Value at Risk

The term value in risk (or English value at risk , abbreviated VaR) refers to a measure of risk for the exposure of a portfolio in finance . It is the quantile of the loss function: The value at risk for a given probability level indicates the amount of loss that will not be exceeded within a given period with this probability.

General

The value at risk is now a standard risk measure in the financial sector. In the meantime, the measure is also used in industrial and trading companies for risk measurement.

An asset at the value at risk of EUR 10 million with a holding period of one day and a confidence level of 97.5% means that the potential loss of the risk position under consideration from one day to the next has a probability of 97.5% will not exceed EUR 10 million.

Definition and characteristics

The VaR at a confidence level is the - quantile of the loss function, which measures the negative change in value of a risk position over the holding period. The VaR is therefore a measure of the risk of loss. ${\ displaystyle 1- \ alpha}$ ${\ displaystyle (1- \ alpha) \!}$

The random variable describes the loss function of the portfolio over the period under consideration. A loss is positive in the loss function, and a gain is negative. The associated distribution function is denoted by. The VaR at a given confidence level is then defined as follows using the generalized inverse distribution function ( quantile function ): ${\ displaystyle X}$ ${\ displaystyle F_ {X}}$ ${\ displaystyle 1- \ alpha}$

{\ displaystyle {\ mathit {VaR}} _ {1- \ alpha} (X) = F_ {X} ^ {- 1} (1- \ alpha) = \ inf {\ big \ {} x \ in \ mathbb {R}: F_ {X} (x) \ geq 1- \ alpha {\ big \}}}

.

The VaR can also be defined using the profit function instead of the loss function: The random variable describes the profit function of the portfolio over the period under consideration and the associated distribution function is denoted by. The VaR at the confidence level is then given by ${\ displaystyle Y: = - X}$ ${\ displaystyle F_ {Y}}$ ${\ displaystyle 1- \ alpha}$

{\ displaystyle {\ mathit {VaR}} _ {1- \ alpha} (X) = - \ inf {\ big \ {} y \ in \ mathbb {R}: F_ {Y} (y)> \ alpha { \ big \}}}

.

If the distribution function of (and thus also of ) is strictly monotonic, then applies ${\ displaystyle X}$ ${\ displaystyle Y}$

{\ displaystyle {\ mathit {VaR}} _ {1- \ alpha} (X) = - {\ mathit {VaR}} _ {\ alpha} (Y) = - F_ {Y} ^ {- 1} (\ alpha)}

.

Both a higher confidence level and a longer holding period imply ( ceteris paribus ) a higher VaR. The VaR is monotonic , homogeneous and translation-invariant , but not subadditive .

Areas of application

The VaR can be used to measure different types of risk. The risk of an equity portfolio, an interest portfolio or even a credit portfolio can be described with the help of the VaR, whereby the economic interpretation of the key figure is always the same. Likewise, the VaR could be measured for mixed portfolios made up of several different asset classes, provided that the joint probability distribution of the mixed portfolio can be calculated. In practice, however, this often fails because the interdependencies between the various asset classes cannot be modeled (e.g. because no correlation coefficients are known).

Practical use of Value at Risk

In principle, the VaR can be used for every stochastically modelable risk. In practice, however, there are mostly specific applications.

Market price risk models

Value-at-risk models were originally developed to measure market price risks and for this purpose have become widely used as “market price risk models ”. Market price risk models are used to measure the risk of individual trading portfolios (see Trading ) as well as to measure risk at the overall bank level, in particular to measure the risk of interest rate changes . What all market price risk models have in common is that they basically refer to risks that can be more or less liquidly traded on the financial markets using appropriate instruments.

The different approaches are all based on

to describe the drivers relevant for the market price risks of a portfolio with a stochastic model and
from this to determine the quantile of the future possible changes in value of the portfolio under consideration.

The drivers of the market price risk are the market prices that determine the portfolio value, i.e. share prices, exchange rates, interest rates, etc. (the so-called risk factors). These are included in the stochastic modeling with the fluctuation ranges ( volatility ) of future changes and the relationships ( correlation ) between the changes in various risk factors. The corresponding values for fluctuation ranges and correlations are usually estimated on the basis of historical changes in market prices.

With the help of valuation models and information about the 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 ; One example is the present value formula , which indicates the value of a bond as a function of market interest rates. It should be noted here that market price risk models are usually not based on an accounting, but rather a market price-oriented or present value concept. Depending on the model approach, this step immediately gives the quantile of the change in value, i.e. the VaR, or a distribution function for changes in portfolio values from which the VaR can be determined.

The following approaches are most commonly used in practice:

Variance-covariance approach : This term is often used synonymously with the more correct term “delta normal approach” and corresponds to the original VaR model from JP Morgan. The stochastics of the risk factors (volatilities and correlations) are described by a covariance matrix , assuming multivariate, normally distributed changes in the risk factors. The portfolio information flows in in the form of sensitivities, ie the first derivations of the portfolio value according to the risk factors. Since the delta-normal approach can only depict 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 enables a simple analysis of diversification and hedge effects between the portfolio components.
The analytical delta-gamma approach and the Cornish-Fisher approximation , which allow the consideration of non-linear financial instruments, also fall under the variance-covariance approach . A common disadvantage of all variance-covariance approaches is the assumption of normal distribution, which neglects the observed leptokurtic distribution (“fat tails”, see curvature (statistics) ) of market price changes.

With Monte Carlo simulation , a specific approach to be referred to market price risk models. In this process, several thousand random changes in market prices are generated - usually based on the covariance matrix of historical market price changes - and converted into changes in portfolio values. The VaR can be determined from the distribution of changes in portfolio values generated in this way. In contrast to the delta normal approach and the delta gamma method, financial instruments with a strongly non-linear payout profile can also be included in the VaR calculation. Disadvantages are the high computational effort and the normal distribution assumption usually used here as well.

The historical simulation differs from the aforementioned approaches in that it does not use a parameterized model of the risk factors (hence the “non-parametric approach” as opposed to “parametric approaches” such as the two aforementioned methods). Rather, historical changes in market prices are used directly to evaluate the current portfolio. With a historical observation period of 251 days, for example, you get 250 changes in all risk factors, which are converted into 250 possible future changes in value of the current portfolio using the position information and the valuation models. This gives a non-parametric distribution function for changes in the portfolio value, from which the VaR can be read. The advantages of historical simulation are the simple implementation, the simple aggregation of risk figures across different portfolios and IT systems and the fact that no assumptions are made about the distribution function. Disadvantages are a certain instability of the estimator due to the normally low number of calculated future changes in portfolio values and - at least in theory - the lack of subadditivity of the calculated VaR.

Scaling of the daily VaR for different holding periods

In the case of market price risks, the following scaling is often used to convert from the one-day VaR to the value-at-risk to a longer holding period, which for the sake of simplicity assumes that the one-day change in value is normally distributed with the expected value zero:

{\ displaystyle {\ mathit {VaR}} _ {1- \ alpha} ^ {T} = {\ mathit {RP}} \ cdot \ sigma \ cdot {\ sqrt {T}} \ cdot Q_ {SNV} (1 - \ alpha)}

It is

${\ displaystyle {\ mathit {RP}}}$ the risk position (usually the current market value of the investment under consideration),
${\ displaystyle \ sigma}$ the associated daily volatility,
${\ displaystyle T}$ the associated holding period,
${\ displaystyle \ alpha}$ the probability that the calculated loss will be exceeded
${\ displaystyle {\ mathit {Q_ {SNV} (1- \ alpha)}}}$ the quantile of the standard normal distribution.

Example: An investment of 100,000 euros has a daily volatility of 5.8% and a holding period of 5 days. For results: ${\ displaystyle \ alpha = 5 \, \%}$

{\ displaystyle {\ mathit {VaR}} _ {0.95} ^ {5} = {100,000} \ cdot 0 {,} 058 \ cdot {\ sqrt {5}} \ cdot 1 {,} 645 \ approx 21,327 {,} 84}

Interpretation: With a probability of 95%, a loss of 21,327.84 euros will not be exceeded with a holding period of 5 days.

However, this scaling presupposes that the daily value changes are not only normally distributed, but are also distributed identically and temporally independently. This excludes volatility that varies over time , for example in the form of GARCH effects that are common in financial markets . To take these phenomena into account, the Monte Carlo simulation method described above must be used.

Credit risk models

Credit risk models that use the value-at-risk approach differ mainly in how the loss distribution of the loans is modeled. There are essentially the following three types of models:

Default models only differentiate between default and non-default of a loan. The best known calculation methods are:
- The IRB formula according to Basel II . This model essentially only uses the normal distribution.
- CreditRisk + from Credit Suisse Financial Products. Failures are modeled using the Poisson distribution . The correlation of the failures is taken into account by means of the gamma distribution . This results in a negative binomial distribution overall .
- Ratio calculandi periculi determines the loss distribution using a generalization of the binomial distribution. The retail portfolio is approximated by a normal distribution using Moivre-Laplace's theorem . Systematic macroeconomic aspects are mapped on the model side by decomposing the default probabilities.
Migration models (mark-to-market models) take into account not only the defaults, but also the change in the value of a loan if the debtor's creditworthiness improves or deteriorates. The best known calculation methods are:
- CreditMetrics from JP Morgan. The many different ways in which the creditworthiness of individual customers can change are calculated using the Monte Carlo method.
- The model from KMV . The possible default of a loan is modeled using a put option. The value of this option can be calculated using the Black-Scholes model .
- McKinsey’s CreditPortfolioView uses logistic regression to calculate default probabilities with the help of macroeconomic variables.
Spread models are essentially market risk models. They measure the risk that results from a change in market opinion on the creditworthiness of a debtor (credit spread). The same methods are available for the calculation as for the market risk models.

The use of value at risk to model credit risks, unlike market risks, has the following problems (with the exception of the spread models):

Credit relationships usually last for years and default events are relatively rare. This means that historical data is often insufficient for estimating statistical parameters. It is therefore practically impossible to control the quality of the risk values using so-called backtesting .
The loss distribution of a loan portfolio is not normally distributed. Rather, it is usually a question of skewed distributions. This makes statistical modeling more difficult, as very high losses can occur with it in rare cases.

Other uses

Tracking VaR is related to market price risk . In contrast to normal market price risk, the tracking VaR does not indicate the quantile of an absolute change in portfolio value, but the quantile of the deviation in the portfolio return relative to a specified benchmark. Tracking VaR is particularly important in asset management.

There are also stochastic models for operational risks with which an attempt is made to forecast the quantile of future losses from operational risks . With the Solvency Ordinance and the capital adequacy requirement for operational risks at banks, these models have become more important (so-called AMA models , see below).

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

The frequency or frequency distribution indicates the probability with which a certain number of loss events from operational risks will occur in a defined period (e.g. one year).
The loss amount distribution indicates the probability that a given event will cause a loss of a certain amount.

The two distributions can be estimated from historical data or determined using expert estimates. In a Monte Carlo simulation, the two distributions are combined to form an overall loss distribution that indicates the probabilities that the sum of all losses will have a certain level in the forecast period. The VaR at the desired confidence level can then be read off from this distribution as the corresponding quantile.

Applications

Corporate management

Credit institutions use the value at risk instrument for daily risk control and monitoring, to determine risk-bearing capacity and to allocate equity across business areas.

In the case of market price risks in particular, the VaR has established itself as a means of daily risk control and monitoring. It is used less at the level of individual traders or trading desks, but more at a more highly aggregated level. What is important here is that the VaR methodology can be used to easily and transparently aggregate different types of market price risks and make them comparable, so that the risk measurement and risk limitation of entire trading departments can rely heavily on a single key figure.

When determining and monitoring the risk-bearing capacity, the results of various VaR models (for market price risks, credit risks, etc.) can be aggregated in order to obtain an overall risk. Since it is currently hardly possible to model all the different types of risk together, generally general assumptions have to be made for the correlations between the types of risk. This overall risk is compared to a risk coverage capital (usually a size based on the equity). If, for example, the total risk is calculated for the 99.95% quantile and a holding period of one year and is currently covered by the risk cover, this would mean in this model that the losses from all risks over a year only have a probability of 0, 05% above the risk cover amount and therefore the bank's probability of survival for the next year is 99.95%. The bank can then adjust its risk level so that the probability of survival corresponds precisely to its target rating (see rating agency ). Because of the uncertainties in the modeling, however, additional risk buffers are usually taken into account.

In the course of the equity allocation, VaR models can be used to determine risk figures and thus the need for risk cover (equity) for individual business areas. With the equity allocated in this way, equity costs can be charged to the business segments in the course of the business segment accounting and risk-adjusted measures of success ( e.g. RAROC or EVA ) can be determined.

Banking prudential application

(see banking supervision )

In the course of the KWG -Novelle 1998 changes made to the principle I allowed German banks for the first time, used for internal Bank management value-at-risk models also used to calculate the prudential capital requirements for market risk in the trading book to be used. The VaR calculated for the capital backing had to be calculated for a holding period of 10 days and a confidence level of 99% and be based on a historical observation period of at least 250 trading days. In addition to these quantitative requirements, Principle I formulated numerous qualitative requirements for integration into the bank's risk management system, for ongoing review of the VaR model (so-called backtesting) and for the consideration of crisis scenarios ( stress tests ). The provisions of Principle I were essentially adopted unchanged in the Solvency Regulation .

When using the IRB approach , the calculation formula for the capital backing for credit risks in accordance with the Solvency Regulation is also based on a VaR model. In the IRB approach (IRB stands for "internal rating based approach"), banks use self-developed risk classification procedures (rating procedures) to estimate up to three risk parameters that describe the credit risk of the individual exposures (in the basic IRB approach this is the probability of default , in the advanced IRB approach, the default loss rate and the level of exposure in the event of default ). The Solvency Ordinance specifies a formula for converting these parameters into capital backing, which is based on a credit risk model (see also the IRB formula ).

With the entry into force of the Solvency Regulation, banks must for the first time also back operational risks (business risks) with regulatory capital. One method of capital adequacy is the use of so-called advanced measurement approaches ( AMA models from Advanced Measurement Approach). These represent, so to speak, a VaR model for operational risks. These are used to calculate the 99.9% quantile of the distribution of losses from operational risks over a period of one year (corresponds to the holding period).

What all three procedures have in common is that they may only be used upon application and with approval from BaFin , although approval is usually preceded by a review by the banking supervisory authority .

weaknesses

The value-at-risk concept is not free from weaknesses. The following are particularly important for market price risk models:

In order to have a sufficiently large database of historical observations, only a short holding period (one to ten days) is usually used to determine the risk-determining market price changes. This also restricts the value at risk forecast horizon to this short period. By assuming a suitable distribution assumption (e.g. the root-T rule ), the forecast horizon can be mathematically extended, but the reliability of the value-at-risk calculated in this way then depends on the validity of the distribution assumption.
The value-at-risk concept assumes liquid markets , i.e. markets in which one's own position can be sold or hedged without any significant influence on the market price (see market liquidity risk and market impact ).

General weak points are:

The value at risk is not a subadditive and therefore not a coherent risk measure. It is therefore possible that the sum of the VaR values of sub-portfolios is smaller than the VaR value of the entire portfolio. Potential diversification effects that could reduce risk are therefore not always taken into account. However, it is controversial whether subadditivity is desirable in practice. If subadditivity is desired, the expected shortfall would be a possible option.

The value-at-risk concept (like other forecasting methods) assumes that events in the (near) future will behave like events in the past. This assumption is particularly wrong if a crisis phase arises after a long period of calm. In order to remedy this deficiency, stress tests are also often calculated.

It is often uncritically assumed that the underlying data are normally distributed. In practice, however, extreme events are often observed more frequently than the normal distribution suggests. This weak point can be remedied if more realistic probability distributions are used instead of the normal distribution.

Due to its design, the value-at-risk approach does not provide any information about the average extent of damage in all unfavorable scenarios beyond the quantile limit. For this purpose, there is the expected shortfall risk measure , which looks at precisely this average.

The disadvantage of the value-at-risk approach is often cited as the fact that it is not suitable for determining the maximum loss. However, this disadvantage is usually of little relevance in practice, since it is usually not one of the company's goals to determine or control the theoretically possible maximum loss. Normally there can be no perfect security; a viable company must also take a minimum of risk. A practice-oriented risk measurement must therefore be based on scenarios that have a certain minimum probability of occurrence.

Web links

Value-at-Risk. ( Memento of March 4, 2012 in the Internet Archive ) in the Risk Glossary
VaR calculation method (variance-covariance model, historical simulation, Monte Carlo simulation) (PDF file; 287 kB)
VaR at Investopedia
pdf on the VaR, its problems and exp. shortfall (93 kB)

Individual evidence

^ Robert Schwarz: Credit Risk Models. Working Paper Series of the University of Applied Sciences of bfi Vienna
↑ Roland Eller, Walter Gruber, Markus Reif (eds.): Handbook of credit risk models and credit derivatives. Schäffer-Poeschel Verlag, Stuttgart 1999, ISBN 3-7910-1411-0 .
↑ Christian Cech: The IRB formula. Working Paper Series of the University of Applied Sciences of bfi Vienna.
↑ csfb.com
↑ Ratio calculandi periculi - an analytical approach to determining the loss distribution of a loan portfolio. (= Dresden contributions to quantitative methods. No. 58/12). Technische Universität Dresden, 2012. ( online ( memento from April 24, 2016 in the Internet Archive ))
^ The benchmark for understanding credit risk. on: defaultrisk.com
↑ See e.g. B. Thomas Cloud: Risk Management. Oldenbourg Wissenschaftsverlag, 2008, ISBN 978-3-486-58714-2 , p. 58 ff.
^ H. Rau-Bredow: Bigger Is Not Always Safer: A Critical Analysis of the Subadditivity Assumption for Coherent Risk Measures . In: Risks . 7, No. 3, 2019, p. 91. doi : 10.3390 / risks7030091 .
↑ J. Dhaene, MJ Goovaerts, R. Kaas: Economic capital allocation derived from risk measures. In: North American Actuarial Journal. 7, 2003, pp. 44-56.
^ MHA Davis: Consistency of risk measures estimates, Working Paper. Imperial College, London 2014. (abstract)

[1] Robert Schwarz: Credit Risk Models. Working Paper Series of the University of Applied Sciences of bfi Vienna

[2] Roland Eller, Walter Gruber, Markus Reif (eds.): Handbook of credit risk models and credit derivatives. Schäffer-Poeschel Verlag, Stuttgart 1999, ISBN 3-7910-1411-0 .

[3] Christian Cech: The IRB formula. Working Paper Series of the University of Applied Sciences of bfi Vienna.

[4] sfb.com

[5] Ratio calculandi periculi - an analytical approach to determining the loss distribution of a loan portfolio. (= Dresden contributions to quantitative methods. No. 58/12). Technische Universität Dresden, 2012. ( online ( memento from April 24, 2016 in the Internet Archive ))

[6] The benchmark for understanding credit risk. on: defaultrisk.com

[7] See e.g. B. Thomas Cloud: Risk Management. Oldenbourg Wissenschaftsverlag, 2008, ISBN 978-3-486-58714-2 , p. 58 ff.

[Rau-Bredow-8] H. Rau-Bredow: Bigger Is Not Always Safer: A Critical Analysis of the Subadditivity Assumption for Coherent Risk Measures . In: Risks . 7, No. 3, 2019, p. 91. doi : 10.3390 / risks7030091 .

[9] J. Dhaene, MJ Goovaerts, R. Kaas: Economic capital allocation derived from risk measures. In: North American Actuarial Journal. 7, 2003, pp. 44-56.

[10] MHA Davis: Consistency of risk measures estimates, Working Paper. Imperial College, London 2014. (abstract)