The newly underwritten are not average
Picture two men, both exactly aged 50. One bought a life policy last week: he filled in a health questionnaire, gave blood, and a doctor signed off. The other is just a random fifty-year-old plucked from the street. The earlier guides in this rung handed you a single life table keyed only to age — it would treat these two men as identical. But they are not. The man who just *passed* underwriting has, by the very act of passing, proven he was not visibly ill that week. For the next several years he will die noticeably *less often* than his street-corner twin.
This is not a fluke; it is a predictable consequence of selection — the same idea you met when underwriting first appeared on this ladder. Underwriting is a filter, and a filter that screens out the sick leaves behind a group that is temporarily *healthier than its age alone would suggest*. Actuaries call the window during which this freshness still shows the select period, typically two to fifteen years depending on the product and the country. After it, the memory of having been screened fades, and the life behaves like everyone else of that age.
Reading a select-and-ultimate table
To capture two dimensions, actuaries widen the life table into a grid. The select-and-ultimate table keeps the old age column down the side, but adds columns for *duration since selection*: the rate for someone just selected at age x, for someone one year past selection, two years, and so on, up to the end of the select period. Once you are past it, all those paths merge into a single ultimate column that depends on age alone — exactly the kind of plain life table you already know. The whole structure is just an ordinary table with a temporary, healthier 'on-ramp' grafted onto its left edge.
Age x | q[x] q[x]+1 q[x]+2 | q(x+3) (ultimate) ------+--------------------------+------------------ 50 | 0.0019 0.0026 0.0035 | 0.0045 51 | 0.0022 0.0030 0.0040 | 0.0051 [x] = selected at x; [x]+t = t yrs after selection; q(x) = ultimate
Notice two things in that little grid. First, read *across* a row and the rate climbs — the select discount wears off as the years since underwriting accumulate. Second, a freshly selected 50-year-old (q[50] = 0.0019) is far safer than a 50-year-old who was selected three years earlier and has now slid into the ultimate rate (q(50) ≈ 0.0045, more than double). Same age, very different price. Charging both the same premium would either overprice the fresh customer out of the market or quietly subsidise him at the insurer's expense — and competitors who split the two will eat your lunch.
Where do the select rates come from? From the same craft of table construction you met earlier, applied to the insurer's *own* policyholder experience and then smoothed by graduation. The general-population life tables a government statistician publishes are *aggregate* — they blend the just-screened and the long-since-screened together — so they are the wrong tool for pricing a brand-new policy. An insurer that priced from population tables would systematically under-charge, because its actual lives are healthier than the population for those crucial first years.
From one death to many doors out
So far a life has had exactly one way to leave the table: death. But real policies leak out through several doors. A term-life customer can die — or he can simply *stop paying* and lapse. A disability-income policyholder can recover, become disabled, or die. A pension member can retire, withdraw, become disabled, or die before retiring. The moment a life can exit in more than one way, you need a multiple-decrement model: a life table that tracks several competing 'causes of decrement' at once, each with its own force pulling people out.
These decrements compete, and that competition is the subtle part. If lapses are heavy, fewer people are left alive-and-in-force to die *of natural causes within the policy* — not because the underlying force of mortality changed, but because they exited first by another door. So the death rate you *observe in a multiple-decrement world* (a 'dependent' rate) is lower than the death rate that would apply if death were the *only* exit (the 'independent' rate). Confusing the two is a classic, expensive blunder: you cannot just bolt a standalone lapse table onto a standalone mortality table and expect the arithmetic to add up.
Multiple-decrement models are themselves a special case of something more general and more powerful: the multiple-state Markov model. Here you draw the life as a set of *states* — boxes labelled Healthy, Disabled, Lapsed, Dead — with arrows for the transitions allowed between them, each arrow carrying its own intensity (a force, just like the force of mortality, but for that particular move). A plain life table is the two-state case: Alive → Dead, one arrow, no way back. Disability insurance needs Healthy ⇄ Disabled (recovery is possible!) → Dead. The model is just an honest map of every door, and every door's traffic rate.
The table is built — now the ground shifts under it
Every table we have discussed so far describes the *past*: it is fitted to deaths that already happened. But a pension or a whole-life policy may pay out sixty years from now, and humans keep living longer. Across the twentieth century, mortality at most ages fell decade after decade — better food, sanitation, antibiotics, fewer smokers, better heart medicine. This downward drift is mortality improvement, and an actuary who ignores it will price annuities and pensions as if people died on last decade's schedule, then watch the payments run years longer than the reserves assumed.
So actuaries *project* the table forward: they take a base table and apply improvement factors — small annual percentage reductions in each rate, often tapering at older ages — to push it into the future. This is exactly where the direction of error matters. For a life insurer, people living longer is *good* (the death benefit is paid later, or never within the term). For an annuity or pension provider it is *bad* — every extra year of life is another year of payments. The very same improvement is a windfall on one side of the ledger and a slow-bleeding loss on the other. This is longevity risk, and it is mostly one-directional and systemic: it hits a whole book at once.
Be honest about what a projection is: a forecast, and forecasts of the far future are humbling. Past improvement was not steady — it stalled, then surged when new medicine arrived. No one in 1980 modelled the future scarring of a pandemic, or the recent flattening of life expectancy in some rich countries. So the improvement assumption is one of the *most uncertain* numbers an actuary chooses, which is why good practice never uses a single point. Actuaries run scenarios, hold capital for the adverse one, and revisit the assumption every valuation. A projection is a disciplined guess, dressed in arithmetic — never a promise.
Tying the rung together
Step back and see the whole rung. You began with the survival function and the force of mortality — the smooth, continuous heartbeat of how a life decays. You discretised that into the life table, the workhorse the whole industry actually computes with. This guide added the two refinements that turn a textbook table into something an insurer can bet money on: a *second dimension* (time since selection) so freshly underwritten lives are priced honestly, and a *web of states* so a life can lapse, fall ill, recover, or retire rather than merely die.
And we closed on the honest part: the table is fitted to the past, the ground keeps shifting, and the single most important number — how fast mortality will improve — is also the least knowable. That tension never resolves; it is simply *managed*, with scenarios and capital, every single year. Hold that humility. It is the difference between an actuary and a calculator.