From: Eric S. Raymond <esr@thyrsus,com>
Date: Mon, 5 Apr 2004 03:09:54 +0000 (+0000)
Subject: End of Chapter III,
X-Git-Tag: 1.0~32
X-Git-Url: http://thyrsus.com/gitweb/?a=commitdiff_plain;h=a443e527c90d0e4a9339963906ae5bd88c8a8d66;p=mad.git

End of Chapter III,
---

diff --git a/doc/mad-manual.xml b/doc/mad-manual.xml
index b1c0c4d..5564736 100644
--- a/doc/mad-manual.xml
+++ b/doc/mad-manual.xml
@@ -4396,13 +4396,68 @@ M<subscript>x</subscript>, N<subscript>x</subscript>, and
 D<subscript>x</subscript> in these formulas occur most often in
 combination, as in the computation of P<subscript>x</subscript>.
 Assuming a population of some initial size (at x =
-b<subscript>0</subscript>) (here 1,000,000), &scriptl;<subscript>x</subscript>
-is the number living a t age x (so that &scriptl;<subscript>bo</subscript> =
-1,000,000), q<subscript>x</subscript> is the mortality
-<emphasis>rate</emphasis>, and d<subscript>x</subscript> is the number
-of deaths at age x.</para>
+b<subscript>0</subscript>) (here 1,000,000),
+&scriptl;<subscript>x</subscript> is the number living a t age x (so
+that &scriptl;<subscript>bo</subscript> = 1,000,000),
+q<subscript>x</subscript> is the mortality <emphasis>rate</emphasis>,
+and d<subscript>x</subscript> is the number of deaths at age x. Thus
+d<subscript> x</subscript> =
+q<subscript>x</subscript>&middot;&scriptl;<subscript>x</subscript>.
+The quantity D<subscript>x</subscript> is computed by the formula
+D<subscript>x</subscript> =
+&scriptl;<subscript>x</subscript>(1+i)<superscript>-x</superscript>.
+It can be used, for example, to compute the cost of term insurance,
+since C<subscript>x</subscript>/D<subscript>x</subscript> is the
+premium for one-year term insurance of $1 at age x.</para>
+
+<para> The sums M<subscript>x</subscript> and
+N<subscript>x</subscript> are obtained by the formula</para>
 
-<!-- FIXME: TEXT FOR PAGES III.42-43 GOES HERE -->
+<mediaobject>
+  <imageobject>
+    <imagedata fileref='equation8.png'/>
+  </imageobject>
+</mediaobject>
+
+<para>We note that for some w. we always have
+q<subscript>x</subscript> = l, so that l<subscript>w+1</subscript> = 0
+(since &scriptl;<subscript>w+1</subscript> =
+&scriptl;<subscript>w</subscript>-d<subscript>w</subscript> =
+&scriptl;<subscript>w</subscript>-&scriptl;<subscript>w</subscript> =
+0), therefore D<subscript>w+1</subscript> =0,
+d<subscript>w+1</subscript> = 0,
+C<subscript>w+1</subscript> = 0, and the sums for
+M<subscript>x</subscript> and
+N<subscript>x</subscript> are actually finite sums.</para>
+
+<para>The three most useful quantities computed here are
+(1)P<subscript>x</subscript> =
+M<subscript>x</subscript>/N<subscript>x</subscript>, which is the
+annual premium payable for an entire life for $1 of whole life
+insurance, (2)A<subscript>x</subscript> =
+M<subscript>x</subscript>/D<subscript>x</subscript>, which is the
+single permium payable at age x for $1 of whole life insurance, and 
+(2)a<subscript>x</subscript> =
+N<subscript>x</subscript>/D<subscript>x</subscript>, which is the
+present value at age x of a whole life annuity of $1, first payment
+at age $x.</para>
+
+<para>Printing of results is under the control of an input variable
+PRINT.  Certain relationships must hold between some independently
+computed values, and these are used as checks on the compuatation:</para>
+
+<para role='centered'>
+M<subscript>b<subscript>0</subscript></subscript> +
+N<subscript>b<subscript>0</subscript>+1</subscript> =
+N<subscript>b<subscript>0</subscript>+1</subscript> =
+N<subscript>b<subscript>0</subscript></subscript>/(1+i)</para>
+<!-- beginpage III.43 -->
+
+<para role='centered'>P<subscript>x</subscript> =
+1/a<subscript>x</subscript> - i/(1+i)</para>
+
+<para>These cannot be expected to come out exactly equal, because of
+round-off, but they should differ by very little.</para>
 
 <!-- beginpage III.44 -->
 <para><emphasis>Flow diagram:</emphasis></para>