VIEWS: 7 PAGES: 51 POSTED ON: 11/4/2012 Public Domain
Market Risk 1. Definition of Market Risk 2. Interest Rate Risk: Cash Flow Risk 3. Motivations to quantify Market Risk 4. Calculating Market VAR 5. VAR and Confidence Level 6. VAR and Diversification Effect 7. Multiperiod VAR 8. Historical VAR 9. Backtesting the VAR 10. Portfolio Toolkit 11. Portfolio VAR 12. Pros and Cons of VAR Measures 13. Market Risk Regulation 14. Stress Testing Defining Market Risk Interest rate shifts Foreign exchange shifts Reduction in the value of A/L, Commodity price shifts securities, etc. portfolios, etc. Equity price shifts Market risk affects : Trading books mark-to-market Banking book Derivative trading mark-to-market Credit derivative trading mark-to-market Interest Rate Risk Shifts in the term structure of interest rate have two impacts A cash flow effect Market value effect To manage cash flow risk, banks can calculte the GAP: NII GAP x i Interest Rate Risk: Cash flow risk ASSETS LIABILITIES RSA RSL 400 @ 8% 600 @ 6% FRA FRL 600 @ 10% 400 @ 8% Interest income = 32 + 60 = 92 Interest expenses = 36 + 32 = 68 NII = 24 Interest Rate Risk: Cash flow risk ASSETS LIABILITIES RSA RSL 400 @ 10% 600 @ 8% FRA FRL 600 @ 10% 400 @ 8% Interest income = 40 + 60 =100 Interest expenses = 48 + 32 = 80 NII = 20 Interest Rate Risk: Cash flow risk Interest rates increase by 2% such that NII decreases by $4 million. Using the GAP: NII $200 x 0.02 NII $4 For all RSA: duration and convexity are used to capture the impact of interest rate. Motivations to Quantify Market Risk Required by many internal instances such as the Chief Risk Officer, various risk management committees, etc, to track at any time the exposure of the bank (or other FI’s) with respects to market risk. Limit setting. Risk components of performance evaluation. Economic capital allocation across the different business units. Required for regulatory purposes (Since Basel 98). Calculating Market VAR Recognized as the best measure to capture market risk. VAR measure? How much could the FI loose on a given position, with an x% probability of occurrence over a given time frame? Example : A one-day VAR of $20 million at a probability of 1%, (or equivalently, at a confidence level of 99%) means that one day out of 100, the bank could loose $20 million or more Statistical Methodology Calculating Market Var Graphically : Two possible Value-at-Risk measures VAR-measure Probability no 1 distribution 1% 0 Expected Return profits VAR-measure return profits no 2 Calculating Market VAR (Normal density function) Example : A European bank currently has a $120 million in foreign exchange position. What is the one-day VAR at a confidence level of 95% (5% probability of occurrence) knowing that the current $US/€ is 1.20 ? Based on an historical sample, assume that the mean and standard deviation of the daily Δ% $US/€ exchange rate are (continuously compounding): r 0.10% and ( r ) 0.55% Assume further that the exchange rate return (Δ% $US/€) follows a normal probability distribution function Calculating Market VAR (Normal density function) • VAR measure no 1: In this case, the deviation is calculated from the mean. Therefore: We look for Probr VAR % 1 c%. Thus, Probz z c % Probz -1.65 5% r - r (r) -1.65 5% Prob Probr -1.65 (r) r 5% Probr -1.65 0.0055 0.0010 5% Calculating Market VAR (Normal density function) Example (cont’d): Prob r 0.008075 5% where VAR 0.8075 % The VAR can also be expressed in Euro: 1 VAR in Euro -0.008075 $US 120 million 1.20 $US/E VAR in Euro Euro 807 500 Calculating Market VAR (Normal density function) • VAR measure no 2: In this case, the VAR is calculated only using the standard deviation We look for Prob r VAR % 1 c%. Thus, Prob z z c % Prob z -1.65 5% Prob r -1.65 (r) 5% Prob r -1.65 0.0055 5% Calculating Market VAR (Normal density function) Example (cont’d): Prob r 0.009075 5% where VAR 0.9075 % The VAR can also be expressed in Euro: 1 VAR in Euro -0.009075 $US 120 million 1.20 $US/E VAR in Euro Euro 907 500 Calculating Market VAR (Normal density function) If the probability of occurrence is set at 1%, under the VAR measure no 2, we obtain : (2.33 0.0055 ) Euro 100 million Euro 1 281 500 The Var measure no 2 closely depends on the standard deviation measure: Var N ( x) -1 VAR and Confidence Level Probability of occurence High Low Less conservative: More conservative: Low capital charges High capital charges Increase in VAR value Often, the level of confidence at which the Var is calculated is closely related to the credit rating. For instance, a AA bank with an S&P one-year probability of default of 0.03% might choose a corresponding level of confidence of 99.97%. VAR and Diversification Effect Example : A European bank has a position in a USD bond with 6 year to maturity, annual coupon of 8% and a yield to maturity of 12%. The price of the bond is $835.54 while its duration is 4.88 years. The bank holds a $120 million (market value) position in that bond. Given a spot exchange rate of $1.20/€ 1: $120 million Euro 100 million 1.20 There are two sources of risk: the interest rate and the currency risk. However, we can calculate individual VAR’s, the first captures the interest risk while the second captures the currency risk VAR and Diversification Effect • Interest rate VAR: ΔP Δy D P (1 y) Δy rBond, y D (1 y) σ(Δy) σ(rBond, y ) D (1 0.12) • Assuming (y) 0.001 4.88 0.001 σ(rBond, y ) 0.43% 1.12 VAR and Diversification Effect • Interest rate VAR at 1% (normal density function, measure no. 2): (2.33 0.0043 $ 120 million ) VAR Euro 1001900 1.20 $ /Euro • Currency VAR at 1% (normal density function, measure no. 2, see previous example): - Euro 1 281 500 VAR and Diversification Effect Total VAR = Interest rate VAR + currency VAR = € 1001900 + € 1 281 500 = € 2 283 400 But that approach does not account for the diversification effect: The return on the total position is : rbond rbond,y rbond,ex %∆ in the bond price %∆ in the bond price resulting from shifts in resulting from shifts in $/€ US interest rate exchange rate VAR and Diversification Effect Thus: σ 2 (rbond ) σ 2 (rbond,y rbond,ex ) σ 2 (rbond ) σ 2 (rbond,y ) σ 2 (rbond,ex ) 2 rbond,y ,rbond,ex σ(rbond,y )σ(rbond,ex ) σ(rbond ) σ 2 (rbond ) σ 2 (rbond,y ) σ 2 (rbond,ex ) 2 rbond,y ,rbond,ex σ(rbond,y )σ(rbond,ex ) VAR and Diversification Effect Assuming r bond,y rbond,ex 0.5 rp 0.43%2 0.55%2 2 0.5 0.43% 0.55% 0.00007239 0.8508% Then, the total VAR is (normal density function): VAR in Euro 2.33 0.008508 Euro 100 million VAR in Euro Euro 1 982 418 VAR and Diversification Effect The gain from diversification are: € 2 283 400 - € 1 982 418 = € 300 982 No diversification Diversification effect effect Correlation effect When there are more than one risk factor involves, the correlation parameter, , is responsable for a risk diversification effect. To see that, rework the previous example with 0, 1 and 1 and look at what happens to the Var calculation. Multiperiod VAR Multiple-day VAR based on the one-day VAR : n-day VAR = n½ x one-day VAR Historical VAR The previous approach is based on the assumption of a normal probability distribution function May be vaild for diversified portfolios with independant risk factors Unlikely to be valid in many situations (data frequency, type of instruments, etc.) Historial VAR : 1) Identification of an historical sample for the risk factors, at a given frequency. 2) Application of those changes to the current risk factors. Computation of the corresponding position. 3) Sorting of the payoff vector and identification of the appropriate percentile on the empirical distribution. Historical VAR An example : A European bank has a position in a USD bond with 6 years to maturity, annual coupon of 8% while the yield to maturity is 12%. The price of the bond is $ 835.54 while its duration is 4.88 years. The bank holds a $120 million position in that bond. The exposure is : $120 million Euro 100 million 1.20 Current data : interest rate exchange rate duration 12% 1.20 4.88 Historical VAR The historical changes in the risk factors : n = 250 ∆% $/Euro ∆ 10-year rate January 2, 2006 - 0.521% 0.02% January 3, 2006 - 0.4405% 0.02% January 4, 2006 - 0.0256% -0.04% ▪ ▪ ▪ ▪ ▪ ▪ ▪ ▪ ▪ December 23, 2006 - 0.563% 0.03% December 24, 2006 - 1.004% -0.05% Historical VAR The return on this position : rbond rbond, y rbond,ex y D %$ /E 1 y 4.88 y %$ /E 1 0.12 On January 2, 2006 : rbond 4.357 0.0002 0.00521 rbond 0.6081% Euro variation in the bond position: Euro 100 million 0.00682 Euro 608 074 .108 Historical VAR The procedure is repeated for every day of the historical sample : Day Euro gains/losses on the bond January 2, 2006 - 608 074.108 January 3, 2006 - 507 812.391 January 4, 2006 149 032.398 ▪ ▪ ▪ ▪ ▪ ▪ December 23, 2006 - 694 284.502 December 24, 2006 - 786 181. 722 The vector of gains/losses is then sorted from the highest to the lowest : Rank Scenario Euro gains/losses on the bond 1 96 1 691 388.41 2 231 1 625 185.38 3 127 1 535 152.77 ▪ ▪ ▪ ▪ ▪ ▪ 238 191 - 1 360 529.24 ▪ ▪ ▪ 240 120 - 1 737 629.56 Historical VAR The 1% VAR is : Rank Scenario Euro gains/losses on the bond 238 191 - 1 360 529.24 The 5% VAR is : Rank Scenario Euro gains/losses on the bond 228 12 - 989 650.223 Historical VAR: Confidence Interval The historical Var measure corresponds to a quantile of the empirical distribution. Kandal and Stuart (1972, c.f. Hull 2007 p. 220) derive an estimate of the corresponding standard error: 1 q(1 q) std(HVar) f ( x) n where q : quantile of the distribution n : number of observations f ( x) : postulated density function of the HVar Assumed f (x) is the normal probability density function. Then, the confidence interval at a 95% confidence level is: prob HVar - std(Hvar) 1.96 HVar HVar std(Hvar) 1.96 95% Historical VAR: Confidence Interval Based on the previous example: 1 0.05 0.95 std(HVar) 81278 .69 1.773087 e 07 240 The value of f ( x) 1.756E 07 comes from Excel assuming that the mean payoff is 0 and its std is $595860.22 and using the following command: NORMINV(0.05,0,595860.22)=-980102.84 NORMDIST(-980102.84,0, 595860.22,False)=1.736E-07 Then, the confidence for the HVar is: $989650.22- 81278.49 1.96 HVar prob 95% $989650.22 81278.49 1.96 Historical VAR: Confidence Interval Remark: It is also possible to produce a confidence interval via bootstrapping: Since n 250 , it is possible to sample 250 000 scenarios of 1000 250 samples For each sample, calculate the HVar. With all the HVar estimates, rank them from the Highest to the lowest This provides an empirical distribution of Var based on 1000 HVar estimates. Using quantiles, we know that: prob HVar (at 97.5%) HVar HVar (at 2.5%) 95% Historical VAR Advantages : Method completely nonparametric, no distributional assumption Easy to apply Captures complex volatility and correlation effects Disavantages : Method is very sensitive to the choice of historical data Historical sample may not account for structural events In some cases, the data availability is a problem Backtesting the Var Example: N=600 observations 99%-day VAR. Exception occurs at 1% of the days 0.01 x 600 = 6 is the expected number of exceedances Actual number of exceedances is 8. Is this statistically significant? Define x, a bernoulli variable, the number of exceedances. Given the large number of observations, we can use the central limit theorem such that x pN z N(0,1) p(1 p)N Hypothesis testing: H o : x pN 6 H1 : x 8 86 2 z 0.82 0.01(1 0.01)600 2.43 At a 5% level of confidence, the critical value is 1.95. Therefore, we cannot not reject H0. Portfolio Toolkit Consider a portfolio composed of one stock (for a value of $6000) and one bond for (for a value of $4000). The value of the portfolio is: $10000 $6000 $4000 p sb p s b p s b p p p s s b b rp p s p b rp ws rs wb rb This last equation says that the return on the portfolio, r p , is a weighted average of the return on the stock, rs , and the bond, 2 rb. Notice that wi 1, with: i 1 $6000 $4000 ws 60% and wb 40% $10000 $10000 Portfolio Toolkit The return on a portfolio is given by: m m rp x i ri with x i 1 i 1 i 1 where ri , is the return on asset i, xi , the proportion of asset i in the portfolio and m , the number of assets in the portfolios. The corresponding variance of the portfolio is (say m 1000) 1000 1000 1000 V(rp ) x i (ri ) x i x j ij (ri ) (r j ) 2 i 1 i 1 j 1 i j Portfolio VAR Under the normality assumption: Var % p N -1 (rp ) but the term (r ) V(r ) requires the p p calculation of (1000 x 999)/2 terms ij. The problem is impractical! Portfolio VAR Consider the quadratic approximation: s 2s s F 0.5 2 F 2 F F where s and F are respectively the value of an asset and a market factor. Consider a portfolio of 500 stocks and 500 bonds: 500 500 rp w si rsi w bi rbi i 1 i 1 Portfolio VAR From portfolio theory, we know that: rsi 0 1 rm si Such that the second derivative is equal to 0. In this case, F=market index. For bonds, based on duration and convexity, we have: b i 2 bi b i y 0.5 y 2 y y 2 b i b i y 2bi 1 y 0.5 y 2 bi y b i y 2 b i rbi D m oiy 0.5C xr y 2 in this case, F=y. Portfolio VAR The 500 beta measures can be aggregated to generate the portfolio beta. The 500 duration and convexity measures can be aggregated to generate the portfolio duration and convexity. Therefore after a few manipulations: rp p rm Dmopy 0.5C xrp y 2 Portfolio VAR Under an historical simulation, the previous equation is applied directly. Under the assumption that the returns on the portfolio are drawn from a normal distribution, we can use the variance equation of the portfolio: 0.5 p (rm ) D m op (y ) 0.25C xrp (y ) 2 2 2 2 2 2 2 2 p D m opcov(rp,rm) (rp ) 2 p C xrp cov(rp,y 2 ) 2D m opC xrp cov(y, y 2 ) Portfolio Var Exemple: In class Pros and Cons of VAR measures Pros : Provide a risk measure that is easly aggregated Captures netting effects and risk diversification Consistent measure across risk factors, securities, etc. Direct translation into risk limits Facilitate the communication amongst managers Cons : Require a choice of methodology. Different approaches may lead to different VAR estimates VAR measures are not made to capture extreme, unlikely events. Other approaches are required Estimates only valid over short horizion Market Risk-VAR Deutsche Bank Market Risk Regulation Basel 1998 : Banks must have capital charges to cover exposures with respect to market risk within the trading book, and with respect to the credit risk involved with the counterparties in the trading book. The regulation affects both on- and off- balance sheets. Large banks are allowed to use their own-internal model based on the VAR methodology. Other banks must use the standard model proposed by BIS. Market Risk Regulation Capital charges : Credit risk of individual issuers K (10 day VAR, 1%) 4 (10 day specific risk VAR) May be changed by the regulator. K = 3 (minimum value) Total capital = 0.08 x (Credit risk RWA + Market risk RWA) where market risk RWA = 12.5 x capital charges above Market Risk Regulation: Basel 98 Capital charges for specific risks (%) Example : Internal Standardized Portfolio of 100 model approach 1 € bonds diversified AAA 0.26 1.6 across AA 0.77 1.6 industries A 1.00 1.6 BBB 2.40 1.6 BB 5.24 8 B 8.45 8 CCC 10.26 8 Source: Crouhy, Galai and Mark (2001) Market Risk Regulation The Group of 30 policy’s recommendations (1993) : Series of recommendations on price risk management for dealers, end-users of derivatives, legislators and regulators. Recommendation 2 : Mark-to-market process Recommendation 5 : Measuring market risk using a VAR metho. Recommendation 6 : Importance of stress testing Recommendation 8 : Banks should have an independant risk management function Stress Testing VAR Market Stress Risk testing Standard environments Extreme events Stress testing: assessments of potential losses under specific scenarios Scenarios from the Derivative Policy Group (1995) : Shifts in the yield curve ± 100 bps Twist risk of ± 25 bps Etc. Basel requires stress-testing in addition to Var calculation Stress Testing Example: Stress testing at BMO: Stress-testing with respect to interest rate at BMO Year 2003 Year 2002 Economic Value Earnings sensitivity Economic Value Earnings sensitivity sensitivity (12 months) sensitivity (12 months) 100 bps increase - $ 202.3 millions $ 10.8 millions - $ 152.7 millions $ 1.1 millions 100 bps decease $ 142.7 millions - $ 17.6 millions $ 123.8 millions - $ 0.1 millions 200 bps increase - $ 431.8 millions $ 431.8 millions - $ 354.1 millions - $ 3.9 millions 200 bps decease - $ 181.2 millions - $ 61.2 millions - $ 180 millions - $ 70.7 millions Source: Bank of Montreal, annual report, 2003