Value at Risk

How much do you stand to lose, over a certain period and with a certain probability?

Assume that:• Everything assumed in the (VaR) calculations/process is true• All approximations made are accurate• The future follows the past and whatever risk you are analyzing only exists for the specified (certain) period

Its efficacy depends on how the tool is put to use.

Is the one number sufficient by itself to completely capture the risk in a position?Whether the users of VaR understand the limitations (of the tool) and the implications of those limitations?

Risk management is concerned with extreme events or large deviations from what is expected. The most common tool used for measuring the above is variance, an average (of sorts) of all the deviations from the mean. Although it is the key tool used in calculating VaR, it is not the most appropriate. Higher order factors that measure symmetry or length & thickness of tails would be more accurate.

VaR uses data from all events to evaluate the impact of extreme events. VaR is forced to do this because by definition, extreme events do not occur frequently enough to generate sufficient data. The downside is that extreme events have much higher means and variances. This means that if VaR (somehow) did use extreme events, it would lead to a much higher Value at Risk estimate.

Given that the object of risk management is to understand risk exposures and neutralize them, there is a strong emphasis on supplementing VAR with scenario analysis or sensitivity testing.

Value at Risk (VAR) is a market risk measurement approach that uses historical market trends and volatilities to estimate the likelihood that a given portfolio’s losses will exceed a certain amount.

VAR measures the largest loss likely to be suffered on a portfolio or a position over a holding period (usually 1 to 10 days) with a given probability (confidence level).

Assuming a 99% confidence level, a VAR of 1 million US dollars means that the there is only a one percent chance that losses will exceed the 1 million US dollar figure over the next ten days. It is also “fashionable” to refer to this loss as the one day in 100 days loss.

There are three primary methods used for calculating Value at Risk (VAR).a. Variance /Covariance methodb. Historical simulation methodc. Monte Carlo simulation method

All methods have a common base but then diverge in how they actually calculate Value at Risk (VAR). They also have a common problem in assuming that the future will follow the past. This shortcoming is normally addressed by supplementing any VAR figures with appropriate sensitivity analysis and/or stress testing.

VAR calculation follows five steps:• Identification of positions for Value at Risk• Identification of risk factors affecting valuation of positions.• Assignment of probabilities (or statistical distribution) to possible risk factors

values.• Creation of pricing functions for positions as a function of values of risk factors.• Calculation of Value at Risk (VAR)

Variance Covariance Approach This method assumes that the daily returns follow a normal distribution. From the distribution of daily returns we estimate the standard deviation . The daily Value at Risk VaR is simply a function of the standard deviation and the desired confidence level. In the Variance-Covariance VaR method the underlying volatility may be calculated either using a simple moving average (SMA) or an exponentially weighted moving average (EWMA).

This approach is generally utilized if it is believed that the daily returns during the look back period have followed normal market conditions.

The SMA approach places equal importance to all returns in the series whereas the EWMA approach places greater emphasis on returns of more recent durations.

The method gets its name from the variance-covariance matrix of securities that is used to calculate Value at Risk (VaR).

The method starts by calculating the standard deviation and correlation for the risk factor and then uses these values to calculate the standard deviations and correlation for the changes in the value of the individual securities that form the position. If price, variance and correlation data is available for individual securities then this information is used directly. The values are then used to calculate the standard deviation of the portfolio.Value at Risk (VaR) for a specific confidence interval is then calculated by multiplying the standard deviation by the appropriate normal distribution factor.

In some cases a method equivalent to the variance covariance approach is used to calculate VAR. This method does not generate the variance covariance matrix and uses the following approach:1. Separate the portfolio in a long side and a short side.2. Calculate the return series for the long side and the short side.3. Use the return series to calculate the correlation and variances for the long and

short sides4. Use the results in (3) to calculate the VaR.

The modified approach can be used where, due to the nature of the institutions strategies, a number of positions would net close to zero on a portfolio basis and also where the set of securities employed is so large that a variance – covariance approach would have significant resource/time requirements.

Historical Simulation Method for calculating Value at Risk (VaR)

It applies the historical (100 days) changes in price levels to current market prices in order to generate a hypothetical data set. The data set is then ordered by the size of gains/losses. Value at Risk (VAR) is the value that is equaled or exceeded the required percentage of times (1, 5, 10).

Historical simulation is a non-parametric approach of estimating VAR, i.e. the returns are not subjected to any functional distribution. VAR is estimated directly from the data without deriving parameters or making assumptions about the entire distribution of the data. This methodology is based on the premise that the pattern of historical returns is indicative of future returns

Monte Carlo Simulation for calculating Value at Risk (VaR)

The hypothetical data set used is generated by a statistical distribution rather than historical price levels. The assumption is that the selected distribution captures or reasonably approximates price behavior of the modeled securities

A Monte Carlo simulator uses random numbers to simulate the real world.

A Monte Carlo VaR model using the following sequence of steps1. Generate randomly simulated prices2. Calculate daily return series3. Repeat the steps in the historical simulation method described below

Implementing Value at Risk (VAR)The objective of a Value at Risk (VAR) implementation is to perform daily VAR analysis of positions within a portfolio. Such a process would be the first step in shifting the current emphasis from calculating VaR to managing VAR. Within the process the focus should be on:

• Positions with low coverage levels.• Positions with VAR beyond a set threshold.• Positions with significant VAR changes.• VAR analysis for the Desk. (All clients, All accounts, All positions)

Calculating VAR

Our sample portfolio that we will use for calculating Value at Risk (VAR) consists of the following 4 items:100 shares of ONGC5 barrels of Crude Oil1 foreign exchange denominated asset with market value of USD 10 on 5th March 2010.100 units of 3-year RF with issue date of 19th February 2009 and coupon rate of 11.25%. This means that the outstanding term of the bond is 1.96 years.

We are calculating Portfolio Value at Risk (VAR) on the 5th of March 2010 at the end of the day

The following steps are common to all the above mentioned Value at Risk (VAR) approaches:Step P1: Determine look back period for Value at RiskDetermine the period over which the risk is to be evaluated. For illustration purposes let us assume a look back period of 5 days from 1st March 2010 to 5th March 2010. In practice window lengths cover a wider duration, such as 6 months, 1 year, etc.Step P2: Obtain the daily times series data for each risk factor for the determined look back period for Value at Risk

ONGC WTI FX Rf 1 yr % Rf 2 year %1/3/2010 100.3 4799.36 47.85 8.27 8.332/3/2010 100.41 4806.63 47.87 8.27 8.333/3/2010 100.28 4799.40 47.86 8.26 8.324/3/2010 99.07 4735.55 47.8 8.25 8.325/3/2010 99.18 4730.89 47.7 8.25 8.33

Adjustments to original time series data for Value at Risk

The interest rate risk factors, need to be adjusted to take into account the portfolio’s exact exposure to this risk factor. The rate series will first be interpolated based on the outstanding term to maturity of the bond and then will be converted into a price series which will be used in the actual VAR calculation:

The detailed steps to this process are :Determine the outstanding term of the bond. This is calculated as follows:

Calculate an interpolated rf rate series for this outstanding term. Interpolation will be done using the following formula:T= outstanding term =1.96 years

T1=rounded down value of the outstanding term = 1 yearT2= T1+1 =2 years

Rf interpolated = (T2 –T)*Rf 1 year + (T –T1)*Rf 2 year / (T2-T1)

= (2-1.96) *8.25+(1.96-1)*8.33/(2-1)

1/3/2010 8.3276

2/3/2010 8.3276

3/3/2010 8.3176

4/3/2010 8.3172

5/3/2010 8.3268

Rf interpolated

Calculate the price of the bond at each data point usingSettlement date = Revaluation date = 5th March 2010Maturity Date =19th March 2012Coupon Rate = 11.25%Yield = Rf_interpolated (%) applicable to that data pointRedemption value = 100Coupon payment frequency = 2Basis = Actual/365

where:t = number of days from settlement to next coupon date.E = number of days in coupon period in which the settlement date falls.N = number of coupons payable between settlement date and redemption dateA = number of days from beginning of coupon period to settlement date.The resulting bond price series works out to:

3/1/2010 105.3777

3/2/2010 105.3777

3/3/2010 105.3968

3/4/2010 105.3976

3/5/2010 105.3793

Calculate a return series from the time series data for Value at Risk

A return series is derived from the given time series data by taking the natural logarithm of the ratio of successive prices /rates

ONGC WTI FX Rf prices

3/2/2010 0.1096% 0.1514% 0.0418% 0.0000%

3/3/2010 -0.1296% -0.1504% -0.0209% 0.0181%

3/4/2010 -1.2140% -1.3394% -0.1254% 0.0007%

3/5/2010 0.1110% -0.0985% -0.2094% -0.0174%

Calculate a return series for the portfolioIn order to evaluate the VaR for the portfolio, the return series for the portfolio will be required. This is derived by calculating a weighted average return series using the individual return series for each instrument in the portfolio. This will be calculated as follows:Calculate the weights of the respective instruments in the portfolio on the revaluation date, where weight = value of the instrument÷total value of the portfolio. The weights for the portfolio are given below:

ONGC 100 Shares 9918.00 20.29%WTI 5 barrels 23654.4 48.39%FX 100 asset of USD 100 4770 9.76%3 yr Rf 100 Units 10537.9 21.56%

48880.36 100.00%

The return series for the sample portfolio

3/2/20103/3/20103/4/20103/5/2010

0.0996%-0.0972%-0.9066%-0.0493%