{"id":332,"date":"2014-12-31T16:03:09","date_gmt":"2014-12-31T16:03:09","guid":{"rendered":"http:\/\/frees.pajarel.net\/?page_id=332"},"modified":"2015-02-20T19:28:59","modified_gmt":"2015-02-21T01:28:59","slug":"special-cases","status":"publish","type":"page","link":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/actuarial-mathematics\/emerging-costs\/6-deferred-acquisition-expenses-and-modified-premium-reserves\/special-cases\/","title":{"rendered":"Special Cases"},"content":{"rendered":"<p><strong>Special Case 1. Ordinary Whole Life<\/strong><\/p>\n<p>Here, take \\(j = \\omega &#8211; x\\) and \\(_1 P_{[x]}^n = v _1 q_{[x]} = A_{[x]:\\overline{1|}}^1\\). Thus,<br \/>\n\\( A_{[x]:\\overline{1|}}^1+ P^{FPT} a_x =P_x \\ddot{a}_x \\) and<br \/>\n\\begin{eqnarray*}<br \/>\nP^{FPT} &#038;=&#038; \\frac{P_x \\ddot{a}_x &#8211; A_{[x]:\\overline{1|}}^1}{a_x} = \\frac{A_x &#8211; A_{[x]:\\overline{1|}}^1}{a_x} \\<br \/>\n&#038;=&#038; \\frac{_1 E_x A_{x+1}}{ _1 E_x \\ddot{a}_{x+1}} = \\frac{A_{x+1}}{\\ddot{a}_{x+1}} = P_{x+1} .<br \/>\n\\end{eqnarray*}<\/p>\n<p><strong>Special Case 2. \\(n\\)-pay, \\(m\\)-year Endowment<\/strong><\/p>\n<p>Here, take \\(j = n\\). Then, use the same logic in Special Case 1 to check that \\(P^{FPT} = _{n-1} P_{x+1:\\overline{m-1|}}.\\) Further,<br \/>\n\\begin{eqnarray*}<br \/>\nV_k^{FPT} &#038;=&#038; A_{x+k:\\overline{m-k|}} &#8211; _{n-1} P_{x+1:\\overline{m-1|}} \\ddot{a}_{x+k:\\overline{m-k|}}<br \/>\n= ~_{k-1}^{n-1} V_{x+1:\\overline{m-1|}},<br \/>\n\\end{eqnarray*}<br \/>\nthat is, a reserve for a life age \\((x+1)\\) at duration \\(k-1\\) for an \\(n-1\\)-pay, \\(m-1\\)-year endowment policy.<\/p>\n<p><div class=\"alignleft\"><a href=\"https:\/\/users.ssc.wisc.edu\/~ewfrees\/actuarial-mathematics\/emerging-costs\/6-deferred-acquisition-expenses-and-modified-premium-reserves\/full-preliminary-term-approach\/\" title=\"Full Preliminary Term Approach\">&#9668 Previous page<\/a><\/div><div class=\"alignright\"><a href=\"https:\/\/users.ssc.wisc.edu\/~ewfrees\/actuarial-mathematics\/universal-life\/\" title=\"Universal Life\">Next page &#9658<\/a><\/div><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Special Case 1. Ordinary Whole Life Here, take \\(j = \\omega &#8211; x\\) and \\(_1 P_{[x]}^n = v _1 q_{[x]} = A_{[x]:\\overline{1|}}^1\\). Thus, \\( A_{[x]:\\overline{1|}}^1+ P^{FPT} a_x =P_x \\ddot{a}_x \\) and \\begin{eqnarray*} P^{FPT} &#038;=&#038; \\frac{P_x &hellip;<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":326,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"jetpack_post_was_ever_published":false},"jetpack_sharing_enabled":true,"jetpack_shortlink":"https:\/\/wp.me\/P8cLPd-5m","acf":[],"_links":{"self":[{"href":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-json\/wp\/v2\/pages\/332"}],"collection":[{"href":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-json\/wp\/v2\/comments?post=332"}],"version-history":[{"count":3,"href":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-json\/wp\/v2\/pages\/332\/revisions"}],"predecessor-version":[{"id":1672,"href":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-json\/wp\/v2\/pages\/332\/revisions\/1672"}],"up":[{"embeddable":true,"href":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-json\/wp\/v2\/pages\/326"}],"wp:attachment":[{"href":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-json\/wp\/v2\/media?parent=332"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}