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Value at Risk By A.V. Vedpuriswar

Value at Risk By A.V. Vedpuriswar. June 14, 2014. Introduction. VAR tells us the maximum loss a portfolio may suffer at a given confidence interval for a specified time horizon.

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Value at Risk By A.V. Vedpuriswar

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  1. Value at RiskBy A.V. Vedpuriswar June 14, 2014

  2. Introduction • VAR tells us the maximum loss a portfolio may suffer at a given confidence interval for a specified time horizon. • If we can be 95% sure that the portfolio will not suffer more than $ 10 million in a day, we say the 95% VAR is $ 10 million.

  3. Computing Value at Risk • VaR is the product of : • Value of portfolio • Z factor ( depends on confidence level) • Volatility • Time ( VaR scales in proportion to square root of time)

  4. Computing Value at Risk • The usual practice is to calculate daily volatility by observing the daily opening and closing prices of the portfolio over a period of time, say 6 months. • Then we obtain the volatility for the actual period under consideration by multiplying by the square root of the time. • Thus the volatility for 5 trading days will be that for one day multiplied by √5.

  5. How Banks disclose VaR

  6. VaR at UBS Ref : Company Annual report

  7. VaR at UBS Ref : Company Annual report

  8. VaR at UBS Ref : Company Annual report

  9. VaR at UBS Ref : Company Annual report

  10. VaR at UBS Ref : Company Annual report

  11. VaR at Credit Suisse Ref : Company Annual report

  12. VaR at Credit Suisse Ref : Company Annual report

  13. VaR at Goldman Sachs Ref : Company Annual report

  14. Problem • Average revenue = $5.1 million per day • Total no. of observations = 254. Std dev = $9.2 million • Confidence level = 95% • No. of observations < - $10 million = 11 • No. of observations < - $ 9 million = 15 • Find 95% VAR. • Solution • The point such that the no. of observations to the left = (254) (.05) = 12.7 • (12.7 – 11) /( 15 – 11 ) = 1.7 / 4 ≈ .4 • So required point = - (10 - .4 x 1) = - $9.6 million • VAR = E (W) – (-9.6) = 5.1 – (-9.6) = $14.7 million • If we assume a normal distribution, • Z at 95% ( one tailed) confidence interval = 1.645 • VAR = (1.645) (9.2) = $ 15.2 million 13

  15. Problem • The VAR on a portfolio using a one day time horizon is USD 100 million. What is the VAR using a 10 day horizon ? • Solution • Variance scales in proportion to time. • So variance gets multiplied by 10 • And std deviation by √10 • VAR = 100 √10 = (100) (3.16) = 316 • (σN2 = σ12 + σ22 ….. = Nσ2) • Note; This approach is valid only when the daily variances are independent. 14

  16. Problem • If the daily VAR is $12,500, calculate the weekly, monthly, semi annual and annual VAR. Assume 250 days and 50 weeks per year. • Solution • Weekly VAR ＝ (12,500) (√5) ＝ 27,951 • Monthly VAR ＝ ( 12,500) (√20) ＝ 55,902 • Semi annual VAR ＝ (12,500) (√125) ＝ 139,754 • Annual VAR ＝ (12,500) (√250) ＝ 197,642 15

  17. Variance Covariance Method

  18. Problem A fund has a portfolio consisting of 40% fixed income and 60% equity. The estimated 95% annual VAR assuming 250 trading days for the entire portfolio was $ 1,367,000 based on the portfolio’s market value of $ 12,500,000. The correlation between bond and stock returns is 0.The annual loss on the equity part of the portfolio is expected to exceed $ 1,153,000 5% of the time. What will be the daily expected loss that will be exceeded 5 % of the time for the bond portfolio? Solution • 1,367,000^2 = 1,153,000^2 + x^2 • X = $ 734,357 • Daily VAR = 734357/√250 • = 46,445

  19. Problem Consider a bond position of $ 10 million, a modified duration of 3.6 years and an annualized bond volatility of 2%.Calculate the 10 day 99% VAR assuming 252 business days in a year. Solution • 10 day volatility = .02X√(10/252)=.003984 • So 99% VAR = 2.33 X 3.6 X 10,000,000 X .003984 • = $ 334,186

  20. Problem Consider a position consisting of a $100,000 investment in asset A and a $100,000 investment in asset B. Assume that the daily volatilities of both assets are 1% and that the coefficient of correlation between their returns is 0.3. What is the 5-day 99% VaR for the portfolio? Solution • The standard deviation of the daily change in the investment in each asset is $1,000. The variance of the portfolio’s daily change is • 1,0002 + 1,0002 + 2 x 0.3 x 1,000 x 1,000 = 2,600,000 • The standard deviation of the portfolio’s daily change is $1,612.45. • The standard deviation of the 5-day change is • 1,612.45 x √5 = $3,605.55 • From the tables of N(x) we see that Z = 2.33. • The 5-day 99 percent value at risk is therefore 2.33 x 3,605.55 = $8,401. Ref : John C Hull, Options, Futures and Other Derivatives

  21. We have a portfolio of $10 million in shares of Microsoft. We want to calculate VAR at 99% confidence interval over a 10 day horizon. The volatility of Microsoft is 2% per day. Problem • Solution • σ = 2% = (.02) (10,000,000) = $200,000 • Z (p = .01) = Z (p =.99) = 2.33 • Daily VAR = (2.33) (200,000) = $ 466,000 • 10 day VAR = 466,000 √10 = $ 1,473,621 20 Ref : John C Hull, Options, Futures and Other Derivatives

  22. Problem • Consider a portfolio of $5 million in AT&T shares with a daily volatility of 1%. Calculate the 99% VAR for 10 day horizon. • Solution • σ = 1% = (.01) (5,000,000) = $ 50,000 • Daily VAR = (2.33) (50,000) = $ 116,500 • 10 day VAR = $ 116,500 √10 = $ 368,405 Ref : John C Hull, Options, Futures and Other Derivatives 21

  23. Problem • Now consider a combined portfolio of AT&T and Microsoft shares. Assume the returns on the two shares have a bivariate normal distribution with the correlation of 0.3. What is the portfolio VAR.? • Solution • σ2 = w12σ12 + w22 σ22 + 2 ῤw1 w2σ1σ2 • = (200,000)2 + (50,000)2 + (2) (.3) (200,000) (50,000) • σ = 220,277 • Daily VAR = (2.33) (220,277) = 513,129 • 10 day VAR = (513,129) √10 = $1,622,657 • Effect of diversification = (1,473,621 + 368,406) – (1,622,657) = 219,369 22 Ref : John C Hull, Options, Futures and Other Derivatives

  24. Problem Consider a portfolio made up of 5 year 5 % coupon government bonds. The bonds are trading at $ 100. The historical annual volatility is 1 %. Expected YTMs are normally distributed with zero mean and volatility of 1%. Calculate the 95% one year VAR. Solution Worst YTM = actual YTM + 1.65 x Volatility = 5 + 1.65 x 1 = 6.65% If YTM is 6.65%, bond price will be 93.1708So the VAR is 100-93.17 = $ 6.83

  25. Problem Consider the following single bond of $10 million, a modified duration of 3.6 yrs and annualized yield of 2%. Calculate the 10 day holding period VaR of the position with a 99% confidence interval, assuming there are 252 days in a year. Solution VAR = $10,000,000* 0.02*3.6* √10/252 * 2.33 = $334,186

  26. Problem Assume that a risk manager wants to calculate VAR for an S&P 500 futures contract using the historical simulation approach. The current price of the contract is 935 and the multiplier is 250. Given the historical price data shown below for the previous 300 days, what is the VAR of the position at 99% using the historical simulation methodology? Returns: -6.1%,-6%,-5.9%,-5.7%, -5.5%, -5.1%..........4.9%, 5%, 5.3%, 5.6%, 5.9%, 6% Solution The 99% return among 300 observations would be the 3rd worst observation among the returns. Among the returns given above -5.9% is the 3rd worst return, the 99% VAR for this position is therefore (935)*250* (0.059) = $13,791.

  27. Problem • Consider the portfolio of an American trader with two foreign currencies, Canadian dollar and euro. These two currencies are uncorrelated and have a volatility against the dollar of 5% and 12% respectively. The portfolio has $2 million invested in CAD and $1 million in Euro. What is the portfolio VAR at 95% confidence level? • Solution • Variance of the portfolio return • = {(2 (.05)}2 + {(1) (.12)}2 =.01 + .0144 = .0244 • Std devn = √.0244 = $ .156205 million • VAR = (1.65) (156,205) = $257,738 • VAR for Canadian dollar part = $ (1.65) (.05) (2) million = $165,000 • VAR for Euro part = $ (1.65) (.12) (1) million = $ 198,000 • Undiversified VAR = $ 363,000 • Thus the diversified VAR is significantly lower.

  28. Problem Suppose we increase the Canadian dollar position by $10,000. What is the marginal VAR? • Solution • Variance = {(2.01) (.05)}2 + {(1) (.12)}2=.0101 + .0144= .0245 • σ= √.0245 = $.1565 million • VAR = (1.65) (156,500) = $ 258,225 • Marginal VAR = 258,225 – 257,738 = $ 487

  29. Problem An American trader owns a portfolio of options on the US dollar-sterling exchange rate. The delta of the portfolio is 56.0. The current exchange rate is $/£ 1.5000. Derive an approximate linear relationship between the change in the portfolio value and the percentage change in the exchange rate. If the daily volatility of the exchange rate is 0.7%, estimate the 10-day 99% VaR. Solution • The approximate relationship between the daily change in the portfolio value, ΔP, and the daily change in the exchange rate, ΔS, is ΔP = 56 ΔS • For a unit change in £, $ will change by 1.5. It follows that • ΔP = 56 x 1.5 Δx • Or ΔP = 84 Δx • The standard deviation of Δx equals the daily volatility of the exchange rate, or 0.7 percent.The standard deviation of ΔP is therefore 84 x 0.007 = $ 0.588. • So the 10-day 99 percent VaR for the portfolio is • 0.588 x 2.33 x √10 = $ 4.33 for an investment of £1. Ref : John C Hull, Options, Futures and Other Derivatives

  30. Problem Some time ago a company entered into a forward contract to buy £1 million for $1.5 million. The contract now has 6 months to maturity. The daily volatility of a 6-month zero-coupon sterling bond (when its price is translated to dollars) is 0.06% and the daily volatility of a 6-month zero-coupon dollar bond is 0.05%. The correlation between returns from the two bonds is 0.8. The current exchange rate is $/ £ 1.53. Calculate the standard deviation of the change in the dollar value of the forward contract in 1 day. What is the 10-day 99% VaR? Assume that the 6-month interest rate in both sterling and dollar is 5% per annum with continuous compounding. Solution • The contract is a long position in a sterling bond plus a short position in a dollar bond. • The value of the sterling bond is 1.53e-0.05x0.5 or $1.492 million. • The value of the dollar bond is 1.5e-0.05x0.5 or $1.463 million. • The variance of the change in the value of the contract in one day is : • 1.4922 x 0.00062 + 1.4632 x 0.00052 – 2 x 0.8 x 1.492 x 0.0006 x 1.463 x 0.0005 = 0.000000288 • The standard deviation is therefore $0.000537 million. • The 10-day 99% VaR is $0.000537 x √10 x 2.33 = $0.00396 million Ref : John C Hull, Options, Futures and Other Derivatives

  31. Problem Consider the contract on the dollar/euro exchange rate (EC) traded on the CME. The notional amount is 125,000 Euros. Assume that the annual volatility is 12% and the current price is $1.05 per Euro. Find daily VAR at 99% confidence level. Solution  VAR = (2.33) [(.12) / √252] × (125,000 × 1.05)   = $ 2310 30

  32. Problem • Consider a portfolio with a one day VAR of $1 million. Assume that the market is trending with an auto correlation of 0.1. Under this scenario, what would you expect the two day VAR to be? • Solution • V2 = 2σ2 (1 + ῤ) • = 2 (1)2 (1 + .1) = 2.2 • V = √2.2 = 1.4832

  33. Auto correlation over longer periods Consider a period of 5 days. We assume the first day’s movement will have an impact on the second day's movement through the correlation coefficient. The first day’s movement will affect the third day’s movement through the square of the correlation coefficient and so on. Then the combined variance will be: σ2 + σ2 + σ2 + σ2 + σ2 + (2) () σ2 + (2) ()2σ2 + (2) ()3σ2 + (2) ()4σ2 + (2) () σ2 + (2) ()2 σ2 + (2) ()3 σ2 + (2) () σ2 + (2) ()2σ2+ (2) () σ2 = 5 σ2 + (8) () σ2 + (6) ()2σ2 + (4) ()3σ2 + (2) ()4σ2

  34. Solution • σ = 0 .1;  = .3 • Variance = (5) (.1)2 + (4) (2) (.3) (.1)2 • + (3) (2) (.3)2 (.1)2 + (2) (2) (.3)3 (.1)2 • + (2) (.3)4 (.1)2 • = .05 + .024 + .0054 + .00108 + .000162 • = .080642 • Volatility = .284 Problem Consider a portfolio with standard deviation of daily returns of 0.1 and autocorrelation of 0.3. Calculate the 5 day volatility.

  35. Monte Carlo Simulation

  36. What is Monte Carlo VAR? • The Monte Carlo approach involves generating many price scenarios (usually thousands) to value the assets in a portfolio over a range of possible market conditions. • The portfolio is then revalued using all of these price scenarios. • Finally, the portfolio revaluations are ranked to select the required level of confidence for the VAR calculation.

  37. Generate Scenarios • The first step is to generate all the price and rate scenarios necessary for valuing the assets in the relevant portfolio, as well as the required correlations between these assets. • There are a number of factors that need to be considered when generating the expected prices/rates of the assets: • Opportunity cost of capital • Stochastic element • Probability distribution

  38. Opportunity Cost of Capital • A rational investor will seek a return at least equivalent to the risk-free rate of interest. • Therefore, asset prices generated by a Monte Carlo simulation must incorporate the opportunity cost of capital.

  39. Probability Distribution • Monte Carlo simulations are based on random draws from a variable with the required probability distribution, usually the normal distribution. • The normal distribution is useful when modeling market risk in many cases. • But it is the returns on asset prices that are normally distributed, not the asset prices themselves. • So we must be careful while specifying the distribution.

  40. Calculate the Value of the Portfolio • Once we have all the relevant market price/rate scenarios, the next step is to calculate the portfolio value for each scenario. • For an options portfolio, depending on the size of the portfolio, it may be more efficient to use the delta approximation rather than a full option pricing model (such as Black-Scholes) for ease of calculation. • Δ (Option) = Δ(ΔS) • Thus the change in the value of an option is the product of the delta of the option and the change in the price of the underlying.

  41. Other approximations • There are also other approximations that use delta, gamma (Γ) and theta (Θ) in valuing the portfolio. • By using summary statistics, such as delta and gamma, the computational difficulties associated with a full valuation can be reduced. • Approximations should be periodically tested against a full revaluation for the purpose of validation. • When deciding between full or partial valuation, there is a trade-off between the computational time and cost versus the accuracy of the result. • The Black-Scholes valuation is the most precise, but tends to be slower and more costly than the approximating methods.

  42. Reorder the Results • After generating a large enough number of scenarios and calculating the portfolio value for each scenario: • the results are reordered by the magnitude of the change in the value of the portfolio (Δportfolio) for each scenario • the relevant VAR is then selected from the reordered list according to the required confidence level • If 10,000 iterations are run and the VAR at the 95% confidence level is needed, then we would expect the actual loss to exceed the VAR in 5% of cases (500). • So the 501st worst value on the reordered list is the required VAR. • Similarly, if 1,000 iterations are run, then the VAR at the 95% confidence level is the 51st highest loss on the reordered list.

  43. Formula used typically in Monte Carlo for stock price modelling

  44. Advantages of Monte Carlo • We can cope with the risks associated with non-linear positions. • We can choose data sets individually for each variable. • This method is flexible enough to allow for missing data periods to be excluded from the VAR calculation. • We can incorporate factors for which there is no actual historical experience. • We can estimate volatilities and correlations using different statistical techniques.

  45. Problems with Monte Carlo • Cost of computing resources can be quite high. • Speed can be slow. • RandomNumbers may not be all that random. • Pseudo random numbers are only a substitute for true random numbers and tend to show clustering effects. • Monte Carlo often assumes normal distribution. • But it can be performed with alternative distributions. • Results (value at risk estimate) depend critically on the models used to value (often complex) financial instruments.

  46. Historical Simulation

  47. Introduction • Unlike the Monte Carlo approach, it uses the actual historical distribution of returns to simulate the VAR of a portfolio. • Real data plus ease of implementation, have made historical simulation a very popular approach to estimating VAR. • Historical simulation avoids the assumption that returns on the assets in a portfolio are normally distributed. • Instead, it uses actual historical returns on the portfolio assets to construct a distribution of potential future portfolio losses. • This approach requires minimal analytics. • All we need is a sample of the historic returns on the portfolio whose VAR we wish to calculate.

  48. Steps • Collect data • Generate scenarios • Calculate portfolio returns • Arrange in order.

  49. What is VAR (90%) ? 10% of the observations, i.e, (.10) (30) = 3 lie below -7 So VAR = -7

  50. Advantages and Disadvantages of Historical simulation • Advantages • Simple • No normality assumption • Non parametric • Disadvantages • Reliance on the past • Length of estimation period • Weighting of data

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