{"id":892,"date":"2015-01-17T14:42:06","date_gmt":"2015-01-17T20:42:06","guid":{"rendered":"http:\/\/www.ssc.wisc.edu\/~jfrees\/?page_id=892"},"modified":"2015-02-20T17:52:44","modified_gmt":"2015-02-20T23:52:44","slug":"examples","status":"publish","type":"page","link":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/actuarial-mathematics\/policy-values\/888-2\/examples\/","title":{"rendered":"Examples"},"content":{"rendered":"

Example 1<\/strong>
\nConsider at 25-year endowment to a select life age [30]. The insured amount is 100,000. The insurer incurs initial expenses of 2,000 plus 50% of first year premium, renewal expenses of 2.5% of premiums. Benefits are payable immediately. Use the Illustrative Mortality Table with \\(i=5\\%\\).<\/p>\n

a) Provide an expression for the gross future loss random variable.<\/p>\n

Solution<\/em>
\n(i) The present value of benefit outgo is \\(100,000 v^{\\min(T[30],25) } \\).<\/p>\n

(ii) The present value of gross premium income is \\( G \\ddot{a}_{\\overline{\\min(K[30],25)|}} \\).<\/p>\n

(iii) The present value of expenses is \\(2000+0.025 G \\ddot{a}_{\\overline{\\min(K[30],25)|}} + 0.475 G\\)<\/p>\n

b) Calculate the expense-augment (gross) annual premium.<\/p>\n

Solution<\/em>
\n(i) The EPV of benefit outgo is
\n\\begin{eqnarray*}
\n&=& 100,000 \\bar{A}_{\\overline{[30],25|}} \\\\
\n&=& 100,000 \\left( \\frac{i}{\\delta} A_{\\overline{[30],25|}}^{~~1} +A_{\\overline{[30],25|}}^{~~~~~1} \\right)\\\\
\n&=& 29,873.20 .
\n\\end{eqnarray*}<\/p>\n

(ii) The EPV of gross premium income is
\n\\begin{eqnarray*}
\n&=& G \\ddot{a}_{\\overline{[30],25|}} = G (14.73113) .
\n\\end{eqnarray*}<\/p>\n

(iii) The EPV of expenses is
\n\\begin{eqnarray*}
\n&=& 2000 + 0.025 G \\ddot{a}_{[30]:\\overline{25|}} + 0.475 G \\\\
\n&=& 2000 + (0.843278) G .
\n\\end{eqnarray*}
\nNow, setting (i)+(iii)=(ii), we have
\n\\begin{eqnarray*}
\nG &=& \\frac{29,873.20 +2,000}{14.73113 – 0.843278} = 2,295.05 .
\n\\end{eqnarray*}<\/p>\n

Example 2<\/strong>
\nFor a fully discrete whole life insurance of 100,000 on \\(x\\), you are given:
\n(i) Expenses, paid at the beginning of the year, are as follows:<\/p>\n\n\n\n\n\n
Year <\/td>\n\n\t\t\t Percentage of Premium Expenses\n\t\t\t<\/td>\n\n\t\t\t Per 1000 Expenses \n\t\t\t<\/td>\n\n Per Policy Expenses \n\t\t\t<\/td>\n<\/tr>\n
First Year \n\t\t\t<\/td>\n50% <\/td>\n2.0 <\/td>\n150 <\/td>\n<\/tr>\n
2+<\/td>\n4% <\/td>\n0.5 <\/td>\n25 <\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

(ii) \\(i = 0.04\\)<\/p>\n

(iii) \\(\\ddot{a}_x =10.8\\).<\/p>\n

Calculate the expense-loaded premium using the equivalence principle.<\/p>\n

Solution<\/em>
\nEPV Premium = \\(G \\ddot{a}_x = G (10.8) \\)<\/p>\n

EPV Benefit = \\(100000 A_x\\)<\/p>\n

EPV Expenses
\n\\begin{eqnarray*}
\n&=& 0.5 G + 200 + 150
\n+ (0.04 G + 50 + 25) a_x \\\\
\n&=& G (0.5 +0.04 a_x) + 350 + 75 a_x = G (0.892) + 1,085 .
\n\\end{eqnarray*}
\nEquating EPV Premium = EPV Benefit + EPV Expenses yields
\n\\begin{eqnarray*}
\n10.8 G &=& 100000 A_x + 0.892 G + 1085 .
\n\\end{eqnarray*}
\nSolving for G yields
\n\\begin{eqnarray*}
\nG &=& \\frac{100000 A_x + 1085}{10.8 – 0.892} = \\frac{58462 1085}{9.908} = 6009.99 .
\n\\end{eqnarray*}<\/p>\n

Example 3<\/strong><\/p>\n

Consider at 10-year term life policy for 100,000 to a life age (30). Benefits are payable immediately, premiums are payable at the beginning of the year. Use the Illustrative Mortality Table with \\(i=6\\% \\). Expenses are according to the following schedule:<\/p>\n\n\n\n\n\n\n\n
Type<\/td>\n\n\t\t\t Per Policy\n\t\t\t<\/td>\n\n\t\t Per 1000 of Insurance\n\t\t\t<\/td>\n\n\t\t\t Percent of Premium\n\t\t\t<\/td>\n<\/tr>\n
First Year \n\t\t\t<\/td>\n 50<\/td>\n 5.00<\/td>\n 82%<\/td>\n<\/tr>\n
Years 2-5<\/td>\n 6<\/td>\n 0.50<\/td>\n 14.5<\/td>\n<\/tr>\n
Years 6-10<\/td>\n 6<\/td>\n 0.50<\/td>\n 7.0<\/td>\n<\/tr>\n
Claim Settlement \n\t\t\t<\/td>\n 25<\/td>\n0.10 <\/td>\n
\n\t\t\t<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

Calculate G, the gross annual premium.<\/p>\n

Solution<\/em>
\n\\begin{eqnarray*}
\nG \\ddot{a}_{30:\\overline{10|}}&=&
\n100,000 \\bar{A}_{30:\\overline{10|}}^{~1}
\n+ (25+100(0.10)) \\bar{A}_{30:\\overline{10|}}^{~1}
\n+ 550 \\\\
\n&~~~+& 0.82 G + 56 a_{30:\\overline{9|}}
\n+ 0.145 G a_{30:\\overline{4|}}
\n+ 0.07 G ~_{5|5} \\ddot{a}_{30} .
\n\\end{eqnarray*}
\nThus,
\n\\begin{eqnarray*}
\nG \\ddot{a}_{30:\\overline{10|}}&=&
\n\\frac{100,035 \\bar{A}_{30:\\overline{10|}}^{~1}
\n+ 56 \\ddot{a}_{30:\\overline{10|}} + 494}
\n{0.93 \\ddot{a}_{30:\\overline{10|}} – 0.075
\n\\ddot{a}_{30:\\overline{5|}} – 0.675} \\approx \\$385.
\n\\end{eqnarray*}
\n

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Example 1 Consider at 25-year endowment to a select life age [30]. The insured amount is 100,000. The insurer incurs initial expenses of 2,000 plus 50% of first year premium, renewal expenses of 2.5% of …<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":888,"menu_order":0,"comment_status":"closed","ping_status":"open","template":"","meta":{"jetpack_post_was_ever_published":false},"jetpack_sharing_enabled":true,"jetpack_shortlink":"https:\/\/wp.me\/P8cLPd-eo","acf":[],"_links":{"self":[{"href":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-json\/wp\/v2\/pages\/892"}],"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=892"}],"version-history":[{"count":5,"href":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-json\/wp\/v2\/pages\/892\/revisions"}],"predecessor-version":[{"id":1572,"href":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-json\/wp\/v2\/pages\/892\/revisions\/1572"}],"up":[{"embeddable":true,"href":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-json\/wp\/v2\/pages\/888"}],"wp:attachment":[{"href":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-json\/wp\/v2\/media?parent=892"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}