8.2. Risk management process

Chapter 9 The Mechanics of Risk Management We are not bad at all. We simply fail to enumerate all the assumptions underlying our estimates. Paul Rooke Here's a simple test of risk awareness in a project: Review the project plan and ask the leader

Chapter 10 Risk Management Rules It remains to describe only the details. In the previous chapters, enough attention has been paid to the basic concepts, so you can already move on to laying out the general rules of what is meant by risk management.

Chapter 15 The Dynamics of Risk Management Despite the repeated assertions that risk management should be an ongoing activity, you may still have the feeling that it is really being dealt with at the beginning of the project, and then (apart from

The concept of risk presupposes the presence of a risk taker; we will call him the Decision Maker (DM).

Let's say the issue of conducting a financial transaction under conditions of uncertainty is being considered. At the same time, the decision maker has several possible solutions i = 1,2, ..., t, and the real situation is uncertain and can take one of the options j = 1,2,..., n... Let it be known that if the decision maker takes i- e decision, and the situation will j- option, then the income q ij will be received. The matrix Q= (q ij) is called impact matrix (possible solutions).

Let us estimate the size of the risk in this scheme.

Let it be accepted i- e solution. Obviously, if it was known that the real situation would be j-i, then the decision maker would make a decision that gives income q j=. But, i- e decision is made in conditions of uncertainty. This means that the decision maker runs the risk of not getting q j , but only q ij... Thus, there is a real opportunity to receive less income, and this unfavorable outcome can be compared with the risk r ij, the size of which is expedient to estimate as the difference

r ij = q j - q ij . (2.1)

The matrix R = (r ij) is called risk matrix .

Example 2.1 . Using formula (2.1), create a risk matrix

R = (r ij) according to a given matrix of consequences

.

Solution... Obviously, q 1 =
= 8; similarly q 2 = 5, q 3 = 8, q 4 = 12. Therefore, the risk matrix has the form

.

6. Analysis of a related group of decisions in conditions of complete uncertainty

Complete uncertainty means the absence of information about the probabilistic states of the environment (“nature”), for example, about the probabilities of certain variants of a real situation; at best, the ranges of values ​​of the quantities under consideration are known. Recommendations for making decisions in such situations are formulated in the form of certain rules (criteria). Let's consider the main ones.

Criterion (rule) of maximax. This criterion is used to determine the solution variant that maximizes the maximum gains - for example, income - for each variant of the situation. This is the criterion extreme ("pink") optimism , according to which the best solution is the one that gives the maximum payoff equal to
... Considering i- e solution, suggest the best income-generating situation
, and then choose the solution with the largest a i .

Example 2.2. For the consequences matrix in example 2.1, choose a solution option based on the maximax criterion.

Solution. Find a sequence of values
:a 1 =8, a 2 =12, a 3 =10, a 4 =8. Of these, we find the largest: a 2 =12 ... Therefore, the maximax criterion recommends making the second solution ( i=2 ).

Wald's rule (the maximin rule, or the criterion of extreme pessimism). Considering the i-e solution, we will assume that in fact the situation is the worst, i.e. bringing the smallest income: b i = min q ij. But now let's choose a solution i 0 with the greatest ... So, Wald's rule recommends making a decision i 0 such that =
=
.

Example 2.3. For the consequences matrix in example 2.1, choose a solution option according to the Wald criterion.

Solution. In Example 2.1, we have b 1 = 2, b 2 = 2, b 3 = 3, b 4 = 1. Now from these values ​​we choose the maximum b 3 = 3. Hence, Wald's rule recommends making the third solution ( i=3 ).

Savage's Rule (criterion of minimax risk). This criterion is similar to the previous Wald criterion, but the decision maker makes a decision based not on the consequences matrix Q, but on the risk matrix R = (r ij). According to this criterion, the best solution is the one in which the maximum risk value will be the smallest, i.e. equal
... Considering the i-e solution, a situation of maximum risk is assumed r i =
and choose the solution i 0 with the smallest =
=
.

Example 2.4. For the initial data in example 2.1, choose a solution option in accordance with the Savage criterion.

Solution... Considering the risk matrix R, we find a sequence of values ​​r i =
: r 1 = 8, r 2 = 6, r 3 = 5, r 4 = 7. Of these values ​​we choose the smallest one: r 3 = 5. Hence, Savage's rule recommends making the third solution ( i=3 ). Note that this coincides with the Wald criterion selection.

Hurwitz rule (weighing pessimistic and optimistic approaches to the situation). According to this criterion, a solution option is selected, at which the maximum of the expression c i = (λminq ij + (1 - λ) maxq ij) is achieved, where 0 λ 1. Thus, this criterion recommends to be guided by some average result between extreme optimism and extreme pessimism... For λ = 0, the Hurwitz criterion coincides with the maximax criterion, and for λ = 1 it coincides with the Wald criterion. The λ value is selected from subjective (intuitive) considerations.

Example 2.5. For the consequences matrix shown in example 2.1, choose the best solution based on the Hurwitz criterion at λ = 1/2.

Solution. Considering the consequence matrix Q by rows, for each i we calculate the values ​​c i = 1 / 2minq ij + 1 / 2maxq ij. For example, with 1 = 1 / 22 + 1 / 28 = 5; are similarly found with 2 = 7; c 3 = 6.5; with 4 = 4.5. The largest is with 2 = 7. Consequently, the Hurwitz criterion for a given λ = 1/2 recommends choosing the second option ( i=2 ).

The company is their assessment. In real business practice, especially in cases of capital investment projects, the level of uncertainty is high. And it is not possible to use probabilistic or statistical methods of assessment measures due to the lack of initial information and the uniqueness of the situation. At such times, a methodology can come to the rescue, which uses the so-called "payoff matrix" and "risk matrix" using elements of game theory.

Introductory Elements of Game Theory

Game theory is a branch of applied mathematics that was developed in the 20s and 40s of the last century. The ideas and methods of the theory were developed thanks to the works of the American mathematician John von Neumann. Game theory refers to modeling the optimal solution in the face of uncertainty. Targeted actions of several parties are considered, each of which has its own interests. The parties in their interaction conflict with each other, since their goals are multidirectional. Therefore, this scientific theory, used in practical problems, is a methodology for finding solutions based on rational actions and conflicting interests.

In this article, we turn to the local economic section of the application of game theory in the so-called "game with nature" model. Let's say we are considering such an economic system as an enterprise. It contains a management system (subject of management), immersed in the business environment of a real business. The subject of management influences objects with the help of alternative solutions in conditions of uncertainty. This leads to a change in the state of the controlled system to one degree or another.

When searching for a solution, the control system does not have complete information about the state of the environment, although the uncertainty is not absolute. The subject of management certainly has some information, and empirical assumptions are made. Therefore, in game theory, the problem of finding a solution is called a “game with nature”, where the subject of control is a “player”, alternative solutions are “strategies”, and the function F (x, y) is a “payoff function of the subject”.

The payoff function matrix of game theory

The payoff function is usually set in a matrix form, which is shown above. X is the set of the player's strategic findings, and Y is the set of emerging states of the business environment (the nature of the business). It is necessary to find the best alternative for the control system, in other words, the optimal strategy. For this, in theory, there are several methods, the main of which involves a certain set of actions.

A hypothesis is put forward about the manifestation of the environment, which allows one to carry out a single assessment of each of the alternative strategies. Finding the maximum numerical estimate for a given hypothesis for one strategy allows us to consider it optimal. Setting the estimation algorithm and its implementation for each strategy allows comparing strategies in pairs and all together due to the formulated criterion.

For game theory, the following criteria have been developed based on the approaches of their authors or on the essence of the method:

• maximax criterion;
• Wald's test (V);
• Savage criterion (S);
• Hurwitz criterion (G);
• Laplace criterion (L).

Multi-criteria risk assessment methodology

Practically all variants of approaches for the advancement of basic hypotheses for evaluating the decision generated by the control system, including in the field of risk assessment, are expressed in the methodological criteria of “playing with nature”. For the convenience of considering the essence of the criteria, we concretize the field of application of the theory. Suppose that by the options Аi we mean the planned directions of activity of some business units, and instead of the parameters of the business environment Sj we take the project alternatives Пj (for example, in regional directions). The payoff values ​​will constitute the income for the line of activity implemented through the corresponding project. Let's localize an example of consideration to the subject area of ​​risk assessment of possible solutions. To make a decision, the analyzed information is concentrated in the payment matrix and the risk matrix.

Payment Matrix Model for Decision Making under Uncertainty

Above is the payoff matrix, which is also called the nature play matrix. The most difficult thing for preparing and making a decision is the construction of this matrix. The cost of filling in the matrix cells is high; it will not be possible to correct the incorrect result in the future. According to the methodology, the payment matrix is ​​transformed into a risk matrix.

Risk values ​​are calculated on the assumption that the decision maker on projects has the completeness of the information and chooses the alternative that corresponds to the best value for the state of the environment (project). In this case, the risk is 0. If he does not know about the best option, then the risk value is determined as the difference between the best option and the one selected in the Pj column.

Risk Matrix Model for Decision Making under Uncertainty

The criteria for "playing with nature" are ideologically constructed from the author's assumption of what level of optimism should be implemented to select an environmental alternative for each of the winnings. In addition, some criteria suggest making a choice among the options for risk values ​​according to the established rules. In other words, we first select the value of the win or risk for each line according to the established principle. And then from the selected values, according to the second rule, we find an option for making a decision. This methodology invites the subject of management (decision-makers) to consider several, if not all, criteria used for the analysis, bring them down into a single table and subject them to a thorough analysis. As a rule, it is logically possible to find a leading solution. Below is a summary methodological table of selection criteria.

A set of basic criteria for the "playing with nature" model

An example of risk assessment in a game theory model

For the purpose of a deeper understanding of the decision-making procedure based on the model under consideration, we will fill our example with digital data, perform calculations and a little analysis. Let's say that a company produces cosmetic products in three directions A1, A2 and A3. The task is to deploy three projects of regional product promotion, each with its own specifics of the implementation environment P1, P2 and P3. The estimated profitability values ​​for the options are collected in the payoff (income) matrix. On the basis of its data, the risk values ​​are calculated, which are then placed in the risk matrix.

Revenue and risk matrices with digital sample data

Applying the maximax criterion, we will select the maximum values ​​of income in projects for each of the considered areas. For direction A1, the maximum value is 90 units, for A2 - 80, for A3 - 70. Among them, A1 has the highest value, therefore, according to this criterion, the optimal solution is behind the specified direction of activity.

In the same way, using the rules of the Wald and Savage criteria, we enumerate the profitable values ​​and risk values ​​for j on the principles of maximini and minimax, respectively. According to both criteria, the direction A2 is chosen as the optimal strategy. Finally, we use the Hurwitz criterion, applying a subjective risk ratio of 0.5. The calculation of values ​​is presented below.

Enumeration of profitable values ​​in the example according to the Hurwitz criterion

All the results of enumeration and selection according to the applied criteria are summarized in one table. A2 is in the lead. Nevertheless, we introduce two more additional criteria to test the correctness of the hypothesis about the optimality of this choice. The first of them is based on comparing the calculated values ​​of the standard deviations from the average income values ​​for each of the studied areas of activity. The second additional criterion implies the application of the Hurwitz principle, but applied to the risk matrix. Essentially, both additional analyzes confirmed that the choice should be made in favor of A2.

Summary table of the application of various decision criteria under conditions of uncertainty

The ancient Greek mathematician Archimedes is credited with the catch phrase: "Give me a fulcrum and I will turn the world upside down." Those considered above with the use of elements of game theory are not distinguished by filigree precision. But in some cases, accuracy is not needed. The subject of management, represented by the company's top management, in conditions of uncertainty, especially in project activities, needs to streamline the totality of ideas about the composition of possible threats. This is sufficiently facilitated by the technique presented to your attention. It is quite possible to use it as a support in order to deliberately move the solution of the problem of choice off the ground and launch the project that carries the best strategic potential.