Risk measures for income streams

A new measure of risk is introduced for a sequence of random incomes adapted to some filtration. This measure is formulated as the optimal net present value of a stream of adaptively planned commitments for consumption. The new measure is calculated by solving a stochastic dynamic linear optimization problem which, for finite filtrations, reduces to a deterministic linear program. We analyze properties of the new measure by exploiting the convexity and duality structure of the stochastic dynamic linear problem. The measure depends on the full distribution of the income process (not only on its marginal distributions) as well as on the filtration, which is interpreted as the available information about the future. The features of the new approach are illustrated by numerical examples. Department of Statistics and Decision Support Systems, Universitaetsstrasse 5, University of Vienna, 1090 Wien-Vienna, Austria; e-mail: georg.pflug@univie.ac.at Department of Management Science and Information Systems and RUTCOR, Rutgers University, 94 Rockefeller Rd, Piscataway, NJ 08854, U.S.A.; e-mail rusz@rutcor.rutgers.edu 1 1 Motivation Since the seminal work of Markowitz it is well understood that consequences of economic activity with uncertain success must be judged in two different and well distinguished dimensions. The mean refers to the average result among a set of possible scenarios, while the risk dimension describes the possible variation of the results under varying scenarios. In the Markowitz model the risk is measured by the variance of the outcome (cf. [8, 9]). In the mean–risk setting the decision maker is faced with a two-objective situation: he/she wants to maximize the mean return and to minimize the risk at the same time. As for all multi-objective situations, there is in general no uniquely defined best decision, which is optimal in both dimensions and one has to seek for compromises. The set of solutions which are Pareto-efficient in the sense of these two objectives is called the mean–risk efficient frontier. In some models for optimal decision making the two dimensions are often mixed by introducing a nondecreasing concave utility function. Risk aversion, i.e. the degree of taking the risk dimension into account, can be modeled by the negative curvature of the utility function. However, it is highly desirable to clearly separate the two dimensions and to make the compromising strategy as transparent as possible, and the efficient frontier approach provides such a transparency. In the first step, the efficient frontier is calculated for a given decision problem and the non-dominated decisions are identified. In the second step, the compromise decision may be chosen among the efficient candidates. There is a vast literature on one-period decision models using several notions of measuring risk (see, e.g., [1, 7, 13, 14, 19]). In the multiperiod situation, however, most proposals focus on the risk contained in the terminal wealth (see [3, 11, 12] and the references therein). The purpose of this paper is to propose a risk measure for multiperiod models which incorporates the risk contained in intermediate incomes. Suppose that I1, . . . , IT is a stream of random incomes. A simple but inappropriate way of defining the multiperiod risk would be to look at the marginal variables separately and fabricate a combined risk measure as a combination of the univariate risk measures. The distinction can be made clear by advocating an example which goes back to Philippe Artzner. Suppose that a coin is thrown three times. In situation 1, a reward of 1 is paid if the coin shows more heads than tails. In situation 2, the same reward is paid if the last throw shows head. Do the two situations reflect the same risk for the decision maker? If the whole experiment is done in a few seconds, one is inclined to say ‘yes’. But suppose that the throw of the coin happens just once a year. Then in situation 1 the decision maker knows her income one year ahead, which is a clear advantage over situation 2. Thus situation 1 should turn out to be less risky than 2, although their income variables have identical marginals. We shall return to this example in section 7. To valuate the entire income stream and not just the terminal wealth appears to be appropriate in many models. For instance, pension funds promise a income streams to their clients. Since the rights emerging from a pension fund membership are not bequeathable, clients are not interested in the terminal wealth at some future moments of time, at which they may not be able to consume it. At least in Europe, pension funds are 2 only administrators and not owners of the funds. They are primarily interested in high management fees, which come from a large number of customers. Customers can only be attracted if a good income stream can be guaranteed. Thus it is in the own interest of a pension fund to keep an eye on the customer’s income stream process (see, e.g., [10, 17]). Besides that, the income stream risk must also be considered in other cases, where the primal investment is just made for the purpose of getting the income at later periods. Real options, or loans to companies are good examples here. The paper is organized as follows. After an introductory section about one-period measures, we introduce our concept of a multiperiod measure in section 3. Its properties are analyzed in section 4, and section 5 contains explicit linear programming models for the case of finitely many scenarios. In section 6 we consider mean–risk models for our measures and compare them to models based on the terminal wealth distribution. Illustrative examples are contained in section 7. All calculations can be done by standard linear programming packages. 2 The one-period case Let I be a random income variable defined on some probability space (Ω, F , P ). The risk contained in I is caused by the lack of information about its exact value. A variable, but predictable value of I is riskless. If a natural catastrophe, e.g. a flood, were completely predictable, there would be no risk and no company would insure against it. If a decision maker were clairvoyant, he/she would face no risk since he/she would see the future in a deterministic way and would be able to adapt to it. For us, normal humans, some but not all information about the future may be available. The amount of information available may be expressed in terms of some σ-algebra F ⊆ F . The extreme cases are the clairvoyant (F = F) and the totally uninformed (F = F0 = {Ω, ∅}). The ultimate goal of engaging in risky enterprises with uncertain income opportunities is consumption. Consumption, however, can only be realized after deciding about the amount one wants to commit for this purpose (to buy a house, a car etc.). Suppose that the decision maker decides to commit an amount a. In this case, he/she risks not achieving this decided target, since I may be less than a. However, he/she may insure against the shortfall event, i.e. the event that I 1. The costs for insurance decrease the possible consumption. If, on the other hand, some surplus is left after consumption, this surplus is discounted by a factor d < 1, since saving does not provide the same satisfaction as the consumption committed for. The Expected Net Present Value (ENPV) of the consumption and savings is therefore E(a + d[I − a] − q[I − a]−). A rational decision maker maximizes the ENPV with respect to the available information We use the notation [x] = max(x, 0) and [x]− = max(−x, 0).