deferred annuity-certain
Sometimes a stream of payments does not begin right away but waits a while first. Think of someone aged 50 who buys an income that will start paying at age 65: nothing happens for fifteen years, then a regular annuity kicks in. When a level series of certain (guaranteed, not life-dependent) payments is delayed before it starts, the arrangement is a deferred annuity-certain.
Valuing it is simply a two-step move. First value the annuity as if it began at its actual start date, then discount that whole value back across the deferral period to today. If an annuity-immediate of n payments is deferred m periods, its present value is v to the power m times a-angle-n. Equivalently, it equals the present value of an (m + n)-period annuity minus the present value of an m-period annuity — the long annuity minus the part you are not entitled to. Either route gives the same answer.
Deferred annuities are everywhere in retirement planning and pensions: you accumulate money during working years, then a stream of income begins at retirement. The certain version (a fixed term, no dependence on survival) is the interest-theory building block; the life-contingent cousin, where payments continue only while a person is alive, is treated in a separate field. The key practical point is the deferral factor v to the m — forgetting to discount across the waiting period is a common and expensive slip.
At 5 percent, an annuity paying 1,000 at the end of each year for 10 years but not starting until 5 years from now is worth v to the fifth times 1,000 times a-angle-10, about 0.7835 times 7,722, roughly 6,051 today.
Value the annuity at its start date, then discount the whole bundle back across the deferral period.
Deferring an annuity changes only when payments start, not how many there are; the deferral period itself produces no payments, so do not accidentally count it as paying years.