Life Contingencies School Level Maths Proof 3

- Given that:

apv_(x)_n_death benefit of 1 payable instantly on death = integral (0,n) v^t * t_p_x * mu(x + t) dt

and

apv_(x)_n_life annuity of 1 per year payable continuously = integral (0,n) v^t * t_p_x dt, show that

apv_(x)_n_death benefit of 1 payable instantly on death = 1 – v^n * n_p_x + ln(v) * apv_(x)_n_life annuity of 1 per year payable continuously.

Method

Looking at the integral (0,n) v^t * t_p_x * mu(x + t) dt,

t_p_x = l(x + t)/l(x) and mu(x + t) = – l’(x + t)/l(x + t) and so t_p_x * mu(x + t) = – l’(x + t)/l(x), which is -t_p_x differentiated with respect to t, and so, integrating by parts, this integral becomes

[-t_p_x * v^t] (0,n) – integral (0,n) -ln(v) * v^t * t_p_x dt

= – v^n * n_p_x + 0_p_x * v^0 + ln(v) * integral (0,n) v^t * t_p_x dt

= 1 – v^n * n_p_x + ln(v) * integral (0,n) v^t * t_p_x dt

= 1 – v^n * n_p_x + ln(v) * apv_(x)_n_life annuity of 1 per year payable continuously.