How do you treat in a realistic model the premiums and reserves?

death benefits = continuous
premiums = discrete

In a realistic model where death benefits are continuous and premiums are discrete, and for the calculation of prospective reserves, the following steps can be taken:

1. Defining Variables:

   • ${�}_{�}^{\circ }(�)$
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   bk∘​(t): Represents the death benefit function shifted by $�$ years, where $0\le �\le -�-�-�$.
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   • This accounts for the timing of death benefits in the prospective reserve calculation.
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   • $��$: Represents the prospective reserve.
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   • ${�}_{�}({�}_{�}^{\circ })$: Represents the net single premium for insurance payable at the moment of death with the shifted death benefit function ${�}_{�}^{\circ }$.
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   • ${\ddot{�}}_{�}^{(�)}({�}_{�}^{\circ })$: Represents the present value of future premiums under the shifted premium function ${�}_{�}^{\circ }$.
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   • $�(�)$ and $�(�)$: Coefficients related to the payment mode.
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2. Prospective Reserve Calculation:

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   • The prospective reserve $��$ can be calculated as the difference between the net single premium ${�}_{�}({�}_{�}^{\circ })$ and the present value of future premiums:
   • $��=[\frac{�}{�}\cdot {�}_{�}({�}_{�}^{\circ })-(�(�)\cdot {\ddot{�}}_{�}({�}_{�}^{\circ })-�(�)\cdot {\ddot{�}}_{�}(\mathrm{\Delta }({�}_{�}^{\circ })))]$
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   • $\frac{�}{�}\cdot {�}_{�}({�}_{�}^{\circ })$: Represents the present value of future death benefits discounted at the interest rate.
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   • $�(�)\cdot {\ddot{�}}_{�}({�}_{�}^{\circ })$: Represents the present value of future premiums.
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   • $�(�)\cdot {\ddot{�}}_{�}(\mathrm{\Delta }({�}_{�}^{\circ }))$: Represents any adjustments in premiums over time.
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   • Subtracting the present value of future premiums from the present value of future death benefits gives the prospective reserve.

3. 
   

4. Practical Considerations:

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   • This approach accounts for the timing of death benefits and premiums in calculating the prospective reserve.
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   • The present value calculations involve discounting future cash flows at an appropriate interest rate.
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   • Coefficients $�(�)$ and $�(�)$ are determined based on the payment mode of the premiums (e.g., annual, semi-annual, quarterly) and any adjustments required.
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   • This method ensures that sufficient funds are reserved to meet future obligations while taking into account the timing and nature of cash flows in a realistic model.

The calculation of prospective reserves in a realistic model with continuous death benefits and discrete premiums involves discounting future cash flows and accounting for the timing of benefits and premiums. This approach ensures that appropriate reserves are maintained to meet future obligations under the insurance policy.