CDO Surfaces Dynamics

Transcription

1 D Surfaces Dynamics Barbara Choroś-Tomczyk Wolfgang Karl Härdle stap khrin Ladislaus von Bortkiewicz Chair of Statistics C..S.E. - Center for pplied Statistics and Economics Humboldt-Universität zu Berlin

2 K H R I N H E R D L E Motivation 1-1 itraxx over Time 8 8 Spread 6 4 Spread Time Time Figure 1: Spreads of itraxx tranches, Series 5, maturity 5 (left) and (right) years, data from Tranches: 1, 2, 3, 4, 5. CD Surfaces Dynamics

3 K H R I N H E R D L E Motivation 1-2 itraxx Spread Surface Spread 2 Spread Tranche 2 5 Time to Maturity 5 15 Tranche 2 5 Time to Maturity Figure 2: Spreads of tranches of all series observed on 2899 (left) and (right). CD Surfaces Dynamics

4 K H R I N H E R D L E Motivation 1-3 Research Goals Modelling the dynamics of CD surfaces spread surfaces base correlation surfaces pplications in trading CD Surfaces Dynamics

5 K H R I N H E R D L E Motivation 1-4 Dynamic Semiparametric Factor Model pplications: 1. Implied volatility surfaces in M. R. Fengler, W. Härdle and E. Mammen, JFE (27) and B. Park, E. Mammen, W. Härdle, and S. Borak, JS (29) 2. Risk neutral densities in E. Giacomini, W. Härdle, and V. Krätschmer, St (29) 3. Limit order book in W. Härdle, N., Hautsch, and. Mihoci, JEF (212) 4. Variance swaps in K. Detlefsen and W. Härdle, QF (213) 5. fmri images in. Myšicková, S. Song, P. Majer, P. Mohr, H. Heekeren, W. Härdle, Psychometrika (213) CD Surfaces Dynamics

6 K H R I N H E R D L E utline 1. Motivation 2. CDs 3. DSFM 4. Empirical Study 5. pplications 6. Conclusions CD Surfaces Dynamics

7 K H R I N H E R D L E CDs 2-1 Risk Transfer CD Surfaces Dynamics

8 K H R I N H E R D L E CDs 2-2 itraxx Europe static portfolio of 125 equally weighted CDS on European entities; Sectors: Consumer (3), Financial (25), TMT (2), Industrials (2), Energy (2), uto (); New series of itraxx Europe issued every 6 months (March and September) and the underlying reference entities are reconstituted; Maturities: 3Y, 5Y, 7Y, Y. Valuation CD Surfaces Dynamics

9 K H R I N H E R D L E CDs 2-3 Large Pool Gaussian Copula Model Default times are modelled from the Gaussian vector (X 1,..., X d ) : X i = ρy + 1 ρz i, where Y (systematic risk factor), {Z i } d i=1 (idiosyncratic risk factors) are i.i.d. N(, 1). ssume that: obligors have the same default probability p and LGD, one dependence parameter ρ, d is large. The cdf of the portfolio loss equals { 1 ρφ 1 (x) Φ 1 } (p) P( L x) = Φ. ρ CD Surfaces Dynamics

10 K H R I N H E R D L E CDs 2-4 Correlation s Types Compound correlation ρ(l j, u j ), j = 1,..., J. Compound Correlation.4 Equity Mezzanine Junior Mezzanine Super Senior Senior Tranches Figure 3: Implied correlation smile in the Gaussian one factor model, CD Surfaces Dynamics

11 K H R I N H E R D L E CDs 2-5 Correlation s Types Base correlation (BC) ρ(, u j ), j = 1,..., J. Represent the expected loss E{L (lj,u j )} as a difference: E{L (lj,u j )} = E ρ(,uj ){L (,uj )} E ρ(,lj ){L (,lj )}, j = 2,..., J. of the expected losses of two fictive tranches (, u j ) and (, l j ). Bootstrapping process: E{L (,3%) } is traded on the market, E{L (3%,6%) } = E ρ(,6%) {L (,6%) } E ρ(,3%) {L (,3%) }, E{L (6%,9%) } = E ρ(,9%) {L (,9%) } E ρ(,6%) {L (,6%) },... CD Surfaces Dynamics

12 K H R I N H E R D L E CDs 2-6 Base Correlations over Time 1 Series 5 Maturity 5 1 Series 5 Maturity ρ t ρ t Time Time Figure 4: BC of itraxx tranches, Series 5, maturity 5 (left) and (right) years, data from Tranches: 1, 2, 3, 4, 5. CD Surfaces Dynamics

13 K H R I N H E R D L E DSFM 3-1 Base Correlation Surfaces 1 1 BC.5 BC Time to Maturity Tranche Time to Maturity Tranche Figure 5: Implied base correlations on day 2899 (left) and (right). CD Surfaces Dynamics

14 K H R I N H E R D L E DSFM 3-2 Dynamic Semiparametric Factor Model L Y t,k = m (X t,k ) + Z t,l m l (X t,k ) + ε t,k = Zt ψ(x t,k ) + ε t,k l=1 Y t,k k X t,k m l Z t,l ψ(x t,k ) log-spreads and Z-transformed BC on day t, t = 1,..., T intra-day numbering of BCs on day t, k = 1,..., K t two-dimensional vector of the tranche seniority and the time-to-maturity factor functions, time invariant, nonparametric estimation time series, l =,..., L, dynamic behavior tensor B-spline basis coefficient matrix CD Surfaces Dynamics

15 K H R I N H E R D L E DSFM 3-3 Estimation Using an iterative algorithm: (Ẑt, Â) = arg min T K t Z t, t=1 k=1 { } 2 Y t,k Zt ψ(x t,k ) Selection of L, the numbers of spline knots R 1, R 2 and the orders of splines k 1, k 2 by maximising the explained variance criterion: EV(L, R 1, r 1, R 2, r 2 ) = 1 where m is an empirical mean surface. T { Kt t=1 k=1 Y t,k } 2 L l=1 Z t,lm l (X t,k ) T Kt t=1 k=1 {Y t,j m (X t,k )} 2, CD Surfaces Dynamics

16 K H R I N H E R D L E DSFM 3-4 DSFM without the Mean Factor Reduce the number of factors estimated in the iterative algorithm by first subtracting the empirical mean m and then fitting the DSFM: Y t,k = m (X t,k )+ L Z t,l m l (X t,k )+ε t,k = m (X t,k )+Zt ψ(x t,k )+ε t,k, l=1 where m l are new factor functions, l = 1,..., L. CD Surfaces Dynamics

17 K H R I N H E R D L E Empirical Study 4-1 Data Series 2- Maturities 5, 7, Y 4 days between data points Year 3Y 5Y 7Y Y ll Table 1: Number of observed values of itraxx tranches in the period CD Surfaces Dynamics

18 K H R I N H E R D L E Empirical Study 4-2 DSFM for Z-transformed-BC EV.93 EV R 2 r 2.99 EV L CD Surfaces Dynamics Figure 6: Proportion of the explained variance as a function of R 2 (up left) with r 2 = 2, as a function of r 2 (up right) with R 2 =, as a function of L (down) for L = 1, L = 2, L = 3, r 1 = 2 and R 1 = 5.

19 K H R I N H E R D L E Empirical Study 4-3 DSFM w/o Mean F. for Z-transformed-BC m.5 m Tranche τ Tranche τ m 2 Z t Tranche τ Time Figure 7: Estimated factor functions and loadings (Ẑt,1, Ẑt,2). CD Surfaces Dynamics pp

20 K H R I N H E R D L E Empirical Study 4-4 DSFM Estimation Results For DSFM for both data types Ẑt,1 is a slope-curvature factor Ẑt,2 is a shift factor Model Log-Spr Z-BC DSFM.16.4 DSFM w/o mean f Table 2: Mean squared error of the in-sample fit. CD Surfaces Dynamics

21 K H R I N H E R D L E Empirical Study 4-5 DSFM without the mean factor Fit Figure 8: In-sample fit of the models to data on 2899 and CD Surfaces Dynamics

22 K H R I N H E R D L E pplications 5-1 Curve Trades So, how can I make money with this? Combine tranches of different time to maturity, see Felsenheimer et al. (24) and Kakodkar et al. (26): Flattener sell protection on a long-term tranche, buy protection on a short-term tranche Example: sell protection on Y 3-6% and buy on 5Y 6-9% utlook: bullish long-term, bearish short-term Steepener opposite trade CD Surfaces Dynamics

23 K H R I N H E R D L E pplications 5-2 Curve Trades F la tte n e r S te e p e n e r S p re a d S p re a d 5 Y 7 Y 1 Y 5 Y 7 Y 1 Y τ τ Figure 9: Mechanism of a flattener and a steepener startegy. Current spread curve, expectation of the future spread curve, indication of the direction of change. CD Surfaces Dynamics

24 K H R I N H E R D L E pplications 5-3 JP Morgan Trading Loss, May 212 J.P. Morgan s flattener bought 5Y CDX IG 9 index, sold Y CDX IG 9 index in a 3:1 ratio. The final loss reached $6.2 billion. CD Surfaces Dynamics

25 K H R I N H E R D L E pplications 5-4 Flattener Sell protection at s 1 (t ) for the period [t, T 1 ] and buy protection at s 2 (t ) for [t, T 2 ], T 1 > T 2. t t for l = 1, 2: T l MTM l (t )= β(t, t) [s l (t ) te{f l (t)} E{L l (t) L l (t t)}]=. t=t 1 t t > t, the market quotes s l ( t) and MTM l ( t) = {s l (t ) s l ( t)} β( t, t) te{f l (t)}. t= t 1 T l CD Surfaces Dynamics

26 K H R I N H E R D L E pplications 5-5 Curve Trade positive MTM means a positive value to the protection seller. If the protection seller closes the position at time t, then receives from the protection buyer MTM l ( t). Flattener-trader aims to maximize the total MTM value PL( t) = MTM 1 ( t) MTM 2 ( t). CD Surfaces Dynamics

27 K H R I N H E R D L E pplications 5-6 Risk in Curve Trades If one buys 5Y 6-9% and sells Y 6-9%, then the trade is hedged for default until the maturity of the 5Y tranche. Defaults that emerge from Y 6-9% are covered by 5Y 6-9% till it expires. Series differ in the composition of the collateral. If one buys 5Y 6-9% and sells Y 3-6%, then these tranches provide protection of different portion of portfolio risk. If there is any default in Y 3-6%, then we must deliver a payment obligation and incur a loss. CD Surfaces Dynamics

28 K H R I N H E R D L E pplications 5-7 Empirical Study Idea Use DSFM to forecast spread and BC surfaces Calculate forecasted MTM surfaces Recover those tranches that maximise P&L Remarks Because of many missing data and short data histories, the standard econometric methods cannot be used for the forecasting. Consider trades that generate no or a positive carry the spread of the long tranche doesn t exceed the spread of the short tranche. Do not account for default payments (no data of historical defaults in itraxx), do not account for the positive carry. CD Surfaces Dynamics

29 K H R I N H E R D L E pplications 5-8 Forecasting with DSFM in Rolling Windows Let Y t be log-spreads or Z-transformed-BC. Consider a rolling window of w = 25. Estimate the DSFMs using {Y ν } t ν=t w+1 for t = w,..., T h. s a result, we get T w + 1 times m = ( m,..., m L ) and Ẑ t = (Ẑt,,..., Ẑt,L) of length w. Compute h-day forecast of the factor loadings using VR. Due to the fixed issuing scheme, X t+h,k is not forecasted. Calculate the forecast Ŷt+h from the forecast Ẑt+h. Transform Ŷt+h suitably to get ŝ(t + h) or ˆρ(t + h). CD Surfaces Dynamics

30 K H R I N H E R D L E pplications 5-9 Forecasting MTM Surfaces For predicted {ŝ k (t), ˆρ k (t)}, t =w +h,..., T, k =1,..., K t, compute MTM k (t), where the initial spread s k (t ) is observed on t =t h. Figure : MTM surfaces on 2899 (left) and (right) calculated using one-day spread and BC predictions obtained with the DSFM. CD Surfaces Dynamics

31 K H R I N H E R D L E pplications 5- Transaction Costs Calculate the ask (bid) spread by increasing (reducing) the observed spread by the following percentage: Maturity Y Y Y Table 3: verage bid-ask spread excess over the mid spread as a percentage of the mid spread for Series 8 during the period CD Surfaces Dynamics

32 K H R I N H E R D L E pplications 5-11 Trading Strategies Construct a curve trade 1. Fit and forecast the DSFM models to spreads and BC. 2. Calculate h-day forecasts of the MTM surfaces. 3. Recover which two tranches optimize a given strategy. Strategies restrict the choice to a flattener (or a steepener) with 1. a fixed tranche and fixed maturities, 2. a fixed tranche and all maturities, 3. all tranches and fixed maturities, 4. all tranches and all maturities (no restrictions), or allow to combine flatteners and steepeners. CD Surfaces Dynamics

33 K H R I N H E R D L E pplications 5-12 Backtesting Consider the time horizons h = 1, 5, 2 days. For the tranches that optimize a given strategy, check the corresponding historical market spreads, calculate the resulting MTM values, and the realised P&L. CD Surfaces Dynamics

34 K H R I N H E R D L E pplications 5-13 Mean of Daily Gains in Percent DSFM DSFM without the mean factor Strategy 1 day 1 week 1 month 1 day 1 week 1 month LZ Z LZ Z LZ Z LZ Z LZ Z LZ Z FS-llT-llM FS-T2-llM FS-T3-llM FS-T4-llM FS-T5-llM F-T2-llM F-T3-llM F-T4-llM F-T5-llM S-T2-llM S-T3-llM S-T4-llM S-T5-llM F-llT F-llT F-llT S-llT S-llT S-llT Table 4: Calculations based on predictions of log-spreads and Z-transformed BCs marked as LZ; based only on Z-transformed BCs marked as Z. CD Surfaces Dynamics

35 K H R I N H E R D L E pplications 5-14 Investor s Strategy Follow a certain strategy over a year and constantly rebalance the portfolio. t t enter an optimal (according to the DSFM) curve trade for h-day horizon. t t + h chose: 1. keep the current position for the next h-days, 2. close the current position and enter a new one. ssume a margin of % of your notional. Every time the position is closed, add to the margin the realized P&L. If margin, quit the trade. CD Surfaces Dynamics

36 K H R I N H E R D L E pplications 5-15 Investor s Strategy Cumulated PL in % Time Cumulated PL in % Time CD Surfaces Dynamics Cumulated PL in % Time Figure 11: Daily cumulated P&L over one year Rebalancing after: 1 day (upper left), 1 week (upper right), 1 month (lower). Calculations based on the DSFM predictions of logspreads and Z-transformed BCs. pp

37 K H R I N H E R D L E Summary 6-1 Conclusions Investigated evolution over time of tranche spread surfaces and base correlation surfaces using the DSFM. Empirical study is conducted using an extensive data set of 49,52 observations of itraxx Europe tranches in Proposed a modification to the classic DSFM. Both DSFMs successfully reproduce the dynamics in data. Used DSFM in constructing the curve trades. nalysed the performance of 43 strategies that combine different positions, tranches, and maturities. Backtesting showed high daily gains of the resulting curve trades. CD Surfaces Dynamics

38 References C. Bluhm and L. verbeck Structured Credit Portfolio nalysis, Baskets and CDs Chapman & Hall/Crc Financial Mathematics Series, 26 J. Felsenheimer, P. Gisdakis, and M. Zaiser DJ itraxx: Credit at its best! Credit derivatives special HVB Corporates & Markets, 24 M. R. Fengler, W. K. Härdle, and E. Mammen semiparametric factor model for implied volatility surface dynamics Journal of Financial Econometrics, 27 C. Gourieroux and J. Jasiak Dynamic factor models Econometric Reviews, 21. Kakodkar, S. Galiani, J. G. Jónsson, and. Gallo Credit derivatives handbook, guide to the exotics credit derivatives market Technical report Merrill Lynch, 26 B. Park, E. Mammen, W. K. Härdle, and S. Borak Dynamic Semiparametric Factor Models Journal of the merican Statistical ssociation, 29

39 D Surfaces Dynamics Barbara Choroś-Tomczyk Wolfgang Karl Härdle stap khrin C K H R I N H E R D L E Ladislaus von Bortkiewicz Chair of Statistics C..S.E. - Center for pplied Statistics and Economics Humboldt-Universität zu Berlin

40 K H R I N H E R D L E ppendix. CD Modelling Introduction 7-1 Default Consider a CD with a maturity of T years, J tranches, and a pool of d entities. Define a loss variable of i-th obligor until t [t, T ] as I i (t) = 1(τ i < t), i = 1,..., d, where τ i is a time to default variable F i (t) = P(τ i t) = 1 exp t t λ i (u)du and λ i is a deterministic intensity function. Talk CD Surfaces Dynamics

41 K H R I N H E R D L E ppendix. CD Modelling Introduction 7-2 Portfolio Loss The proportion of defaulted entities in the portfolio at time t is given by L(t) = 1 d I i (t), t [t, T ]. d i=1 The portfolio loss at time t is defined as L(t) = LGD L(t), where LGD is a common loss given default. CD Surfaces Dynamics

42 K H R I N H E R D L E ppendix. CD Modelling Introduction 7-3 Tranche Loss The tranche loss at time t is defined as where L j (t) = 1 u j l j {L u (t, u j ) L u (t, l j )}, L u (t, x) = min{l(t), x} for x [, 1]. The outstanding notional of the tranche j is given by Γ j (t) = 1 u j l j {Γ u (t, u j ) Γ u (t, l j )}, where Γ u (t, x) = x L u (t, x) for x [, 1]. CD Surfaces Dynamics

43 K H R I N H E R D L E ppendix. CD Modelling Introduction 7-4 Valuation of CD 1. Premium leg T PL j (t ) = β(t, t)s j (t ) t E{Γ j (t)} t=t 1 2. Default leg T DL j (t ) = β(t, t) E{L j (t) L j (t t)} t=t 1 This leads to: s j (t ) = T t=t 1 β(t, t) E{L j (t) L j (t t)} T. t=t 1 β(t, t) t E{Γ j (t)} CD Surfaces Dynamics

44 K H R I N H E R D L E ppendix. CD Modelling Introduction 7-5 Equity Tranche The equity tranche is quoted in two parts: 1. an upfront fee α payed at t, 2. a running spread of 5 BPs. The premium leg is calculated as PL 1 (t ) = α(t ) + T t=t 1 β(t, t) 5 t E{Γ 1 (t)}. The upfront payment given in percent is equal α(t ) = T t=t (β(t, t ) [E{L 1 (t) L 1 (t t)}.5 t E{Γ 1 (t)}]). CD Surfaces Dynamics

45 K H R I N H E R D L E ppendix. CD Modelling Introduction 7-6 Copula For a distribution function F with marginals F X1..., F Xd. There exists a copula C : [, 1] d [, 1], such that for all x i R, i = 1,..., d. F (x 1,..., x d ) = C{F X1 (x 1 ),..., F Xd (x d )} CD Surfaces Dynamics

46 K H R I N H E R D L E ppendix. CD Modelling Introduction 7-7 Copula for CDs The vector of default times (τ 1,..., τ d ) has a (risk-neutral) joint cdf F (t 1,..., t d ) = P(τ 1 t 1,..., τ d t d ) for all (t 1,..., t d ) R d +, where τ i F i. From the Sklar theorem, there exists a copula such that F (t 1,..., t d ) = C{F 1 (t 1 ),..., F d (t d )} and determines the default dependency of the credits. CD Surfaces Dynamics

47 K H R I N H E R D L E ppendix. CD Modelling Introduction 7-8 Monte Carlo Simulation pproach Define a trigger variable as U i = p i (τ i ) U[, 1], i = 1,..., d. The ith obligor survives until t < T if and only if τ i t or U i p i (t). The joint and marginal distributions of the triggers satisfy: C{ p 1 (t),..., p d (t)} = P{U 1 p 1 (t),..., U d p d (t)}, P{U i p i (t)} = p i (t). CD Surfaces Dynamics

48 K H R I N H E R D L E ppendix. CD Modelling Introduction 7-9 Monte Carlo Simulation pproach The time to default variable is calculated as τ i = inf{t t : p i (t) U i } τ i = p 1 i (U i ). ssuming constant intensities compute τ i = (log U i )/λ i. CD Surfaces Dynamics

49 K H R I N H E R D L E ppendix. CD Modelling Introduction 7- Large Pool pproach for Linear Factor Models Default times are calculated from a vector (X 1,..., X d ) X i = ρy + 1 ρz i, where Y (systematic risk factor), {Z i } d i=1 (idiosyncratic risk factors) are i.i.d. ssume that obligors have the same default probability p and LGD, one dependence parameter ρ, d is large. Talk CD Surfaces Dynamics

50 K H R I N H E R D L E ppendix. CD Modelling Introduction 7-11 Large Pool pproximation Computations are simplified significantly when the portfolio loss distribution is approximated: { F 1 X P(L x) = 1 F (p) ρf 1 Z (x) } Y, 1 ρ where X i F X, Z i F Z, Y F Y. CD Surfaces Dynamics

51 K H R I N H E R D L E ppendix. CD Modelling Introduction 7-12 Gaussian Copula Model The factors Y and {Z i } d i=1 are i.i.d. N(, 1). Thus, X i N(, 1) The cdf of the portfolio loss equals { 1 ρφ 1 (x) Φ 1 } (p) P( L x) = Φ. ρ Default times are given by τ i = F 1 i {Φ(X i )}. CD Surfaces Dynamics

52 K H R I N H E R D L E ppendix. CD Modelling Introduction 7-13 NIG Model Factors: Y NIG (α, β, βγ2 α 2, γ3 α 2 ( 1 ρ Z i NIG α, ρ ), γ = α 2 β 2, 1 ρ β, ρ Because of the stability under convolution ( α X i NIG ρ, β ρ, 1 βγ 2 ρ α 2, 1 γ 3 ) ρ α 2 Default times are given by τ i = F 1 i {NIG (1/ ρ) (X i )}. 1 ρ βγ 2 1 ρ ρ α 2, γ 3 ) ρ α 2. = NIG (1/ ρ). CD Surfaces Dynamics

53 K H R I N H E R D L E ppendix. CD Modelling Introduction 7-14 Double-t Model Define X i = νy 2 ρ Y + νz 2 1 ρ Z i, i = 1,..., d, ν Z ν Y where Y and Z i are t distributed with ν Y and ν Z DoF respectively. The t distribution is not stable under convolution: X i are not t distributed and the copula is not a t copula, X i F X has to be computed numerically. Default times are computed as τ i = F 1 i {F X (X i )}. CD Surfaces Dynamics

54 K H R I N H E R D L E ppendix. CD Modelling Introduction 7-15 Large Pool pproach for rchimedean Copulae d-dimensional rchimedean copula C : [, 1] d [, 1] is C(u 1,..., u d ) = φ{φ 1 (u 1 ) + + φ 1 (u d )}, u 1,..., u d [, 1], where φ {φ : [; ) [, 1] φ() = 1, φ( ) = ; ( 1) j φ (j) ; j = 1,..., } is a copula generator. Each φ is a Laplace transform of a cdf of a positive random variable Y F Y φ(t) = e tw df Y (w), t. CD Surfaces Dynamics

55 K H R I N H E R D L E ppendix. CD Modelling Introduction 7-16 Large Pool pproach for rchimedean Copulae If X i, i = 1,..., d, i.i.d. U[, 1] and Y s Laplace transform ( ) is φ, then the rchimedean Copula C is a joint cdf of U i = φ log X i Y. Conditional on the realisation of Y, U i are independent. The large pool approximation of the loss distribution is { } log(1 x) P( L x) = F Y φ 1. ( p) For the Gumbel copula C(u 1,..., u d ; θ) = exp { d i=1 F Y is an α-stable distribution with α = 1/θ. ( log u i ) θ } θ 1, CD Surfaces Dynamics

56 K H R I N H E R D L E ppendix. CD Modelling Introduction 7-17 Correlation s Types Compound correlation ρ(l j, u j ), j = 1,..., J. Compound Correlation.4 Equity Mezzanine Junior Mezzanine Super Senior Senior Tranches Figure 12: Implied correlation smile in the Gaussian one factor model, CD Surfaces Dynamics

57 K H R I N H E R D L E ppendix. CD Modelling Introduction 7-18 Correlation s Types Base correlation (BC) ρ(, u j ), j = 1,..., J. Represent the expected loss E{L (lj,u j )} as a difference: E{L (lj,u j )} = E ρ(,uj ){L (,uj )} E ρ(,lj ){L (,lj )}, j = 2,..., J. of the expected losses of two fictive tranches (, u j ) and (, l j ). Bootstrapping process: E{L (,3%) } is traded on the market, E{L (3%,6%) } = E ρ(,6%) {L (,6%) } E ρ(,3%) {L (,3%) }, E{L (6%,9%) } = E ρ(,9%) {L (,9%) } E ρ(,6%) {L (,6%) },... CD Surfaces Dynamics

58 K H R I N H E R D L E ppendix. CD Modelling Introduction 7-19 Base Correlations ETL Base Correlation Equity Super Senior Senior Mezzanine Mezzanine Junior ρ Tranches Figure 13: Expected loss of the equity tranche calculated using the Gaussian copula model with a one-year default probability computed from the itraxx index Series 8 with 5 years maturity (left) and the base correlation smile (right) on Talk CD Surfaces Dynamics

59 K H R I N H E R D L E ppendix B 8-1 DSFM for Log-Spreads.2 m m Tranche τ Tranche τ m 2 Z t Tranche τ Time Figure 14: Estimated factor functions and loadings (Ẑt,1, Ẑt,2). CD Surfaces Dynamics Talk

60 K H R I N H E R D L E ppendix B 8-2 DSFM without the Mean Factor for Log-Spreads.2 m 5 m Tranche τ Tranche τ m 2 Z t Tranche τ Time Figure 15: Estimated factor functions and loadings (Ẑt,1, Ẑt,2). CD Surfaces Dynamics Talk

61 K H R I N H E R D L E ppendix B 8-3 DSFM for Z-transformed-BC.2 m.2 m Tranche τ Tranche τ 8 3 m Tranche τ 8 Z t Time Figure 16: Estimated factor functions and loadings (Ẑt,1, Ẑt,2). CD Surfaces Dynamics Talk

62 K H R I N H E R D L E ppendix B 8-4 Investor s Strategy Final PL in % 2 Final PL in % 2 Final PL in % Time Time CD Surfaces Dynamics Time Figure 17: Combined flatteners and steepeners from all tranches and all maturities. Closing profits after one year. Rebalancing after: 1 day (upper left), 1 week (upper right), 1 month (lower). Calculations based on the DSFM predictions of logspreads and Z-transformed BCs. Talk