## Measurement and quantification of risk

Risk is a category of probability, therefore in the process of estimating uncertainty and quantifying the degree of risk, probability calculations are used.

Based on the probabilities, the standard risk characteristics are calculated. Consider the main ones.

1. ** Math expectation ** (mean expected value, M) is the weighted average of all possible outcomes, where the probabilities of achieving them are used as scales:

where - result (event or outcome, for example, the amount of income); is the probability of getting the result .

Thus, the mathematical expectation is a generalized quantitative characteristic of the expected result.

2. An important characteristic determining the measure of the variability of a possible result is the ** variance (D) ** - the weighted average of the squares of the deviation of the random variable from its mathematical expectation (i.e., the deviations of the actual results from the expected ones):

and very closely related ** standard deviation, ** defined from the expression:

The standard deviation indicates the degree of variability in the possible outcomes of the project and, therefore, the degree of risk; while more risky investments give a greater value of this value.

Both the variance and the standard deviation are * absolute risk measures * and are measured in the same physical units in which the variable is measured.

**3.**

**coefficient of variation**(V) is often used to analyze the measure of variability, which is the ratio of the standard deviation to the mathematical expectation:

The coefficient of variation is * the relative value. * Therefore, it is possible to compare the variability of the signs expressed in different units of measurement.

4. ** Correlation coefficient ** (R) shows the relationship between variables, consisting in changing the average value of one of them depending on the change of the other:

where .

This indicator varies from -1 to +1. A positive correlation coefficient means a positive relationship between the quantities, and the closer to <1> R is to unity, the stronger this relationship. * R * = 1 means that between * x * 1 and * x * 2 the link is linear. Since many random factors influence the formation of the expected result, it is naturally a random variable.

One of the characteristics of the random variable * X * is the ** law of the distribution of its probabilities. **

The nature, type of distribution reflects the general conditions arising from the nature and nature of the phenomenon, and features that affect the variation of the indicator under study (the expected result).

As practice shows, the so-called * normal distribution is most often used to characterize the distribution of socio-economic phenomena. *

From the course of probability theory and mathematical statistics it is known that a normally distributed random variable is continuous and its differential distribution function has the form

where y = * f * (X) determines the probability distribution density for each point * X. *

The graph of the normal distribution function is described by the so-called * normal curve * (the Gauss curve - Figure 6.3). An important property of the graph of the differential function of the normal distribution is that the area bounded by a normal curve and the axis * X, * is always equal to one.

Using the density function of the normal distribution allows us to calculate the frequency (probability) of the appearance of a random variable.

To estimate the probability of a random variable falling into a certain interval, use the integral probability density function Φ (λ).

where * f * (t) is the differential function of the normal distribution.

** Fig. 6.3. ** Normal distribution graph

The probability of a random variable falling into the interval (α, β) is determined as follows:

The above indicators are the starting point used to quantify risk using both statistical methods and other approaches that use probability theory.

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