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@cff.gu.se or alexander.herbertsson@econoics.gu.se Financial Risk, Chalers University of Technology, Göteborg Sweden April 25, 2017 Alexander Herbertsson (Univ. of Gothenburg) Financial Risk: Credit Risk, Lecture 1 April 25, / 26

2 Main goal of lectures and content of today s lecture The ain goal of coing three lectures is to study the loss distribution for a credit portfolio The loss distribution is used to copute risk easures such as Value-at-Risk etc. Today s 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 Alexander Herbertsson (Univ. of Gothenburg) Financial Risk: Credit Risk, Lecture 1 April 25, / 26

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 April 25, / 26

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 three lectures focuses only on default dependency risk. Alexander Herbertsson (Univ. of Gothenburg) Financial Risk: Credit Risk, Lecture 1 April 25, / 26

5 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 three lectures focuses only on static credit portfolio odels, Alexander Herbertsson (Univ. of Gothenburg) Financial Risk: Credit Risk, Lecture 1 April 25, / 26

6 Literature The slides for the coing three lectures are rather self-contained, but ore details on certain topics can be found in the lecture notes. The content of the lecture today and the next lecture is partly based on aterials presented in: 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) Risk and portfolio analysis - principles and ethods by Hult, Lindskog, Haerlid and Rehn. (Springer) Alexander Herbertsson (Univ. of Gothenburg) Financial Risk: Credit Risk, Lecture 1 April 25, / 26

7 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 In the next two lectures we will study three different ixed binoial odels. discuss Value-at-Risk and Expected shortfall in a ixed binoial odels. Correlations etc in ixed binoial odels. Alexander Herbertsson (Univ. of Gothenburg) Financial Risk: Credit Risk, Lecture 1 April 25, / 26

8 The binoial odel for independent defaults Consider a hoogeneous credit portfolio odel with obligors where each obligor can default up to fixed tie T, and have the sae constant credit loss l. 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 (1) We assue that X 1,X 2,...X are i.i.d, that is they are independent with identical distribution. Furtherore P[X i = 1] = p so 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. i=1 X i Alexander Herbertsson (Univ. of Gothenburg) Financial Risk: Credit Risk, Lecture 1 April 25, / 26

9 The binoial odel for independent defaults, cont. Since X 1,X 2,...X are i.i.d with P[X i = 1] = p we see that N = i=1 X i is binoially distributed with paraeters and p, i.e. N Bin(,p). Hence, we have 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 April 25, / 26

10 The binoial odel for independent defaults, cont The portfolio credit loss distribution in the binoial odel bin(50,0.1) probability nuber of defaults Alexander Herbertsson (Univ. of Gothenburg) Financial Risk: Credit Risk, Lecture 1 April 25, / 26

11 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 is due to the independence assuption for X 1,X 2,...X. To see this, recall that the variance 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 = Var(X i ) = p(1 p) (2) i=1 where the second equality is due the independence assuption. i=1 Alexander Herbertsson (Univ. of Gothenburg) Financial Risk: Credit Risk, Lecture 1 April 25, / 26

12 The binoial odel for independent defaults, cont. Furtherore, by Chebyshev s inequality we have that for any rando variable X, and any c > 0 it holds P[ X E[X] c] Var(X) c 2. Exaple: if p = 5% and = 50 then Var(N ) = 50p(1 p) = and E[N ] = 50p = 2.5. So with p = 5% and = 50, the probability of say, 6 ore or less losses than expected, is saller or equal than 6.6%, since by Chebyshev P[ N 2.5 6] = 6.6%. Next we show that the deviation of the fractional nuber of defaults in the portfolio, N, fro the constant p = E[ ] N, goes to zero as. So N converges towards a constant as (the law of large nubers). Alexander Herbertsson (Univ. of Gothenburg) Financial Risk: Credit Risk, Lecture 1 April 25, / 26

13 Independent defaults and the law of large nubers By applying Chebyshev s inequality to N together with Equation (2) we get [ ] N P p ε Var( ) N 1 Var(N ε 2 = 2 ) ε 2 = p(1 p) 2 ε 2 = p(1 p) ε 2 Thus, P [ N p ε] 0 as for any ε > 0. This result is called the weak law of large nubers For our credit portfolio it eans that the fractional nuber of defaults in N the portfolio, i.e., converges (in probability) to the constant p, i.e 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 April 25, / 26

14 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 fractional nuber of defaults in the portfolio N converges to the constant p alost surely. It is an epirical fact, observed any ties in the history, that defaults tend to cluster and N have often values uch bigger than 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, which iplies that N does not converges to a constant with probability 1, when. Alexander Herbertsson (Univ. of Gothenburg) Financial Risk: Credit Risk, Lecture 1 April 25, / 26

15 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 April 25, / 26

16 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] = YE[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 April 25, / 26

17 The ixed binoial odel The binoial odel can be extended to the ixed binoial odel which randoizes the default probability, allowing for stronger dependence. The ixed binoial odel works as follows: Let Z be a rando variable (discrete or continuous) and let p(x) [0,1] be a function such that the rando variable p(z) is well-defined. 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. 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) 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. Alexander Herbertsson (Univ. of Gothenburg) Financial Risk: Credit Risk, Lecture 1 April 25, / 26

18 The ixed binoial odel, cont Let F(x) and p be the distribution and ean of the rando variable p(z), that is, F(x) = P[p(Z) x] and E[p(Z)] = p. (3) If for exaple Z is a continuous rando variable on R with density f Z (z) then p is given by p = E[p(Z)] = p(z)f Z (z)dz. (4) 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). 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)] = p where the last equality is due to (3). Alexander Herbertsson (Univ. of Gothenburg) Financial Risk: Credit Risk, Lecture 1 April 25, / 26

19 The ixed binoial odel, cont One can show that (see in the lecture notes) Var(X i ) = p(1 p) and Cov(X i,x j ) = E [ p(z) 2] p 2 = Var(p(Z)) (5) 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. 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 Alexander Herbertsson (Univ. of Gothenburg) Financial Risk: Credit Risk, Lecture 1 April 25, / 26 i=1 X i

20 The ixed binoial odel, cont. Hence, P[N = k] = E[P[N = k Z]] = E [( ] )p(z) k (1 p(z)) k k so if Z is a continuous rando variable on R with density f Z (z) then ( ) P[N = k] = p(z) k (1 p(z)) k f Z (z)dz. (7) k Furtherore, because 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 ) (8) 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 ). (9) So inserting (5) in (9) we get that Var(N ) = p(1 p)+( 1) ( E [ p(z) 2] p 2). (10) Alexander Herbertsson (Univ. of Gothenburg) Financial Risk: Credit Risk, Lecture 1 April 25, / 26 (6)

21 The ixed binoial odel, cont. 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 (10) 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 = Var(p(Z)) as (11) Note that in the case when p(z) is a constant, say p, so that p = p. we are back in the standard binoial loss odel and E [ p(z) 2] ( ) p 2 = p 2 p 2 N = 0 so Var 0 as i.e. the fractional nuber of defaults in the portfolio converge to the constant p = p as portfolio size tend to infinity (law of large nubers.) Alexander Herbertsson (Univ. of Gothenburg) Financial Risk: Credit Risk, Lecture 1 April 25, / 26

22 The ixed binoial odel, cont. So in the ixed binoial odel, we see fro (11) that the law of large nubers do not hold, i.e. Var ( N ) does not converge to 0. Consequently, the fractional nuber of defaults in the portfolio N converge to a constant as. does not This is due to the fact that X 1,X 2,...X, are not independent. The dependence aong 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 (12) Since a.s convergence iplies convergence in distribution (12) iplies that for any x [0,1] we have [ ] N P x P[p(Z) x] when. (13) Alexander Herbertsson (Univ. of Gothenburg) Financial Risk: Credit Risk, Lecture 1 April 25, / 26

23 The ixed binoial odel, cont. Note that (13) can also be verified intuitive fro (12) by aking the following observation. Fro (12) we have that [ ] { N P x 0 if p(z) > x Z as 1 if p(z) x that is, Next, recall that [ N P x ] Z 1 {p(z) x} as. (14) [ ] [ [ N P x N = E P x ]] Z so (14) in (15) renders [ ] N P x E [ ] 1 {p(z) x} = P[p(Z) x] = F(x) as where F(x) = P[p(Z) x], i.e. F(x) is the distribution function of the rando variable p(z). (15) Alexander Herbertsson (Univ. of Gothenburg) Financial Risk: Credit Risk, Lecture 1 April 25, / 26

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 fractional nuber of defaults N 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. (16) 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 April 25, / 26

25 Exaples of ixing distributions (next two lectures) Exaple 1: A ixed binoial odel with p(z) = Z where Z is a beta distribution, Z Beta(a,b) and by definition of a beta distribution it holds that P[0 Z 1] = 1 so that p(z) [0,1]. Exaple 2: Another possibility for ixing distribution p(z) is to let p(z) be a logit-noral distribution. This eans that 1 p(z) = 1+exp( (µ+σz)) where σ > 0 and Z is a standard noral. Note that p(z) [0,1]. Exaple 3: The ixed binoial odel inspired by the Merton odel (will be discussed coing lectures) with p(z) given by ( N 1 ( p) ) ρz p(z) = N (17) 1 ρ where Z is a standard noral and N(x) is the distribution function of a standard noral distribution. Furtherore, ρ [0,1] and p = P[X i = 1]. Note that p(z) [0,1]. Alexander Herbertsson (Univ. of Gothenburg) Financial Risk: Credit Risk, Lecture 1 April 25, / 26

26 Thank you for your attention! Alexander Herbertsson (Univ. of Gothenburg) Financial Risk: Credit Risk, Lecture 1 April 25, / 26