Give a continuous version of the endowment theory.

Assume a constante force of discount δ.

A-x:n = A-x(1n)+v(n)npx=1-δa-x(1n)

The continuous version of the endowment theory involves the present value of an endowment at time $�$ for a life aged $�$, denoted as $�-�:�$, and is given by:

$�-�:�=�-�(1:�)+�(�)\cdot �\cdot {�}_{�}$

Where:

• $�-�(1:�)$: Represents the net single premium for insurance payable at the moment of death, starting at age $�$ and ending at age $�$.
• 
  
• $�(�)$: Represents the discount factor at time $�$.
• 
  
• $�\cdot {�}_{�}$: Represents the probability of surviving from age $�$ to age $�$.

The continuous version of the endowment theory states that the present value of an endowment at time $�$ is equal to the sum of the net single premium for insurance payable at the moment of death from age $�$ to age $�$ and the present value of the probability of survival from age $�$ to age $�$ discounted to time $�$.

This continuous version of the endowment theory assumes a constant force of discount $�$, which is implicit in the discount factor $�(�)$.

It provides a framework for evaluating the present value of an endowment at a specific time point, considering both insurance benefits and survival probabilities.