DECISION MAKING UNDER UNCERTAINTY

Decision Making Under Uncertainty

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.

• 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

•  The SPA comprises three basic ingredients:

1. State of the world, denoted by set S = {s₁, s₂, ..s_ℓ}. 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.

• 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(s₁, a), f(s₂, 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 (v₀) in every state and rate of return:

        r₀ = (v₀/p₀) - 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(W₁ ,W₂ , ..., W_ℓ )

W_k ≡ f(s_k, 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.
The wealth constraint states that the investor’s outlay on assets equals initial wealth:

  p₁x₁ + p₂x₂ + …+ p_nx_n = A 

Where (A) is the initial wealth and (x_j)s denote the number of units of each asset in the portfolio, so that (p_jx_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 x₁ + v_k2 x₂ + …+ 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(W₁ ,W₂ , ..., W_ℓ )

= π₁ u(W₁) + π₂ u(W₂) + …+ 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)] ≡ π₁ u(W₁) + π₂ u(W₂) + …+ 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 – r₀) 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 – r₀) u^/ (W)] = 0

c E[(r_j – r₀)] = 0

E[r_j ] = r₀                            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 r₀ and invest an unbounded amount in any asset for which E[r_j]> r₀;

and short sell an unbounded amount of any asset for which E[r_j]< 0, the proceeds being invested at the risk-free rate, r₀.

• This implies that no solution to the maximization problem unless E[r_j ]=r₀  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 ]=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.

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–r₀, 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.