Fundamental Actuarial Mathematics/Premium and Policy Value Calculation for Long-Term Insurance Coverages

Learning objectives

The Candidate will be able to use and explain the premium and policy value calculation processes for long-term insurance coverages.

Learning outcomes

The Candidate will be able to:

1. Identify the future loss random variables associated with whole life, term life, and endowment insurance, and with term and whole life annuities, on single lives.
2. Calculate premiums based on the equivalence principle, the portfolio percentile principle, and for a given expected present value of profit, for the policies in 1.
3. Calculate and interpret gross premium, net premium and modified net premium policy values for the policies in 1.
4. Calculate the effect of changes in underlying assumptions (e.g., mortality and interest).
5. Apply the following methods for modelling extra risk: age rating; constant addition to the force of mortality, constant multiple of the rate of mortality.

Introduction

There are two main job roles for actuaries, namely

• Template:Colored em (calculating Template:Colored em: prices of insurance products) and
• Template:Colored em or Template:Colored em (calculating Template:Colored em or Template:Colored em, which are indeed the same thing)

The two main topics (Template:Colored em and Template:Colored em (or Template:Colored em)) discussed in this chapter are directly related to these two job roles.

Premiums

In previous chapter, we have studied Template:Colored em and Template:Colored em, and their Template:Colored em. Now, those ideas will be combined for calculating Template:Colored em here. Intuitively, it appears that we can simply set the actuarial present value of an insurance/life annuity as its price, which is required to be paid by the insured/annuitant at its issue. However, in practice, the products are purchased by Template:Colored em, instead of just a single payment at issue (time 0) (if this is the case, we refer the premium to as Template:Colored em made at time 0). To be more specific, we usually use Template:Colored em of Template:Colored em to purchase insurance products. Template:Colored exercise Now, to actually Template:Colored em the premium, we need to have some Template:Colored em, or Template:Colored em that Template:Colored em the way of calculation. Otherwise, different people may have different opinions on how to calculate the premium. Of course, in reality, the premium of insurance products are not just calculated by a single Template:Colored em. The calculation is much more complex than what we discuss here, since there are many factors that can affect the premium in practice, and also there are many opinions from different stakeholders to be considered. So, setting a "right" premium is not an easy task in practice. Hence, pricing actuaries are needed for calculating premiums.

Before stating the principle for calculating premiums, let us define a term that is the basis for calculating premiums, namely the insurer's loss. Template:Colored definition Template:Colored remark Symbolically, we can write $$L_0=Z-PY$$ where $Z$ is the p.v.r.v. of the benefits, and $PY$ is the p.v.r.v. of the annuity of premiums ($Y$ is the p.v.r.v. of the life annuity with unit payment, and $P$ is the amount of each premium paid in the life annuity). Then, based on $L_0$, we can introduce various principles for calculating premiums. Intuitively, the insurer should avoid having losses, and hence do not want $L_0$ to be positive. This is the main idea in the following principle. Template:Colored definition Template:Colored exercise In the Template:Colored em, even if the probability for having a positive loss is very small, the Template:Colored em of the loss is Template:Colored em considered. For example, having a probability 0.01 to have a loss of $1 trillion should be more problematic than having a probability of 0.5 to have a loss of $100, right? This suggests that apart from the Template:Colored em of having positive losses, the Template:Colored em of the losses also matters. When both Template:Colored em and Template:Colored em are involved, what do you think of? Template:Colored definition Template:Colored remark Since under Template:Colored em, the calculations of premiums are quite simple, it will be used for premium calculation in the following, unless otherwise specified.

As we have mentioned previously, the concepts related to Template:Colored em and Template:Colored em are applied here for calculating premiums. As a result, there are not many "new" concepts involved in the calculation of premiums for purchasing Template:Colored em and Template:Colored em.

Fully continuous premiums

Let us first consider the premiums for purchasing a Template:Colored em with benefit of 1 payable Template:Colored em, issued to a life aged $x$. Unless otherwise specified, we will assume the premiums will be paid in the same Template:Colored em as in the insurance product. In this case, the payment pattern of the premiums is paying Template:Colored em, since the benefit for the insurance is paid Template:Colored em.

So, we have mentioned how should the Template:Colored em be determined. How about the Template:Colored em of the payments? Of course, the payments Template:Colored em cease when the insured/annuitant dies (it makes no sense for a dead person to continue paying premiums, right?). But, the payments Template:Colored em also cease Template:Colored em the insured/annuitant dies, if it is specified in the terms of the insurance product. For instance, for a whole life insurance, perhaps premiums are only payable for the first 10 years after the policy issue.

In this case, the insurer's loss is $$L_0=v^T-\overline{P}{\overline{a}}_{\bar{T}|},T>0$$ where $\overline{P}$ is the continuous level annual premium. By equivalence principle, we then have $$𝔼[L_0]=0⟹{\overline{A}}_x-\overline{P}{\overline{a}}_x=0⟹\overline{P}=\frac{{\overline{A}}_x}{{\overline{a}}_x}.$$ In this case, the premium $\overline{P}$ can be denoted by $\overline{P}({\overline{A}}_x)$ (the notation specifies that this premium corresponds to continuous whole life insurance). Template:Colored exercise Template:Colored example For other types of insurance products, the formulas for the annual premium $\overline{P}$ are similar. So, some of them are summarized in the following table.

Summary

Insurance product name	$Z$	$Y$	$L_0=Z-\overline{P}Y$	$\overline{P}=\frac{𝔼[Z]}{𝔼[Y]}$
Whole life insurance	$v^T,T\ge 0$	${\overline{a}}_{\bar{T}|},T\ge 0$	$v^T-\overline{P}{\overline{a}}_{\bar{T}|},T\ge 0$	$\overline{P}({\overline{A}}_x)=\frac{{\overline{A}}_x}{{\overline{a}}_x}$.
$n$-year term life insurance	$\{\begin{array}{cc}{v}^T,& T\le n;\\ 0,& T>n\end{array}$	$\{\begin{array}{cc}{\overline{a}}_{\bar{T}|},& T\le n;\\ {\overline{a}}_{\bar{n}|},& T>n\end{array}$	$\{\begin{array}{cc}{v}^T-\overline{P}{\overline{a}}_{\bar{T}|},& T\le n;\\ -\overline{P}{\overline{a}}_{\bar{n}|},& T>n\end{array}$	$\overline{P}({\overline{A}}_{x:\bar{n}|}^1)=\frac{{\overline{A}}_{x:\bar{n}|}^1}{{\overline{a}}_{x:\bar{n}|}}$
$n$-year endowment insurance	$\{\begin{array}{cc}{v}^T,& T\le n;\\ v^n,& T>n\end{array}$	$\{\begin{array}{cc}{\overline{a}}_{\bar{T}|},& T\le n;\\ {\overline{a}}_{\bar{n}|},& T>n\end{array}$	$\{\begin{array}{cc}{v}^T-\overline{P}{\overline{a}}_{\bar{T}|},& T\le n;\\ v^n-\overline{P}{\overline{a}}_{\bar{n}|},& T>n\end{array}$	$\overline{P}({\overline{A}}_{x:\bar{n}|})=\frac{{\overline{A}}_{x:\bar{n}|}}{{\overline{a}}_{x:\bar{n}|}}$
$n$-year pure endowment	$\{\begin{array}{cc}0,& T\le n;\\ v^n,& T>n\end{array}$	$\{\begin{array}{cc}{\overline{a}}_{\bar{T}|},& T\le n;\\ {\overline{a}}_{\bar{n}|},& T>n\end{array}$	$\{\begin{array}{cc}-\overline{P}{\overline{a}}_{\bar{T}|},& T\le n;\\ v^n-\overline{P}{\overline{a}}_{\bar{n}|},& T>n\end{array}$	$\overline{P}(A_{x:\bar{n}|}^1)=\frac{A_{x:\bar{n}|}^1}{{\overline{a}}_{x:\bar{n}|}}$ or $\overline{P}({}_n{E}_x)=\frac{{}_n{E}_x}{{\overline{a}}_{x:\bar{n}|}}$
$h$-payment years whole life insurance	$v^T,T\ge 0$	$\{\begin{array}{cc}{\overline{a}}_{\bar{T}|},& T\le h;\\ {\overline{a}}_{\bar{h}|},& T>h\end{array}$	$\{\begin{array}{cc}{v}^T-\overline{P}{\overline{a}}_{\bar{T}|},& T\le h;\\ v^T-\overline{P}{\overline{a}}_{\bar{h}|},& T>h\end{array}$	${}_h\overline{P}({\overline{A}}_x)=\frac{{\overline{A}}_x}{{\overline{a}}_{x:\bar{h}|}}$
$h$-payment years $n$-year term life insurance ($h<n$)	$\{\begin{array}{cc}{v}^T,& T\le h;\\ v^T,& h<T\le n;\\ 0,& T>n\end{array}$	$\{\begin{array}{cc}{\overline{a}}_{\bar{T}|},& T\le h;\\ {\overline{a}}_{\bar{h}|},& h<T\le n;\\ {\overline{a}}_{\bar{h}|},& T>n\end{array}$	$\{\begin{array}{cc}{v}^T-\overline{P}{\overline{a}}_{\bar{T}|},& T\le h;\\ v^T-\overline{P}{\overline{a}}_{\bar{h}|},& h<T\le n;\\ -\overline{P}{\overline{a}}_{\bar{h}|},& T>n\end{array}$	${}_h\overline{P}({\overline{A}}_{x:\bar{n}|})=\frac{{\overline{A}}_{x:\bar{n}|}^1}{{\overline{a}}_{x:\bar{h}|}}$
$h$-payment years $n$-year endowment insurance ($h<n$)	$\{\begin{array}{cc}{v}^T,& T\le h;\\ v^T,& h<T\le n;\\ v^n,& T>n\end{array}$	$\{\begin{array}{cc}{\overline{a}}_{\bar{T}|},& T\le h;\\ {\overline{a}}_{\bar{h}|},& h<T\le n;\\ {\overline{a}}_{\bar{h}|},& T>n\end{array}$	$\{\begin{array}{cc}{v}^T-\overline{P}{\overline{a}}_{\bar{T}|},& T\le h;\\ v^T-\overline{P}{\overline{a}}_{\bar{h}|},& h<T\le n;\\ v^n-\overline{P}{\overline{a}}_{\bar{h}|},& T>n\end{array}$	${}_h\overline{P}({\overline{A}}_{x:\bar{n}|})=\frac{{\overline{A}}_{x:\bar{n}|}}{{\overline{a}}_{x:\bar{h}|}}$
$n$-year deferred whole life annuity	$\{\begin{array}{cc}0,& T\le n;\\ {\overline{a}}_{\overline{T-n}|},& T>n\end{array}$	$\{\begin{array}{cc}{\overline{a}}_{\bar{T}|},& T\le n;\\ {\overline{a}}_{\bar{n}|},& T>n\end{array}$	$\{\begin{array}{cc}-\overline{P}{\overline{a}}_{\bar{T}|},& T\le n;\\ {\overline{a}}_{\overline{T-n}|}-\overline{P}{\overline{a}}_{\bar{n}|},& T>n\end{array}$	$\overline{P}({}_{n|}{\overline{a}}_x)=\frac{{}_{n|}{\overline{a}}_x}{{\overline{a}}_{x:\bar{n}|}}=\frac{{}_n{E}_x{\overline{a}}_{x+n}}{{\overline{a}}_{x:\bar{n}|}}$

Fully discrete premiums

Now, let us consider the Template:Colored em insurance products, where the premiums are also discretely, rather than continuously paid. However, a difference here is that the Template:Colored em of the premiums is not Template:Colored em the same as that for the insurance products. In particular, we assume that the premiums are always made Template:Colored em of each year unless otherwise specified. As a result, the premiums form a Template:Colored em.

First, let us consider the case for a whole life insurance with benefit of 1 payable at the end of the year of death. Then, the insurer's loss is $$v^{K+1}-P{\ddot{a}}_{\overline{K+1}|}$$ where $P$ is the level annual premium. By equivalence principle, we then have $$𝔼[L_0]=0⟹A_x-P{\ddot{a}}_x=0⟹P=\frac{A_x}{{\ddot{a}}_x}.$$ In this case, we denote the premium $P$ by $P_x$, which is in the "same form" as the notation $A_x$. For other types of insurance products, the formulas for premium are developed similarly. Some of them are summarized below.

Summary

Insurance product name	$Z$	$Y$	$L_0=Z-PY$	$P=\frac{𝔼[Z]}{𝔼[Y]}$
Whole life insurance	$v^{K+1},K=0,1,\dots$	${\ddot{a}}_{\overline{K+1}|},K=0,1,\dots$	$v^{K+1}-P{\ddot{a}}_{\overline{K+1}|},K=0,1,\dots$	$P_x=\frac{A_x}{{\ddot{a}}_x}$.
$n$-year term life insurance	$\{\begin{array}{cc}{v}^{K+1},& K=0,1,\dots ,n-1;\\ 0,& K=n,n+1,\dots \end{array}$	$\{\begin{array}{cc}{\ddot{a}}_{\overline{K+1}|},& K=0,1,\dots ,n-1;\\ {\ddot{a}}_{\bar{n}|},& K=n,n+1,\dots \end{array}$	$\{\begin{array}{cc}{v}^{K+1}-P{\ddot{a}}_{\overline{K+1}|},& K=0,1,\dots ,n-1;\\ -P{\ddot{a}}_{\bar{n}|},& K=n,n+1,\dots \end{array}$	$P_{x:\bar{n}|}^1=\frac{A_{x:\bar{n}|}^1}{{\ddot{a}}_{x:\bar{n}|}}$
$n$-year endowment insurance	$\{\begin{array}{cc}{v}^{K+1},& K=0,1,\dots ,n-1;\\ v^n,& K=n,n+1,\dots \end{array}$	$\{\begin{array}{cc}{\ddot{a}}_{\overline{K+1}|},& K=0,1,\dots ,n-1;\\ {\ddot{a}}_{\bar{n}|},& K=n,n+1,\dots \end{array}$	$\{\begin{array}{cc}{v}^{K+1}-P{\ddot{a}}_{\overline{K+1}|},& K=0,1,\dots ,n-1;\\ v^n-P{\ddot{a}}_{\bar{n}|},& K=n,n+1,\dots \end{array}$	$P_{x:\bar{n}|}=\frac{A_{x:\bar{n}|}}{{\ddot{a}}_{x:\bar{n}|}}$
$n$-year pure endowment	$\{\begin{array}{cc}0,& K=0,1,\dots ,n-1;\\ v^n,& K=n,n+1,\dots \end{array}$	$\{\begin{array}{cc}{\ddot{a}}_{\overline{K+1}|},& K=0,1,\dots ,n-1;\\ {\ddot{a}}_{\bar{n}|},& K=n,n+1,\dots \end{array}$	$\{\begin{array}{cc}-P{\ddot{a}}_{\overline{K+1}|},& K=0,1,\dots ,n-1;\\ v^n-P{\ddot{a}}_{\bar{n}|},& K=n,n+1,\dots \end{array}$	$P_{x:\bar{n}|}^1=\frac{A_{x:\bar{n}|}^1}{{\ddot{a}}_{x:\bar{n}|}}$
$h$-payment years whole life insurance	$v^{K+1},K=0,1,\dots$	$\{\begin{array}{cc}{\ddot{a}}_{\overline{K+1}|},& K=0,1,\dots ,h-1;\\ {\ddot{a}}_{\bar{h}|},& K=h,h+1,\dots \end{array}$	$\{\begin{array}{cc}{v}^{K+1}-P{\ddot{a}}_{\overline{K+1}|},& K=0,1,\dots ,h-1;\\ v^{K+1}-P{\ddot{a}}_{\bar{h}|},& K=h,h+1,\dots \end{array}$	${}_h{P}_x=\frac{A_x}{{\ddot{a}}_{x:\bar{h}|}}$
$h$-payment years $n$-year term life insurance ($h<n$)	$\{\begin{array}{cc}{v}^{K+1},& K=0,1,\dots ,h-1;\\ v^{K+1},& K=h,h+1,\dots ,n-1;\\ 0,& K=n,n+1,\dots \end{array}$	$\{\begin{array}{cc}{\ddot{a}}_{\overline{K+1}|},& K=0,1,\dots ,h-1;\\ {\ddot{a}}_{\bar{h}|},& K=h,h+1,\dots ,n-1;\\ {\ddot{a}}_{\bar{h}|},& K=n,n+1,\dots \end{array}$	$\{\begin{array}{cc}{v}^{K+1}-P{\ddot{a}}_{\overline{K+1}|},& K=0,1,\dots ,h-1;\\ v^{K+1}-P{\ddot{a}}_{\bar{h}|},& K=h,h+1,\dots ,n-1;\\ -P{\ddot{a}}_{\bar{h}|},& K=n,n+1,\dots \end{array}$	${}_h{P}_{x:\bar{n}|}=\frac{A_{x:\bar{n}|}^1}{{\ddot{a}}_{x:\bar{h}|}}$
$h$-payment years $n$-year endowment insurance ($h<n$)	$\{\begin{array}{cc}{v}^{K+1},& K=0,1,\dots ,h-1;\\ v^{K+1},& K=h,h+1,\dots ,n-1;\\ v^n,& K=n,n+1,\dots \end{array}$	$\{\begin{array}{cc}{\ddot{a}}_{\overline{K+1}|},& K=0,1,\dots ,h-1;\\ {\ddot{a}}_{\bar{h}|},& K=h,h+1,\dots ,n-1;\\ {\ddot{a}}_{\bar{h}|},& K=n,n+1,\dots \end{array}$	$\{\begin{array}{cc}{v}^{K+1}-P{\ddot{a}}_{\overline{K+1}|},& K=0,1,\dots ,h-1;\\ v^{K+1}-P{\ddot{a}}_{\bar{h}|},& K=h,h+1,\dots ,n-1;\\ v^n-P{\ddot{a}}_{\bar{h}|},& K=n,n+1,\dots \end{array}$	${}_{h}P(A_{x:\bar{n}|})=\frac{A_{x:\bar{n}|}}{{\ddot{a}}_{x:\bar{h}|}}$
$n$-year deferred whole life annuity	$\{\begin{array}{cc}0,& K=0,1,\dots ,n-1;\\ {\ddot{a}}_{\overline{K+1-n}|},& K=n,n+1,\dots \end{array}$	$\{\begin{array}{cc}{\ddot{a}}_{\overline{K+1}|},& K=0,1,\dots ,n-1;\\ {\ddot{a}}_{\bar{n}|},& K=n,n+1,\dots \end{array}$	$\{\begin{array}{cc}-P{\ddot{a}}_{\overline{K+1}|},& K=0,1,\dots ,n-1;\\ {\ddot{a}}_{\overline{K+1-n}|}-P{\ddot{a}}_{\bar{n}|},& K=n,n+1,\dots \end{array}$	$P({}_{n|}{\ddot{a}}_x)=\frac{{}_n{E}_x{\ddot{a}}_{x+n}}{{\ddot{a}}_{x:\bar{n}|}}$

m-thly payment premiums

Of course, for the insurance products in the previous section, the premiums need not be payable annually. In general, they can be payable $m$ times for every policy year. We can similarly use Template:Colored em to determine the amount of each premium in this case.

Let us first consider the case for whole life insurance with unit benefit payable at the end of year of death. Suppose the premiums are payable in $m$-thly installments at the Template:Colored em of each $m$-thly period . In this case, the insurer's loss is $$L_0=v^{K+1}-P^{(m)}{\ddot{a}}_{\overline{K+1}|}^{(m)}$$ where $P^{(m)}$ is the level Template:Colored em premium payable $m$-thly, that is, the Template:Colored em amount of premium paid at the beginning of each $m$-thly period is $P^{(m)}/m$. (This is similar to the case for the interest rate where $i^{(m)}$ is the Template:Colored em annual interest rate, while the Template:Colored em interest rate for each $m$-thly period is $i^{(m)}/m$.) By equivalence principle, we can similarly get $P^{(m)}=\frac{A_x}{{\ddot{a}}_x^{(m)}}$. In this case, we denote $P^{(m)}$ by $P_x^{(m)}$.

The following is a summary for the formulas for $P^{(m)}$ of some other types of discrete insurance products. Template:Todo We can also apply this idea to insurances payable at the moment of death. For instance, when the above whole life insurance is instead the Template:Colored em one, then we have $$L_0=v^T-P^{(m)}{\ddot{a}}_{\overline{K+1}}^{(m)}.$$ Similarly, we have $P^{(m)}=\frac{{\overline{A}}_x}{{\ddot{a}}_x^{(m)}}$ by equivalence principle. In this case, we denote $P^{(m)}$ by $P^{(m)}({\overline{A}}_x)$. The following is a summary for the formulas for $P^{(m)}$ of some other types of discrete insurance products. Template:Todo Of course, apart from the aforementioned insurance products, we can also apply the idea of equivalence principle for calculating the premiums for insurances where benefits are varying, other types of life annuities, etc. Also, the insurance products can be irregular, and the premium payments can be irregular as well. In those cases, there are no "formulas" for calculating the amount of premiums directly. But, we can always use the equivalence principle for the calculations.

Accumulation-type benefits

In practice, apart from the death benefits from the insurance products, some of the premiums paid may be Template:Colored em when death occurs. In particular, for $n$-year deferred whole life annuity, the annuitant will get Template:Colored em from the life annuity itself if he dies during the deferral period. But when some of the premiums paid are refunded when death occurs in the deferral period, then the annuitant will at least get Template:Colored em if he dies during the deferral period. So, this may be better for the annuitant (but of course, in exchange for this, it is natural to expect that the premiums required will be higher).

For the refund of premiums, depending on the terms, the amount of the refund may or may not consider the Template:Colored em. To be more specific, when the amount of premiums refunded is determined at some time $t$, we may use the Template:Colored em of the premiums paid at time $t$ with a certain interest rate (possibly different from the interest rate used for the calculation of actuarial present values), or simply Template:Colored em interest rate.

In the following, we will discuss the development of formulas for these benefits when the premiums are payable Template:Colored em (at the beginning of each year), and the refund will be made at the Template:Colored em of year of death. We can develop similar formulas when the premium are payable Template:Colored em or Template:Colored em, with the refund to be made at different time points.

Let us first develop a model of a special $n$-year term insurance issued to a life aged $x$, where the benefit of ${\ddot{s}}_{\overline{k+1}|j}$ (evaluated at interest rate $j$) is payable at the end of year $k<n$ (time $k+1$) when death occurs in year $k$ (no benefit if death does not occur within $n$ years), and then apply this model for the refunds of premiums. Graphically, the situation looks like

   *-----*----------------------*
   |     |              |  die  |          ..
  "1"   "1"            "1"  |   v benefit: s k+1|j evaluated at interest rate j
---*-----*-----...------*-------*-----
   0     1     ...      k      k+1
"1": hypothetical "benefits" made at various time points (yet to be realized until death) (they may be interpreted as premiums paid in practice, and then they are not hypothetical in those cases)

Now, let us consider some simple cases first.

Case 1: interest rate $j=0$. Then, the benefit is ${\ddot{s}}_{\overline{k+1}|j}=\underset{k+1\text{ times}}{\underbrace{1+1+\cdots +1}}=k+1$.

Case 2: interest rate $j=i$ ($i$ is the interest rate used for calculating actuarial present value). Then, the benefit is ${\ddot{s}}_{\overline{k+1}|j}={\ddot{s}}_{\overline{k+1}|i}$.

In case 1, the APV of the p.v.r.v. for this insurance is simply given by $(IA{)}_x$ by considering the definition of $(IA{)}_x$. In case 2, the APV of the p.v.r.v. for this insurance is ${\ddot{a}}_{x:\bar{n}|i}-{}_n{E}_x{\ddot{s}}_{\bar{n}|i}$. The formula for case 2 will be proven later in this section. For now, let us give an intuitive explanation to this formula in the following:

Indeed, when $j=i$, the special insurance is very similar to a $n$-year Template:Colored em (get a payment of 1 when the life still survives at the beginning of each of the first $n$ years), in the sense that Template:Colored em, when death occurs in year $k+1$, the value of the benefit provided by the special insurance is ${\ddot{s}}_{\overline{k+1}|i}$.

On the other hand, for the $n$-year life annuity-due (assume $k<n$), the value of the benefits at time $k+1$ is also ${\ddot{s}}_{\overline{k+1}|i}$ (accumulate each of the $k+1$ survival benefits to time $k+1$). So, the APV of the benefits of the special insurance and the life annuity-due are the same. However, we have made an important assumption in the process: $k<n$, i.e. death occurs before year $n$. But this is not necessarily the case. The life can survive for at least $n$ years, right?

So we need to consider this situation also. In the situation where the life survives for at least $n$ years, the life annuity-due provides a benefit of ${\ddot{s}}_{\bar{n}|i}$ at time $n$, but the special insurance will not provide anything. Hence, we need to subtract ${}_n{E}_x{\ddot{s}}_{\bar{n}|i}$ (APV of that "extra" benefit, obtained by actuarially discounting the benefit to time 0) from the APV of the $n$-year life annuity-due ${\ddot{a}}_{x:\bar{n}|i}$ to get the APV of the p.r.r.v. of this special insurance.

Now, let us formally define the present value random variable involved in the model of the special insurance: $$Z=\{\begin{array}{cc}{v}_i^{K+1}{\ddot{s}}_{\overline{K+1}|j},& K=0,1,\dots ,n-1;\\ 0,& K=n,n+1,\dots .\end{array}$$ After that, we can derive a formula for the APV: $$\begin{array}{cc}𝔼[Z]& =\sum _{k=0}^{n-1}{v}_i^{k+1}{\ddot{s}}_{\overline{k+1}|j}{}_k{p}_x{q}_{x+k}\\ & =\sum _{k=0}^{n-1}((1+i{)}^{-(k+1)}\cdot \frac{(1+j{)}^{k+1}-1}{d_j}{}_k{p}_x{q}_{x+k})\\ & =\frac{1}{d_j}\sum _{k=0}^{n-1}[{\left(\frac{1+i}{1+j}\right)}^{-(k+1)}{}_k{p}_x{q}_{x+k}-v_i^{k+1}{}_k{p}_x{q}_{x+k}]\\ & =\frac{1}{d_j}\left[\sum _{k=0}^{n-1}\right(1+\underset{i_{*}}{\underbrace{\frac{i-j}{1+j}}}{)}^{-(k+1)}{}_k{p}_x{q}_{x+k}-\sum _{k=0}^{n-1}{v}_i^{k+1}{}_k{p}_x{q}_{x+k}]\\ & =\frac{1}{d_j}[\sum _{k=0}^{n-1}{v}_{*}^{k+1}{}_k{p}_x{q}_{x+k}-A_{x:\bar{n}|i}^1]\\ & =\frac{1}{d_j}[A_{x:\bar{n}|i_{*}}^1-A_{x:\bar{n}|i}^1].\\ \end{array}$$ where $d_j$ is the discount rate equivalent to the interest rate $j$, i.e., $d_j=\frac{j}{1+j}$, $v_{*}=\frac{1}{1+i_{*}}$, and $v_i=\frac{1}{1+i}$ ($i_{*}$ and $i$ are added to the APV notations for insurances so that we can identify which interest rate we are using for evaluating the APV's), assuming $j\ne 0$. Template:Colored remark Through this formula, we can prove the formula in case 2 above (APV is ${\ddot{a}}_{x:\bar{n}|}-{}_n{E}_x{\ddot{s}}_{\bar{n}|}$):

Proof. When $j=i$, we have $i_{*}=\frac{i-j}{1+j}=0$. So the APV of the special insurance is (notice that ${}_k{p}_x{q}_{x+k}={}_{k|}{q}_x$) $$\frac{\sum _{k=0}^{n-1}{}_{k|}{q}_x-A_{x:\bar{n}|}^1}{d}=\frac{{}_n{q}_x-A_{x:\bar{n}|}^1}{d}=\frac{1-{}_n{p}_x-A_{x:\bar{n}|}+v^n{}_n{p}_x}{d}=\frac{1-A_{x:\bar{n}|}}{d}-v^n{}_n{p}_x\cdot \frac{(1+i{)}^n-1}{d}={\ddot{a}}_{x:\bar{n}|}-{}_n{E}_x{\ddot{s}}_{\bar{n}|}.$$

$◻$

Incorporating expenses

Previously, we have not considered expenses. Now, we will discuss the situation where expenses are incorporated to the premium calculations in this section. As we have mentioned, such premiums calculated are called Template:Colored em. To calculate gross premiums, we need to include expenses in the insurer's loss $L_0$. Since the expenses are to be paid by the insurer, the p.v.r.v. of expenses are Template:Colored em to $L_0$, that is, we now have $$L_0=Z+\text{p.v.r.v. of expenses}-PY.$$ When we use the equivalence principle, we have $$P𝔼[Y]=𝔼[Z]+𝔼[\text{p.v.r.v. of expenses}].$$ The expenses may be incurred from the cost of claiming benefits, commissions, etc. Template:Colored example

Reserves (or policy values)

In the section about premiums, we often use the equivalence principle to calculate premiums, which requires the expected value of the insurer's losses Template:Colored em (time 0) to be zero. But, after a period of time, say at time $t$, this expected value may Template:Colored em be zero anymore, since the "$Z$" and "$Y$" at this time $t$, are different from the "$Z$" and "$Y$" at time 0. Particularly, the "$Z$" and "$Y$" at time $t$ are considering the benefits/payments Template:Colored em, and the benefits/payments from time $0$ to time $t$ are not considered. Graphically, it looks like

 not considered       discount to time t ==> "Y at time t"
   <-------------> <-------->           
   P  P ...      P P P ...  P  benefit <-- discount to time t ==> "Z at time t"
---*---------------*-----------*-------
   0               t           die
   |--------------->        
    Assuming survival to time t

Template:Colored remark We may want the expected value to be still zero at time $t$, and in order for the insurer's loss Template:Colored em to still have a zero expected value (so that there is still equivalence between the financial obligations for the policyholder and the insurer at this time point), a "balancing item" may be needed. To determine what the balancing item should be, let us consider the following two cases:

1. The expected value of the insurer's losses at time $t$ is positive. This means the insurer expects a prospective loss from the policy (since the future benefits paid are expected to be greater than the future premiums received). Then, the insurer should Template:Colored em an amount of money, so that the insurer can "encounter" the losses.
2. On the other hand, if the expected value of the insurer's losses at time $t$ is negative, then this means the insurer expects a prospective gain from the policy. So, the insurer can have a "negative reserve" (hypothetically) for that policy and still be able to encounter the losses.

Template:Colored remark From these, we can observe that in case 1, the insurer should spare an amount of money for the reserve for that policy (increase in financial obligation for the insurer), and in case 2, the insurer can hypothetically take away a sum of money from the policy (decrease in financial obligation for the insurer). Through these changes in the financial obligation for the insurer, there can still be an equivalence between the financial obligations for the policyholder and the insurer.

These lead us to the following definitions. Template:Colored definition Template:Colored remark Template:Colored definition Template:Colored remark Symbolically, if the policy is issued to a life aged $x$, then the net premium reserve is $𝔼[L_t^n|T>t]$, and the gross premium reserve $𝔼[L_t^g|T>t]$ (for gross loss), for the continuous case. (For discrete case, we use "$L_k^n$" ($L_k^g$) and "$K\ge k$") Template:Colored remark By definition, to calculate the conditional expectation $𝔼[L_t^n|T>t]$, we need to consider the conditional distribution of $L_t^n$ given $T>t$. It may appear to be complicated. But, we can prove that the conditional distribution of $T_x-t$ ($T_x-t$ gives the prospective lifetime with respect to $t$, which should be involved in $L_t^n$) given that $T_x>t$ is indeed the same as the unconditional distribution of $T_{x+t}$.

Proof. Consider the survival functions of the two distributions. First, for the conditional distribution of $T_x-t$ given that $T_x>t$, $$\mathbb{P}(T_x-t>s|T_x>t)=\frac{\mathbb{P}(T_x>s+t)}{\mathbb{P}(T_x>t)}=\frac{\mathbb{P}(T_0>x+s+t|T_0>x)}{\mathbb{P}(T_0>x+t|T_0>x)}=\frac{\mathbb{P}(T_0>x+s+t)/\mathbb{P}(T_0>x)}{\mathbb{P}(T_0>x+t)/\mathbb{P}(T_0>x)}=\frac{\mathbb{P}(T_0>x+s+t)}{\mathbb{P}(T_0>x+t)}.$$ On the other hand, for the unconditional distribution of $T_{x+t}$, $$\mathbb{P}(T_{x+t}>s^{\prime })=\mathbb{P}(T_0>x+s^{\prime }+t|T_0>x+t)=\frac{\mathbb{P}(T_0>x+s^{\prime }+t)}{\mathbb{P}(T_0>x+t)}.$$ These show the survival functions of the two distributions are the same (provided that all the conditional probabilities involved are defined). Hence, the two distributions are the same.

$◻$

This result gives us an alternative and often more convenient method to calculate the conditional expectation $𝔼[L_t^n|T_x>t]$:

1. replace all "$T_x-t$" by "$T_{x+t}$" and remove the condition "$T_x>t$"
2. calculate the unconditional expectation, which equals the value of conditional expectation since the distributions involved are the same

Notice that we can also apply this similarly to the discrete case where $K_x$ is involved, since $K_x$ is just defined as $\lfloor T_x\rfloor$, and we can have a similar alternative method for calculations.

Fully continuous reserves

First, let us consider the whole life insurance with unit benefit, the simplest case. In this case, we have $$L_t^n=v^{T-t}-\underset{\text{notation}}{\underbrace{\overline{P}({\overline{A}}_x)}}{\overline{a}}_{\overline{T-t}|}.$$ To understand this, let us consider the following diagram.

        v^{T-t}
           <--------1 future benefit
---*-------*--------*---
   0       t        T
           ^       (die)
   Pa_{T-t}|
           |--------|
               P       future premiums

Then, the reserve, denoted by ${}_t\overline{V}({\overline{A}}_x)$ ("$V$" corresponds to the "v" in "policy value"), is by definition $𝔼[L_t^n|T>t]$. Template:Colored exercise Template:Colored example Template:Colored example Now, let us consider the $n$-year term life insurance with unit benefit. In this case, the prospective net loss is different.

When $t<n$, $$L_t^n=\{\begin{array}{cc}{v}^{T-t}-\overline{P}({\overline{A}}_{x:\bar{n}|}^1){\overline{a}}_{\overline{T-t}|},& 0<T-t<n-t;\\ -\overline{P}({\overline{A}}_{x:\bar{n}|}^1){\overline{a}}_{\overline{T-t}|},& T-t\ge n-t.\end{array}$$ When $t=n$, there are no future premiums or benefits made, so $L_t^n=0$.

When $t>n$, the insurance has ended, so it is not meaningful to consider the reserve for it anymore. (Indeed, if we follow our definition, for other insurance products with finite term, the reserve at such time point must be zero since there will be no premiums or benefits after time $t$. It is therefore meaningless to consider such reserves .)

Then, the reserve, denoted by ${}_t\overline{V}({\overline{A}}_{x:\bar{n}|}^1)$, is $$𝔼[L_t^n|T>t]=\{\begin{array}{cc}{\overline{A}}_{x+t:\overline{n-t}|}^1-\overline{P}({\overline{A}}_{x:\bar{n}|}^1){\overline{a}}_{x+t:\overline{n-t}|},& t<n;\\ 0,& t=n.\end{array}$$ (by considering the alternative method)

For the $n$-year endowment insurance with unit benefits, the prospective net loss is different again.

When $t<n$, $$L_t^n=\{\begin{array}{cc}{v}^{T-t}-\overline{P}({\overline{A}}_{x:\bar{n}|}^1){\overline{a}}_{\overline{T-t}|},& 0<T-t<n-t;\\ v^{n-t}-\overline{P}({\overline{A}}_{x:\bar{n}|}^1){\overline{a}}_{\overline{T-t}|},& T-t\ge n-t.\end{array}$$ When $t=n$, we have $L_t^n=1$ (there is only one benefit made at time $n$, namely the unit survival benefit made. So, the value is exactly one.). Then, the reserve, denoted by ${}_t\overline{V}({\overline{A}}_{x:\bar{n}|})$, is $$𝔼[L_t^n|T>t]=\{\begin{array}{cc}{\overline{A}}_{x+t:\overline{n-t}|}-\overline{P}({\overline{A}}_{x:\bar{n}|}){\overline{a}}_{x+t:\overline{n-t}|},& t<n;\\ 1,& t=n.\end{array}$$ (again by considering the alternative method)

To summarize, the reserves of the above policies and also some more other policies (with unit benefits) are tabulated below.

Summary

Insurance product name	net premium reserve at time $t$
Whole life insurance	${}_t\overline{V}({\overline{A}}_x)={\overline{A}}_{x+t}-\overline{P}({\overline{A}}_x){\overline{a}}_{x+t}$
$n$-year term life insurance	${}_t\overline{V}({\overline{A}}_{x:\bar{n}|}^1)=\{\begin{array}{cc}{\overline{A}}_{x+t:\overline{n-t}|}^1-\overline{P}({\overline{A}}_{x:\bar{n}|}^1){\overline{a}}_{x+t:\overline{n-t}|},& t<n;\\ 0,& t=n.\end{array}$
$n$-year endowment insurance	${}_t\overline{V}({\overline{A}}_{x:\bar{n}|})=\{\begin{array}{cc}{\overline{A}}_{x+t:\overline{n-t}|}-\overline{P}({\overline{A}}_{x:\bar{n}|}){\overline{a}}_{x+t:\overline{n-t}|},& t<n;\\ 1,& t=n.\end{array}$
$h$-payment years whole life insurance	$\underset{h}{\overset{}{t}}\overline{V}({\overline{A}}_x)=\{\begin{array}{cc}{\overline{A}}_{x+t}-{}_h\overline{P}({\overline{A}}_x){\overline{a}}_{x+t:\overline{h-t}|},& t<h;\\ {\overline{A}}_{x+t},& t\ge h.\end{array}$
$h$-payment years $n$-year term life insurance ($h<n$)	$\underset{h}{\overset{}{t}}\overline{V}({\overline{A}}_{x:\bar{n}|}^1)=\{\begin{array}{cc}{\overline{A}}_{x+t:\overline{n-t}|}^1-{}_h\overline{P}({\overline{A}}_{x:\bar{n}|}^1){\overline{a}}_{x+t:\overline{h-t}|},& t<h;\\ {\overline{A}}_{x+t:\overline{n-t}|}^1,& h\le t<n;\\ 0,& t=n.\end{array}$
$h$-payment years $n$-year endowment insurance ($h<n$)	$\underset{h}{\overset{}{t}}\overline{V}({\overline{A}}_{x:\bar{n}|})=\{\begin{array}{cc}{\overline{A}}_{x+t:\overline{n-t}|}-{}_h\overline{P}({\overline{A}}_{x:\bar{n}|}){\overline{a}}_{x+t:\overline{h-t}|},& t<h;\\ {\overline{A}}_{x+t:\overline{n-t}|},& h\le t<n;\\ 1,& t=n.\end{array}$
$n$-year pure endowment	${}_t\overline{V}({}_n{E}_x)=\{\begin{array}{cc}{}_{n-t}{E}_{x+t}-\overline{P}({}_n{E}_x){\overline{a}}_{x+t:\overline{n-t}|},& t<n;\\ 1,& t=n.\end{array}$
$n$-year deferred whole life annuity	${}_t\overline{V}({}_{n|}{\overline{a}}_x)=\{\begin{array}{cc}{}_{n-t|}{\overline{a}}_{x+t}-\overline{P}({}_{n|}{\overline{a}}_x){\overline{a}}_{x+t:\overline{n-t}|},& t\le n;\\ {\overline{a}}_{x+t},& t>n.\end{array}$

Template:Colored example Template:Colored example Template:Colored exercise Template:Colored exercise The premium-difference and paid-up insurance formulas can also be developed similarly for other types of policies. However, we seldom use the formulas themselves for the actual calculation since these formulas can be derived in just a single step, and thus we are not necessary to use such formulas.

Recall that in financial mathematics, to determine the outstanding balance for a loan at a certain time point, we have Template:Colored em and Template:Colored em methods. Indeed, the definition of reserves are prospective. Can we calculate the reserves using a retrospective way? There are actually retrospective formulas for calculating premiums, as illustrated in the following exercise. Template:Colored exercise Template:Colored remark Indeed, the equality of prospective reserve and retrospective reserve under such conditions also applies to other types of policies. Template:Colored proposition

Proof. Let $L_{0,t}$ be the sum of p.v.r.v. of benefits and expenses from time 0 to $t$ less the p.v.r.v. of future premiums from time 0 to $t$. Then, the retrospective reserve is symbolically $$\frac{-𝔼[L_{0,t}]}{{}_t{E}_x}.$$ Also, the insurer's loss at policy issue is $$L_0=L_{0,t}+v^t{L}_t^g\mathrm{𝟏}\{T_x>t\}.$$ The equality holds since the same basis is used for all three random variables (the basis for $L_0$ is the same as the basis for premiums because they are related by equivalence principle).

Then, by equivalence principle, we have $$\begin{array}{cccc}& & 𝔼[L_0]& =0\\ & \Rightarrow & 𝔼[L_{0,t}]+v^t𝔼[L_t^g\mathrm{𝟏}\{T_x>t\}]& =0\\ & \Rightarrow & 𝔼[L_{0,t}]& =-v^t𝔼[L_t^g\mathrm{𝟏}\{T_x>t\}]\\ & \Rightarrow & 𝔼[L_{0,t}]& =-v^t\mathbb{P}(T_x>t)𝔼[L_t^g|T_x>t]& (\text{result in probability})\\ & \Rightarrow & 𝔼[L_{0,t}]& =-\underset{{}_t{E}_x}{\underbrace{v^t{}_t{p}_x}}𝔼[L_t^g|T_x>t]\\ & \Rightarrow & \underset{\text{retrospective}}{\underbrace{\frac{-𝔼[L_{0,t}]}{{}_t{E}_x}}}& =\underset{\text{prospective}}{\underbrace{𝔼[L_t^g|T_x>t]}}.\\ \end{array}$$

$◻$

Fully discrete reserves

Similar to the case for continuous reserves, we often use the "alternative method" to calculate the discrete reserves (i.e., $𝔼[L_k^n|K_x\ge k]$): the conditional distribution of $K_x-k$ given that $K_x\ge k$ is the same as the unconditional distribution of $K_{x+k}$ ($k$ is a nonnegative integer).

Proof. Consider the survival functions of the two distributions. First, for the conditional distribution of $K_x-k$ given that $K_x\ge k$, $$\mathbb{P}(K_x-k\ge m|K_x\ge k)=\frac{\mathbb{P}(K_x\ge k+m)}{\mathbb{P}(K_x\ge k)}=\frac{\mathbb{P}(T_x\ge \lfloor k+m\rfloor )}{\mathbb{P}(T_x\ge \lfloor k\rfloor )}=\frac{\mathbb{P}(T_x\ge k+m)}{\mathbb{P}(T_x\ge k)}.$$ ($m$ and $k$ are integers) For the unconditional distribution of $K_{x+k}$, $$\mathbb{P}(K_{x+k}\ge m^{\prime })=\mathbb{P}(T_{x+k}\ge \lfloor m^{\prime }\rfloor )=\mathbb{P}(T_{x+k}\ge m^{\prime })=\mathbb{P}(T_x\ge k+m^{\prime }|T_x\ge k)=\frac{\mathbb{P}(T_x\ge k+m^{\prime })}{\mathbb{P}(T_x\ge k)}.$$ ($m^{\prime }$ and $k$ are integers) These show that the two distributions are the same.

$◻$

Similarly, this result gives us an alternative and often more convenient method to calculate the conditional expectation $𝔼[L_k^n|K_x\ge k]$:

1. replace all "$K_x-k$" by "$K_{x+k}$" and remove the condition "$K_x\ge k$"
2. calculate the unconditional expectation, which equals the value of conditional expectation since the distributions involved are the same

Let us consider the whole life insurance first. We have $$L_k^n=v^{K-k+1}-P_x{\ddot{a}}_{\overline{K-k+1}|}.$$ Hence, the reserve, denoted by ${}_k{V}_x$, is $$𝔼[L_k^n|K_x\ge k]=𝔼[v^{K_x-k+1}-P_x{\ddot{a}}_{\overline{K_x-k+1}|}|K_x\ge k]=𝔼[v^{K_{x+k}+1}]-P_x𝔼[{\ddot{a}}_{\overline{K_{x+k}+1}|}]=A_{x+k}-P_x{\ddot{a}}_{x+k}.$$ Template:Colored exercise We can develop formulas for other types of policies, and a summary of the formulas for reserves is tabulated below.

Summary

Insurance product name	net premium reserve at time $k$
Whole life insurance	${}_k{V}_x=A_{x+k}-P_x{\ddot{a}}_{x+k}$
$n$-year term life insurance	${}_k{V}_{x:\bar{n}|}^1=\{\begin{array}{cc}{A}_{x+k:\overline{n-k}|}^1-P(A_{x:\bar{n}|}^1){\ddot{a}}_{x+k:\overline{n-k}|},& k<n;\\ 0,& k=n.\end{array}$
$n$-year endowment insurance	${}_k{V}_{x:\bar{n}|}=\{\begin{array}{cc}{A}_{x+k:\overline{n-k}|}-P(A_{x:\bar{n}|}){\ddot{a}}_{x+k:\overline{n-k}|},& k<n;\\ 1,& k=n.\end{array}$
$h$-payment years whole life insurance	$\underset{h}{\overset{}{k}}{V}_x=\{\begin{array}{cc}{A}_{x+k}-{}_{h}P(A_x){\ddot{a}}_{x+k:\overline{h-k}|},& k<h;\\ A_{x+k},& k\ge h.\end{array}$
$h$-payment years $n$-year term life insurance ($h<n$)	$\underset{h}{\overset{}{k}}{V}_{x:\bar{n}|}^1=\{\begin{array}{cc}{A}_{x+k:\overline{n-k}|}^1-{}_{h}P(A_{x:\bar{n}|}^1){\ddot{a}}_{x+k:\overline{h-k}|},& k<h;\\ A_{x+k:\overline{n-k}|}^1,& h\le k<n;\\ 0,& k=n.\end{array}$
$h$-payment years $n$-year endowment insurance ($h<n$)	$\underset{h}{\overset{}{k}}{V}_{x:\bar{n}|}=\{\begin{array}{cc}{A}_{x+k:\overline{n-k}|}-{}_{h}P(A_{x:\bar{n}|}){\ddot{a}}_{x+k:\overline{h-k}|},& k<h;\\ A_{x+k:\overline{n-k}|},& h\le k<n;\\ 1,& k=n.\end{array}$
$n$-year pure endowment	${}_k{V}_{x:\bar{n}|}^1=\{\begin{array}{cc}{}_{n-k}{E}_{x+k}-P({}_n{E}_x){\ddot{a}}_{x+k:\overline{n-k}|},& k<n;\\ 1,& k=n.\end{array}$
$n$-year deferred whole life annuity	${}_{k}V({}_{n|}{\ddot{a}}_x)=\{\begin{array}{cc}{}_{n-k|}{\ddot{a}}_{x+k}-P({}_{n|}{\ddot{a}}_x){\ddot{a}}_{x+t:\overline{n-k}|},& k\le n;\\ {\ddot{a}}_{x+k},& k>n.\end{array}$

Template:Colored remark Template:Colored example Template:Colored example Template:Colored remark We can also develop premium-difference formula and paid-up insurance formula for discrete endowment insurance similarly: Template:Colored exercise Template:Colored exercise We can develop a recursion relation for discrete insurances: Template:Colored proposition Template:Colored remark

Proof. Split the prospective net loss at time $k$: $L_k^n$, into two parts (i.e., $L_k^n$ is the sum of these two parts):

1. p.v.r.v. of benefits at time $k+1$ minus p.v.r.v. of premiums at time $k$ (denoted by $A$)
2. p.v.r.v. of benefits at time $k+2,k+3,\dots$ minus p.v.r.v. of premiums at time $k+1,k+2,\dots$ (denoted by $B$)

Thus, we have $$L_k^n=A+B.$$ Taking expectation, we have $${}_{k}V=𝔼[A]+𝔼[B].$$ First, $𝔼[A]=v{q}_{x+k}{b}_{k+1}-P_k$. Second, to be more explicit about the condition incorporated in $L_k^n$, we can write $𝔼[B|K_x\ge k]$ in place of $𝔼[B]$: $$\begin{array}{cc}𝔼[B|K_x\ge k]& =\frac{𝔼[B\mathrm{𝟏}\{K_x\ge k\}]}{\mathbb{P}(K_x\ge k)}\\ & =\frac{𝔼[B\mathrm{𝟏}\{K_x\ge k+1\}]}{\mathbb{P}(K_x\ge k)}& (\text{the first term in }B\text{ is at time }k+1,\text{so the change does not affect the expectation})\\ & =\frac{𝔼[B\mathrm{𝟏}\{K_x\ge k+1\}]}{\mathbb{P}(K_x\ge k+1)}\cdot \frac{\mathbb{P}(K_x\ge k+1)}{\mathbb{P}(K_x\ge k)}\\ & =𝔼[B|K_x\ge k+1]\cdot \mathbb{P}(K_x\ge k+1|K_x\ge k)\\ & =p_{x+k}𝔼[B|K_x\ge k+1]\\ & =p_{x+k}𝔼[v{L}_{k+1}^n|K_x\ge k+1]\\ & =v{p}_{x+k}𝔼[L_{k+1}^n|K_x\ge k+1]\\ & =v{p}_{x+k}{}_{k+1}V.\\ \end{array}$$ The result follows.

$◻$

To explain the proof more intuitively, consider the following diagram:

                                                                       
                                                                            *
                  |-------------------- ...                                  \
                  |         b ... b ...                                       \
                  |P_{k+1}  P ... P ...     <==== covered by _{k+1} V          *    covered by _k V
                  |------------------- ....                                   /
    P_k          b_{k+1} <=== not covered by _{k+1}V                         *
    _k V        _{k+1} V                                                     
-----*------------*----------------------
     k           k+1                   time

• Consider the policy value at time $k$: ${}_{k}V$. It is the APV of future benefits minus the APV of future premiums.
• We can split the future benefits and future premiums into two parts:

1. benefit at time $k+1$ and premium at time $k$
2. benefits at time $k+2,k+3,\dots$ and premiums at $k+1,k+2,\dots$

• For the second part, they are incorporated by the policy value at time $k+1$ ${}_{k+1}V$. But of course the policy value at time $k+1$ gives the value at time $k+1$, but not the value at time $k$ (which is what we want). Hence, we need to actuarially discount ${}_{k+1}V$ back to time $k$ (multiply $v{p}_x$).
• To incorporate the first part, we add the APV of death benefit at time $k+1$ ($v{b}_{k+1}{q}_{x+k}$), and then subtract the premium at time $k$ ($P_k$).