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On the regulator insurer interaction in structural model

VIEWS: 3 PAGES: 38

									         On the regulator-insurer-interaction in a
                    structural model

                              An Chen

                        University of Amsterdam

                        joint with Carole Bernard


            Internal Models in Risk Management Seminar
           December 12 2007, Amsterdam, The Netherlands



Bernard, Chen                      On the regulator-insurer-interaction in a structural model   1/38
                     Motivation and introduction
                                    Model setup
                   Moving to a dynamic approach
                                     Conclusions


Introduction


        Increasing number of insolvent insurance companies:

            £ Garantie Mutuelle des Fonctionnaires in France in 1993,
              Mannheimer Leben in 2003

            £ Nissan Mutual Life in 1997, Chiyoda Mutual Life
              Insurance Co. and Kyoei Life Insurance Co. in 2000,
              Tokyo Mutual Life Insurance in 2001

            £ First Executive Life Insurance Co. in 1991, Conceso Inc.
              in 2002

   ⇒ New accounting standards


   Bernard, Chen                                   On the regulator-insurer-interaction in a structural model   2/38
                     Motivation and introduction
                                    Model setup
                   Moving to a dynamic approach
                                     Conclusions


IASB/FASB: Risk-oriented Approach


        Assets and liabilities at “Market value” or “Fair value”.
           1   If you have reliable market prices use these.
           2   If you have no market prices, but are able to replicate, then the
               price of the replicating portfolio is the price of the instrument.
           3   If neither 1 or 2 applies, make a model to price the instrument
               as consistently as possible.

        “Risk-based marketing of products” have to include credit
        risk, market risk and operational risk.




   Bernard, Chen                                   On the regulator-insurer-interaction in a structural model   3/38
                     Motivation and introduction
                                    Model setup
                   Moving to a dynamic approach
                                     Conclusions


Solvency II: Risk-based Solvency Requirement


        Risk-based regulation focuses on downside risk

               “Should be compatible with valuation standards”.

               “Likely to require insurance companies to use measures such as
               probability of ruin, Value-at-Risk or Tail Value-at-risk”.

               “Should contain incentives for insurance companies to measure
               and properly manage risks”.

               Solvency II emphasizes the importance of how to develop new
               regulatory methods and tools in order to reduce the threat of
               an insolvency and to better protect the policy holders.



   Bernard, Chen                                   On the regulator-insurer-interaction in a structural model   4/38
                     Motivation and introduction
                                    Model setup
                   Moving to a dynamic approach
                                     Conclusions


Importance of regulation

        In reality, the collapse of many insurance companies is closely
        related to the insufficient regulation, e.g. the fall of First
        Executive Life Insurance Co. (C.f Schulte; 1991).
        Whilst the insurance company tries to maximize returns for its
        equity holder, the regulatory authorities are responsible for
        protecting the policyholders.
        In the most of the existing literature, the regulators act very
        passively.
   ⇒ Necessary to highlight the role of the regulators and
     particularly the interaction between the regulator and the
     insurer!!

   Bernard, Chen                                   On the regulator-insurer-interaction in a structural model   5/38
                       Motivation and introduction
                                      Model setup
                     Moving to a dynamic approach
                                       Conclusions


Objectives


        Given a model for participating contracts under default risk in
        the standard (static) framework proposed in prior literature.

               Study how to set regulatory rules (barrier level) to reach
               regulation objectives (e.g given by a maximum default
               probability)

                   £ Chapter 7 Bankruptcy Procedure
                   £ Chapter 11 Bankruptcy Procedure

        Robustness of this approach when we move to a dynamic
        framework.



   Bernard, Chen                                     On the regulator-insurer-interaction in a structural model   6/38
                     Motivation and introduction
                                    Model setup
                   Moving to a dynamic approach
                                     Conclusions


Literature on default modelling (structural approach)

   Insurance                                             Finance

   Briys and de Varenne                                  Merton (1974, 1989)
   (1994, 1997a, 2001)

   Grosen and Jørgensen (2000, 2002)                     Black and Cox (1976)

   Bernard, Le Courtois                                  Briys and de Varenne (1997b)
   and Quittard-Pinon (2005b, 2006b)                     Longstaff and Schwartz (1995)
                                                         Collin-Dufresne and Goldstein (2001)

                                                         Morellec (2001), Moraux (2004)
   Chen and Suchanecki (2007)                                c
                                                         Fran¸ois and Morellec (2004)
                                                         Broadie, Chernov & Sundaresan (2004)
                                                         Galai, Raviv and Wiener (2003)

   Bernard, Chen                                   On the regulator-insurer-interaction in a structural model   7/38
                     Motivation and introduction
                                    Model setup
                   Moving to a dynamic approach
                                     Conclusions


Agenda


                      √
   1    Introduction ( )

   2    Default risk modelling and how to set regulation rules to
        achieve certain regulation objectives in a static framework

   3    A dynamic framework to investigate the
        regulator-insurer-interaction

   4    Conclusions




   Bernard, Chen                                   On the regulator-insurer-interaction in a structural model   8/38
                     Motivation and introduction
                                    Model setup        Chapter 7 Bankruptcy Procedure
                   Moving to a dynamic approach        Chapter 11 Bankruptcy Procedure
                                     Conclusions


Capital structure and notations

        Initial capital structure of the life insurance company

                                Assets           Liabilities
                                 A0           E0 ≡ (1 − α)A0
                                                 L0 ≡ αA0
                                  A0                A0

          E0                :    initial investment of the equity holder
          L0                :    initial investment of the liability holder
          Lt = L0 e gt      :    the guarantee amount at t
          g                 :    minimum interest rate guarantee
          δ                 :    participation rate


   Bernard, Chen                                   On the regulator-insurer-interaction in a structural model   9/38
                     Motivation and introduction
                                    Model setup         Chapter 7 Bankruptcy Procedure
                   Moving to a dynamic approach         Chapter 11 Bankruptcy Procedure
                                     Conclusions


Payoff to the liability holder (given no premature default)

        At the maturity date T the payoff to the liability holder is:
                    
                     AT ,
                                                                        if AT < LT
                    
                                                                                     LT
         ψL (AT ) =   LT ,                                               if LT ≤ AT ≤
                                                                                       α
                    
                    
                     L + δ(αA − L ),                                            LT
                     T       T   T                                      if AT >
                                                                                  α
        More compactly,


         ψL (AT ) =             LT           + δ[αAT − LT ]+                          −[LT − AT ]+ ,
                        fixed payment               bonus (call option) shorted put option



   Bernard, Chen                                   On the regulator-insurer-interaction in a structural model   10/38
                     Motivation and introduction
                                    Model setup         Chapter 7 Bankruptcy Procedure
                   Moving to a dynamic approach         Chapter 11 Bankruptcy Procedure
                                     Conclusions


Early default (Grosen and Jørgensen (2002))

        An exponential barrier with η as regulation parameter by which the
        regulator can control the strictness of the barrier trigger:

                                     Bt = ηLt = η L0 e gt

        The default and liquidation occur if there exists such a t ∈ [0, T ]
        that At ≤ Bt . And

                              τ = inf{t ∈ [0, T ] : At ≤ Bt }
        gives the liquidation time.

        The rebate payment to the liability holder is

                         ΘL (τ ) = (η ∧ 1) L0 e g τ = (η ∧ 1)Aτ .

   Bernard, Chen                                   On the regulator-insurer-interaction in a structural model   11/38
                     Motivation and introduction
                                    Model setup         Chapter 7 Bankruptcy Procedure
                   Moving to a dynamic approach         Chapter 11 Bankruptcy Procedure
                                     Conclusions


Regulation objectives

        An appropriate regulation level η

          £ to give an acceptable level of default

                                     min η > 0 P(τ ≤ T ) ≤ ε

          £ to protect the policyholders by maximizing their expected cash
            flow given default


                   max η > 0 E min{η, 1}Lτ e r (T −τ ) |τ ≤ T ≥ γLT

               with γ ∈ [0, 1]. P and E are the real world measure and the
               expectation taken under P.


   Bernard, Chen                                   On the regulator-insurer-interaction in a structural model   12/38
                     Motivation and introduction
                                    Model setup         Chapter 7 Bankruptcy Procedure
                   Moving to a dynamic approach         Chapter 11 Bankruptcy Procedure
                                     Conclusions


Default probability

        The firm’s assets value is assumed to follow a geometric
        Brownian motion under the real world measure P

                                     dAt = At (µ dt + σ dWt )
                                                                       σ2
        Default probability: with µ = µ − g −
                                  ˆ                                    2


                                                                            −2µ
                                                                              ˆ
                                 ln( ηL00 ) − µT
                                     A        ˆ                   A0        σ2           ln( ηL00 ) + µT
                                                                                             A        ˆ
          P(τ ≤ T ) = N                  √                 +                      N              √
                                      σ T                         ηL0                         σ T

        Assume that with η ε it holds P(τ ≤ T ) = ε, then only
        η < η ε leads to a default probability smaller than ε.


   Bernard, Chen                                   On the regulator-insurer-interaction in a structural model   13/38
                        Motivation and introduction
                                       Model setup         Chapter 7 Bankruptcy Procedure
                      Moving to a dynamic approach         Chapter 11 Bankruptcy Procedure
                                        Conclusions


Illustration: default probability




                        σ               10%                     15%               20%
                    P(τ ≤ T )        0.00257218               0.07269           0.239842

   Cumulative default probabilities with parameters: A0 = 100; L0 =
         80; T = 20; η = 0.5; µ = 0.04; r = 0.03; g = 0.01.




    Bernard, Chen                                     On the regulator-insurer-interaction in a structural model   14/38
                       Motivation and introduction
                                      Model setup         Chapter 7 Bankruptcy Procedure
                     Moving to a dynamic approach         Chapter 11 Bankruptcy Procedure
                                       Conclusions


Illustration: optimal regulation level η ε

                ε      η ε ; σ = 0.10          η ε ; σ = 0.15               η ε ; σ = 0.20
               0               0                       0                            0
              0.01        0.595660                0.306855                     0.148879
              0.02        0.655581                0.359548                     0.185358
              0.03        0.694975                0.396648                     0.212528
              0.04        0.725144                0.426470                     0.235245
              0.05        0.749929                0.451935                     0.255261
              0.06         0.77114                0.474452                     0.273434
              0.07        0.789786                0.494819                     0.290258
              0.08        0.806489                0.513537                     0.306044
              0.09        0.821664                0.530945                     0.321006
              0.10        0.835603                0.547280                     0.335295
          η ε for given ε and diverse σ-values with parameters:
       A0 = 100; L0 = 80; T = 20; µ = 0.04; r = 0.03; g = 0.01.
    Bernard, Chen                                    On the regulator-insurer-interaction in a structural model   15/38
                      Motivation and introduction
                                     Model setup         Chapter 7 Bankruptcy Procedure
                    Moving to a dynamic approach         Chapter 11 Bankruptcy Procedure
                                      Conclusions


Illustration: optimal regulation level


                                  0.8

                                                               Σ 0.10
                                  0.6                          Σ 0.15
                     Optimal ΗΕ


                                                               Σ 0.20

                                  0.4


                                  0.2


                                    0
                                        0.02        0.04       0.06        0.08          0.1
                                                           Ε


             Trade-off between η ε and ε with parameters
       A0 = 100; L0 = 80; T = 20; µ = 0.04; r = 0.03; g = 0.01.

    Bernard, Chen                                   On the regulator-insurer-interaction in a structural model   16/38
                      Motivation and introduction
                                     Model setup                Chapter 7 Bankruptcy Procedure
                    Moving to a dynamic approach                Chapter 11 Bankruptcy Procedure
                                      Conclusions


Expected cash flows given liquidation

        The expected conditional payoff is given by


                   E (η ∧ 1) L0 e g τ e r (T −τ ) 1{τ <T }
                                   P{τ ≤ T }

                                                          (µ)2 +2(r −g )σ 2
                                                           ˆ
                                                                                                                      
                                                 µˆ                                   ηL
                                         ηL0        −                              ln( A 0 ) −   (µ)2 + 2(r − g )σ 2 T
                                                                                                  ˆ
                                  rT             σ2             σ2                      0
               =   (η ∧ 1) L0 e                                               N                    √
                                                                                                                      
                                                                                                                       
                                          A0                                                      σ T

                              µ+
                              ˆ        µ2 +2(r −g )σ 2
                                       ˆ
                                                                                                   
                                                                 ηL
                       ηL0                                    ln( A 0 ) +     (µ)2 + 2(r − g )σ 2 T
                                                                               ˆ
                                          σ2                       0
                   +                                     N                      √                         P(τ ≤ T )
                                                                                                   
                                                                                                    
                        A0                                                     σ T



         η γ : the optimal regulation level which makes the above
        expected value equal to γ LT


   Bernard, Chen                                         On the regulator-insurer-interaction in a structural model        17/38
                      Motivation and introduction
                                     Model setup         Chapter 7 Bankruptcy Procedure
                    Moving to a dynamic approach         Chapter 11 Bankruptcy Procedure
                                      Conclusions


Illustration: optimal regulation level η γ


              γ      η γ ; σ = 0.10           η γ ; σ = 0.15               η γ ; σ = 0.20
             0.70       0.607954                 0.584077                     0.566748
             0.75       0.643793                 0.619084                     0.601250
             0.80       0.678647                 0.653348                     0.635153
             0.85       0.712546                 0.686897                     0.668484
             0.90       0.745526                 0.719758                     0.701264
             0.95       0.777624                 0.751958                     0.733516
             1.00       0.808877                 0.783522                     0.765261
            η γ for diverse γ and σ-values with parameters:
       A0 = 100; L0 = 80; T = 20; µ = 0.04; r = 0.03; g = 0.01.



    Bernard, Chen                                   On the regulator-insurer-interaction in a structural model   18/38
                      Motivation and introduction
                                     Model setup         Chapter 7 Bankruptcy Procedure
                    Moving to a dynamic approach         Chapter 11 Bankruptcy Procedure
                                      Conclusions


Illustration: optimal regulation level η γ


                                   0.8

                                                 Σ 0.10
                                  0.75           Σ 0.15
                     Optimal ΗΓ


                                                 Σ 0.20
                                  0.7

                                  0.65

                                   0.6

                                    0.7   0.75   0.8       0.85       0.9      0.95         1
                                                             Γ


          Expected cash flow given liquidation with parameters
       A0 = 100; L0 = 80; T = 20; r = 0.03; µ = 0.04; g = 0.01.

    Bernard, Chen                                   On the regulator-insurer-interaction in a structural model   19/38
                     Motivation and introduction
                                    Model setup         Chapter 7 Bankruptcy Procedure
                   Moving to a dynamic approach         Chapter 11 Bankruptcy Procedure
                                     Conclusions


Optimal regulation level


        If η γ makes the above expected value equal to γ LT , then

                                              η ≥ ηγ
        provides in expectation that the payoff given liquidation is not
        smaller than a γ percentage of the guaranteed payment.

        A regulator has to set a regulation level in the area of

                                         ηγ ≤ η ≤ ηε
        in order to achieve a plausible DP and a reasonable ECGL.



   Bernard, Chen                                   On the regulator-insurer-interaction in a structural model   20/38
                     Motivation and introduction
                                    Model setup         Chapter 7 Bankruptcy Procedure
                   Moving to a dynamic approach         Chapter 11 Bankruptcy Procedure
                                     Conclusions


Problem


        Usually the regulator cannot achieve both of the objectives
        simultaneously.
          £ Few intersection areas of η (η γ ≤ η ≤ η ε ) can be found

        In practice, most of the regulators stick to the first objective,
        i.e. to control the default probability under a certain
        constraint.

        Assumption: focus on the first goal




   Bernard, Chen                                   On the regulator-insurer-interaction in a structural model   21/38
                        Motivation and introduction
                                       Model setup         Chapter 7 Bankruptcy Procedure
                      Moving to a dynamic approach         Chapter 11 Bankruptcy Procedure
                                        Conclusions


Under Chapter 11: Standard and cumulative Parisian
options


        Example down–and–out option:

                   Standard Parisian barrier option: option is knocked out if
                   the underlying asset value stays consecutively below the
                   barrier for a time longer than some pre–specified time
                   window d before the maturity date.

                   Cumulative Parisian barrier option: option is knocked out
                   if the underlying asset value spends until maturity in
                   total d units of time below the barrier.




   Bernard, Chen                                      On the regulator-insurer-interaction in a structural model   22/38
                        Motivation and introduction
                                       Model setup         Chapter 7 Bankruptcy Procedure
                      Moving to a dynamic approach         Chapter 11 Bankruptcy Procedure
                                        Conclusions


Standard Parisian options

        Standard Parisian barrier options.

                   The default clock starts ticking when the asset price
                   process breaches the default barrier and the clock is reset
                   to zero if the firm recovers from the default.

                   Successive defaults are possible until one of these
                   defaults lasts d units of time.

           ⇒ The default clock is memoryless, i.e., earlier defaults
             which may last a very long time but not longer than d
             do not have any consequences for eventual subsequent
             defaults.


   Bernard, Chen                                      On the regulator-insurer-interaction in a structural model   23/38
                     Motivation and introduction
                                    Model setup         Chapter 7 Bankruptcy Procedure
                   Moving to a dynamic approach         Chapter 11 Bankruptcy Procedure
                                     Conclusions


Standard Parisian options


        An early liquidation occurs when the following technical
        condition is satisfied:

                −                               A
               TB = inf t > 0              t − gB,t 1{At <Bt } > d                       ≤T

        with
                              A
                             gB,t = sup{s ≤ t|As = Bs },
                A
        where gB,t denotes the last time before t at which the value
        of the assets A hits the barrier B.



   Bernard, Chen                                   On the regulator-insurer-interaction in a structural model   24/38
                                   Motivation and introduction
                                                  Model setup         Chapter 7 Bankruptcy Procedure
                                 Moving to a dynamic approach         Chapter 11 Bankruptcy Procedure
                                                   Conclusions


Illustration: Parisian down and out option
         Asset evolution




                            S0
                                                                                       d
                           K L




                                                                                             ΤL               T
                                                                    Time

   Bernard, Chen                                                 On the regulator-insurer-interaction in a structural model   25/38
                        Motivation and introduction
                                       Model setup         Chapter 7 Bankruptcy Procedure
                      Moving to a dynamic approach         Chapter 11 Bankruptcy Procedure
                                        Conclusions


Cumulative Parisian options


        Cumulative Parisian barrier options:

                   The default clock is not reset to zero when a firm
                   emerges from default, but it is only halted and restarted
                   when the firm defaults again.

           ⇒ A full memory default clock, since every single moment
             spent in default is remembered and affects further
             defaults by shortening the maximum allowed length of
             time that the company can spend in default without
             being liquidated.



   Bernard, Chen                                      On the regulator-insurer-interaction in a structural model   26/38
                     Motivation and introduction
                                    Model setup            Chapter 7 Bankruptcy Procedure
                   Moving to a dynamic approach            Chapter 11 Bankruptcy Procedure
                                     Conclusions


Cumulative Parisian option

        An early liquidation when the following condition holds:
                                                  T
                            Γ−,B =
                             T                        1{At ≤Bt } dt ≥ d,
                                              0

        where Γ−,B denotes the occupation time of the process
                T
        describing the value of the assets {At }t∈[0,T ] below the barrier
        B during [0, T ].
        If τ denoted as the premature liquidation date and it implies:
                                          τ
                        −,b
                       Γτ :=                  1{τ ≤T } 1{At ≤Bt } dt = d.
                                      0



   Bernard, Chen                                      On the regulator-insurer-interaction in a structural model   27/38
                     Motivation and introduction
                                    Model setup         Chapter 7 Bankruptcy Procedure
                   Moving to a dynamic approach         Chapter 11 Bankruptcy Procedure
                                     Conclusions


Default probability



  The probability of default can be written


        in terms of a Laplace inverse (standard Parisian case)

        or with a closed-form expression (cumulative Parisian case)




   Bernard, Chen                                   On the regulator-insurer-interaction in a structural model   28/38
                       Motivation and introduction
                                      Model setup         Chapter 7 Bankruptcy Procedure
                     Moving to a dynamic approach         Chapter 11 Bankruptcy Procedure
                                       Conclusions


Illustration: optimal regulation in standard case

                ε      η ε ; σ = 0.10          η ε ; σ = 0.15               η ε ; σ = 0.20
              0.01          0.6536                  0.3528                       0.1795
              0.02          0.7178                  0.4132                       0.2236
              0.03          0.7603                  0.4554                       0.2558
              0.04          0.7922                  0.4896                       0.2837
              0.05          0.8202                  0.5182                       0.3075
              0.06          0.8443                  0.5431                       0.3293
              0.07          0.8659                  0.5665                       0.3499
              0.08          0.8827                  0.5875                       0.3673
              0.09          0.8991                  0.6076                       0.3855
              0.10          0.9156                  0.6274                       0.4019
   η ε in case of standard Parisian option for given default probability
   constraint ε for diverse σ-values with parameters: A0 = 100; L0 =
           80; T = 20; µ = 0.04; r = 0.03; g = 0.01; d = 0.5.
    Bernard, Chen                                    On the regulator-insurer-interaction in a structural model   29/38
                      Motivation and introduction
                                     Model setup         Chapter 7 Bankruptcy Procedure
                    Moving to a dynamic approach         Chapter 11 Bankruptcy Procedure
                                      Conclusions


Illustration: optimal regulation in cumulative case

               ε      η ε ; σ = 0.10          η ε ; σ = 0.15               η ε ; σ = 0.20
             0.01          0.6332                  0.3376                       0.1697
             0.02          0.6966                  0.3949                       0.2107
             0.03          0.7382                  0.4355                       0.2413
             0.04          0.7700                  0.4678                       0.2670
             0.05          0.7962                  0.4857                       0.2894
             0.06          0.8188                  0.5201                       0.3098
             0.07          0.8384                  0.5422                       0.3287
             0.08          0.8560                  0.5625                       0.3464
             0.09          0.8720                  0.5814                       0.3632
             0.10          0.8869                  0.6000                       0.3792
      η ε in case of cumulative Parisian option for given default
  probability constraint ε for diverse σ-values with parameters: A0 =
    100; L0 = 80; T = 20; µ = 0.04; r = 0.03; g = 0.01; d = 0.5.
   Bernard, Chen                                    On the regulator-insurer-interaction in a structural model   30/38
                     Motivation and introduction
                                    Model setup         Chapter 7 Bankruptcy Procedure
                   Moving to a dynamic approach         Chapter 11 Bankruptcy Procedure
                                     Conclusions


Critique on the static framework

        Optimal regulation level is based on “fixed-volatility-rule”.

        Whenever the insurance company follows an investment
        strategy with a non-constant vola, such a “fixed sigma rule”
        becomes completely useless.

        The insurance company can avoid intervention by switching
        the vola of its risk management strategy...

                                 G&J          Standard Parisian                  Cumulative Parisian
          Volatility Level      0.0752             0.0817                            0.07945

                                Volatility Level
        A0 = 100; L0 = 80; T = 20; µ = 0.04; r = 0.03; g = 0.01; d = 0.5.

   Bernard, Chen                                   On the regulator-insurer-interaction in a structural model   31/38
                     Motivation and introduction
                                    Model setup
                   Moving to a dynamic approach
                                     Conclusions


Strong assumptions




        The assets of the company are lognormally distributed.
        The interest rate is constant.
        The volatility is constant, i.e. the insurer follows a portfolio
        with a constant volatility.




   Bernard, Chen                                   On the regulator-insurer-interaction in a structural model   32/38
                     Motivation and introduction
                                    Model setup
                   Moving to a dynamic approach
                                     Conclusions


Initial parameter setting


        Two possible initial scenarios:

               σ0 = σL = 10% and µL = 4%;

               σ0 = σH = 20% and µH = 5%.

               Other parameters:
               α = α0 = 0.4, A0 = 100, L0 = 40, T = 20,
               r = 0.03, g = 0.01.
        Goal: Maximize the expected rate of returns, given a
        maximum default probability ε.


   Bernard, Chen                                   On the regulator-insurer-interaction in a structural model   33/38
                     Motivation and introduction
                                    Model setup
                   Moving to a dynamic approach
                                     Conclusions


Volatility switching


        At the end of each year t = ti , i = 1, · · · , T , managers face
        three different situations:
           1   Case 1: At < Bt . Bankruptcy is declared.
           2   Case 2: At ≥ Bt and σ = σH . Compute at time t, the default
               probability when there is no switching until T . If this default
               probability is above ε, then regulators reduce the level of the
               volatility, otherwise they do not intervene.
           3   Case 3: At ≥ Bt and σ = σL . The managers decide to switch
               to σH in order to increase their expected payment, keeping
               satisfying that the default probability is below ε.



   Bernard, Chen                                   On the regulator-insurer-interaction in a structural model   34/38
                  Motivation and introduction
                                 Model setup
                Moving to a dynamic approach
                                  Conclusions




                                                σ0 = σL = 10%                 σ0 = σH = 20%
                                                   µL = 4%                       µH = 5%
       Default Probability                        3.8 10−4 %                      10.8%
 Policyholders Expected Return                       1.06                          1.25
 Shareholders Expected Return                        1.34                          2.00
       Default probability and expected returns in a static setting.

                                                  σ0 = σL = 10%                σ0 = σH = 20%
                                                     µL = 4%                      µH = 5%
       Default Probability                            0.62%                        0.64%
Policyholders Expected Payment                     1.40 (+32%)                   1.15 (-8%)
Shareholders Expected Payment                      1.69 (+26%)                   1.86 (-7%)
Default probability and expected returns in case of dynamic approach.



Bernard, Chen                                   On the regulator-insurer-interaction in a structural model   35/38
                     Motivation and introduction
                                    Model setup
                   Moving to a dynamic approach
                                     Conclusions


Conclusions


        In the static framework of Black and Scholes, our study shows
        how the regulator sets the regulatory rule to control the
        default probability.
        We show that if the insurance company follows a risk
        management strategy, it can significantly change the risk
        exposure of the company, and that it should thus be taken
        into account by the regulators.
        ⇒ The regulator-insurer-interaction is highlighted.
        ⇒ We illustrate limitation of the previous models where a
        constant volatility is assumed.


   Bernard, Chen                                   On the regulator-insurer-interaction in a structural model   36/38
                  Motivation and introduction
                                 Model setup
                Moving to a dynamic approach
                                  Conclusions




     We believe that the value of the risk management of the
     company should be included in the risk exposure estimation
     and the market value of liabilities as well.




Bernard, Chen                                   On the regulator-insurer-interaction in a structural model   37/38
                  Motivation and introduction
                                 Model setup
                Moving to a dynamic approach
                                  Conclusions




        THANK YOU FOR YOUR ATTENTION!




Bernard, Chen                                   On the regulator-insurer-interaction in a structural model   38/38

								
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