With benefit function b constant over each year the expression from A-x(b)=Ax(b~) reduces to?

 When the benefit function

$�$ is constant over each year, the expression from $�-�(�)={�}_{�}({�}^{\sim })$ simplifies as follows:

Given:

• ${�}_{�}(�)$ represents the net single premium for insurance payable at the moment of death with a constant benefit function $�$.
• ${�}^{\sim }$
• 
  
b∼ represents the continuous equivalent of the constant benefit function $�$.

Explanation:

1. Continuous Equivalent of Constant Benefit Function:

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   • When the benefit function $�$ is constant over each year, its continuous equivalent ${�}^{\sim }$ is a step function that remains constant between each integer age.
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2. Net Single Premium Equality:

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   • The expression ${�}_{�}(�)={�}_{�}({�}^{\sim })$ indicates that the net single premium for insurance payable at the moment of death remains the same whether we use the discrete constant benefit function $�$ or its continuous equivalent ${�}^{\sim }$.
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   • This equality holds true because the net single premium calculation takes into account the present value of future benefits, mortality rates, and discounting factors.
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   • As long as the present value of future benefits is the same, the net single premium remains unchanged regardless of whether the benefit function is constant or continuous.

3. Simplified Calculation:

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   • Using the continuous equivalent ${�}^{\sim }$ simplifies the calculation process, especially for mathematical operations such as integration, differentiation, and manipulation.
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4. Practical Application:

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   • This simplification aids actuaries and insurance professionals in modeling and analyzing insurance products with constant benefit structures.
   • It ensures consistency and accuracy in premium calculations, allowing for easier comparison and evaluation of different insurance policies.

In summary, the expression ${�}_{�}(�)={�}_{�}({�}^{\sim })$ illustrates that the net single premium for insurance payable at the moment of death remains the same whether the benefit function is represented as a discrete constant function $�$ or its continuous equivalent ${�}^{\sim }$.

This simplifies calculations and ensures consistency in premium evaluation for insurance products with constant benefit structures.