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Value at Risk By A.V. Vedpuriswar June 14, 2014

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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. 1

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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) 2

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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. 3

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How Banks disclose VaR 4

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VaR at UBS 5 Ref : Company Annual report

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VaR at UBS 6 Ref : Company Annual report

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VaR at UBS 7 Ref : Company Annual report

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VaR at UBS 8 Ref : Company Annual report

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VaR at UBS 9 Ref : Company Annual report

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VaR at Credit Suisse 10 Ref : Company Annual report

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VaR at Credit Suisse 11 Ref : Company Annual report

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VaR at Goldman Sachs 12 Ref : Company Annual report

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13 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. Problem 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 = - ( x 1)=- $9.6 million VAR = E (W) – (-9.6) = 5.1 – (-9.6) = $14.7 million If we assume a normal distribution, Z at 95% ( one tailed) confidence interval = VAR = (1.645) (9.2) =$ 15.2 million

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14 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 ( σ N 2 = σ σ 2 2 ….. = Nσ 2 ) Note; This approach is valid only when the daily variances are independent.

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15 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

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16 Variance Covariance Method

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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 = /√250 = 46,445 17

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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)= So 99% VAR = 2.33 X 3.6 X 10,000,000 X = $ 334,186 18

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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, , x 0.3 x 1,000 x 1,000 = 2,600,000 The standard deviation of the portfolio’s daily change is $1, The standard deviation of the 5-day change is 1, x √5 = $3, From the tables of N(x) we see that Z = The 5-day 99 percent value at risk is therefore 2.33 x 3, = $8, Ref : John C Hull, Options, Futures and Other Derivatives

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20 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 Ref : John C Hull, Options, Futures and Other Derivatives

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21 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

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22 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 σ 2 = w 1 2 σ w 2 2 σ ῤw 1 w 2 σ 1 σ 2 =(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, ,406) – (1,622,657) = 219,369 Ref : John C Hull, Options, Futures and Other Derivatives

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23 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 x Volatility = x 1 = 6.65% If YTM is 6.65%, bond price will be So the VAR is = $ 6.83

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24 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

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25 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% %, 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.

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26 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 of5% and 12% respectively. The portfolio has $2 million invested in CAD and $1 million in Euro. What is the portfolio VAR at 95% confidence level? Solution Variance of the portfolio return = {(2 (.05)} 2 + {(1) (.12)} 2 = =.0244 Std devn = √.0244=$ 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.

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

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Problem An American trader owns a portfolio of options on the US dollar-sterling exchange rate. The delta of the portfolio is The current exchange rate is $/£ 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 OrΔP = 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 = $ So the 10-day 99 percent VaR for the portfolio is x 2.33 x √10= $ 4.33 for an investment of £1. 28 Ref : John C Hull, Options, Futures and Other Derivatives

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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 $/ £ 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 : x x – 2 x 0.8 x x x x = The standard deviation is therefore $ million. The 10-day 99% VaR is $ x √10 x 2.33 =$ million 29 Ref : John C Hull, Options, Futures and Other Derivatives

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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) = $

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31 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 V 2 =2 σ 2 (1 + ῤ) =2 (1) 2 (1 +.1)=2.2 V=√2.2=

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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: σ 2 + σ 2 + σ 2 + σ 2 + σ 2 + (2) ( ) σ 2 + (2) ( ) 2 σ 2 + (2) ( ) 3 σ 2 + (2) ( ) 4 σ 2 + (2) ( ) σ 2 + (2) ( ) 2 σ 2 + (2) ( ) 3 σ 2 + (2) ( ) σ 2 + (2) ( ) 2 σ 2 + (2) ( ) σ 2 = 5 σ 2 + (8) ( ) σ 2 + (6) ( ) 2 σ 2 + (4) ( ) 3 σ 2 + (2) ( ) 4 σ 2

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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 = = Volatility= Problem Consider a portfolio with standard deviation of daily returns of 0.1 and autocorrelation of 0.3. Calculate the 5 day volatility.

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34 Monte Carlo Simulation

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35 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.

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36 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

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37 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.

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38 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.

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39 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.

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40 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.

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41 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.

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42 Formula used typically in Monte Carlo for stock price modelling

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43 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.

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44 Problems with Monte Carlo Cost of computing resources can be quite high. Speed can be slow. Random Numbers 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.

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45 Historical Simulation

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46 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.

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

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48 % ReturnsFrequencyCumulative Frequency What is VAR (90%) ? 10% of the observations, i.e, (.10) (30) = 3 lie below -7 So VAR = -7

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49 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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50 Comparison of different VAR modeling techniques

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51 Simulation vs Variance Covariance methods Simulation approaches are preferred by global banks due to: – flexibility in dealing with the ever-increasing range of complex instruments in financial markets – the advent of more efficient computational techniques in recent years – the falling costs in information technology However, the variance-covariance approach might be the most appropriate method for many smaller firms, particularly when : – they do not have significant options positions – they prefer to outsource the data requirement component of their risk calculations to a company such as RiskMetrics – significant savings can often be made by using outsourced volatility and correlation data, compared to internally storing the daily price histories required for simulation techniques

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52 Model Validation

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53 Basel Committee Standards (1) Banks that prefer to use internal models must meet, on a daily basis, a capital requirement that is the higher of either: – the previous day's value at risk – the average of the daily value at risk of the preceding 60 business days multiplied by a minimum factor of three VAR must be computed on a daily basis. A one-tailed confidence interval of 99% must be used. The minimum holding period should be 10 trading days. The minimum historical observation period should be one year.

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54 Banks should update their data sets at least once every three months. Banks can recognize correlations within broad risk categories. Provided the relevant supervisory authority is satisfied with the bank's system for measuring correlations, they may also recognize correlations across broad risk factor categories. Banks' internal models are required to accurately capture the unique risks associated with options and option-like instruments. The Basel Committee has also specified qualitative factors that banks must meet before they are permitted to use internal models. Basel Committee Standards(2)

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55 The Basel Committee prescribes an increase in capital requirements if, based on a sample of 250 observations (a one- year observation period), the VAR model underpredicts the number of exceptions (losses exceeding the 99% confidence level). For such purposes, three 'zones' have been distinguished by the Committee. Green Zone : 0-4 exceptions Yellow zone : 5-9 exceptions Red zone : 10 or more exceptions Basel Committee Standards(3)

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56 Problem Based on a 90% confidence level, how many exceptions in back testing a VAR model should be expected over a 250 day trading year? Solution 10% of the time loss may exceed VAR So no. of observations = (.10) (250) = 25

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Problem We are currently feeding a model with 600 days of data. The VAR confidence level is 99%. Nine exceptions are observed. Should we reject the model? Suppose it had been 12. Would we reject the model? Solution 1 – Binomdist (8, 600,.01, True) =.152 So we cannot reject the model at 5% significance level. 1 – Binomdist (11, 600,.01, True) =.019 So we would reject the model at 5% significance level Ref : John C Hull, Options, Futures and Other Derivatives

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Problem We back test a VAR model with 1000 days of data. The VAR confidence level is 99%. 17 exceptions are observed. Should the model be rejected at 5% significance level? Solution The probability of the VAR being exceeded on a given day = =.01 The probability of the VAR being exceeded on 17 days or more =1 – Binomdist (16, 1000,.01, True) =2.64% 2.64% < 5% So the model should be rejected. Ref : John C Hull, Options, Futures and Other Derivatives

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Problem A $10 million one year loan has a 1.25% probability of default. If there is no default, profit of $ 0.2 million will be made. If there is a default, the recovery can be anything from 0 to the full loan value. Calculate the 99% VAR and conditional VAR. Solution Default zone = 1.25% starting from 98.75% going up to 100%. Loss at 99% point =[.25/1.25] X 10 = $ 2 million. So the 99% VAR is $ 2million. Expected shortfall = {2 + 10]/2 = $ 6 million 59

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60 Stress Testing

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61 Introduction Stress testing involves analysing the effects of exceptional events in the market on a portfolio's value. These events may be exceptional, but they are also plausible. And their impact can be severe. Historical scenarios or hypothetical scenarios can be used.

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62 Two approaches to Stress testing Single-factor stress testing (sensitivity testing) involves applying a shift in a specific risk factor to a portfolio in order to assess the sensitivity of the portfolio to changes in that risk factor. Multiple-factor stress testing (scenario analysis) involves applying simultaneous moves in multiple risk factors to a portfolio to reflect a risk scenario or event that looks plausible in the near future.

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63 Extreme Value Theory EVT is a branch of statistics dealing with the extreme deviations from the mean of statistical distributions. The key aspect of EVT is the extreme value theorem. According to EVT, given certain conditions, the distribution of extreme returns in large samples converges to a particular known form, regardless of the initial or parent distribution of the returns. This distribution is characterized by three parameters – location, scale and shape (tail). The tail parameter is the most important as it gives an indication of the heaviness (or fatness) of the tails of the distribution. The EVT approach is very useful because the distributions from which return observations are drawn are very often unknown. EVT does not make strong assumptions about the shape of this unknown distribution.

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Gaussian Copulas Consider variables, V1 and V2 that are not normally distributed. Map the two variables on to a normal distribution. Apply correlation Create bivariate normal distribution. 64

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Illustration Consider 2 variables that have a uniform distribution. Using Gaussian copula, and assuming a copula correlation of 0.3, define a correlation structure

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, , , , , 00, , , ,

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Illustration xpercentileZ Z Correlation coefficient = 0.5

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VAR – Cash flow mapping Problem Consider a long position in a $1 million Treasury bond. Maturity:0.8 years Coupon:10% payable semiannually Annualized yield & volatility 3 Month6 Month1 Year Annualised yield Volatility Correlations between daily returns 3 Month6 Month1 Year 3 month month year Explain how mapping can be done while calculating VaR. Ref : John C Hull, Options, Futures and Other Derivatives

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Solution The current position involves the following: Cash flow of $50,000 in.3 years Cash flow of $1,050,000 in.8 years So 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 years = (.3) (12) =3.6 months. 3 month interest rate=5.50% 6 month interest rate=6.00%

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Effective interest rate for 3.6 months zero coupon bond = (.6/3)(.5) = 5.6% =.056 Present value ==49,189 Volatility = =.068%. Let us allocate to a 3 month bond and 1 - to a 6 month bond. Then we can write: 2 = 1 2 Here =.068 1 =.06 2 =.10 =.90 or = 2 (.06) 2 + (1- ) 2 (.10) (.9)( ) (1- )(.06)(.10) Putting =.7603, LHS =.00462, RHS = = So we can write the position as equivalent to $ (.7603) (49,189) =$37,397 in 3 month bond $ (.2397) (49,189)=$11,791 in 6 month bond

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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: =.066 Present value of cash flows = =$997,662 Volatility = [.1 + (3.6/6)(0.1) ] =0.16 If is the position in the 6 month bond and (1- ) in the 12 month bond, 2 = 2 (1- ) 2 (1- ) 1 2 Or (.16) 2 = 2 (.1) 2 + (1- ) 2 (.2) (.7) ( ) (1- ) (.1)(.2) LHS=.0256 Put = ; RHS = ≈.0256 So the position is equivalent to (.3203) (997,662)=$319,589 in 6 month bond (.6797) (997,662)=$678,074 in 12 month bond

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We 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 =.8 3 month bond 37, , 397 6 month bond 11,791319,589331,380 12 month bond --678,074678,074

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Let 1, 2, 3 be the volatilities of the 3 month, 6 months, 12 months bonds and 12, 13, 23 be the respective correlations. 2 = 12 1 23 2 13 1 3 = [(37,397) 2 (.06) 2 + (331,380) 2 (.10) 2 + (678,074) 2 (.20) 2 + (2) (37,397) (331,380) (.06) (.10) (.90) + (2) (331,380) (678,074) (.10) (.20) (.70) + (2) (37,397) (678,074) (.06) (.20) (.60)] x =≈2,628,536 ==$ 10 day 99% VAR = x 10 x 2.33 =$11,946

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