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 amount and the time 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 not 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: present-value random variable.

Definition. (Present-value random variable) The present-value random variable $Z=b_{T}v_{T}$, in which $b_{T}$ and $v_{T}$ are benefit function and discount function of $T$ ($T=T_{x}$), respectively.

In the models discussed in this section, 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, it is important to understand the $n$-year term life insurance model clearly.

We often want to know the expected value 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:

Definition. (Actuarial present value) The actuarial present value (APV) is $\mathbb {E} [Z]$, the expected value of the present-value random variable.

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

Definition. ($n$-year term life insurnace) A $n$-year term life insurance provides a payment if the insured dies within the $n$-year term, starting from the time of policy issue, and does not provide any payment otherwise.

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

In this case, the actuarial present value, denoted by ${\bar {A}}_{x:{\overline {n}}|}^{1}$, is $\mathbb {E} [Z]=\int _{0}^{n}v^{t}f_{x}(t)\,dt=\int _{0}^{n}v^{t}{}_{t}p_{x}\mu _{x+t}\,dt$. Here, we use the result $f_{x}(t)={}_{t}p_{x}\mu _{x+t}$ from the Fundamental Actuarial Mathematics/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 at the moment 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 before $n$ years passed from the time of policy issue.

Example. Consider a 10-year term life insurance with benefit of 1, issued to a life aged 60. Suppose the pdf of $T_{x}$ is $f_{x}(t)={\frac {1}{100}},\quad 0\leq t\leq 100$, and the annual force of interest (the force of interest in the following will be annual unless otherwise specified) is (a constant) $\delta =0.05$. Then,

• The APV is denoted by ${\bar {A}}_{60:{\overline {10|}}}^{1}$
• The discount function is $v^{t}=e^{-\delta t}=e^{-0.05t}$ (here we use a result from Financial Math FM)
• The value of APV is thus $\int _{0}^{10}e^{-0.05t}\cdot {\frac {1}{100}}\,dt={\frac {1}{100(-0.05)}}[e^{-0.05t}]_{0}^{10}=-{\frac {1}{5}}(e^{-0.5}-1)=0.0787$.

• This means it is "expected" that benefit with present value of 0.0787 will be paid for each insurance issued.
• It may be intuitive that the insurance should also cost 0.0787 when the life aged 60 purchases it immediately after reaching age 60. However, this may not necessarily be the case, since there is also some cost for handling the purchase of insurance, and also the insurance company selling it will also want to make some profits. As a result, the resultant cost may be higher than this APV.

Exercise.

1 What is the value of APV if the benefit of the insurance is 1 million instead of 1?

	0.0787
	787
	78700
	787000

2 Will the APV increase, decrease, or remain unchanged when the annual (constant) force of interest increases to a higher value?

	Increase.
	Decrease.
	Remain unchanged.

3 Will the APV increase, decrease, or remain unchanged when the term of the insurance is longer?

	Increase.
	Decrease.
	Remain unchanged.

Remark.

• If the force of interest is not constant, the calculation will be more complicated.
• Thus, for simplicity, the force of interest will be assumed to be constant unless otherwise specified in the following.

Example. Amy, aged 30, purchases a 20-year term life insurance with benefit 1,000,000 from XYZ life insurance company. Suppose the survival distribution for Amy at aged $x$ is $S_{x}(t)=e^{-0.05t}$, and the force of interest is 7% (constant).

(a) What is the expression for the APV of this life insurance?

(b) Suppose the premium of this policy is the APV of this life insurance plus 1000. Calculate the premium of this policy.

Solution:

(a) $1000000{\bar {A}}_{30:{\overline {20}}|}^{1}$.

(b) First, given the survival function, we know that $_{t}p_{30}=e^{-0.05t}$. Also, the force of mortality is $\mu =0.05$ which is constant (so we are actually assuming constant force of mortality). The premium is

$$1000000{\bar {A}}_{30:{\overline {20}}|}^{1}+1000=1000000\int _{0}^{20}e^{-0.07t}{}_{t}p_{30}\mu _{30+t}\,dt+1000=50000\int _{0}^{20}e^{-0.12t}\,dt+1000={\frac {50000}{-0.12}}[e^{-0.12t}]_{0}^{20}+1000\approx 379867.52.$$

Exercise.

1. Suppose that XYZ life insurance company finds out that Amy is actually a smoker, and thus the survival function for Amy at aged $x$ is adjusted to $S_{x}(t)=e^{-0.08t}$. Following the assumptions made above, calculate the new premium of this policy.

Solution

The new premium is

$$80000\int _{0}^{20}e^{-0.07t}e^{-0.08t}\,dt+1000={\frac {80000}{-0.15}}[e^{-0.15t}]_{0}^{20}+1000\approx 507780.23.$$

Notice that the premium increases quite a lot.

2. Following the assumption in the previous question, calculate the probability that no benefit is paid to Amy within these 20 years.

Solution

No benefit is paid to Amy within these 20 years is equivalent to Amy lives for these 20 years. Thus, the probability is $_{20}p_{30}=e^{-0.05(20)}\approx 0.368$.

3. After 10 years, Amy still survives and she decides to withdraw from the insurance. According to the withdrawal policy of XYZ life insurance company, when the policyholder withdraws from a life insurance policy, the company will return 50% of the premium of the remaining part of the insurance concerned, that is, the premium charged when that policyholder purchases a life insurance in that length (length of the remaining part) at the withdrawal time point. Calculate the return amount in this case, following the assumption in previous questions.

Solution

The return amount is

$$0.5{\bar {A}}_{40:{\overline {10}}|}=0.5\cdot 80000\int _{0}^{10}e^{-0.07t}e^{-0.08t}\,dt={\frac {40000}{-0.15}}[e^{-0.15t}]_{0}^{10}\approx 207165.29.$$

4. Suppose that Amy unfortunately dies 1 year after the withdrawal from insurance, so no benefit is paid to Amy. Calculate the value of the loss of Amy, at the time point where Amy is at age 30, defined by the sum of the value of her net cash outflow, and the value of the benefit she will have if she does not withdraw from the insurance.

Solution

The value of the loss is

$$507780.23-207165.29e^{-0.07(10)}+1000000e^{-0.07(11)}\approx 867918.046.$$

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:

Definition. (Whole life insurance) A whole life insurance provides a payment following the death of the insured (at any time in the future).

For whole life insurance, we have $b_{T}=1,\quad t\geq 0$, $v_{T}=v^{T},\quad T\geq 0$, and thus $Z=v^{T},\quad t\geq 0$. The APV, denoted by ${\bar {A}}_{x}$, is $\int _{0}^{\infty }v^{t}{}_{t}p_{x}\mu _{x+t}\,dt$.

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.

Example. (Formula of ${\bar {A}}_{x}$ with constant force of interest and force of mortality) Given the force of interest is $\delta$ and the force of mortality is $\mu$, ${\bar {A}}_{x}={\frac {\mu }{\mu +\delta }}$.

Proof. When the force of mortality is $\mu$, $_{t}p_{x}={\frac {S_{0}(x+t)}{S_{0}(x)}}={\frac {e^{-\int _{0}^{x+t}\mu \,ds}}{e^{-\int _{0}^{x}\mu \,ds}}}={\frac {e^{-(x+t)\mu }}{e^{-x\mu }}}=e^{-\mu t}$. Thus,

$${\bar {A}}_{x}=\int _{0}^{\infty }v^{t}{}_{t}p_{x}\mu _{x+t}\,dt=\int _{0}^{\infty }e^{-\delta t}{}_{t}p_{x}(\mu )\,dt=\mu \int _{0}^{\infty }e^{-\delta t}e^{-\mu t}\,dt=\mu \int _{0}^{\infty }e^{(-\delta +\mu )t}\,dt={\frac {\mu }{-(\delta +\mu )}}[e^{(-\delta +\mu )t}]_{0}^{\infty }={\frac {\mu }{-(\delta +\mu )}}(0-1)={\frac {\mu }{\mu +\delta }}.$$

$\Box$

Remark.

• We can see that when $\delta =0$, then ${\bar {A}}_{x}=1$ for each nonzero $\mu$ ($\mu$ should not be zero, or else the life will never die (the survival function is always equal to 1)).
• This is intuitive, since for whole life insurance, the benefit must be paid eventually (since the life will die eventually), and if the interest rate is always zero, then the present value of the benefit is always one. Thus, its expected value is also one.

n-year pure endowment

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

Definition. ($n$-year pure endowment) The $n$-year pure endowment provides a payment at the end of the $n$ years when the insured survives for $n$ years from the time of policy issue, and does not provide any payment otherwise.

For $n$-year pure endowment, we have $b_{T}={\begin{cases}0,&T\leq n;\\1,&T>n\end{cases}}$, $v_{T}=v^{n},\quad T\geq 0$ . Hence, $Z={\begin{cases}0,&T\leq n;\\v^{n},&T>n.\end{cases}}$. The APV, denoted by $A_{x:{\overline {n}}|}^{\;\;1}$, is $\int _{n}^{\infty }v^{n}f_{x}(t)\,dt=v^{n}\int _{n}^{\infty }f_{x}(t)\,dt=v^{n}\mathbb {P} (T_{x}>n)=v^{n}{}_{n}p_{x}$.

Since this notation may look a bit clumsy, there is an alternative notation: $_{n}E_{x}$.

In the notation, there is no ${\bar {}}$ 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 at the end of $n$th year (or time $n$).

n-year endowment insurance

The $n$-year endowment insurance is not purely 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.

Definition. ($n$-year endowment insurance) A $n$-year endowment insurance provides an payment either following the death of the insured when the insured dies within $n$ years, or at the end of $n$ years when the insured survives for $n$ years.

For $n$-year endowment insurance, we have $b_{T}=1,\quad T\geq 0$, $v_{T}={\begin{cases}v^{T},&T\leq n;\\v^{n},&T>n\end{cases}}$, and thus $Z={\begin{cases}v^{T},&T\leq n;\\v^{n},&T>n.\end{cases}}$ 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.

Definition. ($m$-year deferred whole life insurance) A $m$-year deferred whole life insurance provides a payment when the insured dies at least $m$ years following policy issue.

For $m$-year deferred insurance, we have $b_{T}={\begin{cases}0,&T\leq m;\\1,&T>m\end{cases}}$, $v_{T}=v^{T},\quad T\geq 0$ , and thus $Z={\begin{cases}0,&T\leq m;\\v^{T},&T>m.\end{cases}}$

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

m-year deferred n-year term life insurance

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

Definition. ($m$-year deferred $n$-year term life insurance) A $m$-year deferred $n$-year term life insurance provides a payment when the insured survives for at least $m$ years following policy issue, and then dies within the coming $n$ years. The insurance does not provide any payment otherwise.

Remark.

• That is, the insurance provides payment when the insured dies between $m$ years and $m+n$ years following policy issue, and does not provide payment otherwise.

For such insurance, we have $b_{T}={\begin{cases}0,&T\leq m;\\1,&m<T\leq m+n;\\0,&T>m+n\end{cases}}$, $v_{T}=v^{T},\quad T\geq 0$ . Thus, $Z={\begin{cases}0,&T\leq m;\\v^{T},&m<T\leq m+n;\\0,&T>m+n.\end{cases}}$

The APV, denoted by $_{m|n}{\bar {A}}_{x}$, is $\int _{m}^{m+n}v^{t}{}_{t}p_{x}\mu _{x+t}\,dt$, 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.

Definition. (Annually increasing whole life insurance) An annually increasing whole life insurance provides payment of 1, 2, ... at the moment of death during the first, second year, etc., respectively.

For the annually increasing whole life insurance, we have $b_{T}=\lfloor T\rfloor +1,\quad T\geq 0$ , $v_{T}=v^{T},\quad T\geq 0$, and thus $Z=(\lfloor T\rfloor +1)v^{T},\quad T\geq 0$. 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 $\int _{0}^{\infty }(\lfloor t\rfloor +1)v^{t}{}_{t}p_{x}\mu _{x+t}\,dt$. In practice, since it is difficult to integrate "$\lfloor t\rfloor$" directly, 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: $\int _{0}^{\infty }(\lfloor t\rfloor +1)v^{t}{}_{t}p_{x}\mu _{x+t}\,dt=\int _{0}^{1}v^{t}{}_{t}p_{x}\mu _{x+t}\,dt+\int _{1}^{2}2v^{t}{}_{t}p_{x}\mu _{x+t}\,dt+\int _{2}^{3}3v^{t}{}_{t}p_{x}\mu _{x+t}\,dt+\dotsb$, 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:

Definition. (Annually increasing $n$-year term life insurance) An annually increasing $n$-year insurance provides payment of 1, 2, ..., $n$ at the moment of death during the first, second, ..., $n$th year respectively.

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

• $b_{T}={\begin{cases}\lfloor T\rfloor +1,&T\leq n;\\0,&T>n\end{cases}}$ and
• $v_{T}=v^{T},\quad t\geq 0$.

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

Graphically, the benefit function looks like:

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

The APV, denoted by $(I{\bar {A}})_{x:{\overline {n}}|}^{1}=\int _{0}^{n}(\lfloor t\rfloor +1)v^{t}{}_{t}p_{x}\mu _{x+t}\,dt$. 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.

Definition. (Annually decreasing $n$-year term life insurance) An annually decreasing $n$-year term life insurance provides payment of $n$, $n-1$, ... , 1 at the moment of death during the first, second year, ..., $n$th year.

The benefit function is $b_{T}={\begin{cases}n-\lfloor T\rfloor ,&T\leq n;\\0,&T>n.\end{cases}}$, the discount function is $v_{T}=v^{T},\quad T\geq 0$. Thus, $Z={\begin{cases}(n-\lfloor T\rfloor )v^{T},&T\leq n;\\0,&T>n.\end{cases}}$ 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 $\int _{0}^{n}(n-\lfloor t\rfloor )v^{t}{}_{t}p_{x}\mu _{x+t}\,dt$. We have "$D$" instead of "$I$" in the notation to reflect that the insurance is decreasing rather than increasing.

m-thly increasing whole life insurance

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

Definition. ($m$-thly increasing whole life insurance) A $m$-thly increasing whole life insurance provides payment of $1/m$, $2/m$, ... at the moment of death during the first $1/m$ of year, second $1/m$ of year, etc., respectively.

The benefit function is $b_{T}={\frac {\lfloor mT\rfloor +1}{m}},\quad T\geq 0$, the discount function is $v_{T}=v^{T},\quad T\geq 0$, and thus $Z={\frac {\lfloor mT\rfloor +1}{m}}v^{T},\quad T\geq 0$. 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 $\int _{0}^{\infty }{\frac {\lfloor mt\rfloor +1}{m}}v^{t}{}_{t}p_{x}\mu _{x+t}\,dt$. The "$I^{(m)}$" in the notation reflects that the insurance is increasing $m$-thly.

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.

Definition. (Continuously increasing whole life insurance) A continuously increasing whole life insurance provides a payment of $t$ when the time of death is $t$.

• The benefit function is $b_{T}=T,\quad T\geq 0$, and
• the discount function is $v_{T}=v^{T},\quad T\geq 0$, and
• thus, $Z=Tv^{T},\quad T\geq 0$.

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.

Example. (Alternative formula for APV of continuously increasing whole life insurance) Show that $({\bar {I}}{\bar {A}})_{x}=\int _{0}^{\infty }{}_{s|}{\bar {A}}_{x}\,ds$.

Proof.

$${\begin{aligned}({\bar {I}}{\bar {A}})_{x}&=\int _{0}^{\infty }{\color {darkgreen}t}v^{t}{}_{t}p_{x}\mu _{x+t}\,dt\\&=\int _{0}^{\infty }v^{t}{}_{t}p_{x}\mu _{x+t}{\color {darkgreen}\int _{0}^{t}\,ds}\,dt\\&=\int _{0}^{\infty }\int _{0}^{t}v^{t}{}_{t}p_{x}\mu _{x+t}\,ds\,dt&{\text{since }}v^{t}{}_{t}p_{x}\mu _{x+t}{\text{ is independent from }}s\\&=\int _{0}^{\infty }{\color {blue}\int _{s}^{\infty }v^{t}{}_{t}p_{x}\mu _{x+t}\,dt}\,ds&{\text{by changing the expression of the integration region}}\\&=\int _{0}^{\infty }{}_{s|}{\bar {A}}_{x}\,ds.\\\end{aligned}}$$

$\Box$

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.

Definition. ($m$-thly increasing $n$-year term life insurance) A $m$-thly increasing $n$-year term life insurance provides payment of $1/m$, $2/m$, ... at the moment of death during the first $1/m$ of year, second $1/m$ of year, ... , $mn$th $1/m$ of year respectively.

Definition. (Continuously increasing $n$-year term life insurance) A continuously increasing $n$-year term life insurance provides a payment of $t$ when the time of death is $t\leq n$.

For $m$-thly increasing $n$-year term life insurance, the benefit function is $b_{T}={\begin{cases}{\frac {\lfloor mT\rfloor +1}{m}},&T\leq n;\\0,&T>n\end{cases}}$, the discount function is $v_{T}=v^{T},\quad T\geq 0$, and thus $Z={\begin{cases}{\frac {\lfloor mT\rfloor +1}{m}}v^{T},&T\leq n;\\0,&T>n.\end{cases}}$ 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 $b_{T}={\begin{cases}T,&T\leq n;\\0,&T>n\\\end{cases}}$, the discount function is $v_{T}=v^{T},\quad T\geq 0$. Thus, $Z={\begin{cases}Tv^{T},&T\leq n;\\0,&T>n.\end{cases}}$ 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 decreasing 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 rule of moments.

Theorem. (Rule of moments) For insurances with one payment with unit amount (i.e. payment of 1), then $\mathbb {E} [Z^{j}]$ when the force of interest is $\delta _{t}$ (a function of $t$) equals $\mathbb {E} [Z]$ when the force of interest is $j\delta _{t}$ ($j=2,3,\dotsc$) .

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

$$\mathbb {E} [Z^{j}]=\mathbb {E} [v^{jT}]=\int _{0}^{\infty }{\big (}v^{T}{\big )}^{j}{}_{t}p_{x}\mu _{x+t}\,dt=\int _{0}^{\infty }\left(e^{-\int _{0}^{t}\delta _{s}\,ds}\right)^{\color {darkgreen}j}{}_{t}p_{x}\mu _{x+t}\,dt=\int _{0}^{\infty }e^{-{\color {darkgreen}j}\int _{0}^{t}\delta _{s}\,ds}{}_{t}p_{x}\mu _{x+t}\,dt,=\int _{0}^{\infty }e^{-\int _{0}^{t}{\color {darkgreen}j}\delta _{s}\,ds}{}_{t}p_{x}\mu _{x+t}\,dt,$$

which is $\mathbb {E} [Z]$ when the force of interest is $j\delta _{t}$.

$\Box$

Remark.

• When the benefit payment is not 1, we cannot apply this rule directly. We can see from the proof that the benefit payment needs to satisfy $b_{T}^{j}=b_{T}$ for this rule to hold. In this case, we have $Z^{j}=(b_{T}v_{T})^{j}=b_{T}v_{T}^{j}$. When $b_{T}$ is nonzero (benefit payment should not be zero for an insurance), when $b_{T}=1$, this is satisfied, and for other (positive) values of $b_{T}$, this is not satisfied.
• For values of benefit payment other than one, we need to perform some procedure so that we can still use the rule of moments, as follows:

• When the benefit payment is $b_{T}$ ($\neq 1$), we have $\mathbb {E} [Z^{j}]=\mathbb {E} [(b_{T}v_{T})^{j}]=b_{T}^{j}\mathbb {E} [v_{T}^{j}]$.
• After that, the random variable inside the expectation is $Z$ with payment 1, and we can use the rule of moments.

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 .

Example. Recall that ${\bar {A}}_{x}={\frac {\mu }{\mu +\delta }}$ under the constant force of interest $\delta$ and constant force of mortality $\mu$. By rule of moments, we have $^{2}{\bar {A}}_{x}={\frac {\mu }{\mu +2\delta }}$.

Exercise.

Let $Z$ be the present-value random variable corresponding to the above actuarial present value. What is $\operatorname {Var} (Z)$?

	${\frac {\mu }{\mu +2\delta }}$
	${\frac {\mu ^{2}}{\mu +2\delta }}$
	${\frac {\mu (\mu +\delta )^{2}-\mu ^{2}(\mu +2\delta )}{(\mu +2\delta )(\mu +\delta )^{2}}}$
	${\frac {\mu (\mu +\delta )^{2}+\mu ^{2}(\mu +2\delta )}{(\mu +2\delta )(\mu +\delta )^{2}}}$

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	$1,\quad T\geq 0$	$v^{T},\quad T\geq 0$	$v^{T},\quad T\geq 0$	${\bar {A}}_{x}$
$n$-year term life insurance	${\begin{cases}1,&T\leq n;\\0,&T>n\\\end{cases}}$	$v^{T},\quad T\geq 0$	${\begin{cases}v^{T},&T\leq n;\\0,&T>n\end{cases}}$	${\bar {A}}_{x:{\overline {n}}|}^{1}$
$n$-year pure endowment	${\begin{cases}0,&T\leq n;\\1,&T>n\\\end{cases}}$	$v^{n},\quad T\geq 0$	${\begin{cases}0,&T\leq n;\\v^{n},&T>n\end{cases}}$	${\bar {A}}_{x:{\overline {n}}|}^{\;\;1}$ or $_{n}E_{x}$
$n$-year endowment insurance	$1,\quad T\geq 0$	${\begin{cases}v^{T},&T\leq n;\\v^{n},&T>n\\\end{cases}}$	${\begin{cases}v^{T},&T\leq n;\\v^{n},&T>n\end{cases}}$	${\bar {A}}_{x:{\overline {n}}|}$
$m$-year deferred whole life insurance	${\begin{cases}0,&T\leq n;\\1,&T>n\\\end{cases}}$	$v^{T},\quad T\geq 0$	${\begin{cases}v^{T},&T\leq n;\\0,&T>n\end{cases}}$	$_{m|}{\bar {A}}_{x}$
$m$-year deferred $n$-year term life insurance	${\begin{cases}0,&T\leq m;\\1,&m<T\leq m+n;\\0,&T>m+n,\\\end{cases}}$	$v^{T},\quad T\geq 0$	${\begin{cases}0,&T\geq 0;\\v^{T},&m<T\leq m+n;\\0,&T>m+n\\\end{cases}}$	$_{m|n}{\bar {A}}_{x}$
Annually increasing whole life insurance	$\lfloor T\rfloor +1,\quad T\geq 0$	$v^{T},\quad T\geq 0$	$(\lfloor T\rfloor +1)v^{T},\quad T\geq 0$	$(I{\bar {A}})_{x}$
Annually increasing $n$-year term life insurance	${\begin{cases}\lfloor T\rfloor +1,&T\leq n\;\\0,&T>n\\\end{cases}}$	$v^{T},\quad T\geq 0$	${\begin{cases}(\lfloor T\rfloor +1)v^{T},&T\leq n\;\\0,&T>n\\\end{cases}}$	$(I{\bar {A}})_{x:{\overline {n}}|}^{1}$
Annually decreasing $n$-year term life insurance	${\begin{cases}n-\lfloor T\rfloor ,&T\leq n\;\\0,&T>n\\\end{cases}}$	$v^{T},\quad T\geq 0$	${\begin{cases}(n-\lfloor T\rfloor )v^{T},&T\leq n\;\\0,&T>n\\\end{cases}}$	$(D{\bar {A}})_{x:{\overline {n}}|}^{1}$
$m$-thly increasing whole life insurance	${\frac {\lfloor mT\rfloor +1}{m}},\quad T\geq 0$	$v^{T},\quad T\geq 0$	${\frac {\lfloor mT\rfloor +1}{m}}v^{T},\quad T\geq 0$	$(I^{(m)}{\bar {A}})_{x}$
$m$-thly increasing $n$-year term life insurance	${\begin{cases}{\frac {\lfloor mT\rfloor +1}{m}},&T\leq n;\\0,&T>n\end{cases}}$	$v^{T},\quad T\geq 0$	${\begin{cases}{\frac {\lfloor mT\rfloor +1}{m}}v^{T},&T\leq n;\\0,&T>n\end{cases}}$	$(I^{(m)}{\bar {A}})_{x:{\overline {n}}|}$
Continuously increasing whole life insurance	$T,\quad T\geq 0$	$v^{T},\quad T\geq 0$	$Tv^{T},\quad T\geq 0$	$({\bar {I}}{\bar {A}})_{x}$
Continuously increasing $n$-year term life insurance	${\begin{cases}T,&T\leq n;\\0,&T>n\end{cases}}$	$v^{T},\quad T\geq 0$	${\begin{cases}Tv^{T},&T\leq n;\\0,&T>n\end{cases}}$	$({\bar {I}}{\bar {A}})_{x:{\overline {n}}|}$

Insurance payable at the end of the year of death

In practice, it is uncommon that the benefit is paid at the moment 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 end of the year of death, but $K$ gives the time at the beginning 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 discrete rather than continuous (since $K$ is discrete), and thus to calculate the actuarial present value (expected value of $Z$) here, we need to use summation 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 $\mathbb {P} (K_{x}=k)={}_{k}p_{x}q_{x+k}$.

For $n$-year term life insurance,

• 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 $v_{K+1}=v^{K+1},\quad K=0,1,\dotsc$ .

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

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

Remark.

• You may notice that a notation for $n$-year pure endowment is $A_{x:{\overline {n}}|}^{\;\;1}$, in which there is no "${\bar {}}$" under previous assumption already. So, if we follow our convention here, then the notation is exactly the same under this discrete assumption.

• Thus, this may cause some ambiguity when only the notation is presented. Hence, in this case, we usually specify clearly which assumption is to be used, and then use the notation, which can help us to differentiate between these two APV's.

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	$1,\quad K=0,1,\dotsc$	$v^{K+1},\quad K=0,1,\dotsc$	$v^{K+1},\quad K=0,1,\dotsc$	$A_{x}$
$n$-year term life insurance	${\begin{cases}1,&K=0,1,\dotsc ,n-1;\\0,&K=n,n+1,\dotsc \\\end{cases}}$	$v^{K+1},\quad K=0,1,\dotsc$	${\begin{cases}v^{K+1},&K=0,1,\dotsc ,n-1;\\0,&K=n,n+1,\dotsc \end{cases}}$	$A_{x:{\overline {n}}|}^{1}$
$n$-year pure endowment	${\begin{cases}0,&K=0,1,\dotsc ,n-1;\\1,&K=n,n+1,\dotsc \\\end{cases}}$	$v^{n},\quad K=0,1,\dotsc$	${\begin{cases}0,&K=0,1,\dotsc ,n-1;\\v^{n},&K=n,n+1,\dotsc \end{cases}}$	$A_{x:{\overline {n}}|}^{\;\;1}$ or $_{n}E_{x}$ (the latter one is the same as the continuous one)
$n$-year endowment insurance	$1,\quad K=0,1,\dotsc$	${\begin{cases}v^{K+1},&K=0,1,\dotsc ,n-1;\\v^{n},&K=n,n+1,\dotsc \\\end{cases}}$	${\begin{cases}v^{K+1},&K=0,1,\dotsc ,n-1;\\v^{n},&K=n,n+1,\dotsc \end{cases}}$	$A_{x:{\overline {n}}|}$
$m$-year deferred whole life insurance	${\begin{cases}0,&K=0,1,\dotsc ,n-1;\\1,&K=n,n+1,\dotsc \\\end{cases}}$	$v^{K+1},\quad K=0,1,\dotsc$	${\begin{cases}v^{K+1},&K=0,1,\dotsc ,n-1;\\0,&K=n,n+1,\dotsc \end{cases}}$	$_{m|}A_{x}$
$m$-year deferred $n$-year term life insurance	${\begin{cases}0,&K=0,1,\dotsc ,m-1;\\1,&K=m,m+1,\dotsc ,m+n-1;\\0,&K=m+n,m+n+1,\dotsc \\\end{cases}}$	$v^{K+1},\quad K=0,1,\dotsc$	${\begin{cases}0,&K=0,1,\dotsc ,m-1;\\v^{K+1},&K=m,m+1,\dotsc ,m+n-1;\\0,&K=m+n,m+n+1,\dotsc \\\end{cases}}$	$_{m|n}A_{x}$
Annually increasing whole life insurance	$K+1,\quad K=0,1,\dotsc$	$v^{K+1},\quad K=0,1,\dotsc$	$(K+1)v^{K+1},\quad K=0,1,\dotsc$	$(IA)_{x}$
Annually increasing $n$-year term life insurance	${\begin{cases}K+1,&K=0,1,\dotsc ,n-1\;\\0,&K=n,n+1,\dotsc \\\end{cases}}$	$v^{K+1},\quad K=0,1,\dotsc$	${\begin{cases}(K+1)v^{K+1},&K=0,1,\dotsc ,n-1\;\\0,&K=n,n+1,\dotsc \\\end{cases}}$	$(IA)_{x:{\overline {n}}|}^{1}$
Annually decreasing $n$-year term life insurance	${\begin{cases}n-K,&K=0,1,\dotsc ,n-1\;\\0,&K=n,n+1,\dotsc \\\end{cases}}$	$v^{K+1},\quad K=0,1,\dotsc$	${\begin{cases}(n-K)v^{K+1},&K=0,1,\dotsc ,n-1\;\\0,&K=n,n+1,\dotsc \\\end{cases}}$	$(DA)_{x:{\overline {n}}|}^{1}$

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 continuously increasing.

For the $m$-thly increasing $n$-year term/whole life insurance, 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 years as units. Thus, $K$ is in the unit of years, 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.

Remark.

• We should notice that for the insurances where benefit is paid at the moment of death, there is no such "insurance payable $m$-thly", since those insurances are by assumption payable continuously already.

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 another random variable, say $J$, to consider which interval 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 $m$-th of a year lived in the year of death, before death. Using this definition of $J$, we know that $J$ can only take values of $0,1,\dotsc ,m-1$.

Exercise. You may notice that $J$ cannot take the value of $m$. Why?

Solution

When $J=m$, it means that the person lives for $m$ "$m$-th of a year" in his year of death, i.e. he lives for the whole year in his year of death. This means that he does not even die in his "year of death"! Hence, this cannot be the case.

Remark.

• Suppose the person lives for the whole year $k$, and then die in the following $m$-thly interval in year $k+1$. Then, in this case, we do not have $K=k$ and $J=m$. Instead, we have $K=k+1$ and $J=0$.

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 $1,\quad K=0,1,\dotsc ,{\text{ and }}J=0,1,\dotsc ,m-1$;
• 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 $v^{K+{\frac {J+1}{m}}},\quad K=0,1,\dotsc ,{\text{ and }}J=0,1,\dotsc ,m-1$.

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 $\mathbb {P} (K=k{\text{ and }}J=j)=\mathbb {P} \left({\text{the person dies between time }}k+{\frac {j}{m}}{\text{ and }}k+{\frac {j+1}{m}}\right)=\mathbb {P} \left(k+{\frac {j}{m}}\leq T\leq k+{\frac {j+1}{m}}\right)={}_{k+{\frac {j}{m}}}p_{x}\;_{\frac {1}{m}}q_{x+k+{\frac {j}{m}}}$, 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.

Exercise. (Another expression of $A_{x}^{(m)}$) Show that $A_{x}^{(m)}=\sum _{k=0}^{\infty }v^{k}{}_{k}p_{x}\left(\sum _{j=0}^{m-1}v^{\frac {j+1}{m}}{}_{\frac {1}{m}}q_{x+k+{\frac {j}{m}}}\;_{\frac {j}{m}}p_{x+k}\right).$

Solution

Proof.

$$A_{x}^{(m)}=\sum _{k=0}^{\infty }\sum _{j=0}^{m-1}{\color {blue}v^{k+{\frac {j+1}{m}}}}{\color {red}{}_{k+{\frac {j}{m}}}p_{x}}\;_{\frac {1}{m}}q_{x+k+{\frac {j}{m}}}=\sum _{k=0}^{\infty }\sum _{j=0}^{m-1}{\color {blue}v^{k}v^{\frac {j+1}{m}}}{\color {red}{}_{k}p_{x}{}_{\frac {j}{m}}p_{x+k}}\;_{\frac {1}{m}}q_{x+k+{\frac {j}{m}}}=\sum _{k=0}^{\infty }\underbrace {{\color {blue}v^{k}}{\color {red}{}_{k}p_{x}}} _{{\text{independent from }}j}\left(\sum _{j=0}^{m-1}{\color {blue}v^{\frac {j+1}{m}}}{\color {red}{}_{\frac {j}{m}}p_{x+k}}\;_{\frac {1}{m}}q_{x+k+{\frac {j}{m}}}\right)$$

$\Box$

Remark.

• Intuitively, we can interpret this formula as follows:

• For each time $k$, only considering the upcoming year, i.e. from time $k$ to $k+1$, the APV at time $k$ is $\sum _{j=0}^{m-1}v^{\frac {j+1}{m}}{}_{\frac {1}{m}}q_{x+k+{\frac {j}{m}}}\;_{\frac {j}{m}}p_{x+k}$
• Graphically,

v^{(j+1)/m}
     <-----------------*
                 death |
                    |  | benefit
                    v  |    
-----*----...----*-----*--------...---------*
     k    ...  k+j/m  k+(j+1)/m       k+1
     ------------>  q
      _{j/m} p _{x+k}

• To consider the "whole" APV, we need to sum this term over all $k$, discount these APV's to time 0, and also consider the probabilities for surviving to time $k$. As a result, the APV is $\sum _{k=0}^{\infty }v^{k}{}_{k}p_{x}\left(\sum _{j=0}^{m-1}v^{\frac {j+1}{m}}{}_{\frac {1}{m}}q_{x+k+{\frac {j}{m}}}\;_{\frac {j}{m}}p_{x+k}\right)$.

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 $m$-thly increasing $n$-year term life insurance. 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.

Example. (Alternative formula of $(DA)_{x:{\overline {n}}|}^{1}$) From the above table, we can deduce that

$$(DA)_{x:{\overline {n}}|}^{1}=\sum _{k=0}^{n-1}(n-k)v^{k+1}{}_{k}p_{x}q_{x+k}.$$

We can also express $(DA)_{x:{\overline {n}}|}^{1}$ as:

$$(DA)_{x:{\overline {n}}|}^{1}=\sum _{j=0}^{n-1}A_{x:{\overline {n-j}}|}^{1}.$$

An intuitive explanation for this expression is illustrated in the following diagrams:

              benefit amount at different time point
n          1
n-1        1     1
.          .     .  .
.          .     .     .
.          .     .        .
2          1     1           1
1          1     1           1     1
      *----*-----*----...----*-----*--
      0    1     2    ...   n-1    n
"sum up" (considering discounting and survival probabilities) each "vertical bar" gives APV

n          1                            A x:1
n-1        1     1                      A x:2
.          .     .  .                     .
.          .     .     .                  .
.          .     .        .               .
2          1     1           1         A x:n-1
1          1     1           1     1   A x:n
      *----*-----*----...----*-----*--
      0    1     2    ...   n-1    n
sum up all A (which consider discounting and survival probabilities already) ("horizontal bar") gives APV.

Exercise. Show that

$$(DA)_{x:{\overline {n}}|}^{1}=\sum _{j=0}^{n-1}A_{x:{\overline {n-j}}|}^{1}$$

formally and mathematically.

Solution

Proof.

$${\begin{aligned}(DA)_{x:{\overline {n}}|}^{1}&=\sum _{k=0}^{n-1}{\color {darkgreen}(n-k)}v^{k+1}{}_{k}p_{x}q_{x+k}\\&=\sum _{k=0}^{n-1}{\color {darkgreen}\left(\sum _{j=0}^{n-k-1}1\right)}v^{k+1}{}_{k}p_{x}q_{x+k}&({\text{trick}})\\&=\sum _{j=0}^{n-1}\sum _{k=0}^{n-j-1}v^{k+1}{}_{k}p_{x}q_{x+k}&({\text{changing summation order }}{\color {darkgreen}{\textit {suitably}}})\\&=\sum _{j=0}^{n-1}A_{x:{\overline {n-j}}|}^{1}&({\text{definition}}).\end{aligned}}$$

$\Box$

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.

Proposition. (Recursion relations of APV's)

1. $A_{x}=vq_{x}+vp_{x}A_{x+1}$.
2. $A_{x:{\overline {n}}|}^{1}=vq_{x}+vp_{x}A_{x+1:{\overline {n-1}}|}^{\;\;1}$.
3. $A_{x:{\overline {n}}|}=vq_{x}+vp_{x}A_{x+1:{\overline {n-1}}|}$ (a simple corollary from 2.).
4. $_{m|n}A_{x}=vp_{x}{}_{m-1|n}A_{x+1}$.
5. $(IA)_{x:{\overline {n}}|}^{1}=A_{x:{\overline {n}}|}^{\;\;1}+vp_{x}(IA)_{x+1:{\overline {n-1}}|}^{\;\;1}$.
6. $(DA)_{x:{\overline {n}}|}^{1}=nvq_{x}+vp_{x}(DA)_{x+1:{\overline {n-1}}|}^{\;\;1}$.
7. $(IA)_{x}=A_{x}+vp_{x}(IA)_{x+1}$.

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

1.:

$${\begin{aligned}A_{x}&=\sum _{k=0}^{\infty }v^{k+1}{}_{k}p_{x}q_{x+k}\\&=v\cdot \underbrace {{}_{0}p_{x}} _{=1}\cdot q_{x}+\sum _{k=1}^{\infty }v^{k+1}{}_{k}p_{x}q_{x+k}&({\text{split the `}}k=0{\text{' term out, and }}{}_{0}p_{x}=\mathbb {P} (T_{x}\geq 0)=1)\\&=vq_{x}+\sum _{k'=0}^{\infty }{\color {blue}v^{k'+2}}{\color {red}{}_{k'+1}p_{x}}q_{x+1+k'}&(k'=k-1)\\&=vq_{x}+{\color {blue}v}{\color {red}p_{x}}\sum _{k'=0}^{\infty }{\color {blue}v^{k'+1}}{\color {red}{}_{k'}p_{x+1}}q_{x+1+k'}\\&=vq_{x}+vp_{x}A_{x+1}&({\text{definition}}).\end{aligned}}$$

5.:

$${\begin{aligned}(IA)_{x:{\overline {n}}|}^{1}&=\sum _{k=0}^{n-1}(k+1)v^{k+1}{}_{k}p_{x}q_{x+k}\\&=\sum _{k=0}^{n-1}v^{k+1}{}_{k}p_{x}q_{x+k}+\sum _{k=0}^{n-1}k{\color {blue}v^{k+1}}{\color {red}{}_{k}p_{x}}q_{x+k}&({\text{split the sum}})\\&=A_{x:{\overline {n}}|}^{1}+{\color {blue}v}{\color {red}p_{x}}\sum _{k=0}^{n-1}k{\color {blue}v^{k}}{\color {red}{}_{k-1}p_{x+1}}q_{x+k}\\&=A_{x:{\overline {n}}|}^{1}+vp_{x}\sum _{k=1}^{n-1}kv^{k}{}_{k-1}p_{x+1}q_{x+k}&({\text{`}}k=0{\text{' term is zero}})\\&=A_{x:{\overline {n}}|}^{1}+vp_{x}\sum _{k'=0}^{n-2}(k'+1)v^{k'+1}{}_{k'}p_{x+1}q_{x+k}&(k'=k-1)\\&=A_{x:{\overline {n}}|}^{1}+vp_{x}(IA)_{x+1:{\overline {n-1}}|}^{\;\;1}&({\text{definition}}).\end{aligned}}$$

$\Box$

Actuarial discounting

One of the key ideas in the intuitive explanations to these recursive relations is the concept of actuarial discounting. 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 $A_{x:{\overline {1}}|}^{1}$ 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 $A_{x:{\overline {1}}|}^{1}+A_{x+1:{\overline {n-1}}|}^{\;\;1}$ 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 discounting factor, where the effect of interest is considered. On the other hand, in this context, apart from 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 given that 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

$$A_{x:{\overline {n}}|}^{1}=vq_{x}+vp_{x}A_{x+1:{\overline {n-1}}|}^{\;\;1}.$$

In financial mathematics, when we want to discount back $n$ years, we multiply the discount factor to the power $n$. However, in this case, when we want to "actuarially" discount back $n$ years, we are not 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 ($A_{x:{\overline {n}}|}^{1}$). 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 $vp_{x}$) 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 $x$ ($_{m|n}A_{x}$) with respect to 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 $vp_{x}$).

Exercise. Try to explain recursion relation 1 and 5 intuitively.

Solution

• recursion relation 1: we first extract 1-year term life insurance ($vq_{x}$) from the whole life insurance, and then the remaining part of the insurance is a whole life insurance with respect to ($x+1)$ ($A_{x+1}$). After that, we actuarially discount the remaining part back 1 year (multiply $vp_{x}$).
• recursion relation 5: we extract benefit of 1 from each original benefit to get a $n$-year term life insurance (with benefit of 1) ($A_{x:{\overline {n}}|}$). Then, the remaining part of insurance is an annually increasing $n-1$-year term life insurance with respect to $(x+1)$ ($(IA)_{x+1:{\overline {n-1}}|}^{\;\;1}$). After that, we actuarially discount the remaining part back 1 year (multiply $vp_{x}$).

Exercise. Using similar arguments, we can obtain many recursion relations other than those mentioned above. Can you construct another recursion relation for $A_{x}$ that involves $A_{x:{\overline {n}}|}^{1}$, and explain it intuitively?

Solution

$$A_{x}=A_{x:{\overline {n}}|}^{1}+v^{n}{}_{n}p_{x}A_{x+n}.$$

In this case, we extract $n$-year, instead of 1-year, term life insurance from the whole life insurance. Then, the remaining part of the insurance is a whole life insurance with respect to $(x+n)$. After that, we need to actuarially discount this part of insurance back $n$ years (multiply $v^{n}{}_{n}p_{x}$).

Example.

(a) Show that $(IA)_{x:{\overline {n}}|}=vq_{x}+vp_{x}(A_{x+1:{\overline {n-1}}|}^{\;\;1}+(IA)_{x+1:{\overline {n-1}}|}^{\;\;1})$.

(b) Given that $i=0.04,p_{x}=0.9,(IA)_{x:{\overline {n}}|}=3.52,A_{x+2:{\overline {n-2}}|}=0.125$, calculate $(IA)_{x+1:{\overline {n-1}}|}^{\;\;1}$.

Solution:

(a)

Proof. Since $A_{x:{\overline {n}}|}=vq_{x}+vp_{x}A_{x+1:{\overline {n-1}}|}$, we have

$$(IA)_{x:{\overline {n}}|}=A_{x:{\overline {n}}|}+vp_{x}(IA)_{x+1:{\overline {n-1}}|}^{\;\;1}=vq_{x}+vp_{x}A_{x+1:{\overline {n-1}}|}+vp_{x}(IA)_{x+1:{\overline {n-1}}}^{\;\;1}=vq_{x}+vp_{x}(A_{x+1:{\overline {n-1}}|}^{\;\;1}+(IA)_{x+1:{\overline {n-1}}|}^{\;\;1}).$$

$\Box$

(b) Notice that we are given $A_{x+2:{\overline {n-2}}|}$ instead of $A_{x+1:{\overline {n-1}}}$. Thus, we cannot apply the equation derived in (a) directly. However, it is simple to deduce the value of $A_{x+1:{\overline {n-1}}}$ from the given information, by considering the following recursion relation:

$$A_{x+1:{\overline {n-1}}}=vq_{x}+vp_{x}A_{x+2:{\overline {n-1}}|}\implies A_{x+1:{\overline {n-1}}}=(1.04)^{-1}(1-0.9)+(1.04)^{-1}(0.9)(0.125)\implies A_{x+1:{\overline {n-1}}}\approx 0.2043.$$

Now, we can apply the equation in (a) to get

$$3.52=1.04^{-1}(0.1)+1.04^{-1}(0.9)(0.2043+(IA)_{x+1:{\overline {n-1}}}^{\;\;1})\implies (IA)_{x+1:{\overline {n-1}}|}^{\;\;1}\approx 3.752.$$

Exercise. Given that $i=0.07,\ell _{x}=100-x\quad (0\leq x\leq 100),\;_{5|5}A_{30}=0.063$, calculate $_{10|5}A_{25}$.

Solution

Since $\ell _{x}=100-x$, ${}_{t}p_{x}={\frac {100-x-t}{100-x}}=1-{\frac {t}{100-x}}$ where $0\leq x\leq 100-t$ and $0\leq t\leq 100-x$.

We can develop a new recursion relation as follows:

$$_{10|5}A_{25}=v^{5}{}_{5}p_{25}{}_{5|5}A_{30}$$

(possibly using some intuitive arguments).

Hence,

$$_{10|5}A_{25}=(1.07)^{-5}\left(1-{\frac {5}{75}}\right)(0.063)\approx 0.0419.$$

Exercise.

(a) Given that $A_{x}=0.012,A_{x+1}=0.01,v=0.9$, calculate $p_{x}$.

Solution

By the recursion relation, we have

$$A_{x}=vq_{x}+vp_{x}A_{x+1}\implies 0.012=0.9(1-p_{x})+0.9p_{x}(0.01)\implies p_{x}\approx 0.9966.$$

(b) Given that $A_{x-1}=0.012,A_{x}=0.01,v=0.9,{}_{2}p_{x-1}=0.994$, calculate $p_{x}$.

Solution

By the recursion relation, we have

$$A_{x-1}=vq_{x-1}+vp_{x-1}A_{x}\implies 0.012=0.9(1-p_{x-1})+0.9p_{x-1}(0.01)\implies p_{x-1}\approx 0.9966.$$

Thus,

$$p_{x-1}p_{x}={}_{2}p_{x-1}\implies 0.9966p_{x}=0.994\implies p_{x}\approx 0.997$$

Now, you may ask that there are some similar recursion relations holds for continuous 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 continuous 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 rate of change of the APV.

Recursion relations in continuous case: differential equations

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

Example. (Differential equation for ${\bar {A}}_{x}$) Recall that we can write

$${\bar {A}}_{x}=\int _{0}^{\infty }v^{t}{}_{t}p_{x}\mu _{x+t}\,dt.$$

(a) Show that ${\bar {A}}_{x}={\frac {1}{{}_{x}p_{0}v^{x}}}\int _{x}^{\infty }v^{y}{}_{y}p_{0}\mu _{y}\,dy$.

(b) Show that $\int _{x}^{\infty }v^{y}{}_{y}p_{0}\mu _{y}\,dy=\int _{-\infty }^{-x}v^{-u}{}_{-u}p_{0}\mu _{-u}\,du$.

(c) Hence, show that ${\frac {d}{dx}}{\bar {A}}_{x}=(\mu _{x}+\delta ){\bar {A}}_{x}-\mu _{x}$.

Solution:

(a)

Proof.

$${\begin{aligned}{\bar {A}}_{x}&=\int _{0}^{\infty }{\color {blue}v^{t}}{\color {red}{}_{t}p_{x}}\mu _{x+t}\,dt\\&={\frac {1}{{\color {red}{}_{x}p_{0}}{\color {blue}v^{x}}}}\int _{0}^{\infty }{\color {blue}v^{x+t}}{\color {red}{}_{x+t}p_{0}}\,dt&({\color {red}{}_{x+t}p_{0}={}_{x}p_{0}{}_{t}p_{x}})\\&={\frac {1}{{}_{x}p_{0}v^{x}}}\int _{x}^{\infty }v^{y}{}_{y}p_{0}\mu _{y}\,dy&(y=x+t).\\\end{aligned}}$$

$\Box$

(b)

Proof.

$${\begin{aligned}\int _{x}^{\infty }v^{y}{}_{y}p_{0}\mu _{y}\,dy&=-\int _{-x}^{-\infty }v^{-u}{}_{-u}p_{0}\mu _{-u}\,du&(u=-y)\\&=\int _{-\infty }^{-x}v^{-u}{}_{-u}p_{0}\mu _{-u}\,du.\end{aligned}}$$

$\Box$

(c)

Proof.