Lecture Notes: Value at Risk (VAR) 11/30/2018 BF330: Risk Management

What is Value at Risk? Value at Risk is a method for calculating and controlling exposure to market risk. It measures the volatility of an institution's assets - the greater the volatility, the greater is the risk of a loss. More formally, we can define VAR as a single number (currency amount) which estimates the maximum expected loss of a portfolio over a given time horizon (the holding period) and at a given confidence level. Less formally, VAR is a currency amount, say X dollars, where the chance of losing more than X dollars is, for example, 5% over some future time period. This is a statement of probability, so VAR calculations cannot be relied upon with certainty. 11/30/2018 BF330: Risk Management

Calculating Value at Risk VAR calculates the expected maximum loss of a portfolio as a result of an adverse change in risk factors (for example, interest rates, exchange rates and stock prices). The VAR estimate is dependent on a specified holding period, confidence level, volatility and, usually, correlation among the variables. Broadly speaking, therefore, the calculation of VAR involves the following steps: Step 1 - Determine the holding period The holding period used depends on the underlying assets and the underlying activities. For example, foreign exchange dealers are often interested in calculating the amount they might lose in a 1-day period. Therefore, in measuring the VAR of an active trading portfolio of liquid instruments, 1-day VAR is probably appropriate. 11/30/2018 BF330: Risk Management

The longer the holding period, the higher the VAR estimate will be. Cont’d On the other hand, participants in more illiquid markets, or regulators, may be interested in estimating market risk over longer time horizons. In such cases, a 1-week, 10-day or even monthly holding period may be appropriate. The longer the holding period, the higher the VAR estimate will be. 11/30/2018 BF330: Risk Management

Cont’d The Basel Committee on Banking Supervision recommends that institutions use a minimum holding period of 10 days for the purposes of calculating their regulatory capital requirement. Institutions using shorter holding periods (typically one day) can scale up to 10 days by the square root of the time period required. For example, a 10-day VAR estimate will be √10 (or 100.5) times larger than the corresponding 1-day VAR estimate. The important point is that the holding period should relate to the time period over which changes can occur in the portfolio 11/30/2018 BF330: Risk Management

Step 2 - Select the confidence level The confidence level is used to select the degree of certainty associated with the VAR estimate. For example, if a bank needs 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 of these days, the bank expects to lose more than the VAR estimate. The higher the confidence level, the higher the VAR amount will be. Many financial institutions use different numbers, but generally confidence levels between 95% and 99% are popular. A 95% confidence level implies that the VAR estimate will be exceeded about once a month, assuming that a year contains about 252 trading days. The Basel Committee on Banking Supervision proposes that institutions use a confidence level of 99%, which implies that only two to three breaches of the VAR estimate occur during the year. 11/30/2018 BF330: Risk Management

Confidence Intervals Confidence Levels: No. of Standard deviations 90% 1.282 95% 1.645 97.5% 1.960 99% 2.326 99.99% 3.719 11/30/2018 BF330: Risk Management

Step 3 – Create a Probability Distribution of Likely Returns Several methods can be used to create a probability distribution of returns for an asset or portfolio. The easiest to understand and the one most frequently used in VAR models is the normal distribution. 11/30/2018 BF330: Risk Management

Step 4 – Determine Correlations Between Assets Financial instruments are not generally independent of each other. Correlation measures the extent to which the value of one variable is related to the value of another variable. For example, the value of one currency may be correlated to movements in the value of another currency and the value of real estate stocks tends to be correlated with changes in interest rates. Correlation between assets impacts on the risk of a portfolio. Therefore, the degree of correlation between assets (and markets) is a vital consideration for portfolio managers who wish to reduce risk through diversification 11/30/2018 BF330: Risk Management

Step 5 - Calculate the volatility of the portfolio VAR is basically a measure of the volatility of an institution's assets - the greater the volatility, the greater the chance of a loss. The simplest measure of volatility is the standard deviation of the portfolio. Standard deviation measures the dispersion of the observations (returns) in a distribution around the mean value – the higher the standard deviation, the greater the volatility of the asset. The standard deviation of a portfolio is simply the square root of the variance. Volatility Forecasts for VAR Volatility forecasts are particularly important in the context of value at risk as they are key inputs into the variance-covariance and Monte Carlo approaches for calculating VAR. 11/30/2018 BF330: Risk Management

Cont’d There are several methods for forecasting volatility, of which the simplest is the sample variance. Other methods of forecasting volatility include: Exponentially weighted moving average (EWMA) GARCH (generalized autoregressive conditional heteroscedasticity) technique Implied volatility 11/30/2018 BF330: Risk Management

Step 6 - Calculate the VAR estimate Assume you are trying to calculate the 1-day VAR of a portfolio at a 95% confidence level on a one-tail basis (that is, looking at the undesirable side of deviation from the mean performance). Assuming the expected return (mean) of the portfolio (µP) is zero, the value at risk is expressed as: 11/30/2018 BF330: Risk Management

Example: The calculation of VAR is then really a question of finding the appropriate volatility of the portfolio. Let's look at an example. You work in a UK bank and you are holding a USD10 million foreign exchange position. You want to calculate a 95% value at risk over a 1 day time period (holding period). The ruling exchange rate is assumed at £1.344/$1while annual volatility is estimated at 15%. How do you go about doing this? 11/30/2018 BF330: Risk Management

Zero Mean Assumption An expected return of zero is a common assumption when modeling VAR over a short holding period, such as one day. It is justified by the fact that the mean return is typically several orders of magnitude smaller than the standard deviation. The zero mean assumption means that VAR estimates will be biased upwards, but this bias should be negligible except in the case of very long holding periods. In fact, some research suggests that, owing to estimation errors arising from not knowing the true mean return, allowing for a non-zero mean return may actually reduce the accuracy of VAR estimates. 11/30/2018 BF330: Risk Management

Usefulness of VAR as a Measure of Market Risk One of the most useful features of VAR as a measure of risk is the simplicity of the end result. The fact that a measure of market risk can be conveyed to shareholders and senior management in non-technical terms ('x amount of dollars over one-day with 99% probability', and so on) makes VAR an extremely powerful risk management tool. Other benefits include: Unlike some measures of risk (such as sensitivity measures), VAR is not limited to focusing on the risk associated with individual instruments. It can aggregate risks associated with different instruments within a portfolio. VAR promotes more efficient allocation of resources by encouraging financial institutions to avoid being over-exposed to one source of risk. VAR is important for performance evaluation in a trading environment. The natural instinct of most traders is to take on additional risk, but VAR can help 11/30/2018 BF330: Risk Management

Cont’d quantify this risk and so contribute to the establishment of position limits for traders. VAR is helpful for market regulators who wish to ensure that financial institutions do not go bust. By exposing the risk profile of such institutions, regulators can assess the risk and calculate the appropriate capital requirement to cushion this risk. 11/30/2018 BF330: Risk Management

Drawbacks of VAR as a Measure of Market Risk Perhaps the major drawback with VAR is the assumption in many models (variance-covariance and some Monte Carlo simulations) that portfolio returns are normally distributed. All market participants understand that from time to time there are unusual or extreme events in the market that are not captured by a normal distribution. When such events occur, VAR calculations may underestimate the true value at risk. Therefore, relying on the assumption of normally distributed returns is dangerous when there are extreme movements in the market. Other drawbacks include: Some VAR models use historical return data in their calculations. The assumption in such models is that the past is a reliable guide to the future, which is not always the case. 11/30/2018 BF330: Risk Management

Contd Some VAR models (variance-covariance approaches) are unsuitable for portfolios containing options due to the non-linear behavior of options. In other words, the ratio of change in the option value with respect to changes in the underlying asset value (delta) is not constant. Having said this, while the scope is limited, there are approximating methods in variance-covariance models to estimate the VAR of portfolios with optionality (although most sophisticated institutions will use simulation techniques when options are involved). There may be difficulties associated with both the capturing and the reliability of data. If the captured data is unreliable, VAR models are worthless. Some VAR systems, particularly Monte Carlo simulations, are costly and can prove difficult to set up. VAR does not always give a consistent method for calculating market risk; different methods can produce different results on a daily basis. 11/30/2018 BF330: Risk Management

Contd Finally, it is important to note that VAR in itself is not risk management. It is a tool for measuring market risk and is therefore part of a complete range of activities and duties involved in managing and minimizing a financial institution's risk exposure. 11/30/2018 BF330: Risk Management

Advantages of VAR Simplicity of end result Can aggregate risks associated with different instruments within a portfolio Promotes more efficient allocation of resources Important for performance evaluation Helpful for market regulators 11/30/2018 BF330: Risk Management

Disadvantages of VAR Assumption in many models that portfolio returns are normally distributed Some models depend on the past being a reliable guide to the future Some approaches are unsuitable for portfolios containing options May be difficulties with the reliability of data Some VAR systems are costly and can prove difficult to set up Does not always give a consistent method for calculating market risk Only represents one aspect of effective risk management 11/30/2018 BF330: Risk Management