Chapter Four:
Decision Making Under Uncertainty
(Econ 512): Economics of Financial Markets
Dr. Reyadh Faras
Risk and Uncertainty
Knight (1921):
 Risk: applies to events for which objective
probabilities can be assigned.
 Uncertainty: : applies to events for which
objective probabilities cannot be assigned, or for
which it would not make sense to assign them.
 Keynes (1936):
 A game of chance is Risky because, although the
outcome of any one trial is unknown in advance,
repetition of the game a large number of times
enables observed relative frequencies to be
interpreted sensibly as objective probabilities.
Econ 512
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 Uncertain events are those that cannot be repeated
in any controlled way, thus rendering the
calculation of relative frequencies difficult, if not
impossible.
 Insurance markets aside, most financial markets
involve uncertainty rather than risk, in the sense
that relative frequencies are not readily available
to estimate probabilities.
 Most applications in finance permit the estimation
of probabilities from past data or other info.
 Financial analysis is located somewhere a long the
spectrum between two polar extremes, one allows
for calculating frequencies while the other doesn’t.
Econ 512
Dr. Reyadh Faras
4.1 The Stat-preference Approach (SPA)
4.1.1: Modelling Uncertainty
 The SPA comprises three basic ingredients:
1) State of the world, denoted by set S = {s1, s2, ..sℓ}.
It describes some contingency that could occur.
2) Actions, describe all relevant aspects of the
decision that are made before the state of the
world is revealed. They describe choice of assets.
3) Consequences, express the outcomes of an action
corresponding to each state of the world. They
are represented by a list, each element of which is
the total value of the portfolio in the
corresponding state.
Econ 512
Dr. Reyadh Faras
 If (c) denotes a consequence and (a) denotes an
action, then the three components of the theory
are related by a function of (sk) and (a) such that
c = f(sk, a). Function f(.,.) maps states and actions
into the space of consequences.
 In portfolio selection, the function links the
amount of each asset held (the action) to each
asset’s payoff in every state, and hence to the
consequence (terminal wealth).
 In the state-preference model, each individual has
a utility function the value of which serves to rank
all the possible consequences.
Econ 512
Dr. Reyadh Faras
 u = U(f(s1, a), f(s2, a), …, f(sℓ, a))
 The function U(.,.,.,.) differs across individuals.
 The individual’s decision problem is to maximize
utility, u , by choosing a feasible action which is a
portfolio that satisfies the individual’s wealth
constraint, and other constraints.
 At date (0), today, investors have to make
decisions not knowing which of the six states will
occur at date (2).
 At date (1), it becomes known that one of the
events has occurred.
 Finally at date (2) the state is revealed.
Econ 512
Dr. Reyadh Faras
 For simplicity, it is assumed that investors make
decisions with respect to a single future date.
 The payoff on the (n) assets in the (ℓ) possible state
can be arranged in a payoff array.
 Rows correspond to states and columns to assets.
 vkj is the payoff to a unit of asset j if state k occurs
Assets
1
2
...
n
State 1
v11
v12
...
V1n
State 2
v21
v22
...
v2n
State ℓ
vℓ1
vℓ2
…
vℓn
Econ 512
Dr. Reyadh Faras
Let pj denote the price of asset (j) observed today.
Then the rate of return on asset (j) in the state (k)
is defined by:
rkj = (vkj - pj) / pj = (vkj / pj) -1
 The gross rate of return on asset j, Rkj, is:
Rkj ≡ (1+rkj) = vkj / pj
 Risk-free asset has the same payoff in each state, is
denoted with subscript (0), with payoff (v0) in
every state and rate of return:
r0 = (v0/p0) - 1
Econ 512
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 Suppose there is an asset that has a positive payoff
of one unit of wealth in a particular state, k, and
zero in every other state.
 This asset could play the role of an insurance
policy; the purchase of which allows the investor
to offset any adverse consequences in state k, only.
 Assume that one such asset exists for every state.
 Investors can insure against the adverse
consequences of every possible contingency.
 Conditional on the occurrence of any state,
investors could be certain of obtaining a known
payoff, the cost of which is the asset’s price (or
insurance premium).
Econ 512
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 The presence of such an asset for every state is
sufficient for the existence of a complete set of
asset markets.
 Otherwise, asset markets are said to be incomplete
 Completeness is an idealization, and useful as a
benchmark against which more realistic
circumstances can be evaluated.
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4.1.2 Decision making under uncertainty
 Denote terminal wealth as Wk, the investor’s
utility function is defined over the consequences,
Wk , k = 1, 2, ..., ℓ:
u = U(W1 ,W2 , ..., Wℓ )
Wk ≡ f(sk, a)
 U(.) is similar to standard consumer theory, but
in the presence of uncertainty, the investor must
make a decision before the state is revealed. So, he
must weigh up the consequences across all the
conceivable states.
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 The wealth constraint states that the investor’s
outlay on assets equals initial wealth:
p1x1 + p2x2 + …+ pnxn = A
 Where (A) is the initial wealth and (xj)s denote the
number of units of each asset in the portfolio, so
that (pjxj) is the amount of wealth devoted to asset
j= 1, 2, …n.
 The portfolio is linked to terminal wealth via the
payoffs of each asset in each state of the world:
Wk = vk1 x1 + vk2 x2 + …+ vkn xn (k = 1, 2, ..., ℓ)
which is the sum of the payoff of each state
multiplied by the chosen amount of the asset.
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 The result is a portfolio decision in which the
amount of each asset held depends on asset prices,
initial wealth and preferences.
 The analysis can be extended to cover a
multiperiod horizon, generalizing the single period
decision problem described so far.
 In this case, preferences (and utility) depend on
the levels of wealth in all states and all dates.
 The wealth constraint must be modified to reflect
the opportunities for the investor to transfer
wealth from one period to the next.
Econ 512
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4.2 The expected utility hypothesis (EUH)
4.2.1 Assumptions of the EUH
 Probability is completely absent from the analysis
in the state preference model.
 The EUH approach permits a role for probability
(by assigning probabilities to states of the world)
to yield more definite implications.
 By attaching prob. to each state, the EUH enables
a distinction to be drawn between the decision
maker’s beliefs (expressed by probabilities) about
which state will occur and preferences about how
the decision maker orders the consequences of
different actions.
Econ 512
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 The implications of the EUH emerge by imposing
a number of assumptions:
1. Irrelevance of common consequences. The
decision maker orders the actions independently
of the common consequences for states not in the
event.
2. Preferences are independent of beliefs.
Preferences over consequences for the given state
are independent of the state in which they occur.
Decision maker cares only about the consequence
not the label (e.g. k) of the state in which it is
received.
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3. Beliefs are independent of consequences. The
decision maker’s degree of belief about whether a
state will occur is independent of the consequences
in the state.
 Together, the three assumptions imply that:
a) The decision maker acts as if a probability is
assigned to each state.
b) There exists a function that is dependent only on
the outcomes.
c) The decision maker orders the actions according
to the expected value of the utility function.
Econ 512
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 Formally, the EUH implies that:
u = U(W1 ,W2 , ..., Wℓ )
= π1 u(W1) + π2 u(W2) + …+ πℓ u(Wℓ)
where πk is the probability that the investor
assigns to state sk.
 The u(.) is the same for all states, although the
value of its arguments, wk, generally differs across
states.
 But probabilities and u(.) are allowed to differ
across investors.
 Assume u/ (W) > 0, meaning that investors prefer
more wealth to less.
Econ 512
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 The EUH is written as stating that actions are
ordered according to E[u(W)], where E[.] denotes
the operation of summing over the product of
probabilities and utilities.
 The EUH asserts that actions are chosen to
maximize expected utility:
E[u(W)] ≡ π1 u(W1) + π2 u(W2) + …+ πℓ u(Wℓ)
Econ 512
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4.2.3 Portfolio selection in the EUH
 The portfolio selection problem can be stated as:
choose the portfolio of assets to maximize expected
utility subject to the wealth constraint.
 This is the static problem: it does not address the
issues of (a) revising decisions overtime, or (b) the
possibility that the investor wishes to consume
some wealth before the terminal date.
 The analysis is expressed in terms of rates of
return and proportions of initial wealth invested
in assets. Thus, terminal wealth is: W = (1+rP) A
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The Fundamental Valuation Relationship
 Every portfolio that maximizes expected utility
must satisfy a condition called the fundamental
valuation relationship (FVR).
 The FVR is the set of first order conditions for
maximizing expected utility, one for each asset.
 Its general form is: E[(1+rj)H]=1 j=1,2, …, n
 H is a random variable that varies across states.
 If the investor devotes one additional unit of
wealth to asset j.
 The payoff is (1+rj) and the increment to expected
utility is E[(1+rj) u/ (W)].
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 It means: weight the utility increment in each state
by the state’s probability and sum over the states.
 At a maximum of expected utility it is necessary
that the expected utility increment is the same, say
λ, for each asset, so that
E[(1+rj) u/ (W)] = λ
j=1,2, …, n
 If this equation does not hold, then expected utility
can be increased by shifting wealth from those
assets with low values of E[(1+rj) u/ (W)] to those
with high values.
 Only when equality holds for every asset can
expected utility be at a maximum.
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 λ is the increment to expected utility resulting
from a small increase in initial wealth
(i.e. E[u/ (W)]).
 At a maximum of expected utility, the expected
marginal utility of wealth must equal the
increment to expected utility from a small change
in the holding of any asset; otherwise, expected
utility is not at a maximum.
 The FVR in the present of a risk –free asset is:
E[(rj – r0) H] = 0
j=1,2, …, n
 The FVR provides a set of necessary conditions for
a maximum.
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 The 2nd order conditions, together with the FVR,
provide necessary and sufficient conditions that a
solution of the FVR constitutes a maximum of
expected utility.
 The 2nd order conditions amount to the
requirement that u//(W) < 0; that is, that the
investor is risk-averse.
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Risk neutrality
 The case of risk neutrality (u//(W)=0) implies that
the marginal utility of wealth is independent of
wealth – say u/(W) = c, a positive constant.
 This means that:
E[(rj – r0) u/ (W)] = 0
c E[(rj – r0)] = 0
E[rj ] = r0
j=1,2, …, n
 The equation involves no choice variable of the
individual, it either holds or it does not.
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 If it doesn’t hold, the investor can borrow at r0
and invest an unbounded amount in any asset for
which E[rj]> r0;
 and short sell an unbounded amount of any asset
for which E[rj]< 0,
 the proceeds being invested at the risk-free rate,
r0.
 This implies that no solution to the maximization
problem unless E[rj ]=r0 holds; (i.e. the expected
return on every asset equals the rate of return on
the risk-free asset).
Econ 512
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 In such an equilibrium, risk neutral investors are
indifferent about which asset to hold.
 It may seem that a world of uncertainty with risk
neutral investors would look rather like a world of
certainty (asset payoffs not differ across states).
 But, E[rj ]=r0 involves an expectation.
 Exactly one state will be realized, and almost
surely the actual excess return for asset j will be
negative or positive (not zero).
 The expectation may be equal to r0 but the actual
outcome, when the state is revealed, may will
differ. In this case risk is present (future is
unknown) even if investors choose to ignore it.
Econ 512
Dr. Reyadh Faras
4.4 The Mean-variance Model
4.4.1 The Mean-variance approach to decision making
 The EUH remains a general rule for decision
making unless a specific form is assumed for the
von Neumann-Morgenstern utility function.
 One form is that u(.) is quadratic in wealth.
Expected utility can be written as a function of the
expected value (mean) and the variance (or
standard deviation) of terminal wealth.
 Hence the name “mean-variance analysis” for a
framework that greatly facilitates the construction
of optimal portfolios.
Econ 512
Dr. Reyadh Faras
 Denote the expected value of terminal
wealth by E[W] and its variance by
var[W] ≡ E[(W – E[W])2].
 If u(.) is quadratic, the expected value of
u(W) is a function of E[W] and var[W]:
E[u(W)] = F(E[W], var[W])
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4.4.3 The FVR in the Mean-variance model
 The FVR can be written as:
μj – r 0 = μP – r 0 j= 1, 2, …, n
(4.15)
σjP / σP
σP
 Where (σjP) denotes the covariance between the
rate of return on asset j and the rate of return on
the portfolio as a whole.
 The ratio (σjP / σP) equals the increment to risk (as
expressed by σP) associated with an incremental
change to the proportion of asset j in the portfolio.
Econ 512
Dr. Reyadh Faras
 Thus, the FVR states that each asset’s expected
rate of return in excess of the risk-free rate, μj–r0,
per unit of its contribution to overall risk, σjP / σP,
is the same for all assets – a necessary condition
for a mean-variance optimum.
 Investor’s preferences (attitudes to risk) do not
appear in expression (4.15).
 The equalities depend only on beliefs (expressed in
terms of means, variances and covariances of
assets’ rates of return).
 This implies that there is a role of preferences
separate from the role of beliefs.
Econ 512
Dr. Reyadh Faras
 For a mean-variance investor, portfolio selection is
an outcome of a two-stage process:
1. . Choose a portfolio that satisfies the FVR
conditions, (4.15). This portfolio takes a very
special form, consisting of only two special assets,
which are themselves portfolios of assets.
. If a risk-free asset is available, one of the two
special assets can be chosen to comprise just the
risk-free asset alone.
. Investor preferences are not relevant in
constructing the special assets; their composition
depends only on means, variances and covariances
that can be estimated from observed data.
Econ 512
Dr. Reyadh Faras
2. According to investor preferences, choose the
optimal portfolio that optimizes these preferences
(i.e. that reaches the highest feasible indifference
curve).
 The practical importance of this approach is that
it is often reasonable to assume that the first
stage is the same for all investors who have the
same information, while investors can be allowed
to possess their own, unique preferences
(expressing attitudes to risk) in the second stage.
Econ 512
Dr. Reyadh Faras