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 Template:Colored em and the Template:Colored em 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 Template:Colored em 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: Template:Colored em. Template:Colored definition In the models discussed in Template:Colored em, 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, Template:Colored em.

We often want to know the Template:Colored em 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: Template:Colored definition 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

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

In this case, the actuarial present value, denoted by ${\overline{A}}_{x:\bar{n}|}^1$, is $𝔼[Z]=\underset{0}{\overset{n}{\int }}{v}^t{f}_x(t)dt=\underset{0}{\overset{n}{\int }}{v}^t{}_t{p}_x{\mu }_{x+t}dt$. Here, we use the result $f_x(t)={}_t{p}_x{\mu }_{x+t}$ 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".
• $\stackrel{}{¯}$ (on top of $A$) means this APV is "in a continuous manner" (since the benefit is paid Template:Colored em of death, the APV is "somehow continuous").
• $x$ means the insured is aged $x$ at the time of policy issue.
• $:\bar{n}|$ means the term of the life insurance is $n$ years.
• $1$ on top of $x$ (and "before" $:\bar{n}|$) means the benefit of amount 1 is paid to the insured (aged $x$ at the time of policy issue), if he dies Template:Colored em $n$ years passed from the time of policy issue.

Whole life insurance

For whole life insurance, it can be interpreted as a "$\mathrm{\infty }$-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: Template:Colored definition For whole life insurance, we have $b_T=1,t\ge 0$, $v_T=v^T,T\ge 0$, and thus $Z=v^T,t\ge 0$. The APV, denoted by ${\overline{A}}_x$, is $\underset{0}{\overset{\mathrm{\infty }}{\int }}{v}^t{}_t{p}_x{\mu }_{x+t}dt$.

In the notation, we omit the "$1$" and "$:\bar{n}|$" 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. Template:Colored example

n-year pure endowment

The $n$-year pure endowment is similar to the $n$-year term life insurance in some sense, but the Template:Colored em in the $n$-year term life insurnace is replaced by the Template:Colored em (i.e. benefit is paid when the insured survives for, but not dies within, $n$ years) in the $n$-year pure endowment. Template:Colored definition For $n$-year pure endowment, we have $b_T=\{\begin{array}{cc}0,& T\le n;\\ 1,& T>n\end{array}$, $v_T=v^n,T\ge 0$ . Hence, $Z=\{\begin{array}{cc}0,& T\le n;\\ v^n,& T>n.\end{array}$. The APV, denoted by $A_{x:\bar{n}|}^1$, is $\underset{n}{\overset{\mathrm{\infty }}{\int }}{v}^n{f}_x(t)dt=v^n\underset{n}{\overset{\mathrm{\infty }}{\int }}{f}_x(t)dt=v^n\mathbb{P}(T_x>n)=v^n{}_n{p}_x$.

Since this notation may look a bit clumsy, there is an alternative notation: ${}_n{E}_x$.

In the notation, there is no $\stackrel{}{¯}$ 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 "$:\bar{n}|$", since the benefit (if exists) is paid Template:Colored em the end of $n$th year (or time $n$).

n-year endowment insurance

The $n$-year endowment insurance is not Template:Colored em 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. Template:Colored definition For $n$-year endowment insurance, we have $b_T=1,T\ge 0$, $v_T=\{\begin{array}{cc}{v}^T,& T\le n;\\ v^n,& T>n\end{array}$, and thus $Z=\{\begin{array}{cc}{v}^T,& T\le n;\\ v^n,& T>n.\end{array}$ 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 ${\overline{A}}_{x:\bar{n}|}^1+A_{x:\bar{n}|}^1$. Such APV is denoted by ${\overline{A}}_{x:\bar{n}|}$.

In the notation, we can see that there is a $\stackrel{}{¯}$ 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. Template:Colored definition For $m$-year deferred insurance, we have $b_T=\{\begin{array}{cc}0,& T\le m;\\ 1,& T>m\end{array}$, $v_T=v^T,T\ge 0$ , and thus $Z=\{\begin{array}{cc}0,& T\le m;\\ v^T,& T>m.\end{array}$

The APV, denoted by ${}_{m|}{\overline{A}}_x$, is $\underset{m}{\overset{\mathrm{\infty }}{\int }}{v}^t{}_t{p}_x{\mu }_{x+t}dt$, 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, "$:\bar{n}|$" is omitted.

m-year deferred n-year term life insurance

In a similar manner, we can also defer a $n$-year term life insurance. Template:Colored definition Template:Colored remark For such insurance, we have $b_T=\{\begin{array}{cc}0,& T\le m;\\ 1,& m<T\le m+n;\\ 0,& T>m+n\end{array}$, $v_T=v^T,T\ge 0$ . Thus, $Z=\{\begin{array}{cc}0,& T\le m;\\ v^T,& m<T\le m+n;\\ 0,& T>m+n.\end{array}$

The APV, denoted by ${}_{m|n}{\overline{A}}_x$, is $\underset{m}{\overset{m+n}{\int }}{v}^t{}_t{p}_x{\mu }_{x+t}dt$, 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. Template:Colored definition For the annually increasing whole life insurance, we have $b_T=\lfloor T\rfloor +1,T\ge 0$ , $v_T=v^T,T\ge 0$, and thus $Z=(\lfloor T\rfloor +1)v^T,T\ge 0$. Graphically, the value of benefit function $b_T$ is illustrated below:

b_T  =       1    2     3
            /\    /\    /\
           /  \  /  \  /  \
          /    \/    \/    \
        --*-----*-----*-----*-----
          0     1     2     3  ...

The APV, denoted by $(I\overline{A}{)}_x$, is $\underset{0}{\overset{\mathrm{\infty }}{\int }}(\lfloor t\rfloor +1)v^t{}_t{p}_x{\mu }_{x+t}dt$. In practice, since it is difficult to integrate "$\lfloor t\rfloor$" Template:Colored em, we will split the integration interval to $[0,1),[1,2),\dots$, 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: $\underset{0}{\overset{\mathrm{\infty }}{\int }}(\lfloor t\rfloor +1)v^t{}_t{p}_x{\mu }_{x+t}dt=\underset{0}{\overset{1}{\int }}{v}^t{}_t{p}_x{\mu }_{x+t}dt+\underset{1}{\overset{2}{\int }}2v^t{}_t{p}_x{\mu }_{x+t}dt+\underset{2}{\overset{3}{\int }}3v^t{}_t{p}_x{\mu }_{x+t}dt+\cdots$, 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 $I\cdot {\overline{A}}_x$.

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: Template:Colored definition For this insurance, we have a slightly different benefit function:

• $b_T=\{\begin{array}{cc}\lfloor T\rfloor +1,& T\le n;\\ 0,& T>n\end{array}$ and
• $v_T=v^T,t\ge 0$.

Thus, $Z=\{\begin{array}{cc}(\lfloor T\rfloor +1)v^T,& T\le n;\\ 0,& T>n.\end{array}$

Graphically, the benefit function looks like:

b_T  =       1    2     3    ...       n
            /\    /\    /\            / \
           /  \  /  \  /  \          /   \
          /    \/    \/    \        /     \
        --*-----*-----*-----*-------*------*-----
          0     1     2     3  ... n-1     n

The APV, denoted by $(I\overline{A}{)}_{x:\bar{n}|}^1=\underset{0}{\overset{n}{\int }}(\lfloor t\rfloor +1)v^t{}_t{p}_x{\mu }_{x+t}dt$. We have "$x:\bar{n}|$" 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. Template:Colored definition The benefit function is $b_T=\{\begin{array}{cc}n-\lfloor T\rfloor ,& T\le n;\\ 0,& T>n.\end{array}$, the discount function is $v_T=v^T,T\ge 0$. Thus, $Z=\{\begin{array}{cc}(n-\lfloor T\rfloor )v^T,& T\le n;\\ 0,& T>n.\end{array}$ 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 $(D\overline{A}{)}_{x:\bar{n}|}^1$, equals $\underset{0}{\overset{n}{\int }}(n-\lfloor t\rfloor )v^t{}_t{p}_x{\mu }_{x+t}dt$. We have "$D$" instead of "$I$" in the notation to reflect that the insurance is Template:Colored em rather than increasing.

m-thly increasing whole life insurance

This is a "more frequent" version of the annually increasing whole life insurance. Template:Colored definition The benefit function is $b_T=\frac{\lfloor mT\rfloor +1}{m},T\ge 0$, the discount function is $v_T=v^T,T\ge 0$, and thus $Z=\frac{\lfloor mT\rfloor +1}{m}{v}^T,T\ge 0$. 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 $(I^{(m)}\overline{A}{)}_x$, is $\underset{0}{\overset{\mathrm{\infty }}{\int }}\frac{\lfloor mt\rfloor +1}{m}{v}^t{}_t{p}_x{\mu }_{x+t}dt$. The "$I^{(m)}$" in the notation reflects that the insurance is increasing Template:Colored em.

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.

• The benefit function is $b_T=T,T\ge 0$, and
• the discount function is $v_T=v^T,T\ge 0$, and
• thus, $Z=T{v}^T,T\ge 0$.

Its APV, denoted by $(\overline{I}\overline{A}{)}_x$, is $\underset{0}{\overset{\mathrm{\infty }}{\int }}t{v}^t{}_t{p}_x{\mu }_{x+t}dt$. There is a "$\stackrel{}{¯}$" in the notation, since the increase is continuous. Template:Colored example

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

For continuously increasing $n$-year whole life insurance, the benefit function is $b_T=\{\begin{array}{cc}T,& T\le n;\\ 0,& T>n\end{array}$, the discount function is $v_T=v^T,T\ge 0$. Thus, $Z=\{\begin{array}{cc}T{v}^T,& T\le n;\\ 0,& T>n.\end{array}$ Hence, the APV, denoted by $(\overline{I}\overline{A}{)}_{x:\bar{n}|}$, is $\underset{0}{\overset{n}{\int }}t{v}^t{}_t{p}_x{\mu }_{x+t}dt.$

Of course, we can also have $m$-thly/continuously Template:Colored em 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 $𝔼[Z]$. How about the higher moments: $𝔼[Z^2],𝔼[Z^3],\dots$? We will provide a convenient way (for some type of insurance) below to calculate those moments by using the actuarial present value ($𝔼[Z]$), namely Template:Colored em. Template:Colored theorem

Proof. We assume the insurance concerned is a whole life insurance of benefit 1. For other types of insurance (that satisfy the conditions, i.e. the benefit is a single payment of 1), we can just change the integration region below suitably, and we will get the same result.

For whole life insurance, with force of interest ${\delta }_t$, we have $$𝔼[Z^j]=𝔼[v^{jT}]=\underset{0}{\overset{\mathrm{\infty }}{\int }}(v^T{)}^j{}_t{p}_x{\mu }_{x+t}dt=\underset{0}{\overset{\mathrm{\infty }}{\int }}{\left(e^{-\underset{0}{\overset{t}{\int }}{\delta }_{s}ds}\right)}^j{}_t{p}_x{\mu }_{x+t}dt=\underset{0}{\overset{\mathrm{\infty }}{\int }}{e}^{-j\underset{0}{\overset{t}{\int }}{\delta }_{s}ds}{}_t{p}_x{\mu }_{x+t}dt,=\underset{0}{\overset{\mathrm{\infty }}{\int }}{e}^{-\underset{0}{\overset{t}{\int }}j{\delta }_{s}ds}{}_t{p}_x{\mu }_{x+t}dt,$$ which is $𝔼[Z]$ when the force of interest is $j{\delta }_t$.

$◻$

Template:Colored remark 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 $2^{}{\overline{A}}_x$ to denote $𝔼[Z^2]$ in which $Z$ is the present-value random variable for the whole life insurance . Template:Colored example

Summary of insurance types

Summary

Insurance name	benefit function $b_T$	discount function $v_T$	present-value random variable $Z$	APV notation
Whole life insurance	$1,T\ge 0$	$v^T,T\ge 0$	$v^T,T\ge 0$	${\overline{A}}_x$
$n$-year term life insurance	$\{\begin{array}{cc}1,& T\le n;\\ 0,& T>n\end{array}$	$v^T,T\ge 0$	$\{\begin{array}{cc}{v}^T,& T\le n;\\ 0,& T>n\end{array}$	${\overline{A}}_{x:\bar{n}|}^1$
$n$-year pure endowment	$\{\begin{array}{cc}0,& T\le n;\\ 1,& T>n\end{array}$	$v^n,T\ge 0$	$\{\begin{array}{cc}0,& T\le n;\\ v^n,& T>n\end{array}$	${\overline{A}}_{x:\bar{n}|}^1$ or ${}_n{E}_x$
$n$-year endowment insurance	$1,T\ge 0$	$\{\begin{array}{cc}{v}^T,& T\le n;\\ v^n,& T>n\end{array}$	$\{\begin{array}{cc}{v}^T,& T\le n;\\ v^n,& T>n\end{array}$	${\overline{A}}_{x:\bar{n}|}$
$m$-year deferred whole life insurance	$\{\begin{array}{cc}0,& T\le n;\\ 1,& T>n\end{array}$	$v^T,T\ge 0$	$\{\begin{array}{cc}{v}^T,& T\le n;\\ 0,& T>n\end{array}$	${}_{m|}{\overline{A}}_x$
$m$-year deferred $n$-year term life insurance	$\{\begin{array}{cc}0,& T\le m;\\ 1,& m<T\le m+n;\\ 0,& T>m+n,\end{array}$	$v^T,T\ge 0$	$\{\begin{array}{cc}0,& T\ge 0;\\ v^T,& m<T\le m+n;\\ 0,& T>m+n\end{array}$	${}_{m|n}{\overline{A}}_x$
Annually increasing whole life insurance	$\lfloor T\rfloor +1,T\ge 0$	$v^T,T\ge 0$	$(\lfloor T\rfloor +1)v^T,T\ge 0$	$(I\overline{A}{)}_x$
Annually increasing $n$-year term life insurance	$\{\begin{array}{cc}\lfloor T\rfloor +1,& T\le n\\ 0,& T>n\end{array}$	$v^T,T\ge 0$	$\{\begin{array}{cc}(\lfloor T\rfloor +1)v^T,& T\le n\\ 0,& T>n\end{array}$	$(I\overline{A}{)}_{x:\bar{n}|}^1$
Annually decreasing $n$-year term life insurance	$\{\begin{array}{cc}n-\lfloor T\rfloor ,& T\le n\\ 0,& T>n\end{array}$	$v^T,T\ge 0$	$\{\begin{array}{cc}(n-\lfloor T\rfloor )v^T,& T\le n\\ 0,& T>n\end{array}$	$(D\overline{A}{)}_{x:\bar{n}|}^1$
$m$-thly increasing whole life insurance	$\frac{\lfloor mT\rfloor +1}{m},T\ge 0$	$v^T,T\ge 0$	$\frac{\lfloor mT\rfloor +1}{m}{v}^T,T\ge 0$	$(I^{(m)}\overline{A}{)}_x$
$m$-thly increasing $n$-year term life insurance	$\{\begin{array}{cc}\frac{\lfloor mT\rfloor +1}{m},& T\le n;\\ 0,& T>n\end{array}$	$v^T,T\ge 0$	$\{\begin{array}{cc}\frac{\lfloor mT\rfloor +1}{m}{v}^T,& T\le n;\\ 0,& T>n\end{array}$	$(I^{(m)}\overline{A}{)}_{x:\bar{n}|}$
Continuously increasing whole life insurance	$T,T\ge 0$	$v^T,T\ge 0$	$T{v}^T,T\ge 0$	$(\overline{I}\overline{A}{)}_x$
Continuously increasing $n$-year term life insurance	$\{\begin{array}{cc}T,& T\le n;\\ 0,& T>n\end{array}$	$v^T,T\ge 0$	$\{\begin{array}{cc}T{v}^T,& T\le n;\\ 0,& T>n\end{array}$	$(\overline{I}\overline{A}{)}_{x:\bar{n}|}$

Insurance payable at the end of the year of death

In practice, it is uncommon that the benefit is paid Template:Colored em 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 Template:Colored em of the year of death, but $K$ gives the time at the Template:Colored em 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 Template:Colored em rather than continuous (since $K$ is discrete), and thus to calculate the actuarial present value (expected value of $Z$) here, we need to use Template:Colored em instead of integration.

For notations, the notations here are highly similar to the one under previous assumption, except that the $\stackrel{}{¯}$ 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 $\mathbb{P}(K_x=k)={}_k{p}_x{q}_{x+k}$.

For $n$-year term life insurance,

• the benefit function is $b_{K+1}=\{\begin{array}{cc}1,& K=0,1,\dots ,n-1;\\ 0,& K=n,n+1,\dots \end{array}$, and
• the discount function is $v_{K+1}=v^{K+1},K=0,1,\dots$ .

Thus, the present-value random variable $Z=\{\begin{array}{cc}{v}^{K+1},& K=0,1,\dots ,n-1;\\ 0,& K=n,n+1,\dots \end{array}$

As a result, the APV is $𝔼[Z]=\sum _{k=0}^{n-1}{v}^{k+1}\mathbb{P}(K_x=k)=\sum _{k=0}^{n-1}{v}^{k+1}{}_k{p}_x{q}_{x+k}$, and as we mentioned, its notation is basically the same as previous one, except that the $\stackrel{}{¯}$ is removed. That is, the notation is $A_{x:\bar{n}|}^1$. Template:Colored remark

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:

Summary

Insurance name	benefit function $b_{K+1}$	discount function $v_{K+1}$	present-value random variable $Z$	APV notation
Whole life insurance	$1,K=0,1,\dots$	$v^{K+1},K=0,1,\dots$	$v^{K+1},K=0,1,\dots$	$A_x$
$n$-year term life insurance	$\{\begin{array}{cc}1,& K=0,1,\dots ,n-1;\\ 0,& K=n,n+1,\dots \end{array}$	$v^{K+1},K=0,1,\dots$	$\{\begin{array}{cc}{v}^{K+1},& K=0,1,\dots ,n-1;\\ 0,& K=n,n+1,\dots \end{array}$	$A_{x:\bar{n}|}^1$
$n$-year pure endowment	$\{\begin{array}{cc}0,& K=0,1,\dots ,n-1;\\ 1,& K=n,n+1,\dots \end{array}$	$v^n,K=0,1,\dots$	$\{\begin{array}{cc}0,& K=0,1,\dots ,n-1;\\ v^n,& K=n,n+1,\dots \end{array}$	$A_{x:\bar{n}|}^1$ or ${}_n{E}_x$ (the latter one is the same as the continuous one)
$n$-year endowment insurance	$1,K=0,1,\dots$	$\{\begin{array}{cc}{v}^{K+1},& K=0,1,\dots ,n-1;\\ v^n,& K=n,n+1,\dots \end{array}$	$\{\begin{array}{cc}{v}^{K+1},& K=0,1,\dots ,n-1;\\ v^n,& K=n,n+1,\dots \end{array}$	$A_{x:\bar{n}|}$
$m$-year deferred whole life insurance	$\{\begin{array}{cc}0,& K=0,1,\dots ,n-1;\\ 1,& K=n,n+1,\dots \end{array}$	$v^{K+1},K=0,1,\dots$	$\{\begin{array}{cc}{v}^{K+1},& K=0,1,\dots ,n-1;\\ 0,& K=n,n+1,\dots \end{array}$	${}_{m|}{A}_x$
$m$-year deferred $n$-year term life insurance	$\{\begin{array}{cc}0,& K=0,1,\dots ,m-1;\\ 1,& K=m,m+1,\dots ,m+n-1;\\ 0,& K=m+n,m+n+1,\dots \end{array}$	$v^{K+1},K=0,1,\dots$	$\{\begin{array}{cc}0,& K=0,1,\dots ,m-1;\\ v^{K+1},& K=m,m+1,\dots ,m+n-1;\\ 0,& K=m+n,m+n+1,\dots \end{array}$	${}_{m|n}{A}_x$
Annually increasing whole life insurance	$K+1,K=0,1,\dots$	$v^{K+1},K=0,1,\dots$	$(K+1)v^{K+1},K=0,1,\dots$	$(IA{)}_x$
Annually increasing $n$-year term life insurance	$\{\begin{array}{cc}K+1,& K=0,1,\dots ,n-1\\ 0,& K=n,n+1,\dots \end{array}$	$v^{K+1},K=0,1,\dots$	$\{\begin{array}{cc}(K+1)v^{K+1},& K=0,1,\dots ,n-1\\ 0,& K=n,n+1,\dots \end{array}$	$(IA{)}_{x:\bar{n}|}^1$
Annually decreasing $n$-year term life insurance	$\{\begin{array}{cc}n-K,& K=0,1,\dots ,n-1\\ 0,& K=n,n+1,\dots \end{array}$	$v^{K+1},K=0,1,\dots$	$\{\begin{array}{cc}(n-K)v^{K+1},& K=0,1,\dots ,n-1\\ 0,& K=n,n+1,\dots \end{array}$	$(DA{)}_{x:\bar{n}|}^1$

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 Template:Colored em increasing.

For the Template:Colored em, 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 Template:Colored em as units. Thus, $K$ is in the unit of Template:Colored em, 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. Template:Colored remark

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 Template:Colored em, say $J$, to consider Template:Colored em 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 Template:Colored em lived in the year of death, before death. Using this definition of $J$, we know that $J$ can only take values of $0,1,\dots ,m-1$. Template:Colored exercise 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:

• the benefit function is $1,K=0,1,\dots ,\text{ and }J=0,1,\dots ,m-1$;
• the discount function is $v^{K+\frac{J+1}{m}},K=0,1,\dots \text{ and }J=0,1,\dots ,m-1$;
• the present-value random variable is $v^{K+\frac{J+1}{m}},K=0,1,\dots ,\text{ and }J=0,1,\dots ,m-1$.

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 $\mathbb{P}(K=k\text{ and }J=j)=\mathbb{P}(\text{the person dies between time }k+\frac{j}{m}\text{ and }k+\frac{j+1}{m})=\mathbb{P}(k+\frac{j}{m}\le T\le k+\frac{j+1}{m})={}_{k+\frac{j}{m}}{p}_x{}_{\frac{1}{m}}{q}_{x+k+\frac{j}{m}}$, 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 $$A_x^{(m)}=\sum _{k=0}^{\mathrm{\infty }}\sum _{j=0}^{m-1}{v}^{k+\frac{j+1}{m}}{}_{k+\frac{j}{m}}{p}_x{}_{\frac{1}{m}}{q}_{x+k+\frac{j}{m}}.$$ 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. Template:Colored exercise Template:Colored remark 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 Template:Colored em. The benefit function is $K+\frac{J+1}{m},K=0,1,\dots ,n-1,\text{ and }J=0,1,\dots ,m-1$, the discount function is $v^{K+1},K=0,1,\dots$, and thus the present-value random variable is $(K+\frac{J+1}{m})v^{K+1},K=0,1,\dots ,n-1,\text{ and }J=0,1,\dots ,m-1$. It follows that the APV is $$(I^{(m)}A{)}_{x:\bar{n}|}=\sum _{k=0}^{n-1}\sum _{j=0}^{m-1}(k+\frac{j+1}{m})v^{k+1}{}_{k+\frac{j}{m}}{p}_x{}_{\frac{1}{m}}{q}_{x+k+\frac{j}{m}},$$ 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. Template:Colored proposition

Proof. We will only prove 1. and 5., and the remaining relations will be explained in an intuitive manner after this proof.

1.: $$\begin{array}{cc}{A}_x& =\sum _{k=0}^{\mathrm{\infty }}{v}^{k+1}{}_k{p}_x{q}_{x+k}\\ & =v\cdot \underset{=1}{\underbrace{{}_0{p}_x}}\cdot q_x+\sum _{k=1}^{\mathrm{\infty }}{v}^{k+1}{}_k{p}_x{q}_{x+k}& (\text{split the `}k=0\text{' term out, and }{}_0{p}_x=\mathbb{P}(T_x\ge 0)=1)\\ & =v{q}_x+\sum _{k^{\prime }=0}^{\mathrm{\infty }}{v}^{k^{\prime }+2}{}_{k^{\prime }+1}{p}_x{q}_{x+1+k^{\prime }}& (k^{\prime }=k-1)\\ & =v{q}_x+v{p}_x\sum _{k^{\prime }=0}^{\mathrm{\infty }}{v}^{k^{\prime }+1}{}_{k^{\prime }}{p}_{x+1}{q}_{x+1+k^{\prime }}\\ & =v{q}_x+v{p}_x{A}_{x+1}& (\text{definition}).\end{array}$$

5.: $$\begin{array}{cc}(IA{)}_{x:\bar{n}|}^1& =\sum _{k=0}^{n-1}(k+1)v^{k+1}{}_k{p}_x{q}_{x+k}\\ & =\sum _{k=0}^{n-1}{v}^{k+1}{}_k{p}_x{q}_{x+k}+\sum _{k=0}^{n-1}k{v}^{k+1}{}_k{p}_x{q}_{x+k}& (\text{split the sum})\\ & =A_{x:\bar{n}|}^1+v{p}_x\sum _{k=0}^{n-1}k{v}^k{}_{k-1}{p}_{x+1}{q}_{x+k}\\ & =A_{x:\bar{n}|}^1+v{p}_x\sum _{k=1}^{n-1}k{v}^k{}_{k-1}{p}_{x+1}{q}_{x+k}& (\text{`}k=0\text{' term is zero})\\ & =A_{x:\bar{n}|}^1+v{p}_x\sum _{k^{\prime }=0}^{n-2}(k^{\prime }+1)v^{k^{\prime }+1}{}_{k^{\prime }}{p}_{x+1}{q}_{x+k}& (k^{\prime }=k-1)\\ & =A_{x:\bar{n}|}^1+v{p}_x(IA{)}_{x+1:\overline{n-1}|}^1& (\text{definition}).\end{array}$$

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Actuarial discounting

One of the key ideas in the intuitive explanations to these recursive relations is the concept of Template:Colored em. 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:

• 1-year term life insurance issued to a person aged $x$, and
• $n-1$-year term life insurance issued to a person aged $x+1$,

as illustrated in the above graph. The corresponding APV's to these two insurances are $A_{x:\bar{1}|}^1$ and $A_{x+1:\overline{n-1}|}^1$ respectively.

Of course, we cannot simply regard the APV of $n$-year term life insurance issued to the life aged $x$ as $A_{x:\bar{1}|}^1+A_{x+1:\overline{n-1}|}^1$ directly, since $A_{x+1:\overline{n-1}|}^1$ 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 Template:Colored em, where the effect of interest is considered. On the other hand, in this context, Template:Colored em 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 Template:Colored em 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 $v{p}_x$ 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 $A_{x:\bar{n}|}^1=𝔼[Z]$, 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: $$𝔼[Z]=𝔼[Z|K=0]\mathbb{P}(K=0)+𝔼[Z|K=1,2,\dots ]\mathbb{P}(K=1,2,\dots ).$$ Notice that the event $\{K=0\}$ means the person dies within first year, and the event $\{K=1,2,\dots \}$ means the person lives to age $x+1$. After noticing these, we know that $𝔼[Z|K=0]=A_{x:\bar{1}|}^1$, 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, $𝔼[Z|K=0]$ 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, $𝔼[Z|K=1,2,\dots ]$ is essentially the same as the APV of a $n-1$-year term life issued to a person aged $x+1$, since given that $K=1,2,\dots$, the benefit can only possibly be made at age $x+2,x+3,\dots ,x_n$. 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 $\mathbb{P}(K=1,2,\dots )$. 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 $\mathbb{P}(K=1,2,\dots )=p_x$. Thus, we can obtain the recursion relation $$A_{x:\bar{n}|}^1=v{q}_x+v{p}_x{A}_{x+1:\overline{n-1}|}^1.$$

In financial mathematics, when we want to discount back $n$ years, we multiply the discount factor Template:Colored em. However, in this case, when we want to "actuarially" discount back $n$ years, we are Template:Colored em multiplying $v{p}_x$ 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 ($v{q}_x$) from the $n$-year term life insurance ($A_{x:\bar{n}|}^1$). Then, the remaining part of insurance is a $n-1$-year term life insurance with respect to $(x+1)$ ($A_{x+1:\overline{n-1}|}^1$). After that, we actuarially discount by 1 year (multiply $v{p}_x$) 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 $v{p}_x$) to get the pure endowment at LHS (${}_n{E}_x$). Thus, we have ${}_n{E}_x=v{p}_x{}_{n-1}{E}_{x+1}$, and the relation follows.
• recursion relation 4: we consider the $m$-year deferred $n$-year term life insurance issued to a life aged $x$ (${}_{m|n}{A}_x$) Template:Colored em 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 $v{p}_x$).
• recursion relation 6: we first extract the "first year part" ($nv{q}_x$) 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 $v{p}_x$).
• 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 $v{p}_x$).

Now, you may ask that there are some similar recursion relations holds for Template:Colored em 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 Template:Colored em insurances directly, which is theoretical.

For instance, we have ${\overline{A}}_x={\overline{A}}_{x:\bar{1}|}+v{p}_x{\overline{A}}_{x+1}$, 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 ${\overline{A}}_x$, ${\overline{A}}_{x:\bar{1}|}$, $p_x$ and $v$ to get ${\overline{A}}_{x+1}$, 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 Template:Colored em of the APV.

Recursion relations in continuous case: differential equations

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

Incorporating selection ages to insurances

We have discussed the idea of Template:Colored em in the previous chapter. We can also use the selection ages for the ages in the insurances. For example, we can have $A_{[x]},{\overline{A}}_{[x]+1:\bar{n}|},{}_n{E}_{[x]},$ etc. However, it is more common to calculate the Template:Colored em 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. Template:Colored example

Connections between continuous and discrete insurances

We have discussed continuous insurances where the benefit is payable at the moment of death ("$\overline{A}$") 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 Template:Colored em insurances. Template:Colored theorem

Proof. Suppose for the continuous insurances, the benefit function is a function of $K$, so we can write $b_T=b(K)$ where $b$ is a function of $K$, and the discount function is $v^T$. Also, assume UDD. Then, the present-value random variable for the continuous insurance $$Z=b_T{v}_T=b(K)v^T=b(K)v^{K+1}{v}^{S-1}$$ where $T=K+S$. Under UDD assumption, $K$ and $S$ are independent. Hence, the function of $K$ and the function of $S$ are also independent. Thus, applying expectation on $Z$, the APV of the continuous insurance equals $$𝔼[Z]=𝔼[b(K)v^{K+1}]𝔼[v^{S-1}]$$ by independence. Observe that the function $b(K)$ is the benefit function of the corresponding discrete insurance (this is an important observation) . Hence, $𝔼[b(K)v^{K+1}]$ is the APV of the discrete counterpart. Now, it suffices to show that $𝔼[v^{S-1}]=\frac{i}{\delta }$, which is actually true since $$\begin{array}{ccc}𝔼[v^{S-1}]& =\underset{0}{\overset{1}{\int }}{v}^{s-1}\cdot \frac{1}{1-0}ds& (S\text{ follows the uniform distribution on }(0,1))\\ & =\underset{0}{\overset{1}{\int }}(1+i{)}^{1-s}ds\\ & =-\underset{1}{\overset{0}{\int }}(1+i{)}^{u}du& (u=1-s)\\ & =-{\left[\frac{(1+i{)}^u}{\ln (1+i)}\right]}_{u=1}^{u=0}\\ & =-\frac{1}{\delta }+\frac{1+i}{\delta }& (\delta =\ln (1+i))\\ & =\frac{i}{\delta }.\end{array}$$

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Introduction to life annuities

In the previous discussion about life insurance, the benefit payment is Template:Colored em. Now, we will discuss another type of product, namely Template:Colored em, which is Template:Colored em.

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

Similar to life insurance, Template:Colored em may or may not have a term. For life annuity with term, we call such life annuities as Template:Colored em (since the payments are only made Template:Colored em 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 Template:Colored em.

In financial mathematics, we have learnt about various types of annuities, where the payments are Template:Colored em made (not contingent on survival). Hence, they can be called Template:Colored em. 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 Template:Colored em 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, Template:Colored em are often involved, where the payments begin at retirement, and to purchase such annuities, the workers contribute payments, in the form of Template:Colored em, 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 Template:Colored em 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 $({\overline{a}}_{\bar{n}|}=\frac{1-v^n}{\delta })$. This will be useful for the following discussions related to the Template:Colored em.

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

Whole life annuity

Template:Colored definition The present-value random variable for Template:Colored em is thus $$Y={\overline{a}}_{\bar{T}|},T\ge 0.$$ Template:Colored remark Hence, the APV of this annuity, denoted by ${\overline{a}}_x$ (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 $$𝔼[Y]=\underset{0}{\overset{\mathrm{\infty }}{\int }}{\overline{a}}_{\bar{t}|}{}_t{p}_x{\mu }_{x+t}dt.$$ Apparently, computing this integral can be somewhat complicated since "${\overline{a}}_{\bar{t}|}=\frac{1-v^t}{\delta }$" 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. Template:Colored example We can apply the idea in the previous exercise to general cases, and such formula derived from the idea is called Template:Colored em (CPT). In general, the Template:Colored em is given by $$\text{APV}=\underset{0}{\overset{\mathrm{\infty }}{\int }}\underset{\text{actuarial discounting to time zero}}{\underbrace{v^t\times \mathbb{P}(\text{payments are being made at time }t)}}\times \underset{\text{`amount' of payment at time }t}{\underbrace{(\text{payment rate at time }t)dt}}.$$ Template:Colored example 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 Template:Colored em $Y$ with $Z$ (present-value random variable for life insurance). Template:Colored proposition

Proof. We have

• ${\overline{a}}_x=𝔼\left[\frac{1-v^T}{\delta }\right]=\frac{𝔼[1-v^T]}{\delta }=\frac{1-𝔼[v^T]}{\delta }=\frac{1-{\overline{A}}_x}{\delta }$ (${\overline{A}}_x=𝔼[v^T]$ by definition);
• $\mathrm{Var}(Y)=\mathrm{Var}\left(\frac{1-v^T}{\delta }\right)=\frac{\mathrm{Var}(1-v^T)}{{\delta }^2}=\frac{\mathrm{Var}(-v^T)}{{\delta }^2}=\frac{(-1{)}^2\mathrm{Var}(v^T)}{{\delta }^2}=\frac{{\mathrm{\backslash color}}^2{\overline{A}}_x-({\overline{A}}_x{)}^2}{{\delta }^2}$.

$◻$

n-year temporary life annuity

Template:Colored definition From this definition, we know that the present-value random variable for $n$-year temporary life annuity is $$Y=\{\begin{array}{cc}{\overline{a}}_{\bar{T}|},& T<n;\\ {\overline{a}}_{\bar{n}|},& T\ge n.\end{array}$$ It follows that the APV of this annuity, denoted by ${\overline{a}}_{x:\bar{n}|}$, is $$𝔼[Y]=\underset{0}{\overset{n}{\int }}{\overline{a}}_{\bar{t}|}{}_t{p}_x{\mu }_{x+t}dt+\underset{n}{\overset{\mathrm{\infty }}{\int }}{\overline{a}}_{\bar{n}|}{}_t{p}_x{\mu }_{x+t}dt.$$ Since $$\underset{n}{\overset{\mathrm{\infty }}{\int }}{\overline{a}}_{\bar{n}|}{}_t{p}_x{\mu }_{x+t}dt={\overline{a}}_{\bar{n}|}\underset{n}{\overset{\mathrm{\infty }}{\int }}{f}_x(t)dt={\overline{a}}_{\bar{n}|}\mathbb{P}(T>n)={\overline{a}}_{\bar{n}|}{}_n{p}_x,$$ we can also write the APV as $$\underset{0}{\overset{n}{\int }}{\overline{a}}_{\bar{t}|}{}_t{p}_x{\mu }_{x+t}dt+{}_n{p}_x{\overline{a}}_{\bar{n}|}.$$ But we often use the Template:Colored em for calculating the APV instead: $${\overline{a}}_{x:\bar{n}|}=\underset{0}{\overset{n}{\int }}{v}^t{}_t{p}_{x}dt.$$ (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. Template:Colored proposition Template:Colored exercise Recall that we have the following relationship between ${\overline{a}}_{\bar{n}|}$ and ${\overline{s}}_{\bar{n}|}$ in financial mathematics: $${\overline{s}}_{\bar{n}|}={\overline{a}}_{\bar{n}|}(1+i{)}^n=\frac{{\overline{a}}_{\bar{n}|}}{v^n}$$ (we accumulates the present value ${\overline{a}}_{\bar{n}|}$ to the end of $n$ years (multiply $(1+i{)}^n$) to get the accumulated value at the end of the $n$th year ${\overline{s}}_{\bar{n}|}$). 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 Template:Colored em in financial mathematics and Template:Colored em.

In this context, we call the value of APV at the end of $n$th year as Template:Colored em (AAV). For the AAV of the $n$-year temporary life annuity, it is denoted by ${\overline{s}}_{x:\bar{n}|}$, and is given by $${\overline{s}}_{x:\bar{n}|}=\frac{{\overline{a}}_{x:\bar{n}|}}{v^n{}_n{p}_x}$$ (Notice that this equation is similar to the above relationship between ${\overline{a}}_{\bar{n}|}$ and ${\overline{s}}_{\bar{n}|}$). We can interpret the above equation as "actuarially discounting (multiply $v^n{}_n{p}_x$) the AAV (${\overline{s}}_{x:\bar{n}|}$) to time zero gives the APV (${\overline{a}}_{x:\bar{n}|}$)". 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 Template:Colored em 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 Template:Colored em the retirement, which should be quite desirable. Template:Colored definition The present-value random variable is hence $$Y=\{\begin{array}{cc}0,& T<n;\\ v^n{\overline{a}}_{\overline{T-n}|},& T\ge n.\end{array}$$ Thus, its actuarial present value, denoted by ${}_{n|}{\overline{a}}_x$, is $$𝔼[Y]=\underset{n}{\overset{\mathrm{\infty }}{\int }}{v}^n{\overline{a}}_{\overline{t-n}|}{}_t{p}_x{\mu }_{x+t}dt.$$ Using current payment technique, we then have $${}_{n|}{\overline{a}}_x=\underset{n}{\overset{\mathrm{\infty }}{\int }}{v}^t{}_t{p}_{x}dt$$ (there are only payments starting from time $n$ (the end of $n$th year)). Template:Colored example

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. Template:Colored definition The present-value random variable of the $n$-year certain and life annuity is $$Y=\{\begin{array}{cc}{\overline{a}}_{\bar{n}|},& T\le n;\\ {\overline{a}}_{\bar{T}|},& T>n.\end{array}$$ Thus, the APV of this annuity, denoted by ${\overline{a}}_{\overline{x:\stackrel{‾}{n}|}}$, is $$𝔼[Y]=\underset{0}{\overset{n}{\int }}{\overline{a}}_{\bar{n}|}{}_t{p}_x{\mu }_{x+t}dt+\underset{n}{\overset{\mathrm{\infty }}{\int }}{\overline{a}}_{\bar{t}|}{}_t{p}_x{\mu }_{x+t}dt={\overline{a}}_{\bar{n}|}\underset{0}{\overset{n}{\int }}{}_t{p}_x{\mu }_{x+t}dt+\underset{n}{\overset{\mathrm{\infty }}{\int }}{\overline{a}}_{\bar{t}|}{}_t{p}_x{\mu }_{x+t}dt={}_n{q}_x{\overline{a}}_{\bar{n}|}+\underset{n}{\overset{\mathrm{\infty }}{\int }}{\overline{a}}_{\bar{t}|}{}_t{p}_x{\mu }_{x+t}dt.$$ Using current payment technique, we have $${\overline{a}}_{\overline{x:\stackrel{‾}{n}|}}=\underset{\text{certainly made}}{\underbrace{{\overline{a}}_{\bar{n}|}}}+\underset{n}{\overset{\mathrm{\infty }}{\int }}{v}^t{}_t{p}_{x}dt.$$ In particular, there is no survival probability involved for the "certain" part of this annuity: ${\overline{a}}_{\bar{n}|}$.

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

• $(\overline{I}\overline{a}{)}_x=\underset{0}{\overset{\mathrm{\infty }}{\int }}t{v}^t{}_t{p}_{x}dt$
• $(\overline{I}\overline{a}{)}_{x:\bar{n}|}=\underset{0}{\overset{n}{\int }}t{v}^t{}_t{p}_{x}dt$
• $(I\overline{a}{)}_x=\underset{0}{\overset{\mathrm{\infty }}{\int }}\lfloor t+1\rfloor v^t{}_t{p}_{x}dt$
• $(I\overline{a}{)}_{x:\bar{n}|}=\underset{0}{\overset{n}{\int }}\lfloor t+1\rfloor v^t{}_t{p}_{x}dt$
• $(\overline{D}\overline{a}{)}_{x:\bar{n}|}=\underset{0}{\overset{n}{\int }}(n-t)v^t{}_t{p}_{x}dt$
• $(D\overline{a}{)}_{x:\bar{n}|}=\underset{0}{\overset{n}{\int }}(n-\lfloor t\rfloor )v^t{}_t{p}_{x}dt$

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.

Discrete life annuities

After discussing continuous life annuities, let us discuss Template:Colored em life annuities. Just as the case for annuities-certain in financial mathematics, there are quite many types of discrete life annuities:

• $n$-year temporary/whole life annuity
• payable at the beginning/end of the year (due/immediate)
• payable annually/$m$-thly
• increasing/decreasing annually/$m$-thly

Again, for the life annuities discussed in this section, the amount of each payment is one unless otherwise specified.

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 $$\sum _k^{}y(k){}_k{p}_x{q}_{x+k}(\text{definition}),$$ 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)

Template:Colored definition For whole life annuity-due, the present-value random variable $Y={\ddot{a}}_{\overline{K+1}|},K=0,1,\dots$. The APV, denoted by ${\ddot{a}}_x$ is $\sum _{k=0}^{\mathrm{\infty }}{\ddot{a}}_{\overline{k+1}|}{}_k{p}_x{q}_{x+k}$ by definition. Using current payment technique, we have ${\ddot{a}}_x=\sum _{k=0}^{\mathrm{\infty }}{v}^k{}_k{p}_x$. 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 ${\ddot{a}}_{\bar{K}|}$. This is because the payment is made at the Template:Colored em 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 $1,2,\dots ,k+1$, 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: Template:Colored em, 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. Template:Colored example For the derivations of whole life annuity-immediate, see the following exercise. Template:Colored exercise

n-year temporary life annuity-due (annuity-immediate)

Template:Colored definition 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. Template:Colored exercise

n-year deferred whole life annuity-due (annuity-immediate)

Template:Colored definition Again, let's left the derivation of the present-value random variable and expressions of the APV of these annuities to the following exercise. Template:Colored 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 Template:Colored em 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

Template:Colored definition 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 Template:Colored em within one year does death occur. In the following, we will first develop the present-value random variable and APV for whole life Template:Colored em with $m$-thly payment. Then, we will relate whole life Template:Colored em payable $m$-thly with whole life Template:Colored em payable $m$-thly, so we can use the whole life Template:Colored em with $m$-thly payment as a Template:Colored em for the calculation related to whole life Template:Colored em with $m$-thly payment.

For whole life annuity-due with $m$-thly payment, the present-value random variable is $$Y={\ddot{a}}_{\overline{K+\frac{J+1}{m}}|}^{(m)}=\sum _{j=0}^{mK+J}\frac{1}{m}{v}^{j/m}=\frac{1-v^{K+\frac{J+1}{m}}}{d^{(m)}},K=0,1,\dots ,\text{ and }J=0,1,\dots ,m-1.$$ 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 Template:Colored em and the following method to get an expression of the APV, denoted by ${\ddot{a}}_x^{(m)}$. Template:Colored exercise For the Template:Colored em, we have $${\ddot{a}}_x^{(m)}=\frac{1}{m}\sum _{j=0}^{\mathrm{\infty }}{v}^{j/m}{}_{j/m}{p}_x.$$ 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 ${\ddot{a}}_x$, Template:Colored em. Template:Colored example 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. Template:Colored exercise Now, we discuss whole life annuity-immediate with $m$-thly payment. As you may expect, the present-value random variable is quite similar: $$Y=a_{\overline{K+\frac{J}{m}|}}^{(m)},K=0,1,\dots ,\text{ and }J=0,1,\dots ,m-1.$$ 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 $$a_x^{(m)}=\frac{1}{m}\sum _{j=1}^{\mathrm{\infty }}{v}^{j/m}{}_{j/m}{p}_x.$$ Thus, we can relate $a_x^{(m)}$ to ${\ddot{a}}_x^{(m)}$, as follows. Template:Colored example

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 Template:Colored em, 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" Template:Colored em payments. This is the key idea for developing the Template:Colored em and Template:Colored em.

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 Template:Colored em life annuities-immediate and Template:Colored em life annuities-due.

Let us first consider Template:Colored em. 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 Template:Colored em at a certain annual rate $r$ such that $r{\overline{a}}_{\bar{1}|}=1$ (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 $r{\overline{s}}_{\overline{t-k}|}$ (retrospective), and the remaining $r{\overline{a}}_{\overline{1-t}|}$ 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 $$r=\frac{1}{{\overline{a}}_{\bar{1}|}}=\frac{\delta }{1-v}=\frac{\delta }{d},$$ (recall that $v=1-d$ from a result in financial mathematics) the actuarial present value of this Template:Colored em, denoted by ${\ddot{a}}_x^{\{1\}}$ is $${\ddot{a}}_x^{\{1\}}=\frac{\delta }{d}{\overline{a}}_x.$$ 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 $$r{\overline{a}}_{\overline{1-t}|}=\frac{\delta }{d}{\overline{a}}_{\overline{1-t}|},$$ 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 Template:Colored em at a certain Template:Colored em rate $r^{\prime }$ such that $r^{\prime }{\overline{a}}_{1/m}=1/m$ (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 $r^{\prime }{\overline{s}}_{\overline{t-k}|}$ (retrospetive), and the remaining $r^{\prime }{\overline{a}}_{\overline{1/m-t}|}$ is unearned (prospective). Since $$r^{\prime }=\frac{1/m}{{\overline{a}}_{\overline{1/m}|}}=\frac{\delta /m}{1-v^{1/m}}=\frac{\delta }{m(1-v^{1/m})},$$ and $${(1-\frac{d^{(m)}}{m})}^{-m}=v^{-1}⟹1-\frac{d^{(m)}}{m}=v^{1/m}⟹d^{(m)}=m(1-v^{1/m}),$$ we have $$r^{\prime }=\frac{\delta }{d^{(m)}}.$$ 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 $${\ddot{a}}_x^{\{m\}}=\frac{\delta }{d^{(m)}}{\overline{a}}_x.$$ Template:Colored exercise For the Template:Colored em, 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 $r{\overline{s}}_{\overline{1/m}|}=1/m$ (so that each stream of payments has the same Template:Colored em 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 $r{\overline{s}}_{\overline{t-k}|}$ (retrospective), and the remaining $r{\overline{a}}_{\overline{1/m-t}|}$ is unearned (prospective). Since $$r=\frac{1/m}{{\overline{s}}_{\overline{1/m}|}}=\frac{\delta /m}{(1+i{)}^{1/m}-1}=\frac{\delta }{m((1+i{)}^{1/m}-1)}=\frac{\delta }{i^{(m)}},$$ the actuarial present value of the Template:Colored em (for whole life annuity-immediate with $m$-thly payment), denoted by ${\stackrel{\circ }{a}}_x^{(m)}$ (remark: when $m=1$, the "$(m)$" is omitted in this case), is $${\stackrel{\circ }{a}}_x^{(m)}=\frac{\delta }{i^{(m)}}{\overline{a}}_x.$$ 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. Template:Colored exercise Template:Colored example Similarly, we have ${\ddot{a}}_{x:\bar{n}|}^{\{m\}}$ and ${\stackrel{\circ }{a}}_{x:\bar{n}|}^{(m)}$, which correspond to $n$-year temporary life annuities. Using analogous arguments, we can show that

• ${\ddot{a}}_{x:\bar{n}|}^{\{m\}}=\frac{\delta }{d^{(m)}}{\overline{a}}_{x:\bar{n}|}$ and
• ${\stackrel{\circ }{a}}_{x:\bar{n}|}^{(m)}=\frac{\delta }{i^{(m)}}{\overline{a}}_{x:\bar{n}|}$.

(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.)

Recursion relations

Similar to the case for insurances, we can also develop recursion relations for Template:Colored em. For life annuities, the Template:Colored em is quite important for the development. This is because when we use current payment technique, we can "split" the potential future Template:Colored em in some ways, similar to the case for insurances where the Template:Colored em 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)

• ${\overline{a}}_x={\overline{a}}_{x:\bar{n}|}+v^n{}_n{p}_x{\overline{a}}_{x+n}$ (split the potential payments to two parts: payments in first $n$ years (${\overline{a}}_{x:\bar{n}|}$) and payments afterward (${\overline{a}}_{x+n}$). Then, actuarially discount (multiply $v^n{}_n{p}_x$) the payments afterward to age $x$.
• ${\ddot{a}}_x={\ddot{a}}_{x:\bar{n}|}+v^n{}_n{p}_x{\ddot{a}}_{x+n}$ (similar to the first one)
• $a_x^{(m)}=a_{x:\bar{n}|}^{(m)}+v^n{}_n{p}_x{a}_{x+n}^{(m)}$ (similar to the first one)
• ${\ddot{a}}_x^{\{m\}}={\ddot{a}}_{x:\bar{n}|}^{\{m\}}+v^n{}_n{p}_x{\ddot{a}}_{x+n}^{\{m\}}$ (similar to the first one)
• $(I\ddot{a}{)}_{x:\bar{n}|}={\ddot{a}}_{x:\bar{n}|}+v{p}_x(I\ddot{a}{)}_{x+1:\bar{n}|}$ ("extract" an unit payment for each of the payments to form an life annuity-due (${\ddot{a}}_{x:\bar{n}|}$), and then the remaining payments form "$(I\ddot{a}{)}_{x+1:\bar{n}|}$", and we need to actuarially discount (multiply $v{p}_x$) 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. ${\overline{a}}_{[x]}$. The idea involved is quite similar to the case for insurances: we often determine appropriate values for selection ages from life tables. Template:Nav