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Re: [lmi] select tables terminology

From: Greg Chicares
Subject: Re: [lmi] select tables terminology
Date: Mon, 15 Feb 2016 11:57:27 +0000
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On 2016-02-15 00:58, Vadim Zeitlin wrote:
> On Sun, 14 Feb 2016 12:41:11 +0000 Greg Chicares <address@hidden> wrote:
> GC> On 2016-02-13 23:56, Vadim Zeitlin wrote:
> GC> > On Sat, 13 Feb 2016 23:02:13 +0000 Greg Chicares <address@hidden> wrote:
> GC> [...]
> GC> > GC> select tables...not consecutive: Do you simply mean that they're 
> select
> GC> > GC> and ultimate, e.g.
> GC> > GC> 
> GC> > GC>   ---select-- ult.
> GC> > GC>   a0 a1 a2 a3  x4
> GC> > GC>   b1 b2 b3 b4  x5
> GC> > GC>   c2 c3 c4 c5  x6
> GC> > GC>                x7
> GC> [...]
> GC> >  Sorry, I was speaking about the ages: if you remember, we decided that 
> the
> GC> > code had to check that they're consecutive, but this is not actually the
> GC> > case for the ages in the first column of select tables as there is a 
> jump
> GC> > of select_period after max_select_age (generally speaking; there is no 
> jump
> GC> > if max_select_age + select_period == max_age).
> GC> 
> GC> In my example above, arbitrarily assuming minimum age to be zero:
> GC>  - select ages are in [0,2]
> GC>  - select period is 4 years
> GC>  - maximum age is eight
> GC> so
> GC>   2 + 4 < 8
> GC> and if we write the ages around three sides of the table:
> GC> 
> GC>     0  1  2  3     /
> GC>    ---select-- ult.
> GC> 0  a0 a1 a2 a3 x4  4
> GC> 1  b1 b2 b3 b4 x5  5
> GC> 2  c2 c3 c4 c5 x6  6
> GC> 7              x7  7
> GC> 
> GC> then they proceed consecutively across the top and then down the right 
> side;
> GC> but the left side has a jump.
>  Sorry to bother you again,

I wanted to invite such questions.

> but I just wanted to ask if there are any
> established terms for the different parts of the table above, i.e.:
> - The 0..3 in the header row (I just call them "column#" and "ultimate
>   column label")?

The number 0..3 in the header row is called "duration". By convention,
it always begins at zero.

The numbers {a0..a3}..{c2..c5} below the "duration" labels are called
"select rates".

The ultimate column label doesn't have a name; it's just an ultimate
column label. The numbers x4..x7 below it are called "ultimate rates".
Their cardinality generally differs from that of a0..a3.

> - The ages on the right hand side of the column (4..7) (which I inventively
>   call "right hand side age")?

"Ultimate age".

> - If there is a term for the RHS ages, should some special term be also
>   used for the LHS ones?

"Select age".

>  I'm asking about this mostly to make the error messages more clear and,
> possibly, also to make things more clear in the code comments as well.

Let me elaborate just a little more, to explain why these terms are used.

Suppose you're age 45 (when we have to pick an arbitrary age, that's the
one we pick) and you want to buy a life insurance policy. The insurer
(i.e., the insurance company) generally asks you a number of questions
about your health, and may require a medical examination; using that
information, its "underwriting" staff decides whether to issue a policy,
or to decline to issue. That decision is called "selection"; your current
age is the "age at selection", which is "select age" 45 and "duration" 0.

On that day, you've successfully passed underwriting selection, which
indicates that you're (probabilistically) healthier than the general
population and can be expected to live longer. The "effect of selection"
on mortality is generally quite strong in the early durations (perhaps
a decrease of one half the mortality rate), and "wears off" after a
number of years--commonly, twenty years, in which case, if you survive
to age 65, the effect of selection is considered negligible, and you
are regarded as having the same probability of death as someone issued
at, say, age 25, forty years before. Thus--'q(x)' being the probability
that, having survived to your Xth birthday, you will die before your
(1+X)th--your q(65) is the same as that of the person issued at age 25,
which we'd write thus:
  q([45]+20) = q([25]+40)
Everything following q is normally written as a subscript without
parentheses, so we'd normally write something that looks like this:
  q[45]+20 = q[25]+40
The square brackets are never omitted: they indicate select age, which
the following additive term indicates duration. An expression like
'q[45]+5' indicates which row and column contain your 'q' for that year;
in the example given, we'd move down to "select age" 45 in the table and
read across to the sixth column (labelled "5" because it's origin zero).
For 'q[45]+20' in a twenty-year-select table, that would take us to the
twenty-first column: the "ultimate" column. For 'q[45]+25', we'd go to
that "ultimate" column and move five rows down.

There is some age to which the probability of survival is negligible;
that age is symbolized by a greek lower-case omega. In the twentieth
century, this was commonly assumed to be 100, so the highest select
age would be 99.

There is often a minimum age and a maximum age at which a particular
policy may be issued: for example, ages 18 and 80. A mortality table
for such a policy would have select ages 18..80 (i.e., [18,80] in
interval notation). No rates would exist for any lower age. Rates must
exist for higher ages, because the person who bought the policy (the
"insured") may live to age omega-1. If the table is ten-year select,
and omega is 100, then there are ten select columns [0,9] and an
ultimate column for "attained age" [18,99].

Such a table is called a "select and ultimate" table. It is possible
to have a "pure select" table, for which the effect of selection is
assumed never to become negligible, in which case no ultimate column
would exist. It is also possible to disregard the effect of selection,
resulting in an "attained age" table whose mortality rates constitute
a single column vector.

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