Future Lifetime Random Variable

Study guides: Exam FAM, Exam ALTAM

The Future Lifetime Random Variable $T_x$ is the time from now until the death of a single life now aged $x$ , written $(x)$ . Its distribution gives every survival and mortality probability for that life, and its integer part, the curtate future lifetime $K_x = \lfloor T_x \rfloor$ , counts the complete years that $(x)$ will still live. It is the random variable behind the survival model for individual lives in Dickson, Hardy & Waters (AMLCR, chapter 2), and every benefit on a single life depends on $T_x$ or $K_x$ .

${}_tp_x = \Pr(T_x > t) = S_x(t)$

${}_{u|t}q_x = {}_up_x - {}_{u+t}p_x$

$f_x(t) = {}_tp_x\,\mu_{x+t}$

$\mathring{e}_x = E[T_x] = \int_0^{\infty} {}_tp_x\,dt$

$e_x = E[K_x] = \sum_{k=1}^{\infty} {}_kp_x$

Symbols: ${}_tq_x = 1 - {}_tp_x$ is the probability that $(x)$ dies within $t$ years, and ${}_{u|t}q_x$ is the probability that $(x)$ survives $u$ years and then dies within the following $t$ years. When $t = 1$ the subscript is dropped, so $p_x$ and $q_x$ are one-year probabilities. $\mu_{x+t}$ is the force of mortality at age $x+t$ , and $f_x$ is the density of $T_x$ .
One life, one model: The lifetimes at different ages come from one lifetime $T_0$ , conditioned on survival to age $x$ , so ${}_tp_x = S_0(x+t)/S_0(x)$ and ${}_{t+u}p_x = {}_tp_x\,{}_up_{x+t}$ . In a select model the distribution also depends on the age at selection, which the notation shows as ${}_tp_{[x]+s}$ . A joint life status combines the future lifetimes of two such lives, $T_x$ and $T_y$ .
Curtate lifetime and moments: $K_x$ takes the value $k$ with probability ${}_kp_x\,q_{x+k} = {}_{k|}q_x$ , for $k = 0, 1, 2, \ldots$ . The second moments are $E[T_x^2] = 2\int_0^\infty t\,{}_tp_x\,dt$ and $E[K_x^2] = \sum_{k \ge 1}(2k-1)\,{}_kp_x$ , and each variance is the second moment less the squared mean. Capping the lifetime at $n$ years gives the term expectations $\mathring{e}_{x:\overline{n}|} = \int_0^n {}_tp_x\,dt$ and $e_{x:\overline{n}|} = \sum_{k=1}^n {}_kp_x$ .
Under UDD: With the uniform distribution of deaths fractional age assumption, $T_x = K_x + R$ , where $R$ is uniform on $(0,1)$ and independent of $K_x$ . It follows that $\mathring{e}_x = e_x + \frac12$ and $\text{Var}(T_x) = \text{Var}(K_x) + \frac{1}{12}$ .

Probabilities and Moments from a Survival Function A newborn's lifetime has survival function $S_0(x) = \left(1 - \frac{x}{100}\right)^{1/2}$ for $0 \le x \le 100$ . For a life aged 40, find ${}_{20}p_{40}$ , ${}_{10|10}q_{40}$ , $\mu_{60}$ , $\mathring{e}_{40}$ and the standard deviation of $T_{40}$ .
Answer The future lifetime of $(40)$ has ${}_tp_{40} = S_0(40+t)/S_0(40) = \left(1 - \frac{t}{60}\right)^{1/2}$ for $0 \le t \le 60$ , so $\mu_{40+t} = -\frac{d}{dt}\ln {}_tp_{40} = \frac{1}{2(60-t)}$ .
 $\begin{align*} {}_{20}p_{40} &= \left(\tfrac{40}{60}\right)^{1/2} \\ &= 0.81650 \\ {}_{10|10}q_{40} &= {}_{10}p_{40} - {}_{20}p_{40} \\ &= 0.91287 - 0.81650 \\ &= 0.09637 \\ \mu_{60} &= \frac{1}{2(60 - 20)} \\ &= 0.0125 \end{align*}$ 
Substituting $u = 1 - t/60$ turns both integrals into integrals of powers of $u$ .
 $\begin{align*} \mathring{e}_{40} &= \int_0^{60}\left(1 - \tfrac{t}{60}\right)^{1/2}dt \\ &= 60 \cdot \tfrac{2}{3} \\ &= 40 \\ E\left[T_{40}^2\right] &= 2\int_0^{60} t\left(1 - \tfrac{t}{60}\right)^{1/2}dt \\ &= 2(60)^2\left(\tfrac{2}{3} - \tfrac{2}{5}\right) \\ &= 1{,}920 \\ \text{Var}(T_{40}) &= 1{,}920 - 40^2 \\ &= 320 \end{align*}$ 
The standard deviation is $\sqrt{320} = 17.89$ years. The force of mortality grows without limit as $t$ approaches 60, because no one in this model lives past age 100.

Curtate Lifetime at the End of a Table A life table ends at age 100, with $q_{97} = 0.3$ , $q_{98} = 0.5$ and $q_{99} = 1$ . For a life aged 97, find the distribution of $K_{97}$ , $e_{97}$ and $\text{Var}(K_{97})$ . Then, assuming UDD, find $\mathring{e}_{97}$ and $\text{Var}(T_{97})$ .
Answer The survival probabilities are $p_{97} = 0.7$ , ${}_2p_{97} = 0.7(0.5) = 0.35$ and ${}_3p_{97} = 0$ . So $K_{97}$ takes the values 0, 1 and 2 with probabilities $0.30$ , $0.35$ and $0.35$ .
 $\begin{align*} e_{97} &= 0.7 + 0.35 \\ &= 1.05 \\ E\left[K_{97}^2\right] &= 1(0.7) + 3(0.35) \\ &= 1.75 \\ \text{Var}(K_{97}) &= 1.75 - 1.05^2 \\ &= 0.6475 \end{align*}$ 
The direct sum $0(0.30) + 1(0.35) + 4(0.35)$ gives the same second moment. Under UDD, the part of the year lived in the year of death is uniform and independent of $K_{97}$ .
 $\begin{align*} \mathring{e}_{97} &= 1.05 + 0.5 \\ &= 1.55 \\ \text{Var}(T_{97}) &= 0.6475 + \tfrac{1}{12} \\ &= 0.7308 \end{align*}$ 

How it appears on the exam

FAM gives a survival function, a force of mortality or a life table, and asks for ${}_tp_x$ , ${}_tq_x$ , ${}_{u|t}q_x$ , $\mu_{x+t}$ or the density $f_x(t)$ for a life at a stated age, using select notation where the table is select.

Moments: A question can also ask for $\mathring{e}_x$ , $e_x$ , a term expectation such as $\mathring{e}_{x:\overline{n}|}$ , or a variance. When only integer-age values are given, the question states UDD or a constant force between integer ages.

Single lives: Every present value random variable in FAM's long-term topics is a function of $T_x$ or $K_x$ for one life, such as a death benefit paid at time $T_x$ or at the end of year $K_x + 1$ .

ALTAM combines two future lifetimes in joint life models and asks whether $T_x$ and $T_y$ can be treated as independent, which sources of dependence make them not so, and how the models allow for it.

Exam traps

Curtate versus complete: The sum for $e_x$ starts at $k = 1$ , while the integral for $\mathring{e}_x$ starts at 0. Starting the sum at $k = 0$ gives $E[K_x + 1]$ , which is one year too large.

The second moment: The second moment of $T_x$ is $2\int_0^\infty t\,{}_tp_x\,dt$ . Dropping the factor 2, or squaring $\mathring{e}_x$ , gives a variance that is far too small or zero.

Deferred probabilities: ${}_{u|t}q_x$ equals ${}_up_x\,{}_tq_{x+u}$ . Writing ${}_up_x\,{}_tq_x$ applies the mortality of age $x$ to a life that is now $u$ years older.

Conditioning on age $x$ : From a newborn survival function, ${}_tp_x = S_0(x+t)/S_0(x)$ , not $S_0(t)$ . The two agree only under a constant force of mortality.