deferred and guaranteed life annuities

Two common twists on the basic life annuity make pensions work the way real people need them. A deferred life annuity does not start paying right away; it waits m years (the accumulation phase) and then pays for life — exactly what a pension is during your working years, before retirement. A guaranteed (or 'with period certain') life annuity addresses a different fear: that you might pay in, then die almost immediately and get almost nothing back. It promises that payments will continue for at least a guaranteed number of years — say 10 — even if you die early, with payments going to your estate or beneficiary for the rest of that guaranteed period.

Both are valued by combining building blocks you already have. An m-year deferred whole life annuity-due, written m|ä_x, is the whole life annuity minus the temporary annuity for the deferral years: m|ä_x = ä_x - ä_{x:m}, or equivalently the m-year pure endowment times the annuity at the older age, m|ä_x = mE_x times ä_{x+m}. A life annuity with an n-year guarantee splits into two independent pieces that simply add: an n-year annuity-certain (paid no matter what, because those years are guaranteed) plus an n-year deferred life annuity (the genuinely life-contingent payments that only happen if you survive past the guarantee). For example, a 10-year-guaranteed life annuity = annuity-certain for 10 years + 10|ä_x.

These features are the bread and butter of retirement product design, and they explain why two annuities with the same monthly income can have very different prices. A guarantee period raises the price (you are buying away some of the 'die early, lose everything' risk), while a deferral period lowers it (payments are pushed into the discounted, lower-survival-probability future). A common misconception is that a guarantee makes an annuity 'safer' in an investment sense; what it really does is transfer value from the long-lived to the short-lived members of the pool, at the cost of a lower income for the same premium.