EC371 Economic Analysis of Asset Prices
Topic #3: Decision-making Under Uncertainty
R. E. Bailey
Department of Economics
University of Essex
Outline
Contents
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State Preference Approach
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1.1
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Decision-making under uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . .
The Expected Utility Hypothesis
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2.1
Assumptions of the EUH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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2.2
Fundamental Valuation Relationship . . . . . . . . . . . . . . . . . . . . . . . . . .
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Behavioural Alternatives to EUH
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Mean-variance Analysis
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4.1
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Remarks on M-V Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Reading: Economics of Financial Markets, chapter 4
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1.1
State Preference Approach
Decision-making under uncertainty
Modelling uncertainty
• Three ingredients:
1. States: Uncertainty is represented by states of the world: S = {s1 , s2 , . . . , s` },
2. Actions: Investors take actions – buy portfolios of securities. Action: a = (x1 , x2 , . . . xn ).
3. Consequences: Actions have consequences – outcomes that differ across states: c =
f (sk , a)
• Consequences are valued according to preferences.
• Preferences: U = U (f (s1 , a), f (s2 , a), . . . , f (s` , a)).
• Preferences may be “state dependent”.
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Decision making under uncertainty
• Terminal wealth: Wk (k is the state).
• Preferences: U = U (W1 , W2 , . . . , W` ).
• Constraint: p1 x1 + p2 x2 + · · · + pn xn = A
• Payoffs: Wk = vk1 x1 + vk2 x2 + · · · + vkn xn , where vkj is the payoff to security j in state k.
• State-preference approach: wide applicability, few predictions.
• Notice that ‘probability’ is not needed anywhere.
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The Expected Utility Hypothesis
The Expected Utility Hypothesis (EUH)
• EUH distinguishes between beliefs and preferences.
• Beliefs are about which state will occur (reflect information).
• Preferences are represented by the utility of wealth.
• Argument for EUH: the implication of ‘reasonable’ assumptions – if the the assumptions are
plausible (‘true’?), EUH must follow
• But what is the evidence? – Much less convincing than the assumptions – in which case some
assumptions cannot be ‘reasonable’
• Who cares? – most finance models rely on the EUH
2.1
Assumptions of the EUH
Why bother with ‘assumptions’?
The reason is to provide a justification for where the EUH “comes from” – i.e., why it makes sense.
If each of the underlying axioms seems reasonable, then there implication should be reasonable too.
The “Savage axioms”
• EUH can be derived from various sets of assumptions
• One set of assumptions: the “Savage axioms”:
1. Conditional preferences Ordering of actions (portfolio choices) conditional on any event
(a subset of states) is independent of the outcome in all other events.
2. Preferences are independent of beliefs The ordering of outcomes (terminal wealth) for
any state is independent of the state in which each outcome occurs. – ‘label’ attached to
the state is irrelevant.
Interpretation: essentially that the payoff encapsulates all that is relevant for the decisionmaker when evaluating the outcome. In the context of portfolio investment, it means any
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characteristic of the asset other than payoff is irrelevant. If the investor cares, e.g. about
whether the asset represents an ‘ethical’ company, this would violate the axiom. To take
model such preferences, it is necessary to broaden the outcome to include factors other
than the payoff – could be done but it’s messy.
3. Beliefs are independent of preferences Beliefs about which state will occur do not depend
on the outcome in any state.
Interpretation: essentially this axiom excludes wishful thinking, i.e. because an asset
will yield a big payoff if the winter is mild does not affect whether the investor believes
that a mild winter is more, or less, likely to occur.
If decision makers practise wishful thinking, then the EUH would have to be replaced –
but replaced with what?
Implications of the assumptions
• With a technical (continuity) condition, the assumptions imply:
(a) individuals act as if probabilities are assigned to states;
(b) there exists a von Neumann-Morgenstern utility function which depends only on outcomes (terminal wealth);
(c) individuals order actions according to the expected value of the von Neumann-Morgenstern
utility function.
• Formally: the EUH implies that:
U
≡ U (W1 , W2 , . . . , W` )
= π1 u(W1 ) + π2 u(W2 ) + · · · + π` u(W` )
where πk is the probability assigned to state sk .
u(·) is the von Neumann-Morgenstern utility function.
Sometimes the von Neumann-Morgenstern utility function is called the ‘Bernoulli’ utility
function, but mostly the ‘Bernoulli’ utility function is defined as u(W ) = ln W , i.e., a particular functional form, because that’s what Daniel Bernoulli (1700–1782) used in his pioneering
research on probability.
Remarks on the EUH
1. Is the EUH ‘realistic’? Doubts have been expressed about both the axioms and evidence
(i.e. it can be argued that some axioms are implausible descriptions of behaviour or – more
importantly – that evidence is contrary to the predictions of EUH.
Irrespective of whether the EUH provides an accurate description of observed behaviour,
the EUH is often proposed as a normative basis for behaviour – i.e. the EUH is a model
how any sane investor should behave. In this context the EUH is commonly asserted as a
benchmark criterion for ‘rational’ decision making, especially when objective probabilities
are assumed (see below).
You may be frustrated about what is the ‘right’ approach to all this. The answer is that
there is no ‘right’ approach. For EC371 you should be able to explain the issues, in a dispassionate, balanced way – your personal view is ultimately your own (irrelevant for academic
purposes).
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2. EUH implies that decision-makers act as if they assign probabilities to events. Are probabilities subjective or objective?
An alternative approach to the EUH: a common approach is to assume, without explanation,
the existence of probabilities known to decision-makers. Various sets of axioms can then be
proposed from which the EUH can be derived.
The difficulty is that this approach is silent about where the probabilities come from – they
are just taken as given at the outset (some theorists would say “probabilities are primitives” in
this theory). Often these probabilities are called ‘objective’, ‘known’ or ‘true’. But none of
these words suggest how the probabilities are to be found – where do they come from? The
theory is silent.
A way forward used in practice is to estimate the probabilities from data, the estimates being expressed as sample means, standard deviations, correlations, etc, of assumed probability
distributions.
What all this means is that which ever approach is taken it is necessary to ‘model’ –
i.e. make assumptions about – how the probabilities are to be inferred from samples of data.
Some models are, no doubt, more accurate representations of the world. But there is no escape
from the need to make assumptions (to build models).
As a matter of terminology, ‘decision making under risk’ commonly assumes objective
probabilities, while ‘decision making under uncertainty’ commonly assumes subjective probabilities. This is just a convention.
3. Attitude to risk:
u00 (W ) < 0 ←→ risk aversion.
u00 (W ) = 0 ←→ risk neutral.
u00 (W ) > 0 ←→ risk loving.
4. Popular functional forms. The most popular is: Constant Relative Risk Aversion (CRRA):
u(W ) = W 1−γ /(1 − γ) with γ > 0,
u(W ) = ln W for γ = 1
The parameter γ is known as the ‘index of relative risk aversion’ or just ‘risk aversion coefficient’. You should be able to show that γ = −W · u00 (W )/u0 (W ).
There are other popular functional forms but only one other is needed in EC371 – the quadratic,
which can be used as one justification (not the only one) for mean-variance analysis.
2.2
Fundamental Valuation Relationship
Portfolio selection in the EUH
• EUH assumes that investors choose portfolios, {x1 , x2 , . . . , xn } to maximize E[u(W )] subject to the wealth constraint.
• Fundamental valuation relationship (FVR): a necessary condition for maximizing Expected
Utility: E[(1 + rj )H] = 1
for every asset j
The FVR is just the set of First Order Conditions, FOC, from maximizing expected utility
subject to the wealth constraint. Like all FOC, they are satisfied at a maximum but a solution
of the FOC may, or may not, be a maximum.
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• What is H? Depends on the context (i.e. assumptions about vN-M function). You will find
that for intertemporal planning H takes a slightly different form from that analysed in this
note.
• Here: E[(1 + rj )u0 (W )] = λ so that H ≡ u0 (W )/λ. – we’ll justify this in the next screen.
• Second order condition: risk aversion, i.e. u00 (W ) < 0. (The purpose of this is to ensure
that a solution of the FOC is indeed a maximum. This is the real reason why risk aversion is
assumed: so that the FVR is meaningful.)
• Special case: for risk neutral investors an optimal portfolio exists only if: E[rj ] = r0
1, 2, . . . , n
Justifying the FVR
• Start from any optimum portfolio.
• Invest one (small) unit of wealth (say $1) in any asset j.
• Increment in wealth: (1 + rj ).
• Increment in utility: (1 + rj )u0 (W ).
• Increment in expected utility: E[(1 + rj )u0 (W )].
• Increment to expected utility must be the same for every asset
– otherwise, the initial portfolio could not have been an optimum.
• Formally, E[(1 + rj )u0 (W )] = λ
– where λ is the same value for every asset.
• Rearrange: E[(1 + rj )u0 (W )/λ] = 1, for every j.
• Hence E[(1 + rj )H] = 1 where H = u0 (W )/λ
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Behavioural Alternatives to EUH
Behavioural Alternatives to EUH
z
z(W )
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Sq
z∗
←− loss
W∗
gain −→
W
The value function, z(W ), in prospect theory.
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for j =
• Example 1: Prospect Theory1
1. Replace u(W ) with a ‘value function’, z(W )
2. Replace probabilities with ‘decision weights’, sometimes known as ‘value weights’, they
do not satisfy the rules for probability, e.g. the weights may not add to one. )
Both the value function and the decision weights are chosen as replacements in order to express
behaviour that diverges from EUH predictions, as revealed by experimental results. While
these results tend to be fairly similar among different experiments, there is nothing to guarantee
that they must hold – they are fragile assumptions to build into a model.
The detailed properties of the value function and the decision weights in prospect theory
will not be needed for the final exam in EC371 but may be useful for a term paper; a recommended reference is Levy, H. The Capital Asset Pricing Model in the 21st Century, ch. 9.
• Example 2: Regret Theory
Another alternative to the EUH, and closely related to Prospect Theory is ‘Regret Theory’2 .
Regret theory, unlike the EUH, allows decision-makers’ preferences (hence their choices) to
depend on different actions corresponding to the same state. That is, if an investor takes a particular action (e.g., portfolio choice) then when the state is revealed (payoffs of the portfolio),
then the investor may regret not having taken a different action (chosen a different portfolio) –
most importantly, the investor’s initial decision may be affected by the possibility that he/she
will regret not having made a different decision when the state is revealed.
Thus for example, if an investor chosing between a safe investment (savings account)
and a risky investment (stocks and shares) feels in advance that he/she will regret having
chosen stocks and shares in the event that the stock market goes down may choose the saving
account, even though the chance is high that the stock market will not go down and may
rise substantially – the investor fears the regret that will ensue from losing out on the risky
investment (paying less attention to the prospect of gain).
• No one of the replacements for EUH is generally supported – while the weaknesses of the EUH
are well known, different weaknesses also afflict the alternatives that have been proposed.
Sometimes the EUH is used as a benchmark (i.e. a criterion) for what it means to be ‘rational’.
There is no consensus about this either.
• Behavioural finance: excuses or predictions?
Behavioural finance accounts can be very useful as explanations after the event; they can give
a hint as to what has gone wrong with more conventional accounts. They are less persuasive,
however, in explanations in advance, not least because there are so many different behavioural
models from which to choose.
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Mean-variance Analysis
Mean-variance Analysis
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Prospect theory was pioneered by Daniel Kahneman and Amos Tversky in their classic paper “Prospect Theory: an
Analysis of Decision Under Risk”, Econometrica, vol. 47(2), March 1979, pp. 263–291. This paper is not required reading
for EC371.
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See Bleichrodt, H. & P.P. Wakker “Regret Theory: A Bold Alternative to the Alternatives”, The Economic Journal,
vol. 125, March 2015, pp. 493–532. This reference is optional but may be of relevance for term paper preparation.
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• Mean-variance (MV) analysis assumes objective function: G(µP , σP2 ), where µP = expected
rate of return on portfolio P σP2 = variance of rate of return on portfolio P .
• MV assumption: each investor selects a portfolio to maximize G(·, ·).
• Example: G(µP , σP2 ) = µP − ασP2 , where α > 0 represents investor’s preferences (aversion
to risk)
• G(·, ·): expresses an investor’s preferences – allowed to differ among investors
• µP , σP2 express investors’ beliefs – sometimes assumed to be the same for all investors
4.1
Remarks on M-V Analysis
Remarks on MV analysis
1. Why assume MV? – because it’s tractable
This means that it’s possible without too much effort to obtain predictions from the model.
Even more importantly in practice, MV enables optimal portfolio proportions to be calculated
– the reason is that the conditions that must be satisfied for a portfolio to be optimal are linear,
thus easily solvable both in principle and numerically.
2. Justifications:
(a) Quadratic utility in the EUH.
(b) Normal distributions of returns in the EUH.
(c) Directly, just that it seems plausible
N OTE: the application of (a) and (b) involves some technicalities that go beyond EC371. For
a detailed treatment see: Levy, H. The Capital Asset Pricing Model in the 21st Century, ch. 4.
3. What’s so special about MV analysis?
– symmetric distributions: ignore skewness and kurtosis.
This means that there may be some aspects of the patterns of rates of return that are ignored
in MV – for instance, if ‘thick tails’ (rare occurrences of extemely high or low returns) are
important, MV could give misleading predictions, or erroneous prescriptions for optimal portfolios.
4. MV objective often expressed using rates of return, not wealth.
There is an implicit assumption here: that the level of initial wealth does not affect optimal
portfolio proportions – in other words, if two investors both have the same MV objective
function but one has twice as much wealth as the other, the proportion of that wealth invested
in any asset is the same for both investors. Whether this is plausible is debatable.
5. Indifference curves are usually drawn in µP , σP space.
Notice the difference between variance, σP2 , and standard deviation, σP . It’s just more convenient to draw the diagrams with σP . Nothing of substance is affected.
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µP 6
Utility
increases
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Utility increases
σP
Indifference curves in µP , σP space
• Uncontroversial assumptions:
expected return, µP , is a ‘good’, standard deviation, σP , is a ‘bad’.
• Assume indifference curves are ‘convex from below’. Why? Sensible predictions
Interpretation: if an investor did not behave in accordance with the assumption, then his/her
behaviour would exhibit patterns of behaviour that are seldom observed, in particular ‘plunging’ entire initial wealth in just one asset. Such behaviour obviously can occur but seems
sufficiently unusual that it is plausible to exclude.
Summary
Summary
1. State-preference analysis provides a general framework but makes few testable predictions.
2. EUH makes sense of probability but is still too general for most purposes.
3. Mean-Variance is a practical alternative but is more likely to be contradicted by evidence.
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