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December 12, 2022 at 4:34 pm #23698
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) = (ii(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) = (ii(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) – (ii(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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