3. Simulation
3.4 Investment Strategy
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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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