An annuity-certain pays a fixed amount per period for a fixed number of periods, independent of mortality. They serve as the deterministic building blocks for life annuities.
where \(d = 1 - v\) and \(\delta = \ln(1+i)\). Three flavours: immediate (pays at end of each year), due (pays at start), continuous.
A continuous stream of 1 per year is paid as long as (x) is alive. The present-value r.v. is \(Y = \bar a_{\overline T|} = (1 - v^T)/\delta\). The APV is
\[ \bar a_x = E[Y] = \int_0^{\infty} v^t\,{}_tp_x\,dt = \int_0^{\infty} e^{-\delta t}\,{}_tp_x\,dt. \]This identity is the most-used relation in actuarial mathematics — every premium and every reserve formula will return to it.
Take \(\mu = 0.04\), \(\delta = 0.06\) (from Unit 2). Compute \(\bar a_x\) two ways.
Direct: \(\bar a_x = \int_0^\infty e^{-0.06 t} e^{-0.04 t}\,dt = \int_0^\infty e^{-0.10 t}\,dt = 1/0.10 = 10.\)
Via identity: \(\bar A_x = 0.40\) from Unit 2, so \(\bar a_x = (1 - 0.40)/0.06 = 10.\)
Both agree. So ₹1 per year for life is worth ₹10 in APV.
Pays 1 at times \(0, 1, 2, \ldots\) while (x) is alive:
\[ \ddot a_x = \sum_{k=0}^{\infty} v^k\,{}_kp_x = E\!\left[\ddot a_{\overline{K+1}|}\right]. \]Key identity:
\[ \ddot a_x = \frac{1 - A_x}{d}, \qquad A_x = 1 - d\,\ddot a_x. \]The "−1" removes the time-0 payment of the annuity-due.
Identity: \(\ddot a_{x:\overline n|} = (1 - A_{x:\overline n|})/d.\)
The contract pays the annuity only after surviving the deferral period.
Payments are guaranteed for \(n\) years (regardless of death) and continue thereafter for life if the annuitant is still alive. Used heavily in pension plans.
A life age 65 buys a 10-year-guaranteed life annuity with annual payment ₹1 lakh. Given \(\ddot a_{\overline{10}|} = 7.802\) at \(i = 0.05\), \({}_{10}E_{65} = 0.50\), \(\ddot a_{75} = 9.20\). Compute the purchase price.
\(\ddot a_{\overline{10}|}^{65} = 7.802 + 0.50 \times 9.20 = 7.802 + 4.60 = 12.402.\)
Purchase price = ₹1,00,000 × 12.402 = ₹12.40 lakh.
For (65), suppose \(A_{65} = 0.42\), \(i = 0.05\) so \(d = 0.05/1.05 = 0.04762\).
\(\ddot a_{65} = (1 - 0.42)/0.04762 = 0.58/0.04762 = 12.18.\)
An immediate annuity \(a_{65} = 12.18 - 1 = 11.18\). The continuous version under UDD is \(\bar a_{65} = \ddot a_{65} - 0.5 \approx 11.68\).
Most real-world annuities pay monthly (\(m = 12\)). The actuarial present value of an annuity-due paying \(1/m\) at times \(0, 1/m, 2/m, \ldots\) is denoted \(\ddot a_x^{(m)}\).
Under UDD:
\[ \ddot a_x^{(m)} \;\approx\; \ddot a_x - \frac{m-1}{2m}. \]For monthly payments (\(m = 12\)), the correction is \(11/24 \approx 0.458\) — so \(\ddot a_x^{(12)} \approx \ddot a_x - 0.458\).
From Example 2 above, \(\ddot a_{65} = 12.18\). The monthly annuity-due value:
\(\ddot a_{65}^{(12)} \approx 12.18 - 0.458 = 11.72.\)
A monthly pension of ₹10,000 starting at age 65 has APV \(10{,}000 \times 12 \times 11.72 / 12 = \text{₹}1{,}17{,}200\) per ₹1 lakh of annual income — but with monthly payment timing baked in.
An annuity-due pays in advance; the \(m\)thly version splits a single yearly payment into \(m\) equal pieces, and on average advances each piece by \((m-1)/(2m)\) of a year less than the original yearly-due payment. The interest forgone on that "advance" is exactly the correction term.
Pays 1 at the start of each year while both lives are alive:
\[ \ddot a_{xy} = \sum_{k=0}^{\infty} v^k\,{}_kp_{xy} = \frac{1 - A_{xy}}{d}. \]Stops on the first death.
Pays 1 at the start of each year while at least one life is alive:
\[ \ddot a_{\overline{xy}} = \ddot a_x + \ddot a_y - \ddot a_{xy}. \]Stops only on the second death. Inclusion-exclusion applied to annuities.
Pays 1 to (y) only after (x) dies, as long as (y) is alive:
\[ \ddot a_{x|y} = \ddot a_y - \ddot a_{xy}. \]This is the classic "widow's pension": husband (x) buys a contract that pays his wife (y) only after he dies.
Couple aged (65, 60). \(\ddot a_{65} = 12.5\), \(\ddot a_{60} = 14.8\), \(\ddot a_{65:60} = 10.9\).
"Joint-and-100%-survivor" pays full benefit until the second death:
\(\ddot a_{\overline{65:60}} = 12.5 + 14.8 - 10.9 = 16.4.\)
A "50% survivor" benefit (half pension after first death): \(\ddot a_{65:60} + 0.5 \cdot (\ddot a_{\overline{65:60}} - \ddot a_{65:60}) = 10.9 + 0.5(16.4 - 10.9) = 13.65\).
Same couple. The reversionary annuity for the wife (60) after the husband (65) dies:
\(\ddot a_{65|60} = \ddot a_{60} - \ddot a_{65:60} = 14.8 - 10.9 = 3.9.\)
So a ₹1-per-year widow's pension costs only ₹3.9 in APV — the husband is expected to survive long enough that the wife collects relatively few payments.
| Quantity | Identity |
|---|---|
| Continuous whole-life | \(\bar a_x = (1 - \bar A_x)/\delta\) |
| Annuity-due | \(\ddot a_x = (1 - A_x)/d\) |
| Annuity-immediate | \(a_x = \ddot a_x - 1\) |
| Temporary annuity-due | \(\ddot a_{x:\overline n|} = (1 - A_{x:\overline n|})/d\) |
| Deferred annuity | \({}_{m|}\ddot a_x = {}_mE_x \cdot \ddot a_{x+m}\) |
| \(m\)thly (UDD) | \(\ddot a_x^{(m)} \approx \ddot a_x - (m-1)/(2m)\) |
| Last-survivor | \(\ddot a_{\overline{xy}} = \ddot a_x + \ddot a_y - \ddot a_{xy}\) |
| Reversionary | \(\ddot a_{x|y} = \ddot a_y - \ddot a_{xy}\) |