Give the consequence of a constante interest force µ(x):=µ

 Given the provided conditions:

• Constant interest force: $�(�):=�$
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• Survival probability: $���={�}^{-��}>0$ (where $���$ denotes the probability of survival from age $�$ to age $�+�$)
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• Force of interest: $�$
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• Net single premium: $�-�={\int }_0^{\mathrm{\infty }}{�}^{-��}\cdot {�}^{-��}��=\frac{1}{�+�}$
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• Expectation of future lifetime: $�{�}^{\mathrm{°}}={\int }_0^{\mathrm{\infty }}�����={\int }_0^{\mathrm{\infty }}{�}^{-��}��=\frac{1}{�}$
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• Net premium reserve: $��=�-�+�-�-�-�+�=0$

Let's analyze the consequences of these conditions:

1. Net Single Premium ($�-�$):

   • $�-�={\int }_0^{\mathrm{\infty }}�(�)\cdot ���\cdot {�}_{�}(�)��={\int }_0^{\mathrm{\infty }}{�}^{-��}\cdot {�}^{-��}\cdot ���=\frac{�}{�+�}$
   • This represents the net present value of future benefits, discounted at the force of interest $�$.
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2. Net Premium Reserve ($��$):

   • $��=�-�+�-�-�-�+�=0$
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   tV=A−x+t−P−a−x+t=0
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   • This equation represents the relationship between the net premium reserve $��$, the net single premium $�-�$, and the premium $�$.
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   • It indicates that the net premium reserve should be zero when considering the net single premium and the premium.

These consequences indicate how the net single premium and the net premium reserve are related in the context of constant interest force and survival probabilities.

The net premium reserve should balance out with the net single premium and the premium to ensure financial equilibrium in the insurance policy.