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INSTITUTE OF ECONOMIC STUDIES Faculty of Social Sciences of Charles University Value at Risk Lecturer’s Notes No. 3 Course: Portfolio Theory and Investment Management Teacher: Oldřich Dědek VIII. VALUE AT RISK MEASURE value at risk (VaR) is a method that attempts to provide a single number summarising the total market risk in a portfolio of financial assets (contrary to a huge number of partial risk measures such as “Greek letters analysis”) VaR is intended to express the expected maximum loss (worst loss) over a target horizon within a given confidence level 8.1 Building blocks of VaR - probability distribution rates of return on a portfolio R W /W0 can be summarised by a random ~ variable R whose future outcomes are subject to a given probability distribution ~ density function: f ( x) P( x R x ) ~ distribution function: F ( X ) P( R X ) X f ( x)dx mean (expected value): xf ( x)dx (first moment – location) variance (volatility): 2 (x ) 2 f ( x)dx (second moment – dispersion) Probability distribution f (x ) R C (1-C) % of all probability 0 C % of all probability - target horizon (holding period) target horizon is a period of time over which changes in portfolio’s value are measured it is a matter of subjective judgment; ideally it should correspond to the longest period needed for orderly portfolio liquidation 2 relationship between VaRs for different holding periods T-day VaR = 1-day VaR T formula is exactly true if changes in portfolio value on successive days have independent identical normal distribution with mean zero - confidence level confidence level is a percentage number that determines how far the portfolio should be protected against unexpected losses (i.e. Basel II uses 99 % confidence level) a confidence level C determines a cut-off point C (quantil) such that the area to its right captures C % of all probability and consequently the area to its left captures a residual (1-C) % of all probability ~ ~ P( R C ) C % or equivalently P( R C ) (1 C) % - definition of VaR VaR for a given confidence level is defined as the monetary loss relative to the mean; it is a difference between the expected change in the portfolio’s value and the change that corresponds to the cut-off rate of return determined by selected confidence level W0 … initial portfolio’s value (monetary amount) … expected rate of return (in percent) W0 …expected change in portfolio’s value (monetary amount) C … cut-off rate of return for given confidence level (in percent) CW0 … loss corresponding to the cut-off rate of return (monetary amount) value at risk = W0 CW0 ( C )W0 VaR is sometimes defined as the absolute monetary loss VaR W0 (1 C )W0 CW0 ♦ A portfolio has a value 100 mil USD and its average monthly return is 3.5 %. There is only a 5 % chance that the portfolio will fall by more than 1.7 % over a month. Therefore the value at risk for 95 % confidence level is 3 VaR(95 %) = [0.035 (0.017)] 100 mil 0.06 mil USD The amount 60 000 USD represents a monthly underperformance relative to the average outcome that happens with 5 % probability. ♦ VaR serves a number of purposes - information reporting (senior managers are informed about the risks run by trading and investment operations) - resource allocation (setting position limits for trades) - performance evaluation (benchmark for proper evaluation of risk exposures) - prudential regulation (maintaining minimum levels of capital reserves against financial risks) 8.2 Historical approach to the VaR measurement [see also the excel file dpe02 Value at Risk] historical approach assumes that the observed frequency distribution approximates quite well the true probability distribution for a given confidence level C one can find a cut-off point C such that the number of occurrences exceeding this value is C % (or the number of occurrences below this value is (1-C) %) VaR is computed according to definition Frequency distribution and VaR 140 95 0.012 120 100 80 5 % of occurrence 60 40 20 0 -0,035 -0,025 -0,015 -0,005 0,005 4 0,015 0,025 0,035 ♦ An investment company is earning an average daily income 5.1 mil USD. The track record of 260 previous daily observations indicates that the number of cases showing a daily loss 9.6 mil USD and more accounts for 10 % of all observations. Under assumption that the observed frequency distribution realistically describes the future distribution of earning possibilities, the value at risk at 90 % confidence level is following VaR(90 %) = 5.1 – (-9.6) = 14.7 mil USD ♦ disadvantages of historical approach: - possible lack of a representative number of observations - past behaviour need not be a good predictor of future behaviour 8.3 Parametric approach to the VaR measurement [see also the excel file dpe02 Value at Risk] VaR computation can be simplified considerably if the probability distribution can be assumed to be normal - normal distribution of rates of return normal distribution (bell-shaped distribution) is fully characterised by its first two moments: mean and volatility ~ P( N ) (x )2 exp dx 2 2 2 1 66 % of distribution is between and 95 % of distribution is between 2 and 2 2 5 2 - standard normal distribution SND is a normal distribution with a mean of zero and variance of unity ~ every normal distribution N with parameters and can be changed into a standard ~ normal distribution using a transformation ~ ~ R - relationship between cut-off points ~ if C is the cut-off point of a standard normal distribution then C C is the ~ corresponding cut-off point of the associated normal distribution N (for convenience C is defined as positive number) ~ N ~ ~ 1 C P( C ) P C P( N C ) ~ P( N C ) - computation formula for the parametric VaR VaR W0 CW0 [( ( C )]W0 C W0 if the time horizon for which the VaR is measured differs from the unitary horizon over which the volatility is measured then VaR CW0 t Most frequent cut-off points Confidence level 90 % Cut-off point 1.282 95 % 1.645 99 % 2.326 disadvantages of normal distribution: - perfect symmetry of normal distribution does not fit some types of risky profiles (particularly if fails to reflect asymmetries of credit risk) - „thin tails“ of normal distribution tend to underestimate probability of occurrence of extreme outcomes 6 ♦ Compute the VaR (for confidence level 99 % and holding period 10 days) for each of the two shares that display following properties: Share A Value (mil USD) 10 One day volatility (%) 2 B 5 1 The value of the cut-off point for chosen confidence level is 99 2.326 . Hence VaR(A) 2.326 0.02 10 10 1.47 mil USD VaR(B) 2.326 0.01 5 10 0.37 mil USD ♦ diversified and undiversified VaR ~ if rates of return of individual risky assets Ri are normally distributed than the rate of ~ return of a portfolio R P that is composed of these assets is also normally distributed (a linear combination of normal distributions is also normal distribution) N N W ~ ~ R P i Ri , i i , W P Wi WP i 1 i 1 formula for the portfolio’s VaR can be rewritten in terms of individual VaRs VaR P C PWP0 WP0 N i 1 i 1 N 2 C N i2 i2 2 C2 i j i j ij i 1 j i 1 iWi 0 2 C iWi 0 C jW j0 ij N 2 C N i 1 j i 1 N VaR i 1 N i 2 N 2 N VaR VaR i 1 j i 1 i j ij portfolio VaR that takes into account a correlation structure among constituent assets is called diversified VaR (notation is DVaR) DVaR of a portfolio can be compared with undiversified VaR of the portfolio (notation is UVaR) which is the sum of VaRs of constituent assets 7 N UVaR P VaR i i 1 UVaR coincides with DVaR only in case of perfect positive correlations among risky assets ♦ Compare the DVaR and UVaR for a two-share portfolio from the previous example if the correlation coefficient that measures correlation between the two shares is 0.3. UVaR P 1.47 0.37 1.84 mil USD DVaR P 1.47 2 0.37 2 2 1.47 0.37 0.3 1,62 mil USD The diversification effect decreases the VaR of the portfolio by 1.84 – 1.62 = 0.22 mil USD. ♦ 8.4 Monte Carlo simulation [see also the excel file dpe02 Value at Risk] Monte Carlo method repeatedly simulates a random process for the financial variable the method is extensively used where complex financial instruments cannot be priced by applying analytical formulas - choice of a particular stochastic model a variable follows a stochastic process if its values change over time in an uncertain way Markov property means that the only present value of a variable is relevant for predicting the future Wiener process (Brownian motion) is a particular type of Markov stochastic process with a mean change of zero and a variance rate of unity geometric Brownian motion (for a price of share) S t 1 S t t t t St t … expected return per unit of time (expected component or drift) t t … stochastic component ( t can be seen as random drawing from standard normal distribution) t … volatility per unit of time 8 Monte Carlo simulation is prone to model risk (if the stochastic process chosen in unrealistic so will be the estimated VaR - creating random numbers a computerised random number generator uses a uniform distribution over the interval [0,1] a random drawing form uniform distribution must be transformed into a random drawing from standard normal distribution (by using the inverse function to the distribution function of the standard normal distribution) 1 2 0 1 1 0 2 0 0 0 random numbers are generated as many times as necessary, number of iterations should reflect the trade-off between accuracy and computation cost - computation of VaR random number are used for creating a simulated frequency distribution next step in computing VaR is identical with the historical approach to the VaR measurement Cholesky factorisation Cholesky factorisation is a method for adding an observed correlation structure into the simulated random process random numbers that simulate stochastic behaviour of individual components of a portfolio must preserve a given correlated behaviour of these components ~1 ,~2 ,,~N a set of original random variables that are generated by a computer program (mutually independent standard normal distributions) 9 ~1 , ~2 ,, ~N a set of transformed random numbers that respect correlation structure among components of a multi-asset portfolio Cholesky transformation is a triangular system of equations: ~ a ~ 1 11 1 ~2 a 21~1 a 22~2 .................................... ~N a N 1~1 a N 2~2 a NN ~N where coefficients a ij are found in an iterative way that uses following properties: 1 var( ~i ) var( ai1~1 aii~i ), i 1,, N ij cov(~i , ~j ) cov( ai1~1 aii~, a j1~1 a jj~ j ), i j var(~i ) 1, cov(~i ,~ j ) 0 i 1,, N , i j computation of Cholesky factors for two-asset portfolio: 1 var( ~1 ) var( a11~1 ) a112 var(~1 ) a112 a11 1 2 2 1 var(~2 ) var( a21~1 a22~2 ) a21 a22 cov(~1 , ~2 ) cov( a11~1 , a 21~1 a 22~2 ) cov( a11~1 , a 21~1 ) cov( a11~1 , a 22~2 ) a11a 21 var(~1 ) a11a 21 a 21 2 a 22 1 a 21 1 2 random numbers that preserve the correlation structure are the product of the following transformation ~ ~ 1 1 ~2 ~1 1 2 ~2 8.5 Linear and nonlinear risk partial approach to the VaR measurement (so called particle finance) tries to decompose the total risk exposure to individual risk components linear products – a linear relationship exists between movements in the underlying risk factor and the profit and loss profile of the dependent variable nonlinear products – the relationship between the profit and loss profile of the dependent variable can be represented by a curve (smooth versus kinked non-linearity) 10 value at risk of linear products - relationship between the price of forward contract and the price of the underlying asset (see the handout Futures) f 1 (r d )T S 1 rT Linear risk of a forward contract f f1 S f0 S0 S1 - relationship between the percentage change in the price of zero-coupon bond and the yield to maturity P 1 (1 r ) T dP d (1 r ) T P 1 r delta of the product is the sensitivity of the value of the financial instrument V with respect to the change in the price of the underlying risky factor F V F linear products have constant delta hence V F var(V ) var( F ) 2 var(F ) V F if F is normally distributed then V is also normally distributed and VaR of the linear product is VaR(V ) C V ( C F ) VaR(F ) the formula for the linear VaR is frequently used as an approximation of a true (nonlinear) VAR 11 ♦ A bond portfolio has a value of 10 million USD and modified duration of 3.8 years. Interest rates are currently 10 % and there is a 5 % chance that interest rates will go above 11.2 % at the end of the year. What would be the VaR for the portfolio at 95 % confidence level? VaR 3.8 (0.112 - 0.1) 10000000 456000 USD ♦ value at risk of nonlinear products examples: - relationship between the price of coupon bond and the yield to maturity (duration-convexity approximation) P P D 1 r K r 2 1 r 2 - relationship between the price of stock option and the price of an underlying asset (delta-gamma approximation) C S 12 S 2 Nonlinear kinked risk of an option contract C 0 X S if risky factor F is normally distributed then the value of the product V need not be normally distributed and the parametric approach to the VaR assessment is only an approximation of true risk exposure the formula of nonlinear VaR can be derived in the following way V F 12 F 2 12 var(V ) var( F 12 F 2 ) 2 var(F ) ( 12 ) 2 var(F 2 ) 2( 12 ) cov(F , F 2 ) 2 var(F ) 12 ( var(F )) 2 the term cov(F , F 2 ) is zero if variable F is normally distributed (all odd moments are zero for symmetrical probability distribution) it can be shown for normal distribution that var(F 2 ) 2 var 2 (F ) VaR of smooth nonlinear product VaR (V ) C V V CV 2 F2 12 ( F2 ) 2 ♦ A British exporter opened a long position in sterling currency put options to hedge future conversion of dollar revenues into pounds. The option portfolio has the actual value of 1200000 GBP. The delta of one currency option is 0.44 and the gamma is 3.1. The exchange rate dollar-sterling volatility is 3.3 %. What is the VaR of the option portfolio at 95 % confidence level? VaR 1.654 1200000 0.44 2 0.0332 0.5 3.12 0.0334 29047 GBP ♦ computation of VaR in case of kinked (broken) nonlinearity must be approximated by simulation techniques like Monte Carlo simulation 13 IX. CREDIT RISK MEASUREMENT credit risk refers to potential losses that are experienced when i) counterparties are unwilling or unable to fulfil their contractual obligations (i.e. repayment of a bond or coupons of the bond, repayment of loan or charged interest); ii) debtors are downgraded by credit agencies iii) credit spreads are widened due to external events beyond debtor’s control (i.e. increased political uncertainty, a higher macroeconomic instability) rating agencies provide ratings that describe the creditworthiness of debtors Moody’s: Aaa, Aa (Aa1, Aa2, Aa3), A (A1, A2, A3), Baa, Ba, B, Caa, Ca, C, D S&P: AAA, AA (AA+, AA-), A (A+, A-), BBB, BB, B, CCC, CC, C, D 9.1 Expected default loss on bonds credit spread is an excess of the yield of the risky bond over the yield of the risk-free bond zero rates extracted from risk-free (AAA) government bond yields z1 , z 2 ,, zT zero rates extracted form bond yields of given credit rating X: 1X , 2X ,, TX , where X = AA, A, BBB, and so on spread curve: 1X 1X z1 ,, TX TX zT Zero yield curves for different classes of credit risk BBB Yield BBB A AA AA AAA Maturity 14 empirical patterns: - the spread increases as the rating declines - the spread tends to increase faster with maturity for lower credit ratings than for higher credit ratings present value of cost of default is assumed to be equal to the excess of the price of a bond of a given credit rating over the price of the risk-free bond that has the same maturity and pays the same coupon price of risky zero-coupon T-year bond: P M (1 TX ) T price of risk-free zero-coupon T-year bond: Q M (1 zT ) T loss from default: L (1 zT ) T QP 1 1 (1 ( zT TX )) T X T Q (1 T ) 1 (1 T ( zT TX )) T TX ♦ A five year zero-coupon Treasury bond yields 5 % and five year yield of zero-coupon corporate bond yields 6 %. Q 100 100 78.35, P 74.73 5 (1 0.05) (1 0.06) 5 Present value of loss from default is L 78.35 74.73 4.62 % 5 (0.06 0.05) 5 % 78.35 ♦ extracting default losses from swap rates swap quotes can be used for extracting default losses on the assumption that LIBOR curve is the risk-free curve fixed rate coupon Investor LIBOR Swap dealer by holding the bond and buying the swap an investor is still exposed to the credit risk 15 that the bond may default but is compensated for that risk by receiving an extra yield over the risk-free rate ( = coupon – fixed rate) the present value of that compensation can be interpreted as the loss from default ♦ An investor owns a fixed-rate corporate bond that is currently worth par and pays a coupon of 6 % per year. The LIBOR yield curve is flat at 4 % with semiannual compounding. The swap fixed rate is 4.5 % paid semiannually. The compensation for the bond’s credit risk is thus 150 basis points for five years paid semiannually. 10 1 100 0.015 1 2 1 0.0665 6.65 % loss from default = 100 t 1 (1 2 0.045) t ♦ Investor has the opportunity to buy five year bond with coupon 7.5 % together with the five year swap paying fixed rate 6 %. The yield curve for zero rates is flat at the level 4.5 %. 9.2 Risk-neutral probability of default risk-neutral environment means that expected return required by all investors on all investment is the risk-free interest rate (consequence of the efficient market hypothesis) risk-neutral probability is the probability that a risky bond will default during the its life in risk-neutral environment (as opposed to empirical probability of default) - expected pay-off of the bond at its maturity qT R M (1 qT ) M qT … risky-neutral probability that the bond will default at maturity T R… recovery rate (proportion of the claimed amount in the event of a default) M … face value of the bond - no arbitrage condition in a risk-neutral world P q R M (1 qT ) M M T Q (qT R 1 qT ) T (1 T ) (1 zT ) T - probability of default 16 qT (1 TX ) T T TX 1 QP 1 1 1 R P 1 R (1 zT ) T 1 R ♦ A ten year corporate bond rated BBB has a spread 220 bp over the ten year Treasury bond. Historical recovery rate for bonds of this risky class is 40 %. The risk-neutral probability that the bond defaults at maturity is therefore q10 10 0.022 0.37 37 % 1 0.4 ♦ loss given default (LGD) is an estimation of the percentage loss in the event of default LGD = 1 – R T TX expected loss = qT (1 R) (1 qT ) 0 qT LGD LGD T T 1 R ♦ Maturity 1 Riskless return (%) 5.00 Risky return (%) 5.25 Spread (pcp) 0.25 Expected loss (%) 0.25 2 5.00 5.50 0.50 1.00 3 5.00 5.70 0.70 2.10 ♦ 9.3 Historical probabilities of default historical (empirical) probabilities are estimated from historical data and they show default experience of bonds through time that started with a certain credit rating marginal mortality rates (MMR) are probabilities that a bond of a given credit rating will default in a given year of its life MMR 1X = total value of grade X bonds defaulting in year 1 of issue total value of grade X bonds outstanding in year 1 of issue MMR X2 = total value of grade X bonds defaulting in year 1 of issue total value of grade X bonds outstanding in year 1 of issue 17 MMRs are computed as a weighted average of yearly values over some sample of years MMR Xt MMR Xt, j w j , w j are relative issue sizes in different years j Rating Maturity 1 2 3 … … 15 AAA 0.00 0.00 0.04 … … 0.67 AA 0.01 0.04 0.10 … … 1.39 A 0.04 0.12 0.21 … … 2.59 BBB 0.24 0.55 0.89 … … 6.85 … … … … … … … CCC 25.26 34.79 42.16 … … 66.12 MMR 3BBB 0.89 probability that the bond that was issued with the rating BBB will default in the third year of its life is 0.89 % MMRs of high quality bonds tend to increase with the passage of time while MMRs of low quality bonds tend to decrease with the passage of time survival rates (SR) are probabilities that the bond of a given credit rating will not default in a given year of its life SR Xt 1 MMR Xt cumulative mortality rates (CMR) are probabilities that the bond of a given credit rating will default over a given period T CMR TX 1 SR 1X SR X2 SR TX historical default probabilities are significantly less than those derived from an analysis of bond prices explanation: - investors want to be compensated for other types of risks (i.e. liquidity risk, unforeseen bad scenarios, etc.) - risk-neutral environment may not correspond to the actual market conditions (an expected return should be discounted by a higher than a risk-free rate) 18 9.4 Credit at Risk [see also the excel file dpe03 Credit at Risk] credit at risk (CaR) denotes the value-at-risk approach applied to the measurement of credit risk application of parametric formula CaR C W 0 , in which denotes volatility of an underlying asset duet to the credit risk qualifications: credit returns are typical with skewed returns (higher probability of downgrading than upgrading) and a fat downside tail (large losses from extreme credit events) normal distribution need not be the appropriate description of credit events and the parametric formula is only an approximate measure of the credit risk skewed distribution with fat tail symmetrical distribution with thin tails CreditMetrics is a registered trademark of J. P. Morgan calculation of credit risk exposure of each individual bond a) credit transition matrix contains historical probabilities of credit migration over a time horizon (mostly one year) probabilities that a bond with a given credit quality changes its credit ratings (upgrade, downgrade, remains the same) BBB AAA AA A BBB BB B CCC D 0.02 0.33 5.95 86.93 5.30 1.17 0.12 0.18 b) calculation of a zero credit curve for each credit rating level application of a standard bootstrap method across the credit rating spectrum 19 Rating 1 2 3 A 6,25 6,75 7,5 BB 9,5 9,7 10,0 c) determination of bond values for each credit migration ♦ A three year BBB bond pays coupon of 8 %. If the bond is upgraded to the rating A, its price will be PA 8 8 108 101.49 2 1.0625 1.0675 1.0753 If the bond is downgraded to the rating BB, its price will be PBB 8 8 108 3 95.1 2 1.095 1.097 1.1 In a similar manner the bond value for each credit migration can be determined BBB AAA AA A BBB BB B CCC D 104.0 103.0 101.5 100.0 95.1 87.0 75.0 58.0 ♦ d) measuring the expected value and volatility of each bond due to credit risk E ( P) p AAA PAAA p AA PAA pCCC PCCC p D PD 2 ( P) p AAA ( PAAA E ( P)) 2 p D ( PD E ( P)) 2 p X … probability of changing the rating to the grade X PX … bond price at the rating X a bond value for the rating D (default) can be obtained from empirically observed recovery rates e) calculation of the bond CaR given a confidence level C the standard formula of VaR (now called CaR) can be applied CaR C P P 0 20 ♦ Compute the credit at risk of a bond portfolio whose nominal value is 20 mil USD and standard deviation of its price fluctuation is 2.68 %. The confidence level is 95 %. CaR 1.645 0.0268 20 000 000 881720 USD ♦ calculation of credit risk exposure of a two-bond portfolio a) calculation of portfolio’s end-values preparing a table of all possible combined values of the two bond portfolio at year end combined value is the sum of individual values number of possible combined values is N×N, where N is the number of credit classes First bond Second bond AAA AA .......... CCC D P1(AAA) P1(AA) .......... P1(CCC) P1(D) .......... .......... AAA P2(AAA) AA P2(AA) ……. ………. CCC P2(CCC) .......... D P2(D) .......... 210.81 209.48 209.50 .......... 208.16 .......... .......... .......... .......... b) calculation of joint probabilities of portfolio’s end-values First bond Second bond AAA p2(AAA) AA p2(AA) ……. ………. AAA AA .......... CCC D p1(AAA) p1(AA) .......... p1(CCC) p1(D) .......... .......... 0,0000 0,0000 0,0001 .......... 0,0005 .......... .......... .......... .......... CCC p2(CCC) .......... D p2(D) .......... 21 0,0012 0,0013 0,0018 0,0019 if probabilities of credit migration of the two bonds are independent on each other then joint probabilities of portfolio’s end-values are simply the product of individual probabilities c) measuring the expected value and volatility of each bond due to credit risk E ( P Q) p AAA, AAA ( PAAA Q AAA ) p AAA, AA ( PAAA Q AA ) p D , D ( PD QD ) 2 ( P Q) p AAA, AAA ( PAAA QAAA E( P Q)) 2 pD,D ( PD QD E( P Q)) 2 CaR C ( P 0 Q 0 ) d) estimating correlation between bond prices independent transition probabilities are not likely for issues from the same or related industries or if issues are exposed to common macroeconomic factors correlations are estimated by means of multifactor models of stock returns for individual borrowers (stock returns are observed if issuer’s stock is publicly traded) ♦ An issuer of A-rated bond is a universal bank (sensitive to the business climate in banking and insurance industries) and an issuer of BB-rated bond is a chemical company (sensitive to the business climate in the chemical industry). Estimated sensitivities of the two issues to industries’ returns are following R A 0.15 RBANK 0.74 RINS RBB 0.9 RCHEM Estimations of inter-industries’s correlations are following CHEM , BANK 0.08 CHEM , INS 0.16 Estimated proxy correlation between the bond issues is following ( A, BB ) (0.15RBANK 0.74 RINS ,0.9 RCHEM ) 0.15 0.9 ( RBANK , RCHEM ) 0.74 0.9 ( RINS , RCHEM ) 0.15 0.9 0.08 0.74 0.9 0.16 0.1174 ♦ 22 e) computing credit rating thresholds it is assumed that there is a series of threshold levels for changes in firm’s asset value that determine the firm’s credit rating Credit rating thresholds for the BB issue D B CCC AD ACCC BBB BB AB ABBB Bb AA A AA AAA AAA AAAA AX is a threshold rate of return that triggers the change in rating (downgrade or upgrade) W W AD the BB bond will default AD W W ACCC the BB bond will be downgraded to the grade CCC AB W W ABBB the BB bond will retain its rating AAA W W AAAA the bond will be upgraded to the grade AA AAAA W W the bond will be upgraded to the highest grade AAA threshold levels can be easily determined under assumption that the probabilities of ~ underlying changes in the firm’s value W / W follow a normal distribution N ( , ) W ~ N ( , ) has a normal distribution (with mean and volatility ) W Z W / W follows a standard normal distribution (with mean zero and unitary volatility) ( ) P( z ) is the cumulative standard normal distribution function finding the threshold level for the default p D PW W AD P Z AD A P Z D PZ D D 23 D corresponds to the area under the curve of standard normal distribution whose cumulative probability is equal to the probability of default D can be found from the inverse function to the standard normal distribution in a similar way the threshold level for the grade CCC can be determined W pCCC P AD ACCC P AD Z ACCC W A A P D Z CCC ( CCC ) ( D ) ♦ A bond is currently rated at level A. Threshold levels for credit migration can be determined in the following way: Migration probabilities Cumulative probabilities Threshold levels D CCC B BB BBB A AA AAA 0.06 0.01 0.26 0.74 5.52 91.05 2.27 0.09 0.06 0.07 0.33 1.07 6.59 97.64 99.91 100 -3.24 -3.19 -2.72 -2.30 -1.51 1.98 3.12 ♦ e) computation of correlated joint probabilities joint probabilities can be found by integrating the density function of bivariate normal distribution number of computations = (number of rating grades)2 P AX W1 W1 AX 1 , AY W2 W2 AY 1 X x X 1 , Y y Y 1 X 1 Y 1 x y f ( x, y)dxdy 1 X 2 Y where f ( x, y) 1 21 2 ( x 1 ) 2 ( x 1 )( y 2 ) ( y 2 ) 2 1 exp 2 2 2 1 2 22 1 2 2(1 ) 1 24 f) CaR of a large bond portfolio it can be shown that the risk of a portfolio of N bonds depends on the risk of each pairwise combination as well as the risk of each bond individually N var( P1 PN ) var( Pi ) 2 cov( Pi , Pj ) i 1 i j var( Pi Pj ) var( Pi ) var( Pj ) 2 cov( Pi , Pj ) N 1 var( P1 PN ) N N var( Pi Pj ) (n 2) var( Pi ) i 1 j i 1 i 1 References Anson M. P.: Credit Derivatives, Frank J. Fabozzi Associates, 1999. Butler C.: Mastering value at risk, Prentice Hall, 1999. Jorion P.: Value at Risk, McGraw-Hill, 1995. Saunders A.: Credit Risk Measurement. John Wiley & Sons, Inc., 1999. 25