Under a constant force of mortality with force mu and force of interest delta, what is A_x?

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Multiple Choice

Under a constant force of mortality with force mu and force of interest delta, what is A_x?

Under a constant force of mortality μ, the probability density of dying at time t after age x is μ e^{-μ t}, and the discount factor for time t is e^{-δ t}. The present value of a unit death benefit paid at time t is e^{-δ t}, so the expected present value is the integral over all future times of these components: ∫_0^∞ e^{-δ t} μ e^{-μ t} dt = μ ∫_0^∞ e^{-(μ+δ) t} dt = μ/(μ+δ). Therefore, A_x = μ/(μ+δ).

Under a constant force of mortality μ, the probability density of dying at time t after age x is μ e^{-μ t}, and the discount factor for time t is e^{-δ t}. The present value of a unit death benefit paid at time t is e^{-δ t}, so the expected present value is the integral over all future times of these components: ∫_0^∞ e^{-δ t} μ e^{-μ t} dt = μ ∫_0^∞ e^{-(μ+δ) t} dt = μ/(μ+δ). Therefore, A_x = μ/(μ+δ).

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