investment theory week 4

Compound Utility Theory(CUT)

(Literature: Paper Zou 3)(Literature: Paper Zou 3)

4-22006 by Liang Zou. All rights reserved.

Two Popular Approaches to Decision Making under Risk• Expected Utility Theory (EUT)

– Von Neumann & Morgenstern (1944)– Many other extensions

• Prospect Theory (PT)– Kahneman & Tversky (1979)– PT has grown into “cumulative prospect

theory”, which blends the initial PT with rank-dependent models.


Expected Utility Theory (EUT)

• EUT says that the objective of decision making under risk is to maximize one’s expected utility:

where w denotes wealth (or consumption) and ps the probability that state s will occur.

s

ss wUpwEU )()(


Prospect Theory (PT)• PT assumes that the utility function (called value

function) depends on gain and loss and has a S-shaped form with a kink at zero (see next slide). Two implications: loss aversion and risk seeking in the loss region.

• Moreover, PT assumes that people use their own decision weights p (through probability transformation) instead of true probability p in calculating the expected value function.

• The objective under PT is to maximize

0 0

( ) ( )s s

s s s sgain loss

E v v gain v loss u d


Assumed value function v in PT

0 gainloss

v


Why is CUT useful?• Experimental evidence against EUT. Although EUT

is a normative theory, its scope of application is limited by people’s systematic violation of its independence axiom.

• Empirical findings that EU models cannot explain: e.g. equity risk premium puzzle.

• PT is empirically more successful. But its complicated mathematical structure makes it difficult to apply. Moreover, why do people distort true probabilities?

• Behavioral finance is another way to get things explained. But its descriptive approach is too flexible to be useful for rational decision making under risk.


Normative, Positive, and Descriptive Theories

• Descriptive theory: purports to explain actual behavior, allowing irrationality.

• Positive theory: purports to explain actual behavior, assuming people are rational.

• Normative theory: purports to advice as to rational decision making.

It is difficult to agree on a separating line between these theories. Roughly speaking, the assumed degree of rationality increases as one moves toward a normative theory; whereas the descriptive generality increases as one moves toward a descriptive theory.


What is rational decision making?

• There is no universal standard for “rationality.” Normative theories typically assume stronger standards of rationality than descriptive theories – at the cost of loosing some degree of empirical generality.

• Intuitively, rational decision making suggests optimizing behavior: making best use of information available and seeking logical consistency in one’s decision rules.


What kind of theory do we like?• It depends on what our problems are,

and how we wish to apply the theory.

• Most of us prefer theories that help us make rational decisions as well as explain economic phenomena.

• A normative theory that is incongruent with reality, or a descriptive theory that has no predictive power, may be entertaining but also can be misleading.


Evidence against EUT

• The common consequence effect

• The common ratio effect

• Simultaneous gambling and insurance

These observations have a common name in PT (and rank-dependent models): “attitude toward probabilities.” But this view begs the important question as to what motivates people to show such an attitude, especially in situations where the true probabilities are clearly stated and objectively given.

We shall see that CUT offers an entirely different, yet appealing, interpretation about behaviors that exhibit attitude toward risk that the shape of utility functions cannot explain.


Example (Allais Paradox)

P1: $ 1m P2:

$ 5m (0.10)

$ 1m (0.89)

$ 0 (0.01)

• According to EUT you should have chosen either A1 and B2, or A2 and B1.

• What have you chosen?



Which gamble do you prefer? Finding: Most prefer P1.


Allais Paradox (continued)

P3:

$ 1m (0.11)

$ 0 (0.89)

Which gamble do you prefer? Finding: Most prefer P4.

P4:

$ 5m (0.10)

$ 0 (0.90)


Example (ABC Paradox)

1

1.2

0.8

1

1.5

0.6

1

1.2

(A) (B) (C)

0.8

0.5

0.5

0.5

0.5

1.5

0.25

0.6

0.25

0.25

0.25

Assume 1=$10,000. Rank these gambles. Finding: Most for C.


Limitations of extant risk models• Ad hoc assumptions of risk measures:

• Variance

• Probability of shortfalls

• Semi-variance

• Expected loss

• Value at risk, etc.

• Why choose mean as reward?

• Lack of theoretical foundation


The Equity Premium Puzzle

• Low Real Risk-free Interest Rate (1%)

• High Real Equity Risk Premium (7%)

Over more than 15 years to date, no rational theory has been found that resolves the puzzle.


Compound Utility Theory (CUT)

• CUT holds if the objective of decision under risk is to maximize one’s compound utility under target (or reference point) :

where u and d are called utility reward and disutility risk, respectively.

( , ) ( , )

( ), ( )s s

s s s sw w

V u d V EU ED

u p U w d p D w


Attractive Features of CUT• It offers a unified approach that is justified by

a set of intuitive axioms.

• It gives a general measure of (utility) reward and (disutility) risk, as well as a simple framework for analyzing nonlinear preferences.

• It resolves the paradoxes and puzzles in the EU paradigm in easy and rational terms.

• It retains the normative appeal of EUT, and is easy to be brought to resolve specific decision problems that call for better understanding of risk and human attitude toward risk.


Development of CUT

• Key Concepts

• Axioms

• Measure of Disutility Aversion

• Eliciting Compound Utility

• Resolving Paradoxes


Foundation of CUT

“Nature has placed mankind under the governance of two sovereign masters, pain and pleasure. It is for them alone to point out what we ought to do, as well as to determine what we shall do. On the one hand the standard of right and wrong, on the other the chain of causes and effects, are fastened to their throne.”

-- Jeremy Bentham (1789)


Key Ideas of CUT

• We can agree on a set of rational criteria for evaluating risk and reward.

• Perceptions of risk and reward fundamentally determine our choice and conduct.

• Numerical representation of preference can be derived via one’s evaluation of risk and reward.

• People generally avoid disutility-risk. Some people may demand increasingly higher marginal compensations of utility-reward for increases of disutility-risk.


An Example of Theory Making

• Suppose we observe the following sequence

1, 2, 5, 10, 17, 26, 37, 50, ...

• What theory can we make about the numbers that will follow?

• Step 1: Assume that the observed pattern will continue.

• Step 2: Derive from the assumption in Step 1 a prediction (hypothesis).


General vs. Specific Theories




• What have you chosen?Consider three theories (hypotheses):

T1: The sequence to follow will be increasing, starting with a number greater than 50.

T2: Same as T1, and the sequence will have alternating odd and even numbers.

T3: The sequence is 1+n2, n=8,9,...

T1 is most general, followed by T2, and T3 is most specific (most easily wrong). But which theory do you like?


Balancing Generality and Specificity• Similar to EUT, CUT seeks a specific theory

that has strong predictive power and easy to apply. Unlike EUT, however, CUT assumes a two-stage decision: first assessing risk and reward, then choosing the choice-object with best risk-reward combination.

• Similar to PT, CUT stresses the role of reference point in one’s psychology. But unlike PT which typically assumes =0, CUT allows to be a parameter of the decision maker which may change across individuals and with the choice environment. More importantly, CUT does not assume probability transformation as PT does.


Choice-objects under risk

• In this course, we focus on choice-objects that are investment strategies.

• Example (price or return distributions of):

• Stocks, bonds, savings,

• Options, futures, derivative contracts

• Mutual funds, hedge funds, portfolios of (any) tradable securities, etc.


Basic Setup• Denote by X the set of outcomes, by the set of

choice-objects that are probability measures on X.

• Partition X and by some X and define X+, X-, + and - accordingly.

• Let denote a probability mixture, i.e., 0.5P 0.5Q means a lottery that assigns a probability of 0.5 for P and 0.5 for Q.

Let denote an abstract "better than" relation

on , e.g., for any , in , could mean

"P is preferred to Q", "P is less risky than Q", etc.

Define , , , as usual.

P Q P Q


Axiom A1: Weak Order

In words, all strategies can be ranked. If P is better than Q and Q is better than Z, then P is better than Z.

Note that we have dropped the target parameter for notational convenience.

The relation is complete and transitive on .

That is, for all , , in ,

a) Either or (complete)

b) If and , then (transitive)

P Q Z

P Q Q P

P Q Q Z P Z


Axiom A2: Continuity

In words, if P is better than Q and Q is better than Z, then a lottery that yields P with almost sure probability or else Z is better than Q. Likewise, Q is better than a lottery that yields Z with almost sure probability or else P.

For all , , in , if P Q Z then

there exist reals 0 < , < 1 such that

(1 ) , (1 )

P Q Z

P Z Q P Z Q


Axiom B1: Partial Independence

In other words, the “better than” relation is not affected by the chance to play another strategy Z. If P is more rewarding (risky) than Q, then a lottery yielding either P or Z is more rewarding (risky) than a lottery yielding either Q or Z (where both lotteries have the same probabilities for Z ).

+For all , , in (respectively, )

and all real number 0 < < 1, if P Q

then (1 ) (1 )

P Q Z

P Z Q Z


Axiom B2: Monotonicity

In other words, If P is more rewarding and less risky than Q, then P is better than Q. If any one of these conditions holds strictly, then P is strictly better than Q.

For all and in , if and

or if and , then .

P Q P Q P Q

P Q P Q P Q


Representation Theorem

( ( ), ( )) ( ( ), ( ))

( ) ( ), ( ) ( )s s

s s s sw w

P Q V u P d P V u Q d Q

u P p U w d P p D w

Axioms A1-A2 and B1-B2 hold true if and only if there exist functions U, D, and V such that for all P and Q in ,

where U and D are monotonic, unique up to a positive ratio scaling, and where V is continuous, monotonically increasing in u and decreasing in d, and is unique up to a positive transformation.


d

u

V(u,d)

Example of a compound utility function.


Definition of : A local measure of disutility aversion

uduV

dduV

du

),(

),(

),(

It measures the required marginal increase in utility-reward that compensates for a marginal increase in disutility-risk along the indifference curve.


Graphical: Indifference Curves

d

uV(u,d)=constant

•


•

d0

u

A

B

C

Increasing preference

•

dp0

up

A

BC

Increasing preference

(a) CUT (b) PT

Comparing CUT with PT


CUT vs. PT: Interpretations

• PT: "People appear to have an attitude toward probabilities, resulting in observed behavior patterns that only probability transformations seem able to explain. In the context of decision under risk where true probabilities are given, the cause for such systematic distortions of true probabilities is unclear."

• CUT: "People generally avoid disutility-risk. Some people may demand increasingly higher marginal compensations of utility-reward for increases in disutility-risk."

• Whereas both statements may explain the same observed (quasiconcave) behavior at a descriptive level, CUT offers a plausible reason for such effects: "Because disutility hurts. And for some people the undesirability of disutility-risk increases more than in proportion to the desirability of utility-reward."


Resolving Allais Paradox

P3

Q

P4

0 d

u

P2

P1

D(0)

U(5)

U(1)

(a)

Q1 P3

Q

P4

0 d

u

P2P1

D(0)

U(5)

D(1)

(b)

Q1


A Single Parameter CU Function

• For parsimony, we shall often focus on the following special CU function:

where U is a usual utility function and

( , ) ( )[(1 ) ] 0aV u d U u d a

max[ ( ) / ( ) 1,0]

max[1 ( ) / ( ),0]s

s

u E U w U

d E U w U


Estimating parameter a

• In order to derive some idea about the values of a, consider the ABC paradox. Assume that the subject evaluates u and d by expected better and worse outcomes respectively under target =1. We then find that

A C B if 0 < a < 0. 69347

C A B if 0. 69347 < a < 0. 69958

C B A if 0. 69958 < a < 0. 70554

B C A if 0. 70554 < a

investment theory week 4

Documents

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