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  1. 1. Basics of Value At Risk
  2. 2. Value At Risk - Introduction • All portfolio management is about risk and return • “Return” is an unambiguous and self explanatory concept, but “risk” is a harder concept to pin down • In equity markets, we can think of risk in terms of volatility or betas or factor loadings; • in fixed-income markets, we have the notions of volatility, duration, and convexity; • while in the context of options, there are the delta, gamma, theta, and other greeks
  3. 3. VAR Introduction • 1990s, a new tool emerged for measuring portfolio risk called Value-at-Risk or VaR, which was explicitly geared towards gauging the adequacy of capital held to meet losses on risky portfolios • The first prominent mention of VaR occurs in a 1993 report of the Group of Thirty titled “Derivatives: Practices and Principles,” which recommended the use of VaR and stress-testing to evaluate the riskiness of portfolios • VaR was introduced by J.P. Morgan’s as part of Risk Metrics system in 1994. • VaR is one of the popular measure of portfolio risk for gauging capital adequacy.
  4. 4. VAR a Probability Measure • VAR is a probability-based measure of loss potential for a company, a fund, a portfolio, a transaction, or a strategy. • Any position that exposes one to loss is potentially a candidate for VAR measurement. • VAR is most widely and easily used to measure the loss from market risk, but it can also be used-subject to much greater complexity-to measure the loss from credit risk and other types of risks. • VAR is the loss that would be exceeded with a given probability over a specific time period.
  5. 5. VAR contd. •Therefore, the loss that would be exceeded with a given probability is a loss that would be expected to occur over a specific time period. •There is a big difference among potential losses that are incurred daily, weekly, monthly, quarterly, or annually. •Potential losses over longer periods should be larger than those over shorter periods.
  6. 6. VAR contd. •The VAR for a portfolio is $1.5 million for one day with a probability of 0.05. •Consider what this statement says: There is a 5 percent chance that the portfolio will lose at least $1.5 million in a single day. •The emphasis here should be on the fact that the loss is a minimum, not a maximum.
  7. 7. VAR contd. •To state the VAR as a maximum, we would say that the probability is 0.95, or that we are 95 percent confident, that the portfolio will lose no more than $1.5 million
  8. 8. Summary on VAR • First, we see that VAR is a loss that would be exceeded. Hence, it is a measure of a minimum loss • Second, we see that VAR is associated with a given probability • It is the loss that would be exceeded with a given probability • Thus we would state that there is a certain percent chance that a particular loss would be exceeded. • Finally, VAR is defined for a specific period of time.
  9. 9. VAR contd. •Value at Risk (VaR) is an attempt to provide a single number summarizing the total risk in a portfolio of financial assets •It has become widely used by corporate treasurers and fund managers as well as by financial institutions. Bank regulators also use VaR in determining the capital a bank is required to keep for the risks it is bearing.
  10. 10. THE VaR MEASURE • When using the value-at-risk measure, an analyst is interested in making a statement of the following form: • I am X percent certain there will not be a loss of more than V dollars in the next N days. • The variable V is the VaR of the portfolio • It is a function of two parameters: the time horizon (N days) and the confidence level (X%). • It is the loss level over N days that has a probability of only (100-X)% of being exceeded.
  11. 11. VAR calculation • When N days is the time horizon and X% is the confidence level, VaR is the loss corresponding to the (100- X)th percentile of the distribution of the gain in the value of the portfolio over the next N days. • Note that, when we look at the probability distribution of the gain, a loss is a negative gain and VaR is concerned with the left tail of the distribution • When we look at the probability distribution of the loss, a gain is a negative loss and VaR is concerned with the right tail of the distribution
  12. 12. The VAR & Expected Shortfall •VaR is an attractive measure because it is easy to understand. •In essence, it asks the simple question ‘‘How bad can things get?’’ •Expected shortfall asks ‘‘If things do get bad, how much can the company expect to lose?’’
  13. 13. A DRAWBACK OF VaR
  14. 14. Illustration - VAR • Assume a Portfolio : $1,000,000 • What is the maximum loss in a single day? • 5% Value At Risk for 1 Day : $12,5000 • What is the meaning? • 95% confident , losses will not exceed $12,500 in a single day • There is a 5% chance that Portfolio Losses will be $12,500 or more • 95% confident maximum losses will be $12,500 • 5% chance minimum losses will be $12,500
  15. 15. Definition of VAR • VAR is the dollar or percentage loss in portfolio (asset) value that will be equalled or exceeded only X Per cent of the time • Daily VAR(5%) of $15,000 indicates that there is a 5% chance that On any given day , the portfolio will experience a loss of $15,000 or more. • Which means that there is a 95% chance on any given day the portfolio will experience either a loss less than $15,000 or gain.
  16. 16. VAR Calculations
  17. 17. Concept of Standard Normal Distribution
  18. 18. Referring Cumulative Z table •For 1 % VAR the appropriate Z value : -2.33 •For 5 % VAR the appropriate Z value : -1.65 •For 10% VAR the appropriate Z value : -1.28 •=NORM.S.INV(99%) 2.33 ; Normsdist(2.33)=99% •=NORM.S.INV(95%) 1.65 : Normsdist(1.65)=95% •=NORM.S.INV(90%) 1.28 ; Normsdist(1.28)=90%
  19. 19. Converting Z value to Probability •Z Value = 1.282 •=NORMSDIST(1.282)= 0.90  90%probability •=NORMSDIST(1.65) =0.95 -> 95% Probability
  20. 20. Example VAR Basics • Portfolio : $100,000 • Average Expected Return : 10% (monthly) • Monthly Standard Deviation : 8% • Calculate Monthly 5% VAR ? • = Mean – (Z value * SD) • =10% (-1.65 *8) • =-3.2% • VAR in $ 100,000 * 3.2%
  21. 21. Example 2 • Portfolio : 17,000,000 • Expected Daily returns : 0.17% • Daily Standard Deviation : 0.13% • Calculate 10% daily VAR • = Mean – (Z value * SD) (note : Z value for 90% = 1.28) • = 0.17 –(1.28 * 0.13) • = 0.0036% • =61,200
  22. 22. Example 3 • Suppose that the gain from a portfolio during six months is normally distributed with a mean of $2 million and a standard deviation of $10 million. • From the properties of the normal distribution, the one-percentile point of this distribution is 2 − 2.326 × 10 or –$21.3 million. • (Mean- (Zvalue * SD) • The VaR for the portfolio with a time horizon of six months and confidence level of 99% is therefore $21.3 million
  23. 23. Important Points on VAR
  24. 24. Important Points on VAR •Value-at-Risk (VaR) is essentially a measure of volatility, specifically how volatile a bank’s assets are •VaR also takes into account the correlation between different sets of assets in the overall portfolio
  26. 26. • The probability assigned to a set of values is given by the type of distribution and, in fact, from a distribution we can determine mean and standard deviation depending on the probabilities pi assigned to each value xi of the random variable X. The sum of all probabilities must be 100%. • From probability values then, the mean is given by:
  27. 27. In the normal distribution, 2.5% of the outcomes are expected to fall more than 1.96 standard deviations from the mean. So, that means 95% of the outcomes would be expected to fall within 1.96 standard deviations. That is, there is a 95% chance that the random variable will fall between 1.96 standard deviations and -1.96 standard deviations. This would be referred to as a ‘two-sided’ (or ‘two-tailed’) confidence interval
  28. 28. VOLATILITY • In financial market terms, volatility is a measure of how much the price of an asset moves each day (or week or month, and so on). • Volatility is important for both VaR measurement and in the valuation of options. • Volatility is important for both VaR measurement and in the valuation of options. • Statistically, volatility is defined as the fluctuation in the underlying asset price over a certain period of time. • Fluctuation is derived from the change in price between one day’s closing price and the next day’s closing price.
  29. 29. historical volatility
  30. 30. Differing Standard Deviations
  31. 31. Differing means around the same standard deviation
  32. 32. Standard Deviation
  33. 33. Differing standard deviations
  34. 34. THE NORMAL DISTRIBUTION AND VaR • Many VaR models use the normal curve to calculate the estimation of losses over a specified time period.
  35. 35. CORRELATION • The correlation between different assets and classes of assets is an important measure for risk managers because of the role diversification plays in risk reduction • Correlation is a measure of how much the price of one asset moves in relation to the price of another asset. • In a portfolio comprised of only two assets, the VaR of this portfolio is reduced if the correlation between the two assets is weak or negative. • The simplest measure of correlation is the correlation coefficient. This is a value between1 andþ1, with a perfect positive correlation indicated by 1, while a perfect negative correlation is given by 1.
  36. 36. • Note that this assumes a linear (straight line) relationship between the two assets. • A correlation of 0 suggests that there is no linear relationship.
  37. 37. Correlation, STD Dev, Mean – with different Observations Observation Govt Bond 1 Govt Bond 2 Govt Bond 3 Govt Bond 4 1 5.35% 11.00% 7.15% 5.20% 2 6.00% 9.00% 7.30% 6.00% 3 5.50% 9.60% 6.90% 5.80% 4 6.00% 13.70% 7.20% 6.30% 5 5.90% 12.00% 5.90% 5.90% 6 6.50% 10.80% 6.00% 6.05% 7 7.15% 10.10% 6.10% 7.00% 8 6.80% 12.40% 5.60% 6.80% 9 6.75% 14.70% 5.40% 6.70% 10 7.00% 13.50% 5.45% 7.20% Mean 6.30% 11.68% 6.30% 6.30% STD Dev 0.006313 0.018967 0.007605 0.006220 Cor with Bond1 0.357617936 -0.758492885 0.933620205
  38. 38. WHAT IS VaR? • VaR is an estimate of an amount of money. • It is based on probabilities, so cannot be relied on with certainty, but is rather a level of confidence which is selected by the user in advance. • VaR measures • the volatility of a company’s assets, and so the greater the volatility, • the higher the probability of loss
  39. 39. VaR is defined as follows: VaR is a measure of market risk. It is the maximum loss which can occur with X% confidence over a holding period of t days.
  40. 40. • VaR is the expected loss of a portfolio over a specified time period for a set level of probability. • So, for example, if a daily VaR is stated as £100,000 to a 95% level of confidence, this means that during the day there is a only a 5% chance that the loss will be greater than £100,000. • VaR measures the potential loss in market value of a portfolio using estimated volatility and correlations. • It is measured within a given confidence interval, typically 95% or 99%. • The concept seeks to measure the possible losses from a position or portfolio under ‘normal’ circumstances
  41. 41. Calculation of a VaR estimate follows four steps 1. Determine the time horizon 2. Select the degree of certainty required, which is the confidence level 3. Create a probability distribution of likely returns for the instrument or portfolio under consideration 4. Calculate the VaR estimate
  42. 42. VAR Redefined VaR is the largest likely loss from market risk (expressed in currency units) that an asset or portfolio will suffer over a time interval and with a degree of certainty selected by the user.
  43. 43. Three main methods for calculating VaR. 1. the correlation method (or variance/covariance method); 2. Historical simulation; 3. Monte Carlo simulation.
  44. 44. Correlation method •This is also known as the variance–covariance, parametric or analytic method. •This method assumes the returns on risk factors are normally distributed, the correlations between risk factors are constant and the delta (or price sensitivity to changes in a risk factor) of each portfolio constituent is constant
  45. 45. Historical simulation method • The historical simulation method for calculating VaR is the simplest and avoids some of the pitfalls of the correlation method. • Specifically, the three main assumptions behind correlation (normally distributed returns, constant correlations, constant deltas) are not needed in this case. • For historical simulation the model calculates potential losses using actual historical returns in the risk factors and so captures the non- normal distribution of risk factor returns.
  46. 46. The attractive features of VaR as a risk metric are as follows • It corresponds to an amount that could be lost with some chosen probability. • It measures the risk of the risk factors as well as the risk factor sensitivities. • It can be compared across different markets and different exposures. • It is a universal metric that applies to all activities and to all types of risk. • It can be measured at any level, from an individual trade or portfolio, up to a single enterprise-wide VaR measure covering all the risks in the firm as a whole
  47. 47. VAR Interpretation •1% 1-day VaR=$2 million, •then we are 99% confident that we would lose no more than $2 million from holding the portfolio for 1 day.
  48. 48. Problem on VAR • Bank Z has investment of Rs 50 Crores in shares of ABC Ltd. • The daily volatility is 2% (SD) • At 99%, confidence level, Calculate • 1. Daily VaR • 2. Weekly VaR(7days) • 3. Monthly VaR(30 days) • 4. Yearly VaR ( 250days)
  49. 49. Solution • Daily Value at Risk = VaR = Volatility X Probability (Z value) • Z value from Normal Distribution table : 2.33 for 1 % • VAR =Volatility * Probability • VAR = Daily SD * Z value for 1 % • = 2% * 2.33 = 4.66% • Daily VAR Amount = 50,00,00,000 * 4.66% = 2,33,00,000
  50. 50. VAR calculation for different periods Daily VAR VAR = Daily SD * Z value for 1 % 2,33,00,000 Weekly VAR Daily VAR * SQRT (7) 61646005.55 Monthly VAR Daily VAR * SQRT (30) 127619355.9 Yearly VAR Daily VAR * SQRT (250) 368405347.4
  51. 51. Explain the difference between value at risk and expected shortfall. •Value at risk is the loss that is expected to be exceeded (100 – X)% of the time in days for specified parameter values, and . Expected shortfall is the expected loss conditional that the loss is greater than the Value at Risk.
  52. 52. VaR for Portfolio – 2 Securities • If a portfolio has 2 stocks , A and B • Wa and Wb are their value weights • Sa and Sb are their standard deviations of the returns • R is the correlation coefficient of their returns, then • Portfolio variance = (WaSa)^2 + (WbSb)^2 + 2*Wa*Wb*r*Sa*Sb • Portfolio standard deviation = sqrt( portfolio variance)
  53. 53. Portfolio VaR : two securities
  54. 54. Problem on VAR •The volatility of a certain market variable is 30% per annum. •Calculate a 99% confidence interval for the size of the percentage daily change in the variable
  55. 55. Solution
  56. 56. VaR using Beta • Calculate 95% 1 year VaR for a 1000 cr portfolio with Beta of 1.5. Expected average volatility of Nifty is 18% p.a. Correlation between the portfolio and the Nifty is 0.75. • Stdev(portfolio) = Beta * stdev(Nifty) / correlation coefficient • = 1.5 * 18% / 0.75 = 36% p.a. • 95% 1 year VaR = 1000 * 1.65 * 0.36 = INR 594 cr
  57. 57. VaR for single stock - recap • VaR must specify • Period • Confidence level • Assume normal distribution for calculating “delta normal” VaR if the underlying stock return is normally distributed • Otherwise use Student distribution or use historical method • Implied Volatility may be used for “delta normal” VaR if available and appropriate
  58. 58. VAR Formula 1VaR for a security/portfolio: n day VaR for K% confidence level = portfolio value * z score * standard deviation of portfolio/security return p.a. * sqrt(n/252) where z score = norm.s.inv(k%)
  59. 59. VaR Formula 2 To find confidence level when z score is given or computed as an intermediate step: k% = norm.s.dist(z, true)