### What is Value At Risk ?

VAR is a method of calculating and controlling exposure to Market Risk.

It measures the volatility of a portfolio of assets.

This was developed in 1993 in response to the collapse of Barings

The greater the volatility, the greater the risk.

VAR expresses risk in terms of a single currency value.

This is one of the key measures that risk managers use to understand the risks in a portfolio and to compare the risks in one portfolio with those in another.

VAR has a number of limitations

We can define VAR as a single number (currency amount) which estimates the **Maximum Expected Loss** of a portfolio over a period and at a given confidence level.

Less formally we can say that VAR is a currency amount where the chance of losing more than that is defined by a given percentage over a given period.

This is a statement of probability.

It measures the maximum expected loss in the value of a portfolio over a target horizon, subject to a specified confidence level.

VAR - Value at Risk

Provides a measure of market exposure

This is a measure of the worst expected loss over a period of time, under normal market conditions given a specified level of confidence.

VAR is a measure of market risk

It is the maximum loss which can occur with X% confidence over a holding period of n days.

VAR measures the loss using estimated volatility and correlation between the market prices of the different instruments.

These calculations are based on the historical price movements for at least the last year.

The most common VaR models assume that the market prices follow a normal distribution.

Risk Factor - The potential price move in an instrument.

The VaR number can only capture risk that can be measured in quantitive terms.

It does not capture any of the following types of risk:

operational risk

liquidity risk

regulatory risk

sovereign risk

Correlation - is a measure of the degree to which a value of one variable is related to the value of another.

Correlation is the key element of many VaR models and is particularly important in measuring of variance (or volatility) of a portfolio)

VaR at confidence level of "c" = negative of the (1-c)-quantile of the P&L distribution

### Advantages of VAR

Simplicity of the end result which can easily be understood by shareholders and senior management because of the non technical terms.

It can aggregate risks associated with different instruments.

Is important for performance evaluation and can be used to establish position limits for traders

Helpful for market regulation

Promotes more efficient allocation of resources

### Disdvantages of VAR

The assumption that portfolio returns are normally distributed

Any unusual or extreme events are not captured by a normal distrbution

When extreme errors occur VAR estimates tend to underestimate the true value at risk

If a model uses historical data, the past is not always a reliable guide for the future

Some models are unsuitable for portfolios that contain options, due to there non linear behaviour. The delts is not constant for options. Most sophisticated institutions will use simulation techniques when options are involved.

The estimate is only as good as the reliability of the data

Different methods for calculating VAR can produce different results

Only represents one aspect of effective risk management.

### Example

A bank estimates its daily VAR as $10m at a 95% level of confidence.

This means that there is a 5% chance that the banks loss will be greater than $10m over a 1 day period.

In this case a 95% level of confidence implies a 5% chance of losses exceeding $10m.

VAR calculates the maximum expected loss as a result of an adverse change in the risk factors.

### Characteristics

The VAR of a portfolio depends on the following:

1) The period (The longer the period, the higher the VAR estimate)

2) The confidence level (The higher the confidence level, the higher the VAR estimate)

3) The volatility

4) The correlation among the variables

### Parametric Models

Asset - Model

Delta - Normal

Delta - Gamma Normal

### Non Parametric Models

Monte Carlo Simulation

Historical Simulation

Stress Scenarios

Delta Approximation

Delta-Gamma Approximation

### Methods of Calculating VAR

There are three common methods of calculating Value at Risk

**Variance-Covariance**- not appropriate for portfolios containing options/ non linear behaviour**Historical Simulation**-**Monte Carlo Simulation**- most flexible and comprehensive method

### How to Calculate VAR

1) Determine the holding period

This depends on the assets and the underlying activities

For example in foreign exchange they are usually interested in a period of 1 day but in less liquid markets that period might be a week or even longer. 10 days is often the longest.

The longer the period, the higher the VAR estimate.

You can scale up a shorter period to a longer period by multiplying by the square root of the time period required.

A 10 day VAR estimate will be sqrt(10) times larger than the corresponding 1 day VAR estimate

2) Select the confidence level

This is used to select the degree of certainty.

If a bank wants to know the expected maximum loss over a period of 99 days out of 100, then it needs to use a confidence level of 99%.

On the 100th day the bank expects to lose more than the VAR estimate.

The higher the confidence level, the higher the VAR estimate

Confidence levels between 95-99% are common.

A 95% confidence level implies that the VAR estimate will be exceeded about once a month

On the assumption that there are **252 trading days** every year.

A 99% confidence level implies that the VAR estimate will be exceeded about two or three times a year.

3) Create a probability distribution

The one most commonly used in VAR models is the Normal Distribution.

A normal distribution peaks at the mean and trails off at the extremes.

4) Determine correlations between assets

Financial instruments are not generally independent of one another.

Correlation measures the extent to which the value of one variable is related to the value of another.

Correlation between assets impacts on the risk of a portfolio.

For more details, refer to the dedicated __Correlation__

5) Calculate the volatility of the portfolio

The simpliest measure of volatility is the standard deviation of the portfolio.

6) Calculate the VAR estimate

VAR = Value of the portfolio * No of standard deviations from the mean to capture 95% of the values * volatility of the portfolio

### Mapping Positions

It is extremely unlikely that you can get the necessary volatilities and correlations for every conceivable instrument for every day of the year.

What happens when a cashflow falls outside of the standard periods (1 month, 6 month, 1 year, etc)

When this happens we use "mapping" to calculate VAR where is would otherwise be impossible

For a good example, refer to Bond Portfolios

1999 - Basel 11 Accord - mandated VaR to be the preferred measure of risk

VaR is computed daily

Confidence level 99%

10-day holding period (can be scaled to 1 day using the sqrt(10) rule)

250 days minimum historic period

Back testing of VaR model conducted quarterly based on a sample of 250 trading days

The multiplier K is set based on backtesting results

Exceptions

Every day we compare the PnL with the VaR.

If the loss in the portfolio is larger than the VaR threshold then it triggers an exception

Stressed VaR (SVaR)

For a given portfolio, Stressed VaR is defined as the maximum value of VaR on any 12 consecutive months of historical observations

Credit Value Adjustment VaR

### Expected Shortful

A new market measure to replace the VaR

What is a Risk Measure?

A risk measure has the following properties and is also called a Coherent Risk Measure:

1 - Translation Invariant

2 - Monotonicity (bounded from below)

3 - positively homogenous

4 - sub additivity

Risk measure Examples:

(only expected shortful and generalised coherent satisfy the 4 properties)

Standard Deviation, Value at Risk, Expected Shortful, Generalized Coherent, Entropy

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