Chapter 4 An Analysis of the Bankruptcy Cost of Insurance Guaranty Fund under
4.2. The Model
4.2.2. Cost of Insurance Guaranty Fund
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before the life insurer encounters financial problems. These precautions can mitigate losses and increase the minimal compensation for policyholders. From risk management the point view, avoiding and resolving life insurer bankruptcy is an important issue in the current insurance market. The purpose of setting up safety funds is to provide certain buffers and risk control mechanisms to minimize the effect of the collapse of the insurance market and maintain stability.
In insurance guaranty funds in Japan, U.S., U.K., Taiwan, and other countries, the compensation for life insurance contracts is based on a given percentage of the policy reserve. When insurance companies become insolvent, supervisory authorities tend to offer a grace period to the insurer. During this grace period, the life insurer can inject capital or perform financial restructuring.
4.2.2. Cost of Insurance Guaranty Fund
This subsection illustrates the valuation methods and the cost structure of the insurance guaranty fund assuming a continuous-time frictionless financial market.
Any market imperfections are ignored. Within this framework, using the equivalent martingale measures, the price process of insurance company assets {At}t∈[0,T] is assumed to follow a geometric Brownian motion
dAt = At(rdt + σdWt),
where r denotes the deterministic interest rate, σ represents the deterministic volatility of the asset price process {At}t∈[0,T], and{Wt}t∈[0,T] is the equivalent Q-martingale.
The asset price dynamic can be solved as the following equation At = A0exp ��r −12σ2� t + σWt�.
The grace period of regulatory authority is predetermined by d. Since the asset value is greater than the regulatory barrier, i.e., At > Bt, and d ≥ T − t, it is impossible to have an excursion below Bt between t and T with a length greater
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than d. Therefore, the value of the Parisian down-and-out call option corresponds to the Black—Scholes price (Black and Scholes, 1973) of a standard European call option. The Parisian option becomes a standard call option when d ≥ T. If we have At> Bt and d = 0, it corresponds to the scenario introduced by Grosen and Jogensen (2002).
Apart from these special cases, the standard Parisian option is priced as follows. In the standard Parisian down-and-out option framework, the final payoff ΨL(AT)is only paid if the following technical condition is satisfied:
TB− = inf�t > 0��t − gB,tA �1{At<Bt} > 𝑑� > 𝑇 (4.1) a new probability measure P defined by the Radon--Nikodym density
dQ
e.g., Zt = Wt+ mt. The following derivation clarifies this equivalence argument:
gB,tA = sup{s ≤ t|As = Bs}
= sup �s ≤ t|A0exp ��r −1
2 σ2� s + σWs� = ηL0egs�
= sup{s ≤ t|Zs = b} = gb,t
Thus, the “excursion of the value of the assets below the exponential barrier Bt = ηL0egt" is an event in which the excursion of the Brownian motion Zt is below a constant barrier b =1σln �ηLA0
0�". This simplifies the entire valuation procedure.
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Under the new probability measure P, the value of the assets At can be expressed as At= A0exp{σZt}exp{gt}. In a complete financial market, the price of a T -contingent claim with the payoff ψL(AT) corresponds to the expected discounted payoff under the equivalent martingale measure Q, e.g.,
EQ�e−rTψL(AT)1{TB−>𝑇}� This can be rephrased as follows:
e−�r+12 m2�TEP�1{Tb>𝑇}ψL(A0exp{σZT}exp{gT})exp{mZT}�
According to guideline of the insurance guaranty fund or guaranty fund in many countries, the compensation for insurance contracts is primarily based on a certain percentage of the reserve. Let be the minimal compensation ratio. If the asset value of the insurance company is less than the minimal compensation liability at contract maturity, the guaranty fund will compensate the policyholder. The cost of the guaranty fund at maturity can be written as [λLT− AT]+. If the insurance company becomes insolvent before maturity, which means that its asset value is lower than the monitor standard, and a longer grace period d is required. In this case, the guaranty fund will compensate the policyholder for the difference between the residue asset value and the minimal compensation liability. The cost of guaranty fund before maturity can be written as EQ�e−rTB−�λLTB−− min�LTB−, ATB−��+1{TB−≤T}�. Therefore, the bankruptcy cost C(0) of the guaranty fund can be determined by:
C(0) = EQ�e−rTB−�λLTB−− min�LTB−, ATB−��+1{TB−≤T}� +EQ�e−rT[λLT− AT]+1{TB−>𝑇}�
In the equation above, the first term on the right represents the cost of bankruptcy when the time of the insurer becoming insolvent is less than the given grace period.
The second term on the right means the cost of bankruptcy at maturity. It can be
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formulated through the following equation.
C(0) = EQ�e−rTB−��LTB−− ATB−�+− (1 − λ)LTB−�+1{TB−≤T}� +PDOP[A0, T; B0, λL0; r, g]
= EQ�e−rTB−�λLTB−− ATB−�+1{TB−≤T}� +PDOP[A0, T; B0, λL0; r, g]
Note that the cost of bankruptcy consists of two parts:
(1) A Parisian down-and-in put option with strike λLTB− when insurer have defaulted before maturity.
(2) A Parisian down-and-out put option with strike λLT;
Parisian options can be valued by many different methods: Andersen and Brotherton-Ratcliffe (1996) used Monte Carlo simulations, Avellaneda and Wu (1999) and Costabile (2002) used lattices, Haber et al. (1999) used partial differential equations, and Stokes and Zhu (1999) used the finite element method and inverse Laplace transforms. Chesney et al. (1997) introduced the idea of using Laplace transforms to price Parisian options. Their study is based on the Brownian excursion theory, with a focus on the Azéma martingale (see Azéma and Yor (1989)) and the Brownian meander. Prices are computed by numerically inverting the Laplace transforms. Bernard et al. (2005) originally proposed employing this methodology.
They approximated the Laplace transforms using negative power functions whose analytical inverse is well known. However, there is no upper boundary for errors due to the inversion.
This study adopts the approach of Labart and Lelong (2009), who proposed a more accurate result using numerical inversion through the Laplace transform technique. In the following subsection, Laplace transform is employed to obtain the pricing formula. The Appendix C discusses these pricing derivations.
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4.3. Numerical analysis
This section employs the inverse Laplace transform in numerical computations to investigate the cost of bankruptcy of the guaranty fund, and compares the results by leverage ratio, asset volatility, and intervention criterion. The leverage ratio is used to calculate the financial leverage of a company and measure its ability to meet financial obligations. The asset volatility measures the variability of the returns on assets held by the firm. The intervention criterion measures the regulatory intensity and forbearance presented through the grace period. These factors are vital in measuring the cost of bankruptcy. Intuitively, as the leverage ratio and asset volatility of the insurer increases and the intervention criterion of government become more intensive, the cost of bankruptcy increases. However, this study presents numerical analysis to fully explore the correlation between these factors.
Figure 4.1 Cost of the guaranty fund given different leverage ratios and grace periods if default happened after maturity
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Figure 4.2 Cost of the guaranty fund given different leverage ratios and grace periods if default happened before maturity
Figure 4.3 The total cost of the guaranty fund given different leverage ratios and grace periods
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Figure 4.4 The total cost of the guaranty fund per liability given different leverage ratios and grace periods
Figures 4.1-4.4 show the bankruptcy costs of the insurance guaranty fund given leverage ratios and grace periods. These figures show the cost of the 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. 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.
Figures 4.1 through 4.4 compare the cost of the guaranty fund based on the effect of the leverage ratio and grace period. In this setting, the leverage ratio represents the ratio of liabilities to assets based on the insurer’s balance sheet. The intervention criterion is set at η=1. The results in figure 4.1 show that as the leverage ratio or the duration of the grace period increases, the cost of bankruptcy of IGF will increase.
This also indicates that the grace period is significantly influenced when the firm maintains a high leverage ratio.
Figure 4.2 shows that the cost of the guaranty fund increases if the insurer increases the leverage ratio. When the duration of the grace period increases, the cost
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of bankruptcy increases initially and then decreases. This behavior can be explained by the collapse probability of the firm, which initially exhibits an increasing trend that turns to a downward trend as the grace period increases.
Figures 4.3 and 4.4 plot the cost of bankruptcy. These figures show a shape similar to that in figure 4.2. When the grace period reaches a certain length, the cost of bankruptcy coverage converges to a stable value. The cost of bankruptcy per underwritten liability in figure 4.4 falls from 0% to 7.24%, indicating that different leverage ratios and grace periods have diverse default costs.
Figure 4.5 Cost of the guaranty fund given different ratios of monitoring and grace periods if default happened after maturity
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Figure 4.6 Cost of the guaranty fund given different ratios of monitoring and grace periods if default happened before maturity
Figure 4.7 The total cost of the guaranty fund with different given different ratios of monitoring and grace periods
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Figure 4.8 The total cost of the guaranty fund per liability given different ratios of monitoring and grace periods
Figures 4.5-4.8 show the bankruptcy costs of the insurance guaranty fund given different leverage ratios and grace periods. These figures show the cost of the 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 compensation ratio is 90%. The volatility of assets is 10% and the time horizon is 20 years.
Figures 4.5 to 4.8 compare the cost of the guaranty fund for different monitoring ratios and grace periods. In this case, the monitoring ratio equals the initial liability and the monitor barrier. The leverage ratio is equal to 0.95, which is the average leverage ratio in the Taiwanese life insurance industry. Figure 4.5 indicates that as the ratio of monitoring decreases or the grace period increases, the cost of bankruptcy.
These results show that the monitoring ratio and the grace period have a similar effect.
In figure 4.6, with a decrease in the grace period, when the monitoring ratio increases, the cost of the guaranty fund increases. With a decreased grace period, the difference between the minimal compensation ratio and ratio of monitoring has a significant effect on the cost of bankruptcy. Figure 4.6 shows that when the monitoring ratio
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increases more than the minimal compensation ratio (or exceeds this ratio), the cost of the guaranty fund decreases.
Figure 4.6 also shows that as the grace period increases, the cost of bankruptcy initially increases, then decreases in the most condition. However, lower monitoring ratio (less than 70%) leads the cost of bankruptcy decreasing as the grace period increasing. This follows the guaranty fund trend as the grace period increases. Figures 4.7 and 4.8 illustrate the cost of bankruptcy. Though similar to figure 4.6, the grace period in these figures increases before the cost of bankruptcy becomes stable. The cost of bankruptcy per written liability in figure 4.8 ranges from 0% to 7.66%. This shows that the monitoring ratio and the grace period have diverse effects.
Figure 4.9 Cost of the guaranty fund given different volatilities and grace periods if default happened after maturity
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Figure 4.10 Cost of the guaranty fund given different volatilities and grace periods if default happened before maturity
Figure 4.11 The total cost of the guaranty fund given different volatilities and grace periods
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Figure 4.12 The total cost of the guaranty fund per liability given different volatilities and grace periods
Figures 4.9-4.12 show the bankruptcy costs of the insurance guaranty fund given 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 compensation ratio is 90%. The trigger for government intervention is 100% of the liability. The time horizon is 20 years.
Figures 4.9 to 4.12 show how the cost of the guaranty fund affects volatility and grace period. In this case, the leverage ratio is 0.95 and monitoring ratio is 1, since 0.95 is the average 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 figure 4.9, if the volatility or the grace period increases, the cost of bankruptcy also increases. Results show that the grace period has a significant influence on the asset volatility increase. In figure 4.10, the volatility increase causes an increase in the cost of the guaranty fund. Figure 4.10 shows that the cost of bankruptcy initially increases and then decreases as the grace period increases. This is
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because the cost of the guaranty fund initially increases and then follows a downward trend as the grace period increases. Figures 4.11 and 4.12 present the cost of bankruptcy. These figures show a similar pattern to that in figure 4.2, but the grace period increases before the cost of bankruptcy stabilizes. The cost of bankruptcy per underwritten liability in figure 4.12 ranges from 0% to 7.24%. This shows the diversity of default costs between volatility and grace period.
To specify the factors is the most important factor affecting the bankruptcy problem to insurance companies. This study compares volatility, monitoring ratio, and leverage ratio interactively. Tables 1 to 9 illustrate the bankruptcy cost per liability value with different parameters setting, where different settings suggesting different monitoring intentions from regulative aspect. This study compares three different categories, tables 1 to 3, tables 4 to 6 and tables 7 to 9, to describe the effect of monitoring ratio 80%, 90%, to 100%. In each category, this study compares the effect of leverage ratio, volatility, and leverage ratio. Comparing these categories, lower monitoring ratio and higher volatility results higher bankruptcy cost. These tables show the coherent result that increasing volatility will lead to the higher bankruptcy cost. Results show that asset volatility has the most significant effect.
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Table 4.2 The bankruptcy cost per liability value with parameters:
𝛂 = 𝟎. 𝟗𝟓(𝐥𝐞𝐯𝐞𝐫𝐚𝐠𝐞 𝐫𝐚𝐭𝐢𝐨); 𝐫 = 𝟐%(𝐫𝐢𝐬𝐤 𝐟𝐫𝐞𝐞 𝐢𝐧𝐭𝐞𝐫𝐞𝐬𝐭 𝐫𝐚𝐭𝐞);
Table 4.3 The bankruptcy cost per liability value with parameters:
𝛂 = 𝟎. 𝟗(𝐥𝐞𝐯𝐞𝐫𝐚𝐠𝐞 𝐫𝐚𝐭𝐢𝐨); 𝐫 = 𝟐%(𝐫𝐢𝐬𝐤 𝐟𝐫𝐞𝐞 𝐢𝐧𝐭𝐞𝐫𝐞𝐬𝐭 𝐫𝐚𝐭𝐞);
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Table 4.4 The bankruptcy cost per liability value with parameters:
𝛂 = 𝟎. 𝟖𝟓(𝐥𝐞𝐯𝐞𝐫𝐚𝐠𝐞 𝐫𝐚𝐭𝐢𝐨); 𝐫 = 𝟐%(𝐫𝐢𝐬𝐤 𝐟𝐫𝐞𝐞 𝐢𝐧𝐭𝐞𝐫𝐞𝐬𝐭 𝐫𝐚𝐭𝐞);
Table 4.5 The bankruptcy cost per liability value with parameters:
𝛂 = 𝟎. 𝟗𝟓(𝐥𝐞𝐯𝐞𝐫𝐚𝐠𝐞 𝐫𝐚𝐭𝐢𝐨); 𝐫 = 𝟐%(𝐫𝐢𝐬𝐤 𝐟𝐫𝐞𝐞 𝐢𝐧𝐭𝐞𝐫𝐞𝐬𝐭 𝐫𝐚𝐭𝐞);
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Table 4.6 The bankruptcy cost per liability value with parameters:
𝛂 = 𝟎. 𝟗(𝐥𝐞𝐯𝐞𝐫𝐚𝐠𝐞 𝐫𝐚𝐭𝐢𝐨); 𝐫 = 𝟐%(𝐫𝐢𝐬𝐤 𝐟𝐫𝐞𝐞 𝐢𝐧𝐭𝐞𝐫𝐞𝐬𝐭 𝐫𝐚𝐭𝐞);
Table 4.7 The bankruptcy cost per liability value with parameters:
𝛂 = 𝟎. 𝟖𝟓(𝐥𝐞𝐯𝐞𝐫𝐚𝐠𝐞 𝐫𝐚𝐭𝐢𝐨); 𝐫 = 𝟐%(𝐫𝐢𝐬𝐤 𝐟𝐫𝐞𝐞 𝐢𝐧𝐭𝐞𝐫𝐞𝐬𝐭 𝐫𝐚𝐭𝐞);
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Table 4.8 The bankruptcy cost per liability value with parameters:
𝛂 = 𝟎. 𝟗𝟓(𝐥𝐞𝐯𝐞𝐫𝐚𝐠𝐞 𝐫𝐚𝐭𝐢𝐨); 𝐫 = 𝟐%(𝐫𝐢𝐬𝐤 𝐟𝐫𝐞𝐞 𝐢𝐧𝐭𝐞𝐫𝐞𝐬𝐭 𝐫𝐚𝐭𝐞);
Table 4.9 The bankruptcy cost per liability value with parameters:
𝛂 = 𝟎. 𝟗(𝐥𝐞𝐯𝐞𝐫𝐚𝐠𝐞 𝐫𝐚𝐭𝐢𝐨); 𝐫 = 𝟐%(𝐫𝐢𝐬𝐤 𝐟𝐫𝐞𝐞 𝐢𝐧𝐭𝐞𝐫𝐞𝐬𝐭 𝐫𝐚𝐭𝐞);
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Table 4.10 The bankruptcy cost per liability value with parameters:
𝛂 = 𝟎. 𝟖𝟓(𝐥𝐞𝐯𝐞𝐫𝐚𝐠𝐞 𝐫𝐚𝐭𝐢𝐨); 𝐫 = 𝟐%(𝐫𝐢𝐬𝐤 𝐟𝐫𝐞𝐞 𝐢𝐧𝐭𝐞𝐫𝐞𝐬𝐭 𝐫𝐚𝐭𝐞);
Negative interest rate spread, ALM mismatch, and morbidity uncertainties are the primary risks relating to the development of the Taiwan life insurance market. If these risks emerge, they might seriously jeopardize the financial stability of companies. For financially distressed life insurers incapable of raising capital and facing possible liquidation, reducing the cost bankruptcy of the insurance guaranty fund through early government disciplinary actions is a critical issue.
This study discusses the cost of the guaranty fund when a life insurance company faces default. Specifically, this study employs the Parisian option to evaluate the cost of the guaranty fund under Chapter 11 bankruptcy procedures. Financial leverage, performance stability, and government intervention are crucial factors influencing the cost of the guaranty fund. The results of this study show that the volatility of investment performance has a greater effect on the cost of bankruptcy per underwritten liability than other factors do. Numerical analysis shows that financial
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leverage and government intervention have a similar effect on the cost of the guaranty fund. This also suggests that the current premium rates (i.e., 10 basis points for non-life insurers and 20 basis points for life insurers) for insurance guaranty fund are inadequate. Hence, financial restructuring should be carefully designed to avoid moral hazard.
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In this appendix, we give the detailed illustration of the learning mechanism and the parameter values used in numerical analysis, and the glossary of the notations.
Appendix A.1
The investor's prior distribution of the initial value of β is assumed to be Gaussian and β is assumed to follow equation (8). Since de t( ) / ( )e t and dL t( ) are following a joint Brownian motion and all the parameters in equation (4), (8), and (9) are linear functions of the unobservable state variable β , the distribution function of β at time t, F xt( )=P(β ≤x F| tI), is Gaussian given the investor's information two mutually independent Wiener processes, where
1 2
The signal process is written as
2
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Applying the theorem 12.3 in Liptser and Shiryayev (2001), we have the following transition equations.
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The process b and v satisfy the following equations
( )
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Appendix A.2 Parameter values used in numerical analysis
Parameter Descriptions Notation Parameter values
Volatility of exchange rate
σe 0.1
Long run mean of the exchange rate return
µe 0.01,0.03,0.05
Volatility of domestic short rate
rd
σ 0.02
Mean-reversion speed of domestic short rate
ad 0.2
Long run mean of domestic short rate
bd 0.05
Volatility of foreign short rate
rf
σ 0.02
Mean-reversion speed of foreign short rate
af 0.2339
Long run mean of foreign short rate
bf 0.08
Volatility of domestic stock index return
Sd
σ 0.2
Volatility of foreign stock index return
Sf
σ 0.2
Risk premium of domestic stock index
Sd
µ 0.05
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Risk premium of foreign stock index
Sf
µ 0.075
Market price of risk of domestic discount bond
rd
λ 0.15
Market price of risk of foreign discount bond
rf
λ 0.2
Correlation coefficient between the exchange rate return
and the domestic short rate e r,d
ρ -0.1
Correlation coefficient between the exchange rate return
and the foreign short rate e r,f
ρ 0.2
Correlation coefficient between the exchange rate return
and the domestic stock index return ρ e S, d -0.1
Correlation coefficient between the exchange rate return
and the foreign stock index return e S, f
ρ 0.3
Correlation coefficient between the domestic short rate
and the foreign short rate r rd,f
ρ -0.3
Correlation coefficient between the domestic short rate and the domestic stock index return r Sd, d
ρ 0.2
Correlation coefficient between the domestic short rate and the foreign stock index return r Sd, f
ρ -0.2
Correlation coefficient between the foreign short rate and the domestic stock index return r Sf,d
ρ -0.2
Correlation coefficient between the foreign short rate and the foreign stock index return r Sf, f
ρ 0.2
Correlation coefficient between the domestic stock index return and the foreign stock index return Sd,Sf
ρ -0.1
Correlation coefficient between β1 and the domestic
short rate β1,rd
ρ -0.15
Correlation coefficient between β1 and the foreign short
rate β1,rf
ρ 0.06
Correlation coefficient between β1 and the exchange rate
return β1,e
ρ 0.01
Correlation coefficient between β2 and the domestic
short rate ρβ2,rd 0.2
Correlation coefficient between β2 and the foreign short
rate β2,rf
ρ -0.1
Correlation coefficient between β2 and the exchange rate
return ρ β2,e 0.07
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Appendix A.3 Glossary of the notations
The following are notations used in this paper to formulate the market structure and the filtering mechanism.
Md: domestic money market account;
Mf: foreign money market account;
Bd: domestic discount bond;
Bf: foreign discount bond;
Sd: domestic equity index;
Sf: foreign equity index;
rd: domestic risk free rate;
ad: mean-reversion speed of domestic short rate;
bd: mean of domestic short rate;
rd
σ : standard deviation of domestic short rate;
Bd
σ : standard deviation of domestic discount bond;
σ : standard deviation of domestic discount bond;