Value at Risk (VAR) is a risk management measure used to calculate potential losses over a given time period at a specified confidence level. There are three key elements - the level of loss, time period, and confidence level. For example, there is a 5% chance losses will exceed $20M over 5 days. VAR does not provide information on potential losses above the VAR level. There are three main methodologies used to calculate VAR - historical simulation, variance-covariance, and Monte Carlo simulation. Each has its own strengths and weaknesses in terms of implementation and ability to capture risk.

1. 09/26/10 Case Study By: Mohit Rathi Dheeraj Lakshay Ahuja Vivek Verma

2. Value At Risk a risk management measure… Value at Risk (VAR) calculates the maximum loss expected (or worst case scenario) on an investment, over a given time period and given a specified degree of confidence. Specifically, VAR is a measure of losses due to “normal” market movements. There are three key elements of VAR- a specified level of loss in value, a fixed time period over which risk is assessed and a confidence level. For example , "there is only a 5% chance that our company's losses will exceed $20M over the next five days". This is the "classic" VaR measure. VaR does not provide any information about how bad the losses might be if the VaR level is exceeded. 09/26/10 Last Updated: September 26, 2010

3. Value At Risk 09/26/10 Mathematical Formula: "Given some confidence level the VaR of the portfolio at the confidence level α is given by the smallest number l such that the probability that the loss L exceeds l is not larger than (1 − α)"

4. History 09/26/10 VAR was well established in quantitative trading groups at several FI’s before 1990,(due to financial events in early 1990s), though it was not standardized. Development was most extensive at J.P. Morgan, which published methodology and gave free access to estimates of necessary underlying parameters in 1994. Two years later, VAR was spun off into an independent, for profit business, now part of Risk Metrics Group. In 1997, US SEC, made it compulsory to disclose all quantitative measures of derivate and present financial statements including VAR.

5. Methodologies of VAR 09/26/10 HISTORICAL SIMULATION VARIANCE-COVARIANCE METHOD MONTE CARLO SIMULATION or STOCHASTIC SIMULATION

6. Historical Simulation 09/26/10 Historical simulation is a simple, theoretical approach that requires relatively few assumptions about the statistical distributions of the underlying market factors. Historical Simulation can be described in terms of five steps:- STEP1 : Identify the basic market factors, and obtain a formula expressing the mark-to-market value of the contract in terms of the market factors. STEP2 : Obtain historical values of the market factors for the last N periods. Contd.

7. 09/26/10 STEP3 : Subject the current portfolio to the changes in market rates and prices experienced on each of the most recent 100 business days, calculating the daily profits and losses that would occur if comparable daily changes in the market factors are experienced and the current portfolio is marked-to-market. STEP4 : Order the mark-to-market profits and losses from largest profit to largest loss. STEP5 : Finally, select the loss which is equaled or exceeded 5 % of the time.

8. Variance and Covariance method 09/26/10 It is based on the assumption that the underlying market factors have a multivariate Normal distribution. A key step in the variance covariance approach is known as "risk mapping." This involves taking the actual instruments and "mapping" them into a set of simpler, standardized positions or instruments. This method requires 4 steps:- STEP1 :Identify the basic market factors and the standardized positions that are directly related to the market factors, and map the contract onto the standardized positions. Contd.

9. 09/26/10 STEP2 : Assume that percentage changes in the basic market factors have a multivariate Normal distribution with means of zero, and estimate the parameters of that distribution. This is the point at which the variance-covariance procedure captures the variability and co-movement of the market factors. STEP3 : Use the standard deviations and correlations of the market factors to determine the standard deviations and correlations of changes in the value of the standardized positions. Contd.

10. 09/26/10 STEP4 : Now with the help of the standard deviations of and correlations between changes in the values of the standardized positions, calculate the portfolio variance and standard deviation using uses standard mathematical results about the distributions of sums of Normal random variables and determine the distribution of portfolio profit or loss.

11. Monte Carlo simulation 09/26/10 This is similar to Historical Simulation, the difference is that rather than carrying out simulation using the observed changes in the market factors over last N periods to generate N hypothetical portfolio profits or losses, one chooses a statistical distribution that is believed to capture or approximate the possible changes in market factors. This also include 5 steps: STEP1 : Identify the basic market factors, and obtain a formula expressing the mark-to-market value of the forward contract in terms of the market factors. Contd.

12. 09/26/10 STEP2 : Determine a specific distribution for changes in the basic market factors, and to estimate the parameters of that distribution. STEP3 : Use pseudo-random generator to generate N hypothetical values of changes in market factors, where N is almost certainly greater than 1000 and perhaps greater than 10,000. These factors are then used to calculate N hypothetical mark-to-market portfolio values. Then from each of the hypothetical portfolio values subtract actual mark-to-market portfolio value to obtain N hypothetical daily profits and losses. Contd.

13. 09/26/10 STEP4&5 : The last two steps are the same as in historical simulation. The mark-to-market profits and losses are ordered from the largest profit to the largest loss, and the value at risk is the loss which is equaled or exceeded 5 percent of the time.

14. Comparison in Methodologies 09/26/10 Contd. Basis of comparison Historical Simulation Variance//Covariance approach Monte-Carlo Simulation Able to capture the risks of portfolios which include options? Yes, regardless of the options content of the portfolio No, except when computed using a short holding period for portfolios with limited or moderate options content Yes, regardless of the options content of the portfolio Easy to implement? Yes, for portfolios for which data on the past values of the market factors are available. Yes, for portfolios restricted to instruments and currencies covered by available "off-the-shelf" software. Yes, for portfolios restricted to instruments and currencies covered by available "off-the-shelf" software

15. 09/26/10 Basis of comparison Historical Simulation Variance/Covariance approach Monte-Carlo Simulation Computations performed quickly? Yes. Yes. No, except for relatively small portfolios. Easy to explain to senior management? Yes. No. No. Produces misleading value at risk estimates when recent past is typical? Yes. Yes, except that alternative correlations/standard deviations may be used. Yes, except that alternative estimates of parameters may be used. Easy to perform "what-if" analyses to examine effect of alternative assumptions? No. Able to examine about correlations/standard deviations . Yes

16. Advantages of VAR 09/26/10 Captures an important aspect of risk in a single number. Easy to understand. It provides a measure of total risk. It is useful for monitoring and controlling risk within portfolio. It can measure risk of many types of financial securities. As a tool, it is very useful for comparing a portfolio with market portfolio.

17. Stress Testing 09/26/10 This involves testing how well a portfolio performs under some of the most extreme market moves seen in the last 10 to 20 years. Back Testing Tests how well VaR estimates would have performed in the past We could ask the question: How often was the actual 10-day loss greater than the 99%/10 day VaR?

18. Alternatives to VAR 09/26/10 Sensitivity Analysis. Cash flow at risk.

19. EXAMPLE 09/26/10 Assuming that agent Rs 1,00,000 portfolio contains Rs. 60,000 worth of Stock X and Rs. 40,000 worth of Stock Y. computing the VaR of the same with 95% confidence level over the coming:- DAY, MONTH and YEAR. We have , wx=0.60 wy=0.40 σx= 0.016284 σy=0.015380 ρ =-0.19055 Contd.

20. 09/26/10 σ ρ =√(0.60) 2 (0.16284) 2 +(0.4) 2 (0.015380) 2 +2(0.6)(0.4)(-0.19055)(0.16284)(0.015380) =0.01144627 =1.144627% The portfolio VAR over agent DAY, V 0 ασ ρ = (100000)(1.645)(0.01144627) =1.88291 Contd.

21. 09/26/10 The portfolio VAR over agent MONTH, The portfolio VAR over a MONTH, V 0 ασ ρ = (100000)(1.645)(0.01144627√22) =8831.638 The portfolio VAR over an YEAR, V 0 ασ ρ = (100000)(1.645)(0.01144627√252) =29890.29

22. Limitations to VAR 09/26/10 Referring to VaR as a "worst-case" or "maximum tolerable" loss. In fact, one expect two or three losses per year that exceed one-day 1% VaR. Making VaR control or VaR reduction the central concern of risk management. It is far more important to worry about what happens when losses exceed VaR. Contd.

23. 09/26/10 Assuming plausible losses will be less than some multiple, often three, of VaR. The entire point of VaR is that losses can be extremely large, and sometimes impossible to define, once you get beyond the VaR point. To a risk manager, VaR is the level of losses at which one stop trying to guess what will happen next, and start preparing for anything. Reporting a VaR that has not passed a backtest. Regardless of how VaR is computed, it should have produced the correct number of breaks in the past. A common specific violation of this is to report a VaR based on the unverified assumption that everything follows a multivariate normal distribution.