Life Contingencies School Level Maths Proof 3

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    Richard Purvey
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    Life Contingencies School Level Maths Proof 3

    1. 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.

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