Appendix Large Homogeneous Portfolio Approximation

Transcription

1 Aendix Large Homogeneous Portfolio Aroximation A.1 The Gaussian One-Factor Model and the LHP Aroximation In the Gaussian one-factor model, an obligor is assumed to default if the value of its creditworthiness is below a re-secified value. The creditworthiness of an obligor is modeled through a latent variable: Z n ¼ ffiffiffi q X þ 1 qx n ; n ¼ 1; 2;...; N; ða:1þ where X is the systemic factor and X n with n ¼ 1; 2;...; N are the idiosyncratic factors; all are assumed to be standard normal random variables with mean zero and unit variance, and q is the correlation between two assets: The nth loan defaulted by time t if: CorrðZ m ; Z n Þ¼q; m 6¼ n: ða:2þ Z n K d n ðtþ; ða:3þ where Kn d ðtþ is the time deendent default barrier. Under the assumtion of the homogenous ool, each asset behaves as the average of the assets in the ool and we can set Kn dðtþ ¼Kd ðtþ for the all n. The default barrier can be chosen such that: PðZ n K d ðtþþ ¼ ðtþ; ða:4þ where ðtþ is the robability of default of a single obligor in the ool by maturity T. It imlies K d ðtþ ¼U 1 ððtþþ. The cumulative ortfolio default rate is given by: PDRðTÞ ¼ XN D n ðtþ N ; ða:5þ F. Camolongo et al., Quantitative Assessment of Securitisation Deals, SringerBriefs in Finance, DOI: / , Ó The Author(s)

2 106 Aendix where D n ðtþ is the default indicator of asset n. The default indicator D n ðtþ equals one (with robability ðtþ) if asset n defaulted by time T and zero otherwise. The exected value of the ortfolio default rate at time T is: E½PDRðTÞŠ ¼ E½ 1 N ¼ 1 N X N X N D n ðtþš E½D n ðtþš ¼ E½D 1 ðtþš ¼ PðD 1 ðtþ ¼1Þ ¼ PðZ 1 K d ðtþþ ¼ ðtþ; ða:6þ where the third equality follows by the homogeneous ortfolio assumtion, and the last equality holds by definition. Thus, under the homogeneous ortfolio assumtion, the ortfolio default rate mean is equal to the individual loan s robability of default ðtþ. The default indicators in (A.5) are correlated and we can not use the Law of Large numbers to derive a limiting distribution. However, conditional on the common factor X, the default indicators are indeendent and we can aly the Law of Large Numbers. Conditional on the common factor, the ortfolio default rate at time T is given by: PDRðT; X ¼ xþ ¼ XN D n ðt; X ¼ xþ ; ða:7þ N where D n ðt; X ¼ xþ is the default indicator of asset n given the systematic factor X. By the Law of Large Numbers, as N tends to infinity we get: PDRðT; X ¼ xþ!e½pdrðt; XÞjX ¼ xš ¼ 1 N X N ðt; xþ ¼ðT; xþ; ða:8þ where ðt; xþ is the default robability for an individual asset given X ¼ x: ðt; xþ ¼PðZ n K d ðtþjx ¼ xþ ffiffiffi ¼ Pð q X 1 qx n K d ðtþjx ¼ xþ ¼ U Kd ðtþ ffiffiffi q x : 1 q ða:9þ

3 Aendix 107 Fig. A.1 a Portfolio default rate versus correlation. Large homogeneous ortfolio aroximation. Correlation between 1 and 90%. Mean default rate: 30%. b Portfolio default rate versus correlation. Large homogeneous ortfolio aroximation. Correlation between 1 and 50%. Mean default rate: 30% 25 It follows that the distribution of PDRðT; XÞ is 1 : F PDRðT;XÞ ðyþ ¼PðPDRðT; XÞ\yÞ ¼ PðXÞ\y ð Þ ¼ P U Kd ðtþ ffiffiffi q X \y 1 q ¼ P X[ Kd ðtþ ffiffiffiffiffiffiffiffiffiffiffi 1 qu 1 ðyþ ffiffiffi : q ða:10þ Using the symmetry of the normal distribution, we get: FPDRðT;XÞ LHP 1 qu 1 ðyþ K d ðtþ ðyþ ¼PðPDRðT; XÞ\yÞ ¼U ffiffiffi ; ða:11þ q where 0% y 100% and K d ðtþ ¼U 1 ððtþþ. Note that the right hand side of (A.11) is indeendent of the systemic factor X. The distribution in (A.11) is sometimes called the Normal Inverse distribution, see, for examle, [1]. Thus, for a reasonably large homogeneous ortfolio, we can use the distribution in (A.11) as an aroximation to the ortfolio default rate distribution. We illustrate in Fig. A.1a, b the ortfolio default rate distribution s deendence on the correlation arameter q, under the assumtion that the default mean is 30%. As can be seen from the lots, under a low correlation assumtion, the PDR distribution will have a bell shaed form, but as the asset correlation increases, the mass of the distribution is shifted towards the end oints of the PDR interval, increasing the likelihood of zero or a very small fraction of the ortfolio defaulting and the likelihood of the whole ortfolio defaulting. This is natural since a very 1 The above convergence is in robability, which imlies convergence in distribution.

4 108 Aendix Fig. A.2 Imlied correlation for different values of mean and coefficient of variation (standard deviation divided by mean) equal to 0.25, 0.5, 0.75, 1.0 and Normal Inverse distribution: Imlied correlation Def. mean = 0.05 Def. mean = 0.10 Def. mean = 0.20 Def. mean = 0.30 Correlation Coefficient of variation (σ/μ) high correlation (close to one) means that the loans in the ool are likely to either survive together or default together. In general, it can be said that the PDR distribution becomes flatter and more mass is shifted towards the tails of the distribution when the default mean is increased. A.2 Calibrating the Distribution The default distribution in (A.11) is a function of the obligor correlation q, and the default robability ðtþ, which are unknown and unobservable. Instead of using these arameters as inuts, it is common to fit the mean and standard deviation of the distribution to the mean and standard deviation, resectively, estimated from historical data (see, for examle, [2, 3]). Let us denote by l cd and r cd the estimated mean and standard deviation, resectively. The mean of the distribution is equal to the robability of default for a single obligor ðtþ, soðtþ ¼l cd. As a result, there is only one free arameter, the correlation q, left to adjust to fit the distribution s standard deviation to r cd, which can be done numerically by minimising r 2 cd Var qðpdrðtþþ, where the subscrit is used to show that the variance is a function of q. Looking at the correlation values given in Fig. A.2 and the density lots in Fig. A.1a, b, one can see that the corresonding default distributions will have very different shaes, ranging from bell shaed curves to very heavy tailed ones, with the mass almost comletely concentrated at zero and one. It is imortant to understand that the behavior of the correlation and the default robability shown in Fig. A.2 should not be taken as a general rule. The grahs show the result of fitting the distribution to means and standard deviations in the distribution s comfort zone, i.e, values that will give good fits. (The root mean

5 Aendix 109 squared error is of the order of magnitude of for the shown results.) For combinations of the default mean and the coefficient of variation that result in an imlied correlation equal to one, the calibration will sto since it cannot imrove the root mean squared error, which in these situations will be much larger than for the values shown in Fig. A.2. References 1. Moody s Investor Service (2003) The Fourier transform method Technical document. Working Paer, 30 Jan Moody s Investor Service (2005) Historical default data analysis for ABS transactions in EMEA. International Structured Finance, Secial Reort, 2 Dec Raynes, S., Rutledge, A.: The analysis of structured securities: recise risk measurement and caital allocation. Oxford University Press, New York (2003)

6 Index A Allocation of rincial ro rata, 6 sequential, 6 Amortising structure, 5 Asset-backed securities definition of, 3 destinctions of, 4 structural characteristics, 5 transaction arties, 4 Assets cashflow modelling, see cashflow modelling, 17, 22 defaulted, 18 delinquent, 17 erforming, 17 reaid, 18 reaid, 18 C Cashflow modelling, 17 assets, 17 examle, 19 homogeneous ortfolio aroach, 18 interest collections, 18, 20 rincial collections, 18, 20 recoveries, 21 available funds, 21 ayment waterfall, 22 examle, 22 Counterarty risk, 11 Credit enhancement, 7 excess sread, see excess sread, 7 external, 8 internal, 7 over-collateralisation, see overcollateralisation, 7 reserve fund, see reserve fund, 7 subordination, see subordination, 7 Credit risk, 8 Cross currency risk, 10 D Default curve, 33 Default distribution, 14, 33 Default models Conditional Default Rate, 34 Default vector, 35 Gamma Portfolio Default model, 44, 45 Generic One-Factor Lévy model, 49 Lévy Portfolio Default model, 43 Logistic model, 36 Normal One-Factor model, 45, 51 E EAL, see exected average life, 14 EL, see exected loss, 15 Elementary effect, 75 Excess sread, 7 Exected average life, 14, 60, 72 Exected loss, 15, 59, 72 G Global rating, 91 F. Camolongo et al., Quantitative Assessment of Securitisation Deals, SringerBriefs in Finance, DOI: / , Ó The Author(s)

7 112 Index I Interest rate risk, 10 Internal rate of return, 59 IRR, see internal rate of return, 59 Issuer, 4 L Large homogeneous ortfolio aroximation, 48, 51 Legal risks, 12 Liquidity risk, 11 Loss allocation, 7 M Market risk, 10 N Normal Inverse distribution, 14, 72 Note redemtion amount, 5 O Oerational risk, 11 Originator, 4 Over-collateralisation, 7 P Pari assu, 6 Payment waterfall searate, 6 combined, 6 Preayment, 9 Preayment curve, 33 Preayment distribution, 33 Preayment models Conditional Preayment Rate, 39 Generalised CPR, 40 Lévy Portfolio Preayment model, 51 Normal One-Factor Preayment model, 51 PSA benchmark, 39 Preayment risk, 9 Priority of ayments, see ayment waterfall, 6 Pro rata, see allocation of rincial, 6 R Rating default definition, 13 definitions of, 13 exected loss, 13 deriving, 14, 28, 72 global, 91 model risk, 60 arameter sensitivity, 61, 64, 86, 88 robability of default, 13 Reinvestment risk, 10 Relative net resent value loss, 15, 28, 73 Relenishment amount, 6 Reserve fund, 7 Revolving structure, 5 RPVL, see relative net resent value loss, 15 S Sensitivity analysis, 73 elementary effects method, 74, 86 global, 74 variance based method, 77, 88 Sensitivity index comuting, 78 first order, 77 second order, 77 total effect term, 77 Sequential, see allocation of rincial, 6 Servicer, 4 Secial urose entity, 4 Secial urose vehicle, 4 Subordination, 7 T Trustee, 5 W WAL, see weighted average life, 15 Weighted average life, 15, 28, 40