2. Literatures Review
2.3 Fair Value Reserve Estimation
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Then the risk margin is estimated as:
For the estimation of the cost of capital rate, QIS5 Calibration paper follows three steps:
(i) Estimate the equity risk premium.
(ii) Adjust the equity risk premium.
(iii) Calibrate it by the market price.
QIS5 follows these three steps and suggests that the cost of capital rate should be set greater than 6%.
2.3 Fair Value Reserve Estimation
Since IFRS4 provides a comprehensive definition of fair value liability along with its components and QIS5 gives techniques for the parameters estimation, we need more specific information for the parameters of fair value reserve estimation. We need the assumptions for liquidity premium and cost of capital.
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We adopt the research report of Hsieh, Tsai, and Kuo (2010)3, which provides a feasible method for Life insurers in Taiwan under the structure of IFRS4 and QIS5.
Hsieh et al. (2010) propose that the fair value reserve should include three basic components according to IFRS4:
, where is the best estimated liability of insurance contracts at time t. is the risk adjustment at time t and denotes the residual margin at time t. Each component is defined as follows:
(a)
The estimation of should be based on the expected cash flows which are marked to market, calculated by the actuarial assumptions of the product and insurer's past experiences. Then the expected cash flows should be discounted at a proper yield curve.
, where n is the policy year and denotes the ith discount factor. The cash flow of year t is expresses as:
Hsieh et al. (2010) fit the yield curve by Smith and Wilson model4, provides a
3
謝明華、蔡政憲、郭維裕，壽險業準備金評估方法之國際發展趨勢研究，
行政院金融監督管理委員會九十九年度委託研究計畫
，2010 年。
4
Smith A. & Wilson, T., 2001, Fitting Yield Curves with Long Term Constraints, Research Notes,
Bacon and Woodrow.
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yield curve of 100 years for Taiwan's life insurers. As we mention in previous sections, it is not necessary for us to use a yield curve with 100 years since we only simulate the insurer's financial statement for 20 years. We therefore only adopt the liquidity premium estimated of the report in our simulation. Hsieh et al. (2010) estimate the liquidity premium by CDS method, Covered bond spread method and Proxy method.
The result is shown in figure 3
Figure. 3 The liquidity premium curve in different methods Resource: Hsieh et al. (2010)
Hsieh et al.(2010) suggest that the CDS method and the Covered bond spread method is not appropriate for the life insurers in Taiwan. Hsieh et al. (2010) use the Proxy method to establish the liquidity premium which is shown in figure 4. And the numerical data is shown in table 3.
(b)
Hsieh et al. suggest that the risk adjustment could be calculated by SCR and the Insurer should estimate SCR by RBC regulation since current Taiwan supervisory regulation is established based on RBC. Hsieh et al. (2010) suggest that the insurer
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Figure 4. The liquidity premium term structure for Taiwan Resource: Hsieh et al. (2010)
Table 3. The adjusted liquidity premium (100%) for 20 Years Maturity Adjusted
Liquidity Premium
Maturity Adjusted Liquidity Premium
1 0.110% 11 0.110%
2 0.110% 12 0.110%
3 0.110% 13 0.110%
4 0.110% 14 0.110%
5 0.110% 15 0.110%
6 0.110% 16 0.088%
7 0.110% 17 0.066%
8 0.110% 18 0.044%
9 0.110% 19 0.022%
10 0.110% 20 0.000%
Resource: Hsieh et al. (2010)
should estimate the Solvency Capital Required for each year t based on C2 (Insurance risk) and C4 (Business risk):
And the risk adjustment at year t is expressed as the combination of SCR and cost of capital rate:
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Hsieh et al. (2010) test several kinds of policy then provide an approximate way for estimating C2 and C4 of each insurance product. For the endowment, Hsieh et al.
(2010) suggest that C2 of the endowment should be estimated as 0.30% of the death benefit, and C4 of the endowment should be estimated as 1.0% of the premium received. Hsieh et al. (2010) also provides a cost of capital for life insurers in Taiwan, which is given as 6%. We therefore adopt these parameters to estimate the fair value of reserve.
(c)
Hsieh et al. (2010) suggests the residual margin should be amortized by the expected claimed paid of each year t. The insurers should amortize the residual margin by four steps:
(i) Calculate the initial residual margin . (ii) Calculate the present value of the claim paid . (iii) Calculate the ratio of the residual margin should be amortized for each
year t by
.
(iv) The residual margin of each year t is therefore: .
3. Simulation
This section is divided into four parts: the first part is how we simulate the market asset prices, including interest model and stock price model. The second part exhibits the policy specifications, including cash flow of the policy, policy
assumptions. The third part is the policy reserve and the reserve duration. The fourth
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part we describe the two investment strategies of the insurers while in the last part we introduce the insurer's activities at each year step by step.
3.1 The Investment Markets
In this part we introduce the stochastic models used to simulate asset prices and insurance activities.
For simplicity, we assume that the insurers can only invest in two kinds of assets:
stock and bond, which are independent to each other. The years to maturity of bonds range from 1 to 20 years. The insurer is therefore able to invest in 21 kinds of securities in the market.
Let denote the differentials of two-dimension Wiener processes including the processes of the one-year spot rate(r) and the return of the stock index(S):
We assume that there is no correlation between these two processes.
(a) Interest Rate Model and Bond Markets
We choose the CIR model to simulate short term interest rate. The CIR model is a mean-reverting process in which the volatility of the short rate is partially related to the square root of the short rate. The discrete CIR model assumes the one-year interest rate (r) follows the stochastic process:
, where is the short rate at time t (t≧0), reflects the speed of the mean reversion, represents the long-term level of the short rate and denotes the volatility of the short rate, which we assume constant.
Notice that the discrete-time CIR model may generate some negative short rates,
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which may not happen in real interest rate market. In this case we take at each time t to avoid generating negative interest rate.
After we have the interest rate for each time t, next we generate the price of zero-coupon bonds with maturity ranged from 1 to 20 years. According to Cox et al.
(1985), a $1 face value zero-coupon bond at time t with maturity T is priced as
, where
,
,
; and(b) Stock markets
We assume that the dynamic of stock price follows geometric Brownian motion as follow:
, where dS denotes the change in stock price, is the stock price expected rate of return. denotes the volatility of the stock index return, which is constant along the simulation period.
Notice that we do not add any interest-rate-adjusted effect in the stock price process. We intend to eliminate the interest rate sensitivity of stocks to avoid that the position of stocks may affect the portfolio duration.
3.2 Policy's Specifications
In this part we introduce the policy's specifications in this simulation, including the cash flows of each year and the assumptions. For simplicity, we only use a
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twenty-year endowment product as the analyzed policy. We assume that the insurer sells the policy only to a fixed number population and the entire insurer's cash flow is generated from this endowment pool during the simulation period. The simulation's length is 20 years, we therefore assume the covered period of the endowment to be 20 years. The insured age is set at 30 before the simulation begins.
(a) Cash Flow of a Twenty-Year Endowment Policy
We assume that the commission, expense and expected premium are assumed to be received or paid out at the beginning of the year while the death benefit and surrender value are incurred at the end of the year. The expected cash flow at time t for the policy at the beginning of its policy year t after the nth premium has been collected with the nth expenses and commission have been paid out, where and , is defined as:
, where is the probability that the policy for the insured with age
remains valid at the beginning of year t (i.e. probability is adjusted by surrender rate). is the probability of the insured with age dying within 1 year, DB denotes the death benefits paid at the end of the year while Pre denotes the premium received at the beginning of each surviving year, which are constant during the policy years. is the surrender rate of the policy in year t,
denotes the surrender value paid at the end of policy year
. is the commission rate for the commission paid at the beginning of the year , represents the fixed expense paid at the beginning of the year and stands the variable expense rate.‧ 國
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The first bracket of the equation is the expected death benefit and surrender value paid at the end of each policy year n. The expected death benefit and surrender value paid out equals the death benefit DB, which is fixed during the policy years, times the conditional probability that the policy is valid for years ( ) and then the insured dies in this year ( . So similarly the expected surrender value is the surrender value paid at the end of policy year ( ) times the probability that the policy is valid for years ( ) and then surrender at this year ( ). The second bracket represents the net premium (i.e. expected premium deduct expected expense and commission) received at the beginning of each policy year n, and the expected net premium is the net premium
times the probability that the insured will survive at the end of the year .
At the beginning of the last year of the policy years, no more premiums will be received because the insured has paid all of the twenty premiums. Neither the
commissions nor the variable expense are paid at this time, instead the insurer should pay the survival benefit. The expected cash flow is therefore adjusted as follow:
(b) Policy assumptions
The assumptions of the variables we use in the policy cash flow estimation are shown in Table 4.
3.3 Reserves
(a) Policy reserves
The policy reserve at policy year n is estimated as present value of the expected
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Table 4. Assumptions about the endowment product Policy
net cash out flow, which is calculated after the nth premium, which can be expressed as:
, where represents the present value of the expected liability along the policy
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years, denotes the discount factor for the expected net cash flow at time t.
Negative represents that the insurer's aggregate expected cash flow is negative and the insurer should prepare the same amount of capital for future claimed, which means the policy reserve . Contrarily, positive net present value means the insurer should not prepare such amount of capital since current regulation do not allow negative reserve value. We take as zero in this case.
Book value reserve
The main difference between book value reserve and fair value reserve is the discount rate used. For the discount factor , the discount rate of each time t remains fixed under book value reserving method. The book value policy reserve's discount factor of time t, denoted as , is therefore expressed as:
, where r is the given discount rate of the policy.
Fair value reserve
If we estimate policy reserves in fair value, the discount rate of each time t has to be consistent with the interest rate market. In previous section we simulate the process of the one-year rate. Since the expected cash flow is measured annually, the discount factor of the fair value policy reserve is therefore expressed as
, where is the one year rate of year i, plus the liquidity premium mentioned in previous section.
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(b) Reserve Duration
According to Fabozzi (1998), we take the modified duration as the measurement of interest rate sensitivity. We assume that the insurers with ALM allocate their asset in order to match their asset duration with liability duration, which is also known as reserve duration.
The reserve duration at the policy year n, which is denoted as , is calculated as follows:
, where t is the time that the expected cash flow incurs.
Similar to reserve estimation, the discount factor can be calculated in fixed discount rate ( ) or market one year rate ( ). For consistent, we denote the duration calculated in fixed discount rate as . The duration calculated in variable discount rate as .
3.4 Investment Strategy
The insurers allocate their assets by their investment strategy. We assume that the there are two investment strategies for the insurers: The first one is buy-and-hold strategy, which assume the insurers simply allocate their capital on the financial assets in a given weighted. They do not change weighted or manage the interest risk during the simulation period.
The second strategy assumes the insurers manage their capital by duration matching. They estimate the liability duration in the beginning of every period, allocate their assets in proper position to match the duration of the asset and the
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liability. This strategy is strongly focused on eliminating the interest risk of insurer's liabilities.
In this section we introduce these two strategies in detail, including the assumptions and the method used for duration matching simulation.
(a) Buy-and-Hold Strategy
This strategy represents the insurers who do not want to eliminate their interest rate risk. The insurer under this strategy purchases financial assets in fixed weighted every year regardless of any interest rate volatility.
We assume that the insurers decide their weighted by Markowitz efficient frontier5. Since there are only two assets with no dependence to each other and there is no risk-free asset in the financial market, the optimization formula is expressed as follows:
, where represents the weighted of stocks and represents the weighted of zero-coupon bonds. denotes the volatility of the stock price and denotes the volatility of the bond price. denotes the return of the stock price and denotes the return of the bond price.
We can therefore have several sets of which provide the minimum portfolio variance. For there are zero-coupon bonds with maturity ranged from 1 to 20 years in bond market, we assume the insurer allocates equally in each kind of
5
Markowitz, H.M. (March 1952). Portfolio Selection. The Journal of Finance 7 (1): 77–91.
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zero-coupon bonds. The weighted matrix of the financial assets is therefore expresses as:
, where denotes the weighted of zero-coupon bonds with one-year maturity (which equals to ). denotes the weighted of weighted of zero-coupon bonds with two-year maturity and so on.
We have to distinguish the items of the financial assets on the balance sheet in order to calculate the shareholder's equity. We assume that the insurer under this strategy recognizes its asset position only as available-for-sell or hold-to-maturity. We then assume that the insurer recognizes its stock position, short-term zero-coupon bond position and half of the long-term bond position as available-for-sale. Thus, the insurer holds half of its long-term zero-coupon bond position as hold-to-maturity items. We define the long-term zero-coupon bonds is the zero-coupon bonds with maturity greater than 10 years. For example, if the initial capital of the insurer in period t is , the insurer will allocate in stocks and in the zero-coupon bonds with maturity i. The available-for-sell position will be:
And the hold-to-maturity position will be:
(b) Duration Matching Strategy
Duration matching strategy is assumed to be used by the insurers with ALM. In
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this strategy the insurer varies its asset position every period in order to match the liability duration. For simplicity we assume that the insurers under this strategy have no hold-to-maturity items. All the assets purchased will be used as hedge tools during the policy years, the insurer therefore recognize them as available-for-sale on the balance sheet.
Under this strategy, the insurer estimates its liability duration in the beginning of every year t. After that the insurer finds an optimization solution to minimize the gap between asset duration and liability duration. We use the linear least-squares approach here, solve least-squares curve fitting problems of the form:
, where is the lower bound of the weighted and is the upper bound of the weighted. These two parameters will be given before the simulation. Insufficient restriction in this equation may leads to infinite solutions. We only choose the solution which varies portfolio positions the less. Notice that in order to compare the pure effect of the allocation, the stock weighted is assumed to be consistent with the buy-and-hold strategy. Under this assumption, the insurer will only vary their bond position, which ensures the only difference between these two strategies is the management of bond portfolio (i.e. the management of the interest risk). And the available-for-sell position under this strategy will be:
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3.5 Insurer's Activities
In this part we introduce the activities of the insurer step by step, and how does the activity affect the insurer's balance sheet. We divided each year t into two periods:
beginning period and end period. Assume that all activities happen only at the beginning or the end of each policy year.
At the beginning of each year t, the insurer receives premiums from the policyholders, simultaneously pays the commissions and the expenses. The insurer increases the reserve listed on the balance sheet accordingly, then makes an asset allocation based on their investment strategy.
At the end of year t, the insurer renews the asset prices marked to the market (i.e.
marked to the price of t+1). Then the insurer pays claims through selling the asset proportionally and deducts the reserve. In follow subsections we describe detailed descriptions of each activity.
We denotes as the value of asset, liability and shareholder's equity on the insurer's balance sheet at the beginning of year t.
And , and as the value of asset, liability and shareholder's equity on the insurer's balance sheet at the end of year t.
(a) At the Beginning Period
The activities of the beginning period includes: collecting premium, paying expenses and commissions, increasing reserve and allocating capitals. At the beginning period, the insurer receives the net premium:
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Then the insurer increases the reserve based on the premium already collected.
= at the beginning period.
The variation in reserve increases the liability value on the balance sheet:
Then the insurer allocates their capital (i.e. premium received) to the financial assets, which are bonds and stock, based on market price at year t. The initial capital of the insurer equals to the net premium:
Then the insurer decides its asset position according to the investment strategy:
, where is the weighted matrix for year t.
The market price the insurer observes at the beginning of the year t is denoted as follow:
, where denotes the stock price at year t, denotes the price of zero coupon bond with one-year maturity at year t, the price of zero coupon bond with two-year maturity at year t, etc.
The insurer then purchases the market assets by the weighted and the market price:
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：
, where is the volume of stocks purchased at the beginning period of year
t,
denotes the volume of zero coupon bonds with one-year maturity, where is the volume of stocks purchased at the beginning period of year