I am currently referring to Life Contingencies by C. W. Jordan, which gives the following reason:
"The select symbol $[x]$ is not used in the $\ell_{x+3}$ column, since the effects of selection do not carry over into the fourth year, and $\ell_{x+3}$ is therefore equally representative of the number of survivors of the $\ell_{[x]}$ lives insured 3 years previously, the $\ell_{[x-1]}$ lives insured 4 years previously, and so on, and this column constitutes an ultimate mortality table."
However, I do not understand the author's reasoning (it's not just a specific part, I am really quite confused about the whole thing). Could someone give me a detailed walk-through of the reasoning used by the author?
Best Answer
To understand the select and ultimate life table, it is helpful to think of it in the context of an example in which lives follow one survival model for some specified amount of time, and then after that time elapses, they follow a different survival model.
To this end, one such example is when a life insurance policy is newly underwritten. In such a case, the insured might be at higher risk of death for some number of years--this is not because the act of insuring them puts them at increased risk, but because the application for insurance suggests that the insured might have reason to believe they are at risk and need coverage. After a few years, though, this selection effect "wears off."
If the select period did not apply, we would have the traditional life table that models the survivorship of the cohort. However, to account for selection, we must expand the life table at each age at enrollment for each year (or period) in which the selection effect occurs. In the table you provided, the selection effect has a duration of $3$ years, so there are three additional columns to describe the survivorship in the select period. The notation $l_{[x]}$ indicates the number of lives surviving if selected at age $x$. Then, by age $x+3$, the select period has ended, and the number of survivors is given by $l_{x+3}$. Subsequently, you read the table down the final column.
Here's how you'd calculate survival probabilities for your table. Suppose $[61]$ is underwritten for a life insurance policy. What is the probability that $[61]$ survives $5$ years? You would simply calculate $$_{5}p_{[61]} = \frac{l_{66}}{l_{[61]}} = \frac{65000}{78000} = 0.833333.$$ What is the probability that $[61]$ survives to $63$? This is $$_{2}p_{[61]} = \frac{l_{[61]+2}}{l_{[61]}} = \frac{73000}{78000} = 0.935897.$$ Notice in both cases we have $l_{[61]}$ in the denominator, but in the first case, we can just write $l_{66}$ whereas in the second, we can neither write $l_{63} = 74000$ nor $l_{[63]} = 71000$, because those don't mean the same thing as $l_{[61]+2}$; the first represents the ultimate number living at age $63$, either because they were $60$ at selection and survived three years, or they were younger than $60$, survived the select period, and however many additional years to reach $60$. The second represents the number of survivors aged $63$ in the cohort when selected--that is to say, when unwritten for insurance. But $l_{[61]+2}$ represents the number alive who were selected at age $61$ and survived $2$ years of the select period.
For a slightly more sophisticated example, suppose as before $[61]$ is underwritten. Given that they have survived the first year, what is the probability that they will survive another $3$ years? Well, this is simply $$_{3}p_{[61]+1} = \frac{l_{65}}{l_{[61]+1}} = \frac{67000}{76000} = 0.881579.$$
But back to your specific question: why, if the select period is only $3$ years, shouldn't there be a column for $l_{[x]+3}$? As you can see from our earlier computations, we were basically reading the table along rows until the final column, and then read down that column, to follow a life that was selected at age $[x]$. If you had a column for $[x]+3$, you couldn't read down that column because, like $l_{[x]}$, $l_{[x]+1}$, and $l_{[x]+2}$, we don't read down those columns to model the survivorship of $[x]$. We have to read across until we reach an ultimate column. Now, that is not to say you couldn't insert a column with $l_{[x]+3}$ there, but then what is $l_{x+4}$? If your select period was only $3$ years, you'd have to make $l_{x+4} = l_{[x+1]+3}$, in which case that $l_{x+4}$ column is redundant.
To make it absolutely clear why that's the case, suppose we had a select period of only one year. Then why would you need three columns to describe a model where survivorship is modified only in the first year of selection?