Fundamental Actuarial Mathematics/Present Value Random Variables for Long-Term Insurance Coverages

Learning objectives
The Candidate will be able to perform calculations on the present value random variables associated with benefits and expenses for long term insurance coverages.

Learning outcomes
The Candidate will be able to:
 * 1) Identify the present value random variables associated with life insurance, endowment, and annuity payments for single lives, based on annual, 1/m-thly and continuous payment frequency.
 * 2) Calculate probabilities, means, variances and covariances for the random variables in 1., using fractional age or claims acceleration approximations where appropriate.
 * 3) Understand the relationships between the insurance, endowment, and annuity present value random variables in 1., and between their expected values.
 * 4) Calculate the effect of changes in underlying assumptions (e.g., mortality and interest).
 * 5) Identify and apply standard actuarial notation for the expected values of the random variables in 1.

Introduction to life insurance
In a life insurance policy, when the insured dies at a time within the coverage of the life insurance, the insurance company needs to pay a benefit for the insured at a certain time. The and the  of benefit payment depends on the terms stated in the contract, and the type of the life insurance. In this chapter, we will discuss some models for the payments of different types of life insurance, and perform some calculations related to them. Also, we will introduce some related actuarial notations.

Insurances payable at the moment of death
In this section, we will model some types of insurances payable at the moment of death, which is associated with the random variable $$T_x$$, and all insurances mentioned in this section are payable at the moment of death unless stated otherwise. In practice, the insurance is payable at the moment of death, since it is infeasible. However, with this theoretical assumption, some calculations can be more convenient.

Before modeling the insurances, we need to introduce a fundamental random variable:.

In the models discussed in, they differ in only the definitions of $$b_T$$ and $$v_T$$, and the model for $$n$$-year term life insurance is the basic one, and other models are just some modifications on this model. Because of this,.

We often want to know the of the present-value random variable so that we would somehow predict how much benefit will be paid, in the present value term. As a result, we have a special name for such expected value:

For simplicity, we will assume the benefit amount is one in the following models unless otherwise specified. If we want to change the benefit amount to other values, then we can just multiply it accordingly. For the insurances for which the benefit amount is varying, we can also change the benefit amounts "together" through appropriate multiplication.

n-year term life insurance
For $$n$$-year term life insurance, we have $$ b_T=\begin{cases}1,& T\le n;\\ 0,&T>n,\end{cases} $$ and $$ v_T=v^T,\quad T\ge 0 $$ ($$v$$ is the discount factor which is $$\frac{1}{1+i}$$, in which $$i$$ is the (annual) effective interest rate). Hence, $$Z=\begin{cases}v^T,&T\le n\\ 0,&T>n\end{cases}$$.

In this case, the actuarial present value, denoted by $$\bar A^1_{x:\overline n|}$$, is $$\mathbb E[Z]=\int_{0}^{n}v^tf_x(t)\,dt=\int_{0}^{n}v^t{}_tp_x \mu_{x+t}\,dt$$. Here, we use the result $$f_x(t)={}_tp_x\mu_{x+t}$$ from the ../Mortality Models chapter.

The notation for the APV may seem weird at first, but after knowing the meaning of each of symbols involved, the notation will make more sense. Indeed, the other notations for the APV in other types of life insurance are constructed in a similar way. The meaning of the symbols involved is as follows:
 * $$A$$ means "a life insurance".
 * $$\bar{}$$ (on top of $$A$$) means this APV is "in a continuous manner" (since the benefit is paid of death, the APV is "somehow continuous").
 * $$x$$ means the insured is aged $$x$$ at the time of policy issue.
 * $$:\overline n|$$ means the term of the life insurance is $$n$$ years.
 * $$1$$ on top of $$x$$ (and "before" $$:\overline n|$$) means the benefit of amount 1 is paid to the insured (aged $$x$$ at the time of policy issue), if he dies $$n$$ years passed from the time of policy issue.

Whole life insurance
For whole life insurance, it can be interpreted as a "$$\infty$$-year term life insurance", and so payment will be eventually provided since no one lives for eternity. To be more precise, its definition is as follows:

For whole life insurance, we have $$b_T=1,\quad t\ge 0$$, $$v_T=v^T,\quad T\ge 0$$, and thus $$Z=v^T,\quad t\ge 0$$. The APV, denoted by $$\bar A_x$$, is $$\int_{0}^{\infty}v^t{}_tp_x\mu_{x+t}\,dt$$.

In the notation, we omit the "$$1$$" and "$$:\overline n|$$" appearing in the notation of APV for $$n$$-year term life insurance., since for whole life insurnace, the "term" does not exist (or is "infinite"). Also, the benefit must be paid eventually, and so there is no need to emphasize the benefit payment, unlike above, where the benefit may or may not be paid, depending on how long the insured live.

n-year pure endowment
The $$n$$-year pure endowment is similar to the $$n$$-year term life insurance in some sense, but the in the $$n$$-year term life insurnace is replaced by the  (i.e. benefit is paid when the insured survives for, but not dies within, $$n$$ years) in the $$n$$-year pure endowment.

For $$n$$-year pure endowment, we have $$b_T=\begin{cases}0,&T\le n;\\ 1,&T>n\end{cases}$$, $$v_T=v^n,\quad T\ge 0$$. Hence, $$Z=\begin{cases}0,&T \le n;\\ v^n,&T>n.\end{cases}$$. The APV, denoted by $$A^{\;\;1}_{x:\overline n|}$$, is $$\int_{n}^{\infty}v^nf_x(t)\,dt=v^n\int_{n}^{\infty}f_x(t)\,dt=v^n\mathbb P(T_x>n)=v^n{}_np_x$$.

Since this notation may look a bit clumsy, there is an alternative notation: $$_n E_x$$.

In the notation, there is no $$\bar{}$$ on top of $$A$$, since the benefit is paid at a fixed timing, so the APV is "not quite in a continuous manner". Also, the "$$1$$" is placed on top of "$$:\overline n|$$", since the benefit (if exists) is paid the end of $$n$$th year (or time $$n$$).

n-year endowment insurance
The $$n$$-year endowment insurance is not endowment. Instead, it is a mixture between $$n$$-year term life insurance and $$n$$-year endowment. That is, both death and survival benefits exist. Because of this, this type of insurance is similar to to whole life insurance, in the sense that benefit must be paid.

For $$n$$-year endowment insurance, we have $$b_T=1,\quad T\ge 0$$, $$v_T=\begin{cases}v^T,&T\le n;\\ v^n,&T>n\end{cases}$$, and thus $$Z=\begin{cases}v^T,& T\le n;\\ v^n,&T>n.\end{cases}$$ We can observe that $$Z$$ in this case is the sum of $$Z$$ for $$n$$-year term life insurance and $$Z$$ for $$n$$-year pure endowment. It follows that the APV is also the sum of the two corresponding APV's, that is, the APV for $$n$$-year endowment insurance is $$\bar A_{x:\overline n|}^1+A_{x:\overline n|}^{\;\;1}$$. Such APV is denoted by $$\bar A_{x:\overline n|}$$.

In the notation, we can see that there is a $$\bar{}$$ on top of $$A$$, since the benefit may be paid at the moment of the death of the insured, so the APV is somehow "in a continuous manner". Also, the "$$1$$" is omitted, with the same reason for whole life insurance: the benefit must be paid.

m-year deferred whole life insurance
As suggested by its name, this type of insurance is a deferred whole life insurance, i.e. the "start" of the insurance takes place in some years after the policy issue. To be more precise, we have the following definition.

For $$m$$-year deferred insurance, we have $$b_T=\begin{cases}0,&T\le m;\\1,&T>m\end{cases}$$, $$v_T=v^T,\quad T\ge 0$$ , and thus $$Z=\begin{cases}0,&T\le m;\\v^T,&T>m.\end{cases}$$

The APV, denoted by $$_{m|}\bar A_x$$, is $$\int_{m}^{\infty}v^t{}_tp_x\mu_{x+t}\,dt$$, which is similar to the one for whole life insurance, except that the lower bound of the integral is replaced by $$m$$. The "$$m|$$" in the notation refers to deferring $$m$$ years, and since the insurance does not have a term, "$$:\overline n|$$" is omitted.

m-year deferred n-year term life insurance
In a similar manner, we can also defer a $$n$$-year term life insurance.

For such insurance, we have $$b_T=\begin{cases}0,& T\le m;\\1,& mm+n\end{cases}$$, $$v_T=v^T,\quad T\ge 0$$. Thus, $$Z=\begin{cases}0,& T\le m;\\v^T,&mm+n.\end{cases}$$

The APV, denoted by $$_{m|n}\bar A_x$$, is $$\int_{m}^{m+n}v^t{}_tp_x\mu_{x+t}\,dt$$, which is similar to the one for the $$n$$-year term life insurance, except that both the lower and upper bound is added by $$m$$. In the notation, $$m|n$$ means that $$m$$ years are deferred and the term is $$n$$ years after the deferral, i.e. the insurance lasts for $$n$$ years after deferred by $$m$$ years. This has the similar meaning as the "$$m|n$$" in "$$_{m|n}q_x$$". Indeed, this is how "$$m|n$$" means in actuarial notations.

Annually increasing whole life insurance
Starting from this section, we will discuss the life insurances for which the benefit amount is varying.

For the annually increasing whole life insurance, we have $$b_T=\lfloor T\rfloor +1,\quad T\ge 0$$ , $$v_T=v^T,\quad T\ge 0$$, and thus $$Z=(\lfloor T\rfloor +1)v^T,\quad T\ge 0$$. Graphically, the value of benefit function $$b_T$$ is illustrated below: b_T =       1    2     3 /\   /\    /\           /  \  /  \  /  \          /    \/    \/    \        --*-*-*-*-          0     1     2     3  ... The APV, denoted by $$(I\bar A)_x$$, is $$\int_{0}^{\infty}(\lfloor t\rfloor+1) v^t{}_tp_x\mu_{x+t}\,dt$$. In practice, since it is difficult to integrate "$$\lfloor t\rfloor$$", we will split the integration interval to $$[0,1),[1,2),\dotsc$$, and multiple integrals are created from this, so that in each of these intervals, $$\lfloor t\rfloor$$ equals an integer only. That is, we split the integral like this: $$\int_{0}^{\infty}(\lfloor t\rfloor+1) v^t{}_tp_x\mu_{x+t}\,dt=\int_{0}^{1}v^t{}_tp_x\mu_{x+t}\,dt+\int_{1}^{2}2v^t{}_tp_x\mu_{x+t}\,dt+\int_{2}^{3}3v^t{}_tp_x\mu_{x+t}\,dt+\dotsb$$, which is an infinite sum.

In the notation, the "$$I$$" stands for "increasing" (annually), and we add a bracket so that the notation is not to be confused with $$I\cdot \bar A_x$$.

Annually increasing n-year term life insurance
When we "strip off" the benefit starting from year $$n+1$$ and onward, we obtain an annually increasing $$n$$-year insurance. To be more precise, we have the following definition:

For this insurance, we have a slightly different benefit function: Thus, $$Z=\begin{cases}(\lfloor T\rfloor+1) v^T,& T\le n;\\ 0,& T>n.\end{cases}$$
 * $$b_T=\begin{cases}\lfloor T\rfloor+1,& T\le n;\\ 0,& T>n\end{cases}$$ and
 * $$v_T=v^T,\quad t\ge 0$$.

Graphically, the benefit function looks like: b_T =       1    2     3    ... n           /\    /\    /\            / \ / \  /  \  /  \          /   \          /    \/    \/    \        /     \        --*-*-*-*---*--*-          0     1     2     3  ... n-1     n

The APV, denoted by $$(I\bar A)_{x:\overline n|}^1=\int_{0}^{n}(\lfloor t\rfloor+1) v^t{}_tp_x\mu_{x+t}\,dt$$. We have "$$x:\overline n|$$" instead of $$x$$ in the notation, since this is a term life insurance.

Annually decreasing n-year term life insurance
Similarly, we have annually decreasing $$n$$-year insurance, but the benefit does not "start" at 1.

The benefit function is $$b_T=\begin{cases}n-\lfloor T\rfloor,& T\le n;\\ 0,& T>n.\end{cases}$$, the discount function is $$v_T=v^T,\quad T\ge 0$$. Thus, $$Z=\begin{cases}(n-\lfloor T\rfloor)v^T,& T\le n;\\ 0,& T>n.\end{cases}$$ Graphically, the benefit function looks like: b_T =       n    n-1   n-2  ... 1           /\    /\    /\            / \           /  \  /  \  /  \          /   \          /    \/    \/    \        /     \        --*-*-*-*---*--*-          0     1     2     3  ... n-1     n The APV, denoted by $$(D\bar A)_{x:\overline n|}^1$$, equals $$\int_{0}^{n}(n-\lfloor t\rfloor)v^t{}_tp_x\mu_{x+t}\,dt$$. We have "$$D$$" instead of "$$I$$" in the notation to reflect that the insurance is rather than increasing.

m-thly increasing whole life insurance
This is a "more frequent" version of the annually increasing whole life insurance.

The benefit function is $$b_T=\frac{\lfloor mT\rfloor+1}{m},\quad T\ge 0$$, the discount function is $$v_T=v^T,\quad T\ge 0$$, and thus $$Z=\frac{\lfloor mT\rfloor+1}{m}v^T,\quad T\ge 0$$. Graphically, the benefit function looks like: b_T =     1/m   2/m   3/m /\   /\    /\           /  \  /  \  /  \          /    \/    \/    \        --*-*-*-*-          0    1/m   2/m   3/m ... The APV, denoted by $$(I^{(m)}\bar A)_x$$, is $$\int_{0}^{\infty}\frac{\lfloor mt\rfloor+1}{m}v^t{}_tp_x\mu_{x+t}\,dt$$. The "$$I^{(m)}$$" in the notation reflects that the insurance is increasing.

Continuously increasing whole life insurance
We can further increasing the frequency and obtain a "continuous" version of the annually increasing whole life insurance. This version is a bit theoretical, and there may not be such kind of insurance in real life. However, the calculation is simpler using this continuous model, as we will see.

Its APV, denoted by $$(\bar I\bar A)_x$$, is $$\int_{0}^{\infty}tv^t{}_t p_x\mu_{x+t}\,dt$$. There is a "$$\bar{}$$" in the notation, since the increase is continuous.
 * The benefit function is $$b_T=T,\quad T\ge 0$$, and
 * the discount function is $$v_T=v^T,\quad T\ge 0$$, and
 * thus, $$Z=Tv^T,\quad T\ge 0$$.

m-thly and continuously increasing n-year term life insurance
We can modify slightly $$m$$-thly/continuously increasing whole life insurance to $$m$$thly/continuously increasing $$n$$-year term life insurance, which is similar to the previous case for modifying annually increasing whole life insurance to annually increasing $$n$$-year term life insurance.

Despite their definitions are quite similar to the whole life insurance counterparts, we still include their definition as follows for completeness.

For $$m$$-thly increasing $$n$$-year term life insurance, the benefit function is $$b_T=\begin{cases}\frac{\lfloor mT\rfloor+1}{m},& T\le n;\\0,&T>n\end{cases}$$, the discount function is $$v_T=v^T,\quad T\ge 0$$, and thus $$Z=\begin{cases}\frac{\lfloor mT\rfloor+1}{m}v^T,& T\le n;\\0,&T>n.\end{cases}$$ Hence, the APV, denoted by $$(I^{(m)}\bar A)_{x:\overline n|}$$, is $$\int_{0}^{n}\frac{\lfloor mt\rfloor+1}{m}v^t{}_t p_x\mu_{x+t}\,dt$$ ("$$:\overline n|$$" is added to the notation to represent the $$n$$-year term).

For continuously increasing $$n$$-year whole life insurance, the benefit function is $$b_T=\begin{cases}T,& T\le n;\\ 0,& T>n\\\end{cases}$$, the discount function is $$v_T=v^T,\quad T\ge 0$$. Thus, $$Z=\begin{cases}Tv^T,&T\le n;\\0,&T>n.\end{cases}$$ Hence, the APV, denoted by $$(\bar I\bar A)_{x:\overline n|}$$, is $$\int_{0}^{n}tv^t{}_t p_x\mu_{x+t}\,dt.$$

Of course, we can also have $$m$$-thly/continuously whole/$$n$$-year term life insurance, and their APV notations are constructed in a similar manner. But, to avoid being repetitive, these insurances are omitted here.

Rule of moments
We have discussed how to calculate the actuarial present value, which is $$\mathbb E[Z]$$. How about the higher moments: $$\mathbb E[Z^2],\mathbb E[Z^3],\dotsc$$? We will provide a convenient way (for some type of insurance) below to calculate those moments by using the actuarial present value ($$\mathbb E[Z]$$), namely.

Because of this result, we introduce some special notation for the higher moment: we add a "$$j$$" in the upper left corner of the notation when we are discussing the $$j$$th moment of $$Z$$, rather than actuarial present value of the concerning insurance. For example, we use $$^2 \bar A_x$$ to denote $$\mathbb E[Z^2]$$ in which $$Z$$ is the present-value random variable for the whole life insurance.

Insurance payable at the end of the year of death
In practice, it is uncommon that the benefit is paid of death. Instead, it is more common to pay the benefit at the end of the year of death (or may be in other time unit, e.g. at the end of month of death, etc.). In this case, we can just modify the benefit function and discount function in the previous section slightly, and then we can calculate the actuarial present value under this assumption similarly.

Under this case, the present-value random variable is defined using $$K$$ ($$K=K_x$$) instead of $$T$$. Since the payment is made at the of the year of death, but $$K$$ gives the time at the  of the year of death. Because of this, the input of benefit function and discount function is $$K+1$$, rather than $$K$$. Then, we have the present-value random variable $$Z=b_{K+1}v_{K+1}$$.

For example, when a life aged 50 dies at the middle between age 53 and 54, then benefit is paid at age 54. Graphically, death benefit paid | |                        v v ---*-*-*-*-* 50   51    52     53    54

Basically, the different insurance models under this assumption are quite similar to the corresponding one under previous assumption, except that the present-value random variable $$Z$$ is now rather than continuous (since $$K$$ is discrete), and thus to calculate the actuarial present value (expected value of $$Z$$) here, we need to use instead of integration.

For notations, the notations here are highly similar to the one under previous assumption, except that the $$\bar{}$$ on top of $$A$$ (if exists) is removed, since the APV is now "in a discrete manner". Again, we assume the amount of benefit is one unless otherwise specified.

Let us discuss the model for $$n$$-year term life insurance under this assumption. For other models, we can just build them from this model, in a similar way as in the previous section.

Before the discussion, we need to recall that the pmf $$\mathbb P(K_x=k)={}_kp_x q_{x+k}$$.

For $$n$$-year term life insurance, Thus, the present-value random variable $$Z=\begin{cases}v^{K+1},&K=0,1,\dotsc,n-1;\\0,&K=n,n+1,\dotsc\end{cases}$$
 * the benefit function is $$b_{K+1}=\begin{cases}1,&K=0,1,\dotsc,n-1;\\0,&K=n,n+1,\dotsc\end{cases}$$, and
 * the discount function is $$v_{K+1}=v^{K+1},\quad K=0,1,\dotsc$$.

As a result, the APV is $$\mathbb E[Z]=\sum_{k=0}^{n-1}v^{k+1}\mathbb P(K_x=k)=\sum_{k=0}^{n-1}v^{k+1}{}_kp_x q_{x+k}$$, and as we mentioned, its notation is basically the same as previous one, except that the $$\bar{}$$ is removed. That is, the notation is $$A^1_{x:\overline n|}$$.

For other types of insurance, they are defined in a similar manner. Since you should be now quite familiar with those types of insurance, we simply summarize most of the definitions for different types of insurance (under the discrete assumption) in the following table:

We should notice that there is no continuously increasing whole life/ $$n$$-year term life insurance under this discrete assumption, since the payment is made discretely, and hence the payment cannot be increasing.

For the, it is quite different from the previous deviation for the continuous one. Let us illustrate the deviation for $$m$$-thly increasing whole life insurance as follows. For the $$m$$-thly increasing $$n$$-year term life insurance, the deviation can be done in a similar manner.

Before discussing $$m$$-thly increasing whole life insurance, we should discuss similar insurances where payment is made $$m$$-thly, and the payment is not varying first.

In financial mathematics, you may have encountered similar situations: annuities where payments are made $$m$$-thly, instead of annually. In that case, we can simply calculate the equivalent $$m$$-thly interest rate corresponding to the annual interest rate, and then treat such annuities as ordinary one (i.e. annuities payable annually), so that the calculation process is the same as the ordinary annuities.

However, in the current context, we also incorporate the survival probabilities in the calculation, and we define the time-until-death random variable using as units. Thus, $$K$$ is in the unit of, and we cannot convert $$K$$ to some other "equivalent $$m$$-thly $$K$$" using simple ways. As a result, we need to develop some methods and formulas for the insurance payable $$m$$-thly.

Consider the following diagram: death benefit | |              v  v --*--...--*--*-...-*-...-*-- x ... x+k  x+k+1/m  ...  x+k+j/m  ... x+k+1 In this type of insurance, the benefit is paid at the end of the $$m$$thly interval where death occurs, instead of at the end of the year where death occurs.

We can observe that when death occurs in any $$m$$-thly interval within one year, say from year $$k$$ to $$k+1$$, the value of $$K$$ is the same, which is $$k$$. Because of this, to perform calculations related to this kind of insurance, we need to introduce, say $$J$$, to consider within one year does death occur.

Recall that $$K$$ represents the number of complete years lived before death. Hence, it is natural to define $$J$$ in a similar manner, namely the number of complete lived in the year of death, before death. Using this definition of $$J$$, we know that $$J$$ can only take values of $$0,1,\dotsc,m-1$$.

Using these definitions, we can define, for example, the benefit, discount function, present-value random variable of whole life insurance payable $$m$$-thly as follows: Since there are two random variables involved, the calculation of APV is more complicated. We first consider the joint pmf of $$K$$ and $$J$$: since $$\mathbb P(K=k\text{ and }J=j)=\mathbb P\left(\text{the person dies between time }k+\frac{j}{m}\text{ and }k+\frac{j+1}{m}\right) =\mathbb P\left(k+\frac{j}{m}\le T\le k+\frac{j+1}{m}\right) ={}_{k+\frac{j}{m}}p_x \;_{\frac{1}{m}}q_{x+k+\frac{j}{m}}$$, the joint pmf of $$K$$ and $$J$$ is $$f_{K,J}(k,j)={}_{k+\frac{j}{m}}p_x\;_{\frac{1}{m}}q_{x+k+\frac{j}{m}}$$. It follows by the definition of expectation that the APV is $$A_x^{(m)}=\sum_{k=0}^{\infty}\sum_{j=0}^{m-1}v^{k+\frac{j+1}{m}}{}_{k+\frac{j}{m}}p_x\;_{\frac{1}{m}}q_{x+k+\frac{j}{m}}.$$ Notice that the APV notation has an extra "$$(m)$$" at the upper-right corner of "$$A$$" to represent that this insurance is payable $$m$$-thly.
 * the benefit function is $$1,\quad K=0,1,\dotsc,\text{ and } J=0,1,\dotsc,m-1$$;
 * the discount function is $$v^{K+\frac{J+1}{m}},\quad K=0,1,\dotsc\text{ and }J=0,1,\dotsc,m-1$$;
 * the present-value random variable is $$v^{K+\frac{J+1}{m}},\quad K=0,1,\dotsc,\text{ and }J=0,1,\dotsc,m-1$$.

After understanding the construction of this insurance, we can construct other types of insurance that are also payable $$m$$-thly similarly. Also, their APV notations also have an extra "$$(m)$$" added at the upper-right corner of "$$A$$".

Besides, we can apply this "$$m$$-thly concept" to the frequency of increasing/decreasing payment. To illustrate this, let us consider the. The benefit function is $$K+\frac{J+1}{m},\quad K=0,1,\dotsc,n-1,\text{ and }J=0,1,\dotsc,m-1$$, the discount function is $$v^{K+1},\quad K=0,1,\dotsc$$, and thus the present-value random variable is $$\left(K+\frac{J+1}{m}\right)v^{K+1},\quad K=0,1,\dotsc,n-1,\text{ and }J=0,1,\dotsc,m-1$$. It follows that the APV is $$(I^{(m)}A)_{x:\overline n|}=\sum_{k=0}^{n-1}\sum_{j=0}^{m-1}\left(k+\frac{j+1}{m}\right)v^{k+1}{}_{k+\frac{j}{m}}p_x\;_{\frac{1}{m}}q_{x+k+\frac{j}{m}},$$ where there is an additional "$$(m)$$" at the upper-right corner of "$$I$$" to represent that the increase is $$m$$-thly.

Recursion relations of APV's
For the insurances payable at the end of the year of death, we can develop recursion relations for them. These recursion relations can be useful when we need to calculate some APV's, given some other APV's and related terms only. Also, these recursion relations can give some insights about the relationship between different APV's in some sense. The following proposition includes some recursion relations, where the their deviations contain similar idea.

Actuarial discounting
One of the key ideas in the intuitive explanations to these recursive relations is the concept of. To understand this, let us consider the intuitive explanation to the above recursion relation 2. Graphically, the idea in the recursion looks like "actuarial discounting" *---  |   \   v    \ ||     A x+1:n-1 |-|                      A x:1= vq_x |--|     A x:n ---*-*...**---  x    x+1   ... x+n-1 x+n   age We "split" the $$n$$-year term life insurance issued to a person aged $$x$$ into two parts: as illustrated in the above graph. The corresponding APV's to these two insurances are $$A^1_{x:\overline 1|}$$ and $$A^{\;\;1}_{x+1:\overline {n-1}|}$$ respectively.
 * 1-year term life insurance issued to a person aged $$x$$, and
 * $$n-1$$-year term life insurance issued to a person aged $$x+1$$,

Of course, we cannot simply regard the APV of $$n$$-year term life insurance issued to the life aged $$x$$ as $$A^1_{x:\overline 1|}+A^{\;\;1}_{x+1:\overline {n-1}|}$$ directly, since $$A^{\;\;1}_{x+1:\overline {n-1}|}$$ is not for a life aged $$x$$. Instead, it is for a life aged $$x+1$$. Hence, adjustment needs to be made on this insurance, and the process of doing this adjustment is called "actuarial discounting".

In financial mathematics, discounting means multiplying the, where the effect of interest is considered. On the other hand, in this context, the interest effect, we also have "survival effect", that is, we also need to consider the survival probabilities of a life.

To be more precise, "$$n-1$$-year term life insurance issued to a person aged $$x+1$$" only takes into effect the person aged $$x$$ actually lives to age $$x+1$$. Otherwise, if the person dies within the first year, there is not any "life aged $$x+1$$". Thus, apart from multiplying the discounting factor, we also need to multiply the "survival factor".

In this case, the discounting factor is $$v$$ since 1 year is discounted back, and the "survival factor" is $$p_x$$ since the person aged $$x$$ is required to live for 1 year for the $$n-1$$-year term life insurance to take into effect, and we need to multiply this probability to ensure that this happens (the multiplication of probability is related to the "conditional" concept in probability). Since we need to multiply both $$v$$ and $$p_x$$ in this case, the term $$vp_x$$ is called "actuarial discount factor" (notice that this is actually the APV of a 1-year pure endowment).

Of course, it may not be convincing if we just say multiplying such probability can do this. Thus, let us explain the theory behind this intuition in the following. Recall that $$A_{x:\overline n|}^1=\mathbb E[Z]$$, where $$Z$$ is the present-value random variable for the $$n$$-year term life insurance issued to the person aged $$x$$. Now, we apply a result in probability ("generalized" law of total probability) to get the following equation: $$\mathbb E[Z]=\mathbb E[Z|K=0]\mathbb P(K=0)+\mathbb E[Z|K=1,2,\dotsc]\mathbb P(K=1,2,\dotsc).$$ Notice that the event $$\{K=0\}$$ means the person dies within first year, and the event $$\{K=1,2,\dotsc\}$$ means the person lives to age $$x+1$$. After noticing these, we know that $$\mathbb E[Z|K=0]=A_{x:\overline 1|}^1$$, since given that $$K=0$$, it is impossible to have the benefit after age $$x+1$$, that is, the only possibility for getting benefit is that a life aged $$x$$ dies within first year. So, $$\mathbb E[Z|K=0]$$ is essentially the same as the APV of a 1-year term life insurance issued to a person aged $$x$$. Also, we know that $$\mathbb P(K=0)=q_x$$ since this is the probability for the person aged $$x$$ to die within first year. On the other hand, $$\mathbb E[Z|K=1,2,\dotsc]$$ is essentially the same as the APV of a $$n-1$$-year term life issued to a person aged $$x+1$$, since given that $$K=1,2,\dotsc$$, the benefit can only possibly be made at age $$x+2,x+3,\dotsc,x_n$$. These are time 1,2,...,$$n-1$$ with respect to a life aged $$x+1$$. Since the benefits in this APV are made with respect to a life aged $$x+1$$, to convert this APV to the APV for a person aged $$x$$, we need to discount back the benefit for 1-year, i.e. multiplying $$v$$. Now, consider the probability $$\mathbb P(K=1,2,\dotsc)$$. Since this probability is the probability for the person aged $$x$$ to live for 1 year, i.e. live to age $$x+1$$, we have $$\mathbb P(K=1,2,\dotsc)=p_x$$. Thus, we can obtain the recursion relation $$A_{x:\overline n|}^1=vq_x+vp_x A^{\;\;1}_{x+1:\overline{n-1}|}.$$

In financial mathematics, when we want to discount back $$n$$ years, we multiply the discount factor. However, in this case, when we want to "actuarially" discount back $$n$$ years, we are multiplying $$vp_x$$ to the power $$n$$. The reason for this is due to the "survival part" of the actuarial discount factor: the probability for a person to survive for $$n$$ years is not $$p_x^n$$ (unless constant force assumption is assumed). Instead, the probability is given by $$_n p_x$$. Hence, when we want to "actuarially" discount back $$n$$ years, we are multiplying $$v^n {}_n p_x$$, which is actually the APV of $$n$$-year pure endowment (in other words, the APV of a particular payment made at time $$n$$ can be obtained by "actuarially" discount it back to time 0. This idea will be useful when we discuss life annuities).

Intuitive explanations
Now, we are ready to explain the recursion relations intuitively.


 * recursion relation 2: we first extract 1-year term life insurance ($$vq_x$$) from the $$n$$-year term life insurance ($$A^1_{x:\overline n|}$$). Then, the remaining part of insurance is a $$n-1$$-year term life insurance with respect to $$(x+1)$$ ($$A^{\;\;1}_{x+1:\overline{n-1}|}$$). After that, we actuarially discount by 1 year (multiply $$vp_x$$) the remaining part of insurance..
 * recursion relation 3: notice that we can split the endowment insurance to term life insurance and pure endowment. For the term life insurance part, it follows from relation 2. For the pure endowment part, the pure endowment at RHS ($$_{n-1}E_{x+1}$$) is with respect to $$(x+1)$$, and thus we actuarially discount it by 1 year (multiply $$vp_x$$) to get the pure endowment at LHS ($$_n E_x$$). Thus, we have $$_n E_x=vp_x {}_{n-1}E_{x+1}$$, and the relation follows.
 * recursion relation 4: we consider the $$m$$-year deferred $$n$$-year term life insurance issued to a life aged $$x$$ ($$_{m|n}A_x$$) a life aged $$x+1$$. Then, from this point of view, the APV is $$_{m-1|n} A_{x+1}$$ (since with respect to a life aged $$x+1$$, such insurance is just deferred for $$m-1$$ years, and its term is still $$n$$ years (from age $$x+1+m-1$$ to $$x+m+n$$). After that, we actuarially discount this insurance back 1 year (multiply $$vp_x$$).
 * recursion relation 6: we first extract the "first year part" ($$nvq_x$$) out. Then, the remaining part is an annually decreasing $$n-1$$-year term life insurance with respect to $$(x+1)$$ ($$(DA)_{x+1:\overline{n-1}}^{\;\;1}$$). After that, we actuarially discount the remaining part back 1 year (multiply $$vp_x$$).
 * recursion relation 7: we first extract benefit of 1 from each original benefit to get a whole life insurance (with benefit of 1) ($$A_x$$). Then, the remaining part of insurance is an annually increasing whole life insurance with respect to $$(x+1)$$ ($$(IA)_{x+1}$$). After that, we actuarially discount the remaining part back 1 year (multiply $$vp_x$$).

Now, you may ask that there are some similar recursion relations holds for insurances. The answer is yes, but since the insurances are continuous, the relations may get more complicated. Also, such relations are generally not quite useful, since often there are more than one type of insurance involved in the relation, and also we often are not given the values of APV of insurances directly, which is theoretical.

For instance, we have $$\bar A_x=\bar A_{x:\overline 1|}+vp_x\bar A_{x+1}$$, which is analogous to the recursion relation 1 for the discrete insurance above. But, this relation is not very helpful actually, since we need to have the values of $$\bar A_x$$, $$\bar A_{x:\overline 1|}$$, $$p_x$$ and $$v$$ to get $$\bar A_{x+1}$$, but often we do not have such information.

Indeed, instead of such relationship of APV between integer ages ($$x$$ vs. $$x+1$$), which is applicable to discrete insurances. To have the idea about relationship of APV between different ages for continuous insurances, it is better for us to consider the of the APV.

Recursion relations in continuous case: differential equations
We can also develop "recursion relations" of APV's of insurances. Since the insurances are continuous, we can use to have some "recursions". To understand this more clearly, let us consider the following example.

Incorporating selection ages to insurances
We have discussed the idea of in the previous chapter. We can also use the selection ages for the ages in the insurances. For example, we can have $$A_{[x]},\bar A_{[x]+1:\overline{n}|},{}_n E_{[x]},$$ etc. However, it is more common to calculate the APV's involving selection ages, since the life table is usually given in a discrete form, so it is often only possible to calculate the discrete, rather than continuous APV's. To calculate the values of these APV's, we usually obtain the value of different terms related to the survival probability from the life table, as discussed in the previous chapter.

Connections between continuous and discrete insurances
We have discussed continuous insurances where the benefit is payable at the moment of death ("$$\bar A$$") and discrete insurances where the benefit is payable at the end of year of death ("$$A$$"), and we know that in practice, discrete insurances are more common than continuous insurances. However, in terms of the calculations, the calculations of APV for continuous insurances (which involve integral) are often more convenient and simple than the calculations of APV for discrete insurances (which involve summation), particularly in the case that there is no special "nice" formula to calculate the summation, where we may actually need to calculate the terms one by one and sum them up, which can be quite complicated and tedious.

This motivates us to find some relationships between the APV of continuous and discrete insurances, so that we can use the APV of continuous insurances (which is often more convenient to calculate) to calculate the APV of discrete insurances, which makes the calculation of the APV of discrete insurances simpler.

Of course, the following relationship has some limitations. Otherwise, if the relationship holds for every insurance with no further assumptions made, then we can simply calculate the APV's of continuous insurances and then apply the relationship, and therefore we do not need to develop the formula for APV's of discrete insurances at all!

Indeed, we need to have the UDD assumption (from previous chapter) to develop such relationship for insurances.

Introduction to life annuities
In the previous discussion about life insurance, the benefit payment is. Now, we will discuss another type of product, namely, which is.

For, a series of payments are made (for continuous life annuities), or at equal intervals (for discrete life annuities), where the  may be years, quarters, months, etc., , that is, the life only receives these payments when he survives. Hence, these payments can be regarded as.

Similar to life insurance, may or may not have a term. For life annuity with term, we call such life annuities as (since the payments are only made  for a certain interval, and even if the life still survives after that interval, no payments will be made). On the other hand, for life annuities without term, they are similarly called.

In financial mathematics, we have learnt about various types of annuities, where the payments are made (not contingent on survival). Hence, they can be called. Many concepts and formulas developed there can apply to life annuities. For example, there are life annuity-immediate, lie annuity-due, continuous life annuity, etc.

Even if is not a life insurance itself, it is still quite important in life insurance operations. For example, we will later see that life insurances are often purchased using a life annuity of premium, instead of a single premium. Also, in retirement plans for workers, are often involved, where the payments begin at retirement, and to purchase such annuities, the workers contribute payments, in the form of, while they are actively working. Such annuities can ensure that the workers still have some stable incomes even after retirement.

Continuous life annuities
Let us first discuss some life annuities payable at the rate of 1 per year (level payment), similar to the case of life insurance. Recall that we have discussed continuously paying $$n$$-year annuities-certain in financial mathematics $$\left(\bar a_{\overline n|}=\frac{1-v^n}{\delta}\right)$$. This will be useful for the following discussions related to the.

For life annuities, the present-value random variable is usually denoted by $$Y$$, instead of $$Z$$.

Whole life annuity
The present-value random variable for is thus $$Y=\bar a_{\overline T|},\quad T\ge 0.$$

Hence, the APV of this annuity, denoted by $$\bar a_x$$ (we use "$$a$$" for life annuity, and similarly we add "$$j$$" to the upper-left corner of the notation when we are discussing the $$j$$th moment of $$Y$$), is $$\mathbb E[Y]=\int_{0}^{\infty}\bar a_{\overline t|}{}_t p_x\mu_{x+t}\,dt.$$ Apparently, computing this integral can be somewhat complicated since "$$\bar a_{\overline t|}=\frac{1-v^t}{\delta}$$" is involved in the integral. Fortunately, there is an alternative, and often more convenient formula for calculating this APV, as shown in the following example.

We can apply the idea in the previous exercise to general cases, and such formula derived from the idea is called (CPT). In general, the is given by $$\text{APV}=\int_{0}^{\infty}\underbrace{v^t\times\mathbb P(\text{payments are being made at time }t)}_{\text{actuarial discounting to time zero}}\times\underbrace{(\text{payment rate at time }t)\,dt}_{\text{`amount' of payment at time }t}.$$

From the previous exercise, we have calculated the variance of the present-value random variable $$Y$$, and the calculation process is a bit complicated. Indeed, there are some alternative methods to calculate the variance (and also expectation) of the present-value random variable $$Y$$, by $$Y$$ with $$Z$$ (present-value random variable for life insurance).

n-year temporary life annuity
From this definition, we know that the present-value random variable for $$n$$-year temporary life annuity is $$Y=\begin{cases}\bar a_{\overline T|},& Tn)=\bar a_{\overline n|}{}_n p_x,$$ we can also write the APV as $$ \int_{0}^{n}\bar a_{\overline t|}{}_t p_x\mu_{x+t}\,dt+{}_n p_x\bar a_{\overline n|}. $$ But we often use the for calculating the APV instead: $$\bar a_{x:\overline n|}=\int_{0}^{n}v^t{}_t p_x\,dt.$$ (there are payments in the first $$n$$ years only, so from the current payment technique, the integration region is from $$t=0$$ to $$t=n$$)

For whole life annuity, we have some results that relate it with life insurance. Naturally, one may ask whether there are some similar results for $$n$$-year temporary life annuity. Indeed, there are similar results.

Recall that we have the following relationship between $$\bar a_{\overline n|}$$ and $$\bar s_{\overline n|}$$ in financial mathematics: $$\bar s_{\overline n|}=\bar a_{\overline n|}(1+i)^n=\frac{\bar a_{\overline n|}}{v^n}$$ (we accumulates the present value $$\bar a_{\overline n|}$$ to the end of $$n$$ years (multiply $$(1+i)^n$$) to get the accumulated value at the end of the $$n$$th year $$\bar s_{\overline n|}$$). Naturally, we will expect a similar kind of relationship should hold for $$n$$-year temporary life annuity. This is indeed true, but of course, there is a difference between the in financial mathematics and.

In this context, we call the value of APV at the end of $$n$$th year as (AAV). For the AAV of the $$n$$-year temporary life annuity, it is denoted by $$\bar s_{x:\overline n|}$$, and is given by $$\bar s_{x:\overline n|}=\frac{\bar a_{x:\overline n|}}{v^n{}_n p_x}$$ (Notice that this equation is similar to the above relationship between $$\bar a_{\overline n|}$$ and $$\bar s_{\overline n|}$$). We can interpret the above equation as "actuarially discounting (multiply $$v^n{}_n p_x$$) the AAV ($$\bar s_{x:\overline n|}$$) to time zero gives the APV ($$\bar a_{x:\overline n|}$$)". Graphically, actuarial discounting: v^n _n p_x *-*    /               \    |                 \    v                  | -                 -  a x:n              s x:n 0                 n   time

n-year deferred whole life annuity
Similar to insurances, we have life annuities. The idea involved is basically the same: the "start" of the life annuity starts some time later than the issue of the life annuity. This kind of life annuity is indeed quite common in real life. For instance, we may purchase a deferred whole life annuity before our retirement, so that we can receive income from the life annuity after our retirement, while we survive. This can ensure a stable income even the retirement, which should be quite desirable.

The present-value random variable is hence $$Y=\begin{cases}0,&T<n;\\ v^n\bar a_{\overline{T-n}|},& T\ge n.\\\end{cases}$$ Thus, its actuarial present value, denoted by $${}_{n|}\bar a_x$$, is $$\mathbb E[Y]=\int_{n}^{\infty}v^n\bar a_{\overline{t-n}|}{}_tp_x\mu_{x+t}\,dt.$$ Using current payment technique, we then have $$_{n|}\bar a_x=\int_{n}^{\infty}v^t{}_tp_x\,dt$$ (there are only payments starting from time $$n$$ (the end of $$n$$th year)).

n-year certain and life annuity
Let us discuss a new kind of concept that does not appear for insurances, and applies to annuities. For this kind of annuity, it is a "mixture" of annuity-certain (studied in financial mathematics) and life annuity.

The present-value random variable of the $$n$$-year certain and life annuity is $$ Y=\begin{cases}\bar a_{\overline n|},&T\le n;\\ \bar a_{\overline T|},& T>n.\end{cases} $$ Thus, the APV of this annuity, denoted by $$\bar a_{\overline{x:\overline n|}}$$, is $$ \mathbb E[Y]=\int_{0}^{n}\bar a_{\overline n|}{}_t p_x\mu_{x+t}\,dt +\int_{n}^{\infty}\bar a_{\overline t|}{}_t p_x\mu_{x+t}\,dt =\bar a_{\overline n|}\int_{0}^{n}{}_t p_x\mu_{x+t}\,dt +\int_{n}^{\infty}\bar a_{\overline t|}{}_t p_x\mu_{x+t}\,dt ={}_n q_x\bar a_{\overline n|} +\int_{n}^{\infty}\bar a_{\overline t|}{}_t p_x\mu_{x+t}\,dt. $$ Using current payment technique, we have $$ \bar a_{\overline{x:\overline n|}}=\underbrace{\bar a_{\overline n|}}_{\text{certainly made}}+\int_{n}^{\infty}v^t{}_t p_x\,dt. $$ In particular, there is no survival probability involved for the "certain" part of this annuity: $$\bar a_{\overline n|}$$.

From the definition, we can observe that when $$T>n$$, there are some payments contingent on survival, and it appears that those payments form a. This is indeed the case, as shown in the following exercise.

Just like annuities-certain, there are also other types of continuous annuities, e.g. annually (continuously) increasing (decreasing) continuous annuities. We can also denote their APV's, and calculate their APV's (using current payment technique) similarly: Of course, the payment patterns can also be a "mixture" of different kinds of patterns discussed, something that is irregular, etc. Then, for those life annuities, one can still calculate their APV using the current payment technique, by considering their present-value random variable from their definitions.
 * $$(\bar I\bar a)_x=\int_{0}^{\infty}tv^t{}_t p_x\,dt$$
 * $$(\bar I\bar a)_{x:\overline n|}=\int_{0}^{n}tv^t{}_t p_x\,dt$$
 * $$(I\bar a)_x=\int_{0}^{\infty}\lfloor t+1\rfloor v^t{}_t p_x\,dt$$
 * $$(I\bar a)_{x:\overline n|}=\int_{0}^{n}\lfloor t+1\rfloor v^t{}_t p_x\,dt$$
 * $$(\bar D\bar a)_{x:\overline n|}=\int_{0}^{n}(n-t)v^t{}_t p_x\,dt$$
 * $$(D\bar a)_{x:\overline n|}=\int_{0}^{n}(n-\lfloor t\rfloor)v^t{}_t p_x\,dt$$

Discrete life annuities
After discussing continuous life annuities, let us discuss life annuities. Just as the case for annuities-certain in financial mathematics, there are quite many types of discrete life annuities: Again, for the life annuities discussed in this section, the amount of each payment is one unless otherwise specified.
 * $$n$$-year temporary/whole life annuity
 * payable at the beginning/end of the year (due/immediate)
 * payable annually/$$m$$-thly
 * increasing/decreasing annually/$$m$$-thly

In general, the present-value random variable of the discrete life annuities is a function of $$K$$, say $$y(K)$$. Then, their actuarial present values have the form of $$\sum_{k}^{}y(k){}_kp_x q_{x+k}\quad (\text{definition}),$$ or using current payment technique, for the annuities with level payments of 1, their actuarial present values have the form of $$ \sum_{k}^{}v^k{}_k p_x. $$ Let us consider some simple cases first.

Whole life annuity-due (annuity-immediate)
For whole life annuity-due, the present-value random variable $$Y=\ddot a_{\overline{K+1}|},\quad K=0,1,\dotsc$$. The APV, denoted by $$\ddot a_{x}$$ is $$\sum_{k=0}^{\infty}\ddot a_{\overline{k+1}|}{}_k p_xq_{x+k}$$ by definition. Using current payment technique, we have $$\ddot a_x=\sum_{k=0}^{\infty}v^k{}_k p_x$$. We can observe that the formula developed by the current payment technique is much simpler.

One may ask that why $$Y$$ is defined to be $$\ddot a_{\overline{K+1}|}$$ instead of $$\ddot a_{\overline K|}$$. This is because the payment is made at the of each year. So, even if the annuitant does not survive the whole year, he will still get the payment at the beginning, so there is a "$$+1$$". To be more precise, suppose the annuitant dies between year $$k$$ and $$k+1$$ (i.e. within year $$k+1$$). Then, $$K=k$$, but the annuitant gets the payment at the beginning of year $$1,2,\dotsc,k+1$$, since he still survives at those time points. Therefore, the annuitant gets $$k+1$$ payments, and hence the present value is $$\ddot a_{\overline{k+1}|}$$.

It may appear that such annuity is advantageous to the annuitant, since the annuitant can get the payment for the whole year at the beginning of that year, even if he does not survive the whole year. In a later section, we will discuss a more "fair" life annuity-due:, which requires the annuitant to refund some part of the payment to the payer of the life annuity if the annuitant does not survive the whole year.

For the derivations of whole life annuity-immediate, see the following exercise.

n-year temporary life annuity-due (annuity-immediate)
Based on the definition, it should not be too hard to derive the present-value random variable and expressions of the APV of these annuities. The details are left to the following exercise.

n-year deferred whole life annuity-due (annuity-immediate)
Again, let's left the derivation of the present-value random variable and expressions of the APV of these annuities to the following exercise.

Life annuities-due (annuities-immediate) with m-thly payments
In practice, the payment frequency of the life annuities may not be annual. Instead, it can be semiannual, quarterly, monthly, etc. Thus, we will develop life annuities-due (annuities-immediate) payable $$m$$-thly here. For the annuities here, we assume the payment is 1 per year, payable in installments of $$1/m$$ at the beginning (or end) of every $$m$$-th of the year. Graphically, Annuity-due: 1/m 1/m   1/m         1/m --*-*-*...*-*- 0    1/m   2/m   ... 1-1/m 1      time Annuity-immediate: 1/m  1/m         1/m   1/m --*-*-*...*-*- 0    1/m   2/m   ... 1-1/m 1      time

We can develop the present-value random variable and APV using similar ideas as in the insurances payable $$m$$-thly: introducing a random variable $$J$$ that represents within one year does death occur. In the following, we will first develop the present-value random variable and APV for whole life with $$m$$-thly payment. Then, we will relate whole life payable $$m$$-thly with whole life  payable $$m$$-thly, so we can use the whole life with $$m$$-thly payment as a  for the calculation related to whole life  with $$m$$-thly payment.

For whole life annuity-due with $$m$$-thly payment, the present-value random variable is $$Y=\ddot a_{\overline {K+\frac{J+1}{m}}\big|}^{(m)}=\sum_{j=0}^{mK+J}\frac{1}{m}v^{j/m}=\frac{1-v^{K+\frac{J+1}{m}}}{d^{(m)}},\quad K=0,1,\dotsc,\text{ and }J=0,1,\dotsc,m-1.$$ Graphically, Annuity-due: death: K=k, J=m-2 | 1/m ...  1/m  1/m   1/m     ... 1/m  v     0         k   k+1/m  k+2/m   ... k+(m-2)/m  k+(m-1)/m  k+1      time It appears that the present-value random variable of this life annuity is quite complicated. Hence, the expression of the actuarial present value by definition will be quite complex, and therefore not useful for the actual calculation.

Thus, we will just use the and the following method to get an expression of the APV, denoted by $$\ddot a_x^{(m)}$$.

For the, we have $$\ddot a_x^{(m)}=\frac{1}{m}\sum_{j=0}^{\infty}v^{j/m}{}_{j/m}p_x.$$ However, the above two expressions can also be be quite difficult to be calculated sometimes. In particular, if we are not given $$A_x^{(m)}$$, it is complicated to calculate $$A_x^{(m)}$$, and then apply the relationship between $$\ddot a_x^{(m)}$$ and $$A_x^{(m)}$$. Also, for the current payment technique, we may the term in the sum may not be "nice", and so there may not be a formula for calculating the sum efficiently. Hence, in the following, we will give a third approach of expressing $$\ddot a_x^{(m)}$$, which relates it with $$\ddot a_x$$,.

Apart from the whole life annuity-due with $$m$$-thly payment, we can define $$n$$-year temporary life annuity-due with $$m$$-thly payment, $$n$$-year deferred whole life annuity-due with $$m$$-thly payment, etc., similarly. Their notations are constructed in a similar manner.

Now, we discuss whole life annuity-immediate with $$m$$-thly payment. As you may expect, the present-value random variable is quite similar: $$Y=a_{\overline{K+\frac{J}{m}\big|}}^{(m)},\quad K=0,1,\dotsc,\text{ and }J=0,1,\dotsc,m-1.$$ Graphically, Annuity-immediate: death: K=k, J=m-2 |    1/m  ....    1/m  1/m   1/m     ... 1/m   v     0     1           k   k+1/m  k+2/m   ... k+(m-2)/m  k+(m-1)/m  k+1      time By current payment technique, the actuarial present value of whole life annuity-immediate with $$m$$-thly payment, denoted by $$a_x^{(m)}$$, is $$a_x^{(m)}=\frac{1}{m}\sum_{j=1}^{\infty}v^{j/m}{}_{j/m} p_x.$$ Thus, we can relate $$a_x^{(m)}$$ to $$\ddot a_x^{(m)}$$, as follows.

Apportionable annuities-due and complete annuities-immediate
Previously, we have mentioned that the life annuities-due are advantageous to the annuitant, and the life annuities-immediate are advantageous to the payer. These cause some "unfairness", and the essential cause of this unfairness is that the annuities payments are, rather than continuous. (We do not have this issue if the payments are made continuously.) Thus, it is natural to address this issue by appropriately "converting" the discrete payments to some "equivalent" payments. This is the key idea for developing the and.

Let us the consider the case for whole life annuities-due (annuities immediate) first. In this case, these two kinds of annuities are addressing the unfairness issues in life annuities-immediate and  life annuities-due.

Let us first consider. Consider the following diagram. $1    |   die v   | -*--*- time k   t    k+1 \   /\    /      \  /  \  /       \/    \/    earned  unearned portion portion Suppose the annuitant receives an annual payment from the life annuity-due at time $$K=k$$ (i.e. at the beginning of year $$k+1$$), and then dies at time $$T=t$$ (within year $$k+1$$). In this case, the annuitant only lives for a portion of the year $$k+1$$, so he should just earn a "portion" of the benefit for the year $$k+1$$. To determine the size of the benefit earned, we assume the payments within each year are earned at a certain annual rate $$r$$ such that $$r\bar a_{\overline 1|}=1$$ (so that each stream of payments has the same present value (i.e. value at the beginning of the year) as the single payment made at the beginning of the year). Thus, in terms of the value at time $$t$$, the annuitant earns $$r\bar s_{\overline{t-k}|}$$ (retrospective), and the remaining $$r\bar a_{\overline{1-t}|}$$ is unearned (prospective). Of course, for the previous years: year 1,2,...,$$k$$, all continuously paid payments in each are earned. Therefore, to summarize, the annuitant can earn the payments at an annual rate $$r$$ continuously until death. Notice that this earning of payment is the same as the earning in whole life continuous annuity. Since the payment rate $$r=\frac{1}{\bar a_{\overline 1|}}=\frac{\delta}{1-v}=\frac{\delta}{d},$$ (recall that $$v=1-d$$ from a result in financial mathematics) the actuarial present value of this, denoted by $$\ddot a_x^{\{1\}}$$ is $$\ddot a_x^{\{1\}}=\frac{\delta}{d}\bar a_x.$$ For the unearned portion, the annuitant should refund to the payer at the moment of death, where the value of the refund payment at that moment (time $$t$$) is $$r\bar a_{\overline{1-t}|}=\frac{\delta}{d}\bar a_{\overline{1-t}|},$$ as suggested above.

We can extend this idea to whole life annuity-due with $$m$$-thly payment. $1/m |  die v   | -*--*- time k   t    k+1/m \   /\    /      \  /  \  /       \/    \/    earned  unearned portion portion In this case, we assume the payments within each $$1/m$$ of year are earned at a certain  rate $$r'$$ such that $$r'\bar a_{1/m}=1/m$$ (so that each stream of payments has the same present value as the single payment of $$1/m$$ made at the beginning). Now it follows that in terms of the value at time $$t$$, the annuitant earns $$r'\bar s_{\overline{t-k}|}$$ (retrospetive), and the remaining $$r'\bar a_{\overline{1/m-t}|}$$ is unearned (prospective). Since $$r'=\frac{1/m}{\bar a_{\overline{1/m}|}}=\frac{\delta/m}{1-v^{1/m}}=\frac{\delta}{m(1-v^{1/m})},$$ and $$\left(1-\frac{d^{(m)}}{m}\right)^{-m}=v^{-1}\implies 1-\frac{d^{(m)}}{m}=v^{1/m}\implies d^{(m)}=m(1-v^{1/m}),$$ we have $$r'=\frac{\delta}{d^{(m)}}.$$ Thus, the actuarial present value of the apportionable annuity-due (for whole life annuity-due with $$m$$-thly payments), denoted by $$\ddot a_x^{\{m\}}$$ is similarly $$\ddot a_x^{\{m\}}=\frac{\delta}{d^{(m)}}\bar a_x.$$

For the, the situation is somehow "reversed". Consider the following diagram. $1/m      $1/m (cannot get) |  die    | v   |     v -*--*- time k   t    k+1/m \   /\    /      \  /  \  /       \/    \/    earned  unearned portion portion We have similar treatments on the payments within each year. We assume the payments within each year are earned continuously at a certain rate $$r$$ such that $$r\bar s_{\overline {1/m}|}=1/m$$ (so that each stream of payments has the same value (i.e. value at the end of the $$m$$th of year) as the single payment made at the end of the $$m$$th of year). Then, in terms of the value at time $$t$$, the annuitant earns $$r\bar s_{\overline{t-k}|}$$ (retrospective), and the remaining $$r\bar a_{\overline{1/m-t}|}$$ is unearned (prospective). Since $$r=\frac{1/m}{\bar s_{\overline{1/m}|}}=\frac{\delta/m}{(1+i)^{1/m}-1}=\frac{\delta}{m((1+i)^{1/m}-1)}=\frac{\delta}{i^{(m)}},$$ the actuarial present value of the (for whole life annuity-immediate with $$m$$-thly payment), denoted by $$\overset{\circ}{a}_x^{(m)}$$ (remark: when $$m=1$$, the "$$(m)$$" is omitted in this case), is $$\overset{\circ}{a}_x^{(m)}=\frac{\delta}{i^{(m)}}\bar a_x.$$ Of course, in this case, the annuitant does not receive any payment for the year of death, and hence is not responsible for "refund". Instead, the payer should be responsible to pay additional payment (the amount of earned payments yet not paid) to the annuitant, at time $$t$$. We call such payment as adjustment payment.

Similarly, we have $$\ddot a_{x:\overline n|}^{\{m\}}$$ and $$\overset{\circ}{a}_{x:\overline n|}^{(m)}$$, which correspond to $$n$$-year temporary life annuities. Using analogous arguments, we can show that (There are no payments after a certain time point for each of the case. So, we divide our arguments by cases. For the case where there are payments, it is similar to the above argument. For the case where there are no payments, the actuarial present value is simply zero.)
 * $$\ddot a_{x:\overline n|}^{\{m\}}=\frac{\delta}{d^{(m)}}\bar a_{x:\overline n|}$$ and
 * $$\overset{\circ}{a}_{x:\overline n|}^{(m)}=\frac{\delta}{i^{(m)}}\bar a_{x:\overline n|}$$.

Recursion relations
Similar to the case for insurances, we can also develop recursion relations for. For life annuities, the is quite important for the development. This is because when we use current payment technique, we can "split" the potential future in some ways, similar to the case for insurances where the of the insurance is split. As a result, we can develop similar recursion relations for life annuities, e.g., (the intuitive explanations is in the brackets)
 * $$\bar a_x=\bar a_{x:\overline n|}+v^n{}_n p_x\bar a_{x+n}$$ (split the potential payments to two parts: payments in first $$n$$ years ($$\bar a_{x:\overline n|}$$) and payments afterward ($$\bar a_{x+n}$$). Then, actuarially discount (multiply $$v^n{}_n p_x$$) the payments afterward to age $$x$$.
 * $$\ddot a_x=\ddot a_{x:\overline n|}+v^n{}_n p_x\ddot a_{x+n}$$ (similar to the first one)
 * $$a^{(m)}_x=a^{(m)}_{x:\overline n|}+v^n{}_n p_x a^{(m)}_{x+n}$$ (similar to the first one)
 * $$\ddot a_x^{\{m\}}=\ddot a_{x:\overline n|}^{\{m\}}+v^n{}_n p_x\ddot a_{x+n}^{\{m\}}$$ (similar to the first one)
 * $$(I\ddot a)_{x:\overline n|}=\ddot a_{x:\overline n|}+vp_x(I\ddot a)_{x+1:\overline n|}$$ ("extract" an unit payment for each of the payments to form an life annuity-due ($$\ddot a_{x:\overline n|}$$), and then the remaining payments form "$$(I\ddot a)_{x+1:\overline n|}$$", and we need to actuarially discount (multiply $$vp_x$$) it back to age $$x$$).

Incorporating selecting ages to life annuities
Of course, we can incorporate selecting ages to life annuities. We just replace some terms related to the survival probability by their values for selection ages. Also, we can similarly put a square bracket in the notation when selecting ages are involved, e.g. $$\bar a_{[x]}$$. The idea involved is quite similar to the case for insurances: we often determine appropriate values for selection ages from life tables.