School Level Maths Proof 9

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    Richard Purvey
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      Given that:

      apv_(x)_n_life annuity due of 1 per year payable once a year = (1 – n_E_x – apv_(x)_n_death benefit of 1 payable at the end of the year of death)/d

      and

      apv_(x)_n_life annuity due of 1 per year payable m times per year = (1 – n_E_x – apv_(x)_n_death benefit of 1 payable at the end of the 1/m year of death)/d(m)

      and

      apv_(x)_n_life annuity due of 1 per year payable m times per year, under the UDD assumption, equals alpha(m) * apv_(x)_n_life annuity due of 1 per year payable once a year – beta(m) * (1 – n_E_x), show that

      apv_(x)_n_death benefit of 1 payable at the end of the 1/m year of death, under the UDD assumption, equals i * apv_(x)_n_death benefit of 1 payable at the end of the year of death/i(m)

      alpha(m) = i*d/(i(m)*d(m))

      beta(m) = (i-i(m))/(i(m)*d(m))

      Method

      For ease, letting;

      W denote apv_(x)_n_life annuity due of 1 per year payable m times per year,

      X denote apv_(x)_n_death benefit of 1 payable at the end of the 1/m year of death,

      Y denote apv_(x)_n_life annuity due of 1 per year payable once a year and

      Z denote apv_(x)_n_death benefit of 1 payable at the end of the year of death,

      the given equations become;

      Y = (1 – n_E_x – Z)/d

      and

      W = (1 – n_E_x – X)/d(m)

      and

      W, under the UDD assumption, equals alpha(m) * Y – beta(m) * (1 – n_E_x)

      This means that

      (1 – n_E_x – X)/d(m), under the UDD assumption, equals alpha(m) * (1 – n_E_x – Z)/d – beta(m) * (1 – n_E_x), meaning that

      1 – n_E_x – X, under the UDD assumption, equals d(m) * alpha(m) * (1 – n_E_x – Z)/d – d(m) * beta(m) * (1 – n_E_x)

      And because alpha(m) = i*d/(i(m)*d(m)) and because beta(m) = (i-i(m))/(i(m)*d(m)), this means that

      1 – n_E_x – X, under the UDD assumption, equals i * (1 – n_E_x – Z)/i(m) – (i-i(m)) * (1 – n_E_x)/i(m), meaning that

      1 – n_E_x – X, under the UDD assumption, equals – i * Z/i(m) + 1 – n_E_x, meaning that

      – X, under the UDD assumption, equals – i * Z/i(m), meaning that

      X, under the UDD assumption, equals i * Z/i(m), meaning, in other words, that

      apv_(x)_n_death benefit of 1 payable at the end of the 1/m year of death, under the UDD assumption, equals i * apv_(x)_n_death benefit of 1 payable at the end of the year of death/i(m)

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