Basic approaches to risk assessment
When considering risk from the point of view of its assessment, it is necessary to solve the following tasks:
 describe all possible future development scenarios that correspond to this risk (possible outcomes of decision making or random events):
 determine the probabilities of each of these variants (random events).
The average expected value (expectation) of the random variable X is expressed in monetary units, denoted by MX and is calculated as a weighted average for all its different values, where the probability of each value is used as the weighting factor. Calculated as
(6.1)where εi is the value of the random variable X in the situation i (i = 1, ..., k) , p i is the probability of the situation i.
Standard deviation is the most common measure of risk level. Determined by the formula
(6.2)where i  the number of options for action (situation development); εi  estimated income (estimated losses) for each of the options; ͞ε is the average expected return (expectation, MX); Pi is the probability that i.
An example of calculating a mathematical expectation is shown in Table. 6.7. There are data on projects A and B and the likely options for the development of the situation.
The calculation of the rootmeansquare deviation is shown in Table. 6.8.
Table 6.7
Estimated project and development scenarios
Option ( i ) 
Project A 
Project In 

Income (ε) 
Probability ( P i) 
Expected Revenue P i ) 
Income, ε 
Probability (P i ) 
Expected return ( P i ) 

Favorable 
600 
0.25 
150 
800 
0.20 
160 
Medium 
500 
0.5 
250 
450 
0.60 
270 
Unfavorable 
200 
0.25 
50 
100 
0.20 
20 
On average (͞ε) 
 
1.0 
450 
 
1.0 
450 
Table 6.8
Standard deviation for projects
Project 
Varie ant 
Income (ε) 

A 
Favorable 
600 
450 
+150 
22,500 
0.25 
5625 

Medium 
500 
450 
+50 
2500 
0.5 
1250 
 

Unfavorable 
200 
450 
250 
62,500 
0.25 
15 625 

Average 
 
450 
 
 
1.0 
22,500 
150 

In 
Favorable 
800 
450 
+350 
122,500 
0.2 
24 500 

Medium 
450 
450 
0 
0 
0.6 
0 
 

Unfavorable 
100 
450 
350 
122,500 
0.2 
24 500 

Average 
 
450 
 
 
1.0 
49 000 
221 
Coefficient of variation is the risk/income ratio of the project. The higher it is, the more risky the project is. The coefficient of variation allows to determine the level of risk if the indicators of the average expected income for projects are different.
where σj  the standard deviation of the project j ; ͞εi  average expected income (expectation, MX) for the project j.
In the world practice of management, various methods of analysis of design risks are used. The most common of these are:
 method of adjusting the discount rate;
 the method of reliable equivalents (reliability coefficients);
 sensitivity analysis of performance criteria (net present value ( NPV ), internal rate of return ( IRR ) , profitability index (PI) , etc.);
 the scenario method;
 analysis of probability distributions of payment flows;
 Decision trees;
 the Monte Carlo method (simulation simulation), etc.
The method of adjusting the discount rate is to introduce a risk premium in the financial model of the project (when calculating the net present value), increasing the discount rate. The allowance is established proceeding from the concept of the account of cost of money in time accepted in the organization. As a rule, the risk is taken into account by the cumulative method on the basis of assignable to different types of risk ranks (in the process of qualitative analysis).
Advantages of this method are in the simplicity of calculations, as well as in intelligibility and accessibility. At the same time, the method has significant drawbacks.
The method of adjusting the discount rate carries the future payments flows to the present moment of time (discounting at a higher rate), but does not give any information about the degree of risk (possible deviations of the results). In this case, the results obtained depend substantially only on the value of the risk premium. It also implies an increase in the risk in time with a constant coefficient, which can hardly be considered correct, since many projects are characterized by the presence of risks in the initial periods, with a gradual decrease in their end of the implementation. Thus, profitable projects that do not involve a significant increase in risk over time can be assessed incorrectly and rejected.
This method does not carry any information about the probabilistic distributions of future payment flows and does not allow them to be evaluated.
Finally, the downside of the simplicity of the method is the significant limitations of the modeling capabilities of the various options, which amounts to analyzing the criteria NPV ( IRR , PI , etc.) from changes in only one indicator  the discount rate.
Despite the mentioned shortcomings, the method of adjusting the discount rate is widely used in practice.
The method of reliable equivalents. The disadvantages of this method are:
 the complexity of calculating the reliability coefficients adequate to the risk at each stage of the project:
 the inability to analyze probability distributions of key parameters.
Sensitivity analysis. Helps determine which risks can have the greatest potential impact on an enterprise's objectives. The sensitivity analysis examines the degree to which the uncertainty of each project element affects the overall results, while the remaining undefined elements remain at their basic (initially estimated) level.
Sensitivity analysis is the simplest and, therefore, the most frequently used quantitative method of risk research. He underlies a number of managerial decisions. So, if the product's price turned out to be a critical factor (as in the example), you can strengthen the marketing program or revise the cost part to reduce the cost of the project. If analysis shows high sensitivity to changes in output, attention should be paid to improving productivity.
Sensitivity analysis has a number of undoubted advantages, including its simplicity, theoretical transparency, the possibility of formalization and the naturalness of the applied mathematical apparatus, the visibility of the interpretation of the results. It is these virtues that determined the wide practical use of this method. However, the analysis of sensitivity is inherent and significant drawbacks. First, this method is expert, i.e. different groups of methods can get different results. Secondly, sensitivity analysis does not take into account the correlation between mutable factors, which can reduce the reliability of the results. Third, the disadvantage is the onefactor nature of this method, i.e. focus on accounting for only one factor, while all others remain unchanged. This does not allow for the dynamic nature of the project.
Scripting Method in general allows you to get a fairly clear picture for various project implementation options, as well as provides information about sensitivity and possible deviations, and the use of software tools such as Excel allows you to significantly increase the effectiveness of such an analysis by virtually unlimited increase in the number of scenarios and the introduction of additional variables.
Analysis of the probability distributions of payment flows. Using this method provides useful information about the expected values of NPV and net cash flows, and analyze their probability distributions. However, the use of this method assumes that the probabilities for all cash flow options are known or can be accurately determined. In fact, in some cases, the probability distribution can be given with a high degree of certainty based on the analysis of past experience in the presence of large volumes of actual data. However, most often such data are not available, therefore the distributions are set based on the assumptions of experts and carry a large share of subjectivity.
Decision trees. This is a diagram that describes the decision making associated with the adoption of one or another of the available alternatives. It combines the risks and costs and benefits of each logical scenario associated with future events and decisions. The use of a decision tree involves identifying which solution will bring the highest expected value to the decision maker if all the uncertain components, costs, results, and future actions are quantified.
The limitation of the practical use of this method is the initial premise that a project must have a clear or reasonable number of development options. The method is especially useful in situations where decisions made at each time point strongly depend on decisions made earlier, and in turn determine scenarios for the further development of events.
Simulation modeling. Monte Carlo simulations ( MonteCarlo Simulation) model for a project with indefinite parameter values and knowing the probability distribution function for the project parameters, as well as the correlation between the parameters, obtain the project profitability distribution. The enlarged scheme of MonteCarlo risk analysis for the investment project is shown in Fig. 6.5.Fig. 6.5. Monte Carlo Risk Analysis Scheme
Practical application of this method has demonstrated wide possibilities of its use in investment design, especially in conditions of uncertainty and risk. This method is especially convenient for practical application by that it is successfully combined with other economic and statistical methods, as well as with game theory and other methods of operations research.
The high risk of the project leads to the need to find ways to artificially reduce it. In the practice of project management, there are three ways to reduce the risk:
 risk distribution between project participants (transfer of a part of the risk to coexecutors);
 insurance;
 reservation of funds to cover unforeseen expenses.
The usual practice of risk distribution is to make the project participant responsible for the risk who is able to count and control the risk better than anyone else. However, in life it often happens that this partner is not financially strong enough to overcome the consequences of the risks. The distribution of risk is realized when developing the financial project plan and contract documents.
Most large projects are characterized by a delay in their implementation, which can lead to an increase in the cost of work that exceeds the initial cost of the project. The way out of this situation is to involve insurance companies in the project.
Providing a reserve for unpredictable expenses is a way to deal with risk, providing for the correlation between potential risks that affect the project cost and the amount of costs needed to overcome project failures. The main problem in creating a reserve for covering unforeseen expenses is to assess the potential consequences of risk.
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