Comput Econ https://doi.org/10.1007/s10614-018-9825-6

Stress Testing for Retail Mortgages Based on Probability Analysis Chang Liu1

· Raja Nassar2

Accepted: 28 May 2018 © Springer Science+Business Media, LLC, part of Springer Nature 2018

Abstract One big problem with stress testing used by banks, regulators, and international financial organization is that the test does not predict occurrence probabilities of certain pre-specified stress scenarios and their consequent loss to be expected, which is, however, the real purpose of stress testing in the first place. As a result, institutes lack information sufficient enough for preserving appropriate resources to hedge risks prompted by these scenarios. In this study we use real life retail mortgages from a Chinese commercial bank and propose a stress testing approach based on probability analysis of different scenarios. This method would provide not only the amount of expected loss, but also that of the loan distributed over the loan classification states: Standard, Special Mention, Substandard, Doubtful, Loss, and Paid-off. Consequently, the bank management would have useful information when making credit operation policy decisions. In addition, the models and algorithms, providing practical risk management tools for banks and regulators, could be implemented on other commercial credit products as well. Keywords Stress testing · Non-stationary Markov chains · Copula · Simulation · Retail mortgages

B

Chang Liu [email protected] Raja Nassar [email protected]

1

Southwestern University of Finance and Economics, 555 Liulin Ave, Chengdu 610041, China

2

Louisiana Tech University, 3000 English Turn, Ruston, LA 71270, USA

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1 Introduction Stress testing has been used by commercial banks, central banks, international financial organizations, and regulators to evaluate robustness of a loan, strength of a financial institute, and even stability of a financial market. According to “Principles for sound stress testing practices and supervision” (2009), stress testing should be implemented on pre-specified scenarios or historical scenarios that give worst case consequences, given the shocks from these two types of scenarios. For pre-specified scenarios, stock market crash and interest rate drop have been used. For historical scenarios, those that happen within 25 years or 75 years should be implemented to estimate worst case consequences in loans, banks and financial markets, as recommended by BIS and International Monetary Funds (IMF). Stress testing has two important implications. First, the scenarios for stress testing input have to be plausible for the test to make any sense. Second, stress testing is a risk management tool based on probability analysis. For instance, a scenario involving a loss of $5,000,000 with an occurance probability of 0.005%, should not be of equal concern as a scenario involving a loss of $5,000,000 and an occurrence probability of 5%. As a result, in selecting a scenario to come up with a meaningful stress test, we need to choose the relevant scenario from the plausible space and provide the risk management stress testing outcome with an accompanying probability of occurence. Only by so doing would stress testing be a practical and complementary approach in risk prediction and hedging. In this study, a non-stationary Markov chain, inspired by Smith et al. (1996), and t-Copula simulation with an empirical marginal distribution, inspired by Genest and Mackay (1986), would be used to estimate the retail mortgage portfolio probability distribution over the loan classification states: Standard, Special Mention, Substandard, Doubtful, Loss, and Paid-off states, with the expected amounts distributed over those states. This study differs from the current stress testing mainly in that it does not take predefined testing scenarios into the simulation directly. It first estimates the occurrence probability for each scenario in order to determine its rationale to be considered in the simulation. In addtion, the approach proposed in this study predicts the distribution of monetary loss over the states of the portfolio. The proposed stress testing excels other tests in that it predicts the probability of occurrence for each scenario. Without the probability of occurrence of a scenario, financial institutions find it very difficult to allocate resources to buffer monetary loss. This study proposes a non-stationary Markov chain model and simulation algorithms based on the t-Copula function in order to estimate the probability of occurrence of a given scenario and to evaluate plausible sets of scenarios to be used in stress testing. As a result, the method could be used by government, regulators, and financial institutions for quantitative risk control and precise solvency management. This paper is organized as follows: Sect. 1 is a brief introduction of background information. In Sect. 2, a literature review of the relevant topic is presented in a chronological order. Section 3 presents the models and algorithms for the estimation of scenario shock magnitude and probability. Section 4 implements the retail mortgage data provided by a Chinese commercial bank. Finally, Sect. 5 concludes this study.

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2 Literature Review According to Berkowitz (2000), stress testing should be implemented by the simulated scenario in a coherent framework, where coherence is defined as follows: The original  scenarios   y  P(X ) are used to build a model, and the final evaluated model y  P 



Xf 



is implemented by using the simulated plausible stress test scenarios X f  

( x 1 , x 2 , . . . , x n )T .This coherent framework is adopted in this study. Extreme value theory, or EVT , has been used by Legras and Soupé (2000) to select multivariate stress scenarios. They proposed to use quantile to measure the probability of certain scenarios. Defining a current portfolio value as P0 and a portfolio value with scenario r as P1 , Breuer and Pistovcak (2004) propose to use a quasi-random method to search the maximum loss (Max Lossε  P0 − P1 (rwc )  max(P0 − P1 (r ))) over a plausible r ∈ε

domain ε, defined as ε  {r ∈ R n |r T · Σ −1 · r ≤ k 2Max , k Max ∈ R+ } where rwc is the worst case scenario and k Max is a parameter measuring the plausibility level. Breuer (2007) propose a way to systematically search for plausible, severe, and suggestive scenarios that could be used in a meaningful stress test. For risk measure in a coherent framework, the maximum loss for a portfolio is defined as Max Loss A P  ess sup A (PC M − P)  inf{m ∈ |PC M − P ≤ m υA · as} where PC M is the current market value of the portfolio,P is the worst possible value of the same portfolio and A is the admissibility domain of the possible scenarios. Intuitively, Breuer (2007)’s model could be understood as the essential supremum set for the difference between the current market value of the portfolio and its value under a shock in the admissible domain A. Letting r  (r1 , r2 , . . . , rn )T be a vector of risk factor scenarios and μ  (μ1 , μ2 , . . . , μn )T a vector of its  mean value, Breuer et al. (2009) suggest using Mahalanobis distance,Maha(r )  (r − μ)T Cov −1 (r − μ) to measure the deviation of the scenario from the normal state, which is represented by its mean. Besides, a similar approach to select the worst case scenario is also reported by Breuer et al. (2008). Several specific scenarios are commonly used for stress testing, including Credit Risk, Interest Rate Risk, Foreign Exchange Risk, Interbank (Solvency) Contagion Risk and Sovereign Risk. Many historical scenarios like Black Monday 1987, Bond market crash 1994, Middle East crisis, and hypothetical ones like Global tightening and Global emerging market crises could be found in “a survey of stress tests and current practice at major financial institutions” (2001). Stress testing now is useful to and popular among financial systems. For instance, European Banking Authority (EBA) has implemented EU-wide stress testing since 2009, aiming to assess the resilience of financial institutions to adverse market developments, as well as contributing to the overall assessment of systemic risk in the EU financial system. Applications and results of stress testing could be found in Macro stress tests of UK banks (Hoggarth et al. 2005), Macro Stress and Worst Case Analysis of Loan Portfolios (Breuer et al. 2008), A Framework for Assessing the Systemic Risk of Major Financial Institutions (Huang et al. 2009), Stress Testing Credit Risk: The Great Depression Scenario (Varotto 2011), Stress Testing in the Investment Process

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(Ruban et al. 2010), Stress Testing in Wartime and in Peacetime (Schuermann 2016), and “2016 EU-wide stress test results” (2016). More general surveys about stress testing practices could be found in Blaschke et al. (2001), Bank for International Settlements (2005), and Sorge (2004). For official banking stress testing guidance or policy direction, the reader is referred to “Principles for sound stress testing practices and supervision” (2009). Attention has been seldom given to the estimation of the probabilities associated with different scenarios, which provide crucial information on practical risk management for commercial banks and regulators. As a result, it is the attempt of the present study to fill this gap by providing an approach to predict the probabilities of occurrence for scenarios of different magnitudes and the expected losses incurred as well as to provide crucial information for credit operation and decision making.

3 The Models and Algorithms To predict the portfolio distribution over pre-specified states under scenarios with different shock magnitudes, a non-stationary Markov chain model can be used to express the evolutionary process of the portfolio. To minimize model error, t-Copula, instead of Logistic Regression as in Smith et al. (1996), would be used to simulate the functional relationship between the transition probabilities Pitj and macro-risk factors, or scenarios, X, X ∈ (x1 , x2 , . . . , xn )T , according to the coherent framework proposed by Berkowitz (2000). Let Θ be a space of plausible, severe, and suggestive scenarios by Breuer (2007). The transition probabilities under different shock scenarios are given by (.|X ∈ Θ), with corresponding occurrence probabilities P˜it+1 (.|X ∈ Θ). Finally, Pit+1 j j let Π be a set of loan states, and the expected portfolio amounts over different states are given by Max {E(Π t+1 )} with the estimation of portfolio amount in different states X ∈Θ

as Max {Π t+1 }, and scenario occurrence probabilities as P˜ t+1 (Π, X ∈ Θ). X ∈Θ

3.1 Non-stationary Markov Chain (NMC) Smith et al. (1996) provide an excellent framework for a non-stationary Markov Chain. In their study, they use logistic regression to link the transition probabilities with macro-economic factors, or risk factors. Predicting the transition probabilities from the risk factors allows one to predict the portfolio distribution. In our study, however, a non-stationary Markov Chain (NMC) model with t-Copula simulation is used. Instead of using any specific analytic model, which may not be valid, we propose using simulation to relate the macro-economic risk factors to the transition probabilities of the NMC. By applying shocks of different scenarios to the risk factors, one can determine the effects on the transition probabilities and hence the portfolio distribution over the five mortgage states in Table 1. Let P t (Π ) be the last date of the observed retail portfolio distribution and PSt+1 D (Π ) be the estimated distribution on the same portfolio under different shock scenarios,

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Stress Testing for Retail Mortgages Based on Probability… Table 1 Portfolio distributions P t (Π ), Π ∈ {A, B, C, D, E, F} The last observed probability

Π ∈ {A, B, C, D, E, F} Definition of the loan states

A:

Standard

Loans with no past due

P t ( A)

Probability of The Standard State

B:

Special Mention

Loans with 1–90 days past due

P t (B)

Probability of The Special Mention State

C:

Substandard

Loans with 91–360 days past due

P t (C)

Probability of The Substandard State

D:

Doubtful

Loans with 361–720 days past due

P t (D)

Probability of The Doubtful State

E:

Loss

Loans with more than 721 days past due

P t (E)

Probability of The Loss State

F:

Paid-off

Loans with the whole balance paid-off

P t (F)

Probability of The Paid-off State

where SD is the magnitude of each shock scenario. Table 1 gives definitions of loan states and portfolio distribution according to one Chinese commercial bank. Using monthly data, the transition matrix could be expressed as:

(1)

Transitions only happen from i ∈ {A, B, C, D, E} to j ∈ {A, B, C, D, E}. This is true since State F is an absorbing state. Once a loan is paid off, it would be written-off from the bank’s books and no future transitions could be recorded from it. Hence,Pit+1 j  0, i  F, j  A, B, C, D, E, F. Becasue banks usually use monthly data to provide the risk report, the transition from a Standard State, defined as loans with no past due days, to a Substandard State, defined as loans with 91–360 days past due, could not happen within 30 days. As a result,PAC  0. The same reasoning applies to other zero transitions in the transition matrix above. Also, in practice, if a loan is more than 90 days past due, banks normally do not change the state back to Normal, or A, immediately after its past due balance

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t +1

Standard P t ( A)

Pijt +1 , i = A, j = A, B, F

Standard P

Special Mention P t ( B)

Pijt +1 , i = B, j = A, B, C , F

Special Mention P t +1 ( B)

Substandard P t (C )

Pijt +1 , i = C , j = C , D, F

Substandard P t +1 (C )

Pijt +1 , i = D, j = D, E , F

Doubtful P t +1 ( D )

Pijt +1 , i = E , j = E , F

Loss P t +1 ( E )

t

Doubtful P ( D ) t

Loss P ( E )

t

Payoff P ( F )

( A)

Payoff P t +1 ( F )

Fig. 1 Transition process

has been paid off unless the whole balance on the account is paid off. In this latter case, however, the loan would transit to the paid off State F. As a result, the transitions C-A, C-B, D-A, D-B,D-C, E-A, E-B, E-C, and E-D are zeros. Based on Smith et al. (1996), the transition process in this Chinese commercial bank could be presented as in Fig. 1.

3.2 t-Copula t-Copula is used to determine the effects of risk factors on the transition probabilities. The risk factors to be considered are those that are correlated with the transition probabilities as determined by the nonparametric Kendall’s Tau, according to Genest and Mackay (1986). According to Demarta and McNeil (2010), multivariate t-Copula is given by: ⎧ 

t ⎪ ⎨ C t Pi j , x1 , x2 , . . . , xn , τ  Cτ,υ F1−1 (x1 ), F2−1 (x2 ), . . . , Fn−1 (xn )  − υ+n 

−1

−1 2 Γ ( υ+n X τ −1 X ⎪ t (x , x , . . . , x )  F1 (x1 ) . . . Fn (xn ) 2 ) √ ⎩ Cτ,υ d X, 1 + 1 2 n υ −∞ −∞ υ Γ ( 2 ) (πυ)n |τ |

(2)

where νi denotes the degrees of freedom for the i-th univariate t-distribution. F −1 (Pi j ) is the inverse of the empirical distribution function for the transition probabilities, Fk−1 (xk ), k  1, 2, . . . , n are the inverses of the empirical distribution functions for the macro-risk factor, υ is a vector of the degrees of freedom for the t-distribution, υ  (υ0 , υ1 , . . . , υk )T , k  1, 2, . . . n, and τ is a nonparametric correlation coefficient matrix. Equation (2) estimates the transition probabilities Pi j by considering the correlation between them and their corresponding risk factors X  (x1 , . . . , xk )T , k  1, 2, . . . , n under different shock scenarios.

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Stress Testing for Retail Mortgages Based on Probability… Algorithm 1 Implementation of multivariate t-Copula simulation Step 1: Calculate Kendall’s τ and its P Value matrix. Then, record the significant elements in the first column with their P-Values less than 5%. Step 2: Use Kendall’s τ and generate t-distributed random variables T  (t, t1 , t2 , . . . , tn )T with degrees of freedom υ. Step 3: Calculate the multivariate cumulative t-distribution function: U  C D F(t, t1 , t2 , . . . , tn ). Step 4: Upload the transition probabilities and their corresponding macro-risk factors in columns to a temporary matrix Temp. Sort each column in matrix Temp and calculate the empirical distribution for each of them. Step 5: Calculate the inverse cumulative distribution functions: F −1 (Pi j ), F1−1 (x1 ), F2−1 (x2 ), . . . , Fn−1 (xn ). Step 6: Calculate the final simulated matrix, M, by mapping the results from Step 5 to Step 3 in such a way that the joint cumulative probability distribution of each elements in M is equivalent to U from Step 3.

The effect of Step 6 in Algorithm 1 is actually a mapping procedure that randomly draws elements from F −1 (Pi j ), F1−1 (x1 ), F2−1 (x2 ), . . . , Fn−1 (xn ) according to the probability law (probability of occurrence and correlation coefficients estimated in Step 2) represented by the multivariate t distribution simulated in Step 3.3.3 Scenario Probability Analysis. To estimate the expected worst case portfolio distribution Max {E(Π t+1 )}, Π ∈ X ∈Θ

{A, B, C, D, E, F}, t-Copula would be used to first estimate the transition prob, and therefore the portfolio distribution PXt+1 abilities Pit+1 ∈Θ (Π ), Π ∈ {A, B, j,X ∈Θ C, D, E, F} under scenarios with different shock magnitudes. Consider the Loss State as an example and let M t be the entire value of the portfolio at the last observation date t, and then the expected loss amount under a certain scenario is given by:   Max E{E t+1 } X ∈Θ    Max E{M t · P t+1 (E)} X ∈Θ    Max M t · P t+1 (E) · P˜ t+1 (E, X ∈ Θ) X ∈Θ   E  t t t+1 t+1 P (E) · P · P˜ (E, X ∈ Θ) , (3)  Max M · X ∈Θ

i E,X ∈Θ

iD

, given in Sect. 3.2.2, is where P t (E) is the observed probability of state E, Pit+1 E,X ∈Θ the transition probabilities from state i ∈ {D, E} to state E and P˜ t+1 (E, X ∈ Θ) is the joint probability of certain scenario and loss amount:

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P˜ t+1 (E, X ∈ Θ)  P˜ t+1 (E|X ∈ Θ) P˜ t+1 (X ∈ Θ) E 



· P˜ t+1 (X ∈ Θ), P˜it+1 E|X ∈Θ

(4)

iD

, i ∈ {D, E} are the scenario probabilities of the transition probabilities while P˜it+1 E|X ∈Θ conditioned on certain macro-risk shocking, and P˜ t+1 (X ∈ Θ) are the occurrence probabilities of certain macro-risk scenarios. Both are given in Sect. 3.2.1. As a result, expected distributions amounts under a certain scenario are given as:  Min E{A

t+1

X ∈Θ

 Max E{B

X ∈Θ

t+1

X ∈Θ

 Max E{C

t+1

Max E{D X ∈Θ

 Min E{F X ∈Θ

}  Max

M · t

}  Max

M · t

}  Max X ∈Θ

t+1

M · t

D 

}  Min M · t

X ∈Θ

˜ t+1

P (B) · Pi B,X ∈Θ · P t+1

(B, X ∈ Θ) ,

(6)

 ˜ t+1

P (C) · PiC,X ∈Θ · P t

t+1

(C, X ∈ Θ) ,

(7)

 ˜ t+1

P (D) · Pi D,X ∈Θ · P t

t+1

(D, X ∈ Θ) ,

iC





C 

(5)

 t

iB





B  iA





X ∈Θ

t+1

iA





X ∈Θ

X ∈Θ



  B   t t t+1 t+1 }  Min M · P ( A) · Pi A,X ∈Θ · P˜ ( A, X ∈ Θ) ,

E 

(8)

 ˜ t+1

P (F) · Pi F,X ∈Θ · P t

t+1

(F, X ∈ Θ) ,

(9)

iA

Note that the worst case for Standard and Paid off states is estimated for the minimum amount. From the discussion above, one sees that actually two sets of estimations need to be performed, namely, the estimation of the magnitude of the consequence from the scenario shock Pit+1 , i ∈ {A, B, C, D, E}, j ∈ {A, B, C, D, E, F} j,X ∈Θ and the estimation of occurrence probabilities of scenarios, P˜ t+1 (Π, X ∈ Θ), Π ∈ {A, B, C, D, E, F}. 3.2.1 Occurrence Probabilities of Certain Macro-Risk Scenarios For each 15 transition probabilities in the transition matrix Pi j of Eq. (1), a particular set of macro-risk factors needs to be found. Let Pitj , i ∈ {A, B, C, D, E}, j ∈ {A, B, C, D , E, F} be any transition probability from state i to state j in the time T  interval (0, t) and X it j ∈ Θ, X it j  xit j,1 , . . . , xit j,m , m  1, . . . , n be a particular set of macro-risk factors that are significantly correlated with Pitj during the same period of time, and we use Eq. (2) and Algorithm 1 to simulate the scenarios with different shock magnitudes with their accompanying probabilities P˜ t+1 (X i j ∈ Θ).

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Fig. 2 Inconsistent estimation of P˜ t+1 (X i j ∈ Θ)

In practice, for each Pitj , i ∈ {A, B, C, D, E}, j ∈ {A, B, C, D, E, F}, there is a corresponding set of significant macro-risk factors simulated. As a result, when one uses Berkowitz (2000)’s coherent stress testing framework, it would be very difficult to make the estimation of scenarios shock magnitudes, measured by the standard deviations from its mean, and their accompanying probabilities, represented by its quantiles, consistent over different Pitj , i ∈ {A, B, C, D, E}, j ∈ {A, B, C, D, E, F}. This is demonstrated in Fig. 2. Algorithm 2 has been developed to overcome this dilemma We can note from Fig. 2 that if we sort each simulated column and find out the magnitudes and the corresponding probabilities, they are not necessarily consistent x t+1

x t+1

x t+1

A A,1 A A,2 E F,m

 sd∼ q5%

 · · ·  sd∼ q5% in terms of coherence. In other words, sd∼ q5% That is, the simulated accompanying probabilities of the magnitudes differ by sets of macro-risk factors, and vary even within each set, which makes the next simulation of transition probabilities very difficult. Therefore, Algorithm 2 is required to get around this issue.

Algorithm 2 Estimating the accompanying probabilities P˜ t+1 (X i j ∈ Θ) Step 1: For each transition probability and its corresponding set of macro-risk factors, run Algorithm 1. Step 2: Sort each column in M1, and store the sorted result in columns in a temporary matrix, T. Step 3: For each column in Matrix T, find the mean and standard deviation for each column. Step 4: For each column in Matrix T, find the corresponding value of quantile 5%, 10%,…, 100%, calculate their distances to the mean as multiples of the standard deviation respectively, and store the distances in a new matrix, M2. Step 5: Output Matrix M2.

As could be seen from Algorithm 2, the procedure to determine the distance to the mean for each time series macro-risk factor obeys the recommendation and policy of BIS (2009). Algorithm 2 will simulate the historical scenarios up to 25 years with shock magnitudes measured by their distances to their means and probabilities measured by quantiles. As a result, Algorithm 2 gives consistent estimates of macro-risk factor scenarios with different shock magnitudes and different accompanying shock probabilities just as demonstrated in Fig. 3.

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Fig. 3 Consistent estimation of P˜ t+1 (X i j ∈ Θ)

As presented in Fig. 3, Algorithm 2 approximates the historical scenario shock magnitudes and put them in corresponding probability categories that are measured by quantiles. For instance, let X tA A ∈ {x tA A,1 , . . . , x tA A,m } be a set of macro-risk factors that are significantly correlated with the transition probability PAt A . Using the data set from time 0 to time t and using Algorithm 2, we can estimate multiples of the standard x t+1

A A,1 deviation for x A A,i , i  1, 2, . . . , m as sd∼ q5% , corresponding to a probability of 0.05. Following the same procedure, we can obtain the magnitudes for the significant macro-risk factors. In other words, we now have 20 historical scenarios with different shock magnitudes, linked to their corresponding probabilities. This finishes the estimation of P˜ t+1 (X i j ∈ Θ), i ∈ {A, B, C, D, E}, j ∈ {A, B, C, D, E, F}. Note that we used X t for the observed macro-risk factors and X t+1 for simulated macro-risk factors scenarios.

3.2.2 Transition Probabilities Under Scenarios with Different Shock Magnitudes First, we assume that the linkage between the loan state transition probabilities Pit+1 j , i  A, B, C, D, E, j  A, B, C, D, E, F and the macro-economic risk factors X t ∈ (x1 , x2 , . . . , xn )tT could be described by the following equation:  Pitj  f (X t )  f (x1 , x2 , . . . , xn )tT , i  A, B, C, D, E, j  A, B, C, D, E, F

(10)

No particular model type would be specified in this study. Instead, simulation would be used to estimate the effects of different shock scenarios on the transition probabilities in order to address issues such as “Model misspecification”. The Function f in Eq. (10) should not be understood as a “real” function since no particular model type is assumed in this study. It is merely a relationship summarized by the link f . As a result, for a given shock scenario, the shift in each macroeconomic risk factor (in terms of a risk factor standard deviation) will determine the shift in the magnitude of the mortgage loan transition probabilities Pit+1 j , i  A, B, C, D, E, j  A, B, C, D, E, F. This shift can be estimated as follows:

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⎧

f (x1 , x2 , . . . , xn )tT |Sc , Sc  Scenario 1 ⎪ ⎪ ⎨ f (x1 , x2 , . . . , xn )tT |Sc , Sc  Scenario2, Pit+1 (Sc)  j ⎪ . . . . . . ⎪

⎩ f (x1 , x2 , . . . , xn )tT |Sc , Sc  Scenario n,

(11)

where i  A, B, C, D, E, j  A, B, C, D, E, F As a result, with shock scenarios of different magnitudes, represented by the distance to their means, the simulated transition probability PAt+1 A,Sc under different scenarios could be given as: ⎧   T ⎪ t+1 , . . . , x t+1 ⎪ f x |Sc1 , Sc1  5% quantile, ⎪ A A,1 A A,m t ⎪ ⎪ ⎪   ⎪ T ⎪ ⎨ , . . . , x t+1 |Sc2 , S2c  10% quantile, f x t+1 t+1 A A,1 A A,m PA A,X ∈Θ  t ⎪ ⎪ . . . . . . ⎪ ⎪  ⎪ T  ⎪ ⎪ t+1 ⎪ , . . . , x |Sc20 , Sc20  100% quantile. x t+1 ⎩ f A A,1 A A,m

(12)

t

When coupled with Matrix M2 from Algorithm 2, as given in Fig. 3, Eq. (12) becomes:

PAt+1 A,X ∈Θ

T  ⎧  x t+1 x t+1 ⎪ A A,1 A A,m t+1 t+1 t+1 T ⎪ f x t+1 ⎪ A A,1 , . . . , x A A,m )t |Sc1  f x A A,1 ± sd∼ q5% , . . . , x A A,m ± sd∼ q5% ⎪ ⎪ ⎪ Tt    ⎪ T ⎪ t+1 t+1 ⎨ t+1 , . . . , x t+1 t+1 ± sd q x A A,1 , . . . , x t+1 ± sd q x A A,m f x |Sc2  f x ∼ ∼ 10% 10% A A,1 A A,m t A A,1 A A,m  t ⎪ ⎪ ⎪ . . . . . . ⎪ ⎪   ⎪ t+1 t+1 T ⎪ x A A,1 x A A,m T ⎪ t+1 t+1 ⎩ f |Sc20  f ((x t+1 x t+1 A A,1 , . . . , x A A,m A A,1 ± sd∼ q100% , . . . , x A A,m ± sd∼ q100% )t t

(13) where m  1, 2, . . . , n and the sign ± is chosen according to the sign of the correlation coefficient τ .. Consequently, a general equation can be expressed as:

Pit+1 j,X ∈Θ 

⎧    T T t+1 t+1 ⎪ t+1 , . . . , x t+1 t+1 ± sd q xi j,1 , . . . , x t+1 ± sd q xi j,m ⎪ ⎪ f x |Sc1  f x ∼ 5% ∼ 5% ⎪ i j,1 i j,m t i j,1 i j,m ⎪ ⎪ ⎪ tT    ⎪ T t+1 ⎪ x xit+1 ⎪ i j,1 j,m t+1 t+1 t+1 t+1 ⎪ xi j,1 , . . . , xi j,m |Sc2  f xi j,1 ± sd∼ q10% , . . . , xi j,m ± sd∼ q10% ⎨ f t

t

⎪ ... ... ⎪ ⎪    T ⎪ T t+1 t+1 ⎪ ⎪ t+1 , . . . , x t+1 t+1 ± sd q xi j,1 , . . . , x t+1 ± sd q xi j,m ⎪ |Sc20  f x f x ⎪ ∼ ∼ ⎪ 100% 100% i j,1 i j,m t i j,1 i j,m ⎪ ⎪ t ⎩ i  A, B, C, D, E, j  A, B, C, D, E, F, i ≤ j, m  1, 2, . . . , n

(14)

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Similarly, t-Copula would be used and Eq. (2) would be correspondingly changed to: C T (x1 , x2 , . . . , xk )      xit+1 xit+1 j,1 j,m −1 t+1 , . . . , F ± sd q (x ± sd q ) ,  Tτ,υ F −1 Pitj , F1−1 xit+1 ∼ ∼ m j,1 i j,m 5% 5% i  A, B, C, D, E, j  A, B, C, D, E, F, i ≤ j, m  1, 2, . . . , n

(15)

and ⎧      

xit+1 xit+1 ⎪ j,1 j,m ⎪ ⎨ C T Pi j , x1 , x2 , . . . , xn , τ  Tτ,υ F −1 Pitj , F1−1 xit+1 , . . . , Fm−1 xit+1 j,1 ± sd∼ q5% j,m ± sd∼ q5% υ+n   −1 − 2

x1

xn ⎪ Γ ( υ+n ) ⎪ √ 2 ⎩ Tτ,υ (x1 , x2 , . . . , xn )  −∞ 1 + υ τυ υ . . . −∞ dυ, Γ ( υ ) (π υ)n |τ | 2

(16) x t+1

i j,m where Pitj is the observed transition probabilities and xit+1 j,m ± sd∼ q5% , i  A, B, C, D, E, j  A, B, C, D, E, F, i ≤ j are the simulated macro-risk factors. We use Algorithm 3 to simulate the transition probabilities Pit+1 j,X ∈Θ under different scenarios and the scenario probabilities of the transition probabilities conditioned on . certain macro-risk shocking, P˜it+1 j|X ∈Θ

˜ t+1 Algorithm 3 Estimation of Pit+1 j,X ∈Θ and Pi j|X ∈Θ xit+1 j,m

Step 1: Run Algorithm 2 and store the output matrix M2 with recorded sd∼ qα%

α ∈ [0%, 100%].

Step 2: Upload the observed transition probabilities Pitj and corresponding simulated macro-risk factors xit+1 j,m

data xit+1 j,m ± sd∼ qα% , i  A, B, C, D, E , j  A, B, C, D, E, F, i ≤ j. Step 3: Use Algorithm 1 with Eq. (2) being changed to Eq. (16). Step 4: Run Step 2 to Step 5 in Algorithm 2 to get a matrix, M3. Step 5: Output Matrix M3.

˜ t+1 The structure of the matrix for PAt+1 A,X ∈Θ and PA A|X ∈Θ is demonstrated in Fig. 4.

˜ t+1 Fig. 4 Structure of matrix M3 for PAt+1 A,X ∈Θ and PA A|X ∈Θ

123

Stress Testing for Retail Mortgages Based on Probability…

As a result, the portfolio distribution P t+1 (Π ), Π ∈ {A, B, C, D, E, F} can be estimated by Eq. (1). Furthermore, the joint probability of macro-risk factor scenarios with different shock magnitudes and portfolio amounts distributed over states P˜ t+1 (Π, X ∈ Θ), Π ∈ {A, B, C, D, E, F} could be approximated by: P˜ t+1 (Π, X ∈ Θ), Π ∈ {A, B, C, D, E, F}  P˜ t+1 (Π |X ∈ Θ) P˜ t+1 (X ∈ Θ)  · P˜ t+1 (X ∈ Θ), P˜it+1  j|X ∈Θ i j∈Π

i  A, B, C, D, E, j  A, B, C, D, E, F, i ≤ j

(17)

Finally, the expected portfolio amounts distributed in Π ∈ {A, B, C, D, E, F} under macro-risk factor scenarios with different shock magnitudes are estimated by Eqs. (3), (5), (6), (7), (8), and (9), respectively. On the other hand, the estimation of the joint probability P˜ t+1 (Π, X ∈ Θ) is actually a dynamic process in the light of the fact that the estimation of the probabilities of certain scenarios with different shock magnitudes P˜ t+1 (X ∈ Θ) depends on the measurement of the distances to the macro-risk factors’ mean values by their standard deviations sd∼ qα% , α ∈ [0%, 100%]. As a significant factor deviates away from its mean, which happens in the deep recession or high inflation economic sessions, the shocking probabilities assigned to the corresponding scenario increase. The result is that a particular portfolio consequence from this macro-risk scenario would be negatively (or positively) influenced with a higher probability. In this sense, the model proposed in this study works similarly as the Markov Regime Switch models in Hamilton (1989) only with more flexibility because no specific model or distribution needs to be assumed.

3.2.3 Plausible Scenario Space Mathematically, because a non-stationary Markov chains model has been used, the plausible scenario space could be expressed as:

Θ∈

⎧ t+1 ⎪ ⎪ 0 ≤ Pi j,Xqα% ,±sd∼ qα% , ≤ 1, ⎪ ⎪ ⎪ E  ⎪ ⎪ ⎨ P t+1  1, iA

i j,Xqα% ,±sd∼ qα% ,

 t+1 ⎪ t+1 ⎪ (Π ) ≤ 1, PXq (Π )  1 0 ≤ PXq ⎪ ⎪ ,±sd q , ∼ α% α% α% ,±sd∼ qα% , ⎪ Π ⎪ ⎪ ⎩ α ∈ [0%, 100%], i  A, B, C, D, E, j  A, B, C, D, E,

(18)

We will use Algorithm 4 to find the plausible scenario space by implementing Eq. (18).

123

C. Liu, R. Nassar Algorithm 4 Find the plausible scenario space xit+1 j,m

Step 1: Run Algorithm 2 and store the output matrix M2 with recorded sd∼ qα%

α ∈ [0%, 100%].

Step 2: Set k = 1and α  0%, Upload the observed transition probabilities Pitj and corresponding xit+1 j,m

simulated macro-risk factors data xit+1 j,m ± sd∼ qα% , i  A, B, C, D, E , j  A, B, C, D, E, F, i ≤ j, k  j + 1, α  α + 5%. Step 3: Use Algorithm 1 with Eq. (2) being changed to Eq. (16) Step 4: Run Step 2 to Step 5 in Algorithm 2 to get a matrix M3. t+1 Step 5: Use Eq. (18) to evaluate Pit+1 and PXq . If pass, go back to Step 2, if j,Xqα% ,±sd∼ qα% α% ,±sd∼ qα% not, go to Step 6.

Step 6: Set k  k − 1, α  α − 5%. Record α and run Step 2 to Step 5 in Algorithm 3. Step 7: Output α and M3.

Algorithm 4 would provide the plausible scenarios of macro-risk factors shock magnitude, measured by the distance to the mean,α, and the portfolio distribution under these plausible scenarios.

4 An Empirical Example The models and algorithms proposed in this study are implemented on retail mortgage portfolio data provided by a Chinese commercial bank. After data cleansing, which includes interpolation of the missing data by a cubic spline and replacement of the outliers by their means plus or minus their three standard deviation values, the transition probabilities Pitj are calculated based on a non-stationary Markov chain. Then, 15 sets of macro-risk factors X it j ∈ {xit j,1 , . . . , xit j,m } , m  1, 2, . . . , n, i  A, B, C, D, E, j  A, B, C, D, E, F, i ≤ j are chosen for each 15 transition probabilities Pitj based on their significant correlations, measured by τ s. With each set of macro-risk factors uploaded to Algorithms 1 and 2, the adjustments table for the risk factors could be calculated. Finally, Algorithm 3 is loaded to estimate the stress test scenario shock magnitudes with their corresponding probabilities. The plausible scenario space, represented by the standard deviations of risk factors, could be estimated by Algorithm 4 and Eq. (18). 4.1 Data A total of 57 consecutive months of retail mortgage portfolio transition probabilities Pitj , t ∈ [0, 57] along with portfolio distribution P t (i), j  A, B, C, D, E, F at the end of each month, from March, 2004 to Nov., 2008, have been provided by a Chinese national commercial bank. Furthermore, 29 national and regional macro-risk factors were collected from a variety of resources and presented in Table 2. To pick up significant macro-risk factors, the p-value ( p ≤ 0.05) for Kendall’s τ is used. For instance, the selection of macro-risk factors that are significantly correlated with the transition probability Doubtful – Losses are illustrated in Table 3.

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Stress Testing for Retail Mortgages Based on Probability… Table 2 National and reginal marco data List

Names

Resources

x1

GDP Increasing rate: National

China ECONOMIC Information NetWork (CEIN) http://www.cei.gov.cn/

x2

M1 Currency: N (National)

CEIN

x3

CPI: N

CEIN

x4

CPI_Living: N

CEIN

x5

Construction Material Price Index:N

CEIN

x6

Housing Sale Index: N

CEIN

x7

Housing Development Index: N

CEIN

x8

Housing Sale Amount:N

CEIN

x9

HPI:N

CEIN

x10

Housing Rental Index:N

CEIN

x11

Community Management Fee Index:N

CEIN

x12

Export:N

CEIN

x13

Metro Family Monthly Income:N

CEIN

x14

Metro Family Disposable Income:N

CEIN

x15

GDP: R (Reginal)

Statistics Bureau

x16

Metro Family Monthly Income:R

CEIN

x17

Metro Family Disposable Income:R

CEIN

x18

Metro Family Living Expenditures: R

CEIN

x19

CPI:R

CEIN

x20

CPI_Food:R

CEIN

x21

CPI_Living:R

CEIN

x22

Food Sale Index:R

CEIN

x23

Construction Material Price Index:R

CEIN

x24

Living Material Price Index:R

CEIN

x25

HPI:R

Statistics Bureau

x26

Housing Rental Index:R

Statistics Bureau

x27

Unemployment: R

CEIN

x28

Fix Asset Investment Price Index: R

CEIN

x29

1-year Prime Mortgage Rate

China Central Bank

Using this method, each transition probability has been associated with its corresponding set of macro-risk factors. 4.2 Scenario Simulation Preparation Berkowitz (2000) proposes a coherent framework for the scenario selection, which is followed in this study. Scenarios are selected based on the following assumption:  τ











y P( X f ), X f ( x 1 , x 2 ,..., x n )T

⊂ {Ωα95% }

τ yP(X ),X (x1 ,x2 ,...,xn )T ⊂ {Ωα95% }

(19)

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C. Liu, R. Nassar Table 3 Macro-risk factor selection for transition probability: D – E Macro risk factors

Kendall’s τ

p value

− 0.3873106

0.0004069

x1

GDP increasing rate: National

x4

CPI_Living: N

0.43320999

0.0000766

x5

Construction Material Price Index:N

0.46877626

0.00001717

− 0.4835011

x6

Housing Sale Index: N

x18

Metro family living expenditures: R

0.37655502

0.00000785

x21

CPI_Living:R

0.36909129

0.00077285

x23

Construction Material Price Index:R

0.46305596

0.00002040

x28

Fix Asset Investment Price Index: R

0.48619213

0.00000777

x29

1-year prime mortgage rate

0.43026491

0.00018135

0.0005433

Equation (19) means that the correlations of risk factors with each individual transition probability are contained with probability 0.95 in the same space for both original data and the simulated data. On the other hand, to make the simulation algorithms more efficient, the transition probabilities need to be classified. More specifically, the transitions are classified as “Nice” and “Bad”, with “Nice” referring to the transition from a lower rating of a loan to a higher rating of a loan and “Bad” referring to the transition from a higher rating of a loan to a lower rating of a loan. Then, if a risk factor is positively correlated with a “Nice” transition, the shift for this simulated risk factor data should be down in order to make the matter worse. Table 4 presnts details of the simulation methods for the risk factors. Table 5 presents the output of macro-risk factors for transition probability Doubtful–Loss (D–E) from Algorithm 2. With the adjustment direction provided in Table 4 being taken into consideration, the above results provide the inputs of simulated scenarios for Algorithm 3 to estimate , i  A, B, C, D, E, shock magnitudes and their accompanying probabilities P˜it+1 j|X ∈Θ j  A, B, C, D, E, F, which are demonstrated in Table 6a, b, taking transition probability Standard - Standard (A–A) as an example: · P˜ t+1 (X ∈ Θ) which is the The elements of Table 6b are estimated by P˜At+1 A|X ∈Θ first row multiplied by the first column in the above matrix, while Table 6a gives the estimation of the transition probabilities, given different scenario shocks, with their accompanying occurrence probabilities given in Table 6b. 4.3 Final Results The output from Algorithm 3 is input into Eq. (4) for obtaining the portfolio amount distributed in Standard, Special Mention, Substandard, Doubtful, Loss, and Paidoff, respectively, with accompanying probabilities under different scenarios. Table 7 presents partial results for the portfolio amount distributed in the Standard state. Figure 5 demonstrates how the distributions change magnitudes and probabilities under different shock scenarios.

123

Bad

Nice

Bad

Bad

Nice

Bad

Nice

Sub-Standard–Doubtful

Sub-Standard–Paid-off

Doubtful–Doubtful

Doubtful–Loss

Doubtful–Paid-off

Loss–Loss

Loss–Paid-off

sd∼ qα% , α ∈ [0%, 100%] is estimated by Algorithms 1 and 2

Bad

Bad

Special Mention–Special Mention

Sub-Standard–Sub-Standard

Nice

Special Mention–Standard

Bad

Nice

Standard–Paid-off

Nice

Bad

Standard–Special Mention

Special Mention–Paid-off

Nice

Standard–Standard

Special Mention–Sub-Standard

Type

Transition probabilities

Table 4 Methods to prepare the simulated risk factor data

x8,x13,x14,x15,x16,x24,x26,x27,x29

x4,x5,x18,x21,x23,x28,x29

x1,x2,x8,x10,x14,x15,x16,x24,x27,x29

x4,x5,x18,x21,x23,x28,x29

x3,x5,x11,x19,x20,x21,x22,x29

x17

x3,x5,x11,x18,x20,x21,x22,x29

x3,x5,x11,x18,x20,x21,x22,x29

x1,x8,x17

x9,x12,x25

x5, x11,x18,x21,x29

x6,x17,x25

x6,x25

x3, x5, x11, x29

x12

Risk Factors with negative Kendall’s τ would be adjusted by + sd∼ qα% , α ∈ [0%, 100%]

None

x1,x6,x25

None

x1,x6,x25

x12

x3,x4,x19,x20,x22,x23,x28

x12,x25

x12,x25

x3,x4,x19,x20,x22,x23,x28

x5,x11,x18,x20,x21,x22,x29

x6,x9,x12,x25

x3, x5, x11, x20, x22,x29

x3,x5, x11, x19,x20, x22,x29

x12

x3, x5, x11, x29

Risk factors with positive Kendall’s τ would be adjusted by -sd∼ qα% , α ∈ [0%, 100%]

Stress Testing for Retail Mortgages Based on Probability…

123

123

…

…

1.545517

− 1.48265

− 1.02568

10%

100%

− 2.13752

− 1.24357

5%

1.387714

…

x t+1 D E,6

x t+1 D E,1

Quantile

2.219647

…

− 1.07869

− 1.77685

x t+1 D E,25

2.250535

…

− 1.20106

− 1.38764

x t+1 D E,4

2.447126

…

− 1.0838

− 1.31919

x t+1 D E,5

2.870199

…

− 1.26062

− 1.68948

x t+1 D E,18

t+1 t+1 Table 5 Output of macro-risk factors scenarios X t+1 D E ∈ {x D E,1 , . . . , x D E,m } for D–E from Algorithm 2

2.066633

…

− 1.36716

− 1.74869

x t+1 D E,21

2.364763

…

− 1.09328

− 1.22377

x t+1 D E,23

2.893084

…

− 0.86891

− 0.86891

x t+1 D E,28

1.653599

…

− 1.08999

− 1.08999

x t+1 D E,29

C. Liu, R. Nassar

Stress Testing for Retail Mortgages Based on Probability… Table 6 (a) Estimation of transition probabilities PAt+1 A,X ∈Θ under different scenarios, (b) estimation of the scenario probabilities of the transition probabilities conditioned on certain macro-risk shocking P˜ At+1 A|X ∈Θ P˜Xt+1 P˜ t+1 α% A A,α%|X α%

5%

10%

15%

……

60%

20%

0.967508

0.9673035

0.9669659

……

0.9675086

40%

0.970863

0.9707183

0.9705200

……

0.9708631

60%

0.974821

0.9747441

0.9746129

……

0.9748214

80%

0.977144

0.9770672

0.976705

……

0.9771445

0.9854567

0.98216773

(a)

100% (b) 20%

1%

2%

3%

……

12%

40%

2.00%

4.00%

6.00%

……

24.00%

60%

3.00%

6.00%

9.00%

……

36.00%

80%

4.00%

8.00%

12.00%

……

48.00%

100%

5.00%

10.00%

15.00%

……

60.00%

Table 7 Estimation of ˜ t+1 ( A, X ∈ Θ) and P P t ( A) · Pit+1 A,X ∈Θ A,B

P˜ t+1 ( A, X ∈ Θ)

 A,B

P t ( A) · Pit+1 A,X ∈Θ

99%

0.98088626

98%

0.97993386

97%

0.98245887

……

……

40%

0.98003167

Figure 5 also demonstrates the relationship between portfolio amounts distributed over states: Standard, Special Mention, Substandard, Doubtful, Loss, and Paid-off. From Fig. 5 Standard: A and Table 7, one could find the following phenomena. First, the portfolio amount distributed in Standard state would be about 98.1%, with a probability of 99% under a plausible scenario of macro-risk factors. In other words, with the probability being 56%, the portfolio in the Standard state would not be less than 98%, given the scenarios simulated according to those that happen in the past four and half years. Second, it is also possible that the portfolio amount in certain states will take on several values with different probabilities, as demonstrated in Fig. 5. This phenomenon is caused by the mapping procedure when estimating the joint probability P˜ t+1 (Π, X ∈ Θ), Π  A, B, C, D, E, F. However, the oscillations as demonstrated in Fig. 5 could be smoothed out by fiting a second-order polynomial to the original simulated data. Finally, with decreasing probabilities, the portfolio in the “Bad” states, including Special Mention, Substandard, Doubtful, and Loss, is increasing, whereas the portfolio in the “Nice” state, Paid-off, is decreasing with the Standard state resembling some kind of “U” shape.

123

C. Liu, R. Nassar

Fig. 5 The amounts of load distributed in different categories with their corresponding probabilities. The Y-axis is the percentage of the loans distributed in each category, such as Standard and Special Mention. The X-axis is the probabilities corresponding to each percentage. For instance, in Standard A, the value of the middle circled part is (86%, 98.4%). It means that with a probability of 86%, the percentage of the loan categorized as Standard is 98.4%

Based on the fact that Macro-risk factors are only available for the past four and half years, the magnitudes for the simulated scenarios may not be severe, or extreme, enough to reach any decision about this particular stress testing results. This approach, however, has demonstrated to some extent its practical value for risk management in this dynamically changing financial environment. In Fig. 6, the Z-axis presents the transition probabilities. X-axis presents the accompanying probabilities of those transition probabilities. Y–axis means the scenarios probabilities of macro factors. Figure 6 presents the relationship of the portfolio distributions that have two components. The first component is macro-risk factors shock probabilities P˜ t+1 (X ∈ Θ) where the macro-risk factors X  (x1 , x2 , . . . , xm )T ∈ Θ, m  1, 2, . . . , n were chosen according to their significance as measured by τ . The second is the accompanying probabilities (defined as the probabilities of those ,i  transition probabilities occurring under different macro factor scenarios) P˜it+1 j|X ∈Θ t+1 A, B, C, D, E, j  A, B, C, D, E, F of the transition probabilities Pi j,X ∈Θ , where ˜ t+1 Pit+1 j,X ∈Θ and Pi j|X ∈Θ are fed into Eq. (4) in order to get the final results. The expected portfolio distribution from Eqs. (3), (5), (6), (7), (8), and (9) is presented in Table 8. The values in Table 8 are millions in CNY. For instance, 0.00015 stands for 0.00015 million in CNY. The total numbers add up to 243 million.

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Stress Testing for Retail Mortgages Based on Probability…

Standard: A

Special Mention: B

Substandard: C

Doubtful: D

Loss: E

Paid-Off: F

Fig. 6 Portfolio distributions obtained from Eq. (4)

Finally, the plausible space for the Transition Probabilities Pit+1 j,X ∈Θ , i  A, B, C, D, E , j  A, B, C, D, E, F is given by Algorithm 4 and Eq. (18) which corresponds to P˜ t+1 (X ∈ Θ) of 60%. This result is also confirmed in Fig. 5 and Table 7. The plausible space for PAt+1 A,X ∈Θ is presented in Table 9.

123

C. Liu, R. Nassar Table 8 The balance of the loans in milions of CNY distributed in each state Π ∈ {A, B, C, D, E, F}

F

E

D

C

B

A

Portfolio distributions in millions of CNY

2.18

0.00015

0.002

0.006

2.25

238.55

Table 9 The plausible space of scenarios for PAt+1 A,X ∈Θ X ∈Θ

x t+1 A A,3

sd∼ q60% 0.2714

x t+1 A A,5

x t+1 A A,6

x t+1 A A,11

x t+1 A A,19

x t+1 A A,20

x t+1 A A,22

x t+1 A A,25

x t+1 A A,29

− 0.0245 0.4264

0.1997

0.1168

0.0996

0.0476

0.0521

− 0.0611

5 Conclusion In this study, a novel approach to estimate the shock magnitudes and shock probabilities of macro-risk factors with regard to a credit portfolio is proposed and implemented on the retail mortgages portfolio of a Chinese commercial bank by using a nonstationary Markov chain model, t-Copula simulation, and Cubic spline interpolation. Also, the plausible scenario space is defined by Eq. (18) and estimated by Algorithm 4. The contribution of this study is three-fold. First, the study proposes a method to estimate the scenario occurrence probabilities and, therefore, the expected loss for each scenario. Thus, the institutions implementing the method would have crucial quantitative risk management tools for their resource allocation to compensate for losses. Second, traditional simulation of accompanying probabilities of risk factors differ by the sets of macro-risk factors and may even vary within each set. Algorithm 2 is designed to address this issue. Third, one significant problem with stress testing is whether the scenarios proposed are plausible or not. This study offers Algorithm 4 to estimate the plausible sets of impact scenarios which make stress testing more reliable and the simulation itself more efficient because unfeasible scenarios are ignored. Despite these advantages, however, this study has two drawbacks. First, because the scenario shock probabilities are estimated from the macro-risk factors’ standard deviations, the time frame of the time series data, has an effect on estimating the standard deviation and as a consequence the shock probability. As both IMF and BIS accept the use of historical scenarios of 25 or 75 years, one can expect a difference in the expected portfolio distribution between 25 and 75 year scenarios. Second, results from the present study provide shock information only in 2-dimensions, magnitude and probability. To be more useful, the risk early warning system, based on stress testing, should at least provide a 3-dimensional signal—magnitude, probability, and shock time framework. In this case, the bank management would have information on how likely a given portfolio would occur, how much loss is expected, and when this would happen before making any strategic decisions. Future research designed to solve this problem is needed. Acknowledgements This study was funded by China Natural Science Fundation (Grant No: 71473204). The authors thank Prof. Dongtao Lin of Sichuan University for copyediting this manuscript.

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