Recent decades world economy regularly falls into the whirlpool financial crises. 1987, 1997, 2008 almost led to the collapse of the existing financial system, which is why leading experts began to develop methods that can be used to control the uncertainty that prevails in financial world. AT Nobel Prizes recent years (obtained for the Black-Scholes model, VaR, etc.) there is a clear trend towards mathematical modeling of economic processes, attempts to predict market behavior and assess its stability.

Today I will try to talk about the most widely used loss prediction technique - Value at Risk (VaR).

The concept of VaR

An economist-friendly explanation of VaR sounds in the following way: "Expressed in monetary units an estimate of the value that losses expected over a given period of time will not exceed with a given probability. In essence, VaR is the amount of loss due to investment portfolio for a fixed period of time, in case some unfavorable event occurs. Under "unfavorable events" we can understand various crises, poorly predictable factors (changes in legislation, natural disasters, ...) that can affect the market. As a time horizon, one, five or ten days are usually chosen, due to the fact that it is extremely difficult to predict the behavior of the market for a longer period. The level of acceptable risk (in fact, the confidence interval) is taken equal to 95% or 99%. Also, of course, the currency in which we will measure losses is fixed.
When calculating the value, it is assumed that the market will behave in a “normal” way. Graphically, this value can be illustrated as follows:

VaR Calculation Methods

Consider the most commonly used methods for calculating VaR, as well as their advantages and disadvantages.

Historical modeling

In historical modeling, we take values already known from past measurements financial fluctuations for a portfolio. For example, we have the behavior of a portfolio over the previous 200 days, based on which we decide to calculate the VaR. Suppose that the next day the financial portfolio will behave the same way as on one of the previous days. Thus, we will get 200 outcomes the next day. Further, we assume that the random variable is distributed according to the normal law, based on this fact, we understand that VaR is one of the percentiles of the normal distribution. Depending on what level of acceptable risk we have taken, we choose the appropriate percentile and, as a result, we obtain the value we are interested in.

The disadvantage of this method is the impossibility of making predictions for portfolios about which we do not have information. A problem may also arise if the portfolio components change significantly in a short period of time.

A good example of calculations can be found at the following link.

Leading Component Method

For each financial portfolio, you can calculate a set of characteristics that help assess the potential of assets. These characteristics are called the leading components and are usually a set of partial derivatives of the portfolio price. To calculate the value of a portfolio, the Black-Scholes model is usually used, which I will try to talk about next time. In a nutshell, the model represents the dependence of the value of a European option on time and on its current value. Based on the behavior of the model, we can evaluate the potential of the option by analyzing the function using classical methods of mathematical analysis (convexity/concavity, increasing/decreasing intervals, etc.). Based on the analysis data, VaRs are calculated for each of the components and the resulting value is built as a combination (usually a weighted sum) of each of the estimates.

Naturally, these are not the only methods for calculating VaR. There are both simple linear and quadratic price prediction models, as well as a rather complex method of variations-covariances, which I did not talk about, but those who are interested can find a description of the methods in the books below.

Criticism of the methodology

It is important to note that when calculating VaR, the hypothesis of normal market behavior is accepted, however, if this assumption were correct, crises would occur once every seven thousand years, but, as we can see, this is absolutely not true. Nassim Taleb, a well-known trader and mathematician, in his books Fooled by Randomness and The Black Swan exposes existing system risk assessment is severely criticized, and also offers his own solution, in the form of using a different risk calculation system based on the lognormal distribution.

Despite criticism, VaR is quite successfully used in all major financial institutions. It should be noted that this approach is not always applicable, which is why other methods were created with a similar idea, but with a different calculation method (for example, SVA).

In response to criticism, modifications to VaR have been developed based either on other distributions or on other calculation methods at the peak of the Gaussian curve. But I will try to talk about this another time.

All currency risk forecasting methods can be conditionally divided into two groups:

*statistical methods based on quantitative analysis
*expert methods based on qualitative analysis
Basis for quantification currency risks the Value at Risk (VaR) method was adopted, which determines the functional relationship between the probability of risk occurrence from external indicators. The VAR method is used by such international institutions as the Bank for International Settlements, banking federation European Community and others to calculate capital adequacy. This technique is used by many European banks for measuring market risks(which includes currency risk) of the bank.

VaR is the amount of loss that, with the probability of a level confidence interval(eg 99%) will not be exceeded. Accordingly, in 1% of events, the loss may exceed VaR. The VaR method is essentially a development of the classical risk measurement method based on the calculation of the standard deviation and the subsequent application of the normal distribution law. Advantages of assessing currency risks using the VaR method has the following advantages, because it allows you to:
*calculate risks for all possible markets
*calculate the risk of losses according to the probability of their occurrence

In general terms, VaR can be defined as a statistical estimate of the maximum loss of an investor's portfolio for a given distribution of market factors over a certain period of time in almost all cases (with the exception of a small percentage of situations).

When calculating VaR, you need to determine the basic elements that affect its value: probabilistic distribution of market factors, confidence interval, i.e. the probability with which losses should not exceed VaR, holding period(holding period). VaR calculation formula:

VAR=k*σ* Y

Where k is the coefficient of a certain confidence interval, Y is the value of the asset, σ is the volatility of the exchange rate.
Volatility is equal to the square root of the variance: a measure of the dispersion of a currency from its average. The next step in calculating the VaR indicator is the choice of a confidence interval, a quantitative characteristic of the forecast accuracy. Each confidence interval has its own coefficient(multiplier k). The most commonly used are 95% confidence interval (coefficient 1.65) and 97.5% (coefficient 1.96) and 99% (coefficient 2.33) intervals. The specified intervals determine the probability of exceeding the calculated VaR.

There are three methods for calculating VaR: the variance-covariance method (analytical), historical modeling, and statistical modeling (called the Monte Carlo method). The most commonly used method for calculating VaR is variance-covariance method(disassembled above). Its widespread use is due to the fact that it is easy to use, and the results of the calculation accuracy are high level. It is possible to use this method only if the studied statistical data correspond to the normal distribution law, which in reality should mean the absence of any significant deviations of price values from the average level. The analytical method for calculating VaR can be implemented on any computer, however, when using it, one must take into account the stationary normal distribution, which makes it unsuitable for Russian conditions.

Value at Risk one of the most common forms of measurement financial risks. Commonly referred to as "VaR".

It is also often called "16:15", he got this name because 16:15 is the time at which he supposedly should lie on the table of the head of the bank's board JPMorgan. (In this bank this indicator was first introduced to improve the efficiency of risk management)

In essence, VaR reflects the size of a possible loss that will not be exceeded within a certain period of time with a certain probability ( also known as the "tolerable risk level""). Those. the largest expected loss that an investor can receive with a given probability within n days

The key parameters of VaR are:

Time horizon - the period of time for which the risk is calculated. (According to the Basel documents - 10 days, according to the Risk Metrics method - 1 day. The most common calculation is with a time horizon of 1 day. 10 days is used to calculate the amount of capital covering possible losses.)
The level of acceptable risk is the probability that losses will not exceed a certain value (According to the Basel documents, the value of 99% is used, in the RiskMetrics system - 95%).
Base currency - the currency in which VaR is calculated

Those. A VaR of X with a time horizon of n days, a 95% acceptable risk level and a base currency of the US dollar would mean that there is a 95% chance that the loss will not exceed X dollars within n days.

Standard for Broker-Dealer Reports on OTC Derivatives Reported to the Commission on Exchanges and securities USA are 2 week period and 99% chance.
The Bank of International Settlements to assess the sufficiency banking capital set the probability to 99% and the period to 10 days.
JP Morgan publishes its daily VaR values at 95% confidence level.
According to a New York University Stern School of Business study, about 60% pension funds The United States uses VaR in its work

An example of calculating VaR in Excel:

Let's take the price history of the asset we are interested in, for example, ordinary shares Sberbank. In the example, I took EOD (EndOfDay) prices for 2010.

Let's calculate the standard deviation of the yield obtained (the formula for calculating the standard deviation for the sample for Microsoft Excel will look like this =SDV.V(C3:C249)):

Assuming an acceptable risk level of 99%, we calculate the inverse normal distribution (quantile) for a probability of 1% (the formula for Excel in our case will look like =NORM.INV(1%, AVERAGE(C3:C249), C250)):

Well, now let's calculate directly the value of the VaR itself. To do this, subtract the estimated value obtained by multiplying by the quantile from the current value of the asset. Therefore, for excel formula will take the form: =B249-(B249*(C251+1))

In total, we got the calculated value VaR = 5.25 rubles. Given our time horizon and the degree of acceptable risk, this means that Sberbank shares will not fall in price by more than 5.25 rubles over the next day, with a 99% probability!

One of the main tasks of financial institutions is to assess market risks that arise as a result of fluctuations (favourable events) in the prices of shares, commodities, exchange rates, interest rates etc. The simplest measure of an investor's dependence on market risks is the change in portfolio capital, i.e. profits or losses arising from movements in asset prices. The most common methodology for assessing market risks today is Risk Cost (Value - at - Risk, VAR). VAR is a summary measure of risk capable of comparing risk across portfolios (eg equity and bond portfolios) and across different financial instruments (eg forwards and options).

The risk value indicator was developed in the late 1980s. and immediately won recognition among the largest participants in the financial market. Subsequently, the value at risk indicator (VAR) became a full-fledged standard for reporting a firm's risk, which could be used within the company itself, as well as reported to investors and regulators.

Over the past few years, VAR has become one of the most popular risk management and control tools in companies of various types. This was due to several reasons. The first reason was undoubtedly the disclosure in 1994 by the largest US investment company of J.P. Morgan risk assessment system Riskmetrics TM and the provision of free use of the database for this system for all market participants. The VAR values obtained using the Riskmetrics TM system are still a kind of benchmark for VAR estimates. The second reason lies in the investment "climate" that prevailed in the late 1990s and was associated with huge losses incurred by financial institutions, in particular, when operating on the derivatives markets (financial market instruments operating on the basis of fixed assets (stocks, bonds etc.)). Table 3.7. indicated the losses suffered by some Western companies and the dates on which they were made public. The third reason , is the decision of bank supervisors to use VARs to determine capital reserves.

Table 3.7.

Losses of large Western companies in 1993-1995

Report date	Company	Losses (in million rubles)

	Metallgesellschaft
	Askin Capital Management
	Procter & Gamble
	Paine Webber Bond Mutual Fund

	Orange County CA

The risk value reflects the maximum possible loss from a change in the value of a financial instrument, portfolio assets, company, which can occur over a given period of time with a given probability of its occurrence. For example, when it is said that the 1-day value at risk is $100,000 with a 95% confidence interval (or a 5% probability of loss), this means that losses in one day exceeding $100,000 can occur no more than than 5% of the time.

In simple terms, the calculation of the value of VAR is carried out in order to conclude a statement of this type: "We are X% sure (with a probability of X%) that our losses will not exceed Y dollars over the next N days." In this sentence, the unknown value Y is VAR. It is a function of 2 parameters: N - time horizon and X - confidence interval (level). So, for example, the standard for broker-dealer reports on transactions with over-the-counter derivatives submitted to the US Securities and Exchange Commission are N equal to 2 weeks and X = 99%. The Bank of International Settlements set X = 99% and N equal to 10 days to assess bank capital adequacy. Company J.P. Morgan publishes his daily VAR values at a 95% confidence level.

To determine the risk value, it is necessary to know the relationship between the size of profits and losses and the probabilities of their occurrence, i.e. probability distribution of profits and losses during the selected time interval. In this case, by set value loss probability, it is possible to unambiguously determine the size of the corresponding loss.

A typical trick is to use a normal probability distribution.

Key parameters in determining risk value – confidence interval and time horizon. Since losses are a consequence of market fluctuations, the confidence interval serves as the boundary that, in the opinion of the portfolio manager, separates “normal” market fluctuations from extreme price spikes in terms of their frequency. Usually the probability of loss is set at 1%, 2.5 or 5% (the corresponding confidence interval is 99%, 97.5 and 95%), however, the risk manager can choose any other value in accordance with the money management strategy, which the company holds.

In addition to the subjective assessment, the confidence interval can also be established by an objective method. To do this, build a graph of the actually observed (empirical) distribution of probabilities of profits and losses and combine it with a graph of the density of the normal distribution. The intersection points of the "tails" of the empirical and normal distributions will set the required confidence interval.

It should be borne in mind that with an increase in the confidence interval, the risk value will increase.

The choice of time horizon depends on how often transactions are made with these assets, as well as on their liquidity. For financial institutions active in the capital markets, the typical settlement period is 1 day, while strategic investors and non-financial companies may use longer periods. In addition, when setting the time horizon, one should take into account the availability of statistics on the distribution of profits and losses for the desired time interval. Along with the lengthening of the time horizon, the indicator of risk value also increases.

The value of risk value is determined based on the properties of the normal distribution. So, if the confidence interval is set at the level of 95%, then the risk value is equal to 1.65 standard deviation of the portfolio. Thus, the value of risk value is calculated according to the following formula:

where Z is the number of standard deviations corresponding to a given confidence interval;

t– time horizon; p– position size vector; Q– covariance matrix of changes in the cost of positions.

It should be noted that the concept of risk value implicitly assumes that the composition and structure of the assessed portfolio of assets will remain unchanged throughout the entire time horizon. Such an assumption is hardly justified for relatively long time intervals, therefore, with each portfolio update, it is necessary to adjust the amount of risk value.

Historically, the VAR risk assessment approach was first recommended by The Global Derivatives Study Group (G30) in 1993 in the Derivatives: Practices and Principles study. In the same year, the European Council in the directive "EEC 6 - 93" ordered the establishment of capital reserves to cover market risks using VAR models. In 1994, The Bank of International Settlements recommended that banks disclose their VAR values. In 1995, the Basel Committee on Banking Supervision suggested that banks use their own VAR estimation models as the basis for calculating capital reserves. Requirements for the amount of reserve capital V were calculated as a maximum of two values: the current value of VAR (VAR t) and the average VAR for the previous 60 days multiplied by a factor between 3 and 4:

Factor value λ depends on the one-day prediction of the model for previous time periods. So, if we denote by K - the number of times when one-day losses exceeded the predicted value of VAR for Last year(or the last 250 trading days), then the following 3 zones are distinguished: "green" zone (K less than or equal to 4), "yellow" zone (K in the range from 5 to 9), "red" zone (K greater than or equal to 10 ). If K lies in the "green" zone, then λ= 3, if in the "yellow" zone, then 3< λ< 4, если в "красной" зоне, то λ =4.

The development and implementation of VAR models is happening at a rapid pace. In investment companies and banks, the VAR methodology can be applied according to at least in 4 directions of activity.

1) Internal monitoring of market risks. Institutional investors can calculate and monitor VAR values by several levels: aggregated portfolio, asset class, issuer, counterparty, trader/portfolio manager, etc. From the point of view of monitoring, the accuracy of estimating the value of VAR goes by the wayside, since in this case the value of the relative rather than the absolute value of VAR is important, i.e. The VAR of a manager or portfolio VAR compared to the VAR of a benchmark portfolio, index, another manager, or the same manager at previous times.

2) External monitoring. VAR allows you to create a view of the market risk of a portfolio without disclosing information about the composition of the portfolio (which can be quite confusing). In addition, regular reports using VAR figures provided to superiors can serve as one of the arguments that the risk taken by the managing managers is within acceptable limits.

3) Monitoring the effectiveness of the hedge. VAR values can be used to determine the extent to which a hedging strategy is meeting its objectives. A manager can evaluate the effectiveness of a hedge by comparing the VARs of portfolios with and without hedge. If, for example, the difference between the two values is small, then the question arises as to whether the hedging is appropriate or whether the hedging is applied correctly.

4) "What - if" analysis of possible trades. The VAR methodology makes it possible to give more freedom and autonomy to the management staff, as it becomes possible to reduce all sorts of bureaucratic procedures associated with the approval of certain transactions (especially with derivatives). This is achieved through the monitoring of transactions (deals) using VAR. For example, senior management can simply set a rule for their broker-dealers of this kind: "No transaction should increase the value of VAR by more than X% of the initial capital" and then not go into the details of each individual trade afterwards.

Thus, companies can use VAR values to create reports for managers, shareholders and external investors, since VAR allows you to aggregate all kinds of market risks into one number that has a monetary value. With the help of the VAR methodology, it becomes possible to calculate the risk assessments of various market segments and identify the most risky positions. VAR estimates can be used to diversify capital, set limits, and evaluate a company's performance. In some banks, the evaluation of traders' operations, as well as their remuneration, is calculated based on the calculation of profitability per VAR unit.

Non-financial corporations can use the VAR technique to assess the riskiness of cash flows and make hedging decisions (protecting capital from adverse price movements). So one of the interpretations of VAR is the amount of uninsured risk that a corporation takes on. Among the first non-financial companies that began to use VAR to assess market risk, one can note the American company Mobil Oil, the German companies Veba and Siemens, and the Norwegian Statoil.

Investment analysts use VAR to evaluate various projects. Institutional investors such as pension funds use VAR to calculate market risk. As noted in a study by the New York University Stern School of Business, about 60% of US pension funds use the VAR methodology in their work.

As already noted, for a given time interval , where t is the current time, and the confidence level p VAR is a loss on the time interval , which will occur with a probability of 1 - p.

Here is a simple example: Let the daily VAR for this portfolio be $2 million at the 95% confidence level. This value of VAR means that in the absence of sharp changes in market conditions, a one-day loss will exceed $2 million in 5% of cases (or once a month, assuming that there are 20 working days in a month).

Mathematically speaking, VAR = VAR t,T is defined as the upper bound of the one-sided confidence interval:

Probability (R t (T)< – VAR}) = 1 – α,

where α is the confidence level, R t (T) is the growth rate of the portfolio's capital over the interval under the "continuous method of interest calculation":

R t (T) = log (V(t+T)/ V(t)),

where V(t+T) and V(t) are the values of the portfolio equity at time t+T and t, respectively. In other words, V(t+T) = V(t) * exp(R t (T)).

Note that R t (T) is random variable and is thus characterized by some probabilistic distribution. The value of VAR is determined from the distribution of portfolio increments as follows:

where F R (x) = Probability (R ≤ x) is the distribution function of the portfolio growth rate, f R (x) is the distribution density R t (T).

The traditional techniques for approximating the distribution R t (T) are:

parametric method;

historical data modeling

Monte Carlo method

scenario analysis

If changes in portfolio equity are characterized by a parametric distribution, then VAR can be calculated in terms of the parameters of this distribution.

Figure 3.19. the density of the normal distribution is presented and the quantile Z 1 - α is indicated. The area under the graph of the density function to the left of Z 1 - α (the area of the "left tail") is equal to 1 - α.

It is assumed that the growth rate of the asset μ= 0. Then VAR=-V t z 1-α σ , where V t is the value of the portfolio's capital at the current time t.

Example 1: Single asset case.

On the next chart 3.20. a histogram of monthly growth rates for the FTSE 100 index from 1988 to 1995 is shown.

To calculate VAR, we use the fact that the probability of the probability in the "left tail" of the normal distribution is a known function of the standard deviation σ, namely, 5% of the probability of the normal distribution is to the left of 1.65 standard deviations from the mean value μ. In this example, we have estimates μ=0.76% and σ=4.58%. Assuming that the current value of the portfolio's capital is £1 million, the value of VAR over a 1 month time interval at a 95% confidence level is:

VAR = 1"000"000 (0.0076 – 1.65  0.0458) = 68 "012 f.st.

Example 2: The case of two assets.

Consider now the previous example of a portfolio consisting of the "FTSE 100 index" (it is assumed that the investor can form his portfolio of stocks in such a way that each stock has the same weight as in the FTSE index - 100. Thus, the increment of such a portfolio will be is equal to the increment of the FTSE index - 100.), but from the point of view of an investor for whom the base currency is the US dollar. Thus, the portfolio now consists of two "assets": stock index, denominated in pounds sterling, and the GBP/USD exchange rate.

Let the current value of the exchange rate be 1.629 dollars per pound. Then the capital of the investment portfolio in US dollars is 1 "000" 000 / 1.629 = $ 613 "874. Thus, the value of the 1-month VAR of the stock index at a 95% confidence level there is:

VAR equity = $613"874  (0.0076 – 1.65  0.045)=$40"915

The estimates of the standard deviation and average GBP/USD exchange rate for the time interval 01/88 - 01/95 are 0.0368 and -0.001 respectively. Thus, the 1-month VAR of the GBP/USD exchange rate is:

VAR forex = $613"874  (– 0.001 – 1.65  0.0368)=$37"888

We are now able to calculate the total VAR of a portfolio, using the fact that the variance of a portfolio of two assets that have a joint normal distribution is equal to the sum of the variances of each asset and the double correlation between those assets multiplied by the standard deviations of the assets:

(VAR portfolio) 2 =(VAR equity) 2 +(VAR forex) 2 +2  ρ  VAR equity  VAR forex ,

where ρ is the correlation coefficient between the growth rates of the FTSE-100 index and the GBP/USD exchange rate. The estimate of ρ is – 0.2136, i.e. the FTSE index - 100 and the GBP/USD rate are inversely correlated. Thus, the 1-month VAR of the portfolio at the 95% confidence level is

Thus, we can expect portfolio losses of more than 8% of initial capital in 5 out of 100 months in the future.

As you can easily see, the VAR of the portfolio turned out to be less than the sum of the VAR of the index and the exchange rate (equal to $78"803). This was a consequence of portfolio diversification: since assets are negatively correlated, losses on one asset are offset by gains on another asset.

In addition, as expected, the VAR value for, for example, an American investor in the FTSE 100 index turns out to be larger compared to the VAR value for a British investor (equal to GBP68"012*1.629=USD41"751) who invests his funds in the same "asset-index". This was a consequence of the additional risk posed by the GBP/USD exchange rate.

In the above examples, the normal distribution was chosen for illustrative purposes only due to the simplicity of the calculations. In practice, as is known, increments in asset prices are said to have heavier "tails" compared to the normal law, i.e. in reality, there are more "extreme" events compared to what one would expect from a normal distribution. VAR, by its nature, just deals with the prediction of events from the "tails" of the distribution (with events from the "left tail" for "long" positions in an asset and with events from the "right tail" for "short" positions in an asset). Such "catastrophic risk" events are well known in the insurance and reinsurance business.

Modeling method according to historical data consists in constructing the distribution of portfolio changes R t (T) according to historical data. In this case, only one hypothesis is made about the distribution of the return on the portfolio's capital: the "future" will behave in the same way as the "past". For example 1, discussed above, we have that the 5% - th quintile of the historical increments of the FTSE - 100 index is - 6.87% (marked by a vertical line on the histogram). Thus, using historical data, we obtain the following VAR for the portfolio from the "FTSE-100 Index":

VAR=GBP 1"000"000 * (- 6.87%)=GBP 68"700

(compare with VAR=GBP 68"012 from example 1).

Monte Carlo Method consists in defining statistical models for portfolio assets and modeling them by generating random trajectories. The VAR value is calculated from a portfolio capital growth rate distribution similar to that shown in the histogram for the FTSE-100 index, but resulting from artificial modeling.

Scenario analysis method examines the effect of a change in portfolio capital depending on changes in the values of risk factors (eg, interest rate, volatility) or model parameters. Modeling occurs in accordance with certain "scenarios". This is how many banks estimate the value of "PV01" of their portfolios with "fixed income" (fixed - income portfolios, i.e. portfolios consisting of instruments "at an interest rate": bonds, forwards on an interest rate, swaps, etc.) , which is calculated as the change in portfolio equity for a parallel shift in the yield curve by 100 basis points.

The use of a particular method should be based on such factors as the quality of the database, the ease of implementation of the method, the availability of high-speed computers, the requirements for the reliability of the results, etc.

I would like to note that the VAR methodology is not a universal way to prevent financial losses. It only helps companies to imagine whether the risks they are exposed to are the risks they are would like to take over or think they have taken over. VAR cannot tell the company manager "how much risk to take", but can only say "how much risk has already been taken". VAR can and should be used not as a substitute for, but in addition to other methods of risk analysis such as, for example, Shortfall-at-Risk(SAR, Average Loss), when they are interested not only limit value of capital, below which one should expect a loss with a certain degree of probability, but also the size of this loss.

As a rule, risk value calculation is accompanied by a detailed analysis of several possible scenarios, modeling of empirical probability distributions and testing the portfolio for resistance to changes in key parameters. The value of risk value, as a generalized assessment of market risk, is needed primarily for making operational decisions by the top management of the company.

The concept of risk value (value at risk -var)

In practice financial management there has always been a need for a unified, prompt and commonly understood assessment of the possible losses in the value of a portfolio of assets on certain period time. The risk value indicator just meets all these requirements. It was developed in the late 1980s. and immediately won recognition among the largest participants financial market. Its popularity was due to the fact that, thanks to a certain simplification, it was accessible to the understanding of managers at all levels of company management. Subsequently, the risk value indicator became a full-fledged standard for information about the risk of a firm, which could be used within the company itself, as well as reported to investors and regulators.

Value at risk (VaR) reflects the maximum possible loss from a change in the value of a financial instrument, portfolio of assets, company, etc., that can occur over a given period of time with a given probability of its occurrence. For example, when they say that the risk value for 1 day. is 100 thousand dollars. United States with 95 confidence interval % (or a loss probability of 5%), this means that losses in one day exceeding $100,000 can occur no more than 5% of the time.

In other words, the risk value is the amount of loss that can be exceeded with a probability of no more than x% [will not be exceeded with a probability of (100 - x)%] over the next n days. To determine the value of risk value, it is necessary to know the relationship between the size of profits and losses and the probabilities of their occurrence; i.e., the probability distribution of profits and losses over a selected time interval. In this case, according to the given value of the loss probability, it is possible to unambiguously determine the size of the corresponding loss. However, the real law of the probability distribution is unknown in most cases, so another, well-studied distribution has to be used as a replacement. A typical trick is to use a normal probability distribution.

It follows from the definition that key parameters in determining the risk value -- confidence interval and time horizon. Since losses are a consequence of market fluctuations, the confidence interval serves as the boundary that, in the opinion of the portfolio manager, separates “normal” market fluctuations from extreme price spikes in terms of their frequency. Usually the loss probability is set at 1%, 2.5 or 5 % (corresponding confidence intervals are 99%, 97.5 and 95%), however, the risk manager may choose any other value in accordance with the money management strategy followed by this company. In particular, in the RiskMetrics system developed by the J. P. Morgan, 5% probability is used. In addition to the subjective assessment, the confidence interval can also be established by an objective method. To do this, build a graph of the actually observed (empirical) distribution of probabilities of profits and losses and combine it with a graph of the density of the normal distribution. The intersection points of the "tails" of the empirical and normal distributions will set the required confidence interval.

It should be taken into account that with an increase in the confidence interval, the risk value will increase: it is obvious that losses that occur with a probability of only 1% will be higher than losses that occur with a probability of 5%.

The choice of time horizon depends on how often transactions are made with these assets, as well as on their liquidity. For financial institutions leading active operations in capital markets, the typical settlement period is 1 day, while strategic investors and non-financial companies may use longer periods. In addition, when setting the time horizon, one should take into account the availability of statistics on the distribution of profits and losses for the desired time interval. Along with the lengthening of the time horizon, the indicator of risk value also increases. It is intuitively clear that possible profits or losses, for example, in 5 days. may have a larger scale than 1 day. In practice, it is believed that over a period of n days, the risk value will be approximately n times greater than for 1 day.

It should be remembered that the concept of risk value implicitly assumes that the composition and structure of the portfolio of assets being assessed will remain unchanged throughout the entire time horizon. Such an assumption is hardly justified for relatively long time intervals, therefore, with each portfolio update, it is necessary to adjust the amount of risk value.

The indicator of risk value, of course, is not the only one and universal tool risk assessments. As a rule, risk value calculation is accompanied by a detailed analysis of several possible scenarios, modeling of empirical probability distributions and testing the portfolio for resistance to changes in key parameters. The value of risk value, as a generalized assessment of market risk, is needed primarily for making operational decisions by the top management of the company.

To calculate the risk value indicator, three different economic and mathematical methods are used: analytical, historical modeling method and Monte Carlo statistical test method. The first of them is parametric and allows one to obtain estimates in a closed form, while the other two represent a kind of mathematical experiment. The initial stage and a necessary condition for the implementation of these methods is the determination of the so-called "market risk factors", i.e. the main prices and interest rates that affect the value of the portfolio. Identification of a limited set of market factors makes it possible to present the price of a financial instrument as a function of these factors and thereby decide main problem quantitative description of the portfolio value.

The definition of market factors involves the "decomposition" of the portfolio financial instruments to simpler ones directly related to market risk factors, and their further consideration as "subportfolios", or positions consisting of such primary instruments. For example, the price forward contract for the supply of one currency in exchange for another depends on three market factors: the spot exchange rate of one currency to another and two interest rates for each of the currencies of the contract. For all instruments included in the portfolio, analytical dependencies should be obtained expressing their present value through market risk factors.

In some cases, when the exact value formula is not known, numerical methods are used to estimate the value of the instrument. This is the most difficult stage, because for a large financial institution the number of such factors can be measured in hundreds. Further steps include species identification and parameter evaluation statistical distribution future values of market factors, using the obtained values and analytical dependencies to identify potential changes in the value of various positions that make up the portfolio, and then ranking and summing changes in the value of all positions to assess the expected changes in the value of the entire portfolio.