How do we evaluate the net single premium when death benefit is paid at the moment of death?

To evaluate the net single premium when the death benefit is paid at the moment of death, we can follow the calculation steps using the formula provided:

Given:

• $�-�(�)={\lim }_{�\to \mathrm{\infty }}{�}^{(�)}(�)$
• ${�}^{(�)}(�)=�(0)\cdot �(\frac{1}{�})(1-\frac{1}{���})+�(\frac{1}{�})\cdot �(\frac{2}{�})(\frac{1}{���}-\frac{2}{���})+\dots$
• $�-�(�)=-{\int }_0^{�-�}�(�)\cdot �(�)��{�}_{�}=-{\int }_0^{�-�}�(�)\cdot �(�)\frac{�}{��}�����=-{\int }_0^{�-�}�(�)\cdot �(�)���{�}_{�}(�)��$
• $�-�(�)=�-�(�)$ where $�(�)=�(�)\cdot {�}_{�}(�)$

Explanation:

1. The first equation establishes that the net single premium $�-�(�)$ is the limit of the expected present value of future benefits as the number of payments, $�$, approaches infinity.

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3. 
   

4. The second equation represents the calculation of ${�}^{(�)}(�)$, which involves summing up the present values of future benefits discounted at different points in time, where $�(\cdot )$ represents the discount factor and ${�}_{�}$ represents the probability of surviving to the next age.

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6. The third equation uses integration to calculate the net single premium $�-�(�)$ by integrating the product of the death benefit function $�(�)$, the discount factor $�(�)$, and the force of mortality ${�}_{�}(�)$ over the interval $[0,�-�]$.

7. 
   

8. The fourth equation shows that the net single premium $�-�(�)$ is equivalent to the net single premium for a different insurance type $�-�(�)$, where $�(�)=�(�)\cdot {�}_{�}(�)$. This equivalence allows for easier calculation and comparison.

These equations provide a method to calculate the net single premium for insurance payable at the moment of death, taking into account the timing of benefits, discounting, and mortality factors.