Quantitative index of regulatory forbearance

Chapter 3 Too Big to Fail or Too Small to Save: Regulatory Forbearance and

3.2. Contract specification and Guarantee Benefit Index

3.2.3. Quantitative index of regulatory forbearance

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excursion of the Brownian motion Z_t is below a constant barrier b =¹_σln �^ηL_A⁰

0�".

Therefore, under the Q measure, the maturity benefit ΨL(A_T) can be expressed as follows:

E_Q�e^−rTΨ_L(A_T)1{T_B⁻>𝑇}�。 (3.9) This can be rephrased as follows:

e^−�r+²¹^m²^�TE_P�1_�T_b⁻_>𝑇�Ψ_L(A₀exp{σZ_T}exp{gT}exp{mZ_T})�， (3.10) where At is the asset process under P which can be expressed as follows:

A_t = A₀exp{σZ_t}exp{gt} (3.11)

3.2.3. Quantitative index of regulatory forbearance

Based on the definition in above subsection, this study develops a guarantee benefit index that expresses the present value of the residual value when liquidation happens before or after maturity.

M(0) = E_Q�e^−rTmin{L_T, A_T}1_{T_B⁻_>𝑇}� + EQ�e^−rT^B⁻min�L_T_B⁻, A_T_B⁻�1{T_B⁻≤T}� = E_Q�e^−rT(L_T− [L_T− A_T]⁺)1{T_B⁻>𝑇}� + E_Q�e^−rT^B⁻min�L_T_B⁻, A_T_B⁻�1{T_B⁻≤T}� = E_Q�e^−rTL_T1_{T_B⁻_>𝑇}� − e^{−�r−g+12m}²^�TE_P�[L0− A₀e^σZ^T]⁺e^mZ^T1_�T_b⁻_>𝑇��

+ E_P�e^−�r+12m²^�T^b⁻exp�mZ_T_b⁻�min�L_T_b⁻, A_T_b⁻�1_�T_b⁻_≤T�� = E_Q�e^−rTL_T1_{T_B⁻_>𝑇}� − PDOP[A0, T; B₀, L₀; r, g]

+ E_P�e^−�r+12m²^�T^b⁻exp�mZ_T_b⁻�min�LT_b⁻, A_T_b⁻�1_�T_b⁻_≤T�� 。 (3.12) Note that the value of the guarantee benefit index includes three parts:

1. A deterministic guaranteed part L_Twhich is paid at maturity when the value of the assets has not stayed below the barrier for a time longer than d;

2. A Parisian down-and-out put option with strike L_T;

3. A rebate paid immediately when the liquidation occurs.

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Before detail calculation, this study presents some mathematical notations, definitions, and propositions.

Definition 3.1 (Laplace transform) The Laplace transform of a function f(t) for all t ≥ 0 is the function f̂ defined by

ℒ{f(t)} = f̂(λ) = � e^+∞ ^−λtf(t)dt

0 ,

Definition 3.2 (Inverse Laplace transform) The inverse Laplace transform of a

Laplace transform f̂ is

ℒ⁻¹�f̂(λ)� = f(t) where t ≥ 0.

Assume the following notations:

θ = √2λ, k1 =1 σ log �

L₀

A0� , D =b − k

√d .

Proposition 3.1 BSP is the Black and Scholes (1973) put option price, which can be expressed as

BSP[A₀, T; L₀; r, g] = E_Q[e^−rT[L_T− A_T]⁺] = L₀e^−(r−g)TN(d₂) − A₀N(d₁)

d_1/2 =ln �AL₀⁰� + �r − g ± 12 σ²� T σ√T

Proof: The result can be obtained following the work in Black and Scholes(1973).

This study denote the Digital Parisian Down and In Call option by DPDIC[A0, T; L0; r, g].

Proposition 3.2 The Laplace transform of DPDIC[A₀, T; L₀; r, g] is given by the following formula

DPDIC� [A0, λ; L₀; r, g] = e^b(θ+m)

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of the standard normal distribution.

Proof: See appendix B.

Proposition 3.3 The Laplace transform of a Parisian Up and In Call option is denoted

by PUIC� [A₀, λ; B₀, L₀; r, g]. This can be expressed by the following formula. For any

Proof: See Labart and Lelong (2009).

Proposition 3.4 The following relationships hold (put call parity of Parisian option) PDOP[A₀, T; B₀, L₀; r, g] = A₀B₀PUOC �1

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Proof: See Labart and Lelong (2009).

Proposition 3.5 ℒ �E_P�e^−�r−g+¹²^m²^�T^b⁻1_�T_b⁻_≤T��= E� �eP ^−�r−g+¹²^m²^�T^b⁻1_�T_b⁻_≤T��

=1 λ

exp ��λ + r − g + 12m²� b�

ψ ��λ + r − g + 12m²� d�

Proof: See Appendix B.

Proposition 3.6

EP�exp�(m + σ)ZT_b⁻�� = e^(m+σ)b�1 − (m + σ)√2πde^(m+σ)² ²^dN�−(m + σ)√d��

Proof: See Appendix B.

Proposition 3.7

Proof: See Appendix B.

Proposition 3.8

Proof: See Appendix B.

A detailed calculation of the guarantee benefit index M(0)appears in the following theorem.

Theorem 3.1 The guarantee benefit index M(0) can be expressed as the following

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Proof: See Appendix B.

In theorem 1, to find M(0), first find the Laplace transform of DPDIC, PUIC, and E_P�e^−�r−g+¹²^m²^�T^b⁻1_�T_b⁻_≤T��. Then, use the numerical method of Laplace inversion to find the M(0).

3.3. Relative and Absolute intervention criterion

Before comparing different intervention criterion, this subsection discusses the guarantee benefit index. The guarantee benefit index can be rewritten as a linear

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function of asset or liability. This means we can easily find the guarantee benefit index per unit asset.

Define the leverage ratio α =^L_A⁰

0, and the guarantee benefit index can be expressed as the following formula:

M(0) = EQ�e^−rTLT1{T_B⁻>𝑇}� − e^{−�r−g+12m}²^�TEP�[L0− A0e^σZ^T]⁺e^mZ^T1_�T_b⁻_>𝑇��

The equation above illustrates the linearity of the guarantee benefit index. This linear relationship can be used to evaluate different intervention criteria. The following context discussion defines two intervention criteria: the relative intervention criterion and the absolute intervention criterion.

Definition 3.3 (Monitoring ratio) The monitoring ratio R is a benchmark set by a regulatory authority intervening in the operation of an insurance company. This ratio satisfies

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where γ is the trigger standard of the government intervention and represents the liability ratio. Therefore, as the minimal capital requirement (the ratio of asset larger than liability) increases, the monitoring ratio will increase. According to the definition of the guarantee benefit index, a higher monitoring ratio increases both the rebate payment the policy holder can receive and the guarantee benefit index.

Definition 3.4 (Relative intervention criterion) Under the relative intervention

criterion, the regulatory authority adopts the ratio of the liability of the insurance company (ηL₀) as a benchmark for triggering the government’s intervention.

According to Definition 3, the monitoring ratio is ηα.

Definition 3.5 (Absolute intervention criterion) Under the absolute intervention

criterion, the regulatory authority adopts the absolute value (L₀− Υ) as a benchmark for triggering the government’s intervention. Let Υ = βL₀, and β will fluctuate as L0 increases or decreases. According to Definition 3, the monitoring ratio is α(1 − β).

After defining the relative intervention criterion and the absolute intervention criterion, it is important to know under what conditions these two styles of intervention criteria have the same M(0). This is easy to show as the following formula:

ηα = α(1 − β) ⇒ β = 1 − η (3.16) which means

Υ = (1 − η)L0 (3.17) i.e., if (3.17) hold, for two kinds of intervention criteria, regulatory authority uses the same intervention standards.

Definition 3.6 (Average regulatory intensity) The average regulatory intensity

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represents the guarantee benefit index per asset value.

Indeed, the average regulatory intensity of relative intervention criterion is ^M^R⁽⁰⁾

A₀ , and the average regulatory intensity of absolute intervention criterion is ^M^A⁽⁰⁾

A₀ .

Definition 3.7 (The principal of fairness) The principal of fairness means the average

regulatory intensity should not affect by the firm size for all insurance companies having the same financial conditions, including volatility, minimal guarantee interest rates, and leverage ratio.

Corollary 3.1 Under the same regulatory intensity, relative intervention criterion is

more suitable to the principal of fairness than absolute intervention criterion.

Discussion: First consider the condition of one company. In Fig. 2, the average regulatory intensity of the relative intervention criterion is fixed, and will not change as the initial liability fluctuates. In contrast, the average regulatory intensity of the absolute intervention criterion floats, and changes based on initial liability. On the other hand, when a regulatory authority adopts the absolute intervention criterion, the larger insurance companies will suffer more attentive by a regulatory authority, but this kind of situation will not happen in relative intervention criterion.

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Figure 3.2 Relationship between 𝐀_𝟎 and average regulatory intensity ^𝐌^𝐑^(𝟎)

𝐀𝟎 ：blue,

𝐌^𝐀(𝟎)

𝐀𝟎 ：red, with parameters 𝛂 = 𝟎. 𝟗𝟓; 𝐫 = 𝟎. 𝟎𝟐; 𝐠 = 𝟎. 𝟎𝟏𝟓; 𝐝 = 𝟎. 𝟓; 𝛈 = 𝟎. 𝟗; 𝚼 = 𝟓𝟎; 𝐓 = 𝟐𝟎; 𝛔 = 𝟎. 𝟏

The following discussion discusses the conditions of two companies. For Company 1, assume its initial asset level is A¹₀ and its initial liability is L¹₀. The term M₁^R(0) is the guarantee benefit index of relative intervention criterion for Company 1 and M₁^A(0) is the guarantee benefit index of absolute intervention criterion for Company 1. For Company 2, adopt a similar definition. If we fix other factors, i.e., these two insurance companies have the same r, g, and σ. Based on the definition of relative intervention criterion, the leverage ratio α larger cause the higher monitoring ratio and the higher guarantee benefit index M^R(0). Indeed, α1 = _A^L⁰¹

01 > ^L_A⁰²

02 = α2 ⇒

M₁^R(0)

A₀¹ >^M²^R_A⁽⁰⁾

02 . This shows that the insurance company with a higher leverage ratio

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will experience more stringent monitoring from the regulatory authority. For the condition of α₁ = α₂, even if these insurance company has different assets and liabilities, we can still to monitor them under the same supervisory standard.

Under the absolute intervention criterion, we fix other factors, i.e., these two insurance companies have the same α, r, g, and σ, the monitoring ratio α(1 − β) raise as the asset increasing. If A¹₀ is larger than A²₀, then ^M¹^A⁽⁰⁾

A₀¹ >^M²^A_A⁽⁰⁾

02 . In this situation, the regulatory authority will be more concerned about insurance companies with a high asset value, even if these insurance companies have the same leverage ratio, guaranteed interested rate, and other factors.

The discussion above shows that the relative intervention criterion is a better fit with the principal of fairness. The intervention criterion should not only focus on the asset size of the insurance company, but also pay more attention to the leverage ratio.

3.4. Numerical analysis

This section first shows why the pricing method of Labart and Lelong (2009) is more accurate than that of Bernard et al. (2005b). Numerical results clearly define the guarantee benefit index, and it react the force of regulatory and compare different intervention criterion. Finally, the scenario analysis in this study expresses the guarantee benefit index under different market conditions.

3.4.1. Comparative numerical method

Chesney et al. (1997) expressed the Parisian down and in call option as follows:

For K > 𝐽,

PDIC[x, T; J, K; r, q] = e^−(r+12m²^)T� e^∞ ^my(xe^σy− K)h₁(T, y)dy

β(x)

for K ≤ J,

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strike price, J is the barrier price, r is riskless interest rate, q is the dividend rate, and T is the maturity of the option.

h1(T, y) and h₂(T, y) are integrable functions with no closed form solution.

Labart and Lelong (2009) found the Laplace transform of different types of Parisian options using the following formula:

f(t) = 1

2πi�^a+i∞e^stf̂(s)ds

a−i∞

where a, t > 0.

We can approximate the integration above using the trapezoidal rule. Labart and Lelong (2009) proved that this approximation is bounded.

Bernard et al. (2005b) and Labart and Lelong (2009) presented the similar numerical results for Parisian options. However, only Labart and Lelong (2009) proved the bounded solution. Therefore, this study adopts the Labart and Lelong (2009) method to obtain accurate results.

3.4.2. Appropriateness of the guarantee benefit index

The guarantee benefit index should be an increasing function of the monitoring

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ratio and a decreasing function of the grace period. Indeed, a stricter intervention criterion increases the security of the asset the policy holder can receive, and vice versa. Therefore, the guarantee benefit can represent the regulatory intensity in some way.

Table 3.2 The guarantee benefit index M(0) with parameter:

𝐀_𝟎 = 𝟏𝟎𝟎; 𝐋_𝟎 = 𝟗𝟓; 𝛂 = 𝟎. 𝟗𝟓; 𝐫 = 𝟎. 𝟎𝟐; 𝐠 = 𝟎. 𝟎𝟏𝟓; 𝛔 = 𝟎. 𝟎𝟓。

Table 3.2 illustrates that the guarantee benefit index is an increasing function of the monitoring ratio and is a decreasing function of the grace period. A longer grace period and a lower monitoring ratio induce a lower guarantee benefit index. This means the guarantee benefit index reflects the regulatory intensity.

In table 3.2, further shows that the guarantee benefit index converges to a fixed value as the grace period lengthens or the monitoring ratio decreases. This condition represents the guarantee payment of the regulatory authority not intervenes the insolvency companies of long time. In addition, the guarantee benefit index affected by monitoring ratio sensitively as a shorter grace period and affected by grace period sensitively as a higher monitoring ratio. This explains the regulatory authority

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adopting conservative action which means a higher monitoring ratio and shorter grace period can protect the policy holder’s rights, but if one of these two active is over relaxation which means the grace period too long and the monitoring ratio too low will lead the effect of the other active lower.

3.4.3. Scenario analysis

This subsection discusses two problems:

1. The effects of the riskless rate and minimal guarantee rate.

2. The effects of asset volatility.

3. The effects of different intervention criteria.

3.4.3.1. The effects of riskless interest rate and minimal guarantee interest rate

The interest rate is usually the most important factor affecting the premium. This study uses the guarantee benefit index to analyze the solvency of life insurance companies under different riskless rates and minimal guarantee rates. Tables 3.3 and 3.4 show the situation in which the riskless interest rate is equal to or less than the minimal guarantee interest rate, i.e., r = 𝑔 and r < 𝑔. The guarantee benefit index is higher if r < 𝑔. Thus, the situation r < 𝑔 generates more concern for regulatory authorities. Adopting the guarantee benefit index as a quantitative index of regulatory forbearance gives some feedback on the increasing probability of insolvency.

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Table 3.3 The guarantee benefit index of the condition 𝐫 = 𝒈 with parameters:

𝐀𝟎 = 𝟏𝟎𝟎; 𝐋𝟎= 𝟗𝟓; 𝛂 = 𝟎. 𝟗𝟓; 𝐫 = 𝟎. 𝟎𝟐; 𝐠 = 𝟎. 𝟎𝟐; 𝛔 = 𝟎. 𝟎𝟓

Table 3.4 The guarantee benefit index of the condition 𝐫 < 𝑔 with parameters:

𝐀_𝟎= 𝟏𝟎𝟎; 𝐋_𝟎= 𝟗𝟓; 𝛂 = 𝟎. 𝟗𝟓; 𝐫 = 𝟎. 𝟎𝟏𝟓; 𝐠 = 𝟎. 𝟎𝟐; 𝛔 = 𝟎. 𝟎𝟓

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Tables 3.3 and 3.4 illustrate a higher guarantee benefit index than Table 3.2. This means the regulatory authority should enhance its monitoring of the insurance company, if there is the lower riskless interest and the higher minimal guarantee rate in the future. Tables 3.2 to 3.4 show that a lower riskless interest rate induces a higher guarantee benefit index. This suggests that the regulatory authorities should focus on the interest rate condition of the insurance company.

3.4.3.2. The effect of asset volatility

The asset volatility is also an important factor affecting the solvency of insurance company. This study uses the guarantee benefit index to analyze the solvency of life insurance companies under different asset volatilities. Table 3.5 and 3.6 show the guarantee benefit index with different asset volatilities. The value of guarantee benefit index concentrates in lower volatility situation, which suggesting that the higher volatility usually decrease the guarantee benefit index and guarantee benefit for the policy holder.

Table 3.5 The guarantee benefit index M(0) with parameter:

𝐀_𝟎 = 𝟏𝟎𝟎; 𝐋_𝟎= 𝟗𝟓; 𝛂 = 𝟎. 𝟗𝟓; 𝐫 = 𝟎. 𝟎𝟐; 𝐠 = 𝟎. 𝟎𝟏𝟓; 𝛔 = 𝟎. 𝟏。

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Table 3.6 The guarantee benefit index M(0) with parameter:

𝐀_𝟎 = 𝟏𝟎𝟎; 𝐋_𝟎 = 𝟗𝟓; 𝛂 = 𝟎. 𝟗𝟓; 𝐫 = 𝟎. 𝟎𝟐; 𝐠 = 𝟎. 𝟎𝟏𝟓; 𝛔 = 𝟎. 𝟏𝟓。

3.4.3.3. The effect of different intervention criterion

This subsection uses the average regulatory intensity to explain which relative intervention criteria best fits the principle of fairness. Table 3.7 shows the numerical results of average regulatory intensity for different intervention criteria. The firm size changes the average regulatory intensity in the absolute intervention criterion. This indicates that the firm size will change the regulator’s attitude, even if the other financial factors are the same. This agrees with the findings in the previous section that the relative intervention criterion is the best supervisory intervention criterion.

Table 3.7 shows that if the grace period is long enough, the average regulatory intensity will converge to a steady state. This means that if a regulatory authority uses a tolerance supervisory standard, then the two intervention criteria involve similar degrees of supervision.

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Table 3.7 The average force of regulatory in difference intervention criterions with parameters: 𝛂 = 𝟎. 𝟗𝟓; 𝐫 = 𝟎. 𝟎𝟐; 𝐠 = 𝟎. 𝟎𝟏𝟓; 𝛔 = 𝟎. 𝟎𝟓; 𝛈 = 𝟎. 𝟗; 𝚼 = 𝟓𝟎

Average regulatory intensity in the relative intervention criterion

Average regulatory intensity in the absolute intervention criterion

In the global financial crisis, financial institutions have suffered asset depression in the volatile financial market. To stabilize the financial market, regulatory and supervisory bodies tend to adopt regulatory forbearance in such situation, giving grace period to financial institutions to allow them to reconstruct and reduce the chance of systematic risk in the market. For instance, the U.S. government and The Taiwan Financial Supervisory Committee have injected capital to insurance companies or adopted regulatory forbearance action to avoid the bankruptcy of insurance companies suffering large losses in the subprime crises in 2008. Therefore, how to select the proper capital intervention criterion to steady the insurance market is critical not only for existing competitors, but also for potential entrants to the market.

This study develops a quantitative index, called the guarantee benefit index, to estimate regulatory forbearance and analyze the effects of guarantee value, asset size,

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and leverage affect on life insurance companies. This guarantee benefit index evaluates the suitability of different intervention criteria. This also evaluates the effects of leverage, grace period, and minimal guarantee interest rate on the guarantee benefit index. The numerical analysis in this study uses the method developed by Labart and Lelong (2009) to obtain more accurate result.

The numerical results of this study are as follows:

(1) Leverage ratio and guarantee benefit: The guarantee benefit index increase as the leverage ratio increases. An increase in liability value increases the leverage ratio.

Therefore, the leverage ratio and guarantee benefit fluctuate in a similar way.

(2) Regulatory forbearance and guarantee benefit: The government’s action to reduce the standard of financial supervisory is called regulatory forbearance. This study uses two regulatory forbearance factors: grace period and monitoring ratio to evaluate different regulatory condition. A longer grace period or lower monitoring ratio induce a lower guarantee benefit index. If the grace period is long enough or the monitoring ratio low enough, the average regulatory force converges to a steady state which means the regulatory condition is too relaxation.

This study uses the guarantee benefit index to evaluate the effects of different intervention criteria. Numerical results indicate that the relative intervention criteria is an appropriate supervisory intervention standard. The regulatory authority should more concentrative on the leverage ratio while focus on the firm sizes.

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Chapter 4 An Analysis of the Bankruptcy Cost of Insurance Guaranty Fund under Regulatory Forbearance

4.1. Introduction and Motivation

The financial crisis of 2008 and its continuing low interest rate environment have heightened the urgency of addressing the issues of negative interest rate in the Taiwanese life insurance industry. Addressing solvency and capital problems is critical for existing life insurers and for potential new entrants to the market. This forces them to be competent, and to adopt the recognized prudent and international standards of a regulation framework. Taiwan opened its insurance market to U.S.

based insurers in 1987 and to all potential foreign insurers in 1994. Due to this market internationalization and liberalization, the Taiwanese life insurance market has seen rapid and continuous growth. Taiwan’s life insurance market is currently ranked the ninth largest in the world based on total premiums received, and the largest in the world by premium penetration (total premiums as a percentage of Gross Domestic Product, GDP) in 2008 by Swiss Re (2009).

Foreign life insurance firms operating in Taiwan have recently been forced to withdraw from the domestic market. This is primarily due to financial troubles plaguing parent groups and worsening negative interest rate differentials. Foreign life insurers such as ING (International Nederland Group N.V.), PCA, Aegon Insurance Group, and several others sold off their branch offices in Taiwan in 2009. This sale was for two major reasons: one was that their parent groups were suffering financial predicaments, and the other involved the worsening negative interest differential, especially in traditional policies. The worsening negative interest differential has seriously damaged the financial sustainability of the life insurance industry. In late 2008, domestic listed life insurance companies were scrambling to inject capital in an

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attempt to raise the level of risk-based capital (RBC). The Financial Supervisory Commission (FSC) relaxed its RBC requirements to decelerate the drop in insurer RBC resulting from the big slump in the domestic stock market. Due to the effect of depreciation in foreign exchange and impaired assets, domestically listed life-insurance companies are facing the pressures of increasing their capital.

To relieve the domestic life insurer solvency capital pressure, the FSC relaxed the method of calculating RBC, allowing insurers to recognize only 50% of their losses from stock investments, rather than the past 100% requirement. However, this convenient measure was only effective until the end of 2008. RBC is the required minimum capital that a life insurer must maintain to cover operating risks. The FSC currently requires insurance companies to publically announce their RBC twice annually, and the ratio of RBC to overall capital cannot be lower than 200%.

Companies with an RBC ratio below 200% must refrain from doling out dividends.

Cathay Life, the largest life insurer in Taiwan, recently decided to issue NT$20 billion (US$600.6 million) in subordinated corporate bonds, NT$15 billion (US$450.45 million) of which was used to purchase class A preferred new shares issued by Cathay Life, under its umbrella.

For financially distressed life insurers who are not capable of raising capital, and possibly facing liquidation, reducing bankruptcy costs through early government intervention is becoming an important issue. Influenced by the global financial crisis, the stock market downturn, and the declining interest rate regime, the high interest rate insurance policies issued by Taiwanese life insurance companies have raised the question of sustaining these insurance companies. Since the 1980s, Europe, Japan, and the United States have seen a long list of defaulted life insurers. Increasing uncertainty in the capital market has dramatically increased the default rate of life insurance companies due to sudden asset depreciation. Thus, how to set up proper

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solvency regulatory guidelines to control the default risk of life insurers has become an urgent topic.

To stabilize the insurance market, a guaranty fund scheme functions as a receiver once the government triggers the exit mechanism. Taiwan’s life and non-life insurance guaranty funds were originally set up in 1993, and have operated for more than 15 years according to the Insurance Act. A recent amendment in 2007 introduced some major changes to the system. First, the two originally independent foundations for life

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