Fundamental Actuarial Mathematics/Mortality Models

Learning objectives
The Candidate will understand key concepts concerning parametric and non-parametric mortality models for individual lives.

Learning outcomes
The Candidate will be able to:
 * 1) Understand parametric survival models, life tables, and the relationships between them.
 * 2) Given a parametric survival model, calculate survival and mortality probabilities, the force of mortality function, and curtate and complete moments of the future lifetime random variable.
 * 3) Identify and apply standard actuarial notation for future lifetime distributions and moments, including select and ultimate functions.
 * 4) Given a life table, calculate survival and mortality probabilities, the force of mortality function, and curtate and complete moments of the future lifetime random variable, using appropriate fractional age assumptions where necessary.
 * 5) Understand and apply select life tables.
 * 6) Identify common features of population mortality curves.

Survival distributions
In the model discussed in this chapter, it describes the length of survival (or time until death) of an individual. Thus, the will be the basic building block.

Age-at-death random variable
In this section, we will discuss a special case for the time-until-death random variable, in which the time until death is applicable to newborn (i.e. people aged zero). We denote this kind of random variable by $$T_0$$. We can observe that $$T_0$$ also represents the age at death, since the age is counted starting from the beginning of life.

Since time-until-death random variable is describing time, it is a continuous random variable. Also, time is nonnegative, so the support (or "domain") of time-until-death random variable is $$[0,\infty)$$.

To describe the time until death for newborn, we need to determine the distribution of $$T_0$$ completely. There are several ways to do this. You should have learnt about cdf and pdf when learning about probability, but may not have learnt about survival function and force of mortality. Thus, we will discuss them here.
 * cumulative distribution function (cdf): $$F_0(t)=\mathbb P(T_0\le t)$$
 * probability density function (pdf): $$f_0(t)=F'_0(t)$$ if $$F_0(t)$$ is differentiable.

Survival function
As suggested by the name " function", we may guess that this function is somewhat related to survival. This is actually true. Take the time-until-death random variable $$T_0$$ as an example, when the newborn survives for, say, $$t$$ units of time, what is its probability? It is $$\mathbb P(T_0>t)$$ (or $$\mathbb P(T_0\ge t)$$, but since $$T_0$$ is continuous, it does not matter). This probability corresponding to the input $$t$$ is actually the survival function, which is defined below:

Force of mortality
In financial mathematics, you should have learnt about, which can be interpreted as the , and it is given by $$\frac{A'(t)}{A(t)}$$ in which the notation has its usual meaning in financial mathematics. Why do we call it force of ? This is because the interest refers to an increase (or positive change) in the amount.

We may guess that the force of is defined similarly, in the sense that it can also be interpreted as the relative rate of change of something. We know that the interest refers to change in amount, but what change does mortality refer to? Since mortality means the state of being susceptible to death, it refers to the (or negative change) in survival rate, and the "higher" the mortality, the larger decrease in survival rate. Recall that the is related to survival rate (probability of surviving for a certain time) in some sense. So, we can make use of survival function to define mortality.

However, there is a difference between force of and force of, namely for force of interest, interest refers to an  in amount, while for force of mortality, mortality refers to a  in survival rate. So the changes are in opposite direction, and thus if we define force of mortality in an exact analogous manner, its value will be negative (the relative rate of change will be negative). To make the force of mortality positive, we can define the as follows:

After that, we will introduce some propositions related to the and.

Future lifetime of a life aged $x$
Now, we our discussion from future lifetime of a life aged zero (a newborn) to a life aged $$x$$ ($$x\ge 0$$). For simplicity of presentation, we denote a life aged $$x$$ by $$(x)$$.

Similarly, we denote the future lifetime of $$(x)$$ by $$T_x$$ (recall that we denote the future lifetime of $$(0)$$ (newborn) by $$T_0$$). We the distribution of $$T_x$$ mathematically (and quite naturally) as the  of $$T_0-x$$,  $$T_0>x$$.

To understand this, consider the following reasoning: Refer to the following timeline: death x      T_x   | - 0        x                 t      T_0 We can observe that $$T_x=T_0-x$$ if $$T_0>x$$ (or $$T_0\ge x$$, but since $$T_0$$ is continuous, it does not matter). So, if $$T_0>x$$, then $$T_x=T_0-x$$.
 * -|---v

On the other hand, if $$T_0x$$.
 * ---v-|

From this definition, we have $$\mathbb P(T_x\le t)=\mathbb P(T_0-x\le t|T_0>x)$$, $$\mathbb P(T_x>t)=\mathbb P(T_0-x>t|T_0>x)$$, etc.. This is quite important since it is the basis for the calculations of probabilities related to $$T_x$$.

For the pdf, cdf and survival function of $$T_x$$, we have similar notations as follows:
 * $$f_x(t)$$: pdf of $$T_x$$
 * $$F_x(t)$$: cdf of $$T_x$$
 * $$S_x(t)$$: survival function of $$T_x$$

In particular, we have some special actuarial notations for the cdf and survival function, as follows: In actuarial notations, "$$q$$" often refers to something related to, while "$$p$$" often refers to something related to. In this context, this holds since $$_tq_x$$ refers to the probability for $$(x)$$ to within $$t$$ time units, and $$_tp_x$$ refers to the probability for $$(x)$$ to  for $$t$$ time units.
 * $$_tq_x=F_x(t)=\mathbb P(T_x\le t)$$
 * $$_tp_x=S_x(t)=\mathbb P(T_x>t)=1-{}_tq_x$$

For simplicity, if $$t=1$$, we write $$_1q_x$$ as $$q_x$$ and $$_1p_x$$ as $$p_x$$.

Using the relationship between $$T_x$$ and $$T_0$$, we can develop some useful formulas for $$_tp_x$$ and $$_tq_x$$, as follows:

We can also express the pdf of $$T_x$$ as follows:

We have a special notation for the probability for $$(x)$$ to ages $$x+t$$ and $$x+t+u$$ ($$x,t,u\ge 0$$), namely $$_{t|u}q_x$$ (we use "$$q$$" here since this is related to death). Thus, we have by definition $$_{t|u}q_x=\mathbb P(t<T_x<t+u))$$. We have the following proposition for another formula of $$_{t|u}q_x$$.

Curtate-future-lifetime of a life aged $x$
The is just like the future lifetime in previous sections, except that it is.

Similarly, we would like to completely determine the distribution of $$K_x$$, as in the case for $$T_x$$. We can do this using cdf or probability mass function (pmf). Its pmf is given by the following proposition.

Life tables
In a life table, the values of $$q_x$$ and other functions for different (integer) ages $$x$$ are tabulated. The values are assumed to be based on the survival distribution discussed in previous sections. In this section, we will discuss more functions appearing in a life table.

In previous sections, we have discussed time-until-death random variable for one person, and we will consider multiple people here. Suppose there are $$\ell_0$$ newborns. Let the indicator function $$\mathbf 1_j(x)=\begin{cases}1,&\text{if life }j\text{ survives to age }x;\\ 0,&\text{otherwise}. \end{cases}$$ Also let $$\mathcal L(x)$$ be the sum of all such indicator functions $$\mathbf 1_j(x)$$, i.e. $$\mathcal L(x)=\sum_{j=1}^{\ell_0}\mathbf 1_j(x)$$. We can interpret $$\mathcal L(x)$$ as the number of survivors to age $$x$$ for the $$\ell_0$$ newborns.

We denote the of $$\mathcal L(x)$$ by $$\ell_x$$.

As a corollary, $$\ell'_x=(\ell_0S_0(x))'=\ell_0S'_0(x)$$ ($$\ell_0$$ is constant with respect to $$x$$). Also, $$\frac{-\ell'_x}{\ell_x}=\frac{-\ell_0S'_0(x)}{\ell_0S_0(x)}=\frac{-S'_0(x)}{S_0(x)}=\mu_x$$. Also, we can use $$\ell_x$$ to calculate probabilities like $$_t p_x$$ and $$_t q_x$$, as follows: $$_t p_x=\frac{S_0(x+t)}{S_0(x)}=\frac{\ell_0S_0(x+t)}{\ell_0S_0(x)}=\frac{\ell_{x+t}}{\ell_x}$$, and thus $$_t q_x=1-{}_t p_x=1-\frac{\ell_{x+t}}{\ell_x}$$. In a later section in which selection age is involved in the life table (select table), we will use these formulas to calculate these probabilities from such life table, to incorporate the effect of selection.

We have discussed about the number of to age $$x$$, and we will discuss the "opposite thing" in the following, namely the number of  to age $$x$$ (i.e. between age 0 and $$x$$), or in general, between age $$x$$ and $$x+n$$.

We denote the of such number of deaths by $$_n d_x$$.

Apart from the life table functions $$\ell_x$$ and $$_n d_x$$ which are related to the expectation of of survivors and deaths respectively, we will also discuss two more life table functions, that is related to the expectation of.

There are two types of expectation of life: one is discrete and another is continuous, and they are called curtate-expectation-of-life and complete-expectation-of-life respectively.

The following are recursion relations for $$\overset{\circ}{e}_x$$ and $$e_x$$, which can be useful when we want to find the complete/curtate-expectation-of-life of $$(x)$$ given the expectation of a life with some other ages, say $$x+1$$ and $$x+2$$.

We will state the recursion relations as a form of proposition, and then prove them formally. After the proof, we will try to give some intuitive explanations about the recursion relation for $$e_x$$.

An intuitive explanation of this recursion relation is as follows:
 * for LHS, $$e_x$$ is the curtate-expectation-of-life of $$(x)$$;
 * for RHS, $$e_{x+1}$$ is the curtate-expectation-of-life of $$(x+1)$$, and we want to "transform" it to the expectation of $$(x)$$. The first step is adding 1 to it, since this is the expectation with respect to $$(x+1)$$, but we want the expectation from the perspective of $$(x)$$, which is 1 year younger. But only this step is not enough, since "$$e_{x+1}$$" assumes the life already lives for $$x+1$$ years, but for $$e_x$$, the life is only assumed to live for $$x$$ years. Hence, we also need to multiply the probability for $$(x)$$ to live for one year, $$p_x$$, to "get to" $$e_{x+1}$$.
 * Now, "the expectation of life from age $$x+1$$ onward" is done through $$e_{x+1}$$. How about "the expectation of life from age $$x$$ to age $$x+1$$"? Indeed, when the life dies within age $$x$$ and $$x+1$$, $$K=0$$. This means such "expectation of life" is zero.

Assumptions for fractional ages
Previously, we have discussed the continuous random variable $$T_x$$ and discrete random variable $$K_x$$. A life table can specify the distribution of $$K_x$$ since the values of $$q_x$$ for different integer $$x$$ can be obtained from the life table. However, the life table is not enough to specify the distribution of $$T_x$$, since we do not know the value of $$q_x$$ when $$x$$ is not an integer. Thus, in order to specify a distribution of $$T_x$$ using a life table, we need to make some assumptions about the fractional (non-integer) ages.

In actuarial science, three assumptions are widely used, namely (UDD) (or linear interpolation),  (or exponential interpolation), and  (or Balducci) assumption (or harmonic interpolation). We will define them each using survival functions, as follows:

Under UDD assumption, we have some "nice" and simple expressions for various probabilities related to mortality. We can obtain those expressions by substituting $$S_0(x+t)$$ by the RHS of the equation mentioned in the assumption.

For the three assumption mentioned, there is a particularly "nice" and simple result for each of them, and we may use those "nice" results for the calculation in practice, rather than applying the definitions. The "nice" result for UDD assumption is mentioned in a previous example: when $$0\le t\le 1$$, $${}_tq_x=t(q_x)$$. The "nice" results for the other two assumptions are as follows:

An interesting result under the UDD assumption is related to the independence of two random variables.

To simplify the notations, from now on, we let $$T=T_x$$ and $$K=K_x$$ unless otherwise specified.

Define a continuous random variable $$S$$ by $$T=K+S,\quad S\in (0,1)$$. That is, $$S$$ is the random variable representing the lived in the year of death of $$(x)$$. For example, if $$S=0.5$$, then $$(x)$$ lives for half year in the year of death.

Then, $$K$$ and $$S$$ are independent under UDD assumption. This is because $$\mathbb P(K=k\text{ and }S\le s)=\mathbb P(k\le T\le k+s)={}_{k}p_{x}{}_{s} q_{x+k}={}_kp_x\cdot s\cdot q_{x+k}=(_{k|}q_x)\cdot s=\mathbb P(K=k)\mathbb P(S\le s)$$ under UDD assumption. Also, we can observe that the cdf of $$S$$ is $$\mathbb P(S\le s)=s=\frac{s-0}{1-0}$$, which is the cdf of uniform distribution with support $$(0,1)$$. This means $$S$$ follows the uniform distribution on $$(0,1)$$ under UDD assumption. Hence, $$\mathbb E[S]=\frac{1}{2}$$ and $$\operatorname{Var}(S)=\frac{1}{12}$$. This gives rise to results under UDD assumptions: These two results give us an alternative way to calculate the mean and variance of $$T$$ where only things are used in the calculation. However, we should be careful that these results hold, so we cannot use these results without UDD assumption.
 * $$\overset{\circ}{e}_x=\mathbb E[T]=\mathbb E[K+S]=\mathbb E[K]+\mathbb E[S]=e_x+\frac{1}{2}$$.
 * $$\operatorname{Var}(T)=\operatorname{Var}(T)\overset{\text{ independence }}{=}\operatorname{Var}(K)+\operatorname{Var}(S)=\left(\sum_{k=1}^{\infty}(2k-1){}_k p_x\right)-(e_x)^2+\frac{1}{12}$$.

Laws of mortality
In this section, we will introduce some simple laws of mortality (i.e. some specified distributions for mortality). Some of these laws may be appropriate to model the human mortality for ages, but it is commonly believed that none of these laws is appropriate to model the human mortality for ages.

Indeed, if we want to model the human mortality using some probability distribution, we may need a of distributions, since the mortality should be distributed in a different manner when human is in different ages, and thus different distributions should be used in different ages. To choose a suitable distribution for some ages, we may investigate the shape of corresponding empirical distribution based on the actual human mortality data, and pick a suitable distribution accordingly. For example, if the mortality increases exponentially for some ages, then we may select a distribution for which the force of mortality also increases exponentially.

In practice, to calculate the probabilities related to human mortality, we usually use life tables for the calculations. This is usually the case for insurance companies. Each insurance company has its own life table, based on the mortality data of, possibly its clients. Since such life table is constructed using the human mortality data based on past experiences, the life table is usually deemed to be more accurate than a specific distribution.

Nevertheless, having a law for mortality allows us to simplify the calculation of the probabilities related to mortality.

Select and ultimate table
When a person purchases a life insurance policy offered by an insurance company, he needs to give some personal information to the insurance company, e.g. some information about his health status. For the insurance company to decide whether it should sell the policy to the person, those information provided by that person is accessed through the process of. For underwriting, the check the information to see whether the risk in insuring that person is appropriate.

Without underwriting, it is likely that people will only purchase life insurance policy when they think they will die soon (e.g. they have a very serious disease), so that they will likely have early claims. In this case, the insurance company may need to pay a large amount of money and suffer a great loss in a short time, and then bankrupt. This shows the necessity of underwriting.

Basically, the "select" in the section title arises from the underwriting process, and when we say an individual is "selected" at age $$x$$, he is underwritten at age $$x$$ (so the most recent information about the individual is known). Since there are some new information about the individual when he is underwritten (or selected), we will expect there is some update in his survival distribution, and therefore his probabilities related to mortality will change as well. Because of this, we need to have some changes in the actuarial notations, depending on the selection age.

In such actuarial notations, we usually add square brackets around the age at selection, and the numbers in the subscript change accordingly. For example, $$q_{25}$$ becomes $$q_{[25]}$$ if the selection age is 25, $$q_{[12]+13}$$ if the selection age is 12.

Since when a person is underwritten long time ago, he may have poorer health condition (e.g. getting older having some new diseases) from the time at which he is underwritten to now, we will intuitively expect that the longer time passed from the time at which a person aged $$x$$ is underwritten, the more likely that person will be die in the coming year. That is, $$q_{[0]+x}>q_{[1]+x}>\dotsb>q_{[x-1]+1}>q_{[x]}$$.

The impact on the survival distribution from the selection age may decrease when the time passed from the selection is longer. Beyond a certain time period, say $$r$$ years, the "$$q$$"'s at the same attained age (i.e. selection age plus the time passed from it to now) but different selection ages will be very close. In other words, $$q_{[x-j]+r+j}\approx q_{[x]+r},\quad 0\le j\le x$$ (the condition on $$j$$ is to ensure that the selection age at LHS $$x-j\ge 0$$). Such $$r$$ years is called the. Because the "$$q$$"'s mentioned above will be very close, all such "$$q$$"'s will only be written as $$q_{x+r}$$, without any square brackets (since the effect of selection is basically "gone", and so the square brackets are gone). For example, if the selection period is 2 years, then $$q_{[10]+2}$$ and $$q_{[5]+7}$$ will be both written as $$q_{12}$$ simply. However, we will not write $$q_{[15]+1}$$ and $$q_{[16]}$$ as $$q_{16}$$, since these two "$$q$$"'s are "quite different" because of the selection impact is still "quite large".

The following are some terms related to life table.
 * An is a life table in which the functions are only given for.
 * A is a life table in which some function involves the age at selection.
 * An is usually appended to a select table as a last column to reflect the setting of select table. The combination of a select table and an ultimate table in such a way is called a  table.

For example, an excerpt of select-and-ultimate table with select period 2 years may look like: The last column is the ultimate table. We can observe that we do not need additional columns for $$q_{x+3},q_{x+4}$$ etc., since we can already get such values in the "$$q_{x+2}$$" value in a different row (with different $$x$$).

Given a select-and-ultimate table, we can do various calculations based on it.