{"id":911,"date":"2015-01-17T15:18:41","date_gmt":"2015-01-17T21:18:41","guid":{"rendered":"http:\/\/www.ssc.wisc.edu\/~jfrees\/?page_id=911"},"modified":"2015-02-20T18:22:01","modified_gmt":"2015-02-21T00:22:01","slug":"8-policies-with-discrete-cash-flows-other-than-annual","status":"publish","type":"page","link":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/actuarial-mathematics\/policy-values\/8-policies-with-discrete-cash-flows-other-than-annual\/","title":{"rendered":"8. Policies with Discrete Cash Flows other than Annual"},"content":{"rendered":"

Policy Valuation<\/strong><\/p>\n

It is common to assume that benefits are payable at the moment of failure and that premiums are payable at the beginning of the \\(m\\)thly period (e.g., \\(m =1, 2, \\mbox{or } 4\\)).<\/p>\n

Traditional actuarial symbols are based on discrete annual cash flows because mortality rates are available no more frequent than annually.<\/p>\n

The “UDD” assumption means uniform distribution of deaths<\/em> within a year. This is not the same as the DeMoivre assumption which is uniform over the lifetime of an individual.<\/p>\n

Recall earlier relations. For a whole life policy with a benefit payable at the end of the \\(m\\)thly period, the expected present value is
\n\\begin{eqnarray*}
\nA_x^{(m)} = \\frac{i}{i^{(m)}} A_x,
\n\\end{eqnarray*}
\nwhere \\(i^{(m)}\\) is the \\(m\\)thly nominal interest rate determined by \\(1+i= \\left(1+\\frac{i^{(m)}}{m}\\right)^m\\). Note that this is an \\(exact\\) relationship under the UDD assumption. As \\(m \\rightarrow \\infty\\), this yields
\n\\begin{eqnarray*}
\n\\bar{A}_x = \\frac{i}{\\delta} A_x .
\n\\end{eqnarray*}
\nFurther,
\n\\begin{eqnarray*}
\n\\ddot{a}_x^{(m)} = \\alpha(m) \\ddot{a}_x – \\beta(m),
\n\\end{eqnarray*}
\nwhere
\n\\begin{eqnarray*}
\n\\alpha(m) = \\frac{id}{i^{(m)} d^{(m)}} \\textrm{ and } \\beta(m) = \\frac{i-i^{(m)}}{i^{(m)} d^{(m)}},
\n\\end{eqnarray*}
\nand \\(d^{(m)}\\) is the \\(m\\)thly nominal discount rate determined by \\(1-d= \\left(1-\\frac{d^{(m)}}{m}\\right)^m\\).
\nAs \\(m \\rightarrow \\infty\\), this yields
\n\\begin{eqnarray*}
\n\\bar{a}_x^{(m)} = \\alpha(\\infty) \\ddot{a}_x – \\beta(\\infty),
\n\\end{eqnarray*}
\nwhere
\n\\begin{eqnarray*}
\n\\alpha(\\infty) = \\frac{id}{\\delta^2} \\textrm{ and } \\beta(\\infty) = \\frac{i-\\delta}{\\delta^2} .
\n\\end{eqnarray*}
\nFor approximations, we often use
\n\\begin{eqnarray*}
\n\\alpha(m) \\approx 1 \\textrm{ and } \\beta(m) \\approx \\frac{m-1}{2m} .
\n\\end{eqnarray*}<\/p>\n

With these assumptions and approximations, one can readily handle common traditional policies. <\/p>\n

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Policy Valuation It is common to assume that benefits are payable at the moment of failure and that premiums are payable at the beginning of the \\(m\\)thly period (e.g., \\(m =1, 2, \\mbox{or } 4\\)). …<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":280,"menu_order":7,"comment_status":"closed","ping_status":"open","template":"","meta":{"jetpack_post_was_ever_published":false},"jetpack_sharing_enabled":true,"jetpack_shortlink":"https:\/\/wp.me\/P8cLPd-eH","acf":[],"_links":{"self":[{"href":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-json\/wp\/v2\/pages\/911"}],"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=911"}],"version-history":[{"count":3,"href":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-json\/wp\/v2\/pages\/911\/revisions"}],"predecessor-version":[{"id":1585,"href":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-json\/wp\/v2\/pages\/911\/revisions\/1585"}],"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=911"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}