Lifetime Annuities and Life Insurance

Long-horizon insurance products—whole life insurance, annuities, endowments—are the backbone of classical actuarial mathematics. Their valuation requires not only knowledge of mortality, but also proper discounting of future payments taking into account the probability of survival. The equivalence principle (E[contributions] = E[benefits]) gives the “fair” net premium that underlies rate-setting. These concepts, developed as early as by Edmond Halley (1693) and generalized in the 20th century, remain relevant—every life insurer applies them.

Actuarial Discounting

Discount factor: $v = 1/(1 + i)$, where $i$ is the annual interest rate. The present value of 1 ruble in $t$ years: $v^t = (1 + i)^{-t}$.

Actuarial notations:

• $_n E_x = v^n \cdot {_n}p_x$ — present value of a payment of 1 ruble in $n$ years conditional on survival. Combines discounting ($v^n$) and probability ($_n p_x$).

This quantity is the “building block” of all life actuarial mathematics.

Whole Life Insurance

Whole Life Insurance. Payment of 1 ruble at the moment of death, whenever it occurs. Present value in the actuarial sense:

$\bar{A}_x = E[v^{T_x}] = \int_0^\infty v^t \cdot \mu(x + t) \cdot {_t}p_x , dt.$

An underline indicates a "continuous" payment (at the moment of death, not at end of year). Without the underline: $A_x = \sum_{k=0}^\infty v^{k+1} \cdot {k}p_x \cdot q{x+k}$ (payment at the end of the year of death).

Term Life Insurance. Payment of 1 ruble upon death within $n$ years, otherwise 0:

$A^1_{x:n|} = \sum_{k=0}^{n-1} v^{k+1} \cdot {k}p_x \cdot q{x+k}$.

Pure Endowment. Payment of 1 ruble upon survival to $x + n$:

$_n E_x = v^n \cdot {_n}p_x$.

Endowment Insurance (combined). Payment of 1 upon death within $n$ years OR upon survival:

$A_{x:n|} = A^1_{x:n|} + {_n}E_x$.

Recurrence Relation

$A_x = v \cdot q_x + v \cdot p_x \cdot A_{x+1}$.

Interpretation: “the present value of whole life = discounted payment if death in the first year + discounted value of whole life at the beginning of next year if survived.” This allows for recursive calculation of $A_x$ for all $x$ according to the mortality table.

Lifetime Annuities (annuities)

Whole Life Annuity-due (annuity at the beginning of each year). Payment of 1 ruble at the start of each year as long as alive:
$\ddot{a}x = \sum{k=0}^\infty v^k \cdot {k}p_x = 1 + v \cdot p_x \cdot \ddot{a}{x+1}$ (recursion).

Whole Life Annuity-immediate (at end of each year): $a_x = \sum_{k=1}^\infty v^k \cdot {_k}p_x = \ddot{a}_x - 1$.

Temporary Annuity: $\ddot{a}{x:n|} = \sum{k=0}^{n-1} v^k \cdot {_k}p_x$.

Deferred Annuity: $_m|\ddot{a}_x = \ddot{a}x - \ddot{a}{x:m|} = v^m \cdot {m}p_x \cdot \ddot{a}{x+m}$ — an annuity beginning in $m$ years (if survived).

Fundamental Identity

$A_x + d \cdot \ddot{a}_x = 1$, where $d = i/(1+i) = i \cdot v$.

This means: “the value of whole life insurance + $d$ × value of whole life annuity = 1.” Alternatively: an annuity (stream of 1 ruble/year as long as alive) + insurance (payment of 1 at death) equals “guaranteed 1 ruble at death + annuities until death,” and this is equivalent to a payment of 1 ruble now under suitable transformations.

Net Premium

Equivalence Principle. Premium $P$ is such that the present value of future premiums equals the present value of future benefits:

$P \cdot \ddot{a}_x = A_x \rightarrow$ $P_x = A_x / \ddot{a}_x$.

This is the net premium (without expenses and profit). The gross premium adds a loading of 15–40% for expenses.

Reserves: $V_t = A_{x+t} - P_x \cdot \ddot{a}_{x+t}$ — present value of future liabilities minus future premiums. Grows over time: the insurer "puts aside" part of the early premiums for payouts in later years.

Numerical Example: Male age 40, $i = 5%$

We use a simplified table: $q_{40} = 0.0030$, $q_{50} = 0.0070$, $q_{60} = 0.0180$, $q_{70} = 0.0420$, $q_{80} = 0.110$.

Step 1. Via recursion $A_x = v \cdot q_x + v \cdot p_x \cdot A_{x+1}$, starting from $A_{110} = 1$, work backwards.

Numerically (Python, simplified Gompertz–Makeham approximation): $A_{40} \approx 0.232$. This means: the insurer will take ~232,000 rubles now to pay out 1 million at death.

Step 2. $\ddot{a}{40} = \frac{1 - A{40}}{d} = \frac{1 - 0.232}{0.05/1.05} = \frac{0.768}{0.0476} \approx 16.13$.

Step 3. $P_{40} = \frac{A_{40}}{\ddot{a}_{40}} = 0.232 / 16.13 \approx 0.01438$. For a 1 million sum insured—the premium is 14,380 rubles/year. This is the net amount; the gross premium (including expenses) is ~17,000–22,000 rubles/year.

Step 4. Reserve $V_{10}$ (for a 40-year-old after 10 years): $A_{50} \approx 0.300$, $\ddot{a}{50} \approx 14.7$. $V{10} = 0.300 - 0.01438 \cdot 14.7 \approx 0.300 - 0.211 = 0.089$. That is, 89,000 rubles for a 1 million coverage.

$V_t$ grows from 0 at policy inception to 1 at the moment of death. The maximum for whole life is around retirement age.

Real Applications

• Life insurance. Russia (SOGAZ, AlfaStrakhovanie Zhizni)—premiums according to Russian mortality tables with conservative rate of 3–4%. Western insurers (Allianz, Manulife)—1–2% (low interests).
• Pension annuities. The retiree gives insurer capital $C$, receives a lifetime annuity $X = C/\ddot{a}{65}$. For a male age 65, $i = 3%$: $\ddot{a}{65} \approx 13.5$, $X \approx C/13.5$.
• Variable Annuities with guarantees (US, JP). Trillion-level liabilities, require dynamic hedging (delta, vega, longevity).
• Reserves under IFRS 17. Modern accounting of insurance obligations requires careful actuarial calculation of $V_t$ for each policy cohort.

Assignment. For a male aged 40, interest rate $i = 5%$, using the real mortality table (HMD Russia 2019): (a) Numerically (Python, recursion by ages 40–110) compute $\bar{A}{40}$ (whole life). (b) Compute $\ddot{a}{40}$ via the identity $\ddot{a}{40} = (1 - \bar{A}{40})/d$. (c) Find the net premium $P_{40}$ for a sum insured of 1 million rubles. (d) Plot the reserve $V_t = A_{40+t} - P_{40} \cdot \ddot{a}_{40+t}$ for $t = 0, 1, ..., 70$. At what age is the reserve maximal? (e) Compare the premium for a female age 40 (using the female table).