MATH 403A, fall 98, CHAPTER 3 Section 3.8

Select and Ultimate Life tables

A select table is a mortality table for a group of people subject to a special circumstance (disability, retirement, etc). The table is bivariat, with [x] being the age at selection, and t is the duration since selection.

\begin{matrix}q_{[x]} & q_{[x]+1} & q_{[x]+2} & q_{[x]+3} & … \\ q_{[x+1]} & q_{[x+1]+1} & q_{[x+1]+2} & q_{[x+1]+3} & … \\ q_{[x+2]} & q_{[x+2]+1} & q_{[x+2]+2} & q_{[x+2]+3} & … \end{matrix}

A select and ultimate table uses the select table for a prescribed period r and truncates the select table after the r + 1 column. For durations beyond r we would use q_{[x−j]+r+j} ∼ q_{[x]+r}. For example, if r = 1 we would have

q_{[x−2]} q_{[x−2]+1} q_{[x−1]+1} q_{[x]+1}
q_{[x−1]} q_{[x−1]+1} q_{[x]+1}
q_{[x]} q_{[x]+1}

A life table in which functions are given only for attained ages is called an aggreagate table (see table 3.3.1 in book). In select and ultimate tables, the last column is a special agregate table that is usually referred to the ultimate table, to reflect the select table setting. For example, let r = 1:

\begin{matrix}q_{[x−2]} & q_{[x−2]+1} & q_{x} \\ q_{[x−1]} & q_{[x−1]+1} & q_{x+1} \\ q_{[x]} & q_{[x]+1} & q_{x+2} \\ q_{[x+1]} & q_{[x+1]+1} & q_{x+3} \end{matrix}

Homework

1. \begin{matrix}x & q_{[x]} & q_{[x]+1} & q_{x+2} \\ 35 & .05 & .08 & .12 \\ 36 & .07 & .10 & .13 \\ 37 & .09 & .11 & .15 \end{matrix}

If l_{[35]} = 1000, find l_{[37]}.

2. For a 2–year select and ultimate mortality table, you are given taht

q_{[x]} = \frac{q_{x}}{3} and q_{[x]+1} = \frac{2q_{x+1}}{3} ,

for all x. You are also given the following portion of the ultimate table:

\begin{matrix}x & q_{x} \\ 40 & .015 \\ 41 & .021 \\ 42 & .030 \\ 43 & .045 \\ 44 & .066 \end{matrix}

Given that l_{[42]} = 5000, find l_{[40]} + 1.

3. Textbook problems: 3.36, 3.39, 3.41.