 # How do you calculate the Value-at-Risk?

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The Value-at-Risk (VaR) is an in the entire financial sector widely used risk measure. It is basically the standard measure for calculating very different risks. Companies also use the VaR to make certain risks measurable – however, only financial risks in the company are quantified with the VaR.

## What exactly does Value-at-Risk indicate?

The statement of the value-at-risk is strictly speaking divided into three parts: It deals with a certain amount of loss, a probability that this amount of loss will not be exceeded and a certain period of time considered in each case.

This can be illustrated better with a practical example:

A portfolio of securities has no losses higher than EUR 11,500 in the course of the next year (=period of time) with a probability of 95 % (=probability or the so-called “confidence level”).

What does this statement mean in detail? The information refers only to the time period under consideration, namely the next 12 months. During this period, losses of more than 11,500 euros could occur in the portfolio, but the probability is only 5%. It can therefore be assumed with a 95 % certainty that the maximum losses will not exceed 11,500 euros.

With the VaR, only the probable maximum loss risk is considered – possible profit risks cannot be calculated with these models and are also completely disregarded. It is only about the risk.

### Use of the VaR

The VaR can be used to indicate the risk of both an equity portfolio and a loan portfolio. The risk of an interest rate portfolio can also be calculated using the VaR. The economic or financial significance always remains the same – but the VaR is always calculated in different ways, depending on the area of risk under consideration.

For the calculation, it is necessary that the given risk is represented mathematically (stochastically) exact by a corresponding distribution function of the values – and since the risk in different areas (e.g. a loan portfolio versus a share portfolio) is composed of different parameters and the values are distributed differently in each case, a mathematical model must be used for each considered risk which represents the respective risk parameters, probabilities and correlations as exactly as possible.

Basically, the calculation always works like this:

VaR (x) = Fx-1(?)

the value-at-risk is calculated from the respective distribution function equation for the confidence level ? for the considered risk value x. It is therefore an inverse distribution function.

(How this distribution function equation may look always depends on the stochastic value distribution considered in each case, as already mentioned).

Mathematically, this would also mean:

VaR ?(x) = inf{x|Fx(x) >= ?}

### Conclusions drawn from the calculation

The implications are obvious: the longer the holding period and the higher the confidence level, the higher the VaR if no other values change (i.e. ceteris paribus, strictly speaking).

## Quantifying market price risks

A so-called market price risk would be, for example, the risk of losses due to changes in share prices (but also exchange rates or interest rates). As we have already seen above, the so-called “drivers”, meaning all relevant factors, must be taken into account. In this case, the risk drivers are, for example, the prices of the equities in a portfolio. In the mathematical equation for the distribution function, the volatility (and the changes in volatility, precisely speaking) within a certain time period must also be taken into account. The correlations, meaning the links between different risk factors, must also be mathematically incorporated into the equation.

The following models have become established for calculating market price risks:

• the variance-covariance approach (the classical way of calculating the VaR, also known as the delta-normal approach), which, however, cannot be used for the calculation of the risks of some financial instruments, such as option risks
• the delta-gamma approach, which is also suitable for option risk calculations
• the Monte-Carlo simulation, which, however, involves a high level of calculation effort

Alternatively, the VaR can also be calculated from historical data alone by simply deriving the distribution directly from past values and projecting it onto the future. This can have advantages, because “real” values are used instead of a mathematical model and the calculation effort is almost completely eliminated – the disadvantage can be that with short holding periods, too few value changes are projected into the future.

A practical calculation example of the VaR (for a market price risk) using a simple approximation model:

We assume the following values:

…a risk position of 100,000 euros (RP)
…a daily volatility ? of 5.8%.
…a holding period T 5 days
…a probability ? of 5 % (the probability with which the maximum loss will be exceeded)

Our VaR is calculated as follows:

VaR = RP * ? * ?T * Qsnv(1-?)

Qsnv is the “quantile standard normal distribution” – i.e. the quarter value in the distribution function.

With the given values, we thus arrive at a VaR of EUR 21,327.84, which will not be exceeded with a probability of 95 % over a holding period of 5 days.

If we extend the holding period, however, the risk increases accordingly, as already mentioned and as can also be seen in the calculation. 