Risk measure

The concept of risk is a collective term for statistical measures, with which it is possible to detect the (total ) risk position of a company.

Introduction

The calculation of risk measures is a subtask in the risk quantification, an assessment of risks by means of a suitable description of density or distribution function ( or historical data) about the effect of the risk and the allocation of risk measures. The aim of quantification, it is first to describe the risks identified quantitatively by appropriate distribution functions (probability distribution). For this, there are two alternative versions:

From the distribution function can be risk measures (such as the standard deviation or the Value at Risk) for comparison of risks derived, even if they are described by different types of distribution functions. The risk measures may relate to individual risks (for example, property, plant damage ), but also on the total risk volume ( approximately relative to profits) of a company. The quantitative assessment of the overall risk position requires an aggregation of individual risks. This is possible, for example by means of Monte Carlo simulation in which the effects of all individual risks are considered in their dependence in the context of planning.

Characterization

Risk measures can be fundamentally different in dimensions for a single risk (ie, a measure of risk in the narrower sense, such as the standard deviation) or dimensions that minimizes the risk of two random variables related to each other (ie, a measure of risk in a broader sense, such as the covariance ).

Risk measures in the strict sense can now be classified in various ways on, firstly, by the position dependence:

• Position-independent risk measures ( such as the standard deviation) quantify the risk and magnitude of the deviations from a target.
• Position-dependent risk measures such as capital requirements, however, are dependent on the amount of the expected value. Frequently such a risk measure as "necessary equity " or "necessary premium " to cover the risk can be considered.

The two types can be partially converted into each other. Does not apply, for example, a position- dependent measure across a random variable, but on a centered random variable, so there is a position- independent measure of risk. As in the calculation of position-dependent risk measures the height of the expected value flows, they can also be interpreted as a kind of risk-adjusted performance measures.

A further distinction of the risk measure is derived from the degree of consideration of information of the underlying distribution. Two-sided risk measures (such as the standard deviation) consider this completely, while the so-called downside risk measures ( such as the VaR and the LPM dimensions) consider only the distribution up to a certain limit.

Survey

Standard deviation

The standard deviation as a measure of risk for an uncertain payment is calculated as

In which

And positive and negative deviations detected equally from the expected value. The (apparent) symmetry and identical importance of opportunities and risks in risk measurement, however, is to relativize. It also seems the intuition and the risk perception of most people to disagree, the risks (possible negative deviations ) rate much higher than the same high chances.

To describe the overall risk scale are used because of the special meaning of possible losses, in particular, so-called " Downside risk measures ", which detect specifically the possible extent of negative deviations. These include, for example, the value at risk, the capital requirements, the LPMs ( lower partial moments ) and the probability of default. They are useful if the risks are not symmetrical and losses are of particular importance.

In particular in the banking and insurance sectors is the VaR as a downside risk measure often use. The VaR takes into account explicitly the - relevant to the KonTraG - consequences of a particularly unfavorable development for the company. The VaR is defined as the amount of damage that in a given period with a specified probability ( " confidence level " ) is not exceeded. Formally, the VaR is the (negative ) quantile of a distribution. The x % quantile of a distribution indicates the threshold, up to which x% of all possible values ​​. Does the VaR is not a " value " but for example on the cash flow one speaks occasionally of " cash flow at risk", which, however, the same risk measure says.

The VaR is positively homogeneous, monotone, translation invariant, in general but not subadditive and hence not coherent. Thus, so let constellations construct in which the VaR of a combined two exposures financial position is higher than the sum of the VaR of the individual items. This contradicts a world characterized by diversification thoughts intuition.

Capital requirements - EKB

The capital requirement EKB ( as a special case of risk capital, RAC ) is a cognate with the VaR, position- dependent risk measure which explicitly refers to the company's earnings. He expresses how much equity (or liquidity) is necessary in order to take realistic risk-related losses of a period. Is calculated as the maximum of zero and the negative quantile of a random variable, which represents the measure of success and the confidence level ( security level ) respectively.

Which applies

Especially for normally distributed with mean and standard payments derogation shall:

Lower Partial Moments - LPM

Among the lower partial moments are understood risk measures that relate as downside risk measure only a portion of the total probability density. You acquire only the negative deviations from a barrier (target ) values ​​here but the entire information of the probability distribution (up to the theoretically possible maximum damage).

The barrier can be, for example, the expected value or even any deterministic target size (for example, planned value ) or a required minimum return. A stochastic benchmark is possible. For example, considering a probability distribution for a return, then, for example, as barriers to the calculation of an LPM possible:

• (nominal capital preservation )
• (real capital preservation )
• ( risk-free interest rate ) and
• ( expected return )

The understanding of risk corresponds to the point of view of an investor, what is the risk of shortfall, the shortfall of reference established by him target ( returns on plan assets, the minimum required rate of return ) in the foreground. This is known exactly why even of shortfall risk measures. In general, a LPM degree of order calculated by

In the case of discrete random variables results in the following relation shown

Herein, the possible values ​​that are smaller than the required limit, the number of these values ​​and the probability of the occurrence of. In the case of continuous random variables is the calculation rule

Order must not necessarily be an integer. Through it is determined whether and how the amount of deviation will be evaluated by the barrier. The higher the risk aversion of an investor, the greater should be selected.

Usually, in practice, three special cases are considered:

• The shortfall probability ( probability of default), ie m = 0

• The shortfall expectation value, that is, m = 1

• The shortfall variance, that is, m = 2

The extent of the risk of falling below the target size is taken into account in various ways. The shortfall probability only the probability of falling below plays a role. When shortfall expectation, however, the average shortfall amount is taken into account and the shortfall variance the mean square underflow height.

The relationship between value at risk and LPM can thereby be described as follows: The value at risk results from the fact that for a certain period, a maximum acceptable shortfall probability, ie a dictated and determined the appropriate minimum yield size of the LPM definition.

The shortfall risk measures can be divided into conditional and unconditional risk measures. While unconditional risk measures (such as the shortfall expected value or the shortfall variance ) suggests that the probability of falling below the barrier aside, flows this in the calculation of the conditional shortfall risk measures ( such as the Conditional Value at Risk) with a.

The Conditional Value at Risk ( CVaR), and its variants Expected Shortfall (ES ) and expected tail loss ( ETL ) are other risk measures.

Let X be a random variable and. Then

It corresponds to the expected value of the realizations on a risky size, which are above the quantile to the level. The CVaR specifies which variation in the occurrence of the extreme case, ie is to be expected when exceeding the VaR. The CVaR therefore considers not only the probability of a "large" deviation ( extreme values ​​), but also the amount of deviation beyond this.

If X has a continuous distribution, the CVaR is positively homogeneous, monotone, subadditive and translation invariant, ie coherent.

Expected Shortfall - ES

In most cases considered ( random variables with continuous densities ) ES and CVaR are identical.

If the distribution function of the random variable jump points, so too has discontinuities.

The ES considered, as is a sudden provide free variant of the CVaR.

ES and CVaR differ only on the set.

Is defined as follows:

It is true.

Capital allocation after Kalkbrener

A company with several divisions represents an ES, it should be allocated to the individual segments. Kalkbrener proposes the following distribution before: is the loss size of the ith segment and the sum of X, ie the loss of size of the company.

Each division of the expectation value of all the damage that he is assigned, which are caused by events for which the total damage exceeds the.

Is there only one division in the company, then there is the ES shown above:

Examples

The Conditional Value at Risk is especially noticeable in the normal distribution case, where the expected value and the variance describes explicitly represent. It is then first for the VaR:

Where the quantile of the standard normal distribution respectively. If we denote by the density of a standard normal random variable, we obtain for the CVaR:

In comparison with thus added on a higher multiplier of the standard deviation is thus.

As related to the value at risk thus arises that the supplement as opposed to the case of the normal distribution here is not additive but multiplicative.

Sometimes the calculation of CVaR (and thus the consideration of all possible extreme damage) in practice is still not meaningful. The damages that result more than once to an insolvency of a company, are ( for the owners ) are not worse than any damage that trigger insolvency.

Drawdown

The drawdown of a financial investment is a measure that describes depending on the configuration the maximum possible loss over a period in the past. Because of this loss calculation for this measure of risk is also applicable to asymmetric distributions. The maximum drawdown is the percentage loss between the highest point and the lowest point of a curve of the values ​​to be considered investments in a given period. Further, an average of the N minimum drawdowns are formed. Purpose, the individual returns are sorted according to their size in the considered period. The smallest are usually negative. Be to form an average of the N -th smallest forms added and divided by N. The number N of values ​​can be chosen freely, where they should move to an appropriate level. Another way to use the drawdown to measure risk is to form a kind of variance. To this end, the N smallest observed yield characteristics of the period are squared, then added and finally the sum of square root.

Sources of information and in-depth information

• Statistics
• Stochastics
• Economic and Social Statistical Key Figures