Under the same framework, what is a_x?

• A. mu/(mu+delta)
• B. 1/(mu+delta)
• C. 1/mu
• D. delta/(mu+delta)

Answer: (B)

A_x represents the present value of a continuous life annuity paying at rate 1 per year for as long as the life aged x is alive, under constant force of mortality μ and constant force of interest δ. The life survives to time t with probability e^{-μ t}, and payments received at time t are discounted by e^{-δ t}. The present value is the expected discounted payments: ∫_0^∞ e^{-δ t} P(T > t) dt = ∫_0^∞ e^{-(μ+δ)t} dt = 1/(μ+δ). Therefore the value is 1/(μ+δ).