆ but not about its comparative statics.
โข How does the behavior of an agent change when we marginally increase his exposure to risk?
โข An old hypothesis (J.M. Keynes) is that people save more when they face greater uncertainty
precautionary saving
โข Two forms: Shape of utility function ๐ขโฒโฒโฒ
Borrowing constraint ๐๐ก โฅ โ๐
Prudence and Pre-cautionary Savings
FIN501 Asset PricingLecture 04 Risk Prefs & EU (69)
Precautionary Savings 1: Prudence
โข Utility maximization ๐ข ๐0 + ๐ฟ๐ข๐ธ0[๐ข(๐1)] Budget constraint: ๐1 = ๐1 + 1 + ๐ (๐0 โ ๐0) Standard Euler equation: ๐ขโฒ ๐๐ก = ๐ฟ 1 + ๐ ๐ธ๐ก ๐ขโฒ ๐๐ก+1
โข If ๐ขโฒโฒโฒ > 0, then Jensenโs inequality implies:
โข1
๐ฟ 1+๐=
๐ธ๐ก ๐ขโฒ ๐๐ก+1
๐ขโฒ ๐๐ก>
๐ขโฒ ๐ธ๐ก ๐๐ก+1
๐ขโฒ ๐๐กโข Increase variance of ๐1 (mean preserving spread)โข Numerator ๐ธ๐ก[๐ข
โฒ ๐๐ก+1 ] increases with variance of ๐๐ก+1
โข For equality to hold, denominator has to increase๐๐ก has to decrease, i.e. savings has to increase precautionary savings
โข Prudence refers to curvature of ๐ขโฒ, i.e. ๐ = โ๐ขโฒโฒโฒ
๐ขโฒโฒ
FIN501 Asset PricingLecture 04 Risk Prefs & EU (70)
โข Does not directly follow from risk aversion, involves ๐ขโฒโฒโฒ Leland (1968)
โข Kimball (1990) defines absolute prudence as
๐ ๐ โ โ๐ขโฒโฒโฒ ๐
๐ขโฒโฒ ๐โข Precautionary saving if any only if prudent.
important for comparative statics of interest rates.
โข DARA โ Prudence๐ โ
๐ขโฒโฒ
๐ขโฒ
๐๐< 0, โ
๐ขโฒโฒโฒ
๐ขโฒโฒ> โ
๐ขโฒโฒ
๐ขโฒ
Precautionary Savings 1: Prudence
FIN501 Asset PricingLecture 04 Risk Prefs & EU (71)
Precautionary Savings 2: Future Borrowing Constraint
โข Agent might be concerned that he faces borrowing constraints in some state in the future
โข agents engage in precautionary savings (self-insurance)
โข In Bewley (1977) idiosyncratic income shocks, mean asset holdings mean ๐ (across individuals) result from individual optimization
mean[a]
-b
r
๐
FIN501 Asset PricingLecture 04 Risk Prefs & EU (76)
โข Asset structure 3:โ No risk free bondโ One risky asset with random gross return ๐
โข Theorem (Rothschild and Stiglitz,1971) : Let ๐ ๐ด, ๐ ๐ต be two return distributions with identical means such that ๐ ๐ต = ๐ ๐ด + ๐, where ๐ is white noise, and let ๐ ๐ด, ๐ ๐ต be the savings out of ๐0 corresponding to the return distributions ๐ ๐ด, ๐ ๐ตrespectively.
If ๐ ๐ โฒ ๐ โค 0 and ๐ ๐ ๐ > 1, then ๐ ๐ด < ๐ ๐ต
If ๐ ๐ โฒ ๐ โฅ 0 and ๐ ๐ ๐ < 1, then ๐ ๐ด > ๐ ๐ต
Precautionary Savings 1: Prudence
FIN501 Asset PricingLecture 04 Risk Prefs & EU (77)
๐ ๐ = โ๐ขโฒโฒโฒ ๐
๐ขโฒโฒ ๐
๐ ๐ ๐ = โ๐๐ขโฒโฒโฒ ๐
๐ขโฒโฒ ๐
โข Theorem: Let ๐ ๐ด, ๐ ๐ต be two return distributions such that ๐ ๐ด SSD ๐ ๐ต,
let ๐ ๐ด and ๐ ๐ต be, respectively, the savings out of ๐0. Then,
๐ ๐ด โฅ ๐ ๐ต โ ๐๐ ๐ โค 2 and conversely,
๐ ๐ด < ๐ ๐ต โ ๐๐ ๐ > 2
Precautionary Savings 1: Prudence
FIN501 Asset PricingLecture 04 Risk Prefs & EU (78)
Overview: Risk Preferences
1. State-by-state dominance
2. Stochastic dominance [DD4]
3. vNM expected utility theorya) Intuition [L4]
b) Axiomatic foundations [DD3]
4. Risk aversion coefficients and portfolio choice [DD5,L4]
5. Uncertainty/ambiguity aversion
6. Prudence coefficient and precautionary savings [DD5]
7. Mean-variance preferences [L4.6]
FIN501 Asset PricingLecture 04 Risk Prefs & EU (79)
Mean-variance Preferences
โข Early research (e.g. Markowitz and Sharpe) simply used mean and variance of return
โข Mean-variance utility often easier than vNM utility function
โข โฆ but is it compatible with vNM theory?
โข The answer is yes โฆ approximately โฆ under some conditions.
FIN501 Asset PricingLecture 04 Risk Prefs & EU (80)
Mean-Variance: quadratic utility
Suppose utility is quadratic, ๐ข ๐ = ๐๐ โ ๐๐2
Expected utility is then๐ธ ๐ข ๐ = ๐๐ธ ๐ โ ๐๐ธ ๐2
= ๐๐ธ ๐ โ ๐ ๐ธ ๐ 2 + var ๐
Thus, expected utility is a function of the mean ๐ธ ๐ , and the variance var ๐ ,only.
FIN501 Asset PricingLecture 04 Risk Prefs & EU (81)
Mean-Variance: joint normals
โข Suppose all lotteries in the domain have normally distributed prized. (independence is not needed). This requires an infinite state space.
โข Any linear combination of jointly normals is also normal.
โข The normal distribution is completely described by its first two moments.
โข Hence, expected utility can be expressed as a function of just these two numbers as well.
FIN501 Asset PricingLecture 04 Risk Prefs & EU (86)
Mean-Variance: small risks
โข Let ๐:โ โ โ be a smooth function. The Taylor approximation is
๐ ๐ฅ
โ ๐ ๐ฅ0 + ๐โฒ ๐ฅ0
๐ฅ โ ๐ฅ01
1!
+ ๐โฒโฒ ๐ฅ0
๐ฅ โ ๐ฅ02
2!+ โฏ
โข Use the Taylor approximation for ๐ธ ๐ข ๐ฅ
FIN501 Asset PricingLecture 04 Risk Prefs & EU (87)
Mean-Variance: small risks
โข Since ๐ธ ๐ข ๐ค + ๐ฅ = ๐ข ๐๐ถ๐ธ , this simplifies
to ๐ค โ ๐๐ถ๐ธ โ ๐ ๐ด ๐คvar ๐ฅ
2
๐ค โ ๐๐ถ๐ธ is the risk premium
We see here that the risk premium is approximately a linear function of the variance of the additive risk, with the slope of the effect equal to half the coefficient of absolute risk.
FIN501 Asset PricingLecture 04 Risk Prefs & EU (88)
Mean-Variance: small risks
โข Same exercise can be done with a multiplicative risk.
โข Let ๐ฆ = ๐๐ค, where ๐ is a positive random variable with unit mean.
โข Doing the same steps as before leads to
1 โ ๐ โ ๐ ๐ ๐คvar ๐
2 where ๐ is the certainty equivalent growth rate, ๐ข ๐ ๐ค =
๐ธ[๐ข ๐๐ค ].
The coefficient of relative risk aversion is relevant for multiplicative risk, absolute risk aversion for additive risk.