{"id":321,"date":"2014-12-31T15:47:23","date_gmt":"2014-12-31T15:47:23","guid":{"rendered":"http:\/\/frees.pajarel.net\/?page_id=321"},"modified":"2015-02-20T19:25:11","modified_gmt":"2015-02-21T01:25:11","slug":"4-annual-profits","status":"publish","type":"page","link":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/actuarial-mathematics\/emerging-costs\/4-annual-profits\/","title":{"rendered":"4. Annual Profits"},"content":{"rendered":"

General Discrete Policy.<\/strong> It is important to monitor the actual experience of a block of business. To this end, we use carats, or “hats”, to denote actual values. We use<\/p>\n

* \\(\\hat{i}_k\\) is the actual rate of return from invested assets,
\n * \\(\\hat{e}_k\\) is the actual annual expenses per contract,
\n * at death, \\(\\hat{q}_{[x]+k}^{(d)}\\) is the realized fraction of deaths,
\n * at withdrawal, \\(\\hat{q}_{[x]+k}^{(w)}\\) is the realized fraction of withdrawals, and
\n * at survival to the end of the year, \\(\\hat{p}_{[x]+k}^{(\\tau)} = 1 – (\\hat{q}_{[x]+k}^{(d)}+\\hat{q}_{[x]+k}^{(w)})\\) is realized fraction that survived.<\/p>\n

With these quantities, the profit during the year (at time \\(k+1\\)) is
\n\\begin{eqnarray*}
\nProfit_{k+1} &= &
\n\\left( _k AS + G_k -\\hat{e}_k\\right) (1+ \\hat{i}_k) – \\hat{q}_{[x]+k}^{(d)} \\left(b_{k+1} + E_{k+1}\\right) \\\\
\n&~~~~~~~~-& \\hat{q}_{[x]+k}^{(w)} ~_{k+1} CV – \\hat{p}_{[x]+k}^{(\\tau)} ~_{k+1} AS \\\\
\n&= &\\left( _k AS + G_k -\\hat{e}_k\\right) (1+ \\hat{i}_k) – \\hat{q}_{[x]+k}^{(d)} \\left(b_{k+1} + E_{k+1} – ~_{k+1} AS\\right) \\\\
\n&~~~~~~~~-& \\hat{q}_{[x]+k}^{(w)} (~_{k+1} CV -~_{k+1} AS) – ~_{k+1} AS
\n\\end{eqnarray*}
\nFor this illustration, we have that assumed premiums \\(G_k\\) and settlement expenses \\(E_k\\) are the same as actual values. We also assume that \\( _k AS\\) represents the terminal (year-end) company obligation (that does not depend on experience). Recall the recursive asset share calculation
\n\\begin{eqnarray*}
\n(~_k AS + G_k – e_k)(1+i_k) &=& q_{[x]+k}^{(d)} \\left(b_{k+1} + E_{k+1} – ~_{k+1} AS \\right) \\\\
\n&~~~~~~~~+& q_{[x]+k}^{(w)} ( _{k+1} CV – ~_{k+1} AS) + ~_{k+1} AS
\n\\end{eqnarray*}
\nSubstituting for \\(_{k+1} AS\\), we may write the profit during the year is
\n\\begin{eqnarray*}
\nProfit_{k+1} &= &
\n\\left( _k AS + G_k -\\hat{e}_k\\right) (1+ \\hat{i}_k) –
\n\\hat{q}_{[x]+k}^{(d)} \\left(b_{k+1} + E_{k+1} – ~_{k+1} AS\\right) \\\\
\n&~~~~~~~~-& \\hat{q}_{[x]+k}^{(w)} (~_{k+1} CV -~_{k+1} AS) – ~_{k+1} AS \\\\
\n&= & \\left( _k AS + G_k \\right) (\\hat{i}_k – i_k) \\\\
\n&~~~~~~~~+& e_k (1+ \\hat{i}_k) – \\hat{e}_k (1+ i_k) \\\\
\n&~~~~~~~~+& \\left(b_{k+1} + E_{k+1} – ~_{k+1} AS\\right) (q_{[x]+k}^{(d)} – \\hat{q}_{[x]+k}^{(d)} ) \\\\
\n&~~~~~~~~+& \\left( _{k+1} CV – ~_{k+1} AS\\right) (q_{[x]+k}^{(w)} – \\hat{q}_{[x]+k}^{(w)} ) .
\n\\end{eqnarray*}
\nThis is one way to decompose the profit into identifiable components, known as the analysis of surplus. Here, we interpret these portions of the profit:<\/p>\n

* \\(\\left( _k AS + G_k \\right) (\\hat{i}_k – i_k)\\) – due to favorable investment experience,
\n * \\(e_k (1+ \\hat{i}_k) – \\hat{e}_k (1+ i_k)\\) – due to favorable expense experience,
\n * \\(\\left(b_{k+1} + E_{k+1} – ~_{k+1} AS\\right) (q_{[x]+k}^{(d)} – \\hat{q}_{[x]+k}^{(d)} )\\) – due to favorable mortality experience, and
\n * \\(\\left( _{k+1} CV – ~_{k+1} AS\\right) (q_{[x]+k}^{(w)} – \\hat{q}_{[x]+k}^{(w)} )\\) – due to favorable withdrawal experience.<\/p>\n

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General Discrete Policy. It is important to monitor the actual experience of a block of business. To this end, we use carats, or “hats”, to denote actual values. We use * \\(\\hat{i}_k\\) is the actual …<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":302,"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-5b","acf":[],"_links":{"self":[{"href":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-json\/wp\/v2\/pages\/321"}],"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=321"}],"version-history":[{"count":5,"href":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-json\/wp\/v2\/pages\/321\/revisions"}],"predecessor-version":[{"id":1665,"href":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-json\/wp\/v2\/pages\/321\/revisions\/1665"}],"up":[{"embeddable":true,"href":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-json\/wp\/v2\/pages\/302"}],"wp:attachment":[{"href":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-json\/wp\/v2\/media?parent=321"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}