As with single life functions, one can readily extend fundamental principles to handle many practical contracts. We indicate how to do so in the discrete, with similar extensions to the continuous case analogous<\/p>\n
We will consider a generic status \\(u\\).
\nVarying Benefits and Payments. Let \\(K\\) be the curtate failure time associated with that status. For an insurance benefit the pays \\(b_{K+1}\\) for failure in the \\(K^{th}\\) period, the EPV is
\n\\begin{eqnarray*}
\n\\sum_{k=0}^{\\infty} b_{k+1} v^{k+1}
\n~_{k|} q_u .
\n\\end{eqnarray*}<\/p>\n
For an annuity benefit that pays \\(\\pi_h\\) at the beginning of each year when the status \\(u\\) has survived, the EPV is
\n\\begin{eqnarray*}
\n\\mathrm{E~} \\left( \\sum_{h=0}^K \\pi_h v^h \\right)=
\n\\sum_{k=0}^{\\infty} \\pi_k v^k ~ _k p_u .
\n\\end{eqnarray*}
\nGeneric Status – Insurance and Annuities. Temporary and deferred annuities, as well as endowment and deferred insurances, can be defined in a fashion similar to single life functions.<\/p>\n
Consider two lives, \\(x\\) and \\(y\\) but let \\(T(y)=n\\) with probability one. Then, for the joint life status, we have
\n\\begin{eqnarray*}
\n\\ddot{a}_{xy} = \\ddot{a}_{x:\\overline{n|}}
\n\\end{eqnarray*}
\nand
\n\\begin{eqnarray*}
\nA_{xy} = A_{x:\\overline{n|}}.
\n\\end{eqnarray*}
\nSimilarly for the last-survivor status,
\n\\begin{eqnarray*}
\n\\ddot{a}_{\\overline{xy}} = \\ddot{a}_{\\overline{x:\\overline{n|}}}
\n=\\ddot{a}_x +\\ddot{a}_{\\overline{n|}} – \\ddot{a}_{x:\\overline{n|}}
\n=\\ddot{a}_{\\overline{n|}} + ~_{n|} \\ddot{a}_x,
\n\\end{eqnarray*}
\nan annuity that is payable for certain for \\(n\\) years and as long as
\n\\(x\\) survives thereafter.<\/p>\n
Generic Status – Multiple Lives. We can allow multiples lives on a contract, say, \\(x\\), \\(y\\), and \\(z\\), in the same fashion. For example, compute the survival function
\n\\begin{eqnarray*}
\n~_t p _{xyz} = ~_t p _x \\times ~_t p _y \\times ~_t p _z
\n\\end{eqnarray*}
\nassuming independence among lives.<\/p>\n
Further, we can one life, say \\(z\\), but let \\(T(z)=n\\) with probability one. Then, for example, the joint life annuity on the triple becomes
\n\\begin{eqnarray*}
\n\\ddot{a}_{xyz} = \\ddot{a}_{xy:\\overline{n|}}
\n\\end{eqnarray*}
\na temporary joint life annuity on the joint status \\(xy\\). In this way, we can introduce temporary and deferred annuities, as well as endowment and deferred insurances, for the joint life status and last-survivor status.
\nTo summarize, here are some examples of different statuses we regularly consider:<\/p>\n
| Status \\(u\\) <\/font>\n\t\t\t<\/td>\n | Description<\/font> \n\t\t\t<\/td>\n | Status \\(u\\)<\/font><\/td>\n | Description<\/font> \n\t\t\t<\/td>\n<\/tr>\n| \\(xy\\) <\/font><\/td>\n | joint-life<\/font><\/td>\n | \\(\\overline{xy:\\overline{n|}}\\)<\/font><\/td>\n | \\(n\\)-year guaranteed<\/font><\/td>\n<\/tr>\n | \\(\\overline{xy}\\) <\/font><\/td>\n | last-survivor<\/font><\/td>\n | \\( ~ _{x:\\overline{n|}}^1\\)<\/font><\/td>\n | contingent<\/font><\/td>\n<\/tr>\n | \\(xyz\\)<\/font><\/td>\n | joint lives <\/font><\/td>\n | \\( ~ _{xy}^1\\)<\/font><\/td>\n | contingent<\/font><\/td>\n<\/tr>\n | \\(xy:\\overline{n|}\\)<\/font><\/td>\n | temporary<\/font><\/td>\n | \\(x|y\\)<\/font><\/td>\n | reversionary<\/font><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/table>\n | |