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Deferred annuities

An annuity whose first payment is delayed: value it as usual, then discount the whole block back by the deferral period.

The formulas

Deferred immediate

$${}_{m|}a_{\overline{n}|} = v^{m} a_{\overline{n}|} = a_{\overline{m+n}|} - a_{\overline{m}|}$$

First payment date

$$t = m + 1$$

for a deferred annuity-immediate

Deferred perpetuity

$${}_{m|}a_{\overline{\infty}|} = \dfrac{v^{m}}{i}$$

Where it comes from

1. Valuing the $n$ payments at time $m$ gives $a_{\overline{n}|}$ - the annuity formula is blind to where its own clock started.
2. Bringing that value back to time 0 multiplies by $v^m$.
3. The subtraction form says the same thing: pay everything for $m+n$ periods, then refund the first $m$ payments.

Worked example

This answer is recomputed from the site’s own interest-theory and probability functions every time the test suite runs, so the page and the mathematics cannot drift apart.

Memory hooks

• Deferral m means the FIRST payment is at time m + 1 for an immediate annuity. Count the gap, not the payment.
• vᵐ·aₙ and a₍ₘ₊ₙ₎ − aₘ are the same number; use whichever the given data makes easier.

Traps

• Off-by-one on the deferral: 'first payment at time 6' is m = 5, not m = 6.
• Discounting by vᵐ⁺ⁿ instead of vᵐ.

Related

• Annuity-immediate
• Perpetuities

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