Is force of mortality monotonic function?

Question

I'm trying to do solve some question from the actuarial exam. The task is to map a function with a graph. In the graph the function is increasing on $[0,80]$ and decreasing on $[80,100]$. The options I can choose from are : $l_x,\ l_x^2,\ u_x,\ l_x u_x$. The first two can't be correct because they are decreasing everywhere. What can we tell about the $u_x$?

Answer 1

The curve in question is

The choices are:

(A) $\mu(x)$

(B) $l_x \mu(x)$

(C) $l_x p_x$

(D) $l_x$

(E) $l_x^2$.

As you noted, we can immediately eliminate (D) and (E) because $l_x$ is monotonically decreasing. We can eliminate (C) for a similar reason, since $p_x$, the survival function of a life aged $x$, is bounded between $0$ and $1$, and it tends to be the case that $p_x < p_y$ for $x > y$ in adulthood, because the probability of surviving one additional year is smaller as the age is greater.

We should also eliminate (A) because a typical force of mortality at the ultimate age should be higher than at younger ages. This leaves (B) as the only candidate for this plot.

What does $l_x \mu(x)$ represent? It is the product of the number of lives still living at age $x$, times the force of mortality experienced by those lives. Consequently this product is proportional to the probability that a newborn life dies at age $x$ (with the constant of proportionality equal to the size of the cohort at birth, $l_0$). So this explains why the plot is increasing up to age $80$ and then curves back down at around $x = 80$, because the number of lives that have died out by then makes $l_x$ small, thus making the product small even when the force of mortality is high. This curve is also known as the curve of deaths. It captures the likelihood of infant mortality, the low likelihood of young and middle-age adult mortality, the gradually increasing likelihood of death at older ages, and importantly, the relatively low likelihood of reaching extreme old age because most people don't make it that far.

Answer 2

In general, $u_x$ isn't monotonic. One counterexample of support $\Bbb R^+$ is $S(x)=\frac{1}{1+x^2},\,u_x=\frac{2x}{1+x^2}$, so $u_x$ peaks at $x=1$. Since $l_x=NS(x)$ if $N$ is the number of people originally, $l_xu_x=-NS^\prime(x)$, which also need not be monotonic.