Method minimum risk. This method was developed in connection with the problems of radar, but can be quite successfully used in problems of technical diagnostics.

Let the parameter x be measured (for example, the vibration level of the product) and, based on the measurement data, it is required to make a conclusion about the possibility of continuing operation (diagnosis - good condition) or about sending the product for repair (diagnosis - faulty condition).

On fig. 1 shows the values ​​of the probability density of the diagnostic parameter x for two states.

Let the control norm for the level of vibrations be set.

In accordance with this norm, they accept:

The sign means that an object with a vibration level x is assigned to a given state.

From fig. 1 it follows that any choice of value is associated with a certain risk, since the curves intersect.

There are two types of risk: the risk of "false alarm", when a serviceable product is considered faulty, and the risk of "missing the target", when a faulty product is considered good.

In theory statistical control these are called provider risk and receiver risk, or type I and type II errors.

Given the probability of a false alarm

and the probability of missing the target

The task of the theory of statistical decisions is to choose the optimal value

The minimum risk method considers the total cost of risk

where is the “price” of a false alarm; - "price" of missing the target; - a priori probabilities of diagnoses (conditions), determined by preliminary

Rice. 1. Probability density of a diagnostic feature

statistical data. The value represents the "average value" of the loss in an erroneous decision.

From the necessary minimum condition

we get

It can be shown that for unimodal distributions condition (23) always ensures the minimum of the value If the cost of erroneous decisions is the same, then

The last relation minimizes the total number of erroneous decisions. It also follows from the Bayes method.

Neumann-Pearson method. This method proceeds from the condition of the minimum probability of skipping a defect at an acceptable level of false alarm probability.

Thus, the probability of a false alarm

where is the allowable false alarm level.

In the one-parameter problems under consideration, the minimum probability of missing the target is achieved when

The last condition determines the boundary value of the parameter (value

When assigning a value, take into account the following:

1) the number of decommissioned products must exceed the expected number of defective products due to the inevitable errors in the condition assessment method;

2) the accepted false alarm value should not, unless absolutely necessary, disrupt normal operation or lead to large economic losses.

Risk avoidance. It is extremely difficult to completely eliminate the possibility of losses, so in practice this means not taking on risk beyond the usual level.

Loss Prevention. An investor may attempt to reduce, but not completely eliminate, specific losses. Loss prevention means the ability to protect yourself from accidents through a specific set of preventive actions. Preventive measures are understood as measures aimed at preventing unforeseen events in order to reduce the likelihood and magnitude of losses. Usually, measures are taken to prevent losses, such as constant control and analysis of information on the securities market; safety of capital invested in securities, etc. Every investor is interested in preventive activities, but their implementation is not always possible for technical and economic reasons and is often associated with significant costs.

Preventive measures, in our opinion, include reporting. Reporting is a systematic documentation of all information related to the analysis and assessment of external and internal risks, with fixation of the residual risk after all risk management measures have been taken, etc. All this information should be entered into certain databases and reporting forms that are easy for investors to use in the future.

Loss minimization. An investor may try to prevent a significant portion of their losses. Loss minimization methods are diversification and limiting.

Diversification is a method aimed at reducing risk, in which the investor invests in different areas(various types of securities, enterprises of various sectors of the economy), so that in case of a loss in one of them, compensate for this at the expense of another area.
Diversification of a portfolio of securities involves the inclusion in the portfolio of various securities with different characteristics (levels of risk, profitability, liquidity, etc.). Possible low incomes (or losses) on one securities will be compensated by high incomes on other securities. The selection of a diversified portfolio requires certain efforts, primarily related to the search for complete and reliable information about the investment qualities of securities. To ensure the stability of the portfolio, the investor limits the size of investments in securities of one issuer, thus achieving a reduction in the degree of risk. When investing in shares of enterprises in various sectors of the national economy, sectoral diversification is carried out.

Diversification is one of the few risk management techniques that any investor can use. Note, however, that diversification reduces only unsystematic risk. And the risk of investing capital is influenced by the processes taking place in the economy as a whole, such as the movement of the bank interest rate, the expectation of an increase or decrease, and so on, and the risk associated with them cannot be reduced by diversification. Therefore, the investor needs to use other ways to reduce risk.

Limiting is the establishment of maximum amounts (limits) for investing capital in certain types of securities, etc. Setting the size of limits is a multi-step procedure, including establishing a list of limits, the size of each of them, and their preliminary analysis. Compliance with the established limits ensures economic conditions to preserve capital, generate sustainable income and protect the interests of investors.

Search for information- this is a method aimed at reducing risk by finding and using the necessary information for an investor to make a risky decision.

The adoption of erroneous decisions in most cases is associated with the absence or lack of information. Information asymmetry, where individual market participants have access to important information, which the rest do not have interested people, prevents investors from behaving rationally and is a barrier to the efficient use of resources and funds.

Obtaining the necessary information, increasing the level of investor information support can significantly improve the forecast and reduce the risk. To determine how much information is needed and whether it is worth buying, one must compare the expected marginal benefits of information with the expected marginal cost of obtaining it. If the expected benefit from the purchase of information exceeds the expected marginal cost, then such information must be acquired. If it is the other way around, then it is better to refuse to buy such expensive information.

Currently, there is a business area called accounting, related to the collection, processing, classification, analysis and presentation various kinds financial information. Investors can use the services of professionals in this business area.

Loss minimization methods are often referred to as risk control methods. The use of all these methods of preventing and reducing losses is associated with certain costs, which should not exceed the possible amount of damage. As a rule, an increase in the cost of preventing a risk leads to a decrease in its danger and the damage caused by it, but only up to a certain limit. This limit occurs when the amount of annual costs of risk prevention and reduction becomes equal to the estimated amount of annual damage from the realization of the risk.

Reimbursement Methods(least cost) losses apply when an investor incurs losses despite efforts to minimize their losses.

Risk transfer. Most often, the transfer of risk occurs through hedging and insurance.

Hedging- this is a system for concluding futures contracts and transactions, taking into account possible future changes in prices, rates and pursuing the goal of avoiding the adverse consequences of these changes. The essence of hedging is the purchase (sale) of futures contracts simultaneously with the sale (purchase) of real goods with the same delivery time and the reverse operation with the actual sale of the goods. As a result, sharp price fluctuations are smoothed out. IN market economy hedging is a common way to reduce risk.

According to the technique of carrying out operations, there are two types of hedging:

Hedging up(purchase hedging or long hedge) is an exchange transaction for the purchase of futures contracts (forwards, options and futures). Hedging for an increase is used in cases where it is necessary to insure against a possible increase in rates (prices) in the future. It allows you to set the purchase price much earlier than the actual asset is purchased.

Down hedging(selling hedge or short hedge) is an exchange transaction for the sale of futures contracts. Downward hedging is used in cases where it is necessary to insure against a possible decrease in rates (prices) in the future.

Hedging can be done using futures contracts and options.

Hedging futures contracts implies the use of standard (in terms of terms, volumes and terms of delivery) contracts for the purchase and sale of securities in the future, circulating exclusively on stock exchanges.

The positive aspects of hedging using futures contracts are:

• availability of an organized market;
• the ability to hedge without taking on significant credit risks. Credit risk is mitigated by efficient offsetting mechanisms offered by the exchange;
• ease of adjusting the size of the hedging position or closing it;
• availability of statistics on prices and trading volumes for available instruments, which allows you to choose the optimal hedging strategy.

The downsides of hedging with futures contracts are:

• inability to use fixed-term contracts of arbitrary size and maturity. Futures contracts are standard contracts, their set is limited, because of this, the basis risk of hedging cannot be made less than a certain specified value;
• the need for commission expenses when concluding transactions;
• the need to divert funds and accept liquidity risk when hedging. The sale and purchase of Standard Contracts require a deposit margin and its subsequent increase in the event of an unfavorable price change.

Hedging helps to reduce the risk from adverse price or exchange rate changes, but does not provide an opportunity to take advantage of favorable price changes. During the hedging operation, the risk does not disappear, it changes its carrier: the investor transfers the risk to the stock speculator.

Insurance is a method aimed at reducing risk by turning accidental losses into relatively small fixed costs. When buying insurance (concluding an insurance contract), the investor transfers the risk to the insurance company, which compensates for various losses and damages caused by adverse events by paying insurance compensation and sums insured. For these services, she receives a fee (insurance premium) from the investor.

The risk insurance regime in an insurance company is established taking into account the insurance premium, additional services provided by the insurance company, and financial position insured. The investor must determine the ratio between the insurance premium and the sum insured that is acceptable to him, taking into account the additional services provided by the insurance company.

If the investor carefully and clearly assesses the balance of risk, then he thereby creates the prerequisites for avoiding unnecessary risk. Every opportunity should be taken to increase the predictability of potential losses so that an investor can have the data they need to explore all of their payout options. And then he will turn to the insurance company only in cases of catastrophic risk, that is, very high in terms of probability and possible consequences.

Transfer of risk control. The investor may entrust control of the risk to another person or group of persons by transferring:

• real property or activities associated with risk;
• responsibility for the risk.

An investor can sell any securities in order to avoid investment risk, can transfer his property (securities, cash, etc.) to trust management of professionals (trust companies, investment companies, financial brokers, banks, etc.), thereby transferring all the risks associated with this property and its management activities. An investor can transfer risk by transferring a certain activity, for example, transferring the functions of finding the optimal insurance coverage and portfolio of insurers to an insurance broker who will deal with this.

Risk distribution is a method in which the risk of possible damage or loss is divided among the participants so that the possible losses of each are small. This method underlies risk financing. The existence of various collective funds, collective investors is based on this method.

The main principle of risk financing is the division and distribution of risk through:

1. preliminary accumulation of financial resources in general funds not related to a specific investment project;
2. organization of the fund in the form of partnership;
3. management of several partnership funds at different stages of development.

Funds risk (venture) financing associated with both the management of individual enterprises and the organization of independent risk-taking firms-investors. The main purpose of such funds is to support start-up science-intensive companies (ventures), which, in case of failure of the entire project, will take on part of the financial losses. Venture capital is used to finance the latest scientific and technical developments, their implementation, the release of new types of products, the provision of services and is formed from the contributions of individual investors, large corporations, government departments, insurance companies, banks.

In practice, risks are not strictly divided according to certain categories, and it is not easy to give precise risk management recommendations, however, we suggest using the following risk management framework.

Risk management scheme:

Each of these risk management methods has its own advantages and disadvantages. The specific method is selected depending on the type of risk. An investor (or a risk specialist) chooses methods to reduce risk that are most capable of influencing the amount of income or the value of his capital. The investor must decide whether it is more profitable to resort to traditional diversification or to use some other risk management method in order to most reliably cover possible losses and infringe on their financial interests to the least extent. A combination of several methods at once may ultimately be the best solution.

From a cost minimization point of view, any risk mitigation method should be used if it requires the least cost. The costs of risk prevention and loss minimization should not exceed the possible damage. Each method should be used as long as the cost of its application does not begin to exceed the return.

Reducing the risk level necessitates technical, organizational measures that require certain, and in many cases even significant costs. And this is not always advisable. Thus, economic considerations set some limits on risk reduction for a particular investor. When deciding on risk reduction, it is necessary to compare a number of indicators related to the costs that provide acceptable level risk and expected effect.

Summarizing the above methods of portfolio risk management, we can distinguish two forms of securities portfolio management:

• passive;
• active.

The passive form of management consists in creating a well-diversified portfolio with a predetermined level of risk and keeping the portfolio unchanged for a long time.

The passive form of securities portfolio management is carried out using the following main methods:

• diversification;
• index method (mirror reflection method);
• portfolio maintenance.

As already noted, diversification involves the inclusion in the portfolio of a variety of securities with different characteristics. The selection of a diversified portfolio requires certain efforts, primarily related to the search for complete and reliable information about the investment qualities of securities. The structure of a diversified portfolio of securities should correspond to certain goals of investors. When investing in shares of industrial companies, sectoral diversification is carried out.

Index Method, or the method of mirror reflection, is based on the fact that a certain portfolio of securities is taken as a standard. The structure of the reference portfolio is characterized by certain indexes. Further, this portfolio is mirrored. Usage this method complicated by the difficulty of selecting a reference portfolio.

Portfolio preservation based on maintaining the structure and maintaining the level general characteristics portfolio. It is not always possible to keep the portfolio structure unchanged, since, given the unstable situation in the Russian stock market buy other securities. In large transactions with securities, a change in their exchange rate may occur, which will entail a change in the current value of assets. A situation is possible when the amount of sale of securities of joint-stock companies exceeds the cost of their purchase. In this case, the manager must sell part of the portfolio of securities in order to make payments to clients who return their shares to the company. Large sales volumes can have a downward effect on a company's stock prices, which negatively affects its financial position.

The essence of the active form of management is permanent job with a portfolio of securities. The basic characteristics of active management are:

• selection of certain securities;
• determining the timing of the purchase or sale of securities;
• constant swapping (rotation) of securities in the portfolio;
• providing net income.

If the interest rate of the Central Bank of the Russian Federation is predicted to decrease, then it is recommended to buy long-term bonds with low income but coupons, the rate of which rises quickly when the interest rate falls. At the same time, short-term bonds with high coupon yields should be sold, since their rate in this situation will fall. If the dynamics of the interest rate reveals uncertainty, then the manager will turn a significant part of the securities portfolio into assets of increased liquidity (for example, into term accounts).

When choosing an investment strategy, the factors that determine the sectoral structure of the investment portfolio are the risk and return on investment. When choosing securities, the factors that determine the return on investment are the profitability of production and the prospects for growth in sales.

Koshechkin S.A. PhD in Economics, International Institute of Economics, Law and Management (MIEPM NNGASU)

Introduction

In practice, an economist in general and a financier in particular very often have to evaluate the effectiveness of a particular system. Depending on the characteristics of this system, the economic meaning of efficiency can be put into various formulas, but their meaning is always the same - it is the ratio of results to costs. In this case, the result has already been obtained, and the costs have been incurred.

But how important are such a posteriori estimates?

Of course, they are of a certain value for accounting, characterize the work of the enterprise over the past period, etc., but it is much more important for a manager in general and a financial manager in particular to determine the effectiveness of the enterprise in the future. And in this case The efficiency formula needs to be slightly adjusted.

The fact is that we do not know with 100% certainty either the value of the result obtained in the future, or the value of potential future costs.

The so-called. "uncertainty", which we must take into account in our calculations, otherwise we will simply get the wrong solution. As a rule, this problem arises in investment calculations when determining the effectiveness of an investment project (IP), when an investor is forced to determine for himself what risk he is ready to take in order to obtain the desired result, while the solution of this two-criteria problem is complicated by the fact that investors' tolerance for risk is individual.

Therefore, the criterion for making investment decisions can be formulated as follows: IP is considered effective if its profitability and risk are balanced in an acceptable proportion for the project participant and formally represented as expression (1):

IP efficiency = (Return; Risk) (1)

By "yield" it is proposed to understand economic category characterizing the ratio of results and costs of IP. IN general view IP profitability can be expressed by formula (2):

Yield =(NPV; IRR; PI; MIRR) (2)

This definition does not conflict with the definition of the term "efficiency", since the definition of the concept of "efficiency", as a rule, is given for the case of complete certainty, i.e. when the second coordinate of the "vector" - risk, is equal to zero.

Efficiency = (Profitability; 0) = Result:Costs (3)

Those. in this case:

Efficiency ≡ Profitability(4)

However, in a situation of "uncertainty" it is impossible to speak with 100% certainty about the magnitude of the results and costs, since they have not yet been obtained, but are only expected in the future, therefore, it becomes necessary to make adjustments to this formula, namely:

P p and P s - the possibility of obtaining a given result and costs, respectively.

Thus, in this situation, a new factor appears - a risk factor, which certainly must be taken into account when analyzing the effectiveness of IP.

Definition of risk

In general, risk is understood as the possibility of some unfavorable event occurring, entailing various kinds of losses (for example, physical injury, loss of property, income below the expected level, etc.).

The existence of risk is associated with the inability to predict the future with 100% accuracy. Based on this, the main property of risk should be singled out: risk occurs only in relation to the future and is inextricably linked with forecasting and planning, and therefore with decision-making in general (the word “risk” literally means “making a decision”, the result of which is unknown). Following the foregoing, it is also worth noting that the categories “risk” and “uncertainty” are closely related and are often used as synonyms.

First, the risk takes place only in those cases when it is necessary to make a decision (if this is not the case, there is no point in taking risks). In other words, it is the need to make decisions under conditions of uncertainty that gives rise to risk; in the absence of such a need, there is no risk.

Second, risk is subjective, while uncertainty is objective. For example, the objective lack of reliable information about the potential volume of demand for manufactured products leads to a spectrum of risks for project participants. For example, the risk generated by uncertainty due to the lack of marketing research for an individual entrepreneur, turns into a credit risk for the investor (the bank financing this individual entrepreneur), and in case of no return of the loan, into the risk of loss of liquidity and further into the risk of bankruptcy, and for the recipient this risk is transformed into the risk of unforeseen market fluctuations., and for each of the participants in the individual entrepreneur, the manifestation of risk is individual both in qualitative and quantitative terms.

Speaking of uncertainty, we note that it can be specified in different ways:

In the form of probability distributions (the distribution of a random variable is known exactly, but it is not known what specific value the random variable will take)

In the form of subjective probabilities (the distribution of a random variable is unknown, but the probabilities of individual events are known, determined by an expert);

In the form of interval uncertainty (the distribution of a random variable is unknown, but it is known that it can take any value in a certain interval)

In addition, it should be noted that the nature of uncertainty is formed under the influence of various factors:

Temporal uncertainty is due to the fact that it is impossible to predict the value of a particular factor in the future with an accuracy of 1;

The uncertainty of the exact values ​​of the parameters of the market system can be characterized as the uncertainty of the market situation;

The unpredictability of the behavior of participants in a situation of conflict of interest also gives rise to uncertainty, etc.

The combination of these factors in practice creates a wide range of different types of uncertainty.

Since uncertainty is a source of risk, it should be minimized by acquiring information, in the ideal case, trying to reduce uncertainty to zero, that is, to complete certainty, by obtaining high-quality, reliable, comprehensive information. However, in practice, as a rule, this cannot be done, therefore, when making a decision under conditions of uncertainty, it should be formalized and the risks posed by this uncertainty should be assessed.

Risk is present in almost all spheres of human life, therefore it is impossible to formulate it precisely and unambiguously, because the definition of risk depends on the scope of its use (for example, for mathematicians, risk is a probability, for insurers it is an object of insurance, etc.). It is no coincidence that there are many definitions of risk in the literature.

Risk is the uncertainty associated with the value of an investment at the end of a period.

Risk is the probability of an unfavorable outcome.

Risk - possible loss caused by the occurrence of random adverse events.

Risk is a possible danger of losses arising from the specifics of certain natural phenomena and activities of human society.

Risk level financial loss expressed a) in the possibility of not achieving the goal; b) in the uncertainty of the predicted result; c) in the subjectivity of the assessment of the predicted result.

The whole set of studied risk calculation methods can be grouped into several approaches:

First approach : risk is estimated as the sum of the products of possible damages, weighted according to their probability.

Second approach : risk is assessed as the sum of risks from decision making and risks external environment(independent of our decisions).

Third Approach : risk is defined as the product of the probability of a negative event occurring by the degree of negative consequences.

All of these approaches have the following drawbacks to varying degrees:

The relationship and differences between the concepts of "risk" and "uncertainty" are not clearly shown;

The individuality of the risk, the subjectivity of its manifestation were not noted;

The range of risk assessment criteria is limited, as a rule, to one indicator.

In addition, the inclusion in risk assessment indicators of such elements as opportunity costs, lost profits, etc., which is found in the literature, according to the author, is inappropriate, because. they are more about return than risk.

The author proposes to consider risk as an opportunity ( R) losses ( L), arising from the need to make investment decisions under conditions of uncertainty. At the same time, it is emphasized that the concepts of "uncertainty" and "risk" are not identical, as is often believed, and the possibility of an adverse event should not be reduced to one indicator - probability. The degree of this possibility can be characterized by various criteria:

The probability of an event occurring;

The amount of deviation from the predicted value (range of variation);

Dispersion; expected value; standard deviation; asymmetry coefficient; kurtosis, as well as many other mathematical and statistical criteria.

Since uncertainty can be specified by its various types (probabilistic distributions, interval uncertainty, subjective probabilities, etc.), and risk manifestations are extremely diverse, in practice one has to use the entire arsenal of the listed criteria, but in the general case, the author suggests using the expectation and standard deviation as the most adequate and well-proven criteria in practice. In addition, it is emphasized that risk assessment should take into account individual risk tolerance ( γ ), which is described by indifference or utility curves. Thus, the author recommends that risk be described by the three parameters mentioned above (6):

Risk = (P; L; γ) (6)

Comparative analysis of statistical risk assessment criteria and their economic nature are presented in the next paragraph.

Statistical risk criteria

Probability (R) events (E)- the ratio of the number TO cases of favorable outcomes total number all possible outcomes (M).

P (E) \u003d K / M (7)

The probability of an event occurring can be determined by an objective or subjective method.

An objective method for determining probability is based on calculating the frequency with which a given event occurs. For example, the probability of getting heads or tails when flipping a perfect coin is 0.5.

The subjective method is based on the use of subjective criteria (judgment of the evaluator, his personal experience, expert estimate) and the probability of an event in this case may be different, being estimated by different experts.

In connection with these differences in approaches, several nuances should be noted:

First, objective probabilities have little to do with investment decisions that cannot be repeated many times, while the probability of getting heads or tails is 0.5 with a significant number of tosses, and for example, with 6 tosses, 5 heads and 1 tails can fall.

Secondly, some people tend to overestimate the likelihood of adverse events and underestimate the likelihood of positive events, while others, on the contrary, i.e. react differently to the same probability (cognitive psychology calls this the context effect).

However, despite these and other nuances, it is believed that the subjective probability has the same mathematical properties as the objective one.

Span variation (R)- the difference between the maximum and minimum value of the factor

R= X max - X min (8)

This indicator gives a very rough estimate of the risk, as it is an absolute indicator and depends only on the extreme values ​​of the series.

Dispersion – the sum of squared deviations of a random variable from its mean value, weighted by the corresponding probabilities.

(9)

Where M(E)– average or expected value (mathematical expectation) of a discrete random variable E is defined as the sum of the products of its values ​​and their probabilities:

(10)

Mathematical expectation is the most important characteristic of a random variable, because serves as the center of its probability distribution. Its meaning lies in the fact that it shows the most plausible value of the factor.

The use of variance as a measure of risk is not always convenient, because its dimension is equal to the square of the unit of measurement of the random variable.

In practice, the results of the analysis are more illustrative if the scatter index of the random variable is expressed in the same units of measurement as the random variable itself. For this purpose, the standard (root mean square) deviation σ(Ε).

(11)

All of the above indicators have one common drawback - they are absolute indicators, the values ​​of which predetermine absolute values original factor. It is much more convenient, therefore, to use the coefficient of variation (CV).

(12)

Definition CV especially evident for cases where the average values ​​of a random event differ significantly.

There are three points to be made regarding the risk assessment of financial assets:

First, in a comparative analysis of financial assets, profitability should be taken as a basic indicator, since the value of income in absolute form can vary significantly.

Secondly, the main indicators of risk in the capital market are dispersion and standard deviation. Since profitability (profitability) is taken as the basis for calculating these indicators, the criterion is relative and comparable for different types of assets, there is no urgent need to calculate the coefficient of variation.

Thirdly, sometimes in the literature the above formulas are given without taking into account the weighting on the probability. In this form, they are suitable only for retrospective analysis.

In addition, the criteria described above were supposed to apply to a normal probability distribution. Indeed, it is widely used in the analysis of risks of financial transactions, because its most important properties (distribution symmetry with respect to the mean, negligible probability of large deviations of a random variable from the center of its distribution, the three-sigma rule) makes it possible to significantly simplify the analysis. However, not all financial operations assume a normal distribution of income (the issues of choosing a distribution are discussed in more detail below). For example, the distribution of probabilities of receiving income from operations with derivative financial instruments (options and futures) is often characterized by asymmetry (skew) with respect to the mathematical expectation of a random variable (Fig. 1).

So, for example, an option to purchase a security allows its owner to make a profit in the event of a positive return and at the same time avoid losses in the event of a negative one, i.e. in effect, the option cuts off the distribution of returns at the point where losses begin.

Fig.1 Probability density plot with right (positive) skewness

In such cases, the use of only two parameters (mean and standard deviation) in the analysis process can lead to incorrect conclusions. The standard deviation does not adequately characterize the risk in case of biased distributions, because it is ignored that most of the volatility is on the “good” (right) or “bad” (left) side of the expected return. Therefore, when analyzing asymmetric distributions, an additional parameter is used - the coefficient of asymmetry (bevel). It is a normalized value of the third central moment and is determined by formula (13):

The economic meaning of the asymmetry coefficient in this context is as follows. If the coefficient has a positive value (positive skew), then the highest returns (right tail) are considered more likely than the lowest ones and vice versa.

The skewness coefficient can also be used to approximate the hypothesis of a normal distribution of a random variable. Its value in this case should be 0.

In some cases, a right-shifted distribution can be reduced to a normal distribution by adding 1 to the expected return and then calculating the natural logarithm of the resulting value. Such a distribution is called lognormal. It is used in financial analysis along with normal.

Some symmetric distributions can be characterized by a fourth normalized central moment – kurtosis (e).

(14)

If the kurtosis value is greater than 0, the distribution curve is more pointed than the normal curve and vice versa.

The economic meaning of kurtosis is as follows. If two transactions have symmetrical distributions of returns and the same averages, the investment with the larger kurtosis is considered less risky.

For a normal distribution, kurtosis is 0.

Choice of distribution of a random variable.

The normal distribution is used when it is impossible to accurately determine the probability that a continuous random variable takes on a particular value. The normal distribution assumes that the variants of the predicted parameter gravitate towards the mean. Parameter values ​​that are significantly different from the average, i.e. located in the "tails" of the distribution, have a low probability of implementation. This is the nature of the normal distribution.

The triangular distribution is a surrogate for the normal distribution and assumes a distribution that increases linearly as it approaches the mode.

The trapezoidal distribution assumes the presence of an interval of values ​​with the highest probability of realization (HPR) within the WFD.

Uniform distribution is chosen when it is assumed that all variants of the predicted indicator have the same probability of realization.

However, when the random variable is discrete rather than continuous, apply binomial distribution And Poisson distribution .

Illustration binomial distribution An example is the throwing of a die. In this case, the experimenter is interested in the probabilities of “success” (falling out of a face with a certain number, for example, with “six”) and “failure” (falling out of a face with any other number).

The Poisson distribution is applied when the following conditions are met:

1. Each small interval of time can be considered as an experience, the result of which is one of two things: either "success" or its absence - "failure". The intervals are so small that there can only be one "success" in one interval, the probability of which is small and unchanged.

2. The number of “successes” in one large interval does not depend on their number in another, i.e. "successes" are randomly scattered over time intervals.

3. The average number of "successes" is constant throughout time.

Usually the Poisson distribution is illustrated by the example of registering the quantity traffic accidents per week on a certain section of the road.

Under certain conditions, the Poisson distribution can be used as an approximation of the binomial distribution, which is especially convenient when the application of the binomial distribution requires complex, laborious and time-consuming calculations. The approximation guarantees acceptable results under the following conditions:

1. The number of experiments is large, preferably more than 30. (n=3)

2. The probability of "success" in each experiment is small, preferably less than 0.1. (p=0.1) If the probability of "success" is high, then the normal distribution can be used for replacement.

3. The expected number of “successes” is less than 5 (np=5).

In cases where the binomial distribution is very laborious, it can also be approximated by a normal distribution with a “continuity correction”, i.e. making the assumption that, for example, the value of a discrete random variable 2 is the value of a continuous random variable in the interval from 1.5 to 2.5.

The optimal approximation is achieved under the following conditions: n=30; np=5, and the probability of “success” p=0.1 (optimal value p=0.5)

The price of risk

It should be noted that in the literature and practice, in addition to statistical criteria, other risk measurement indicators are also used: the amount of lost profits, lost income, and others, usually calculated in monetary units. Of course, such indicators have the right to exist, moreover, they are often simpler and clearer than statistical criteria, however, in order to adequately describe the risk, they must also take into account its probabilistic characteristics.

C risk = (P; L) (15)

L - is defined as the sum of possible direct losses from an investment decision.

To determine the price of risk, it is recommended to use only indicators that take into account both coordinates of the “vector”, both the possibility of an adverse event and the amount of damage from it. As such indicators, the author proposes to use, first of all, the variance, the standard deviation ( RMS-σ) and coefficient of variation ( CV). For the possibility of economic interpretation and comparative analysis of these indicators, it is recommended to convert them into a monetary format.

The need to take into account both indicators can be illustrated by the following example. Assume the probability that a concert for which a ticket has already been bought will take place with a probability of 0.5, it is obvious that the majority of those who have bought a ticket will come to the concert.

Now suppose that the probability of a favorable outcome of an airliner flight is also 0.5, it is obvious that the majority of passengers will refuse to fly.

This abstract example shows that with equal probabilities of an unfavorable outcome, the decisions made will be polar opposites, which proves the need to calculate the "price of risk".

Particular attention is focused on the fact that the attitude of investors to risk is subjective, therefore, in the description of risk, there is a third factor - the investor's tolerance for risk. (γ). The need to take this factor into account is illustrated by the following example.

Suppose we have two projects with the following parameters: Project "A" - profitability - 8% Standard deviation - 10%. Project "B" - profitability - 12% Standard deviation - 20%. The initial cost of both projects is the same - $100,000.

The probability of being below this level will be as follows:

From which it clearly follows that project "A" is less risky and should be preferred to project "B". However, this is not entirely true, since final decision about investing will depend on the degree of risk tolerance of the investor, which can be clearly represented by an indifference curve .

Figure 2 shows that projects "A" and "B" are equivalent for the investor, since the indifference curve unites all projects that are equivalent for the investor. In this case, the nature of the curve for each investor will be individual.

Fig.2. Indifference Curve as a Criterion of Investors' Risk Tolerance.

You can graphically evaluate an individual investor's attitude to risk by the degree of steepness of the indifference curve, the steeper it is, the higher the risk aversion, and vice versa, the more indifferent the attitude towards risk. In order to quantify risk tolerance, the author proposes to calculate the tangent of the slope of the tangent.

The attitude of investors to risk can be described not only by indifference curves, but also in terms of utility theory. The investor's attitude to risk in this case reflects the utility function. The x-axis represents the change in expected income, and the y-axis represents the change in utility. Since, in general, zero income corresponds to zero utility, the graph passes through the origin.

Since the investment decision made can lead to both positive results (income) and negative results (losses), its usefulness can also be both positive and negative.

The importance of using a utility function as a guide for investment decisions is illustrated by the following example.

Suppose an investor is faced with the choice of whether or not to invest his funds in a project that allows him to win and lose $10,000 with the same probability (outcomes A and B, respectively). Assessing this situation from the standpoint of probability theory, it can be argued that an investor with an equal degree of probability can both invest his funds in a project and abandon it. However, after analyzing the utility function curve, we can see that this is not entirely true (Fig. 3)

Figure 3. Utility curve as a criterion for making investment decisions

Figure 3 shows that the negative utility of outcome B is clearly higher than the positive utility of outcome A. The algorithm for constructing a utility curve is given in the next paragraph.

It is also obvious that if the investor is forced to take part in the "game", he expects to lose utility equal to U E = (U B - U A):2

Thus, the investor must be prepared to pay the amount of OS for not participating in this "game".

We also note that the utility curve can be not only convex, but also concave, which reflects the need for the investor to pay insurance on this concave section.

It is also worth noting that the utility plotted along the y-axis has nothing to do with the neoclassical concept of utility. economic theory. In addition, on this chart, the y-axis has an unusual scale, utility values ​​on it are plotted on it as degrees on the Fahrenheit scale.

The practical application of utility theory has revealed the following advantages of the utility curve:

1. Utility curves, being an expression of the individual preferences of the investor, being built once, allow making investment decisions in the future, taking into account his preferences, but without additional consultations with him.

2. The utility function in the general case can be used to delegate the right to make decisions. In this case, it is most logical to use the utility function of top management, since in order to ensure its position in making decisions, it tries to take into account the conflicting needs of all interested parties, that is, the entire company. However, keep in mind that the utility function may change over time, reflecting the financial conditions of a given point in time. Thus, utility theory makes it possible to formalize the approach to risk and thereby scientifically substantiate decisions made under conditions of uncertainty.

Building a utility curve

The construction of an individual utility function is carried out as follows. The research subject is offered to make a series of choices between various hypothetical games, according to the results of which the corresponding points are plotted on the graph. So, for example, if an individual is indifferent to winning $10,000 with complete certainty, or playing the game with a win of $0 or $25,000 with the same probability, then it can be argued that:

U(10.000) = 0.5 U(0) + 0.5 U(25.000) = 0.5(0) + 0.5(1) = 0.5

where U is the utility of the amount indicated in brackets

0.5 - the probability of the outcome of the game (according to the conditions of the game, both outcomes are equivalent)

Utilities of other sums can be found from other games by the following formula:

Uc (C) = PaUa(A) + PbUb(B) + PnUn(N)(16)

Where Nn- the utility of the sum N

Un- the probability of an outcome with the receipt of a sum of money N

The practical application of utility theory can be demonstrated by the following example. Suppose an individual needs to choose one of two projects described by the following data (Table 1):

Table 1

Building a utility curve.

Despite the fact that both projects have the same mathematical expectation, the investor will give preference to project 1, since its utility for the investor is higher.

The nature of risk and approaches to its assessment

Summarizing the above study of the nature of risk, we can formulate its main points:

Uncertainty is an objective condition for the existence of risk;

The need to make a decision is the subjective reason for the existence of risk;

The future is a source of risk;

The amount of losses is the main threat from the risk;

Possibility of losses - the degree of threat from the risk;

The relationship "risk-return" - a stimulating factor in decision-making under conditions of uncertainty;

Risk tolerance is a subjective component of risk.

When deciding on the effectiveness of the IP under uncertainty, the investor solves at least a two-criteria problem, in other words, he needs to find the optimal combination of “risk-return” of the IP. It is obvious that it is possible to find the ideal option "maximum profitability - minimum risk" only in very rare cases. Therefore, the author proposes four approaches for solving this optimization problem.

1. The “maximum gain” approach is that of all the options for investing capital, the option that gives the greatest result is selected ( NPV, profit) at an acceptable risk for the investor (R pr.add). Thus, the decision criterion in a formalized form can be written as (17)

(17)

2. The “optimal probability” approach consists in choosing from the possible solutions the one in which the probability of the result is acceptable to the investor (18)

(18)

M(NPV) - expectation NPV.

3. In practice, the “optimal probability” approach is recommended to be combined with the “optimal volatility” approach. The fluctuation of indicators is expressed by their variance, standard deviation and coefficient of variation. The essence of the strategy of optimal volatility of the result is that of the possible solutions, one is chosen at which the probabilities of winning and losing for the same risky investment of capital have a small gap, i.e. the smallest value of dispersion, standard deviation, variation.

(19)

Where:

CV(NPV) - coefficient of variation NPV.

4. Approach "minimum risk". Of all possible options, the one that allows you to get the expected payoff is selected. (NPV pr.add) with minimal risk.

(20)

Investment project risk system

The range of risks associated with the implementation of IP is extremely wide. There are dozens of risk classifications in the literature. In most cases, the author agrees with the proposed classifications, however, as a result of studying a significant amount of literature, the author came to the conclusion that there are hundreds of classification criteria, in fact, the value of any IP factor in the future is an indefinite value, i.e. is a potential source of risk. In this regard, the construction of a universal general classification of IP risks is not possible and is not necessary. According to the author, it is much more important to determine an individual set of risks that are potentially dangerous for a particular investor and evaluate them, so this dissertation focuses on the tools for quantifying the risks of an investment project.

Let us examine in more detail the risk system of an investment project. Speaking about the risk of IP, it should be noted that it is inherent in the risks of an extremely wide range of areas of human activity: economic risks; political risks; technical risks; legal risks; natural risks; social risks; production risks, etc.

Even if we consider the risks associated with the implementation of only the economic component of the project, their list will be very extensive: the segment of financial risks, risks associated with fluctuations in market conditions, risks of fluctuations in business cycles.

Financial risks are risks arising from the probability of losses due to the implementation financial activities under conditions of uncertainty. Financial risks include:

Risks of fluctuations in the purchasing power of money (inflationary, deflationary, currency)

The inflationary risk of IP is primarily due to the unpredictability of inflation, since an erroneous inflation rate included in the discount rate can significantly distort the value of the IP efficiency indicator, not to mention the fact that the conditions for the functioning of national economy entities differ significantly at an inflation rate of 1% per month (12.68% per year) and 5% per month (79.58% per year).

Speaking about inflationary risk, it should be noted that the interpretation of risk that is often found in the literature as that income will depreciate faster than indexation is, to put it mildly, incorrect, and in relation to IP is unacceptable, because. The main danger of inflation lies not so much in its magnitude as in its unpredictability.

Under the condition of predictability and certainty, even the largest inflation can be easily taken into account in IP either in the discount rate or by indexing the amount of cash flows, thereby reducing the element of uncertainty, and hence the risk, to zero.

Currency risk - risk of loss financial resources due to unpredictable fluctuations in exchange rates. Currency risk can play a cruel joke on the developers of those projects that, in an effort to get away from the risk of unpredictable inflation, rely on cash flows in "hard" currency, as a rule, in US dollars, because even the hardest currency is subject to internal inflation, and the dynamics of its purchasing power in a single country can be very unstable.

It is also impossible not to note the relationship of various risks. For example, currency risk can transform into inflationary or deflationary risk. In turn, all these three types of risk are interconnected with price risk, which refers to the risks of market fluctuations. Another example: business cycle risk is associated with investment risk, interest rate risk, for example.

Any risk in general, and the risk of IP in particular, is very multifaceted in its manifestations and often represents a complex structure of elements of other risks. For example, the risk of market fluctuations is a whole set of risks: price risks (both for costs and for products); risks of changes in the structure and volume of demand.

Fluctuations in market conditions can also be caused by fluctuations in business cycles, etc.

In addition, the manifestations of risk are individual for each participant in a situation associated with uncertainty, as mentioned above.

The versatility of risk and its complex relationships is evidenced by the fact that even a risk minimization solution contains risk.

IP risk (R un) is a system of factors that manifests itself in the form of a complex of risks (threats), individual for each IP participant, both in quantitative and qualitative terms. The IP risk system can be represented in following form (21):

(21)

The emphasis is on the fact that the IP risk is a complex system with numerous interrelations, which manifests itself for each of the IP participants in the form of an individual combination - a complex, that is, the risk of the i-th project participant (Ri) will be described by formula (22):

The matrix column (21) shows that the value of any risk for each project participant also manifests itself individually (Table 2).

table 2

Example of IP risk system.

To analyze and manage the IP risk system, the author proposes the following risk management algorithm. Its content and tasks are presented in Figure 4.

1. Risk analysis usually starts with qualitative analysis, the purpose of which is to identify risks. This goal breaks down into the following tasks:

Identification of the entire range of risks inherent in an investment project;

Description of risks;

Classification and grouping of risks;

Analysis of initial assumptions.

Unfortunately, the vast majority of domestic IP developers stop at this initial stage, which, in fact, is only a preparatory phase of a full-fledged analysis.

Rice. 4. IP risk management algorithm.

2. The second and most difficult phase of risk analysis is quantitative risk analysis, the purpose of which is to measure risk, which leads to the solution of the following tasks:

Formalization of uncertainty;

Risk calculation;

Risk assessment;

Risk accounting;

3. At the third stage, risk analysis is smoothly transformed from a priori, theoretical judgments into practical risk management activities. This happens at the moment when the design of the risk management strategy is completed and its implementation begins. The same stage completes the engineering of investment projects.

4. The fourth stage - control, in fact, is the beginning of IP reengineering, it completes the risk management process and ensures its cyclicality.

Conclusion

Unfortunately, the volume of this article does not allow to demonstrate in full practical use the above principles, in addition, the purpose of the article is to substantiate the theoretical basis for practical calculations, which are detailed in other publications. You can find them at www. koshechkin.narod.ru.

Literature

1. Balabanov I.T. Risk management. M.: Finance and statistics -1996-188s.
2. Bromwich M. Analysis economic efficiency investments: translation from English - M.: -1996-432s.
3. Van Horn J. Fundamentals of financial management: per. from English. (edited by I.I. Eliseeva - M., Finance and Statistics 1997 - 800 p.
4. Gilyarovskaya L.T., Endovitsky Modeling in strategic planning long-term investments // Finance-1997-№8-53-57
5. Zhiglo A.N. Calculation of discount rates and risk assessment.// Accounting 1996-№6
6. Zagoriy G.V. On methods for assessing credit risk.// Money and credit 1997-№6
7. 3ozulyuk A.V. business risk in entrepreneurial activity. Diss. on the competition account Ph.D. M. 1996.
8. Kovalev V.V. “ The financial analysis: Capital Management. Choice of investments. Reporting analysis.” M.: Finance and statistics 1997-512 pp.
9. Kolomina M. Essence and measurement of investment risks. //Finance-1994-№4-p.17-19
10. Polovinkin P. Zozulyuk A. Entrepreneurial risks and their management. // Russian economic journal 1997-№9
11. Salin V.N. Mathematical and economic methodology of analysis risk types insurance. M., Ankil 1997 - 126 pages.
12. Sevruk V. Analysis of credit risk. // Accounting-1993-№10 p.15-19
13. Telegina E. On risk management in the implementation of long-term projects. //Money and credit -1995-№1-p.57-59
14. Trifonov Yu.V., Plekhanova A.F., Yurlov F.F. The choice of effective solutions in the economy under uncertainty. Monograph. Nizhny Novgorod: UNN Publishing House, 1998. 140s.
15. Khussamov P.P. Development of a method for a comprehensive assessment of the risk of investing in industry. Diss. on the competition account PhD in Economics Ufa. 1995.
16. Shapiro V.D. Project management. St. Petersburg; TwoThree, 1996-610s.
17. Sharp W.F., Alexander G.J., Bailey J. Investments: per. from English. -M.: INFRA-M, 1997-1024s
18. Chetyrkin E.M. Financial analysis of industrial investments M., Delo 1998 - 256 pages.

TECHNICAL DIAGNOSIS OF ELECTRONIC DEVICES

UDC 678.029.983

Compiled by: V.A. Pikkiev.

Reviewer

Candidate of Technical Sciences, Associate Professor O.G. Cooper

Technical diagnostics electronic means : guidelines for conducting practical classes in the discipline "Technical diagnostics of electronic means" / Yugo-Zap. state university; comp.: V.A. Pikkiev, Kursk, 2016. 8s.: ill.4, tab.2, app.1. Bibliography: p. 9 .

Guidelines for conducting practical classes are intended for students of the direction of preparation 11.03.03 "Design and technology of electronic means".

Signed for printing. Format 60x84 1\16.

Conv. oven l. Uch.-ed.l. Circulation 30 copies. Order. For free

Southwestern State University.

INTRODUCTION PURPOSE AND TASKS OF STUDYING THE DISCIPLINE.
1. Practical exercise No. 1. The method of the minimum number of erroneous decisions
2. Practice #2: Minimal Risk Method
3. Practice #3: Bayes Method
4. Practice #4: Maximum Likelihood Method
5. Practice No. 5. The minimax method
6. Practice No. 6. Neumann-Pearson Method
7. Practical lesson No. 7. Linear separating functions
8. Practical lesson No. 8. Generalized algorithm for finding a separating hyperplane

INTRODUCTION PURPOSE AND TASKS OF STUDYING THE DISCIPLINE.

Technical diagnostics considers diagnostic tasks, principles of organization of test and functional diagnostic systems, methods and procedures of diagnostic algorithms for checking malfunctions, operability and correct functioning, as well as for troubleshooting various technical objects. The main attention is paid to the logical aspects of technical diagnostics with deterministic mathematical models of diagnosis.

The purpose of the discipline is to master the methods and algorithms of technical diagnostics.

The objective of the course is to train technical specialists who have mastered:

Modern methods and algorithms for technical diagnostics;

Models of objects of diagnostics and malfunctions;

Diagnostic algorithms and tests;

Modeling of objects;

Equipment for element-by-element diagnostic systems;

signature analysis;

Automation systems for diagnosing REA and EVS;

Skills in the development and construction of models of elements.

Provided in curriculum practical classes allow students to form professional competencies of analytical and creative thinking by acquiring practical skills in diagnosing electronic means.

Practical classes include work with applied problems of developing troubleshooting algorithms electronic devices and the construction of control tests in order to further use when simulating the operation of these devices.

PRACTICE #1

METHOD OF THE MINIMUM NUMBER OF ERRONOUS SOLUTIONS.

In reliability problems, the considered method often gives "careless decisions", since the consequences of erroneous decisions differ significantly from each other. Typically, the cost of missing a defect is significantly higher than the cost of a false alarm. If the indicated costs are approximately the same (for defects with limited consequences, for some control tasks, etc.), then the application of the method is fully justified.

The probability of an erroneous decision is defined as

D 1 - diagnosis of good condition;

D 2 - diagnosis of a defective condition;

P 1 -probability 1 diagnosis;

P 2 - probability of the 2nd diagnosis;

x 0 - boundary value of the diagnostic parameter.

From the condition of the extremum of this probability, we obtain

The minimum condition gives

For unimodal (i.e., contain no more than one maximum point) distributions, inequality (4) is satisfied, and the minimum probability of an erroneous solution is obtained from relation (2)

The condition for choosing the boundary value (5) is called the Siegert–Kotelnikov condition (the ideal observer condition). Bayes' method also leads to this condition.

The decision x ∈ D1 is made for

which coincides with equality (6).

The dispersion of the parameter (the value of the standard deviation) is assumed to be the same.

In the case under consideration, the distribution densities will be equal to:

Thus, the obtained mathematical models (8-9) can be used to diagnose ES.

Example

Diagnostics of the health of hard drives is carried out by the number of bad sectors (Reallocated sectors). Western Digital manufactures the “My Passport” hard drive model using the following tolerances: Good discs are considered to have an average value of x 1 = 5 per unit volume and standard deviation σ 1 = 2 . In the presence of a magnetic deposition defect (faulty state), these values ​​are equal to x 2 = 12, σ 2 = 3. The distributions are assumed to be normal.

It is required to determine the limit of the number of bad sectors, above which the hard drive must be removed from service and disassembled (to avoid dangerous consequences). According to statistical data, the faulty state of magnetic deposition is observed in 10% of railways.

Distribution densities:

1. Distribution density for good condition:

2. Distribution density for defective condition:

3. Divide the state densities and equate them to the state probabilities:

4. Let's take the logarithm of this equality and find the maximum number of bad sectors:

This equation has a positive root x 0 = 9.79

The critical number of bad sectors is 9 per volume unit.

Job Options

No. p / p	x 1	σ 1	x 2	σ2

Conclusion: Using this method allows you to make a decision without assessing the consequences of errors, from the conditions of the problem.

The disadvantage is that the indicated values ​​are approximately the same.

The application of this method is common in instrument making and mechanical engineering.

Practice #2

MINIMUM RISK METHOD

The purpose of the work: to study the method of minimal risk for diagnosing the technical condition of ES.

Work tasks:

Explore theoretical basis minimum risk method;

Carry out practical calculations;

Draw conclusions on the use of the method of minimal risk of ES.

Theoretical explanations.

The probability of making an erroneous decision is the sum of the probabilities of a false alarm and a missed defect. If we attribute "prices" to these errors, we get an expression for the average risk.

Where D1 is the diagnosis of good condition; D2 - diagnosis of a defective condition; P1-probability of 1 diagnosis; P2 - probability of the 2nd diagnosis; x0 - boundary value of the diagnostic parameter; C12 - the cost of a false alarm.

Of course, the cost of an error has a conditional value, but it should take into account the expected consequences of false alarms and missing a defect. In reliability problems, the cost of skipping a defect is usually much higher than the cost of a false alarm (C12 >> C21). Sometimes the cost of correct decisions C11 and C22 is introduced, which is taken as negative for comparison with the cost of losses (errors). In the general case, the average risk (the expected loss) is expressed by the equation

Where C11, C22 - the price of the right decisions.

The value x presented for recognition is random and therefore equalities (1) and (2) represent the average value (expectation) of the risk.

Let's find the boundary value x0 from the condition of minimum average risk. Differentiating (2) with respect to x0 and equating the derivative to zero, we first obtain the extremum condition

This condition often determines two values ​​of x0, of which one corresponds to the minimum, the second to the maximum risk (Fig. 1). Relation (4) is a necessary but insufficient condition for a minimum. For the existence of a minimum of R at the point x = x0, the second derivative must be positive (4.1.), which leads to the following condition

(4.1.)

with respect to the derivatives of the distribution densities:

If the distributions f(x, D1) and f(x, D2) are, as usual, unimodal (i.e., contain no more than one maximum point), then for

Condition (5) is satisfied. Indeed, on the right side of the equality there is a positive value, and for x>x1 the derivative f "(x / D1), while for x

In what follows, x0 will be understood as the boundary value of the diagnostic parameter, which, according to rule (5), ensures the minimum average risk. We will also consider the distributions f (x / D1) and f (x / D2) to be unimodal (“one-humped”).

It follows from condition (4) that the decision to assign the object x to the state D1 or D2 can be associated with the magnitude of the likelihood ratio. Recall that the ratio of the probability densities of the distribution of x under two states is called the likelihood ratio.

According to the minimum risk method, the following decision is made about the state of an object that has a given value of the parameter x:

(8.1.)

These conditions follow from relations (5) and (4). Condition (7) corresponds to x< x0, условие (8) x >x0. The value (8.1.) is the threshold value for the likelihood ratio. Recall that the diagnosis D1 corresponds to the serviceable state, D2 - to the defective state of the object; C21 – price of a false alarm; C12 – target skip price (the first index is the accepted state, the second is the actual one); C11< 0, C22 – цены правильных решений (условные выигрыши). В большинстве практических задач условные выигрыши (поощрения) для правильных решений не вводятся и тогда

It often turns out to be convenient to consider not the likelihood ratio, but the logarithm of this ratio. This does not change the result, since the logarithmic function increases monotonically with its argument. The calculation for normal and some other distributions using the logarithm of the likelihood ratio turns out to be somewhat simpler. Let us consider the case when the parameter x has a normal distribution in the serviceable D1 and faulty D2 states. The dispersion of the parameter (the value of the standard deviation) is assumed to be the same. In the considered case, the distribution densities

Introducing these relations into equality (4), we obtain after taking the logarithm

Diagnostics of the performance of flash drives is carried out by the number of bad sectors (Reallocated sectors). Toshiba TransMemory manufactures the “UD-01G-T-03” model using the following tolerances: Drives with an average value of x1 = 5 per unit volume are considered serviceable. We take the standard deviation equal to ϭ1 = 2.

In the presence of a NAND memory defect, these values ​​are x2 = 12, ϭ2 = 3. The distributions are assumed to be normal. It is required to determine the limit for the number of bad sectors above which the hard drive is subject to decommissioning. According to statistics, a faulty condition occurs in 10% of flash drives.

Let's assume that the ratio of the cost of missing the target and a false alarm is , and we will refuse to "reward" the correct decisions (С11=С22=0). From condition (4) we obtain

Task options:

Var.

X 1 mm.

X 2 mm.

b1

b2

Conclusion

The method allows you to estimate the probability of making an erroneous decision is defined as minimizing the extremum point of the average risk of erroneous decisions at the maximum likelihood, i.e. the calculation of the minimum risk of the occurrence of an event is carried out in the presence of information on the most similar events.

PRACTICAL WORK № 3

BAYES' METHOD

Among the methods of technical diagnostics, the method based on the generalized Bayes formula occupies a special place due to its simplicity and efficiency. Of course, the Bayes method has disadvantages: a large amount of preliminary information, "oppression" of rare diagnoses, etc. However, in cases where the volume of statistical data allows the application of the Bayes method, it is advisable to use it as one of the most reliable and effective.

Let there be a diagnosis D i and a simple sign k j that occurs with this diagnosis, then the probability of the joint occurrence of events (the presence of a state D i and a sign k j in an object)

Bayes formula follows from this equality

It is very important to determine the exact meaning of all the quantities included in this formula:

P(D i) is the probability of diagnosis D i determined from statistical data (a priori probability of diagnosis). So, if N objects were previously examined and N i objects had a state D i , then

P(kj/D i) is the probability of the appearance of a feature k j in objects with a state D i . If among N i objects having the diagnosis D i , N ij has a feature k j , then

P(kj) is the probability of the appearance of a feature k j in all objects, regardless of the state (diagnosis) of the object. Let from the total number of N objects the sign k j was found in N j objects, then

To establish a diagnosis, a special calculation of P(k j) is not required. As will be clear from what follows, the values ​​of P(D i) and P(k j /D v), known for all possible states, determine the value of P(k j).

In equality (2), P(D i / k j) is the probability of diagnosis D i after it became known that the object under consideration has a feature k j (posterior diagnosis probability).

The generalized Bayes formula refers to the case when the examination is carried out on the basis of a set of features K, including features k 1 , k 2 , …, k ν . Each of signs k j has m j digits (k j1 , k j2 , …, k js , …, k jm). As a result of the survey, the implementation of the feature becomes known

and the whole complex of features K * . Index * , as before, means a specific value (implementation) of the feature. The Bayes formula for a set of features has the form

where P(D i / K *) is the probability of diagnosing D i after the results of the examination according to the complex of signs K became known; P(D i) – preliminary probability of diagnosis D i (according to previous statistics).

Formula (7) refers to any of n possible states (diagnoses) of the system. It is assumed that the system is in only one of the specified states and therefore

In practical problems, the possibility of the existence of several states A 1 , ..., Ar r is often allowed, and some of them can occur in combination with each other. Then separate states D 1 = A 1 , …, D r = A r and their combinations D r+1 = A 1 /\ A 2 should be considered as different diagnoses D i .

Let's move on to the definition P (K * / D i) . If the set of features consists of n features, then

Where k * j = k js- the category of the sign revealed as a result of the examination. For diagnostically independent signs;

In most practical problems, especially with a large number of features, it is possible to accept the condition of feature independence even if there are significant correlations between them.

The probability of occurrence of a complex of features K *

The generalized Bayes formula can be written

where P(K * / D i) is defined by equality (9) or (10). From relation (12) it follows

which, of course, should be, since one of the diagnoses is necessarily implemented, and the implementation of two diagnoses at the same time is impossible.

It should be noted that the denominator of the Bayes formula for all diagnoses is the same. This allows you to first determine the probabilities of the joint occurrence of the i-th diagnosis and the given realization of the set of features

and then the posterior probability of the diagnosis

To determine the probability of diagnoses using the Bayesian method, it is necessary to compile a diagnostic matrix (Table 1), which is formed on the basis of preliminary statistical material. This table contains the probabilities of feature discharges for various diagnoses.

Table 1

If the signs are two-digit (simple signs "yes - no"), then in the table it is enough to indicate the probability of the appearance of the sign P(k j / D i).

The probability of the absence of a feature P (kj / D i) = 1 − P (kj / D i) .

However, it is more convenient to use the uniform form, assuming, for example, for a two-digit feature P(kj/D) = P(kj 1/D) ; P(kj/D) = P(kj 2/D).

Note that ∑ P (k js / D i) =1 , where m j is the number of bits of feature k j .

The sum of the probabilities of all possible implementations of the feature is equal to one.

The diagnostic matrix includes a priori probabilities of diagnoses. The learning process in the Bayesian method consists in the formation of a diagnostic matrix. It is important to provide for the possibility of refining the table during the diagnostic process. To do this, not only P(k js / D i) values ​​should be stored in the computer memory, but also the following values: N is the total number of objects used to compile the diagnostic matrix; N i - number of objects with diagnosis D i ; N ij is the number of objects with diagnosis D i , examined on the basis of k j . If a new object arrives with a diagnosis D μ , then the previous a priori probabilities of diagnoses are corrected as follows:

Next, corrections to the probabilities of features are introduced. Let a new object with the diagnosis D μ have the rank r of the feature k j . Then, for further diagnostics, new values ​​of the probability of intervals of the attribute k j are accepted for the diagnosis D μ:

The conditional probabilities of signs for other diagnoses do not require adjustment.

Practical part

1. Study the guidelines and get the task.

PRACTICAL WORK № 4

State Committee of the Russian Federation for Fisheries

Federal State Educational

Institution of Higher Professional Education

Kamchatka State Technical University

Department of Math

Coursework by discipline

"Mathematical Economics"

On the topic: "Risk and insurance."

Introduction…………………………………………………………..……………….....3

1.CLASSICAL SCHEME FOR EVALUATION OF FINANCIAL OPERATIONS UNDER UNCERTAINTY ……………..................................................................................................................4 1.1. Definition and essence of risk…………………………………..……………..…...4

1.2. Matrices of consequences and risks……………………………………….……..……6

1.3. Analysis of a related group of decisions under conditions of complete uncertainty………………………………………………………………………......7

1.4. Analysis of a related group of decisions under conditions of partial uncertainty……………………………………………………………………..8

1.5. Pareto optimality…………………………………………………….9

2. CHARACTERISTICS OF PROBABLE FINANCIAL OPERATIONS……..…..…...12

2.1. Quantitative risk assessment………………………………………………..12

2.2. Risk of a single operation………………………………………………………..13 2.3. Some common measures of risk………………………………………….15

2.4. Risk of ruin……………………………………………………………..…16

2.5. Risk indicators in the form of ratios…………………………………………..17

2.6. Credit risk……………………………………………………………….17

3. GENERAL RISK REDUCTION……………………………………….…….18

3.1. Diversification………………………………………………………………18

3.2. Hedging……………………………………………………………………21

3.3. Insurance…………………………………………………………………...22

3.4. Quality risk management………………………………….……….24

Practical part……………………………………………………………...…….27

Conclusion………………………………………………………..………….…. ..29

References…………………………………………….……….……..….30

Applications………………………………………………………….…………..…...31

INTRODUCTION

The development of world financial markets, characterized by the intensification of the processes of globalization, internationalization, liberalization, has a direct impact on all participants in the world economic space, the main members of which are large financial and credit institutions, production and trade corporations. All participants in the world market directly feel the impact of all of the above processes and in their activities must take into account new trends in the development of financial markets. The number of risks arising in the activities of such companies has increased significantly in recent years. This is due to the emergence of new financial instruments actively used by market participants. The use of new instruments, although it makes it possible to reduce the risks assumed, is also associated with certain risks for the activities of financial market participants. Therefore, awareness of the role of risk in the company's activities and the ability of the risk manager to respond adequately and in a timely manner to the current situation, to make the right decision regarding the risk, is becoming increasingly important for the successful operation of the company. To do this, it is necessary to use various insurance and hedging tools against possible losses and losses, the set of which has significantly expanded in recent years and includes both traditional insurance methods and hedging methods using financial instruments.

The effectiveness of the company as a whole will ultimately depend on how correctly this or that tool is chosen.

The relevance of the research topic is also predetermined by the incompleteness of the development of the theoretical basis and classification of financial risk insurance and the identification of its features in Russia.

Chapter 1

OPERATIONS UNDER UNCERTAINTY

Risk – one of the most important concepts accompanying any active human activity. However, this is one of the most obscure, ambiguous and confusing concepts. However, despite its vagueness, ambiguity and confusion, in many situations the essence of risk is very well understood and perceived. These same qualities of risk are a serious obstacle to its quantitative assessment, which in many cases is necessary for the development of theory and practice.

Consider the classical decision-making scheme under uncertainty.

1.1. Definition and essence of risk

Recall that financial is an operation, the initial and final states of which have a monetary value and the purpose of which is to maximize income – difference between final and initial

ratings (or some other similar indicator).

Almost always, financial transactions are carried out under conditions of uncertainty and therefore their result cannot be predicted in advance. Therefore, financial transactions risky : when they are carried out, both profit and loss are possible (or not very large profit compared to what those who carried out this operation hoped for).

The person conducting the operation (making the decision) is called the decision maker. – Face ,

decision maker . Naturally, the decision maker is interested in the success of the operation and is responsible for it (sometimes only to himself). In many cases, the LPR – is an investor who invests money in a bank, in which – then a financial transaction, buying securities, etc.

Definition. The operation is called risky , if it can have several outcomes that are not equivalent for the decision maker.

Example 1 .

Consider three operations with the same set of two outcomes –

alternatives A , IN, which characterize the income received by decision makers. All three

operations are risky. It is clear that the first and second are risky

transactions, since losses are possible as a result of each transaction.

But why should the third operation be recognized as risky? After all, it promises only positive income for decision makers, doesn't it? Considering the possible outcomes of the third operation, we see that we can get an income of 20 units, so the possibility of obtaining an income of 15 units is considered as a failure, as a risk of missing 5 units of income. Thus, the concept of risk necessarily implies risky – who is at risk, who is concerned about the outcome of the operation. The risk itself arises only if the operation may end in outcomes that are not equivalent to him, despite, perhaps, all his efforts to manage this operation.

So, under conditions of uncertainty, the operation acquires one more characteristic – risk. How to evaluate the operation in terms of its profitability and risk? This question is not so easy to answer, mainly because of the versatility of the concept of risk. There are several different ways to make this assessment. Let's consider one of these approaches.

1.2. Impact and risk matrices

Suppose we are considering a financial transaction. It is not clear how it might end. In this regard, an analysis of several possible solutions and their consequences is carried out. So we come to the following general decision-making scheme (including financial ones) under conditions of uncertainty.

Assume that the decision maker considers several possible solutions

i =1, …,n. The situation is uncertain, it is only clear that there is some – one of the options j =1,….,n. If accepted i– there is a solution, but there is a situation j– i, then the firm headed by the decision maker will receive income q ij . Matrix Q =(q ij) is called impact matrix(possible solutions). Let's say we want to evaluate the risk posed by i th decision. We do not know the real situation. But if we knew it, we would choose the best solution, i.e. generating the most income. If the situation j-i, then a decision would be made that would generate income q i=max q ij . So, taking i th decision, we risk getting no q j , but only q ij , those. Adoption i-th decision carries the risk of not getting r ij = q j -q ij is called risk matrix .

Example 2

Let the consequence matrix be

Let's create a risk matrix. We have q 1=max q i1=8, q 2 =5, q 3 =8, q 4=12. Therefore, the risk matrix is

1.3. Analysis of a connected group of decisions under complete uncertainty

The situation of complete uncertainty is characterized by the absence of any additional information (for example, about the probabilities of certain variants of the real situation). What are the rules – recommendations for decision making in this situation?

Wald's rule (rule of extreme pessimism).

Considering i th decision, we will assume that in fact the situation is the worst, i.e. with the least income: a i=min q a 0 with the highest a i0 . So, Wald's rule recommends making a decision i 0 such that a i0 =max a i =max(min q ij). So, in example 2 we have a 1 =2, a 2 =2, a 3 =3, a 4 = 1. Now from the numbers 2, 2, 3, 1 we find the maximum - 3. So, the Wald rule recommends making the 3rd decision.

Savage's rule (minimum risk rule).

When applying this rule, the risk matrix is ​​analyzed R =(r ij). Considering i th decision, we will assume that in fact there is a situation of maximum risk b i=max r ij . But now let's choose a solution i 0 with the least b i0 . So, Savage's rule recommends making a decision i 0 such that b i0=min b i =min(max r ij). So, in example 2 we have b 1 =8, b 2 =6, b 3 =5, b 4=7. Now from the numbers 8, 6 , 5, 7 we find the minimum - 5.

Hurwitz's rule (weighing pessimistic and optimistic approaches to a situation).

A decision is made i, which reaches the maximum

{λ min q ij +(1 – λ max q ij)),

where 0≤ λ ≤1. Meaning λ chosen on a subjective basis. If λ approaching 1 , then the Hurwitz rule approaches the Wald rule, when approaching λ to 0, the Hurwitz rule approaches the “pink optimism” rule (guess for yourself what this means). In example 2, for λ=1/2, the Hurwitz rule recommends the second solution.

1.4. Analysis of a related group of decisions under partial uncertainty

Let us assume that in the scheme under consideration the probabilities are known R j the fact that the real situation develops according to the variant j. This situation is called partial uncertainty. How to make a decision here? You can select one of the following rules.

The rule for maximizing the average expected return.

Income received by the company from the sale i-th solution is a random variable Q i with distribution series. Expected value M [Q i ] and is the average expected return, also denoted Q i . So, the rule recommends making a decision that brings the maximum average expected return. Suppose that in the scheme of example 2, the probabilities are – 1/2, 1/6, 1/6, 1/6.

Then Q 1 =29/6, Q 2 =25/6, Q 3 =7, Q 4=17/6. The maximum average expected return is 7 and corresponds to the third solution.

Average expected risk minimization rule.

The risk of the firm in the implementation i th solution is a random variable R i with distribution series

Expected value M [R i ] and is the average expected risk, also denoted R i . The rule recommends making a decision that entails the minimum average expected risk. Let us calculate the average expected risks for the above probabilities. We get R 1 =20/6, R 2 =4, R 3 =7/6, R 4=32/6. The minimum average expected risk is 7/6 and corresponds to the third solution.

Comment. The difference between partial (probabilistic) uncertainty and complete uncertainty is very significant. Of course, no one considers decision-making according to the rules of Wald, Savage, Hurwitz to be final, the best. But when we begin to evaluate the likelihood of an option, this already assumes the repeatability of the considered decision scheme: it has already happened in the past, or it will be in the future, or it is repeated somewhere in space, for example, in the branches of the company.

1.5. Pareto optimality

So, when trying to choose the best solution, we encountered in the previous paragraph that each solution has two characteristics – average expected return and average expected risk. Now we have an optimization two-criteria problem of choosing the best solution.

There are several ways to formulate such optimization problems.

Let's consider this problem in a general way. Let A - some set of operations, each operation A has two numbers E (A), r (A) (effectiveness and risk, for example) and different operations necessarily differ in at least one characteristic. When choosing the best operation, it is desirable that E there were more and r less.

We will say that the operation A operation dominates b, and designate A >b, If E (A)≥E (b) And r (A)≤r (b) and at least one of these inequalities is strict. At the same time, the operation A called dominant , and the operation b- dominated . It is clear that under no reasonable choice of the best operation, a dominated operation can be recognized as such. Therefore, the best operation must be sought among non-dominated operations. Many of these operations are called set of Pareto or set of Pareto optimality .

There is an extremely important statement.

Statement.

On the Pareto set, each of the characteristics E , r-(single-valued) function is different. In other words, if an operation belongs to the Pareto set, then one of its characteristics can uniquely determine another.

Proof. Let A ,b- two operations from the Pareto set, then r (A) And r (b) – numbers. Let's pretend that r (A)≤r (b), Then E (A) cannot be equal to E (b), since both points A ,b belong to the Pareto set. It is proved that according to the characteristic r E. It is just as easy to prove that by the characteristic E characteristics can be determined r .

Let us continue the analysis of the example given in § 10.2. Consider a graphic illustration. Each operation (solution) ( R, Q) mark as a point on the plane – we put the income up vertically, and the risk – to the right horizontally (Fig. 10.1). We got four points and continue the analysis of example 2.

The higher the point ( R, Q), the more profitable the operation, the point to the right, the more risky it is. So, you need to choose a point above and to the left. In our case, the Pareto set consists of only one third of the operation.

To find the best operation, a suitable weighting formula is sometimes used, which for the operation Q with characteristics ( R, Q) gives one number, by which the best operation is determined. For example, let the weighting formula be f (Q)=2Q–R. Then for the operations (solutions) of example 2 we have: f (Q 1)=2*29/6– 20/6=6,33; f (Q 2)=4,33; f (Q 3)=12,83; f (Q 4)=0.33. It can be seen that the third operation is the best, and the fourth – worst.

Chapter 2. CHARACTERISTICS OF PROBABILISTIC FINANCIAL

OPERATIONS

The financial transaction is called probabilistic , if there is a probability of each of its outcomes. The profit of such an operation – the difference between the final and initial monetary valuations – is a random variable. For such an operation, it is possible to introduce a quantitative risk assessment that is consistent with our intuition.

2.1. Risk Quantification

In the previous chapter, a risky operation was defined as having at least two outcomes that are not equivalent in the decision maker's preference system. In the context of this chapter, instead of the decision maker, you can also use the term "investor" or some similar one, reflecting the interest of the person conducting the operation (perhaps passively) in its success.

When examining the risk of an operation, we come across a fundamental statement.

Statement.

A quantitative assessment of the risk of an operation is possible only with a probabilistic characterization of the set of outcomes of the operation.

Example 1

Consider two probabilistic operations:

Undoubtedly, the risk of the first operation is less than the risk of the second operation. As for which operation the decision maker chooses, it depends on his risk appetite (such issues are discussed in detail in the Appendix to Part 2).

2.2. Single operation risk

Since we want to quantify the riskiness of the operation, and this cannot be done without the probabilistic characteristics of the operation, we will assign probabilities to its outcomes and evaluate each outcome by the income that the decision maker receives from this outcome. As a result, we get a random value Q, which it is natural to call the random income of the operation, or simply casual income . For now, we restrict ourselves to a discrete random variable (d.r.v.):

Where q j - income, and R j – the probability of this income.

An operation and a random variable representing it – random income will be identified if necessary, choosing from these two terms more convenient in a particular situation.

Now you can apply the apparatus of probability theory and find the following characteristics of the operation.

Average expected return – mathematical expectation of r.v. Q, i.e. M [Q ]=q 1 p 1 +…+q n p n , also denoted m Q , Q, the name is also used operation efficiency .

Operation dispersion - dispersion of r.v. Q, i.e. D [Q ]=M [(Q-m Q) 2 ], is also denoted D Q.

Standard deviation r.v. Q, i.e. [ Q ]=√(D [E ]), denoted

Also σ Q.

Note that the average expected return, or the efficiency of the operation, as well as the standard deviation, is measured in the same units as the income.

Recall the fundamental meaning of the mathematical expectation of r.v.

The arithmetic mean of the values ​​taken by the r.v. in a long series of experiments, approximately equal to its mathematical expectation. It is becoming more and more accepted to assess the riskiness of the entire operation by means of the standard deviation of the random value of income Q, i.e. through σ Q. In this book, this is the main quantitative assessment.

So, operation risk called a number σ Q – the standard deviation of the random income of the operation Q. Also denoted r Q.

Example 2.

Let's find the risks of the first and second operations from example 1:

First, we calculate the mathematical expectation of the r.v. Q 1:

T 1 =– 5*0.01+25*0.99=24.7. Now we calculate the variance using the formula D 1 =M [Q 1 2 ]-m 1 2 . We have M [Q 1 2 ]= 25*0.01+625*0.99=619. Means, D 1 =619– (24.7)2=8.91 and finally r 1 =2,98.

Similar calculations for the second operation give m 2 =20; r 2=5. As “intuition suggested”, the first operation is less risky.

The proposed quantitative risk assessment is quite consistent with the intuitive understanding of risk as the degree of dispersion of the outcomes of the operation. – after all, the variance and the standard deviation (the square root of the variance) are the measures of such dispersion.

Other measures of risk.

In our opinion, the standard deviation is the best measure of the risk of an individual operation. In ch. 1 considers the classical decision-making scheme under uncertainty and risk assessment in this scheme. Good to know: Other risk measures. In most cases, these meters – just the probabilities of unwanted events.

2.3. Some common measures of risk

Let the distribution function be known F random income operation Q. Knowing it, you can give meaning to the following questions and answer them.

1. What is the probability that the income of the operation will be less than the specified s. You can ask for – to another: what is the risk of receiving an income less than a given one? Answer: F (s).

2. What is the probability that the operation will be unsuccessful, i.e. her income will be less than the average expected income m ?

Answer: F (m) .

3. What is the probability of losses and what is their average expected size? Or what is the risk of loss and their assessment?

4. What is the ratio of average expected loss to average expected return? The lower this ratio, the lower the risk of ruin if the decision maker has invested all his funds in the operation.

When analyzing operations, the decision maker wants to have more income and less risk. Such optimization problems are called two-criteria. When analyzing them, two criteria are income and risk. – are often “rolled up” into one criterion. Thus, for example, the concept relative risk of surgery . The fact is that the same value of the standard deviation σ Q , which measures the risk of a transaction, is perceived differently depending on the size of the average expected return T Q , so the value σ Q / T Q is sometimes referred to as the relative risk of surgery. Such a risk measure can be interpreted as a convolution of the two-criteria problem

σ Q→min,

T Q→max,

those. maximize the average expected return while minimizing risk.

2.4. Risk of ruin

This is the name of the probability of such large losses that the decision maker cannot compensate and which, consequently, lead to his ruin.

Example 3.

Let the random income of the operation Q has the following distribution series, and losses of 35 or more lead to the ruin of the decision maker. Consequently, the risk of ruin as a result of this operation is 0.8;

The severity of the risk of ruin is estimated precisely by the value of the corresponding probability. If this probability is very small, then it is often neglected.

2.5. Risk indicators in the form of ratios.

If the funds of the decision maker are equal WITH, then in excess of losses At above WITH there is a real risk of ruin. To prevent this attitude TO 1 = At / WITH , called risk factor , limited by a special number ξ 1 . Operations for which this coefficient exceeds ξ1 are considered particularly risky. Often the probability is also taken into account. R losses At and then consider the risk factor TO 2 = R Y/ WITH , which is limited by another number ξ 2 (it is clear that ξ 2 ≤ ξ 1). In financial management, inverse relationships are often used WITH / At And WITH /(RU), which are called risk coverage ratios and which are limited from below by the numbers 1/ ξ 1 and 1/ ξ 2 .

This is the meaning of the so-called Cook's coefficient, equal to the ratio:

Cook's ratio is used by banks and other financial companies. In the role of weights in "weighing" are the probabilities – risks of loss of the respective asset.

2.6. Credit risk

This is the probability of non-repayment of the loan taken on time.

Example 4.

Loan requests statistics are as follows: 10% – government agencies, 30% – other banks and others – individuals. The probabilities of non-repayment of the loan taken are, respectively, as follows: 0.01; 0.05 and 0.2. Find the probability of non-return of the next loan request. The head of the credit department was informed that a message had been received about the default of the loan, but the name of the client was poorly printed in the fax message. What is the probability that this loan will not repay any – is that a bank?

Solution. The probability of non-return is found by the formula of total probability. Let H 1 - the request came from a government agency, H 2 – from the bank H 3 – from an individual and A - non-repayment of the loan in question. Then

R (A)= R (H 1)R H1 A + R (H 2)R H2 A + R (H h) P H3 A = 0,1*0,01+0,3*0,05+0,6*0,2=0,136.

We find the second probability using the Bayes formula. We have

R A H 2 =R (H 2)R H2 A / R (A)= 0,015/0,136=15/136≈1/9.

How in reality all the data given in this example is determined, for example, conditional probabilities R H1 A? By the frequency of loan defaults for the respective group of clients. Let individuals take only 1,000 loans and not return 200. So the corresponding probability R H3 A estimated as 0.2. Relevant data – 1000 and 200 are taken from the information database of the bank.

Chapter 3. GENERAL RISK REDUCTION PRACTICES

As a rule, they try to reduce the risk. There are many methods for this. A large group of such methods is associated with the selection of other operations. Such that the total operation has less risk.

3.1. Diversification

Recall that the variance of the sum of uncorrelated random variables is equal to the sum of the variances. This implies the following statement, which is the basis of the diversification method.

Statement 1.

Let ABOUT 1 ,...,ABOUT n – uncorrelated operations with efficiencies e 1 ,..., e n and risks r 1 ,...,r 2 . Then the operation "arithmetic mean" ABOUT =(ABOUT 1 +...+O n) / P has efficiency e =(e 1 +...+e n)/ n and risk r =√(r 1 2 +…r 2n)/ n .

Proof of this statement – a simple exercise on the properties of mathematical expectation and variance.

Consequence 1.

Let the operations be uncorrelated and a≤ e i and b ≤ r i ≤ c c for everyone i =1,..,n. Then the efficiency of the operation "arithmetic mean" is not less than A(i.e., the smallest of the efficiency of operations), and the risk satisfies the inequality b √ n ≤ r ≤c √ n and thus, with an increase n decreases. So, with an increase in the number of uncorrelated operations, their arithmetic mean has an efficiency from the interval of the efficiencies of these operations, and the risk unambiguously decreases.

This conclusion is called diversification effect(diversity) and is essentially the only reasonable rule for working in the financial and other markets. The same effect is embodied in folk wisdom – "Don't put all your eggs in one basket." The principle of diversification says that it is necessary to carry out a variety of operations that are not related to each other, then the efficiency will be averaged, and the risk will definitely decrease.

You need to be careful when applying this rule. So, it is impossible to refuse uncorrelated operations.

Suggestion 2.

Suppose that among the operations there is a leading one with which all the others are in a positive correlation. Then the risk of the "arithmetic mean" operation does not decrease with an increase in the number of summable operations.

Indeed, for simplicity, we take a stronger assumption, namely, that all operations ABOUT i ; i =1,...,n, just copy the operation O 1 in which – then the scales, i.e. O i = k i O 1 and all coefficients of proportionality k i are positive. Then the operation "arithmetic mean" ABOUT =(O 1 +...+O n)/ n is just an operation O 1 to scale

and risk of this operation

Therefore, if the operations are approximately the same in scale, i.e. k i ≈1, then

We see that the risk of the "arithmetic mean" operation does not decrease with an increase in the number of operations.

3.2. Hedging

In the diversification effect, the decision maker made up a new operation from several available to him. When hedging (from the English. hedge- fence) The decision maker selects or even specially designs new operations in order to reduce the risk by conducting them together with the main one.

Example 1.

According to the contract, the Russian company must receive a large payment from the Ukrainian company in six months. The payment is 100,000 hryvnia (approximately 600 thousand rubles) and will be made in hryvnia. The Russian company has fears that during these six months the hryvnia will fall against the Russian ruble. The company wants to hedge against such a fall and enters into a forward contract with one of the Ukrainian banks to sell 100,000 hryvnia at the rate of 6 rubles. for hryvnia. Thus, no matter what happens during this time with the ruble exchange rate – hryvnia, the Russian company will not incur – for this damages.

This is the essence of hedging. When diversifying, independent (or uncorrelated) transactions were the most valuable. When hedging, operations are selected that are strictly related to the main one, but, so to speak, of a different sign, more precisely, negatively correlated with the main operation.

Indeed, let O 1 – main operation, its risk r 1 , O 2 – some additional operation, its risk r 2 , ABOUT - operation – sum, then the variance of this operation D =r 1 2 +2k 12 r 1 r 2 +r 2 2 , where k- coefficient of correlation between the efficiency of the main and additional operations. This variance can be less than the variance of the main operation only if this correlation coefficient is negative (more precisely: it should be 2 k 12 r 1 r 2 +r 2 2 <0, т.е. k 1 2 <-r 2 /(2r 1)).

Example 2.

Let the decision maker decide to carry out the operation O 1 .

He is advised to undergo surgery at the same time S, which is strictly related to ABOUT. In essence, both operations must be depicted with the same set of outcomes.

Let us denote the total operation by ABOUT, this operation is the sum of operations O 1 and S. Let us calculate the characteristics of operations:

M [O 1 ]=5, D [O 1 ]=225, r 1 =15;

M [S ]=0, D [S ]=25;

M [O ]=5, D [O ]=100, r =10.

The average expected efficiency of the operation remained unchanged, and the risk decreased due to the strong negative correlation of the additional operation. S in relation to the main operation.

Of course, in practice it is not so easy to choose an additional operation that is negatively correlated with the main one, and even with zero efficiency. Usually, a small negative efficiency of the additional operation is allowed, and because of this, the efficiency of the summary operation becomes less than that of the main one. The extent to which a decrease in efficiency per unit of risk reduction is allowed depends on the attitude of the decision maker to risk.

3.3. Insurance

You can consider insurance as one of the types of hedging. Let's clarify some terms.

Policyholder(or insured) – one who is insured.

Insurer - the one who insures.

Sum insured - the amount of money for which the property, life, health of the insured is insured. This amount is paid by the insurer to the insured upon the occurrence of an insured event. The sum insured is called insurance compensation .

Insurance payment paid by the insured to the insurer.

Denote the sum insured ω , insurance payment s, the probability of an insured event R . Assume that the property insured is valued at z. According to the rules of insurance ω≤ z.

Thus, the following scheme can be proposed:

Thus, insurance seems to be the most profitable measure in terms of risk reduction, if not for the insurance payment. Sometimes the insurance payment is a significant part of the sum insured and is a substantial amount.

3.4. Quality risk management

Risk – such a complex concept that it is quite often impossible to quantify it. Therefore, qualitative risk management methods are widely developed, without a quantitative assessment. These include many banking risks. The most important of them – These are credit risk and risks of illiquidity and insolvency.

1. Credit risk and ways to reduce it . When issuing a loan (or loan), there is always a fear that the client will not repay the loan. Preventing non-repayment, reducing the risk of non-repayment of loans – This is the most important task of the credit department of the bank. What are the ways to reduce the risk of loan default.

The department must constantly systematize and summarize information on loans issued and their return. Information on issued loans should be systematized by the amount of loans issued, a classification of customers who took out a loan should be built.

The department (the bank as a whole) must keep the so-called credit history of its customers, including potential ones (i.e. when, where, what loans were taken and how the client returned them). While in our country, most customers do not have their own credit history.

There are various ways to secure a loan, for example, a client pledges something and if the loan is not returned, the bank becomes the owner of the pledge;

The bank should have a clear instruction on issuing a loan (to whom what loan can be issued and for how long);

There should be clear authority to grant credit. Say, an ordinary employee of a department can issue a loan of no more than $1000, loans up to $10,000 can be issued by the head of the department, over $10,000, but not more than $100,000, can be issued by the vice president for finance, and loans over $100,000 are issued only by the board of directors (read the novel by A. Hailey "Changers");

To issue especially large and dangerous loans, several banks unite and jointly issue this loan;

There are insurance companies that insure non-repayment of a loan (but there is a point of view that non-repayment of a loan is not subject to insurance – is the risk of the bank itself);

There are external restrictions on the issuance of loans (for example, established by the Central Bank); say, it is not allowed to issue a very large loan to one client;

2. Illiquidity risks , insolvency and ways to reduce them . It is said that the bank's funds are sufficiently liquid if the bank is able to quickly and without much loss to ensure the payment to its customers of the funds that they have entrusted to the bank on a short-term basis. Illiquidity risk – that's the risk of failing to do so. However, this risk is only for brevity named so, its full name – imbalance risk balance in terms of liquidity .

All assets of the bank according to their liquidity are divided into three groups:

1) first-class liquid funds (cash, bank funds on a correspondent account with the Central Bank, government securities, bills of large reliable companies;

2) liquid funds (expected short-term payments to the bank, some types of securities, some tangible assets that can be sold quickly and without great losses, etc.);

3) illiquid funds (overdue loans and bad debts, many tangible assets of the bank, primarily buildings and structures).

When analyzing the risk of illiquidity, first-class liquid funds are taken into account in the first place.

It is said that a bank is solvent if it is able to pay off all its customers, but this may require some large and lengthy transactions, up to the sale of equipment, buildings belonging to the bank, and so on. The risk of insolvency arises when it is not clear whether the bank will be able to pay.

Bank solvency depends on a lot of factors. The Central Bank sets a number of conditions that banks must comply with in order to maintain their solvency. The most important of them are: limiting the bank's liabilities; refinancing of banks by the Central Bank; reserving part of the bank's funds on a correspondent account with the Central Bank.

The risk of illiquidity leads to possible unnecessary losses for the bank: in order to pay off the client, the bank may have to borrow money from other banks at a higher interest rate than under normal conditions. The risk of insolvency may well lead to the bankruptcy of the bank.

Practical part

Let's assume that the decision maker has the opportunity to make up an operation from four uncorrelated operations, the efficiencies and risks of which are given in the table.

Let us consider several options for compiling operations from these operations with equal weights.

1. The operation is composed of only the 1st and 2nd operations. Then e 12 =(3+5)/2=4;

r 12 = √ (2 2 +4 2)/2≈2,24

2. The operation is composed of only the 1st, 2nd and 3rd operations.

Then e 123 =(3+5+8)/3=5,3; r 123 =√(2 2 +4 2 +6 2)/3≈2,49.

3. The operation is made up of all four operations. Then

e 1– 4 =(3+5+8+10)/4=6,5; r 1– 4 =√(2 2 +4 2 +6 2 +12 2)/4≈ 3,54.

It can be seen that when compiling an operation from an increasing number of operations, the risk grows very slightly, remaining close to the lower limit of the risks of the constituent operations, and the efficiency each time is equal to the arithmetic mean of the components of efficiency.

The principle of diversification is applied not only to averaging transactions carried out simultaneously, but in different places (averaging in space), but also carried out sequentially in time, for example, when repeating one operation in time (averaging in time). For example, a reasonable strategy is to buy shares in a stable company on January 20th of each year. The inevitable fluctuations in the stock price of this company are averaged out due to this procedure, and this manifests the effect of diversification.

Theoretically, the effect of diversification is only positive – efficiency is averaged and risk is reduced. However, efforts to conduct a large number of operations, to track their results, can, of course, negate all the advantages of diversification.

CONCLUSION

This course work considers theoretical and practical issues and problems of risks.

The first chapter discusses the classical scheme for evaluating financial transactions under conditions of uncertainty.

The second chapter provides an overview of the characteristics of probabilistic financial transactions. Financial risks are understood as credit, commercial, exchange transactions risks and the risk of unlawful application of financial sanctions by state tax inspectorates.

The third chapter shows general risk mitigation techniques. Examples of high-quality risk management are given.

Bibliography

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3. Gvozdenko A.A. Financial and economic methods of insurance: Textbook.–M.: Finance and statistics, 2000.–184p.

4. Serbinovsky B.Yu., Garkusha V.N. Insurance Business: Textbook for High Schools. Series “Textbooks, teaching aids” Rostov n / a: “Phoenix”, 2000–384 p.