**Decision Making Under Uncertainty**

Risk and Uncertainty

Risk and Uncertainty

§

**Knight (1921):****applies to events for which objective probabilities can be assigned.**

*Risk:*

*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.

**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.
**The Stat-preference Approach (SPA)**

v

**Modelling Uncertainty**- State of the world, denoted by set S = {s
_{1}, s_{2},_{ }..s_{ℓ}}. It describes some contingency that could occur. - Actions, describe all relevant aspects of the
decision that are made before the state of the world is revealed. They
describe choice of assets.
- 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.

- If (c) denotes a consequence and (a) denotes an action, then the three components of the theory
are related by a function of (s
_{k}) and (a) such that

c = f(s

_{k}, 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.

u = U(f(s

_{1}, a), f(s_{2}, a), …, f(s_{ℓ}, 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
**.**

- 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.
- v
_{kj}is the payoff to a unit of asset j if state k occurs

§ Let p

_{j}denote the price of asset (j) observed today. Then the rate of return on asset (j) in the state (k) is defined by:
r

_{kj }= (v_{kj }- p_{j}) / p_{j }= (v_{kj }/ p_{j}) -1- The gross rate of return on asset j, R
_{kj}, is:

R

_{kj }≡ (1+r_{kj})_{ }= v_{kj }/ p_{j}- Risk-free asset has the same payoff in each state, is denoted with subscript (0),
with payoff (v
_{0}) in every state and rate of return:

r

_{0}= (v_{0}/p_{0}) - 1- 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).
- 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

**Decision making under uncertainty**

- Denote terminal wealth as W
_{k}, the investor’s utility function is defined over the consequences,

W

_{k }, k = 1, 2, ..., ℓ:
u = U(W

_{1},W_{2}, ..., W_{ℓ })
W

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._{k }≡ f(s_{k}, a)The wealth constraint states that the investor’s outlay on assets equals initial wealth:

p

Where (A) is the initial
wealth and (x_{1}x_{1}+ p_{2}x_{2}+ …+ p_{n}x_{n}= A_{j})s denote the number of units of each asset in the portfolio, so that (p

_{j}x

_{j})

_{ }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:

W

_{k }= v_{k1 }x_{1}+ v_{k2 }x_{2}+ …+ v_{kn }x_{n}(k = 1, 2, ..., ℓ)
§ which is the sum of the payoff of each state multiplied by the chosen
amount of the asset.

- 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.

**The Expected Utility Hypothesis (EUH)**

*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
- 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.

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:**

§ The decision maker acts as if a probability is assigned to each state.

§ There exists a function that is dependent only on the outcomes.

§
The decision maker orders the actions according to
the expected value of the utility function.

**Formally, the EUH implies that:**

u

**= U(W**_{1},W_{2}, ..., W_{ℓ })**=**

**π**

_{1 }**u(W**

_{1}) +**π**

_{2 }**u(W**

_{2}) + …+**π**

_{ℓ }**u(W**

_{ℓ})
where π

_{k}is the probability that the investor assigns to state s_{k}.- The u(.) is
the same for all states, although the value of its arguments, w
_{k}, 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.

- 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:

o
E[u(W)] ≡ π

_{1 }u(W_{1}) + π_{2 }u(W_{2}) + …+ π_{ℓ }u(W_{ℓ})

__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+r
_{P}) A

**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+r
_{j})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+r
_{j}) and the increment to expected utility is E[(1+r_{j}) u^{/}(W)]. - 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+r

_{j}) 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+r
_{j}) u^{/}(W)] to those with high values. - Only
when equality holds for every asset can expected utility be at a
maximum.
- 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[(r
_{j}– r_{0}) H] = 0 j=1,2, …, n

- The FVR provides a set of necessary conditions
for a maximum.
- The 2
^{nd}order conditions, together with the FVR, provide necessary and sufficient conditions that a solution of the FVR constitutes a maximum of expected utility. - The 2
^{nd}order conditions amount to the requirement that u^{//}(W) < 0; that is, that the investor is risk-averse.

__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[(r

_{j}– r_{0}) u^{/}(W)] = 0
c E[(r

_{j}– r_{0})] = 0
E[r

_{j}] = r_{0 }j=1,2, …, n- The equation involves no choice variable of the
individual, it either holds or it does not.
- If it doesn’t hold, the investor can borrow at
r
_{0}and invest an unbounded amount in any asset for which E[r_{j}]> r_{0};

and short sell an unbounded amount of any asset for
which E[r

_{j}]< 0, the proceeds being invested at the risk-free rate, r_{0}.- This implies that no solution to the
maximization problem unless E[r
_{j}]=r_{0}holds; (i.e. the expected return on every asset equals the rate of return on the risk-free asset). - 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[r
_{j}]=r_{0}_{ }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 r
_{0}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.

__The Mean-Variance Model__
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.
- 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])

__The FVR in the Mean-variance model__
The
FVR can be written as:

__μ__

_{j}

__– r__

_{ 0 }=

__μ__

_{P}

__– r__

_{ 0 }j= 1, 2, …, n (4.15)

σ

_{jP}/ σ_{P}σ_{P}- Where (σ
_{jP}) denotes the covariance between the rate of return on asset j and the rate of return on the portfolio as a whole. - The ratio (σ
_{jP}/ σ_{P}) equals the increment to risk (as expressed by σ_{P}) associated with an incremental change to the proportion of asset j in the portfolio. - Thus, the FVR states
that each asset’s expected rate of return in excess of the risk-free rate,
μ
_{j}–r_{0}, per unit of its contribution to overall risk, σ_{jP}/ σ_{P}, 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.
- For a mean-variance
investor, portfolio selection is an outcome of a two-stage process:
- . 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.

o . If a risk-free asset is
available, one of the two special assets can be chosen to comprise just the
risk-free asset alone.

o . 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.

- 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.

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