First, recall the simple (non-life-contingent) annuity formulas. An annuity certain pays ₹1 at fixed times regardless of survival.
Relationships: \(\ddot a_{\overline{n}|} = (1 + i)\, a_{\overline{n}|} = a_{\overline{n}|} \cdot (1+i)\). And \(\bar a_{\overline{n}|} \approx \dfrac{i}{\delta}\, a_{\overline{n}|}\) (under UDD).
A life annuity on (\(x\)) pays ₹1 per year (or per period) as long as (\(x\)) is alive. Unlike an annuity certain, payments stop on death. It is the fundamental building block of pensions.
As with insurance, life annuities come in continuous and discrete forms, and in whole-life, temporary (n-year) and deferred variants.
₹1 per year payable continuously as long as (\(x\)) is alive:
Equivalently:
\[ \bar a_x \;=\; \int_0^{\infty} v^t\, {}_tp_x\, dt. \]If \(\mu(x+t) = \mu\) constant:
Note that \(\bar A_x + \delta \bar a_x = 1\) here:
\(\dfrac{\mu}{\mu+\delta} + \delta \cdot \dfrac{1}{\mu+\delta} = \dfrac{\mu + \delta}{\mu + \delta} = 1\) ✓.
₹1 paid at the start of each year, as long as (\(x\)) is alive:
₹1 paid at the end of each year, as long as (\(x\)) is alive at the payment time:
These are the fundamental identities linking insurances and annuities. They reflect that paying ₹1 forever (a perpetuity) decomposes into "₹1 at death" (insurance) + "₹1 stream until death, discounted by interest" (annuity × interest rate).
Often pensions are paid monthly, quarterly etc., not annually. Let \(m\) be the number of payments per year, each of size \(1/m\), so the total annual rate is ₹1.
where \(\alpha(m) = id/(i^{(m)} d^{(m)})\) and \(\beta(m) = (i - i^{(m)})/(i^{(m)} d^{(m)})\). For typical values: \(\alpha(m) \approx 1\) and \(\beta(m) \approx (m-1)/(2m)\), giving
\[ \ddot a_x^{(m)} \;\approx\; \ddot a_x - \dfrac{m - 1}{2m}. \]As \(m \to \infty\) we recover the continuous annuity:
Subtract half a year because payments under \(\ddot a_x\) are at year-start while continuous payments are spread evenly.
\(\mu = 0.02,\;\delta = 0.05\). Compute \(\bar a_x,\; \ddot a_x\), and m-thly variants.
\(\bar a_x = 1/(0.02 + 0.05) = 14.29\) years (continuous whole-life annuity).
\(i = e^{0.05} - 1 = 0.0513;\; d = 0.0488\). \(A_x = \) ... actually, use \(\ddot a_x = (1 - A_x)/d\). First \(A_x\) under discrete (UDD): \(A_x = (\delta/i)\bar A_x = (0.05/0.0513)(0.286) = 0.279\). So \(\ddot a_x = (1 - 0.279)/0.0488 = 14.78\).
\(\bar a_x = 14.29\) and \(\ddot a_x = 14.78\); difference ≈ 0.49 ≈ 0.5 — matches the approximation.
A retired person aged 65 receives a monthly pension of ₹50,000. What lump sum should fund it for life? Assume \(\ddot a_{65} = 11.5\) (from a life table at \(i = 6\%\)).
Annual pension = 50000 × 12 = ₹6,00,000.
For monthly payment: \(\ddot a_{65}^{(12)} \approx \ddot a_{65} - (11/24) = 11.5 - 0.458 = 11.042\).
Lump sum APV = 6,00,000 × 11.042 = ₹66.25 lakh — the present value of a lifetime monthly ₹50 000 pension.
The net premium is set so that the actuarial present value of future premiums equals the APV of future benefits at policy inception:
\[ \text{APV (Premiums in)} \;=\; \text{APV (Benefits out)}. \]This is the foundation of pricing in life insurance.
Premiums paid continuously at rate \(P\) per year for as long as (\(x\)) lives; benefit paid at moment of death.
APV of premiums: \(P \cdot \bar a_x\). APV of benefits: \(\bar A_x\). Equating:
For constant force: \(\bar P = \dfrac{\mu/(\mu+\delta)}{1/(\mu+\delta)} = \mu\). The premium rate equals the force of mortality — a remarkable result.
Premiums paid at the start of each year (annuity-due), benefit at end of year of death.
This is the net annual premium for ₹1 of whole-life cover purchased at age \(x\).
Higher than \(P_x\) because premiums are concentrated in fewer years.
For premium paid m times per year (e.g., monthly, m = 12), each payment is the annual premium divided by m, but the timing changes the APV:
Since \(\ddot a_x^{(m)} < \ddot a_x\) (later payments worth less), \(P^{(m)}_x > P_x\) per year. The monthly premium is \(P^{(m)}_x / 12\).
The gross premium includes loadings for expenses, profit margin, and contingencies on top of the net premium. Extended equivalence principle:
\[ \text{APV (Gross Premiums)} \;=\; \text{APV (Benefits)} \;+\; \text{APV (Expenses)}. \]Common expense categories:
If initial expense = \(e_0\) (one-time), renewal expense = \(e_r\) per year, then:
\[ G \cdot \ddot a_x \;=\; A_x + e_0 + e_r \cdot \ddot a_x. \]Solving for the gross premium \(G\):
\[ G \;=\; \dfrac{A_x + e_0}{\ddot a_x} + e_r \;=\; P_x + \dfrac{e_0}{\ddot a_x} + e_r. \]Suppose at age 35, from a life table: \(A_{35} = 0.150,\; \ddot a_{35} = 17.85\) at \(i = 5\%\). Compute the net annual premium for a ₹10 lakh whole-life policy.
\(P_{35} = 0.150 / 17.85 = 0.008403\) per ₹1 of face amount.
For ₹10 lakh: \(P = 10{,}00{,}000 \times 0.008403 = \text{₹}8{,}403\) per year.
Monthly: with \(\ddot a_{35}^{(12)} \approx 17.85 - 11/24 = 17.39\): \(P^{(12)}_{35} = 0.150/17.39 = 0.00863\). Annual ≈ ₹8 630, monthly ≈ ₹719.
20-year endowment on (40), face ₹5 lakh, \(i = 6\%\). From a life table: \(A_{40:\overline{20}|} = 0.392,\; \ddot a_{40:\overline{20}|} = 12.30\).
\(P = 0.392 / 12.30 = 0.03187\) per ₹1.
For ₹5 lakh: ₹15 935 per year — note this is much higher than whole-life because the policy must accumulate enough to pay ₹5 lakh either at death or at maturity in 20 years.
At any time \(t\) after issue, the insurer holds a policy reserve \(_tV\) — the amount that, together with future premiums, equals future benefits in expectation. For a whole-life policy:
Initially \({}_0V = 0\) (equivalence principle), reserves grow as the insured ages, and at death the reserve plus the year's premium equals the benefit.
Reserves are studied in detail in Advanced Actuarial Statistics.
| Insurance Type | Discrete Premium | Continuous Premium |
|---|---|---|
| Whole-life | \(P_x = A_x / \ddot a_x\) | \(\bar P(\bar A_x) = \bar A_x / \bar a_x\) |
| n-Year Term | \(A^{\,1}_{x:\overline{n}|} / \ddot a_{x:\overline{n}|}\) | \(\bar A^{\,1}_{x:\overline{n}|} / \bar a_{x:\overline{n}|}\) |
| n-Year Endowment | \(A_{x:\overline{n}|} / \ddot a_{x:\overline{n}|}\) | \(\bar A_{x:\overline{n}|} / \bar a_{x:\overline{n}|}\) |
| Whole-life paid for k yrs only | \(A_x / \ddot a_{x:\overline{k}|}\) | — |
| m-thly Whole-life | \(A_x / \ddot a_x^{(m)}\) | — |