Financial advice free

It is often convenient to decompose S into a product set, for instance the product of a set representing time, T and another one, E, representing uncertain events: S =E ×T. What distinguishes a dynamic problem from a static one is the type of the decision-maker’s preferences on consequences: the problem is dynamic if and only if preferences vary according to information arrivals in the future (the distinction static vs dynamic is the object of Part III). Another distinction in decision problems formalisation refers to the type of uncertainties the decision-maker is facing. If the set has a known probability distribution, as in lottery betting, the problem is different from when there is not, as in horse race betting. There again, the distinction is relevant, only if the decision-maker preference order has different
properties in one case from the other. In consumer theory under uncertainty (or Arrow– Debreu’s General Equilibrium model, see Chapter 3), for example, preferences are general enough to encompass both cases. Some versions of this model, however, restrain preferences to the case where they decompose into a utility on certain consequences and a probability on random events, in order to obtain particular results.

The general formulation of a decision problem allows us to forget about the set of actions
in order to concentrate on the consequence function: c  A× S→C. The relevant object of study is the set,     ca · a ∈ A , of functions associating an uncontrolled variable to a particular consequence. Savage (1954) called “acts” those functions that correspond to actions, and the term was kept in the literature; we shall follow this use throughout. Acts
have the same representation as assets with payoffs contingent on future events, or contracts contingent on the realisation of some observable events. Assets and other contingent contracts are the objects of trades in financial markets.

In practice, solving a decision problem by decision theoretic tools relies heavily on the way the problem is described. In particular, the analysis of the set S of uncontrolled variables is crucial. We shall see in Chapter 4 how to use a decision tree to analyse and define the elements of a decision problem. For normative purposes, decision theory provides a basis on which a criterion may be defined and constructed in practice. A criterion is a numerical function on the set of actions such that action a is chosen over action a if and only if the consequences of a are preferred
to the consequences of a. In terms of acts, or assets, the criterion defines a value function. The economic literature calls “certainty equivalent”, or more precisely, “present certainty equivalent”, an act of valuation when future consequences are uncertain. This appellation clarifies that valuation erases the decision problem’s temporal and uncertain aspects. Indeed, the criterion “averages” consequences over time and uncertain events. Obviously, this
averaging may hide the limitations due to the conditions on preferences on which thecriterion was constructed. For example, a simple criterion such as the discounted expected monetary value assumes that the decision-maker uses a probability distribution on the set of events and a discount factor, which should represent its preferences for present consumption.

Get more info:  Economics and Finance of Risk and of the Future by Robert Kast and André Lapied

Games of chance are other examples of decision problems where consequences are contingent on some variables: uncertain outcomes. A lottery player faces the set of all authorised bets on each number (or event, e.g. combination of numbers), under its budget constraint. That is A. For each chosen bet, a set of consequences is defined by the lottery rules: the set of prizes associated with each event (contingent prize). If we call E the set of events on which players can bet, then the lottery’s rules yield a well-defined function: c  A×E→C where ca e is the prize obtained by betting a if event e is realised

This formalisation is adapted to consumption commodities with uncertain qualities as well.
The actions are the commodities, the consequences are the same commodities with the quality that is observed ex post. More generally, contingent assets correspond to this representation: their payoffs are contingent on some future events. As a general rule, a decision’s consequences occur in the future. They are contingent on time and on other events, both phenomena that escape the decision-maker’s control. An asset, in general, is contingent on such variables. Such uncontrolled phenomena may be due to:
• A random mechanism, such as a lottery.
• Other players’ reactions, e.g. in a chess game.
• Both, in a card game for example.
• Error variables: when a mechanism is assumed to have deterministic outcomes. Several error sources can blur the results obtained: flaws in the deterministic theory, reading imprecision, errors in using the device, “collateral damages”, etc.
• Social or physical factors considered by the decision-maker as impossible to forecast precisely: election results, earthquakes, etc.
• Limited ability of the decision-maker to dissociate consequences of a chosen action, as in shipping risks, venture capital, foreign country risk (see Chapter 9), etc. In all cases, we shall be able to formalise the link between actions and consequences by introducing a set S that represents uncontrolled variables, whatever their origin:
c  A×S→C Note that S is often an ill-defined set and is only an artefact introduced for formalisation purposes, in decision theory.4 However, the same formalisation can be used to represent contingent contracts, and, in this case, the set S is obviously well defined by the contracts’ clauses. S can always be “constructed” in the following way: the decision-maker knows the set of consequences associated with each decision a, say Ca. A set of “qualities” s may be
associated with each element of Ca  ca s is the quality of a particular element of Ca. The set S is obtained as the union of the sets of ca  s’s. Qualities may be time, random events, uncertain events (the difference between random and uncertain is the topic of Part II) coming from endogenous and/or exogenous causes and/or other decision-makers’ actions. In any case, the formalisation above is always relevant, even if the set S remains ill-defined, as long as the function c is well defined and induces a criterion on A from the preference order on C.

More: Economics and Finance of Risk and of the Future – Robert Kast and André Lapied