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:�)+�(�)⋅�⋅��

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 �.

  • �⋅��: 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.

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