Value at Risk

The concept of value in risk or English Value at Risk ( VaR ) denotes a measure of risk which indicates not exceed the value of the loss of a particular risk exposure (eg, a portfolio of securities) with a given probability within a given time horizon.

A Value at Risk of EUR 10 million, for a holding period of 1 day and a confidence level of 97.5 % means that the potential loss of the particular risk position from one day to the next with a probability of 97.5 % to the amount of 10 EUR million will not exceed.

The Value at Risk is now a Standardrisikomaß in the financial sector. Meanwhile, the concept is also used in industrial and commercial companies used for the quantification of certain risks (mostly financial risks ).

  • 4.1 Corporate Management
  • 4.2 Prudential application

Definition

The Value at Risk at a confidence level is the quantile of the distribution of the change in value ( gains and losses) to a risk position of the holding period. The VaR is a downside risk measure, which serves only the measurement of potential losses, ie only the "negative end" of the probability distribution is considered.

The change in value of the portfolio over the period under consideration will be described with the random variables is the distribution function. The value at risk given at a confidence level is then as follows on the generalized inverse distribution function defined as:

Both a higher confidence level and a longer holding period implicate ( ceteris paribus ) a higher VaR.

The VaR is a distribution -related risk measure, it is monotone, homogeneous and translation invariant, it is not subadditive.

Applications

The VaR can measure different types of risk. Thus, the risk of an equity portfolio, a portfolio interest or a loan portfolio using the VaR can be described, with the economic interpretation of this indicator is always the same. Also could be used for mixed portfolios that are composed of several different asset classes, VaR can be measured, provided that the joint probability distribution of the mixed portfolios is calculated. In practice, this often fails because the interdependencies between the various asset classes can not be modeled (eg because no correlation coefficients are known ).

The VaR can in principle be applied to any risk stochastically modeled. In practice, however, there are usually specific applications.

Market risk models

Value-at -risk models were originally developed for the measurement of market risk and have gained wide acceptance for this purpose as " market risk models ". Market risk models for risk measurement of individual trading portfolios (see the commercial ) as well as used to measure risk taken by the bank, especially for measuring the interest rate risk. All market risk models have in common that they basically relate to risks that are more or less on appropriate instruments by liquid trading in the financial markets.

The different approaches are all based on

  • To describe the relevant market risk in a portfolio driver with a stochastic model and
  • To determine from the quantile of possible future changes in the value of the considered portfolios.

The drivers of market risk are the portfolio value -determining market prices, ie stock prices, exchange rates, interest rates, etc. ( the so-called risk factors). These go with the fluctuation ( volatility) of a future change and the relationship ( correlation) between the changes of various risk factors in the stochastic modeling. The corresponding values ​​for fluctuation and correlations are usually estimated on the basis of historical market price changes.

With the help of valuation models and information about portfolio composition ( " position") the market price changes must then be converted into portfolio value changes. The valuation models describe the relationship between market prices and the values ​​of the financial instruments in the portfolio; an example is the present value formula that specifies the value of a bond as a function of the market interest rates. It should be noted that market risk models is usually not based on any accounting, but a market price- oriented or net present value concept of value. Depending on the model approach is obtained from this step once the quantile of the change in value, so the VaR, or a distribution function of portfolio value changes from which the VaR can be determined.

The following approaches are used in practice, most commonly:

  • Variance- covariance approach: This term is often used interchangeably with the more correct term " delta-normal approach " and corresponds to the original VaR model of JP Morgan. The stochastics of the risk factors ( volatilities and correlations) is described by a covariance matrix, where one starts from a multivariate normal distribution changes in risk factors. The portfolio information flows in the form of sensitivities, ie the respective first derivatives of the portfolio value by the risk factors. Since the delta-normal approach can only represent linear relationships between risk factors and market prices, it is not suitable for highly non-linear financial instruments such as options. Its advantage is that it is easy to implement and a simple analysis of diversification and hedging effects allows between the portfolio components. Also among the variance- covariance approach fall of the analytical delta-gamma approach and the Cornish -Fisher approximation, which allow the consideration of non- linear financial instruments. A common drawback of all variance- covariance approaches the normal distribution assumption, which neglects the observed leptokurtic distribution ( " fat tails ", see Buckle ( statistics) ) to market price.
  • With Monte Carlo simulation, a specific approach to market risk models is called. Here are - usually based on the covariance matrix of historical market price changes - several 1000 random market price changes generated and converted in portfolio value changes. From the thus generated distribution of portfolio value changes, the VaR can be determined. In contrast to the delta-normal approach and the delta-gamma method as well as financial instruments with strongly nonlinear payoff profile can be included in the VaR calculation. Disadvantages are the high computational cost and the normal distribution assumption commonly used here.
  • The historical simulation differs from the above approaches in that it does not use a parameterized model of the risk factors (hence " non-parametric approach " as opposed to " parametric approaches " as the two aforementioned methods). Historic market price changes are directly used to evaluate the current portfolio. In a historical observation period of 251 days, for example, one obtains 250 changes all risk factors that one converts on the position information and the valuation models in 250 possible future changes in the value of the current portfolio. Thus one obtains a non-parametric distribution function of the portfolio value changes, from which one can read the VaR. Merits of the historical simulation are the ease of implementation, the simple aggregation of risk data across different portfolios and IT systems across and the fact that no assumptions about the distribution function are made. A disadvantage is a certain instability of the estimator due to the usually small number of calculated future portfolio values, and - at least theoretically - the lack of sub-additivity of the calculated VaRs.

Credit risk models

Credit risk models that use the value-at -risk approach, differ mainly in how the loss distribution of loans is modeled. In much there are three types of models:

  • Failure models (default models) differ only between default and non - default of a loan. The most common calculation methods are: The IRB formula to Basel II This model uses only the normal distribution substantially.
  • Credit Risk of Credit Suisse Financial Products .. failures are modeled using the Poisson distribution. The correlation of defaults is taken into account by means of the gamma distribution. Thus resulting in a total negative binomial distribution.
  • Ratio calculandi periculi determine the loss distribution using a generalization of the binomial distribution. The retail portfolio is approximated using the theorem of Moivre -Laplace by a normal distribution. Systematic macroeconomic aspects are mapped model side by decomposition of the failure probabilities.
  • Credit Metrics by J. P. Morgan .. The many different ways that can change the creditworthiness of individual customers are calculated with the Monte Carlo method.
  • The company KMV model. The possible failure of a loan is modeled on a put option. The value of this option can be calculated using the Black-Scholes model.
  • CreditPortfolioView McKinsey used the logistic regression to calculate with the help of macroeconomic variables, the default probabilities.

The use of Value at Risk for the modeling of credit risk has unlike market risk following problems (except here the spread models):

  • Lending relationships usually go on for years and failure events are relatively rare. This historical data material for the estimation of statistical parameters is often insufficient. Therefore, quality control of the risk values ​​over a so-called backtesting is practically impossible.
  • The loss distribution of a loan portfolio is not normally distributed. Rather, it is usually to skewed distributions. This complicates a statistical modeling, since it can also be very high losses occur in rare cases.

Other Applications

Related to market price risk is the concept of tracking VaR. In contrast to the normal market price risk of tracking VaR is not the quantile of absolute portfolio value change, but the quantile of the deviation of the portfolio return relative to a predetermined benchmark. The tracking VaR is particularly in asset management is important.

Also for operational risks exist stochastic models are used to attempt to predict the quantile of future losses from operational risks. These models have the Solvency Regulation and the specified therein capital charge for operational risk in banks increased importance acquired (called AMA models, see below).

A common approach is the so-called loss distribution approach. These two probability distributions are used:

  • The frequency or frequency distribution indicates the probability with which a certain number loss from operational risk events in a defined period (eg one year) occurs.
  • The amount of loss distribution gives the probability that a given event causes a loss of a certain amount.

The two distributions can be estimated from historical data or determined by expert assessments. In a Monte Carlo simulation, both distributions are combined into an aggregate loss distribution that specifies the probability that the forecast period, the sum of all the losses has a certain level. The VaR at the desired confidence level can then be read as the corresponding quantile of this distribution.

Applications

Corporate Management

Credit institutions use across the instrument of the Value at Risk for daily risk management and monitoring, to determine the risk-bearing capacity and the allocation of capital across business units.

In particular, for market price risks, the VaR has established itself as a means of daily risk management and monitoring. He will be used less at the level of individual traders or trading desks, but on higher aggregated level. This comes into play that the VaR methodology simple and transparent types can be aggregated market price risks and made ​​comparable, so that the risk measurement and risk limiting all trading departments may rely heavily on a single measure.

In the determination and monitoring of the risk-bearing capacity, the results of different VaR models ( for market price risks, credit risks, etc.) are aggregated in order to obtain an overall risk. Since there is currently hardly possible, all the different types of risks to be modeled jointly, must be taken for the correlations between risk types usually quite general assumptions. This overall risk is a risk cover (usually a style similar to the equity size ) are compared. Is the total risk, for example, the 99.95 - % quantile and a holding period of one year and just covered by the risk cover, would that mean in this model that the losses from all risks over a year with a probability of 0, 05 % higher than the risk cover and therefore is the probability of survival of the bank for the next year at 99.95%. The bank can then adjust their risk level such that the probability of survival corresponds precisely to their target rating (see rating agency). Because of the uncertainties in the modeling, however, additional risk buffers are normally taken into account.

As part of the allocation of equity VaR models can be used to determine risk numbers and thus demand for risk coverage ( equity) for individual businesses. With the so- allocated capital to business lines can capital costs are expensed in the course of the business segments and it can risk-adjusted measure of success will be determined (eg RAROC or EVA).

Prudential application

(see Banking Supervision )

The analysis in the course of the Banking Act 1998 amendment changing the principle I would allow German banks for the first time, to use used for internal Bank management value-at -risk models also for the calculation of prudential capital requirements for market risk in the trading book. The calculated VaR to measure capital requirements had to be calculated for a holding period of 10 days and a 99% confidence level and based on a historical observation period of at least 250 trading days. In addition to these quantitative requirements of Principle I formulated a number of qualitative requirement for inclusion in the risk management system of the Bank, the ongoing review of the VaR model ( back-testing or back-testing ) and to the consideration of crisis scenarios (stress tests). The provisions of Principle I have not been changed in the Solvency substantially.

The formula for calculating the capital requirement for credit risk under Solvency is the use of the IRB approach, a VaR model as a basis. Under the IRB approach (IRB stands for "internal rating-based approach") use banks proprietary risk classification (rating method ) to estimate up to three risk parameters that describe the credit risk of individual exposures ( in the foundation IRB approach, this is the probability of default, additionally the advanced IRB approach, LGD and the exposure at default ). For the translation of these parameters into a capital charge the Solvency specifies a formula based on a credit risk model (see also IRB formula ).

With the entry into force of the Solvency banks must also underpin operational risks ( operational risk ) with bankaufsichtlichem equity for the first time. A method of capital requirements is the use of so-called Advanced Measurement Approaches (AMA models of the Advanced Measurement Approach). This in a sense a VaR model for operational risk dar. With these, the 99.9 to - % quantile of the distribution of losses from operational risk at a horizon ( corresponding to the holding period) of one year are calculated.

All three methods have in common that they may be used only at the request and with approval by the BaFin, the authorization should normally be preceded by an examination by the Banking Supervision.

Weaken

The value-at -risk concept is not without its weaknesses. Particularly important for market risk models are:

  • In order to have a sufficiently large data base of historical observations, only a short holding period is usually used (one to ten days) for determining the risk-determining market price changes. This also forecast horizon value at risk is limited to this short period. By insinuation of an appropriate distributional assumptions (eg the root - T rule) Namely, the forecast horizon is extended mathematically, so the reliability of the calculated value- at-risk but then depends on the validity of the assumed distribution.
  • The value-at -risk concept requires liquid markets, ie markets in which to sell the own position without significant impact on the market price or secure manner (see market liquidity risk and market impact ).

General weaknesses are:

  • The Value at Risk is a non subadditives and therefore no coherent risk. It is therefore possible that the sum of the VaR values ​​of sub-portfolios is smaller than the value at risk of the overall portfolio. As a solution, here lends itself to the use of the risk measure Conditional Value at Risk.
  • The value- at-risk approach assumes (along with other forecasting methods as well), and events in the (near ) future will behave like the events have behaved in the past. This assumption is especially wrong when a crisis phase is formed after a long quiet period. To remedy this deficiency, stress tests are calculated additionally often.
  • It is often uncritically assumed that the underlying data are normally distributed. In practice, however, extreme events are often more likely to be observed than suggested by the normal distribution. This vulnerability can be resolved if, instead of the normal distribution more realistic probability distributions are used.
  • The value-at -risk approach provides by design no information about the average level of damage all lying beyond the Quantilgrenze unfavorable scenarios. For this purpose, there is the risk measure Conditional Value at Risk ( cvar ), looking straight at this average.
  • It is often cited as a disadvantage of the value-at -risk approach is that it is not suitable to determine the maximum loss. However, this disadvantage is usually of little relevance in practice, since it usually is not one of the objectives of the company, to determine or control the theoretically possible maximum loss. A perfect security can not exist normally; a profitable company must also take a minimum of risk. A practical risk measurement must therefore be based on scenarios, which have a certain minimum level of probability.
Cornish–Fisher expansion Stress testing
799082
de