To develop intuition, it is helpful to begin with a traditional framework. Here, policies have premiums that are payable annually (at the beginning of the year, by convention) and have benefits that are payable at the end of the year of failure.<\/p>\n
Special Case: \\(n\\)-Year Term Insurance.<\/strong> For this policy, level premiums are payable at the beginning of each year up until failure, or \\(n\\) years if sooner. A death benefit of 1 is payable at the end of the year of failure, if within \\(n\\) years. When time point \\(t=h\\) is an integer, the loss variable is Special Case: \\(m\\)-pay, \\(n\\)-Year Endowment Insurance.<\/strong> For this policy, level premiums are payable at the beginning of each year up until failure, or \\(m\\) years if sooner. A benefit of 1 is payable at the end of the year of failure or \\(n\\) years, whichever is sooner.<\/p>\n When the policy is in the premium payment period, time point \\(t=h < m\\). The loss variable is\n\\begin{eqnarray*}\nL_h &=&\n\\left\\{\n\\begin{array}{cc}\nv^{K+1-h} - P \\ddot{a}_{\\overline{K+1-h}|} & 0 \\leq K < m \\\\\nv^{K+1-h} - P \\ddot{a}_{\\overline{m-h}|} & m \\leq K < n \\\\\nv^{n-h} - P \\ddot{a}_{\\overline{m-h}|} & K \\ge n\n\\end{array}\n\\right .\n\\end{eqnarray*}\nThis has expectation, or policy value, \\(_h V = A_{x+h:\\overline{n-h|}} - P_{x:\\overline{n|}} ~ \\ddot{a}_{x+h:\\overline{m-h|}}\\).\n\nWhen the policy is past the premium payment period, time point \\(t=h \\ge m\\). In this case, the loss variable is\n\\begin{eqnarray*}\nL_h &=&\n\\left\\{\n\\begin{array}{cc}\nv^{K+1-h} & h \\leq K < n \\\\\nv^{n-h} & K \\ge n\n\\end{array}\n\\right .\n\\end{eqnarray*}\nThis has expectation, or policy value, \\(_h V = A_{x+h:\\overline{n-h|}} \\).\n\nGeneral Discrete Policy.<\/strong> For a general discrete policy, we have (known) premiums \\(P_h\\) payable at time \\(h\\) and benefits payable at time \\(b_h\\). The loss variable is To develop intuition, it is helpful to begin with a traditional framework. Here, policies have premiums that are payable annually (at the beginning of the year, by convention) and have benefits that are payable at …<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":280,"menu_order":2,"comment_status":"closed","ping_status":"open","template":"","meta":{"jetpack_post_was_ever_published":false},"jetpack_sharing_enabled":true,"jetpack_shortlink":"https:\/\/wp.me\/P8cLPd-ef","acf":[],"_links":{"self":[{"href":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-json\/wp\/v2\/pages\/883"}],"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=883"}],"version-history":[{"count":5,"href":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-json\/wp\/v2\/pages\/883\/revisions"}],"predecessor-version":[{"id":1565,"href":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-json\/wp\/v2\/pages\/883\/revisions\/1565"}],"up":[{"embeddable":true,"href":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-json\/wp\/v2\/pages\/280"}],"wp:attachment":[{"href":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-json\/wp\/v2\/media?parent=883"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
\n\\begin{eqnarray*}
\nL_h &=&
\n\\left\\{
\n\\begin{array}{cc}
\nv^{K+1-h} – P \\ddot{a}_{\\overline{K+1-h}|} & h \\leq K < n \\\\\n0 - P \\ddot{a}_{\\overline{n-h}|} & K \\ge n\n\\end{array}\n\\right .\n\\end{eqnarray*}\nThis has expectation, or policy value, \\(_h V = A_{x+h:\\overline{n-h|}}^{~1} - P_{x:\\overline{n|}}^{1} \\ddot{a}_{x+h:\\overline{n-h|}}\\).\n\nSee the animated display where the value changes by the interest rate<\/strong><\/p>\n
\n\\begin{eqnarray*}
\nL_h &=&
\n\\left\\{
\n\\begin{array}{cc}
\n0 & 0 \\leq K < h \\\\\nb_{K+1} v^{K+1-h} - \\sum_{j=h}^K P_j v^{j-h} & K \\geq h \\\\\n\\end{array}\n\\right .\n\\end{eqnarray*}\nUsing a summation by parts, this has expectation\n\\begin{eqnarray*}\n_h V &=& \\sum_{j=0}^{\\infty}\n\\left(b_{h+j+1} v^{j+1} - \\sum_{k=0}^y P_{h+k} v^k \\right) ~_{j|} q_{x+h} \\\\\n&=& \\sum_{j=0}^{\\infty}b_{h+j+1} v^{j+1} ~_{j|} q_{x+h}\n- \\sum_{j=0}^{\\infty} P_{h+j} v^j ~_j p_{x+h} \\\\\n&=& E (PVFB) - E(PVFP) .\n\\end{eqnarray*} \n\n[previous][next]\n<\/p>\n","protected":false},"excerpt":{"rendered":"