Modeling Contagion Risk and a Tax-Carry-Modeling Contagion Risk and a Tax-Carry-Forward Program in an Insurance Guaranty Forward Program in an Insurance Guaranty

Fund: Fund: The Case of PACICC in CanadaThe Case of PACICC in Canada

Gilles Bernier, Ph.D.Gilles Bernier, Ph.D.Professor of Finance and InsuranceProfessor of Finance and InsuranceFaculty of Business AdministrationFaculty of Business Administration

Laval UniversityLaval UniversityChairholderChairholder

and,and,

Ridha Mahfoudhi, Ph.D.Ridha Mahfoudhi, Ph.D.Chief AnalystChief Analyst

Securitization & ALMSecuritization & ALMNational Bank of CanadaNational Bank of Canada

ARIA, Quebec City, August 7, 2007ARIA, Quebec City, August 7, 2007

www.fsa.ulaval.ca/chaire-www.fsa.ulaval.ca/chaire-industrielleallianceindustriellealliance

ContentContent

Purpose and Research Question.Purpose and Research Question.

Literature Review on Contagion Risk in the Literature Review on Contagion Risk in the Financial Services Industry.Financial Services Industry.

Description of the Basic Model of the Insurance Description of the Basic Model of the Insurance Firm.Firm.

Modeling the Impact of Ex-Post Assessments in a Modeling the Impact of Ex-Post Assessments in a Guarantee Fund such as PACICCGuarantee Fund such as PACICC

With/Without a TCFP.With/Without a TCFP. Optimality Criteria for TCFP.Optimality Criteria for TCFP. Model Calibration and Implementation.Model Calibration and Implementation. Numerical Results and their Interpretation.Numerical Results and their Interpretation. Conclusions.Conclusions.

Purpose and Research QuestionPurpose and Research Question

Contagion risk is a Contagion risk is a source of concernsource of concern for for members of PACICC here in Canada.members of PACICC here in Canada.

PurposePurpose of our research: of our research:– Study how contagion risk can come into play Study how contagion risk can come into play

as a result of ex-post guarantee fund as a result of ex-post guarantee fund assessments.assessments.

Research QuestionResearch Question::– Can a tax-carry-forward program be a plausible Can a tax-carry-forward program be a plausible

solution for the contagion problem, while solution for the contagion problem, while maintaining protection for policyholders and maintaining protection for policyholders and claimants?claimants?

Literature Review on Contagion Literature Review on Contagion Risk in the Financial Services Risk in the Financial Services

IndustryIndustry DefinitionDefinition of contagion risk: of contagion risk:

– Spill over effects of shocks from one or more firms to Spill over effects of shocks from one or more firms to other firms.other firms.

Topic largely studied in banking:Topic largely studied in banking:– Evidence of different transmission mechanisms (liquidity Evidence of different transmission mechanisms (liquidity

and/or asset-quality problems, rumors/panics, etc.) at and/or asset-quality problems, rumors/panics, etc.) at both levels - domestic and international.both levels - domestic and international.

Fewer studies in the insurance literature:Fewer studies in the insurance literature:– Brewer & Jackson (2002) observed evidence of « intra & Brewer & Jackson (2002) observed evidence of « intra &

inter » industry contagion effects in the L&H sector;inter » industry contagion effects in the L&H sector;– Angbazo & Narayanan (1996) also found contagion Angbazo & Narayanan (1996) also found contagion

effects in the P&C sector due to catastrophic and effects in the P&C sector due to catastrophic and regulatory events.regulatory events.

Our Basic Model of the Insurance Our Basic Model of the Insurance Firm Firm

EBIT-based default risk model of an insurance firm that EBIT-based default risk model of an insurance firm that allows for stochastic CF’s and interest rates.allows for stochastic CF’s and interest rates.

Main featuresMain features::

– Revenue and costs are described by a mean-reverting Revenue and costs are described by a mean-reverting Gaussian process (eq 3.1);Gaussian process (eq 3.1);

– Investment portfolio contains short and long-term bonds Investment portfolio contains short and long-term bonds with term structure dynamics as in Vasicek (1977) (eq with term structure dynamics as in Vasicek (1977) (eq 3.2-3.3);3.2-3.3);

– The franchise value is accounted for as the PV of future The franchise value is accounted for as the PV of future economic rents for an identical unlevered firm (eq 3.4 & economic rents for an identical unlevered firm (eq 3.4 & 3.5);3.5);

– Financial leverage is considered through a stationary Financial leverage is considered through a stationary debt structure for which there is an interest charge and a debt structure for which there is an interest charge and a repayment of principal each year (as in Leland & Toft, repayment of principal each year (as in Leland & Toft, 1996). This leads to the firm’s annual net income (eq 1996). This leads to the firm’s annual net income (eq 3.6);3.6);

Our Basic Model of the Insurance Our Basic Model of the Insurance Firm (cont’)Firm (cont’)

Main features:Main features:– The insurer pays dividends out of its cash reserves The insurer pays dividends out of its cash reserves

which includes all other marketable securities;which includes all other marketable securities;

– If the cash reserves fall below zero, then the insurer If the cash reserves fall below zero, then the insurer bankrupts (eq 3.8);bankrupts (eq 3.8);

– Bankruptcy time is random (eq 3.10);Bankruptcy time is random (eq 3.10);

– Ultimately, if the insurer has not defaulted, the firm is Ultimately, if the insurer has not defaulted, the firm is liquidated and SH’s are entitled to a residual claim (eq liquidated and SH’s are entitled to a residual claim (eq 3.11);3.11);

– At any time t, the economic value of the insurance firm At any time t, the economic value of the insurance firm is given by the PV of expected payoffs to all claimants is given by the PV of expected payoffs to all claimants (equity, debt and government) less bankruptcy costs (eq (equity, debt and government) less bankruptcy costs (eq 3.12 to 3.15).3.12 to 3.15).

Modeling the Impact of Ex-Post Modeling the Impact of Ex-Post Assessments in a Guarantee Fund such as Assessments in a Guarantee Fund such as

PACICC PACICC

Extension of basic model without a TCFPExtension of basic model without a TCFP

– Members of the guaranty fund are all identical and similar to the Members of the guaranty fund are all identical and similar to the insurance firm described in our basic model.insurance firm described in our basic model.

– Upon a failure event at time tUpon a failure event at time t00, the fund’s remaining solvent , the fund’s remaining solvent companies become engaged in a sort of loss-recovery program:companies become engaged in a sort of loss-recovery program: The firm we are modeling must pay a periodic amount to The firm we are modeling must pay a periodic amount to

PACICC in order to meet obligations toward clients;PACICC in order to meet obligations toward clients; Size of the amount (h) and time schedule of recovery program Size of the amount (h) and time schedule of recovery program

[t [t11, T, Thh] are negotiated among fund’ s members;] are negotiated among fund’ s members; PACICC becomes a claimant of the failed company and will PACICC becomes a claimant of the failed company and will

receive a fraction of the liquidation proceeds, which will be receive a fraction of the liquidation proceeds, which will be later returned to its solvent members in the form of later returned to its solvent members in the form of dividends;dividends;

Hence, the firm’s cash reserves fall below normal level so that Hence, the firm’s cash reserves fall below normal level so that both leverage and failure risk are increased (eq 4.1 & 4.2).both leverage and failure risk are increased (eq 4.1 & 4.2).

– Under such a default contagion model, the values of both equity Under such a default contagion model, the values of both equity and government claims of the insurance firm will decrease due a and government claims of the insurance firm will decrease due a probability distribution of cash reserves being more skewed.probability distribution of cash reserves being more skewed.

Modeling the Impact of Ex-Post Modeling the Impact of Ex-Post Assessments in a Guarantee Fund such as Assessments in a Guarantee Fund such as

PACICC (cont’)PACICC (cont’) Extension of basic model with a TCFPExtension of basic model with a TCFP

– Under the TCFP, the government agrees to only receive a Under the TCFP, the government agrees to only receive a fraction (fraction (ββ) of regular taxes from solvent members over [t) of regular taxes from solvent members over [t11, , TThh], but it is hopeful to recuperate the difference later at time ], but it is hopeful to recuperate the difference later at time TThh +1: +1: This will reduce the failure risk of the insurance firm caused by This will reduce the failure risk of the insurance firm caused by

PACICC’s extraordinary obligation imposed following the failure PACICC’s extraordinary obligation imposed following the failure event.event.

However, these companies may also fail before the full However, these companies may also fail before the full repayment of residual taxes at Trepayment of residual taxes at Thh +1. So, TCFP is risky to the +1. So, TCFP is risky to the government.government.

On the other hand, the same firms might also face difficulty over On the other hand, the same firms might also face difficulty over [t[t11, T, Thh] for other reasons that could lead to an EBIT ] for other reasons that could lead to an EBIT < 0, so that < 0, so that no regular taxes would have to be paid even without a TCFP.no regular taxes would have to be paid even without a TCFP.

– In this setup, the TCFP can be viewed as an indirect source of In this setup, the TCFP can be viewed as an indirect source of debt financing, where the government has a prior claim over debt financing, where the government has a prior claim over debtholders.debtholders.

– Equations 4.3 and 4.11 formulate both the government and Equations 4.3 and 4.11 formulate both the government and the equity claims under the scenario of a default contagion the equity claims under the scenario of a default contagion model with TCFP.model with TCFP.

Optimality Criteria for TCFPOptimality Criteria for TCFP Decision rule for governmentDecision rule for government::

– Maximize expected utility of future tax revenues.Maximize expected utility of future tax revenues.

Model Calibration Model Calibration Model calibatred using financials Model calibatred using financials (EBIT, TA, CA, TL, Div(EBIT, TA, CA, TL, Div) of 38 ) of 38

members of PACICC over 1997-2005:members of PACICC over 1997-2005:– Median sample value of the panel averages for each variable.Median sample value of the panel averages for each variable.

Model ImplementationModel Implementation

Numerical Results and their Numerical Results and their InterpretationInterpretation

In Table 1 and Figure 1 (varying the firm’s cash reserves and In Table 1 and Figure 1 (varying the firm’s cash reserves and dividend payout), we find that:dividend payout), we find that:

– Introducing ex-post assessments will increase the failure rate Introducing ex-post assessments will increase the failure rate of the insurance firm, thus lowering the value of its claims (E of the insurance firm, thus lowering the value of its claims (E and G)and G) Direct consequence of contagion effectDirect consequence of contagion effect Much lower for high initial cash reserves.Much lower for high initial cash reserves.

– In order to make the TCFP sustainable, it appears that initial In order to make the TCFP sustainable, it appears that initial cash reserves must be high.cash reserves must be high.

– The default rates over time (credit curves) are also indicative The default rates over time (credit curves) are also indicative of a contagion effect due to ex-post assessmentsof a contagion effect due to ex-post assessments Appear to be lower when the TCFP is introduced, Appear to be lower when the TCFP is introduced,

independently from the level of initial cash reserves.independently from the level of initial cash reserves. TCFP adds value by lowering bankruptcy costs. The higher TCFP adds value by lowering bankruptcy costs. The higher

the initial cash reserves, the more solvent the firm is and the initial cash reserves, the more solvent the firm is and the higher will the optimal tax deferral rate (the higher will the optimal tax deferral rate (ββ) be.) be.

Numerical Results and their Numerical Results and their InterpretationInterpretation

In Table 2 and Figure 2 (varying the firm’s debt and the guaranty In Table 2 and Figure 2 (varying the firm’s debt and the guaranty fund’s ex-post assessment), we find that:fund’s ex-post assessment), we find that:

– Again, there is a downward shift in value of the firm’s claims (E Again, there is a downward shift in value of the firm’s claims (E and G) and an upward shift in the default rates over time and G) and an upward shift in the default rates over time (credit curves) following the introduction of ex-post (credit curves) following the introduction of ex-post assessments.assessments.

– TCFP does produce a systematic increase in the government’s TCFP does produce a systematic increase in the government’s claims (G) but it does have a mixed impact on the value of the claims (G) but it does have a mixed impact on the value of the insurer’s equity claim (E):insurer’s equity claim (E): E increases (decreases) when the assessment is low (high)E increases (decreases) when the assessment is low (high) Same effect on E when the insurer’s debt level is highSame effect on E when the insurer’s debt level is high

– The optimal tax deferral rate drops rapidly as the extraordinary The optimal tax deferral rate drops rapidly as the extraordinary obligation imposed by the guaranty fund goes up. obligation imposed by the guaranty fund goes up. Here, the tax authority does not find the option of more Here, the tax authority does not find the option of more

deferral very attractive.deferral very attractive.

Numerical Results and their Numerical Results and their InterpretationInterpretation

In Figure 3, we find that:In Figure 3, we find that:

– The optimal tax deferral rate is highly sensitive to the The optimal tax deferral rate is highly sensitive to the tax authority’s degree of risk aversion:tax authority’s degree of risk aversion: W/r to the insurer’s dividend payout, a higher risk-W/r to the insurer’s dividend payout, a higher risk-

tolerance does not necessarily imply a higher optimal tolerance does not necessarily imply a higher optimal deferral rate.deferral rate.

The tax authority is likely to be more tolerant when The tax authority is likely to be more tolerant when the amount of assessment is lower.the amount of assessment is lower.

A medium level of risk aversion leads to higher tax A medium level of risk aversion leads to higher tax deferral rates.deferral rates.

In Figure 4, we find that the appreciation rate of the In Figure 4, we find that the appreciation rate of the government’s claim due to TCFP:government’s claim due to TCFP:– is not very sensitive to the tax authority’s degree of risk is not very sensitive to the tax authority’s degree of risk

aversionaversion– largely depends on the amount of assessment charged largely depends on the amount of assessment charged

by the guaranty fund and, to a lesser extent, also upon by the guaranty fund and, to a lesser extent, also upon the insurer’s dividend payout.the insurer’s dividend payout.

Figure 3: The Optimal Tax-Deferral Figure 3: The Optimal Tax-Deferral RateRate

Figure 4: Figure 4: The Appreciation Rate of the Government’s Claim The Appreciation Rate of the Government’s Claim

Due to the TCFPDue to the TCFP

ConclusionsConclusions

Overall, our results suggest that:Overall, our results suggest that:– TCFP does effectively reduce the TCFP does effectively reduce the

contagion effect;contagion effect;– TCFP does systematically increase the TCFP does systematically increase the

value of the government’s prior claim;value of the government’s prior claim;– TCFP does not always verify the TCFP does not always verify the

incentive compatibility condition w/r to incentive compatibility condition w/r to equityholders as shown in eq.4.14equityholders as shown in eq.4.14