Future Lifetime Random Variable
For a life aged exactly $x$, the future lifetime $T_x$ is a non-negative random variable. Its distribution is characterized by the survival function:
\[{}_t p_x = P(T_x > t) = S_x(t) = \frac{l_{x+t}}{l_x}\]The probability density function of $T_x$ is:
\[f_{T_x}(t) = -\frac{d}{dt}\,S_x(t) = {}_t p_x \cdot \mu_{x+t}\]The hazard rate (force of mortality) is:
\[\mu_{x+t} = -\frac{d}{dt}\ln S_x(t) = \frac{f_{T_x}(t)}{S_x(t)}\]The moments of future lifetime:
\[E[T_x] = \int_0^\infty {}_t p_x\,dt = \mathring{e}_x\] \[E[T_x^2] = 2\int_0^\infty t\cdot{}_t p_x\,dt\] \[\text{Var}(T_x) = E[T_x^2] - (\mathring{e}_x)^2\]The curtate future lifetime $K_x = \lfloor T_x \rfloor$ (complete years before death) has:
\[P(K_x = k) = {}_k p_x \cdot q_{x+k}, \quad k = 0, 1, 2, \ldots\]Gompertz-Makeham and Parametric Laws
The Gompertz law $\mu_x = Bc^x$ gives survival function:
\[{}_t p_x = \exp\!\left(-\frac{Bc^x(c^t - 1)}{\ln c}\right) = s^{(c^x(c^t - 1))}\]where $s = e^{-B/\ln c}$.
The Makeham law $\mu_x = A + Bc^x$ adds a constant hazard:
\[{}_t p_x = e^{-At} \cdot s^{c^x(c^t-1)}\]The Perks law $\mu_x = (A + Bc^x)/(1 + Dc^x)$ allows for a leveling-off at very old ages (mortality plateau), consistent with observed supercentenarian data.
Weibull hazard: $\mu_x = k x^{n-1}$ gives Weibull survival, useful for disability onset modeling.
| Law | $\mu_x$ | Behavior | Use |
|---|---|---|---|
| Gompertz | $Bc^x$ | Exponential increase | Adult mortality |
| Makeham | $A + Bc^x$ | Constant + exponential | Standard tables |
| Perks | $(A+Bc^x)/(1+Dc^x)$ | Plateau at high ages | Oldest-old |
| Weibull | $kx^{n-1}$ | Power-law | Disability |
Select-and-Ultimate Mortality
Select mortality arises because underwriting selects healthier lives. The select period $s$ is the number of years after selection during which mortality remains lower than the population average.
The two-dimensional hazard $\mu_{[x]+t}$ satisfies:
\(\mu_{[x]+t} \le \mu_{x+t}^{\text{ult}} \quad \text{for } t < s\) \(\mu_{[x]+t} = \mu_{x+t}^{\text{ult}} \quad \text{for } t \ge s\)
The survival function for a life selected at age $x$:
\[{}_t p_{[x]} = \exp\!\left(-\int_0^t \mu_{[x]+r}\,dr\right)\]In pension valuation, select tables matter when new entrants receive enhanced mortality assumptions that grade into standard mortality over a select period of 5-10 years.
Joint Life and Last Survivor
For two lives aged $x$ and $y$, define joint lifetimes $T_x$ and $T_y$ (possibly dependent). The joint-life status fails at the first death; the last-survivor status fails at the second death.
Under independence:
\[{}_t p_{xy} = P(T_x > t, T_y > t) = {}_t p_x \cdot {}_t p_y\] \[{}_t p_{\overline{xy}} = P(\max(T_x, T_y) > t) = {}_t p_x + {}_t p_y - {}_t p_{xy}\]Actuarial present values for joint lives:
\[A_{xy} = \sum_{k=0}^\infty v^{k+1}\,{}_k p_{xy}\,q_{x+k:y+k} \quad \text{(joint-life)}\] \[A_{\overline{xy}} = A_x + A_y - A_{xy} \quad \text{(last-survivor)}\]The reversionary annuity pays while $y$ is alive but $x$ is dead:
\[a_{y|x} = a_y - a_{xy}\]This is essential for spouse’s pension benefits in defined benefit pension schemes.
Dependent Lives via Copulas
Independence between spouses’ lifetimes is unrealistic due to common shock (simultaneous death, e.g., accident) and broken heart syndrome (grief-induced mortality increase). The copula approach models the joint survival function:
\[{}_t p_{xy}^{\text{joint}} = C({}_t p_x, {}_t p_y; \alpha)\]where $C: [0,1]^2 \to [0,1]$ is a copula with dependence parameter $\alpha$.
Common copula families for joint lifetimes:
| Copula | $C(u,v)$ | Dependence |
|---|---|---|
| Frank | $-\frac{1}{\alpha}\ln!\left(1+\frac{(e^{-\alpha u}-1)(e^{-\alpha v}-1)}{e^{-\alpha}-1}\right)$ | Symmetric |
| Clayton | $(\max(u^{-\alpha}+v^{-\alpha}-1,0))^{-1/\alpha}$ | Lower tail |
| Gumbel | $\exp(-[(-\ln u)^\alpha+(-\ln v)^\alpha]^{1/\alpha})$ | Upper tail |
Kendall’s $\tau$ (rank correlation) measures dependence; for Clayton: $\tau = \alpha/(\alpha+2)$.
Studies of married couples (Frees et al., 1996) found $\tau \approx 0.21$ for joint mortality, with Clayton copulas fitting better than Gaussian copulas (evidence of lower-tail dependence — simultaneous early death).
Multiple Decrement Tables
A multiple decrement model tracks a life subject to several competing causes of exit (decrements). For decrements $j = 1, \ldots, m$:
\[\mu_x^{(j)} = \text{cause-}j\text{ hazard (force of decrement)}\] \[\mu_x^{(\tau)} = \sum_{j=1}^m \mu_x^{(j)} \quad\text{(total force)}\] \[{}_t p_x^{(\tau)} = \exp\!\left(-\int_0^t \mu_{x+s}^{(\tau)}\,ds\right) \quad\text{(probability of remaining)}\]The probability of exiting by cause $j$ between ages $x$ and $x+t$:
\[{}_t q_x^{(j)} = \int_0^t {}_s p_x^{(\tau)}\,\mu_{x+s}^{(j)}\,ds\]Example — pension fund decrements: Death ($j=1$), ill-health retirement ($j=2$), withdrawal ($j=3$), normal retirement ($j=4$). The actuary must value the benefit payable under each decrement separately:
\[\text{APV} = \sum_{j=1}^4 \int_0^{n} v^t\,{}_t p_x^{(\tau)}\,\mu_{x+t}^{(j)}\,b_j(t)\,dt\]where $b_j(t)$ is the benefit payable on exit by cause $j$ at time $t$.
Associated single-decrement tables: Under the assumption of independence between decrements (i.e., each cause operates as if it were the only one), the single-decrement probability $q_x^{\prime(j)}$ (in the absence of other decrements) relates to the multiple decrement table by:
\[q_x^{(j)} \approx q_x^{\prime(j)}\left(1 - \tfrac{1}{2}\sum_{k \ne j}q_x^{\prime(k)} + \tfrac{1}{3}\sum_{k\ne j}\sum_{l\ne j,l\ne k}q_x^{\prime(k)}q_x^{\prime(l)} - \ldots\right)\]Under UDD in the associated single-decrement tables, this simplifies to:
\[q_x^{(j)} \approx q_x^{\prime(j)}\left(1 - \tfrac{1}{2}\sum_{k\ne j}q_x^{\prime(k)}\right)\]