The first term on the right represents the bankruptcy cost when the time of the insurer becoming insolvent is within the grace period. The second term on the right means the bankruptcy cost at maturity. It can be formulated through the following equation.
Note that the fair premium consists of two parts:
(1) A Parisian down-and-in put option with strike when insurers have defaulted before maturity.
(2) A Parisian down-and-out put option with strike ;
We employ the inverse Laplace transform in numerical computations to investigate the fair premium, and compare the results by leverage ratio, asset volatility, and intervention criterion. The financial leverage of a company is used to measure its ability to meet financial obligations. While asset volatility measures the investment behavior of the insurer and intervention criterion measures the regulatory intensity and forbearance through the grace period. Intuitively, as the leverage ratio and asset volatility of the insurer increases and the intervention criterion of government become more intensive, the fair premium increases. This study presents numerical results to explore the relationship between these factors.
Finding and Observation
Tables A, B, and C summarize the fair premiums based on various scenarios. The risk free interest rate is set at 2%. In Table A, the trigger point of government intervention is 100% of the liability and the leverage ratio is 95%. The volatilities of asset portfolio form 1% to 5%. In Table B, the trigger points of government intervention are 80%, 85%, 90%, 95%, and 100%. The leverage ratio is 95% and the volatility of asset portfolio is 3%. In Table C, the trigger point of government
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intervention is 100% and the asset portfolio volatility is 3%. The leverage ratios of the insurer are 91% to 95%.
Table A Fair Premium of Insurance Guaranty Fund in Basis Points
ote: when the volatility of an insurer’s asset portfolio changes from 1% to 5%, the fair premiums increase from 0 b.p. to 129 b.p., while the premiums increase almost twelve times at 5% from one-year to five-year grace period.
Grace period (Year)
Volatility
1% 2% 3% 4% 5%
0 0 0 0 0 0
0.25 0 0 0 0 0
0.5 0 0 0 0 0
1 0 0 0 1 8
1.5 0 0 0 5 22
2 0 0 1 10 37
5 0 0 7 39 99
10 0 0 13 55 127
15 0 0 13 56 129
20 0 0 13 56 129
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In Merton (1974), the equity value ( ) of a firm can be described as a European call option with strike price of the debt. According to Black and Schole (1973) pricing formula, the European call option is expressed as follows:
Where
Using the market value of equity and the book value of asset and liability, we can find the implied volatility of the asset value. The following table shows an empirical example of a Taiwan insurance company. The implied volatility is from 1.67% to 12.03% with different maturity and different market condition of the reference insurance company. Comparing the assumptions of the volatility in table A, it can infer the investor’s holding period of the stock is probability about 3 to 4 years.
Table A-1 implied volatility
This table describes the implied volatility of a Taiwan insurance company with different maturity assumption. Data source: Taiwan Economic Journal (TEJ).
date asset value liability value equity value implied volatility T=1 T=2 T=3 T=4 2011/3/31 670,369 638,969 48,661 10.77% 6.93% 5.02% 3.69%
2010/12/31 648,753 616,932 50,865 12.03% 7.85% 5.82% 4.45%
2010/9/30 620,441 593,821 44,321 11.18% 7.26% 5.34% 4.04%
2010/6/30 592,155 571,696 32,175 7.86% 4.83% 3.21% 1.67%
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Table B Fair Premium of Insurance Guaranty Fund in Basis Points
Note: when the ratio of monitoring changes from 100% to 80%, the fair premiums increase from 0 b.p. to 17 b.p.. The premiums converge to 13 b.p. when the length of grace periods increase to 20 years.
Table C Fair Premium of Insurance Guaranty Fund in Basis Points
Note: when the leverage ratio of the insurer changes from 91% to 95%, the fair premiums increase from 0 b.p. to 13 b.p., while the premiums increase more than ten times at 95% insolvency. The determinants include financial leverage, performance stability, and government intervention which impede the fair premium of the TIGF. Regulatory
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forbearance might cause the wrong incentives for the managers and significantly increase the cost of the guaranty fund by increasing the volatility of the insurer’s asset portfolio. Comparing the premium rates, i.e., 10 basis points for life insurers in TIGF, with the results in Table C, it shows that the asset volatility should be controlled less than 3% within five-year grace period.
Figures 2-5 The fair premium given leverage ratios and grace periods
These figures show the cost of the guaranty fund at maturity, before maturity, total cost, and fair premium. The initial asset is assumed to be 100 monetary units. The risk free interest rate is set at 2%.
The guaranteed rate is 1.5% and the minimal compensation ratio is 90%. The trigger point of government intervention is 100% of the liability. The volatility of asset is 10% and the time horizon is 20 years.
Fig 2 Fig 3
Cost of guaranty fund at maturity Cost of guaranty fund before maturity
Fig 4 Fig 5
Total cost of guaranty fund Fair premium
Figures 2-5 show the fair premium given leverage ratios and grace periods. These figures show the cost of the guaranty fund at maturity, before maturity, total cost, and fair premium. The initial asset is assumed to be 100 monetary units. The risk free interest rate is set at 2%. The guaranteed
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rate is 1.5% and the minimal compensation ratio is 90%. The trigger point of government intervention is 100% of the liability. The volatility of asset is 10% and the time horizon is 20 years.
Figures 2 through 5 compare the fair premium based on various leverage ratios and grace periods. In our setting, the intervention criterion is set at =1. The results in Fig. 2 show that as leverage ratio or grace period increases, the fair premiums increase. This indicates that the grace period is significantly influenced when the insurer maintains high leverage ratio.
Figure 3 shows that the fair premiums increase if the insurer increases leverage ratio. When we increase the length of grace periods, the fair premiums first increase and then decrease. This behavior can be explained by the ruin probability of the insurer, which initially exhibits an increasing trend and then turns a downward trend as the grace period increases.
Figures 4 and 5 plot the fair premium. The figures show a shape similarity to those in Fig. 3. When the grace period reaches a certain length, the fair premiums converge to a stable value. The fair premium in Fig. 5 falls from 0% to 7.24%, indicating that different leverage ratios and grace periods have result diverse default costs.
Figures 6-9 The fair premium given different leverage ratios and grace periods
The figures show the fair premium at maturity, before maturity, total cost, and fair premium. The initial asset is assumed to be 100 monetary units and the liability is 95. The risk free interest rate is set at 2%. The guaranteed rate is 1.5% and the minimal compensation ratio is 90%. The volatility of assets is 10% and the time horizon is 20 years.
Fig 6 Fig 7
Cost of guaranty fund at maturity Cost of guaranty fund before maturity
Fig 8 Fig 9
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Total cost of guaranty fund Fair premium
Figures 6 to 9 compare the fair premium for different monitoring ratios and grace periods. The monitoring ratio equals the initial liability and the monitor barrier.
The leverage ratio is set to be 0.95, which nears the average leverage ratio in Taiwan life insurance industry. Figure 6 indicates as the ratio of monitoring decreases or the grace period increases, the fair premium increases. These results show that the monitoring ratio and the grace period have a similar effect. In Fig. 7, with a decrease in the grace period, when the monitoring ratio increases, the cost of the guaranty fund increases. Given a shorter grace period, the difference between the minimal compensation ratio and ratio of monitoring has a significant effect on the fair premiums. Figure 7 shows that when the monitoring ratio increases more than the minimal compensation ratio increases (or exceeds this ratio), the cost of the guaranty fund decreases.
Figure 7 shows that as the grace period increases, the fair premium first increases, and then decreases. This concludes the guaranty fund trend as the grace period increases. Figures 8 and 9 illustrate the fair premium. Though the similarity to Fig. 7, the grace period in these figures increases before the fair premium becomes stable.
The fair premium in Fig. 9 ranges from 0% to 7.66%. This shows that the monitoring ratio and the grace period have diverse effects.
Figures 10-13 The fair premium given different volatilities and grace periods
These figures show the cost of guaranty fund at maturity, before maturity, total cost, and cost of bankruptcy per written liability. The initial asset is assumed to be 100 monetary units and the liability is 95. The risk free interest rate is set at 2%. The guaranteed rate is 1.5% and the minimal
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compensation ratio is 90%. The trigger for government intervention is 100% of the liability. The time horizon is 20 years.
Fig 10 Fig 11
Cost of guaranty fund at maturity Cost of guaranty fund before maturity
Fig 12 Fig 13
Total cost of guaranty fund Fair premium
Figures 10 to 12 show how the fair premium of the guaranty fund affects volatility and grace period. In the sample scenario, the leverage ratio is 0.95 and monitoring ratio is 1, since 0.95 is the normal leverage ratio in the Taiwanese life insurance industry. The liability monitoring value is set at 100% to represent basic minimal compensation for the policyholder. In Fig. 10, if the volatility or the grace period increases, the fair premium increases. Results show that the grace period has a significant influence when the asset volatility increases. In Fig. 11, the volatility increase causes an increase in the fair premium. Figure 11 shows that the premium initially increases and then decreases as the grace period increases. Figures 12 and 13 present the fair premium. These figures show a similar pattern to that in Fig. 3, but the grace period increases before the cost of bankruptcy stabilizes. The fair premiums are in Fig. 13 ranging from 0% to 7.24%. This shows the diversity of default costs due to various asset volatility and grace period.
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By extending our model, we examine the factors to determine their effect on the fair premium. Results show that asset volatility has the most significant effect.
While, leverage ratio and monitoring ratio have minor influence and behave similarly.
Conclusion
This study develops a quantitative benchmark, MGV, to measure regulatory forbearance and the impacts of leverage ratio, grace period and guarantee rate on MGV are investigated. Numerical results suggest that the relative intervention criterion (RIC) is better than the absolute intervention criterion (AIC) in accordance with the fairness principle in financial supervision.
Secondly, we compute the ex ante premiums of the insurance guaranty fund in our basic model in order to investigate the moral hazard problem. The results show that the volatility of investment performance has greater effect on the premiums than other factors. Numerical analysis shows that financial leverage and government intervention have a similar effect on the premiums. The results indicate that the fair premium for TIGF is risk sensitive and hence risk-based premium scheme should be implemented to ease the moral hazard.
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