{"id":516,"date":"2015-01-06T17:38:35","date_gmt":"2015-01-06T17:38:35","guid":{"rendered":"http:\/\/frees.pajarel.net\/?page_id=516"},"modified":"2015-02-20T18:49:44","modified_gmt":"2015-02-21T00:49:44","slug":"1-constant-forces-of-transition","status":"publish","type":"page","link":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/actuarial-mathematics\/multiple-decrement-models\/6-building-mult-decrement-tables-from-associated-single-life-functions\/1-constant-forces-of-transition\/","title":{"rendered":"1. Constant Forces of Transition"},"content":{"rendered":"

One way to build a multi-decrement table from the associated single decrement tables (and vice-versa) is to constant forces of transition.
\nUnder this assumption, transition forces are constant within the year<\/em>,
\n\\begin{eqnarray*}
\n\\mu_{x+t}^{(j)} = \\mu_{x}^{(j)} , \\text{ for } 0 \\leq t \\leq 1 .
\n\\end{eqnarray*}
\nThus,
\n\\begin{eqnarray*}
\n~ _t q_x^{\\prime(j)} = 1- \\exp \\left\\{ – t \\mu_{x}^{(j)}\\right\\} .
\n\\end{eqnarray*}
\nand, with \\(t \\rightarrow 1\\), we have \\(1- q_x^{\\prime(j)} = \\exp(-\\mu_{x}^{(j)})\\). Now
\n\\begin{eqnarray*}
\n~ q_x^{(j)} &=& \\int_0^1 ~ _t p_x^{(\\tau)} \\mu_{x+t}^{(j)}~dt
\n~=~ \\frac{\\mu_{x}^{(j)}}{\\mu_{x}^{(\\tau)}} \\int_0^1 ~ _t p_x^{(\\tau)} \\mu_{x+t}^{(\\tau)} ~dt
\n\\\\
\n&=& \\frac{\\mu_{x}^{(j)}}{\\mu_{x}^{(\\tau)}} ~ q_x^{(\\tau)} ~=~ \\frac{\\ln (1- q_x^{\\prime(j)})}{\\ln (1- q_x^{(\\tau)})} ~ q_x^{(\\tau)} .
\n\\end{eqnarray*}
\nThus, we have
\n\\begin{eqnarray*}
\n1- q_x^{\\prime(j)} &=& \\left( p_x^{(\\tau)}\\right)^{q_x^{(j)}\/q_x^{(\\tau)}} .
\n\\end{eqnarray*}
\nFrom this, we can use multi-decrement probabilities to calculate the associated single decrement tables, and vice-versa.
\nExample<\/strong>
\nIn a triple decrement table, suppose that \\(q_{65}^1 = 0.02\\), \\(q_{65}^2 = 0.02\\), and \\(q_{65}^3 = 0.04\\). Calculate \\(p_{65}^{01}\\).
\nSolution<\/em>
\nFirst, \\(p_{65}^{00} = \\prod_{j=1}^3 ~ p_x^j = (1-0.02)(1-0.02)(1-0.04) = 0.921984\\) so that \\( p_{65}^{0\\bullet}= 1-0.921984 = 0.078016\\).
\nThen, for \\(x=65\\)
\n\\begin{eqnarray*}
\n\\ln (1- q_{65}^1) = \\ln(0.98) &=& \\frac{p_{65}^{01}}{p_{65}^{0\\bullet}}\\ln \\left( p_{65}^{00}\\right) \\\\
\n&=& \\frac{p_{65}^{03}}{0.078016}\\ln \\left( 0.921984\\right),
\n\\end{eqnarray*}
\nwhich implies that \\(p_{65}^{01} = 0.019404\\). <\/p>\n

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One way to build a multi-decrement table from the associated single decrement tables (and vice-versa) is to constant forces of transition. Under this assumption, transition forces are constant within the year, \\begin{eqnarray*} \\mu_{x+t}^{(j)} = \\mu_{x}^{(j)} …<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":514,"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-8k","acf":[],"_links":{"self":[{"href":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-json\/wp\/v2\/pages\/516"}],"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=516"}],"version-history":[{"count":3,"href":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-json\/wp\/v2\/pages\/516\/revisions"}],"predecessor-version":[{"id":782,"href":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-json\/wp\/v2\/pages\/516\/revisions\/782"}],"up":[{"embeddable":true,"href":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-json\/wp\/v2\/pages\/514"}],"wp:attachment":[{"href":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-json\/wp\/v2\/media?parent=516"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}