Objective of the sessionexplain how value at risk is used to measure market risk describe the three methods for calculating VAR list the main advantages and disadvantages of VAR as a

measure of risk

Prerequisite Knowledge

knowledge of statistical concepts - such as distributions, confidence levels, correlation and volatility. Also require knowledge regarding:Interest Calculations Time Value of Money Probability Distributions & Hypothesis Testing Estimating Volatility

Definition of VARValue at risk (VAR) 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.

VAR can be defined 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.

VAR expresses risk in terms of Currency amounts.

The following are the three methods of calculating VAR :Variance-covariance Historical simulationMonte Carlo simulation

Today’s session, we will discuss only the Variance-Covariance approach of calculating VAR.

Variance-CovarianceThis is the method used in Risk Metrics, a service originally developed by bankers at JP Morgan and which is now conducted by an independent company called Risk Metrics.

The approach makes a number of assumptions, including the crucial assumption that the returns on the assets in a portfolio are normally distributed.

Disadvantage of VAR:When there are extreme events in the markets, VAR models that assume that portfolio returns are normally distributed may underestimate the true value at risk.

Other drawbacks include: Some VAR models use historical return data in their

calculations. The assumption in such models is that the past is not always a reliable guide to the future

Some VAR models (variance-covariance approaches) are unsuitable for portfolios containing options due to the non-linear behavior of options.

Disadvantage of VAR:There may be difficulties in 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.

VAR is a powerful tool for understanding and controlling risk. Although it has a number of limitations, and is only one aspect of risk management.In recent years, some of the world's leading financial institutions have suffered huge losses in the financial markets. One implication of this has been the increased monitoring of exposure to market risk.

Other benefits include: 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.

Other benefits: VAR can help quantify this risk and so contribute to

the establishment of position limits for traders.

VAR is helpful for market. By exposing the risk profile of such institutions, regulators can assess the risk and calculate the appropriate capital requirement to cushion this risk.

Less formally, we can say that - 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.

Origins of VARIn a 1993 report, the Group of Thirty (G30) proposed that market risk is best measured using a value at risk approach. The G30 study was headed by Sir Dennis Weatherstone, at the time chairman of JP Morgan. It is reported that, in his capacity as chairman, Weatherstone had a one-page market risk report delivered to him at the close of trading each business day, summarizing the daily earnings at risk (DEaR) for JP Morgan worldwide. The methodology used in the now-famous '4.15 Report' to estimate potential losses has become known as value at risk.

Regulators & VARIn 1996, the Basel Committee on Banking Supervision amended its 1988 Capital Accord to take account of market risks. Included in this amendment was the Committee's decision on the criteria that govern the use of VAR models for determining capital charges.

How VaR quantify Risk?

A bank estimates its daily VAR as USD 10m at a 95% level of confidence means there is a 5% chance that the bank's loss will be greater than USD 10m over a 1-day period.

Calculating Value at RiskVAR 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.

Determine the Holding PeriodThe holding period used depends on the underlying assets and the underlying activities. In measuring the VAR of an active trading portfolio of

liquid instruments, 1-day VAR is probably appropriate. Foreign exchange dealers are often interested in calculating the amount they might lose in a 1-day period

Holding PeriodIn case of, more illiquid markets, Participants or regulators, may be interested in estimating market risk over longer time horizons such as 1-week, 10-day or even monthly holding period may be appropriate.The longer the holding period, the higher the VAR estimate will be.

Holding PeriodThe 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.

If you leave your data as is i.e., "Leave As Is" then the time horizon for both volatility and VaR is the same as the time horizon implied by your data. Thus, given the above data set is monthly then the time horizon for volatility and VaR is monthly. If you would like to convert this to daily you must first select Monthly for your data set and then select Horizon to equal 1-Day. If you use portfolio weights, then the program allocates $1m across the assets given the portfolio weights. You can then rescale VaR to your own portfolio value.

Select the Confidence LevelThe 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.

Confidence Levels & Standard Deviation

The table below shows the number of standard deviations from the mean for a given confidence level.

Create a Probability Distribution of Likely ReturnsSeveral 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.

Real World DistributionsMost studies show that the distribution of returns of financial assets and liabilities is not normal. Due to extreme events in the financial markets (such as stock or bond market crashes), real world distributions tend to exhibit 'fat tails'. In other words, the peak of a real world return distribution is narrower and the tails are fatter than that predicted by the normal distribution.

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. the value of one currency may be correlated to

movements in the value of another currencythe value of real estate stocks tends to be correlated

with changes in interest rates.

Calculate the Volatility of the PortfolioVAR 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.

Find out the Mean, the Variance, and the Standard Deviation.

Find the Mean:

Find the difference of each value from the Mean:

the Standard Deviation is just the square root of Variance, so: Standard Deviation: σ = √21,704 = 147.32... = 147nce

Mean =

600 + 470 + 170 + 430 + 300=

1970

= 3945

5

Variance = 2062 + 762 + (-224)2 + 362 + (-94)2 = 108,520

= 21,7045

5

Impacts of Correlation between assetsCorrelation between assets impacts on the risk of a portfolio.

So, the degree of correlation between assets (and markets) is a vital consideration for portfolio managers who wish to reduce risk through diversification.

Select the Confidence LevelMany financial institutions use different numbers, 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.

Select the Confidence LevelThe 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.

Calculate the Volatility of the PortfolioVAR 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.

Calculate the Volatility 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.

Formula for Standard Deviation:The variance of a portfolio is calculated as follows (the standard deviation is the square root of the variance):

Volatility Forecasts for VARVolatility forecasts are particularly important in the context of value at risk. They are key inputs into the Variance-covariance and Monte Carlo approaches for Calculating VAR. There are several methods for forecasting volatility, of which the simplest is the sample variance.

Volatility Forecasts for VAR

Other methods of forecasting volatility include:Exponentially weighted moving average (EWMA)GARCH (generalized autoregressive conditional

heteroscedasticity) techniqueImplied volatility

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:

Calculation of VAR for a Single-Asset PortfolioSuppose that you have a portfolio consisting only of Microsoft stock with a total market value of USD 25 million. Assume that the annual volatility of Microsoft shares has been around 35% p.a. and that there are 252 trading days in the year.In this case, the 1-day VAR with a 95% confidence level (1.645 standard deviations) is calculated as follows:VAR = USD 25m x 1.645 x 0.35 / (252) 0.5 VAR = USD 906,721This is the same as the point on the cumulative normal distribution curve which represents (1 - a ), where a =

confidence level (in this example, a = 95%) as in the following graph:

One Asset Portfolio Market value of portfolio USD 6,666,666.67

Step 1: Determine holding period (days) 5 Step 2: Select confidence level 99.00%

(no. of Std Deviation = 2.326) Step 3:Create probability distribution of returns Normal

distribution Step 4 Determine correlation between assets n/a Step 5 Annual volatility of portfolio 6.00% Step 6 Calculate the VAR estimate USD 131,055.25 VAR = USD 131,055.25

Calculating VAR Using the Variance-Covariance Approach:Assume that you have a portfolio consisting only of 10,000 shares in ABC Corporation and the share price is USD 2.05. Also assume that the annual volatility of ABC shares has been around 15% p.a. and that there are 252 trading days in the year.

Calculating VAR Using the Variance-Covariance Approach:What is the 1-day VAR at a 95% confidence level (1.645 standard deviations)?VAR = Portfolio value x No. of standard deviations x Volatility/(Days)0.5

VAR = (10,000 shares x USD 2.05) x 1.645 std deviations x 0.15/(252)0.5

VAR = USD 318.6477 or USD 319

Calculation of VAR for a Two-Asset Portfolio

you have diversified your portfolio and now have two stock holdings, Microsoft and Coca-Cola, each with a market value of USD 12.5 million?Assume that the annual volatility of Microsoft is again 35% p.a. and that of Coca-Cola is 28% p.a.Let's calculate VAR at the 95% confidence level in the same way as we did in the example of the one-asset portfolio.

Calculation of VAR for a Two-Asset Portfolio

σp = (ωi2σi

2 + ωj2σj

2 + 2ρijωiσiωjσj)0.5

σp = (0.52 x 28%2 + 0.52 x 35%2 + 2 x 1 x 0.5 x 28% x 0.5 x 35%)0.5

σp = (0.0196 + 0.03063 +0.049) 0.5

σp = 0.315

1-day VAR = USD 25m x 1.645 x 0.315/2520.5

1-day VAR ≈ USD 816,048.9This calculation is lower than the VAR of Microsoft alone (USD 906,721), but it is incorrect unless the price behavior of the two shares is perfectly correlated.

Calculation of VAR for a Two-Asset Portfolio

In the previous example, we assumed that the correlation (ρ) between the share prices of Microsoft and Coca-Cola was perfect. However, if we assume that it's not perfect, that in fact it is 0.2, we will get an even lower value for the 1-day VAR.

With a correlation (ρ) of 0.2,σp = (ωi 2σi 2 + ωj 2σj 2 + 2ρijωiσiωjσj)0.5

σp = ((0.52 x 0.282) + (0.52 x 0.352) + (2 x 0.2 x 0.5 x 0.28 x 0.5 x 0.35))0.5

σp = 0.245

Calculation of VAR for a Two-Asset Portfolio

1-day VAR = USD 25m x 1.645 x 0.245/2520.5 1-day VAR ≈ USD 634,704.72This figure is much lower than the VAR we had when the portfolio was all invested in Microsoft (USD 906,721).The reasons for this lower VAR are that the Coca-Cola stock is less volatile and that the prices of the two stocks do not move together – in other words, they are not highly correlated.This lower VAR figure is therefore the benefit of a diversified portfolio .

Calculating VARA 9-day VAR of a portfolio at a given confidence level is

how many times larger than a 1-day VAR at the same confidence level? : 3

The size of any VAR estimate is related to the square root of time. Therefore, a 9-day VAR should be three times larger than a 1-day VAR.