Financial Risk: Credit Risk, Lecture 1

Transcription

1 Financial Risk: Credit Risk, Lecture 1 Alexander Herbertsson Centre For Finance/Departent of Econoics School of Econoics, Business and Law, University of Gothenburg E-ail: Alexander.Herbertsson@econoics.gu.se Financial Risk, Chalers University of Technology, Göteborg Sweden Noveber 13, 2012 Alexander Herbertsson (Univ. of Gothenburg) Financial Risk: Credit Risk, Lecture 1 Noveber 13, / 36

2 Content of lecture Short discussion of the iportant coponents of credit risk Study different static portfolio credit risk odels. Discussion of the binoial loss odel Discussion of the ixed binoial loss odel Study of a ixed binoial loss odel with a beta distribution Study of a ixed binoial loss odel with a logit-noral distribution A short discussion of Value-at-Risk and Expected shortfall Alexander Herbertsson (Univ. of Gothenburg) Financial Risk: Credit Risk, Lecture 1 Noveber 13, / 36

3 Definition of Credit Risk Credit risk the risk that an obligor does not honor his payents Exaple of an obligor: A copany that have borrowed oney fro a bank A copany that has issued bonds. A household that have borrowed oney fro a bank, to buy a house A bank that has entered into a bilateral financial contract (e.g an interest rate swap) with another bank. Exaple of defaults are A copany goes bankrupt. As copany fails to pay a coupon on tie, for soe of its issued bonds. A household fails to pay aortization or interest rate on their loan. Alexander Herbertsson (Univ. of Gothenburg) Financial Risk: Credit Risk, Lecture 1 Noveber 13, / 36

4 Credit Risk Credit risk can be decoposed into: arrival risk, the risk connected to whether or not a default will happen in a given tie-period, for a obligor tiing risk, the risk connected to the uncertainness of the exact tie-point of the arrival risk (will not be studied in this course) recovery risk. This is the risk connected to the size of the actual loss if default occurs (will not be studied in this course, we let the recovery be fixed) default dependency risk, the risk that several obligors jointly defaults during soe specific tie period. This is one of the ost crucial risk factors that has to be considered in a credit portfolio fraework. The coing two lectures focuses only on default dependency risk. Alexander Herbertsson (Univ. of Gothenburg) Financial Risk: Credit Risk, Lecture 1 Noveber 13, / 36

5 Portfolio Credit Risk is iportant Modelling dependence between default events and between credit quality changes is, in practice, one of the biggest challenges of credit risk odels., David Lando, Credit Risk Modeling, p Default correlation and default dependency odelling is probably the ost interesting and also the ost deanding open proble in the pricing of credit derivatives. While any single-nae credit derivatives are very siilar to other non-credit related derivatives in the default-free world (e.g. interest-rate swaps, options), basket and portfolio credit derivative have entirely new risks and features., Philipp Schönbucher, Credit derivatives pricing odels, p Epirically reasonable odels for correlated defaults are central to the credit risk-anageent and pricing systes of ajor financial institutions., Darrell Duffie and Kenneth Singleton, Credit Risk: Pricing, Measureent and Manageent, p Alexander Herbertsson (Univ. of Gothenburg) Financial Risk: Credit Risk, Lecture 1 Noveber 13, / 36

6 Portfolio Credit Risk is iportant Portfolio credit risk odels differ greatly depending on what types of portfolios, and what type of questions that should be considered. For exaple, odels with respect to risk anageent, such as credit Value-at-Risk (VaR) and expected shortfall (ES) odels with respect to valuation of portfolio credit derivatives, such as CDO s and basket default swaps In both cases we need to consider default dependency risk, but......in risk anageent odelling (e.g. VaR, ES), the tiing risk is ignored, and one often talk about static credit portfolio odels,...while, when pricing credit derivatives, tiing risk ust be carefully odeled (not treated here) The coing two lectures focuses only on static credit portfolio odels, Alexander Herbertsson (Univ. of Gothenburg) Financial Risk: Credit Risk, Lecture 1 Noveber 13, / 36

7 Literature The slides for the coing two lectures are rather self-contained, except for soe results taken fro Hult & Lindskog. The content of the lecture today and the next lecture is partly based on aterials presented in Lecture notes by Henrik Hult and Filip Lindskog (Hult & Lindskog) Matheatical Modeling and Statistical Methods for Risk Manageent, however, these notes are no longer public available, instead see e.g the book Hult, Lindskog, Haerlid and Rehn: Risk and portfolio analysis - principles and ethods. Quantitative Risk Manageent by McNeil A., Frey, R. and Ebrechts, P. (Princeton University Press) Credit Risk Modeling: Theory and Applications by Lando, D. (Princeton University Press) Alexander Herbertsson (Univ. of Gothenburg) Financial Risk: Credit Risk, Lecture 1 Noveber 13, / 36

8 Static Models for hoogeneous credit portfolios Today we will consider the following static odes for a hoogeneous credit portfolio: The binoial odel The ixed binoial odel To understand ixed binoial odels, we give a short introduction of conditional expectations After this we look at two different ixed binoial odels. We also shortly discuss Value-at-Risk and Expected shortfall Next lecture we consider a ixed binoial odel inspired by the Merton fraework. Alexander Herbertsson (Univ. of Gothenburg) Financial Risk: Credit Risk, Lecture 1 Noveber 13, / 36

9 The binoial odel for independent defaults Consider a hoogeneous credit portfolio odel with obligors, and where we each obligor can default up to fixed tie point, say T. Each obligor have identical credit loss at a default, say l. Here l is a constant. Let X i be a rando variable such that { 1 if obligor i defaults before tie T X i = 0 otherwise, i.e. if obligor i survives up to tie T We assue that the rando variables X 1, X 2,... X are i.i.d, that is they are all independent with identical distribution. Furtherore P [X i = 1] = p so that P [X i = 0] = 1 p. The total credit loss in the portfolio at tie T, called L, is then given by L = lx i = l X i = ln where N = i=1 i=1 thus, N is the nuber of defaults in the portfolio up to tie T. Since l is a constant, we have P [L = kl] = P [N = k], so it is enough to study the distribution of N. Alexander Herbertsson (Univ. of Gothenburg) Financial Risk: Credit Risk, Lecture 1 Noveber 13, / 36 i=1 X i (1)

10 The binoial odel for independent defaults, cont. Since X 1, X 2,...X are i.i.d with with P [X i = 1] = p we conclude that N = i=1 X i is binoially distributed with paraeters and p, that is N Bin(, p). This eans that P [N = k] = ( ) p k (1 p) k k Recalling the binoial theore (a + b) = ( k=0 k) a k b k we see that ( ) P [N = k] = p k (1 p) k = (p + (1 p)) = 1 k k=0 k=0 proving that Bin(, p) is a distribution. Furtherore, E[N ] = p since [ ] E[N ] = E X i = i=1 E[X i ] = p. i=1 Alexander Herbertsson (Univ. of Gothenburg) Financial Risk: Credit Risk, Lecture 1 Noveber 13, / 36

11 The binoial odel for independent defaults, cont The portfolio credit loss distribution in the binoial odel bin(50,0.1) probability nuber of defaults lexander Herbertsson (Univ. of Gothenburg) Financial Risk: Credit Risk, Lecture 1 Noveber 13, / 36

12 The binoial odel for independent defaults, cont. The binoial distribution have very thin tails, that is, it is extreely unlikely to have any losses (see figure). For exaple, if p = 5% and = 50 we have that P [N 8] = 1.2% and for p = 10% and = 50 we get P [N 10] = 5.5% The ain reason for these sall nubers (even for large individual default probabiltes) is due to the independence assuption. To see this, recall that the variance of a rando variable Var(X) easures [ the degree of the deviation of X around its ean, i.e. Var(X) = E (X E[X]) 2]. Since X 1, X 2,...X are independent we have that ( ) Var(N ) = Var X i = i=1 Var(X i ) = p(1 p) (2) i=1 where the second equality is due the independence assuption. Alexander Herbertsson (Univ. of Gothenburg) Financial Risk: Credit Risk, Lecture 1 Noveber 13, / 36

13 The binoial odel for independent defaults, cont. Furtherore, by Chebyshev s inequality we have that for any rando varialbe X, and any c > 0 it holds P [ X E[X] c] Var(X) c 2 So if p = 5% and = 50 we have that Var(N ) = 50p(1 p) = and and E[N ] = 50p = 2.5 iplying that having say, 6 ore, or less losses than expected, is saller or equal than 6.6%, since by Chebyshev s inequality P [ N 2.5 6] = 6.6% 36 Hence, the probability of having a total nuber of losses outside the interval 2.5 ± 6, i.e. outside the interval [0, 8.5], is saller than 6.6%. In fact, one can show that the deviation of the average nuber of defaults in the portfolio, N, fro the constant p (where p = E[ ] N ) goes to zero as. Thus, N converges towards a constant as (the law of large nubers). Alexander Herbertsson (Univ. of Gothenburg) Financial Risk: Credit Risk, Lecture 1 Noveber 13, / 36

14 Independent defaults and the law of large nubers By applying Chebyshev s inequality to the rando variable N together with Equation (2) we get [ ] N P p ε Var( N ) 1 Var(N ε 2 = 2 ) p(1 p) p(1 p) ε 2 = 2 ε 2 = ε 2 and we conclude that P [ N p ε] 0 as. Note that this holds for any ε > 0. This result is called the weak law of large nubers, and says that the N average nuber of defaults in the portfolio, i.e., converges (in probability) to the constant p which is the individual default probability. One can also show the so called strong law of large nubers, that is [ ] N P p when = 1 and we say that N converges alost surely to the constant p. In these lectures we write N p to indicate alost surely convergence. Alexander Herbertsson (Univ. of Gothenburg) Financial Risk: Credit Risk, Lecture 1 Noveber 13, / 36

15 Independent defaults lead to unrealistic loss scenarios We conclude that the independence assuption, or ore generally, the i.i.d assuption for the individual default indicators X 1, X 2,... X iplies that the average nuber of defaults in the portfolio N converges to the constant p alost surely. Given the recent credit crisis, the assuption of independent defaults is ridiculous. It is an epirical fact, observed any ties in the history, that defaults tend to cluster. Hence, the fraction of defaults in the portfolio N will often have values uch bigger than the constant p. Consequently, the epirical (i.e. observed) density for N will have uch ore fatter tails copared with the binoial distribution. We will therefore next look at portfolio credit odels that can produce ore realistic loss scenarios, with densities for N that have fat tails, and which not iplies that the average nuber of defaults in the portfolio N converges to a constant with probability 1, when. Alexander Herbertsson (Univ. of Gothenburg) Financial Risk: Credit Risk, Lecture 1 Noveber 13, / 36

16 Conditional expectations Before we continue this lecture, we need to introduce the concept of conditional expectations Let L 2 denote the space of all rando variables X such that E [ X 2] < Let Z be a rando variable and let L 2 (Z) L 2 denote the space of all rando variables Y such that Y = g(z) for soe function g and Y L 2 Note that E[X] is the value µ that iniizes the quantity E [ (X µ) 2]. Inspired by this, we define the conditional expectation E[X Z] as follows: Definition of conditional expectations For a rando variable Z, and for X L 2, the conditional expectation E[X Z] is the rando variable Y L 2 (Z) that iniizes E [ (X Y ) 2]. Intuitively, we can think of E[X Z] as the orthogonal projection of X onto the space L 2 (Z), where the scalar product X, Y is defined as X, Y = E[XY ]. Alexander Herbertsson (Univ. of Gothenburg) Financial Risk: Credit Risk, Lecture 1 Noveber 13, / 36

17 Properties of conditional expectations For a rando variable Z it is possible to show the following properties 1. If X L 2, then E[E[X Z]] = E[X] 2. If Y L 2 (Z), then E[YX Z] = Y E[X Z] 3. If X L 2, we define Var(X Z) as Var(X Z) = E [ X 2 Z ] E[X Z] 2 and it holds that Var(X) = E[Var(X Z)] + Var(E[X Z]). Furtherore, for an event A, we can define the conditional probability P [A Z] as P [A Z] = E[1 A Z] where 1 A is the indicator function for the event A (note that 1 A is a rando variable). An exaple: if X {a, b}, let A = {X = a}, and we get that P [X = a Z] = E [ 1 {X=a} Z ]. Alexander Herbertsson (Univ. of Gothenburg) Financial Risk: Credit Risk, Lecture 1 Noveber 13, / 36

18 The ixed binoial odel The binoial odel is also the starting point for ore sophisticated odels. For exaple, the ixed binoial odel which randoizes the default probability in the standard binoial odel, allowing for stronger dependence. The econoic intuition behind this randoizing of the default probability p(z) is that Z should represent soe coon background variable affecting all obligors in the portfolio. The ixed binoial distribution works as follows: Let Z be a rando variable on R with density f Z (z) and let p(z) [0, 1] be a rando variable with distribution F(x) and ean p, that is F(x) = P [p(z) x] and E[p(Z)] = p(z)f Z (z)dz = p. (3) Let X 1, X 2,...X be identically distributed rando variables such that X i = 1 if obligor i defaults before tie T and X i = 0 otherwise. Furtherore, conditional on Z, the rando variables X 1, X 2,...X are independent and each X i have default probability p(z), that is P [X i = 1 Z] = p(z) Alexander Herbertsson (Univ. of Gothenburg) Financial Risk: Credit Risk, Lecture 1 Noveber 13, / 36

19 The ixed binoial odel Since P [X i = 1 Z] = p(z) we get that E[X i Z] = p(z), because E[X i Z] = 1 P [X i = 1 Z] + 0 (1 P [X i = 1 Z]) = p(z). Furtherore, note that E[X i ] = p and thus p = E[p(Z)] = P [X i = 1] since P [X i = 1] = E[X i ] = E[E[X i Z]] = E[p(Z)] = where the last equality is due to (3). One can show that 1 0 p(z)f Z (z)dz = p. Var(X i ) = p(1 p) and Cov(X i, X j ) = E [ p(z) 2] p 2 = Var(p(Z)) (4) Next, letting all losses be the sae and constant given by, say l, then the total credit loss in the portfolio at tie T, called L, is L = lx i = l X i = ln where N = i=1 i=1 thus, N is the nuber of defaults in the portfolio up to tie T ] Again, since P [ L = kl = P [N = k], it is enough to study N. Alexander Herbertsson (Univ. of Gothenburg) Financial Risk: Credit Risk, Lecture 1 Noveber 13, / 36 i=1 X i

20 The ixed binoial odel, cont. However, since the rando variables X 1, X 2,...X now only are conditionally independent, given the outcoe Z, we have ( ) P [N = k Z] = p(z) k (1 p(z)) k k so since P [N = k] = E[P [N = k Z]] = E [( k) p(z) k (1 p(z)) k] it holds that ( ) P [N = k] = p(z) k (1 p(z)) k f Z (z)dz. (5) k Furtherore, since X 1, X 2,... X no longer are independent we have that ( ) Var(N ) = Var X i = Var(X i ) + Cov(X i, X j ) (6) i=1 i=1 and by hoogeneity in the odel we thus get i=1 j=1,j i Var(N ) = Var(X i ) + ( 1)Cov(X i, X j ). (7) Alexander Herbertsson (Univ. of Gothenburg) Financial Risk: Credit Risk, Lecture 1 Noveber 13, / 36

21 The ixed binoial odel, cont. So inserting (4) in (7) we get that Var(N ) = p(1 p) + ( 1) ( E [ p(z) 2] p 2). (8) Next, it is of interest to study how our portfolio will behave when, that is when the nuber of obligors in the portfolio goes to infinity. Recall that Var(aX) = a 2 Var(X) so this and (8) iply that ( ) N Var = Var(N ) p(1 p) 2 = + ( 1)( E [ p(z) 2] p 2). We therefore conclude that ( ) N Var E [ p(z) 2] p 2 as (9) Note especially the case when p(z) is a constant, say p, so that p = p. Then we are back in the standard binoial loss odel and E [ p(z) 2] p 2 = p 2 p 2 = 0 so Var ( N ) 0, i.e. the average nuber of defaults in the portfolio converge to a constant (which is p) as the portfolio size tend to infinity (this is the law of large nubers.) Alexander Herbertsson (Univ. of Gothenburg) Financial Risk: Credit Risk, Lecture 1 Noveber 13, / 36

22 The ixed binoial odel, cont. So in the ixed binoial odel, we see fro (9) that the law of large nubers do not hold, i.e. Var ( N ) does not converge to 0. Consequently, the average nuber of defaults in the portfolio, i.e. not converge to a constant as. N, does This is due to the fact that the rando variables X 1, X 2,... X, are not independent. The dependence aong the X 1, X 2,... X, is created by Z. However, conditionally on Z, we have that the law of large nubers hold (because if we condition on Z, then X 1, X 2,... X are i.i.d with default probability p(z)), that is given a fixed outcoe of Z then N p(z) as (10) and since a.s convergence iplies convergence in distribution (10) iplies that for any x [0, 1] we have [ ] N P x P [p(z) x] when. (11) Alexander Herbertsson (Univ. of Gothenburg) Financial Risk: Credit Risk, Lecture 1 Noveber 13, / 36

23 The ixed binoial odel, cont. Note that (11) can also be verified intuitive fro (10) by aking the following observation. Fro (10) we have that [ ] { N P θ 0 if p(z) > θ Z as 1 if p(z) θ that is, Next, recall that [ ] N P θ Z 1 {p(z) θ} as. (12) [ ] [ [ ]] N P θ N = E P θ Z so (12) in (13) renders [ ] N P θ E [ ] 1 {p(z) θ} = P [p(z) θ] = F(θ) as where F(x) = P [p(z) x], i.e. F(x) is the distribution function of the rando variable p(z). (13) Alexander Herbertsson (Univ. of Gothenburg) Financial Risk: Credit Risk, Lecture 1 Noveber 13, / 36

24 Large Portfolio Approxiation (LPA) Hence, fro the above rearks we conclude the following iportant result: Large Portfolio Approxiation (LPA) for ixed binoial odels For large portfolios in a ixed binoial odel, the distribution of the average nuber of defaults in the portfolio converges to the distribution of the rando variable p(z) as, that is for any x [0, 1] we have [ ] N P x P [p(z) x] when. (14) The distribution P [p(z) x] is called the Large Portfolio Approxiation (LPA) to the distribution of N. The above result iplies that if p(z) has heavy tails, then the rando variable N will also have heavy tails, as, which then iplies a strong default dependence in the credit portfolio. Alexander Herbertsson (Univ. of Gothenburg) Financial Risk: Credit Risk, Lecture 1 Noveber 13, / 36

25 The ixed binoial odel: the beta function One exaple of a ixing binoial odel is to let p(z) = Z where Z is a beta distribution, Z Beta(a, b), which can generate heavy tails. We say that a rando variable Z has beta distribution, Z Beta(a, b), with paraeters a and b, if it s density f Z (z) is given by f Z (z) = 1 β(a, b) za 1 (1 z) b 1 a, b > 0, 0 < z < 1 (15) where β(a, b) denotes the beta function which satisfies the recursive relation β(a + 1, b) = a β(a, b). a + b Also note that (15) iplies that P [0 Z 1] = 1, that is Z [0, 1] with probability one. Furtherore, since p(z) = Z, the distribution of N converges to the distribution of the beta distribution, i.e [ ] N P x 1 x z a 1 (1 z) b 1 dz as β(a, b) 0 Alexander Herbertsson (Univ. of Gothenburg) Financial Risk: Credit Risk, Lecture 1 Noveber 13, / 36

26 The ixed binoial odel: the beta function, cont. 14 Two different beta densities 12 a=1,b=9 a=10,b=90 10 density x lexander Herbertsson (Univ. of Gothenburg) Financial Risk: Credit Risk, Lecture 1 Noveber 13, / 36

27 The ixed binoial odel: the beta function, cont. If Z has beta distribution with paraeters a and b, one can show that E[Z] = a a + b and Var(Z) = ab (a + b) 2 (a + b + 1). Consider a ixed binoial odel where p(z) = Z has beta distribution with paraeters a and b. Then, by using (5) one can show that ( ) β(a + k, b + k) P [N = k] =. (16) k β(a, b) It is possible to create heavy tails in the distribution P [N = k] by choosing the paraeters a and b properly in (16). This will then iply ore realistic probabilities for extree loss scenarios, copared with the standard binoial loss distribution (see figure on next page). Alexander Herbertsson (Univ. of Gothenburg) Financial Risk: Credit Risk, Lecture 1 Noveber 13, / 36

28 The ixed binoial odel: the beta function, cont The portfolio credit loss distribution in the standar and ixed binoial odel ixed binoial 50, beta(1,9) binoial(50,0.1) probability nuber of defaults lexander Herbertsson (Univ. of Gothenburg) Financial Risk: Credit Risk, Lecture 1 Noveber 13, / 36

29 Mixed binoial odels: logit-noral distribution Another possibility for ixing distribution p(z) is to let p(z) be a logit-noral distribution. This eans that p(z) = exp ( (µ + σz)) where σ > 0 and Z N(0, 1), that is Z is a standard noral rando variable. Note that p(z) [0, 1]. Furtherore, if 0 < x < 1 then p 1 (x) is well defined and given by p 1 (x) = 1 ( ( ) ) x ln µ. (17) σ 1 x The ixing distribution F(x) = P [p(z) x] = P [ Z p 1 (x) ] for a logit-noral distribution is then given by F(x) = P [ Z p 1 (x) ] = N(p 1 (x)) for 0 < x < 1 where p 1 (x) is given as in Equation (17) and N(x) is the distribution function of a standard noral distribution. Alexander Herbertsson (Univ. of Gothenburg) Financial Risk: Credit Risk, Lecture 1 Noveber 13, / 36

30 Correlations in ixed binoial odels Recall the definition of the correlation Corr(X, Y ) between two rando variables X and Y, given by Corr(X, Y ) = Cov(X, Y ) Var(X) Var(Y) where Cov(X, Y ) = E[XY ] E[X] E[Y] and Var(X)) = E [ X 2] E[X] 2. Furtherore, also recall that Corr(X, Y ) ay soeties be seen as a easure of the dependence between the two rando variables X and Y. Now, let us consider a ixed binoial odel as presented previously. We are interested in finding Corr(X i, X j ) for two pairs i, j in the portfolio (by the hoogeneous-portfolio assuption this quantity is the sae for any pair i, j in the portfolio where i j). Below, we will therefore for notational convenience siply write ρ X for the correlation Corr(X i, X j ). Alexander Herbertsson (Univ. of Gothenburg) Financial Risk: Credit Risk, Lecture 1 Noveber 13, / 36

31 Correlations in ixed binoial odels, cont. Recall fro previous slides that P [X i = 1 Z] = p(z) where p(z) is the ixing variable. Furtherore, we also now that Cov(X i, X j ) = E [ p(z) 2] p 2 and Var(X i ) = p(1 p) (18) where p = E[p(Z)]. Thus, the correlation ρ X in a ixed binoial odels is then given by ρ X = E[ p(z) 2] p 2 p(1 p) where p = E[p(Z)] = P [X i = 1] is the default probability for each obligor. Hence, the correlation ρ X in a ixed binoial is copletely deterined by the fist two oents of the ixing variable p(z), that is E[p(Z)] and E [ p(z) 2]. Exercise 1: Show that P [X i = 1, X j = 1] = E [ p(z) 2] where i j. Exercise 2: Show that Var(X i ) = E[p(Z)] (1 E[p(Z)]). (19) Alexander Herbertsson (Univ. of Gothenburg) Financial Risk: Credit Risk, Lecture 1 Noveber 13, / 36

32 Value-at-Risk Recall the definition of Value-at-Risk Definition of Value-at-Risk Given a loss L and a confidence level α (0, 1), then VaR α (L) is given by the sallest nuber y such that the probability that the loss L exceeds y is no larger than 1 α, that is VaR α (L) = inf {y R : P [L > y] 1 α} where F L (x) is the distribution of L. = inf {y R : 1 P [L y] 1 α} = inf {y R : F L (y) α} Note that Value-at-Risk is defined for a fixed tie horizon, so the above definition should also coe with a tie period, e.g, if the loss L is over one day, then we talk about a one-day VaR α (L). In credit risk, one typically consider VaR α (L) for the loss over one year. Note that if F L (x) is continuous, then VaR α (L) = F 1 L (α) Alexander Herbertsson (Univ. of Gothenburg) Financial Risk: Credit Risk, Lecture 1 Noveber 13, / 36

33 Value-at-Risk for static credit portfolios Consider the sae type of hoogeneous static credit portfolio odels as studied previously today, with obligors and where each obligor can default up to tie T. Each obligor have identical credit loss l at a default, where l is a constant. The total credit loss in the portfolio at tie T is then given by L = ln where N is the nuber of defaults in the portfolio up to tie T. Note that the individual loss l is given by ln where N is the notional of the individual loan and l is the loss as a fraction of N (i.e l [0, 1]) By linearity of VaR (see in lecture notes by H&L) we can without loss of generality assue that N = 1, so that l = l, since VaR α (cl) = cvar α (L) Alexander Herbertsson (Univ. of Gothenburg) Financial Risk: Credit Risk, Lecture 1 Noveber 13, / 36

34 Value-at-Risk for static credit portfolios, cont. If p(z) is a ixing variable with distribution F(x) we know that [ ] N P x F(x) as which iplies that P [L x] = P [ N x ] ( x ) F l l as Hence, if F(x) is continuous, and if is large, we have the following approxiation forula for VaR α (L) where L denotes the loss L. VaR α (L) l F 1 (α) (20) Alexander Herbertsson (Univ. of Gothenburg) Financial Risk: Credit Risk, Lecture 1 Noveber 13, / 36

35 Expected shortfall The expected shortfall ES α (L) is defined as and one can show that ES α (L) = E[L L VaR α (L)] ES α (L) = 1 1 α 1 α VaR u (L)du. Hence, for the sae static credit portfolio as on the two previous slides, we have the following approxiation forula for ES α (L) (when is large) ES α (L) l 1 α 1 α F 1 (u)du where L denotes the loss L and where we used (20). Here, F(x) is the continuous distribution of the ixing variable p(z). Alexander Herbertsson (Univ. of Gothenburg) Financial Risk: Credit Risk, Lecture 1 Noveber 13, / 36

36 Thank you for your attention! Alexander Herbertsson (Univ. of Gothenburg) Financial Risk: Credit Risk, Lecture 1 Noveber 13, / 36