{"id":398,"date":"2015-01-05T19:22:39","date_gmt":"2015-01-06T01:22:39","guid":{"rendered":"http:\/\/frees.pajarel.net\/?page_id=398"},"modified":"2015-02-20T18:52:27","modified_gmt":"2015-02-21T00:52:27","slug":"exercise-soa-mlc-293","status":"publish","type":"page","link":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/actuarial-mathematics\/interest-rate-risks-and-simulation\/2-valuation-of-insurances-and-life-annuities\/exercise-soa-mlc-293\/","title":{"rendered":"Exercise: SoA MLC 293"},"content":{"rendered":"

For a fully discrete 3-year term life insurance policy on (60), you are given:
\n* (i) The death benefit is 100,000.
\n* (ii) Mortality follows the Illustrative Life Table.
\n* (iii) The rate of interest is based on the yield curve at \\(t = 0\\).
\nYou are also given the following information about zero coupon bonds based on the yield curve at \\(t = 0\\):<\/p>\n\n\n\n
Years to Maturity<\/td>\n1<\/td>\n2<\/td>\n3<\/td>\n<\/tr>\n
Price of 100 Bond<\/td>\n97.00<\/td>\n92.00<\/td>\n<\/td>\n<\/tr>\n<\/table>\n

Calculate the benefit premium.
\nSolution<\/em>
\nFrom the Illustrative Life Table, we have \\(q_{60} = 0.01376\\), \\(q_{61} = 0.01501\\), and \\(q_{62} = 0.01638\\).
\nFor this 3-year term insurance, the expected present value of future benefits is
\n\\begin{eqnarray*}
\n100000 A_{60:\\overline{3|}}^{~1} &=& 1000000\\left(v_1 q_{60} + v_2 ~_{1|} q_{60} + v_3 ~_{2|} q_{60} \\right) \\\\
\n&=& 1000000\\left((0.97) q_{60} + (0.92) p_{60} q_{61} + (0.87) p_{60}p_{61} q_{62} \\right) \\\\
\n&=& 1000000\\left((0.97)(0.01376) + (0.92)(0.98264)(0.01501) \\right.\\\\
\n& ~& ~~~~~~~ \\left.+ (0.87)(0.98264)(0.98499)(0.01638) \\right) = 4,070.969.
\n\\end{eqnarray*}
\nThe expected present value of an annuity is
\n\\begin{eqnarray*}
\nd\\dot{a}_{60:\\overline{3|}} &=& 1 + v_1 ~ p_{60} + v_2 ~_2 p_{60} \\\\
\n&=& 1 + (0.97)(0.98264) + (0.92)(0.98264)(0.98499)= 2.850374.
\n\\end{eqnarray*}
\nThus, the annual benefit premium is $ 4,070.969\/2.850374= 1,431.74<\/p>\n

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For a fully discrete 3-year term life insurance policy on (60), you are given: * (i) The death benefit is 100,000. * (ii) Mortality follows the Illustrative Life Table. * (iii) The rate of interest …<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":396,"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-6q","acf":[],"_links":{"self":[{"href":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-json\/wp\/v2\/pages\/398"}],"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=398"}],"version-history":[{"count":5,"href":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-json\/wp\/v2\/pages\/398\/revisions"}],"predecessor-version":[{"id":802,"href":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-json\/wp\/v2\/pages\/398\/revisions\/802"}],"up":[{"embeddable":true,"href":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-json\/wp\/v2\/pages\/396"}],"wp:attachment":[{"href":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-json\/wp\/v2\/media?parent=398"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}