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1、Value at RiskBy A V VedpuriswarSeptember 15, 2009What is VAR?VAR summarizes the worst loss over a target horizon that will not be exceeded at a given level of confidence. For example, “under normal market conditions, the most the portfolio can lose over a month is about $3.6 billion at the 99% confi
2、dence level.” The main idea behind VAR is to consider the total portfolio risk at the highest level of the institution.Initially applied to market risk, it is now used to measure credit risk, operational risk and enterprise wide risk. Many banks can now use their own VAR models as the basis for thei
3、r required capital for market risk.12Average revenue = $5.1 million per dayTotal no. of observations = 254. Std dev = $9.2 millionConfidence level = 95%No. of observations - $10 million = 11No. of observations 99%), the normal distribution generally underestimates potential losses. Extreme Value The
4、ory (EVT)57Empirical distributions suffer from a lack of data in the tails. This makes it difficult to estimate VAR reliably. EVT helps us to draw smooth curves through the extreme tails of the distribution based on powerful statistical theory. In many cases the t distribution with 4-6 degrees of fr
5、eedom is adequate to describe the tails of financial data. 6060 Fitting EVT functions to recent historical data is fraught with the same pitfalls as VAR.Once in a lifetime events cannot be taken into account even by powerful statistical tools.So they need to be complemented by stress testing.The goa
6、l of stress testing is to identify unusual scenarios that would not occur under standard VAR models. Stress testing6161The problem with stress testing is the stress needs to be pertinent to the type of risk the institution has. Otherwise, the complex portfolio models banks generally employ may give
7、the illusion of accurate simulation at the expense of substance. 6262During the credit crisis risk models of many banks were unable to predict the likelihood , speed or severity of the crisis.There were several exceptions.Goldman Sachs chief financial officer David Viniar once described the credit c
8、runch as “a 25-sigma event”Why?There was a major paradigm shift.How effective are VAR models? VAR and sub prime6363ExceptionsA few VAR exceptions are expected. A properly working model would still produce two to three exceptions a year. But the existence of clusters of exceptions indicated that some
9、thing was seriously wrong.Credit Suisse reported 11 exceptions at the 99% confidence level in the third quarter, Lehman brothers three at 95%, Goldman Sachs five at 95%, Morgan Stanley six at 95%, Bear Stearns 10 at 99% and UBS 16 at 99%. 6464What window? VAR models failed especially as the environm
10、ent was emerging from a period of relatively benign volatility.The models were clearly reacting not fast enough. What kind of models would have worked best?6565What models work best? With the benefit of hindsight, the type of VAR model that would actually have worked best in the second half of 2007
11、would most likely have been a model driven by a frequently updated short data history. Or any frequently updated short data history that weights more recent observations more heavily than more distant observations.In the wake of the recent credit crisis, there is a strong case for increasing the fre
12、quency of updating. Monthly, quarterly or even weekly updating of the data series would improve the responsiveness of the model to a sudden change of conditions.Problem on VAR cash flow mappingConsider a long position in a $1 million Treasury bond. Maturity:0.8 yearsCoupon:10% payable semiannually A
13、nnualized yield & volatility3 Month6 Month1 YearAnnualised yield5.506.007.00Volatility0.060.100.20Correlations between daily returns3 Month6 Month1 Year3 month1.00.90.66 month0.91.00.71 year0.60.71.0Explain how mapping can be done while calculating VaR,Solution The current position involves the foll
14、owing:Cash flow of $50,000 in .3 yearsCash flow of $1,050,000 in .8 yearsSo the position can be considered a combination of two zero coupon bonds, maturity 0.3, 0.8 years .Let us write the position as equivalent to a combination of standard 3 month, 6 month and 1 year bonds.3 month interest rate=5.5
15、0%6 month interest rate=6.00%.3 years = (.3) (12) =3.6 months.Solution ContEffective interest rate for 3.6 months zero coupon bond = 5.50 + .6/3(.5) = 5.60%Present value =49,189Volatility = =.068%.Let us allocate to a 3 month bond and 1 - of the present value to a 6 month bond.Then we can write: 2=1
16、2 + 22 + 2 12Here = .068 1 = .062 = .10 = .90or .0682 = 2 (.06)2+ (1-)2(.10)2 + 2 (.9)() (1-)(.06)(.10)Solution Cont or .0682 = 2 (.06)2 + (1-)2 (.10)2 + 2(.9) ()(1-)(.06)(.10)Putting = .7603LHS = .00462RHS= .00208 + .00057 + .001968= .00462So we can write the position as equivalent to $ (.7603) (49
17、,189) =$37,399 in 3 month bond$ (.2397) (49,189)=$11,791 in 6 month bondSolution Cont Now consider $1,050,000 received after 0.8 years.It can be considered a combination of 6 month and 12 month positions.Interpolating the interest rate we get: =.066Volatility = .1 + (3.6/6)(0.1) =0.16Present value o
18、f cash flows = = $997,662Let be the position in the 6 month bond and (1-) in the 12 month bond. Then we can write:2 = 2 12 + (1-)2 22 + 2 (1-) 12Or (.16)2 = 2 (.1)2 + (1-)2 (.2)2 + 2 (.7) () (1-) (.1)(.2)LHS=.0256 Put =.320337 We get RHS=.001026 + .01848 + .006096 .0256Solution ContSo the position i
19、s equivalent to (.320337) (997,662) =$319,589 in 6 month bond(.679663) (997,662) =$678,074 in 12 month bondWe can now write the portfolio in terms of 3 month, 6 month, 12 month zero coupon bonds. $50,000$1,050,000 Total t = .3 t = .83 month bond 37,399 - 37,3996 month bond 11,791319,589331,38012 mon
20、th bond -678,074678,074Solution ContLet 1, 2, 3 be the volatilities of the 3 month, 6 months, 12 months bonds and 12, 13, 23 be the respective correlations.Then 2 = 12 + 22 + 32 + 21212 + 22323 + 21313= (37,399)2 (.06)2 + (331,380)2 (.10)2 + (678,074)2 (.20)2 + (2) (37,399) (331,380) (.06) (.10) (.90)+ (2) (331,380) (678,074) (.10) (.20) (.70)+ (2) (37,399) (678,074) (.06) (.20) (.60) x 10-4=5,035,267 + 1,098,127,044 + 18,391,373,980 + 133,847,431+6,291,604,539+365,173,769x10-4Solution Cont2,628,516=$1621.310 day 99% VAR=1621.3 x 10 x 2.33=$11,946
