This presentation captures the methodologies of Value at 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)"

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.

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.

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

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