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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