expected loss over lifetime

Expected Loss Over Lifetime

Professor Daniel RoeschUniversity of Regensburg 15 March 2016

2

The views expressed in the following material are the

author’s and do not necessarily represent the views of

the Global Association of Risk Professionals (GARP),

its Membership or its Management.

Expected Loss Over Lifetime

Steffen Krüger, Toni Oehme, Daniel Rösch, HaraldScheule

Chair of Statistics and Risk Management, Faculty of Business, Economicsand Management Information SystemsUniversität Regensburg

Finance Discipline Group, University of TechnologySydney

March 15, 2016

Contents

1. Motivation

2. Methods

3. Data and Estimation Results

4. Expected Loss Over Lifetime Results

5. Conclusion

Expected Loss Over Lifetime | Daniel Rösch | UR / UTS Sydney March 15, 2016 2 / 34

Motivation

Agenda

1. Motivation

2. Methods



5. Conclusion


Motivation

Credit loss

• Credit loss is determined by

L = D · LR · EAD (1)

where

D : Default indicator (1, if default, 0, else)LR : Loss rate

EAD : Exposure at default

• Here: EAD = 1


Motivation

Expected credit loss

• Expected credit loss is given by

E(L) = E(D) · E(LR) + Cov(D, LR)= E(D) · E(LR|D = 1) (2)= E(D) · E(LGD) (3)

where LGD is the loss (rate) given default.

→ What role does the link between default risk and LGD play?


Motivation

Regulatory Requirements

• BCBS (2005)• Downturn LGD:: ”[...] reflect economic downturn conditions where

necessary to capture the relevant risks.”• ”Under such conditions default rates are expected to be high so that if

recovery rates are negatively related to default rates, LGD parametersmust embed forecasts of future recovery rates that are lower thanthose expected during more neutral conditions.”

• BCBS (2009)• ”[...] there is need to cover substantially longer periods [...] as liquidity

conditions can change rapidly in stressed conditions.”• ”The bank should [...] assess the impact of recession-type scenarios,

including its ability to react over a medium to long time horizon.”

• IASB (2014): Impairment in IFRS 9 Financial Instruments• Lifetime expected credit losses: ”The expected credit losses that result

from all possible default events over the expected life of a financialinstrument.”


Motivation

IASB (2014)Impairment in IFRS 9 Financial Instruments

Stage Impairment requirement Impairment recognition

1 Origination / purchase 12-month expected credit losses2 Significantly increased credit risk Lifetime expected credit losses3 Credit-impaired Lifetime expected credit losses

• Introduction of concept of Lifetime expected credit losses (LEL)

→ Time-dependence of LGD estimates?


Motivation

Illustration of LGD Term Structure


Motivation

Literature on PD and LGD odels

• Default risk• PD models

• Altman (1968), Merton (1974), Gordy (2000), and Campbell et al. (2008)• Survival analysis

• Lee and Urrutia (1996), Shumway (2001), Duffie et al. (2007) and Duffieet al. (2009)

• Loss given default• Carey (1998), Pykhtin (2003), Qi and Yang (2009), Huang and Oosterlee

(2012) and Jankowitsch et al. (2014)

→ No actual dependence investigated


Motivation

Literature on PD/LGD Dependence ModelingThe link between default risk and losses given default

• LGDs are positively correlated with default rates• Frye (2000), Altman et al. (2005) and Acharya et al. (2007)

• ’Jointly’ modeling of default and LGD component• Chava et al. (2011), Bellotti and Crook (2012), Frye and Jacobs Jr

(2012)

• Ignoring dependence and sample selection results in biasedparameter estimates

• Rösch and Scheule (2014)

→ Need for further research


Motivation

This Paper: Two Important Issues andExtensions

1. ”Classical” PD and LGD models are separate modules:• A PD model and a stand-alone LGD model• However: LGD can only be observed conditional on a default• This imposes a sample selection mechanism• As known from early work (eg. Tobin, 1958), this creates inconsistent

LGD estimates, if PD and LGD are correlated and if this not properlyaddressed

2. ”Classical” PD models are one-periodic (eg. have a one-yearforecasting horizon)

• Need to model multi-year/lifetime defaults and the term structure ofLGDs over the lifetime

• As well as interaction (correlation) between default and LGD termstructure


Motivation

Contributions

• We propose a model for expected loss over lifetime (LEL) whichtakes into account

• Dependence between time-to-default and LGD via Copulas• Sample selection

• Derive term structures for PDs and LGDs

• Empirical strategy for estimation

• Estimation and LEL forecasting results for real-world data


Methods

Agenda

1. Motivation

2. Methods



5. Conclusion


Methods

Expected Loss over Lifetime

• ELoL

LEL = E(1{T ≤m} · LGDT · b(T )

), (4)

where m denotes the maturity and b(T ) a discount factor• This is equivalent to

LEL =m∫

0

1∫0

fLGDT ,T (l, t) · l · b(t) dl dt (5)

• Dependence between default time (PD) and LGD is taken intoaccount in two ways:

• Deterministic: using joint covariates• Stochastic: using copulas

• Sample selection is addressed by adjusting the Likelihood


Methods

The Full Model and its Constituents I

• Maximum likelihood estimation• Likelihood:

L(βT , σ, βµ, ϕ, θ) =∏

i:Di=0

( 1 − FTi(ti) )∏

i:Di=1

f(Ti,Yi)(ti, yi)︸︷︷︸πit

(6)

where

πit = c(FTi(ti), FYi(yi)

)· fTi(ti) · fYi(yi) (7)

• PD: Survival (AFT) Model (T , βT , σ )

log Ti = β′T xT

i + σεi, (8)

• LGD: Beta Regression (Y , βµ, ϕ)


Methods

The Full Model and its Constituents II

• Let the LGD be described by the beta distributed random variable Y:

fY (y) = 1B(α, β)yα−1(1 − y)β−1, (9)

with parameters α, β > 0 and beta function B : (0, ∞)2 → R2

• Ferrari and Cribari-Neto (2004)

µ = α

α + βand ϕ = α + β. (10)

µi = 11 + exp(−β′

µxµi ) . (11)

• Copula (θ) with density c(·, ·)


Methods

Introduction to Copulas

Theorem (Special case of Sklar (1959))Let X and Y be univariate continuous random variables with cumulativedistribution functions FX , FY and joint distibution function F(X,Y ).

Then there exists a unique function C : [0, 1]2 → [0, 1] with

F(X,Y )(x, y) = C(FX(x), FY (y)) = C(u, v), u, v ∈ [0, 1], (12)

where C is called copula.

Example (Gaussian copula)

C(u, v) = Φ2(Φ−1(u), Φ−1(v); ρ), u, v ∈ [0, 1], ρ ∈ [−1, 1]. (13)


Methods

Properties of Analysed Copulas

Generator ParameterCopula Cθ(u, v), u, v ∈ [0, 1] φθ(t) space for θ

AMH uv1 − θ(1 − u)(1 − v) log 1 − θ(1 − t)

t [−1, 1)

Clayton (u−θ + v−θ − 1)− 1θ 1

θ(t−θ − 1) (−∞, 0) ∪ (0, ∞)

Frank − 1θ

log(

1 + (e−θu − 1)(e

−θv − 1)(e

−θ − 1)

)− log e

−θt − 1e

−θ−1 (−∞, ∞)

Gaussian Φ2(Φ−1(u), Φ−1(v); θ) - [−1, 1]

Gumbel exp(

−(

(− log(u))θ + (− log(v))θ) 1

θ

)(− log(t))θ [1, ∞)

Joe 1 −(

(1 − u)θ + (1 − v)θ − (1 − u)θ(1 − v)θ) 1

θ − log(1 − (1 − t)θ) [1, ∞)

Product uv − log(t) -

Student’s t t2(t−1(u), t−1(v); θ) - [−1, 1]


Methods

Scatterplot of various copulas

• Rank correlation coefficient Kendall’s τ =

{0, if Product copula,

0.3, else.


Data and Estimation Results

Agenda

1. Motivation

2. Methods



5. Conclusion



Data

• Moody’s Default & Recovery Database (DRD)• Default and recovery data• Lifetime US-corporate bond data with long-term rating• 1982 - 2014• 48,828 observations (2,455 defaults)• Control variables

• Bond-specific (rating, seniority, maturity, face amount, coupon)• Issuer-specific (excess return, market-to-book-ratio,

net-income-to-total-assets, market-cap., liabilities-to-total-assetsindustry)

• Macro-economic (industry production, term spread, downturn andvintage effects)

data

LGD

Fre

quen

cy

0.0 0.2 0.4 0.6 0.8 1.0

0

100

200

300

400



Default Rates and Mean LGDs



Models for Time to Default



Models for LGD



Copula Results

• Implies negative dependence of time-to-default and LGD• Biased parameter estimates otherwise• Best choice (Mc-Fadden R2):

• Log-logistic time-to-default• Frank copula


Expected Loss Over Lifetime Results

Agenda

1. Motivation

2. Methods



5. Conclusion



Analysis of Specific Risk Buckets



LGD Densities (Term Structures)

• Earlier default implies c.p. higher LGD and vice versa• Defaults just after origination by surprising and severe causes• Survival in a first part of maturity demonstrates financial strength



Term Structures for PD and LGD



Model Differences



LEL Predictions for Industrial Bonds


Conclusion

Agenda

1. Motivation

2. Methods



5. Conclusion


Conclusion

Summary

• Link between time-to-default and LGD• Provide general model for Expected Loss over Lifetime• Derive term structures for PDs, LGDs, and ELoL• Negative dependence between time-to-default and LGD after

controlling for covariates

→ Ignoring dependence results in• Biased parameter estimates• Underestimation of risk


References

References I

Acharya, V. V., Bharath, S. T., Srinivasan, A., (2007). Does industry-wide distress affect de-faulted firms? Evidence from creditor recoveries. Journal of Financial Economics85 (3), 787–821.

Altman, E. I., (1968). Financial ratios, discriminant analysis and the prediction of corporatebankruptcy. The Journal of Finance 23 (4), 589–609.

Altman, E. I., Brady, B., Resti, A., Sironi, A., (2005). The Link between Default and Recov-ery Rates: Theory, Empirical Evidence, and Implications. Journal of Business 78 (6),2203–2228.

BCBS, (2005). Guidance on Paragraph 468 of the Framework Document. Bank for Interna-tional Settlements, Basel.

BCBS, (2009). Principles for sound stress testing practices and supervision. Bank for Interna-tional Settlements, Basel.

Bellotti, T., Crook, J., (Jan. 2012). Loss given default models incorporating macroeconomicvariables for credit cards. International Journal of Forecasting 28 (1), 171–182.

Campbell, J. Y., Hilscher, J., Szilagyi, J., (2008). In search of distress risk. The Journal of Fi-nance 63 (6), 2899–2939.

Carey, M., (1998). Credit risk in private debt portfolios. The Journal of Finance 53 (4), 1363–1387.


References

References II

Chava, S., Stefanescu, C., Turnbull, S., (2011). Modeling the Loss Distribution. ManagementScience 57 (7), 1267–1287.

Duffie, D., Eckner, A., Horel, G., Saita, L., (2009). Frailty correlated default. The Journal ofFinance 64 (5), 2089–2123.

Duffie, D., Saita, L., Wang, K., (2007). Multi-period corporate default prediction withstochastic covariates. Journal of Financial Economics 83 (3), 635–665.

Ferrari, S., Cribari-Neto, F., (2004). Beta regression for modelling rates and proportions.Journal of Applied Statistics 31 (7), 799–815.

Frye, J., (2000). Depressing recoveries. Risk 13 (11), 108–111.

Frye, J., Jacobs Jr, M., (2012). Credit loss and systematic loss given default. Journal of CreditRisk 8 (1), 1–32.

Gordy, M. B., (2000). A comparative anatomy of credit risk models. Journal of Banking &Finance 24 (1), 119–149.

Heckman, J. J., (1979). Sample selection bias as a specification error. Econometrica, 153–161.

Huang, X., Oosterlee, C. W., (2012). Generalized beta regression models for random loss-given-default. The Journal of Credit Risk 7 (4), 45–70.

IASB, (2014). IFRS 9 Financial Instruments.


References

References III

Jankowitsch, R., Nagler, F., Subrahmanyam, M. G., (2014). The determinants of recoveryrates in the US corporate bond market. Journal of Financial Economics 114 (1), 155–177.

Lee, S. H., Urrutia, J. L., (1996). Analysis and prediction of insolvency in the property-liabilityinsurance industry: A comparison of logit and hazard models. Journal of Risk andInsurance, 121–130.

Merton, R. C., (1974). On the Pricing of Corporate Debt: The Risk Structure of InterestRates. The Journal of Finance 29 (2), 449–470.

Pykhtin, M., (2003). Unexpected recovery risk. Risk 16 (8), 74–78.

Qi, M., Yang, X., (2009). Loss given default of high loan-to-value residential mortgages.Journal of Banking & Finance 33 (5), 788–799.

Rösch, D., Scheule, H., (2014). Forecasting probabilities of default and loss rates given de-fault in the presence of selection. Journal of the Operational Research Society 65 (3),393–407.

Shumway, T., (2001). Forecasting Bankruptcy More Accurately: A Simple Hazard Model. TheJournal of Business 74 (1), 101–124.

Sklar, M., (1959). Fonctions de répartition à n dimensions et leurs marges. Université Paris8.

Tobin, J., (1958). Estimation of relationships for limited dependent variables. Econometrica26 (1), 24–36.


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expected loss over lifetime

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