The concept of risk value (value at risk -var). Methods for assessing market risk Var risk assessment

In recent decades, the world economy has regularly fallen into a whirlpool of 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 dominates the financial world. In the Nobel Prizes of recent years (received for the Black-Scholes model, VaR, etc.) there is a clear tendency 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 method for predicting losses - Value at Risk (VaR).

Concept of VaR

An economist's understanding of VaR is as follows: “An estimate, expressed in monetary units, of the amount that losses expected during a given period of time will not exceed with a given probability.” Essentially, VaR is the amount of loss on an investment portfolio over a fixed period of time, if some unfavorable event occurs. “Unfavorable events” can be understood as various crises, poorly predictable factors (changes in legislation, natural disasters, ...) that can affect the market. One, five or ten days are usually chosen as the time horizon, due to the fact that it is extremely difficult to predict market behavior over a longer period. The acceptable risk level (essentially a confidence interval) is taken to be 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

Let's consider the most commonly used methods for calculating VaR, as well as their advantages and disadvantages.

Historical modeling

In historical modeling, we take the values of financial fluctuations for the portfolio already known from past measurements. For example, we have the performance of a portfolio over the previous 200 days, based on which we decide to calculate VaR. Let's assume that the next day the financial portfolio will behave the same as on one of the previous days. This way we will get 200 outcomes for 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 select the appropriate percentile and, as a result, we obtain the values that interest us.

The disadvantage of this method is the impossibility of making predictions for portfolios about which we have no information. A problem may also arise if the components of the portfolio 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 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 valuation 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, intervals of increasing/decreasing, etc.). Based on the analysis data, VaR is calculated for each of the components and the resulting value is constructed 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 variation-covariance method, which I did not talk about, but those interested can find a description of the methods in the books below.

Criticism of the technique

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 see, this is absolutely not true. Nassim Taleb, a famous trader and mathematician, in his books “Fooled by Randomness” and “The Black Swan” severely criticizes the existing risk assessment system, and also proposes his solution in the form of using another risk calculation system based on the lognormal distribution.

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

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

Value at Risk- one of the most common forms of measuring financial risks. Commonly referred to as “VaR”.

It is also often called "16:15", it received this name because 16:15 is the time at which it supposedly should lie on the table of the head of the bank’s board JPMorgan. (In this bank, this indicator was introduced for the first time in order to improve the efficiency of working with risks.)

Essentially, VaR reflects the amount of possible loss that will not be exceeded over a certain period of time with some probability ( which is also called the “tolerable risk level”"). Those. the largest expected loss that an investor can receive within n days with a given probability

The key VaR parameters 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. Calculations with a time horizon of 1 day are more common. 10 days are 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 is 99%, in the RiskMetrics system - 95%).
Base currency - the currency in which VaR is calculated

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

The standard for broker-dealer reporting of OTC derivatives transactions to the US Securities and Exchange Commission is a 2-week period and a 99% confidence level.
The Bank of International Settlements To assess the adequacy of bank capital, I set the probability at 99% and a period of 10 days.
JP Morgan publishes its daily VaR values at the 95% confidence level.
According to a study by New York University Stern School of Business, about 60% of US pension funds use VaR in their work

Example of VaR calculation in Excel:

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

Let's calculate the standard deviation of the obtained return (the formula for calculating the standard deviation for a sample for Microsoft Excel will look like =STDEV.B(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.REV(1%, AVERAGE(C3:C249), C250)):

Well, now let’s directly calculate the value of 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 the formula will look like: =B249-(B249*(C251+1))

In total, we received the calculated value of VaR = 5.25 rubles. Taking into account 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!

Risk assessment methods

Types of risks

Risk characterized as the risk of unexpected losses of expected profits, income, property or funds due to random changes in the conditions of economic activity and unfavorable circumstances.

ABOUT There are usually 2 types of risk: systemic And specific risks.

Systemic risk represents the risk of global negative changes in the banking, financial system and the country's economy, affecting the market as a whole.

WITH Systemic risk implies significant losses caused by a decrease in the value of assets, failure of counterparties to fulfill their obligations and disruptions in the operation of payment systems. Within a systemic crisis, risks of various types, independent in a stable situation, show significant correlation.

TO systemic risks include:

interest rate risk— the risk associated with a decrease or increase in the interest rate by the country’s central bank. When interest rates fall, the cost of loans received by companies decreases and their profits increase, which is favorable for the stock market. Conversely, an increase in interest rates has a negative impact on the market.
inflation risk- a type of risk caused by rising inflation. Rising inflation reduces the real profits of companies, which negatively affects the market, and also causes the emergence of another risk - the risk of changes in interest rates.
currency risk- a risk arising from both political and economic factors associated with a sharp change in the exchange rate.
political risk— the threat of a negative impact on the market due to a change of government, government regime, threat of war, etc.

Specific risk(unsystematic or diversifiable risk) is caused by events that relate only to a specific company or issuer, such as management errors, the conclusion of new contracts, the release of new products, mergers, acquisitions, etc.

E These risks are also called “individual security risks” or “unique risks”, since such risks, as a rule, are inherent in the securities of a particular company or, moreover, only in specific financial instruments. The following categories of risks are classified as non-systemic:

risk of loss of liquidity— demand for certain securities may be subject to significant changes, including disappearance for long periods of time;
business risk— the cost of securities (in particular, shares) of any company depends on how successfully the company develops in its chosen direction;
financial risk— the price of a company’s shares may fluctuate depending on the financial policy pursued by its management.
VaR (Value at Risk) risk assessment methods. Market risk. Example calculation in Excel

For example, the degree of financial risk increases if, in financing the company’s activities, its management attaches great importance to the issue of corporate debt;
default risk— the issuer, for various reasons (for example, bankruptcy), may be unable to fulfill its obligations to the holders of its securities on time or at all.

Risk and return. P Essentially, the relationship between risk and return is assessed as follows: the higher the risk, the greater the return the investor expects to receive. In general, long-term investors take on more risk, so they tend to earn higher returns over the long term.

Risk assessment

By “risk assessment” we mean its quantitative measurement. The modern approach to the problem of risk assessment includes two different but complementary approaches:

method of assessing the cost of risk - VaR(Value-at-Risk), based on an analysis of the statistical nature of the market;
method of analyzing the sensitivity of a portfolio to changes in market parameters - Stress or Sensitivity Testing.

VaR risk assessment methodology

VaR is a statistical approach. Methodology VaR has a number of undoubted advantages: it allows you to measure risk in terms of possible losses, correlated with the probabilities of their occurrence; allows you to measure risks in different markets; allows you to aggregate the risks of individual positions into a single value for the entire portfolio, taking into account information about the number of positions, market volatility and the period for maintaining positions.

VaR is a summary measure of risk that can compare risk across different portfolios (for example, portfolios of stocks and bonds) and across different financial instruments (for example, forwards and options).

VaR is a universal method for calculating various types of risk:
— price risk — the risk of changes in the value of the price of a financial asset on the market;
— currency risk — the risk associated with changes in the market exchange rate of the national currency to the currency of another country;

- credit risk - the risk arising from the partial or complete insolvency of the borrower on the loan taken out;

— liquidity risk — the risk associated with the impossibility of selling a financial asset, or with large losses arising from the sale of an asset due to the large difference in the purchase/sale value existing in the market.

WITH the comfort of calculations VaR is a clear and unambiguous answer to the question that arises during financial transactions: What is the maximum loss an investor risks incurring over a certain period of time with a given probability? It follows that the value VaR is defined as the largest expected loss that an investor can receive with a given probability within n days. Key parameters VaR are the period of time for which the risk is calculated and the specified probability that losses will not exceed a certain amount.

D For calculation VaR it is necessary to determine a number of basic elements that influence its value. First of all, this is the probabilistic distribution of market factors that directly affect changes in the prices of assets included in the portfolio. Obviously, to build it, you need some statistics on the behavior of each of these assets over time. If we assume that the logarithms of changes in asset prices follow a normal Gaussian distribution with zero mean, then it is sufficient to estimate only the volatility (i.e., the standard deviation). However, in a real market, the assumption of normal distribution is usually not met. After specifying the distribution of market factors, it is necessary to select a confidence level, that is, the probability with which losses should not exceed VaR. Then you need to determine the holding period for which losses are assessed. Under some simplifying assumptions, it is known that VaR portfolio is proportional to the square root of the position maintenance period. Therefore, it is enough to calculate only the one-day VaR. Then, for example, a four-day VaR will be twice as much.

G In simple terms, calculating the quantity VaR is made to formulate a statement of this type: “We are X% confident (with probability X%) that our losses will not exceed the value of Y over the next N days.” In this position, the unknown quantity Y is VaR.

CALCULATION OF VaR
D To begin, you need to determine the logarithms of one-day changes in stock prices for each position using the formula:

where F is the stock price on the i-th date
Z Then the standard deviation for each position is calculated:

where N is the number of days.
P When calculating the value VaR for a period of more than one day, this expression is also multiplied by the root of the number of days for which it is calculated VaR.
P after this the indicator itself is calculated VaR according to the formula:

Where k— coefficient corresponding to each of the confidence levels of 90%, 95%, 97.5% and 99%;
P— current value of the financial instrument;
N— the number of financial instruments of this position. ABOUT usually calculation VaR produced for confidence levels of 90%, 95%, 97.5% and 99%.
The coefficients corresponding to each of the confidence levels are given in the table:

SEE MORE:

Text of a scientific article on the topic “THE CONCEPT OF RISK VALUE AND ITS APPLICATION IN RISK MANAGEMENT OF NON-FINANCIAL COMPANIES”

The concept of value at risk and its application in risk management of non-financial companies

T.V. Barsukova,

postgraduate student of the Department of Finance, St. Petersburg State University of Economics and Finance (191023, St. Petersburg, Sadovaya St., 21; e-mail: [email protected])

Annotation. The active implementation of a risk management system at enterprises, as well as the experience of financial market participants in this area, contributed to the popularization among non-financial companies of risk assessment methods based on the VaR concept of risk value. In this regard, the question of the feasibility of using this approach for enterprises in the real sector of the economy becomes relevant. This paper examines the scope of application of VaR for both financial and non-financial companies, highlighting the advantages and disadvantages of various methods for calculating the value at risk. It is concluded that the VaR methodology can act as an additional mechanism for risk analysis and is suitable for large enterprises whose activities are largely exposed to market risks.

Abstract. The active adoption of the system of enterprise risk management and also an experience of participants of financial market in this sphere promoted the popularity of methods of risks evaluation based on the concept of value at risk among non-financial companies. Thereupon the question of the expediency of use of these methods for enterprises of the real sector of economy gains urgency. In the present work is examined an area of application of VaR for financial as well as non-financial companies, there are cited advantages and disadvantages of different methods of calculation of value-at-risk. It is concluded that the VaR methodology can serve as an additional mechanism of analysis of risks, and is fit for large enterprises whose activity is exposed to a significant degree to the effects of market risks.

Key words: risk, risk value, assessment, risk management. Keywords: risk, value at risk, evaluation, risk management.

A characteristic feature of many Russian non-financial companies involved in implementing a risk management system at the level of the entire enterprise is the tendency to simplify the models used in risk assessment. Based on foreign experience in the field of risk management, domestic companies use the concept of risk value (Va1ie-a(-^k - VaP), which belongs to the class of statistical models, as the basis for calculating and assessing the level of risks.

The application of this concept is due to the possibility of its use for assessing the return on investments taking into account risk, determining capital adequacy and its diversification, for calculating limits on open positions, as well as for assessing the company’s performance.

This concept, along with financial organizations and institutional investors, is most widespread among large non-financial companies whose activities are associated with global markets for raw materials and capital, export and import operations, and, therefore, are exposed to market risks associated with fluctuations in interest rates and exchange rates , prices for raw materials and securities.

Historically, the application of the concept of value at risk dates back to the late 1980s and early 1990s among large American banks. Appearing in response to the need for a single, quick, easy-to-understand assessment of the total risk of a portfolio of assets, the A&R concept quickly gained popularity among financial market participants. However, before receiving recognition from the environment,

di non-financial corporations, the concept of value at risk has gone through a number of stages:

1993: Commissioned by the Group of Thirty (G30) by J.P. Morgan prepared and published the report “Derivatives: practices and principles”, where the term “Value-at-Risk” first appeared;

1994: J.P. Morgan published and made publicly available on the Internet a description of the RiskMetrics™ risk assessment methodology, on the basis of which it developed the FourFifteen software package for calculating VaR;

1997: the American Securities & Exchange Commission (SEC), in relation to companies reporting to it, approved rules for mandatory disclosure of information about the market value of their financial assets and derivative financial instruments subject to fluctuations in financial markets, where VaR was recognized one of the possible calculation methods.

Thus, the VaR concept has acquired the status of a standard for disclosing information about a company's risk, both for its own purposes and for reporting to investors and regulators.

Among the non-financial companies that were the first to use the VaR approach to assess market risk are the American company Mobil Oil, the German companies Veba and Siemens, and the Norwegian Statoil.

Popularization of the concept among companies in the real sector of the economy entailed the need to develop a corporate version of VaR, which would take into account the specifics of risk management of non-financial corporations.

Economics and Entrepreneurship, No. 6, 2013

radios, in particular the importance of non-financial factors in assessing risks. The first analogues of VaR were proposed in 1999 by the consulting groups RiskMetrics Group in the form of the CorporateManager™ software package and NERA (National Economic Research Associates) in the form of a methodology for calculating cash flow (Cash Flow) under CFaR risk conditions, highlighting the main risk for non-financial corporations as the risk of decline operating cash flows. Among the alternative methods for measuring risk in corporations that have appeared in recent years, one should highlight methods based on the use of regression analysis. Currently, research is ongoing to develop an adequate system for assessing the cost of risk for companies of this type.

In general, VaR is the maximum value of potential losses, expressed in monetary units, from changes in the value of a risky asset or portfolio as a whole over a certain period of time with a given confidence interval. In other words, VaR allows you to calculate how much the value of a position on a financial instrument or a portfolio of instruments may decrease as a result of certain risks (for example, changes in exchange rates, market price fluctuations, stock market volatility) over a specific time period with a certain level of probability . For example, if the risk value for one day is 1 million cu. with a confidence interval of 95%, this means that within one day losses exceeding 1 million cu can occur in no more than 5% of cases.

As can be seen from the definition, the key elements in calculating the value at risk are the time horizon over which the risk is assessed, the confidence interval and the specified level of loss in the value of the asset.

Establishing a time horizon depends on the frequency of transactions with these assets and their liquidity, as well as on the availability of statistical data on the distribution of profits and losses for the selected period. Unlike financial institutions, for which the typical settlement period is 1 day, non-financial companies and strategic investors can record longer periods of time. It is assumed that the composition and structure of the assessed asset portfolio remains unchanged throughout the entire time interval for which VaR is calculated. As the time horizon lengthens, the value at risk will increase.

Value At Risk

In practice, it is believed that over a time interval of n days, the value at risk will be approximately Vn times greater than in one day.

The confidence interval can be determined both on the basis of a subjective assessment of the probability of losses by the risk manager, and by an objective method by identifying the points of intersection of two graphs: the actually observed empirical probability distribution of profits and losses and the density of the normal distribution. In practice, most often trusted

The interval is set at 95%. Supervisors are guided by the 99% level recommended by the Basel Committee on Banking Supervision. As the confidence level increases, the risk value will also increase.

With all the many existing methods for calculating the value of UER, their various modifications and combinations, its calculation is based on three basic economic and mathematical approaches:

Analytical, or covariance, based on the use of variances and covariances of market risks, as well as assumptions about the distribution of returns;

Simulation modeling based on historical data;

Simulation modeling using the Monte Carlo method, or stochastic modeling.

The covariance method is mainly characterized by ease of implementation and relatively low costs for collecting and processing primary data. At the same time, this approach is limited by the need to make assumptions regarding the nature of the distribution of returns on standardized assets prior to the calculation stage. As a rule, the assumption of a normal distribution does not correspond to the actual characteristics of the financial market, which leads to low accuracy of the estimates made.

Unlike the analytical approach, the historical modeling method is not limited by problems associated with making specific assumptions about the nature of the distribution of returns; it has clarity and high accuracy in assessing the risks of nonlinear instruments, but requires an extensive database on all risk factors. This method implicitly assumes the representativeness of historical data in relation to potential future risks, which predetermines difficulties with high volatility of risks in the market, as well as with the emergence of new risks due to the lack of historical data to calculate the value of UER. In addition, with a small amount of historical data, there is a high probability of errors in calculating the value at risk.

The most technically complex and costly in terms of material and time resources

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CHAPTER 6 VALUE AT RISK

General remarks

The VaR (Value at Risk) indicator appeared in the 90s of the last century. Determines the value of a portfolio of financial assets at risk to the investor. The emergence of VaR is due to the fact that in many cases variance cannot be a good indicator of the risk of an asset portfolio.

VaR is a risk indicator showing what maximum damage an investor’s asset or portfolio of assets can suffer during a certain period of time with a given confidence probability.

It is assumed that there is no change in the asset portfolio during the time period for which the valuation is made. The most common period for which VaR is calculated is one day. The longer the period for which VaR is calculated, the greater the number of observations required. Thus, to objectively estimate a one-day VaR, 250 one-day observations are sufficient; to estimate a ten-day VaR, with non-overlapping periods of 10 days, data for almost seven years will be required.

In addition to the technical difficulties of obtaining data over a long period of time, it should be understood that these data will not be representative enough due to the dynamic development of markets.

What risk does VaR measure? According to the calculation methodology, VaR evaluates the possibility of losses caused by market risk, which will manifest itself in changes in the price (and, accordingly, profitability) of financial instruments. It is assumed that the price is capable of reflecting the manifestation of most risk-bearing factors. Therefore, investors tend to look at VaR as a measure of all the risks associated with financial instruments. Some studies indicate that the actual amount of losses may be greater than what VaR estimates, taking into account political risks, liquidity risks, and regulatory risks to which financial assets are exposed.

The second remark is related to the interpretation of profits and losses in VaR, which is considered a priori as a negative factor. Thus, when determining damage with 99% probability, we proceed from the fact that the expected value of the portfolio is not equal to the average, but to almost the maximum possible.

Temporary nature of VaR. In most of its applications, VaR is calculated for short periods of time - one day, week, month. The shorter the period being assessed, the more accurate the VaR estimates. Therefore, this indicator is usually used by companies in the operational management of market risks.

VAR technology is being used for the first time at the 2018 FIFA World Cup: what is it?

Unlike other risk measures such as standard deviation or , which give an idea of some average risk, VaR gives an idea of losses in a specific period

VaR limitations. It is believed that the use of VaR methods may lead to erroneous results due to the following circumstances:

· Distribution of returns. For each VaR indicator, a certain distribution of returns is assumed;

· History is not a very good basis for real forecasts. All VaR forecasts use historical data to some extent. If the period for which historical data was taken was stable, VaR will be small, if unstable, then it will take large values. However, in a market economy, deviations, any deviations, lead to the emergence of mechanisms that restore the disturbed balance. Then, the idea of making judgments about future risks based on former deviations taken into account by the economy looks quite unreliable.

· Non-stationary correlations. VaR estimates depend on correlations between risk sources. Correlation links are usually based on historical data and are voluntary. Since only one correlation matrix is used each time in the calculations, the quality of the estimates depends on how correct the correlation matrix was used.

Advantages of the VaR methodology. Despite well-known criticism, the VaR method is successfully used in the practice of many financial institutions. Among the advantages of this method are the following:

· Using a portfolio approach to considering the structure of assets;

· The calculation of expected profit is determined by real market rates of financial instruments, and not by basic market rates that are of an analytical nature;

· Through the use of correlation matrices, a more reliable assessment of assets and asset portfolios is obtained than using stochastic modeling;

There are two groups of VaR methods: a) analytical or variance-covariance models; b) nonparametric models.

Various VaR models

Parametric VaR model

A model is called parametric if we know the distribution function and distribution parameters of the random variable. In the parametric VaR model, it is assumed that the returns on financial assets follow a certain distribution law, usually normal. Using historical observations, the mean, variance and covariance of returns on financial assets are determined. Based on them, the VaR of a portfolio with a given confidence level is determined using the following formula:

where is the value of the portfolio;

– standard deviation of portfolio returns corresponding to the time period for which VaR is calculated;

– the number of standard deviations corresponding to the given confidence level α.

There are concepts of absolute and relative VaR. Absolute VaR determines the maximum possible amount that an investor can lose over a certain period of time with a given probability. Relative VaR, unlike absolute VaR, is determined relative to the expected return of the portfolio.

In the case when the investor knows the VaR of the assets included in his portfolio, the VaR of the portfolio is determined by the formula:

Where — column vector and row vector VaR of portfolio assets;

– correlation matrix of portfolio assets

If, when determining the VaR of a portfolio, correlations between assets are taken into account, then we are talking about diversified VaR; if correlations are not taken into account, then we are talking about non-diversified VaR. It is the simple sum of the individual VaR of a portfolio's assets.

Since correlations can change over time, along with the diversified VaR indicator, it is advisable to determine the non-diversified VaR, which will show the maximum losses for a given confidence level in the event of unstable correlations or errors in their determination.

The assumption of normal distribution of assets included in the portfolio allows us to transfer the VaR value from one confidence level to another. Let's show it with an example. Let's take and. Let's express it from the first formula and substitute it into the second

Let's take and. Let's express it from the first formula and substitute it into the second

Since VaR is determined based on statistical data over a period of time, it is possible to obtain VaR estimates that are not representative of the population. In this regard, there is a real need to estimate the confidence interval for the standard deviation of the return on an asset portfolio.

The lower () and upper () limits of the confidence interval can be determined by the following formulas:

where are the lower and upper limits of the confidence interval of the standard deviation of the investment portfolio returns

In the event that losses may exceed the VaR value, the investor needs to know what amount of losses he should expect. In this case, use the following ratio:

where is the VaR of portfolio assets at a given confidence probability γ;

– average expected losses, provided that the actual losses of X turn out to be greater than .

The opposite concept in relation to VaR is the concept of EaR (Earnings at Risk), which shows what maximum income can be generated by owning a certain portfolio of financial assets over a certain period of time with a given confidence probability.

When choosing a portfolio, you can rely on the EaR to VaR ratio. The greater this ratio at a certain confidence level, the more preferable the portfolio.

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Risk workshop. Assessing Value at Risk (VaR) using historical modeling

In addition to the standard deviation, investment campaigns calculate a risk indicator such as VaR (Value at Risk). This indicator characterizes the amount of possible loss with a selected probability over a certain period of time. Value-at-Risk is calculated using 3 methods:

Variation/covariance (or correlation or parametric method)
Historical modeling (delta normal method, “manual calculation”)
Calculation using the Monte Carlo method

For risk parameter calculationValue at Riskusing delta normal method, it is necessary to form a sample of the risk factor; it is necessary that the number of sample values be more than 250 (Bank of International Settlements recommendation) to ensure representativeness. Let's take the data on Gazprom stock quotes for the period from January 9, 2007 to July 31, 2008.

Discovered

For Gazprom stock quotes, we calculate the daily return using the formula:

Where: D – daily profitability;
Pi is the current share price;
Pi-1 – yesterday’s stock return.

The correctness of using the Value at Risk method with the delta normal calculation method is achieved by using only risk factors subject to the normal (Gaussian) distribution law.

To check the normality of the distribution of stock returns, you can use the Pearson or Kolmogorov-Smirnov tests.
The formula in Excel will look like this:

LN((C3)/C2)
The result is the following table.

After this, it is necessary to calculate the mathematical expectation of profitability and the standard deviation of profitability for the entire period. Let's use Excel formulas.
Mathematical expectation = AVERAGE(D2:D391)
Standard Deviation =STDEV(D2:D391)

The next step is to calculate the quantile of the normal distribution function. Quantiles are the values of the distribution function (Gaussian function) at given values at which the values of the distribution function do not exceed this value with a certain probability. Quantile reports that losses on Gazprom shares will not exceed 99% with probability.

Quantile is calculated using the formula:
=NORMBR(1%,F2,G2)

To calculate the value of a stock with a 99% probability of the next day, you need to multiply the last (current) value of the stock by the quantile added to one.

Xt+1 – profitability value at the next point in time.

To calculate the value of a stock several days in advance with a given probability, we use the following formula.

Where: Q is the quantile value for the normal distribution of Gazprom shares;
Xt is the value of the stock’s return at the current time;
Xt+1 – the value of the yield deviation at the next point in time;
n is the number of days ahead.

Formulas for calculating VAR for one day VAR(1) and five VAR(5) days ahead are made using the formulas:
X(1) =(F5+1)*C391
X(5) =(ROOT(5)*F5+1)*C391

The calculation of the stock price value with a 99% probability of losses is shown in the figure below.

The obtained values X(1) = 266.06 indicate that over the next day, the Gazprom share price will not exceed the value of 226.06 rubles. with a probability of 99%. And X(5) says that over the next five days, with a 99% probability, the Gazprom share price will not fall below 251.43 rubles.

To calculate Var itself (the amount of possible losses), we calculate the absolute and relative values of losses. The formulas in Excel will be as follows:
=C392-G7 =G11/C392
=C392-G8 =G12/C392

These figures say the following: with a 99% probability, the loss on Gazprom shares will not exceed 7.16 rubles. the next day and the loss on Gazprom shares with a 99% probability will not exceed 21.79 rubles. over the next five days.

Calculation of the indicatorValue at Risk"manually"
Let's create a new worksheet in Excel. In order to determine the Value at Risk values “manually”, you need to find:

Maximum returns for the entire time range = MAX(Sheet1!D3:D392)
Minimum returns for the entire time range =MIN(Sheet1!D3:D392)
Number of intervals (N) = 100
Grouping interval (Int) =(B1-B2)/B3

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VaR(Value-at-Risk) - value at risk. The VaR indicator reflects the maximum possible losses from changes in the value of a financial instrument, portfolio of assets, etc., which can occur over a certain period of time with a given probability. In other words, value at risk is an estimate of the upper bound of the possible losses that a bank can incur over a certain period of time (usually a year), for a certain (specified) level of confidence (for example, 95%).

To determine the value at risk, it is necessary to know the relationship between the volumes of profits and losses and the probabilities of their occurrence, that is, the distribution of the probabilities of profits and losses during the selected time interval. In this case, based on the given values of the probability of losses, the amount of the corresponding damage can be determined. Using the properties of the normal probability distribution, a simple formula for determining VaR is:

VaR = (ασ - μ) А р

Where α — threshold probability value;
σ — standard deviation of the asset’s return (as a percentage of the asset’s value);
μ — average value of the asset’s return (as a percentage of the asset’s value);
A r— asset value.

When determining value at risk, the key parameters are the confidence interval and time horizon. Since losses are a consequence of fluctuations, the confidence interval serves as the line that separates “normal” fluctuations from extreme spikes in the frequency of their occurrence. Typically, the probability of loss is set at 1%, 2.5% or 5% (the corresponding confidence intervals would be 99%, 97.5% and 95%), however, in accordance with the capital management strategy followed by the bank, the risk manager may choose a different value. As the confidence interval increases, the value at risk will increase.

The choice of time horizon depends on how often the asset is used. For banks that have active capital market operations, the typical settlement period is one day, while strategic investors and non-financial companies use other periods. In addition, when setting a time horizon, one should take into account whether there is a statistical distribution of profits and losses for the expected time interval. As the time horizon increases, so does the value at risk. Practice shows that over a period of n days, the value at risk will be approximately n times greater than the VaR calculated for one day.

It is worth remembering that the VaR concept implicitly assumes that the composition and structure of the asset portfolio being valued remains unchanged over the entire time horizon.

This assumption is not sufficiently justified for relatively long time intervals.

What is var in CS GO

Therefore, each time the asset portfolio is updated, it is necessary to adjust the value at risk.

To calculate the value at risk indicator, the following methods are used:

analytical;
historical modeling method;
Monte Carlo method.

The choice of method for calculating the value at risk indicator depends on the composition and structure of the asset portfolio, the availability of statistical data, software, etc.

Analytical (covariance, delta-normal) method is based on the classical theory of a portfolio of financial assets.

It is based on the assumption that changes in market risk factors are normally distributed. This assumption allows us to determine the parameters of the distribution of profits and losses for the entire portfolio. Then, knowing the properties of the law of normal distribution, you can easily calculate the damage that will occur no more often than a given percentage of cases. The analytical method is inferior to simulation methods in the reliability of assessing the risks of asset portfolios consisting of instruments whose value depends on market factors in a non-linear manner, especially over relatively long time horizons.

Historical modeling method relatively simple and most understandable.

It does not rely on probability theory and requires few assumptions about the statistical distributions for market risk factors. As in the analytical method, the values of the portfolio instruments must be previously represented as functions of market risk factors, and the distribution of profits and losses is determined empirically. However, the use of this method requires the availability of time series of values for all market factors used in the calculations, which is not always possible for significantly diversified portfolios.

Monte Carlo method refers to simulation methods. Its main difference from the method of historical modeling is that in the Monte Carlo method a statistical distribution is selected that well approximates changes in observed market factors and an estimate of its parameters is determined. The main difficulty in using the Monte Carlo method is choosing an adequate distribution for each market factor and estimating its parameters.

(See Tolerable risk, Risk management, Risk assessment system, Stress testing, Shock value, Economic capital).

One of the main tasks of financial institutions is to assess market risks that arise due to fluctuations (favorable events) in stock prices, commodities, exchange rates, interest rates, etc. The simplest measure of an investor’s dependence on market risks is the amount of 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 Cost of Risk (Value – at – Risk, VAR). VAR is a summary measure of risk that can compare risk across different portfolios (eg, stock and bond portfolios) and across different financial instruments (eg, forwards and options).

The value at risk indicator was developed in the late 1980s. and immediately gained recognition among the largest participants in the financial market. Subsequently, the Value at Risk (VAR) indicator became a full-fledged standard of information about a firm's risk, which could be used internally and also reported to investors and regulators.

Over the past few years, VAR has become one of the most popular risk management and control tools in various types of companies. This was caused by several reasons. The first reason was, of course, the disclosure in 1994 of the largest US investment company J.P. Morgan risk assessment system Riskmetrics TM and providing free use of the database for this system to all market participants. VAR values obtained using the Riskmetrics TM system are still a kind of standard for VAR assessments. 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 in the derivatives markets (financial market instruments operating on the basis of fixed assets (stocks, bonds) etc.)). In table 3.7. The losses suffered by some Western companies and the dates on which they were made public are indicated. The third reason , is the decision of bank supervisors to use VAR values to determine capital reserves.

Table 3.7.

Losses of large Western companies for 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 losses from changes in the value of a financial instrument, portfolio assets, or company that can occur over a given period of time with a given probability of its occurrence. For example, when the value at risk for 1 day is said to be $100,000 with a 95% confidence interval (or 5% probability of loss), this means that a loss in one day exceeding $100,000 can only occur than in 5% of cases.

In simple terms, a VAR is calculated to make a statement like this: “We are X% confident (with X% probability) that our losses will not exceed $Y over the next N days.” In this sentence, the unknown quantity Y is VAR. It is a function of 2 parameters: N – time horizon and X – confidence interval (level). For example, the standard for broker-dealer reports on transactions in over-the-counter derivatives submitted to the US Securities and Exchange Commission is 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. J.P. Company Morgan publishes its daily VAR values at the 95% confidence level.

To determine the value of risk, it is necessary to know the relationship between the size of profits and losses and the probabilities of their occurrence, i.e. distribution of probabilities of profits and losses during the selected time interval. In this case, based on a given value of the probability of loss, the size of the corresponding loss can be unambiguously determined.

A typical technique is to use the normal probability distribution.

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

In addition to subjective assessment, the confidence interval can also be established by an objective method. To do this, construct a graph of the actually observed (empirical) probability distribution 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 distribution will determine the desired confidence interval.

It should be taken into account that as the confidence interval increases, the risk value indicator will increase.

The choice of time horizon depends on how often transactions with these assets are made, 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 establishing a 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 risk value indicator also increases.

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

Where Z– 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 value of positions.

It should be noted that the concept of value at risk implicitly assumes that the composition and structure of the asset portfolio being valued will remain unchanged over the entire time horizon. This assumption is hardly justified for relatively long time intervals, therefore, each time the portfolio is updated, it is necessary to adjust the value at risk.

Historically, the VAR-based risk assessment approach was first recommended by The Global Derivatives Study Group (G30) in 1993 in the study "Derivatives: Practices and Principles". In the same year, the European Council, in Directive "EEC 6 – 93", mandated 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 models as a basis for calculating capital reserves. The 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 coefficient with a value between 3 and 4:

Factor value λ depends on the model's one-day prediction over previous time periods. So, if we denote by K the number of times that one-day losses exceeded the predicted value of VAR over the last year (or the last 250 trading days), then the following 3 zones are distinguished: the “green” zone (K is less than or equal to 4), the “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 occurring at a rapid pace. In investment companies and banks, the VAR methodology can be applied in at least 4 areas of activity.

1) Internal monitoring of market risks. Institutional investors can calculate and monitor VAR values at several levels: aggregate portfolio, asset class, issuer, counterparty, trader/portfolio manager, etc. From the point of view of monitoring, the accuracy of estimating the value of VAR fades into the background since in this case the value of the relative rather than the absolute value of VAR is important, i.e. A manager's VAR or portfolio VAR compared to the VAR of a reference portfolio, an index, another manager, or the same manager at previous points in time.

2) External monitoring. VAR allows you to create a picture of the market risk of a portfolio without revealing information about the composition of the portfolio (which can be quite confusing). In addition, regular reports using VAR figures provided to management can provide some evidence that the risks taken by the managing managers are 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 stated objectives. A manager can evaluate the effectiveness of a hedge by comparing the VAR values of portfolios with and without a hedge. If, for example, the difference between the two is small, then the question arises as to whether hedging is appropriate or whether hedging is being applied correctly.

4) “What – if” analysis of possible trades. The VAR methodology allows you to give more freedom and autonomy to management personnel, as it becomes possible to reduce all kinds of bureaucratic procedures associated with the approval of certain transactions (especially with derivatives). This is achieved through monitoring transactions (trades) using VAR. For example, senior management may simply set a rule for their broker-dealers of this type: “No transaction should result in an increase in VAR by more than X% of the initial capital” and then not subsequently go into detail about each specific trade.

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

Non-financial corporations can use VAR techniques to assess the riskiness of cash flows and make hedging decisions (protecting capital from adverse price movements). So one interpretation of VAR is the amount of uninsured risk that a corporation assumes. Among the first non-financial companies to use VAR to assess market risk are 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 point in time, and the confidence level p VAR is a loss on a time interval that will occur with probability 1 – p.

Here's a simple example: Let the daily VAR for this portfolio be $2 million at the 95% confidence level. This VAR means that, barring any sudden changes in market conditions, a one-day loss will exceed $2 million 5% of the time (or once a month, assuming there are 20 business days in a month).

In mathematical terms, VAR = VAR t,T is defined as the upper bound of a one-sided confidence interval:

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

where α is the confidence level, R t (T) is the portfolio capital growth rate over the interval with a “continuous method of calculating interest”:

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

where V(t+T) and V(t) are the values of the portfolio capital at times 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 VAR value 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 of R t (T).

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 through the parameters of this distribution.

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

It is assumed that the asset growth rate μ= 0. Then VAR= – V t z 1 – α σ , where V t is the value of the portfolio capital at the current time t.

Example 1: The case of one asset.

The next chart shows 3.20. The histogram of monthly growth rates of the FTSE-100 index from 1988 to 1995 is shown.

To calculate VAR, we use the fact that 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%. Provided that the current value of the portfolio capital is 1 million pounds sterling, the VAR value over a time interval of 1 month 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.

Let us now consider the previous example of a portfolio consisting of the “FTSE 100 index” (it is assumed that the investor can build his portfolio of shares in such a way that each share has the same weight as in the FTSE – 100 index. Thus, the increment of such a portfolio will be 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”: the GBP-denominated stock index 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 the 95% confidence level There is:

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

Estimates of the standard deviation and average exchange rate of GBP/USD 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 portfolio VAR using the fact that the variance of a portfolio of two assets that have a joint normal distribution equals 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 100 index and the GBP/USD exchange 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 the initial capital in 5 out of 100 months in the future.

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

In addition, as one would expect, 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), investing 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, asset price increases have, as they say, heavier “tails” compared to the normal law, i.e. in reality, more "extreme" events are observed compared to what would be expected under a normal distribution. VAR, by its nature, deals with predicting events from the “tails” of the distribution (events from the “left tail” for “long” positions in an asset and events from the “right tail” for “short” positions in an asset). Such "catastrophic risk" events are well known in the insurance and reinsurance business.

Simulation method according to historical data consists of constructing the distribution of portfolio changes R t (T) based on historical data. In this case, only one hypothesis is made about the distribution of return on the portfolio's capital: the “future” will behave the same way as the “past”. For example 1, discussed above, we have that the 5% quantile of 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 estimate for the FTSE 100 index portfolio:

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

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

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

Scenario Analysis Method studies the effect of changes in portfolio capital depending on changes in the magnitude of risk factors (for example, interest rates, volatility) or model parameters. Simulation occurs in accordance with certain “scenarios”. This is how many banks estimate the “PV01” value of their “fixed income” portfolios (fixed – income portfolios, i.e. portfolios consisting of “interest rate” instruments: bonds, interest rate forwards, swaps, etc.) , which is calculated as the change in portfolio equity for a parallel shift in the yield curve of 100 basis points.

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

I would like to note that the VAR methodology is not a universal way to prevent financial losses. It simply helps companies understand whether the risks they are exposed to are the risks they would like to take over or think that 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 replacement, but in addition to other risk analysis methods, such as Shortfall – at – Risk(SAR, Average Loss), when they are interested not only capital limit, below which a loss should be expected with a certain degree of probability, and also the size of this loss.

As a rule, the calculation of the value at risk 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, as a general assessment of market risk, is needed primarily for making operational decisions by the company's top management.