A review of the Markov model of life insurance with a view to surplus

TL;DR

This paper systematically reviews Markov surplus models in life insurance, integrating probabilistic theory with practical UK experience, and extends the models to include dynamic cashflows like reversionary bonuses.

Contribution

It introduces a systematic organization of Markov surplus models with multiple technical bases and expands the definition to include non-contractual cashflows, enhancing model applicability.

Findings

Expanded the definition of 'technical basis' to include non-contractual cashflows.

Developed systematic terms to incorporate dynamic, quasi-contractual cashflows.

Applied models to practical examples like paid-up valuation and reversionary bonuses.

Abstract

We review Markov models of surplus in life insurance based on a counting process following Norberg (1991), uniting probabilistic theory with elements of practice largely drawn from UK experience. First, we organize models systematically based on one and two technical bases, including a suitable descriptive notation. Extending this to three technical bases to accommodate different valuation approaches leads us: (a) to expand the definition of 'technical basis' to include non-contractual cashflows recognized in the associated Thiele equation; and (b) to add new (mainly) systematic terms to the surplus. Making these cashflows dynamic or 'quasi-contractual' covers many real applications, and we give two as examples, the paid-up valuation principle and reversionary bonus on participating contracts.

Tables

Table 1. Table 1: Characteristics of technical bases in terms of boundary conditions satisfied by solutions of Thiele’s differential equation. Solutions are deterministic functions of time t t .

Basis	Name	Type	Solution	Boundary Condition(s)
${ℬ}^P$	Premium	Special Valuation	$V_t^P$	$V_0^P=0,V_n^P=\overline{S}$
${ℬ}^L$	Valuation	Valuation	$V_t^L$	$V_n^L=\overline{S}$
${ℬ}^A$	Accumulation	Accumulation	$W_t^A$	$W_0^A=0$
${ℬ}^M$	Experience	Accumulation	$W_t^M$	See Section 4.3.

Table 2. Table 2: ‘ Θ \Theta ’ notation for stochastic and modeled surplus equations at time t t , with a single (generic) technical basis ℬ Z {\cal B}^{Z} and value of contractual cashflows X ​ ( t ) X(t) .

Name	Notation	Definition	Meas.	Equation
Realized	${\Theta }^{\omega ,\omega }\left(t\right)$	$X\left(t\right)$	${ℱ}_n$	(20)
Stochastic	${\Theta }^{Z,\omega }\left(t\right)$	${\text{E}}_Z\left[X\left(t\right)\mid {ℱ}_t\right]$	${ℱ}_t$	(33), (34)
Stochastic discounted	${\tilde{\Theta }}^{Z,\omega }\left(t\right)$	$v^Z\left(t\right){\text{E}}_Z\left[X\left(t\right)\mid {ℱ}_t\right]$	${ℱ}_t$	(35)
Modeled	${\Theta }^{Z,Z}\left(t\right)$	${\text{E}}_Z\left[{\text{E}}_Z\left[X\left(t\right)\mid {ℱ}_t\right]\mid {ℱ}_0\right]$	${ℱ}_0$	(36), (37)
Modeled discounted	${\tilde{\Theta }}^{Z,Z}\left(t\right)$	${\text{E}}_Z\left[v^Z\left(t\right){\text{E}}_Z\left[X\left(t\right)\mid {ℱ}_t\right]\mid {ℱ}_0\right]$	${ℱ}_0$	(38), (39).

Table 3. Table 3: ‘ Θ \Theta ’ notation for stochastic and modeled surplus equations at time t t , with first-order technical basis ℬ L {\cal B}^{L} and second-order (experience) basis ℬ M {\cal B}^{M} . Value of contractual cashflows is X ​ ( t ) X(t) .

Name	Notation	Definition	Meas.
Stochastic	${\Theta }^{L,\omega }\left(t\right)$	${\text{E}}_L\left[X\left(t\right)\mid {ℱ}_t\right]$	${ℱ}_t$
Stochastic discounted	${\tilde{\Theta }}^{L,\omega }\left(t\right)$	$v^M\left(t\right){\text{E}}_L\left[X\left(t\right)\mid {ℱ}_t\right]$	${ℱ}_t$
Modeled	${\Theta }^{L,M}\left(t\right)$	${\text{E}}_M\left[{\text{E}}_L\left[X\left(t\right)\mid {ℱ}_t\right]\mid {ℱ}_0\right]$	${ℱ}_0$
Modeled discounted	${\tilde{\Theta }}^{L,M}\left(t\right)$	${\text{E}}_M\left[v^M\left(t\right){\text{E}}_L\left[X\left(t\right)\mid {ℱ}_t\right]\mid {ℱ}_0\right]$	${ℱ}_0$.

Table 4. Table 4: Characteristics of actuarial bases.

Name	Definition	Comment on Cashflow Component
Premium	${𝒜}^P=({ℬ}^P,B^P\left(t\right))$	By definition $B^P\left(t\right)\equiv B\left(t\right)$
Valuation	${𝒜}^L=({ℬ}^L,B^L\left(t\right))$	Valuation premium rate ${\tau }_t^L$ is part of $B^L\left(t\right)$
Accumulation	${𝒜}^A=({ℬ}^A,B^A\left(t\right))$	Default is $B^A\left(t\right)=B\left(t\right)$, there may be exceptions
Experience	${𝒜}^M=({ℬ}^M,B^M\left(t\right))$	‘True’ experience, by definition $B^M\left(t\right)\equiv B\left(t\right)$.

Table 5. Table 5: Notation for stochastic and modeled surpluses with cashflows X L t , M ​ ( t ) X^{L_{t},M}(t) allowing for past and future bonus declarations.

Type	Notation	Definition
Stochastic	${\Theta }^{L_t,\omega }\left(t\right)$	${\text{E}}_L\left[X^{L_t,M}\left(t\right)\mid {ℱ}_t\right]$
Stochastic discounted	${\tilde{\Theta }}^{L_t,\omega }\left(t\right)$	$v^M\left(t\right){\text{E}}_L\left[X^{L_t,M}\left(t\right)\mid {ℱ}_t\right]$
Modeled	${\Theta }^{L_t,M}\left(t\right)$	${\text{E}}_M\left[{\text{E}}_L\left[X^{L_t,M}\left(t\right)\mid {ℱ}_t\right]\mid {ℱ}_0\right]$
Modeled discounted	${\tilde{\Theta }}^{L_t,M}\left(t\right)$	${\text{E}}_M\left[v^M\left(t\right){\text{E}}_L\left[X^{L_t,M}\left(t\right)\mid {ℱ}_t\right]\mid {ℱ}_0\right]$.

Equations

$$\phi (t)=\exp \left(-{\int }_0^t({\delta }_r+{\mu }_r)dr\right).$$

$$0={\int }_0^n\phi (r)(P-{\mu }_{r}S)dr-\phi (n)\bar{S}.$$

$$0=\underbrace{\int_{0}^{t}\frac{\varphi(r)}{\varphi(t)}\,(P-\mu_{r}\,S)\,dr}_{\mbox{\scriptsize Retrospective Policy Value = $W_{t}$}}-\underbrace{\int_{t}^{n}\frac{\varphi(r)}{\varphi(t)}\,(\mu_{r}\,S-P)\,dr-\frac{\varphi(n)}{\varphi(t)}\,\bar{S}}_{\mbox{\scriptsize Prospective Policy Value = $V_{t}$}}$$

$$\frac{d}{dt}g(t)={\delta }_{t}g(t)+P-{\mu }_t(S-g(t))$$

$$N(t)=1_{\{T\leq t\}}=\left\{\begin{array}[]{ll}0&\mbox{if }T>t\\ 1&\mbox{if }T\leq t.\end{array}\right.$$

$$J(t)=\mbox{state in ${\cal S}$ occupied at time $t$}\qquad(t\geq 0).$$

$$1_{\{T>t\}}=1_{\{N(t)=0\}}=1_{\{J(t)=1\}}.$$

$$dN(t)=-d\{1_{\{N(t)=0\}}\}=\big{(}1_{\{T\geq t\}}-1_{\{T>t\}}\big{)}.$$

$$B(t)={\int }_0^{t}dB(r)(0\le t\le n).$$

$$dB(t)=\left\{\begin{array}[]{ll}1_{\{N(t)=0\}}\,P_{t}\,dt-S_{t}\,dN(t)&\quad t<n\\[2.15277pt] 1_{\{N(n)=0\}}\,P_{t}\,dt-S_{n}\,dN(n)-1_{\{N(n)=0\}}\,\bar{S}&\quad t=n.\end{array}\right.$$

$$F(t)=\text{mbox}P[T\le t].$$

$${\mu }_t=\underset{dt\to 0}{\lim }\frac{\text{mbox}P[N(t+dt)=1\mid N(t)=0]}{dt}=\underset{dt\to 0}{\lim }\frac{1}{dt}\frac{F(t+dt)-F(t)}{1-F(t)}$$

$$\text{mbox}{E}_{\mathbb{P}}[A\mid {\mathcal{F}}_t]=\text{mbox}{E}_{\mathbb{P}}[A\mid N(t)]\text{mbox}forallA\in \mathcal{F}.$$

$$\text{mbox}{E}_{\mathbb{P}}[dN(t)\mid {\mathcal{F}}_t]=\text{mbox}{E}_{\mathbb{P}}[1_{\{N(t)=0\}}{\mu }_{t}dt\mid {\mathcal{F}}_t]=1_{\{N(t)=0\}}{\mu }_{t}dt$$

$$\frac{d}{dt}g(t)={\delta }_t^{Z}g(t)+P_t-{\mu }_t^Z(S_t-g(t)).$$

$$v^Z(t)=\exp \left(-{\int }_0^t{\delta }_r^{Z}dr\right),p^Z(t)=\exp \left(-{\int }_0^t{\mu }_r^{Z}dr\right),{\phi }^Z(t)=v^Z(t)p^Z(t).$$

$$g(t)$$

$$X(t)$$

$$=$$

$$\mbox{E}\left[\int_{t}^{n}\frac{v(r)}{v(t)}\,d(-B(r))\,\Big{|}\,{\cal F}_{t}\right]$$

$$\text{mbox}E[X(t)\mid {\mathcal{F}}_t]$$

$$=$$

$$1_{\{N(t)=0\}}\,V_{t}=\mbox{E}\left[\int_{t}^{n}\frac{v(r)}{v(t)}\,d(-B(r))\,\Big{|}\,{\cal F}_{t}\right].$$

$$\displaystyle d\big{\{}v(t)\,1_{\{N(t)=0\}}\,V_{t}\big{\}}$$

$$\frac{v(n)}{v(t)}{1}_{\{N(n)=0\}}{V}_n-\frac{v(t)}{v(t)}{1}_{\{N(t)=0\}}{V}_t$$

$$g(t)=\mbox{E}\left[\int_{t}^{n}\frac{v(r)}{v(t)}\,d(-B(r))\,\Big{|}\,{\cal F}_{t},N(t)=0\right]$$

$$W_{t}=\frac{1}{\varphi(t)}\,\mbox{E}\left[\int_{0}^{t}v(r)\,dB(r)\,\Big{|}\,{\cal F}_{0}\right].$$

$$\text{mbox}E[X(t)\mid {\mathcal{F}}_t]$$

$$=$$

Full text

keywords

Actuarial Basis, Life Insurance, Loadings, Surplus, Technical Basis

contact address

†  Department of Actuarial Sciences, The Faculty of Business, Karabük University, Karabük, 78050, Turkey, and Institute of Applied Mathematics, Middle East Technical University, Ankara, 06800, Turkey.

‡  Research Centre for Longevity Risk, Faculty of Economics and Business, University of Amsterdam.

§  Department of Actuarial Mathematics and Statistics, Heriot-Watt University, Edinburgh EH14 4AS, UK, and the Maxwell Institute for Mathematical Sciences, UK.

Corresponding author: Angus Macdonald, A.S.Macdonald@hw.ac.uk.

1 Introduction

1.1 Motivation: From Northampton to Copenhagen

In 1989, Hans Bühlmann designated life insurance actuaries as ‘actuaries of the first kind’ and called them ‘essentially deterministic’ (Bühlmann 1989). The motto of the Institute of Actuaries, certum ex incertis, tells us why; it means ‘certainty out of uncertainty’, and life insurance actuaries were descended from 17th and 18th century probabilists whose key idea was expectation rather than probability (Daston 2023). Their main tools for managing risk were: (a) safe-side margins in premiums and reserves, and: (b) retrospective distribution of any surpluses that emerged. The technical basis used to calculate premiums would build loadings into the rates to be charged, and the technical basis used to calculate reserves then had to stop loadings in future premiums from emerging as surplus too soon. A natural and passive approach was to use the same technical basis for both premiums and reserves, known as the ‘Northampton method’ after the life table originally used by the Equitable Life in the UK (Fisher & Young 1965).

However, life insurance contracts have very long terms indeed. In time, the original technical basis may become archaic, so that using it to calculate the liability seems absurd, raising serious doubts about the balance sheet. During the 19th century, the Northampton method was abandoned by UK companies, in favour of more active valuations on more up-to-date technical bases. Debates on methodology enlivened the early years of the UK actuarial profession (Turnbull 2017). In time for the age of computers, profit-testing emerged (Anderson 1959), in which future loadings were calculated and valued individually and explicitly.

A first step in moving actuaries ‘of the first kind’ beyond deterministic thinking was to express life table probabilities such as ${}_{t}p_{x}$ and actuarial present values such as $\bar{A}_{x}$ as respectively $\mbox{P}[X]$ and $\mbox{E}[Y]$, for well-defined random variables $X$ and $Y$ (the latter involving discounting as well as mortality). Beginning with Hickman (1964), the future lifetime of the canonical ‘individual alive at age $x$’, named $(x)$, was defined to be a random variable $T_{x}$. In the course of only a few generations this has become the usual basis for educating student actuaries (Bowers et al. 1986, Gerber 1990, Dickson et al. 2020). So ${}_{t}p_{x}=\mbox{P}[T_{x}>t]$, the present value of a whole-life benefit of £1 payable immediately on the death of $(x)$ is $v^{T_{x}}$, and by definition $\bar{A}_{x}=\mbox{E}[v^{T_{x}}]$.

The landmark papers Hoem (1969, 1988) introduced actuaries to multiple-state Markov models. The advance from an alive-dead mortality model to multiple states perhaps obscured the advance in thinking represented by modelling a process. A focus on transition probabilities and intensities (often rendered as ${}_{t}p_{x}^{ij}$ and $\mu_{x+t}^{ij}$ in homage to tradition) invited again the question, ‘what is the underlying random object?’ The answer was in Hoem & Aalen (1978), which defined cashflows contingent upon the counting process $N(t)$, counting the number of deaths up to and including time $t$. The seeming triviality of $N(t)$, which takes only the values 0 or 1, may have concealed its virtues. Based on $N(t)$, Aalen (1978) launched a revolution in survival modelling, see Andersen et al. (1993) and Aalen et al. (2022).

Bühlmann (1976) had suggested modelling the surplus under a life insurance contract using a suitable martingale. With $N(t)$ (in its multivariate version) representing underlying stochastic events, Ramlau-Hansen (1988a, b) carried this through in great generality for Markov models. Norberg (1991) explored the definitions of reserves, prospective and retrospective, in terms of conditional expectations with respect to quite general streams of information, not only filtrations111In this work we assume that information is represented by filtrations. See Norberg (1991) for an example in a multiple-state setting where this is not the case.. This work essentially identified technical bases as probability measures. Extensions to handle the distribution of surplus as bonus were added in Ramlau-Hansen (1991) and Norberg (1999, 2001).

Much of this work by what may justly be called the ‘Scandinavian school’ was summarized in the encyclopedia article Norberg (2004). However, all of it assumes what we call ‘Scandinavian-style regulation’. That is, premiums and reserves are calculated on the same safe-side ‘first order’ technical basis, and the experience is represented by another ‘second order’ technical basis. The reserve at outset is zero, and surplus emerges as the difference between the two technical bases. In other words, the Northampton method.

Thus we have two lines of approach, even two traditions: the long history and variety of practice that provoked Bühlmann to define ‘actuaries of the first kind’; and in a more limited setting, the stochastic process approach of the Scandinavian school. Most students today are shown a half-way house between the two, namely life insurance mathematics in terms of the random lifetime $T_{x}$.

1.2 Objectives

• (a)

As well as Norberg (2004), excellent graduate-level books have been written (Møller & Steffensen (2007), Koller (2012), Asmussen & Steffensen (2020)) but the basic counting process framework is still scattered in the literature. Norberg (2004) noted that: “[an] account of counting processes and martingale techniques can be compiled from Hoem & Aalen (1978) and Norberg (1999)” and doing that is where we start.

• (b)

However, Scandinavian-style regulation confines margins (loadings) to the elements of the technical basis, which is too limiting. To describe more realistic regimes we need to extend valuation technical bases to include hypothetical policy cashflows. The main practical effect is to define loadings in the premiums, and to control when these emerge in the surplus.

• (c)

Further, some practical applications require policy cashflows to be modelled dynamically, in which case we call them quasi-contractual. We finish with two such examples; the paid-up valuation principle of Linnemann (2002), and reversionary bonus.

Mention of the stochastic present value $v^{T_{x}}$ in Section 1.1 invites two questions. One is how to stochasticize the discount factor $v$? This would lead us into financial economics. The other is how to generalize the biometric model represented by $T_{x}$? That would lead us to the multivariate version of $N(t)$, the unstated stochastic object underlying Hoem (1969, 1988). Along such lines, Norberg (1992) essentially recast life insurance mathematics into the same form as no-arbitrage financial mathematics. Both questions are subsidiary to our particular aim and to try to address them would simply add a great deal of notation, so apart from a few remarks we leave them aside.

1.3 Plan of the Paper

In Section 2 we start with an elementary model of a balance sheet to fix ideas. Section 3 defines notation for cashflows based on stochastic representations of a lifetime. We leave aside multiple-state models and stochastic assets to focus on the structure of surplus; these extensions follow along clear lines laid down in the literature. Section 4 defines technical bases in terms of Thiele’s differential equation, and the key identification of policy values both as solutions of Thiele’s equation and as conditional expectations. Section 5 defines assets and liabilities in the model balance sheet, leading to the definition of surplus, then in Sections 6 and 7 we introduce surplus based respectively on one or two technical bases, including a systematic notation. Thus far we review material sketched in Norberg (2004) up to and including Scandinavian-style regulation. Section 8 separates premium and valuation technical bases and defines both pure and valuation premiums and the associated loadings as in Haçarız et al. (2024). Section 9 generalizes this by defining an actuarial basis to be a technical basis and associated policy cashflows assumed in the associated Thiele equation. In Section 10 we extend the analysis of surplus to three technical bases, adding a loading term to the systematic component of surplus. To handle cases in practice where the actual policy cashflows are only quasi-contractual, subject to options at the choice of one party, Section 11 introduces a time-indexed family of actuarial bases, and we present two examples, the paid-up valuation principle in Section 12, and reversionary bonus in Section 13. Conclusions are in Section 14.

2 An Elementary Model of a Life Insurer’s Balance Sheet

2.1 A Deterministic Model

To fix ideas, consider a deterministic model of a life insurance policy with term $n$ years, commencing at time $t=0$ on an individual aged $x$, who is hereafter referred to as $(x)$. A sum insured of $S$ is paid immediately on death at time $t$, and a maturity benefit of $\bar{S}$ (possibly zero) is paid on survival for $n$ years. Level premiums are paid throughout the term at rate $P$ per year. We ignore expenses.

The technical basis consists of a force of interest $\delta_{t}$ and force of mortality $\mu_{x+t}$ (which we denote by $\mu_{t}$, taking age $x$ as understood) at time $t\geq 0$. Define $\varphi(t)$ to be a discount factor allowing for survivorship as well as interest:

[TABLE]

The premium rate $P$ is the solution of the equation of value:

[TABLE]

Dividing both sides by $\varphi(t)$ and splitting the integral at time $t$ $(0\leq t\leq n)$ we have:

[TABLE]

in which we identify the assets, or retrospective policy value, denoted by $W_{t}$, and the liabilities, or prospective policy value, denoted by $V_{t}$, equal at all times. Both are ‘per policy in-force at time $t$’, and are solutions of Thiele’s differential equation:

[TABLE]

satisfying the respective boundary conditions $g(0)=0$ and $g(n)=\bar{S}$. Surplus is defined as $W_{t}-V_{t}$ and is uniformly zero. This model is illustrated by panel (a) of Figure 1, which shows the smooth progression of term insurance policy values with $x=40,n=20,(\bar{S}=0)$ on three different technical bases222The contract is a 20-year term assurance for a man age 40, and the technical basis for both (level) premium rate and policy values (referred to as the first-order technical basis) is interest of $\delta=0.05$ per annum and the mortality of the Danish GM82 Males life table, with no expenses..

2.2 Valuation Functions

The valuation problem may now be stated. Let $\mathbf{H}=\{{\cal H}_{t}\}_{t\geq 0}$ represent a flow of information (filtration) available at times $t\geq 0$. We need to choose appropriate asset and liability valuation functions $W_{t}^{*}({\cal H}_{t})$ and $V_{t}^{*}({\cal H}_{t})$ to replace $W_{t}$ and $V_{t}$ from Section 2.1. It is important to realize that functions $W_{t}^{*}$ and $V_{t}^{*}$ can be anything at all that insurers find useful and practical, and regulators allow or require. For example, an older practice was to take assets at book values rather than market values. More recently, perfect matching or a self-financing condition would require that $W_{t}^{*}({\cal H}_{t})=V_{t}^{*}({\cal H}_{t})$ at all times $t$; equation (3) but in a stochastic world. A first step towards achieving mathematical coherence in practice is to find a framework in which practices can be expressed.

2.3 The Stochastic Time-evolution of the Balance Sheet

Panel (b) of Figure 1 illustrates the position half-way through the policy term. Asset values are accumulations of past cashflows; two possible sample paths or scenarios are shown333The stochastic scenarios shown in Figure 1 are merely indicative of stochastic irregularity, based on a large portfolio of policies, they have not been generated by any of the models discussed here.. Liabilities are the in-force policy values, shown on three different technical bases as before. The difference between assets and liabilities at $t=10$ is the accumulated surplus. Panel (c) shows final outcomes at $t=20$ in both stochastic scenarios.

3 Notation: Life History, Cashflow, Interest, Probability, Information

3.1 Representing Life Histories

There are three ways to represent the future lifetime of ($x$), assuming death to be the only stochastic event. We use whichever is convenient, and call it the biometric model.

• (a)

Random Lifetime $T$: The time until death is represented by a non-negative random variable $T$.

• (b)

Counting Process $N(t)$: The life history is represented by the right-continuous counting process $N(t)$ defined as:

[TABLE]

• (c)

State Process $J(t)$: Define the state space $\mathcal{S}=\{1,2\}$ representing ‘alive’ and ‘dead’ statuses and a right-continuous process $J(t)$ as follows:

[TABLE]

Then suppose that $J(t)$ jumps from state 1 to state 2 at time $T$. This extends in an obvious way to a multiple-state model on a larger state space $\mathcal{S}=\{1,2,\ldots,K\}$.

Associated with the event of death is the right-continuous indicator of being alive at time $t$ which is defined in terms of any of the above representations444In this paper we use $1_{\{N(t)=0\}}$ simply because so many expressions are written in terms of $dN(t)$. In multiple-state models the form $1_{\{J(t)=1\}}$ generalizes most easily. as:

[TABLE]

In survival analysis the left-continuous $1_{\{T\geq t\}}=1_{\{N(t^{-})=0\}}=1_{\{J(t^{-})=1\}}$ is used instead because of its rôle in estimators written as stochastic integrals. We have the identities:

[TABLE]

3.2 Representing Contractual Cashflows

We assume a life insurance contract commences at time $t=0$ and has finite term $n$ years. Let $B(t)$ $(0\leq t\leq n)$ be the stochastic cumulative cashflow generated on the interval $[0,t]$ by the contractual payments, with the convention that cashflows to the insurer are positive. Randomness in $B(t)$ derives from the biometric model in Section 3.1, and lump-sum payments at non-random times are denoted by $\Delta B(t)=B(t)-B(t^{-})$. For simplicity we ignore such payments except at $t=n$, in particular we suppose that $B(0)=0$. We assume that $B(t)$ may represented by the Stieltjes integral:

[TABLE]

To make the presentation even simpler we define a rate of premium $P_{t}$ and death benefit $S_{t}$ for $t\in(0,n)$, and maturity benefit $\bar{S}$ at $t=n$. Therefore:

[TABLE]

Traditional actuarial value functions are defined ‘per-policy now in force’. The indicators in equation (10) means cashflows are ‘per policy in force at $t=0$’.

3.3 Representing Interest, Accumulation and Discounting

We assume a left-continuous deterministic force of interest $\delta_{t}$, such that \exp\big{(}\int_{0}^{t}\delta_{r}\,dr\big{)} represents the amount to which $1 invested at time 0 accumulates at time$t$. The discount factor$v(t)$is defined as$\exp\big{(}-\int_{0}^{t}\delta_{r},dr\big{)}$. We comment on stochastic asset returns in Section 3.5.2.

3.4 Distribution Function and Hazard Rate

The distribution function of $T$, denoted by $F(t)$, is:

[TABLE]

The hazard rate (alternatively intensity, force of mortality) denoted by $\mu_{t}$ is then:

[TABLE]

which is assumed always to exist555Boundedness and integrability of transition intensities on closed intervals is enough to rule out ‘explosions’ in models which can jump more than once (Jacobsen 2006). Here, it allows the hazard to be unbounded as $t\to\infty$ (for example, a Gompertz-Makeham model) but rules out models with a finite limiting lifetime ($F(t)=1$ for all $t>\omega$ with $\omega<\infty$). This restriction is easily removed..

3.5 Representing Information

3.5.1 Biometric Risk

We assume a filtered probability space ${\cal S}=(\Omega,{\cal F},\mathbf{F},\mathbb{P})$, where $\mathbf{F}=\{\mathcal{F}_{t}\}_{t\geq 0}$ is a filtration with ${\cal F}={\cal F}_{\infty}=\lim_{t\to\infty}{\cal F}_{t}$. We suppose the points $\omega\in\Omega$ are sample paths of $N(t)$, also called scenarios, and list some assumptions below.

• (a)

The filtration is the natural filtration generated by $N(t)$, {\cal F}_{t}=\sigma\big{(}\{N(s)\}_{0\leq s\leq t}\big{)}, so $N(t)$ and $1_{\{N(t)=0\}}$ are ${\cal F}_{t}$-measureable.

• (b)

The measure $\mathbb{P}$ is that induced by the distribution function $F(t)$ or hazard rates $\mu_{t}$.

• (c)

The Markov property is conferred on the process and its filtration (or not) by $\mathbb{P}$, and is satisfied if:

[TABLE]

• (d)

The martingale property holds:

[TABLE]

equivalently, the process $M(t)$ defined as $N(t)-\int_{0}^{t}1_{\{N(r)=0\}}\,\mu_{r}\,dr$ is an ${\cal F}_{t}$-martingale under $\mathbb{P}$.

3.5.2 Financial Risk

Ideally, the filtration $\mathbf{F}$ would be subsidiary to a larger filtration $\mathbf{H}$ in a stochastic model of financial and biometric risk. If we had a stochastic asset model with natural filtration $\mathbf{G}$, we might combine the underlying spaces in such a way that they support a combined filtration $\mathbf{H}=\mathbf{F}\vee\mathbf{G}$, and define stochastic models by probability measures on ${\cal H}_{\infty}$. This approach has indeed been explored, see for example Asmussen & Steffensen (2020) but little progess has been made beyond assuming financial and biometric risk to be independent666Norberg (1999) provided a stochastic environment by allowing the parameters of financial and biometric models (here, ($\delta_{t},\mu_{t}$)) to be a stochastic process, generating the filtration $\mathbf{G}=\{\mathcal{G}_{t}\}_{t\geq 0}$. Norberg (p.379) assumed that: “$\ldots$ the sample paths of the processes $\delta$ and $\mu_{jk}$ almost surely possess the properties assumed in Section 2 $\ldots$” (namely a Markov model in which the Kolmogorov equations hold, and interest that may be expressed through a discount factor $\exp(-\int\delta_{r}\,dr)$). Then all expectations at time $t$ were conditioned on $\mathcal{F}^{\prime}_{t}\vee\mathcal{G}_{t}$ where $\mathcal{F}^{\prime}_{t}\subseteq\mathcal{F}_{t}$. The implication of conditioning on $\mathcal{G}_{t}$ was that the realized $(\delta_{r},\mu_{r})$ for $r\in[0,t]$ could be treated as if it was deterministic; in other words, carry on as before. Norberg pointed out that “$\ldots$ no particular specification of the marginal distribution [of $(\delta_{r},\mu_{r})$] is needed $\ldots$”.. In the text we assume $\mathbf{G}$ is trivial. Like multiple states, a stochastic asset model would lead to more complex expressions, but the gain in insight would be in a different direction; by omitting it we do not mean to underrate its importance.

4 Technical Bases and Thiele’s Differential Equation

4.1 Defining Technical Bases

We follow earlier authors by defining a generic technical basis denoted by ${\cal B}^{Z}$, to be the pair $(\delta_{t}^{Z},\mu_{t}^{Z})$ of (vector) parameters specifying, respectively, investment and biometric models. The index $Z$ labels different technical bases ${\cal B}^{X},{\cal B}^{Y}$, and so on.

• (a)

Any technical basis ${\cal B}^{Z}$ is assumed to represent (induce) a probability (measure) $\mbox{P}_{Z}$ equivalent to the reference measure $\mathbb{P}$ induced as in Section 3.5.1(b), also a distribution $F_{Z}(t)$ of $T$ (equation (11)); and an expectation operator $\mbox{E}_{Z}$.

• (b)

Given cashflows $S_{t}$ and $P_{t}$, technical basis ${\cal B}^{Z}$ parametrizes Thiele’s differential equation (Hoem 1988) as follows:

[TABLE]

• (c)

The maturity benefit $\bar{S}$ does not appear in equation (15); its rôle is as a boundary condition of certain solutions, see Section 4.2.

• (d)

The quantity $S_{t}-g(t)$ appearing in Thiele’s equation is called the sum at risk.

• (e)

Under ${\cal B}^{Z}$, the process $M^{Z}(t)\equiv N(t)-\int_{0}^{t}1_{\{N(r)=0\}}\,\mu_{r}^{Z}\,dr$ is an ${\cal F}_{t}$-martingale.

4.2 Solutions of Thiele’s Equation: Policy Values, Accumulations and Sums at Risk

It is standard to check the following solutions of Thiele’s equation (15), conveniently expressed in terms of the following discounting/survival factors, see equation (1):

[TABLE]

• (a)

Backward solution: Solving (15) backwards from the boundary value $g(n)=\bar{S}$ gives:

[TABLE]

In this case we usually denote the technical basis by ${\cal B}^{L}$ and call it a valuation basis; denote the solution by $V_{t}^{L}$ and call it a policy value; and denote the associated sum at risk by $R^{L}(t)=S_{t}-V_{t}^{L}$.

• (b)

Forward solution: Solving (15) forwards from the boundary value $g(0)=0$ gives:

[TABLE]

In this case we usually denote the technical basis by ${\cal B}^{A}$, and call it an accumulation basis; denote the solution by $W_{t}^{A}$ and call it an accumulation777Traditionally, $V_{t}^{L}$ and $W_{t}^{A}$ would be called the prospective and retrospective policy values, respectively; see the Appendix of Haçarız et al. (2024) for a discussion of the latter terminology. We follow Linnemann (2002, 2003) in adopting the ‘$W$’ notation for accumulations.; and denote the associated sum at risk by $R^{A}(t)=S_{t}-W_{t}^{A}$.

• (c)

A solution of equation (15) satisfying both $g(0)=0$ and $g(n)=\bar{S}$ satisfies the principle of equivalence; we call the technical basis a special valuation basis. For a given technical basis there may be infinitely many premium rate functions $P_{t}$ satisfying the equivalence principle. In practice we assume there are constraints that identify a unique solution, for example, a level premium rate payable throughout the term.

• (d)

The (partial) classification of technical bases provided by (a) to (c) above is summarized in Table 1, following Haçarız et al. (2024).

4.3 The Experience Technical Basis ${\cal B}^{M}$

We suppose a ‘true’ technical basis exists, which generates the stochastic cashflows $B(t)$, and denote it by ${\cal B}^{M}$ (‘M’ stands for ‘martingale’). We reserve the term ‘experience basis’ exclusively to ${\cal B}^{M}$, and use ${\cal B}^{A}$ to denote any other accumulation basis. When we model the actuary’s actions from a higher standpoint (for example in a model office) we may treat ${\cal B}^{M}$ as a meta-model which we know but the model actuary does not.

5 The Balance Sheet

5.1 A Stochastic Model Balance Sheet

The balance sheet displays the values placed on the assets and the liabilities, and the difference between them is surplus. Barring exact cashflow matching, which would reduce the balance sheet to a tautology, surplus depends on valuation methodology.

In this section and the next, we assume that there is only one technical basis, so without ambiguity we can drop postscripts denoting technical bases and refer to technical basis ${\cal B}=(\delta_{t},\mu_{t})$, discount functions $v(t)$, $p(t)$ and $\varphi(t)$, conditional expectations $\mbox{E}[\,\bullet\mid{\cal F}_{t}\,]$, martingale $M(t)$, policy values $V_{t}$ and accumulations $W_{t}$.

Follow Norberg (1991) and define ${\cal F}_{n}$-measureable $X(t)$ as the value at time $t$ of all cashflows, recalling that $B(0)=0$, and split $X(t)$ into past and future parts:

[TABLE]

where, as usual, we write $\int_{s}^{t}$ to mean $\int_{(s,t]}$ to save space. If we then follow Bühlmann (1976) by defining the propective policy value at time $t$ to be the conditional expectation:

[TABLE]

we get a simple stochastic model of the balance sheet:

[TABLE]

However, we now have two quantities named ‘prospective policy value’; one a deterministic function $V_{t}$ obtained as a solution of Thiele’s equation, the other the ${\cal F}_{t}$-conditional expectation in equation (21). Proposition 1 below shows how they are related.

5.2 Stochastic Interpretation of Policy Values and Accumulations

Proposition 1

If $V_{t}$ is a solution of Thiele’s equation (15) satisfying $V_{n}=\bar{S}$, then:

[TABLE]

Both sides of this equation are ${\cal F}_{t}$-measureable random variables. The proposition says that a stochastic policy value, produced by multiplying deterministic $V_{t}$ by stochastic $1_{\{N(t)=0\}}$, is expressible as an ${\cal F}_{t}$-conditional expectation of discounted future cashflows; equivalently takes constant values on the sets $\{N(t)=0\}$ and $\{N(t)=1\}$.

Proof: Consider d\big{\{}v(t)\,1_{\{N(t)=0\}}\,V_{t}\big{\}};

[TABLE]

Applying the first identity in equation (8) at the middle term above, then adding and subtracting $v(t)\,S_{t}\,dN(t)$ we have:

[TABLE]

Integrate over $r\in[t,n]$, and divide by $v(t)$:

[TABLE]

Note that $V_{n}=\bar{S}$, rearrange, and take conditional expectations $\mbox{E}[\,\bullet\mid\,{\cal F}_{t}\,]$. $\Box$

The converse of Proposition 1, that Thiele’s equation is satisfied by:

[TABLE]

is easily proved by direct calculation (see Norberg (1991)). In the retrospective case we have the following result.

Corollary 1

If $W_{t}$ is a solution of Thiele’s equation satisfying $W_{0}=0$, then:

[TABLE]

Both sides of this equation are ${\cal F}_{0}$-measureable random variables.

Proof: Consider d\big{\{}v(t)\,1_{\{N(t)=0\}}\,W_{t}\big{\}}, and follow the same steps as far as equation (26). Then integrate over $r\in[0,t]$, divide by $v(t)$, note that $W_{0}=0$ and take conditional expectations $\mbox{E}[\,\bullet\mid\,{\cal F}_{0}\,]$. $\Box$

The ‘scale’ of both $V_{t}$ and $W_{t}$ is ‘per policy in-force at time $t$’, which explains the presence of indicator $1_{\{N(t)=0\}}$ in equation (24), and factor $\varphi(t)^{-1}$ in equation (29).

5.3 Asset and Liability Valuation Functions Again

We can now interpret the stochastic balance sheet model of equation (23):

[TABLE]

The prospective policy value is an ${\cal F}_{t}$-measureable random variable, taking the deterministic value $V_{t}$ if the insured individual is alive, and the deterministic value 0 if not.

(Norberg 1991, Section 1A) said: “The concept of prospective reserve is no matter of dispute in life insurance mathematics. It is defined as the conditional expected present value of future benefits less premiums, given its present state.” This is consistent with Bühlmann (1976) and (as we will see) with Scandinavian-style regulation, but it is too limiting, for example it excludes all the approaches used after the Northampton method was abandoned by UK actuaries. Rather, we can now flesh out the asset and liability valuation functions mentioned in Section 2.2. If we are to work with them mathematically we hope that each is a function of a technical basis, a stochastic cashflow function and a flow of information, chosen for the particular purpose. Omitting formal definitions, they might look like this:

[TABLE]

By choosing the two terms on the right-hand side of equation (23) for the functions $W_{t}^{*}$ and $V_{t}^{*}$, we choose Scandinavian-style regulation.

6 One Technical Basis: Introducing Surplus

6.1 Plan of Approach

The definition and analysis of surplus proceeds in two parts.

• (a)

One technical basis. With one technical basis ${\cal B}$ the surplus at time $t$ is given by equation (30). Further discounting to $t=0$ and/or taking ${\cal F}_{0}$-conditional expectations of the surplus shows what projected models of future surplus may be available to the actuary at the inception of the contract. We set out these steps in this section, and in addition define a useful notation for surplus.

• (b)

Two and three technical bases.

• (1)

In Section 7, we model Scandinavian-style regulation with two technical bases, a ‘first-order’ ${\cal B}^{L}$ common to premium and policy value calculations, and a ‘second-order’ ${\cal B}^{M}$ describing the experience (see Section 4.3). Surplus has a systematic or planned component, resulting from a safe-side first-order technical basis.

• (2)

In Section 8 we allow for separate premium, valuation and experience technical bases ${\cal B}^{P},{\cal B}^{L}$ and ${\cal B}^{M}$. Each pair of technical bases defines a different property of the surplus, as described in Haçarız et al. (2024). However some valuation methods (including the historically important net premium method) go beyond choosing a technical basis (meaning that the technical basis ${\cal B}^{L}$ is not the only component of the valuation function $V_{t}^{*}$ in equation (32)). We define an actuarial basis as an enlarged technical basis in Section 9 and the resulting surplus in Section 10.

6.2 Stochastic and Modeled Surplus with One Technical Basis

We suppose the technical basis is a special valuation basis so $V_{0}=0$ and $V_{n}=\bar{S}$.

6.2.1 Stochastic Surplus

The conditional expectation in equation (30) is the stochastic surplus, undiscounted:

[TABLE]

and it is sometimes useful to discount this surplus to $t=0$:

[TABLE]

These are ${\cal F}_{t}$-measurable random variables.

6.2.2 Modeled Surplus

The conditional ${\cal F}_{0}$-expectations of equations (34) and (35) see (Norberg 1999, equation (5.6)) we call the modeled surplus. It may be defined undiscounted:

[TABLE]

or with financial discounting:

[TABLE]

These are ${\cal F}_{0}$-measurable random variables. In particular $W_{t}$, which is the classical retrospective reserve, is an ${\cal F}_{0}$-conditional expectations, despite its name. The fact that equations (37) and (39) are identically zero, which we show in Corollary 2, expresses Norberg’s well-known remark that the classical retrospective reserve is: “$\ldots$ rather a retrospective formula for the prospective reserve” (Norberg 1991).

6.2.3 The Function $\mbox{E}[\,X(t)\mid{\cal F}_{t}\,]$

The object of interest is $\mbox{E}[\,X(t)\mid{\cal F}_{t}\,]$ as a function of $t$. We never consider $\mbox{E}[\,X(t)\mid{\cal F}_{r}\,]$ for $r\not=t$. Note the following.

• (a)

Derivatives of $\mbox{E}[\,X(t)\mid{\cal F}_{t}\,]$ will involve $dX(t)$, where $X(t)$ has the form $\int_{0}^{t}-\int_{t}^{n}$. At this point the two integrands have the same value at their respective upper and lower limits of integration (time $t$) see equation (20). This will not always be so.

• (b)

Although the left-hand side of equation (36) simplifies as $\mbox{E}[\,X(t)\mid{\cal F}_{0}\,]$, this is of interest only for $t=0$, where $\mbox{E}[\,X(0)\mid{\cal F}_{0}\,]=0$ expresses the equivalence principle.

6.2.4 Comments

• (a)

Figure 1 illustrates the idea of learning past history and needing to model only the future. Panel (a) shows ${\cal F}_{0}$-conditional expectations (Section 6.2.2) when there is no history. Panel (b) shows ${\cal F}_{t}$-conditional expectations (Section 6.2.1) as time passes. At time $n$ everything is known; panel (c) shows entire ${\cal F}_{n}$-measurable sample paths.

• (b)

The given definitions of surplus do not imply that $V_{0}=0$ or $V_{n}=\bar{S}$. For example, they do not imply that the policy values in panel (a) of Figure 1 should start and end at zero. It is a matter of assumption in this section that these conditions all hold.

• (c)

If the cashflows $B(t)$ represented a self-financing financial portfolio with $X(n)=0$, not only the ${\cal F}_{0}$-measurable quantities in Section 6.2.2, but all the ${\cal F}_{t}$-measurable quantities in Section 6.2.1 would be identically zero. In other words, the balance sheet would record no more than hedging error.

6.3 Notation for Stochastic and Modeled Surplus

We introduce ‘$\Theta^{Z_{1},Z_{2}}(t)$’ notation888This notation extends the ‘$\Gamma$’ notation from Ramlau-Hansen (1988a) and Haçarız et al. (2024). The change from ‘$\Gamma$’ to ‘$\Theta$’ is for purely typographical reasons, it is easier on the eye with superscripts added. to represent surplus in models following Bühlmann (1976) and Norberg (1991) in choosing conditional expectations as the valuation function.

• (a)

The first superscript place shows the basis used to discount and condition over $(t,n]$.

• (b)

The second superscript place shows the basis of discount and condition over $[0,t]$.

• (c)

Symbol ‘$\omega$’ in the superscript indicates the actual sample path (no conditioning).

• (d)

A tilde ‘$\sim$’ indicates discounting to $t=0$.

• (e)

We include the value of realized surplus $\Theta^{\omega,\omega}(t)=X(t)$ for completeness only.

The differentials $d\Theta^{Z_{1},Z_{2}}(t)$ or $d\tilde{\Theta}^{Z_{1},Z_{2}}(t)$ model the rate of surplus generation at time $t$, being shorthand respectively for a stochastic differential equation and an ordinary differential equation, as we see in the next section. The notation allowing for a single technical basis is summarized in Table 2, where we must now give our single technical basis the generic label $Z$ just to populate the superscripts.

6.4 Surplus With a Single Technical Basis

The main result of this section is the following.

Proposition 2

Let ${\cal B}^{Z}=(\delta_{t},\mu_{t})$ be a generic technical basis, and for brevity also omit the $Z$ postscript from $V_{t}$, $R(t)$ etc.. Then for $t\in(0,n)$:

[TABLE]

Proof: See Proposition 3. $\Box$

${\cal B}^{Z}$ is a special technical basis so $\tilde{\Theta}^{Z,\omega}(0)=\tilde{\Theta}^{Z,Z}(0)=-V_{0}^{Z}=0$, and the terminal net cashflow $\Delta B(n)=-1_{\{N(n)=0\}}\,(\bar{S}-V_{n}^{Z})=0$ so we have the following.

Corollary 2

Either by forming complete differentials in (a) and (c) of Proposition 2, as in (Norberg 1999, Section 5A), or by integrating (b) and (d) directly, we have for $0\leq t\leq n$:

[TABLE]

$\Box$

6.5 Comments

• (a)

Part (b) of Proposition 2 implies Hattendorff’s Theorem (Bühlmann 1976, Ramlau-Hansen 1988b, Norberg 1992) namely that the discounted insurance surpluses (or losses) over non-overlapping periods are uncorrelated.

• (b)

Part (b) of Corollary 2 expresses the classical equality of prospective and retrospective policy values, which is a statement about ${\cal F}_{0}$-measureable quantities, see Section 6.2.2.

• (c)

No boundary values were involved until Corollary 2.

7 Two Technical Bases: Scandinavian-style Regulation

7.1 Two Technical Bases: First-order and Second-order

Consider the two technical bases used under Scandinavian-style regulation.

• (a)

The first-order technical basis denoted by ${\cal B}^{L}$ is a special valuation basis that is intentionally ‘safe-side’ in both interest and mortality. It defines the contractual rate of premium $P_{t}$ by the equivalence principle, see Section 4.2, so by definition $V_{0}^{L}=0$.

• (b)

The second-order technical basis is the experience technical basis ${\cal B}^{M}$ (Section 4.3). Since past cashflows are taken (simplistically) to be accounting facts, this corresponds to the usual practice now of valuing assets at their market value.

7.2 Surplus With Two Technical Bases

The ‘$\Theta$’ notation with two technical bases is summarized in Table 3. Note that $\tilde{\Theta}^{L,\omega}(t)$ is the surplus denoted by $\Gamma_{t}$ in Ramlau-Hansen (1988a), and with a change of sign is $\Gamma_{t}^{L}$ defined in Haçarız et al. (2024), and by equation (5.4) in Norberg (1999). As examples we show the undiscounted stochastic surplus:

[TABLE]

and the discounted modeled surplus:

[TABLE]

To better understand Scandinavian-style regulation, introduce $V_{t}^{M}$, the prospective policy value under ${\cal B}^{M}$, and surplus $\Theta^{M,\omega}(t)$, (Linnemann 2002, 2003):

[TABLE]

Then we can interpret the stochastic surplus using the identity:

[TABLE]

expressing ‘published’ surplus on the left, as ‘realistic’ surplus less a margin in the liability value. This is illustrated in Figure 2, showing policy values and accumulations for a basic endowment contract with unit benefits, on a ‘safe-side’ first-order technical basis ${\cal B}^{L}$ (solid line) and a ‘realistic’ second-order technical basis ${\cal B}^{M}$ (dashed lines)999Linnemann (2002, 2003) gave a detailed analysis of surplus based on Thiele’s equation (in our terms $\Theta^{L,M}(t)$). All surplus was distributed as terminal bonus, and described as an accumulating bonus account; for example: “$\ldots$ bonuses are accumulated to the surviving policyholders in the policyholder’s account, which is given by Thiele’s differential equation $\ldots$” (Linnemann 2003, p.165). The latter defined $\tilde{B}_{r}(t)$ to be the expected accumulation of past surplus, with $\tilde{B}_{r}(t)=0$; $\tilde{B}_{p}(t)$ to be the EPV of future surplus, with $\tilde{B}_{p}(0)=-V_{0}^{M}$; and $\tilde{g}(t)$ to be the rate of emerging surplus ($c^{L,M}(t)$ in our terms, Linnemann’s ‘tilde’ does not mean financial discounting). Then (mixing notations slightly) we have the relations:

$\displaystyle\displaystyle{\frac{d}{dt}\,\tilde{B}_{r}(t)}$ $\displaystyle=$ $\displaystyle(\delta_{t}^{M}+\mu^{M}_{t})\,\tilde{B}_{r}(t)+\tilde{g}(t)\qquad\mbox{\cite[citep]{(\@@bibref{AuthorsPhrase1Year}{linnemann2003}{\@@citephrase{ }}{}, (5.1.3))}}$

(52)

$\displaystyle\displaystyle{\frac{d}{dt}\,\tilde{B}_{p}(t)}$ $\displaystyle=$ $\displaystyle(\delta_{t}^{M}+\mu^{M}_{t})\,\tilde{B}_{p}(t)-\tilde{g}(t)\qquad\mbox{\cite[citep]{(\@@bibref{AuthorsPhrase1Year}{linnemann2003}{\@@citephrase{ }}{}, (4.2.5))}}$

(53)

and the whole system is as illustrated in Figure 2; initially all future surplus is capitalized in $\tilde{B}_{p}(0)=-V_{0}^{M}$, then as time passes it is transformed into realized surplus via (52) and (53) until finally $\tilde{B}_{p}(n)=0$, see the comment following Corollary 3..

We can now complete the description of Figure 1. Panel (d) is the same as panel (b), but representing modeled surplus $\Theta^{L,M}(t)$ for $0\leq t<10$ on two possible technical bases ${\cal B}^{M}$. This can be regarded as the outcome of scenario testing by an actuary at $t=0$, showing two possibilities for ${\cal B}^{M}$ before $t=10$ and three choices of ${\cal B}^{L}$ after $t=10$.

7.3 Surplus With Two Technical Bases

Define the systematic component of surplus, denoted by $c^{L,M}(t)$, to be the rate:

[TABLE]

Proposition 3

For $t\in(0,n)$:

[TABLE]

Proof: (a) This proof follows (Norberg 1999, Section 5A). Differentiate the first term on the right-hand side of equation (47), recalling the definition of $dB(t)$ in equation (10) and the identities in equations (16), then add/subtract $1_{\{N(t)=0\}}\,\delta_{t}^{M}\,V^{L}_{t}\,dt$ to obtain:

[TABLE]

Differentiate the second term on the right-hand side of equation (47):

[TABLE]

The last term in equation (59) and the first term in equation (60) give:

[TABLE]

while adding/subtracting $1_{\{N(t)=0\}}\,\mu_{t}^{M}\,(S_{t}-V_{t}^{L})\,dt$ completes the remaining terms:

[TABLE]

in equation (55).

(b) Follows immediately from (a) and $d\tilde{\Theta}^{L,\omega}(t)=d(v^{M}(t)\,{\Theta}^{L,\omega}(t))$.

(c) Take conditional expectations $\mbox{E}_{M}[\,\bullet\mid{\cal F}_{0}\,]$ of both sides in (a). On the left-hand side, by definition, $\mbox{E}_{M}[\,\Theta^{L,\omega}(t)\mid{\cal F}_{0}\,]=\Theta^{L,M}(t)$, and on the right-hand side the martingale term vanishes, leaving $1_{\{N(t)=0\}}$ as the only stochastic term, with $\mbox{E}_{M}[\,1_{\{N(t)=0\}}\mid{\cal F}_{0}\,]=p_{t}^{M}$.

(d) Similarly to (c), take conditional expectations of both sides of (b) above. $\Box$

Noting the initial conditions $\tilde{\Theta}^{L,\omega}(0)=\tilde{\Theta}^{L,M}(0)=-V_{0}^{L}$, and the terminal net cashflow $\Delta B(n)=-1_{\{N(n)=0\}}\,(\bar{S}-V_{n}^{L})=0$ we have the following.

Corollary 3

Either by forming complete differentials in (a) and (c) of Proposition 3, or by integrating (b) and (d) directly, we have for $0\leq t\leq n$:

[TABLE]

$\Box$**

In particular, if the premium rate $P_{t}$ is fixed as that satisfying the equivalence principle under ${\cal B}^{L}$, but we calculate policy values under ${\cal B}^{M}$ in Corollary 3, we have $\tilde{\Theta}^{M,M}(t)=-V_{0}^{M}$ (that is, a constant), see equation (51) and Figure 2.

7.4 Comments on Proposition 3

• (a)

In general, as we see later, we have cumulative surplus of the form $C(t)=\int_{0}^{t}dC(r)$ and accumulated surplus of the form $\Theta(t)=\int_{0}^{t}v(r)v^{-1}(t)\,dC(r)$.

• (b)

Clearly Proposition 2 is a special case with $c^{Z,Z}(t)=0$ identically.

• (c)

Reduction or simplification of the stochastic surplus by taking ${\cal F}_{0}$-conditional expectations (Norberg 1999, 2001) represents risk management by the law of large numbers, appropriate for an insurer. Møller & Steffensen (2007) and Asmussen & Steffensen (2020) propose instead just dropping the martingale terms. This may represent assumption of the individual stochastic risk by another party, such as a reinsurer.

7.5 Origins and Development

Ramlau-Hansen (1991) introduced two technical bases in the Markov multiple-state model with constant interest. Prospective and retrospective policy values satisfying Thiele’s equations were assumed under both technical bases. His equations (4.6) and the first part of equation (4.14) are respectively more general versions of Corollary 3(b) and (d) above (allowing for statewise decomposition into components).

Proposition 3 is assembled from materials in Norberg (1996, 1999). Part (a) is the same as equation (5.3) in the latter, and part (c) is equivalent to (5.7) (but we do not assume $\tilde{\Theta}^{L,M}(0)=0$). Norberg formalized parts (c) and (d) of the proposition, essentially by calling upon the law of large numbers. As noted in Section 7.4(b), approaches to risk management other than the law of large numbers may lead to an alternative simplification of the stochastic surplus.

Figure 3 gives a graphical overview of the evolution of stochastic process models of life insurance reserves and surplus, based on Scandinavian-style regulation, after Hoem (1969, 1988), and intended to provide context for the $\Theta/\tilde{\Theta}$ notation used here. In passing we note the curious fact that in this literature only Width (1986) and Linnemann (1994, 2002) offer any helpful diagrams other than box-and-arrow depictions of multiple-state models.

8 Three Technical Bases: Valuation Premiums and Loadings

8.1 Three Technical Bases

Without going into historical reasons for separating premium and valuation bases (see for example Cox & Storr-Best (1962), Turnbull (2017)) we may assume that premium-setting and valuation are seen as having different purposes, especially when the premiums were set a long time ago. Introducing a third technical basis, denoted by ${\cal B}^{P}$, for the calculation of premiums means that the equation of value satisfied by the contractual premium rate $P_{t}$ is $\mbox{E}_{P}[X(0)\mid{\cal F}_{0}]=0$.

Surplus (possibly negative) arises from the difference between ${\cal B}^{P}$ and ${\cal B}^{L}$. As we show below, some or all of this surplus may be crystallized at outset into an initial surplus, equal to $-V_{0}^{L}$ if there is no countervailing adjustment of assets. More generally, each pair of technical bases (premium, valuation and experience) induces a relationship in terms of surplus, discussed in full in Haçarız et al. (2024) from which Figure 4 is taken.

8.2 The Net (or Pure) Premium Rate

Let ${\cal B}^{Z}$ be a generic technical basis, and let $\pi_{t}^{Z}$ be the premium rate satisfying Thiele’s equation (15) with boundary conditions $g(0)=0$ and $g(n)=\bar{S}$, under ${\cal B}^{Z}$. We call $\pi^{Z}_{t}$ the net premium rate or pure premium rate for ${\cal B}^{Z}$ and the contractual benefits. Recall that we assume there to be constraints on the form of the premium rate such that $\pi_{t}^{Z}$ is unique, given the contractual benefits, ${\cal B}^{Z}$ and the boundary condition.

Given separate premium and valuation technical bases ${\cal B}^{P}$ and ${\cal B}^{L}$, with pure premium rates $\pi_{t}^{P}=P_{t}$ and $\pi_{t}^{L}$ respectively, $(P_{t}-\pi_{t}^{L})$ is called the net premium loading or pure premium loading. If $(P_{t}-\pi_{t}^{L})>0$ the loading may be regarded as an ‘extra’ rate of premium not required to pay for benefits under ${\cal B}^{L}$. Therefore, when it is received, it should fall intio surplus, which we call loading surplus. The opposite case $(P_{t}-\pi_{t}^{L})<0$, while allowed mathematically, is unsound in practice, because it means that when the contractual premium is received, there must be a charge on surplus in order to make up the required rate of premium $\pi_{t}^{L}$.

8.3 Loadings

Since ${\cal B}^{P}$ and ${\cal B}^{L}$ are both valuation bases, we denote the corresponding solutions of Thiele’s equation by $V_{t}^{P}$ and $V_{t}^{L}$, satisfying $V_{n}^{P}=\bar{S}$ and $V_{n}^{L}=\bar{S}$ respectively. By definition, $V_{0}^{P}=0$, but in general $V_{0}^{L}\not=0$. In fact we have the following proposition.

Proposition 4

[TABLE]

Proof:

[TABLE]

$\Box$

It follows that initial surplus is $-V_{0}^{L}$, and all the pure premium loadings are capitalized at outset. Inspection of the proof of Proposition 3 shows that it holds unaltered, since the valuation pure premium rate $\pi_{t}^{L}$ is irrelevant.

8.4 The Problem of Loadings

Separating valuation and premium technical bases creates the pure premium loadings $P_{t}-\pi_{t}^{L}$. These are not received until the relevant future premiums are paid in future, but they are capitalized and become surplus at outset. This problem preoccupied British actuaries after the Northampton method was abandoned, see Turnbull (2017). In a review of valuation principles, written at about the same time as Thiele formulated his equation, a Scottish actuary said.

“Alike, in calculating rates of premium and in estimating the value of policies, there are three well-known centres round which all the questions we can consider group themselves — the table of mortality to be used, the rate of interest to be assumed, and the method of dealing with the ‘loadings’.” (McCandlish 1876)

Note that ‘loadings’, are mentioned, where today we would probably mention ‘expenses’. Safe-side premiums are larger than they would be based on a ‘realistic’ technical basis. If a valuation then takes a ‘realistic’ view, loadings in future premiums are identified and immediately capitalized as surplus, and if this ‘surplus’ is then disposed of in any form, the very purpose of safe-side premiums would be frustrated.

The Northampton method did not have this defect, but had defects of its own. Most important was lack of realism. This became acute in the case of participating business, because premiums were often loaded well beyond mere considerations of risk, with a view to investment. Apart from that, the simple passage of time might render assumptions made long ago absurd or unsound, especially if they had to be published.

Two basic approaches evolved to ‘solve’ this problem.

• (a)

For participating business only, to include in the valuation future bonuses sufficient to absorb the future loading surpluses that would otherwise be capitalized. This was called a bonus reserve valuation. It became a valuable tool of internal management, but it was unsuited for use in published valuations because any assumed bonuses could be interpreted as a promise.

• (b)

To exclude the pure premium loadings $P_{t}-\pi_{t}^{L}$ from the future premiums valued. That is, in calculating policy values $V_{t}^{L}$ by solving Thiele’s equation backwards from $V_{n}^{L}=\bar{S}$, to substitute the pure premium rate $\pi_{t}^{L}$ for the actual premium rate $P_{t}$.

Methods along the lines of (a) are called as gross premium valuations, and methods along the lines of (b) are called net premium valuations. The latter became important as the basis of life insurance regulation in many jurisdictions. Their main feature, for our purposes, is that policy values are calculated as if certain non-contractual cashflows were payable, namely the net premium rate $\pi_{t}^{L}$ instead of the contractual premium rate $P_{t}$.

8.5 The Valuation Premium Rate

Having raised the possibility of substituting something else for the contractual premium rate $P_{t}$ in calculating policy values, we realize that the pure premium rate $\pi_{t}^{L}$ is not the only possibility. Therefore, let $\tau_{t}^{L}$ be any rate of premium used with valuation technical basis ${\cal B}^{L}$ and substituted for $P_{t}$ in calculating $V_{t}^{L}$. We call $\tau_{t}^{L}$ a valuation premium rate, and we speak of ‘valuing premium rate $\tau_{t}^{L}$’. The two canonical candidates for $\tau_{t}^{L}$ we have already discussed:

• (a)

if we use $\tau_{t}^{L}=P_{t}$ we have a gross premium valuation101010For with-profit business in particular, where premium loadings could include a substantial investment component, this terminology is not primarily related to the inclusion or exclusion of expenses from policy values. By way of contrast see for example (Hoem 1988, p.175): “At this stage we do not distinguish between net and gross premiums and reserves, i.e. we need not specify whether administrative expenses are taken into account or not.” This difference in usage is important.;

• (b)

if we use $\tau_{t}^{L}=\pi_{t}^{L}$ we have a net premium valuation; and

• (c)

under Scandinavian-style regulation, the valuation is both net premium and gross premium, since $\tau_{t}^{L}=\pi_{t}^{L}=P_{t}$.

These are not the only possibilities, $\tau_{t}^{L}$ may differ from both $P_{t}$ and $\pi_{t}^{L}$. Valuing only some fraction of $P_{t}$ is one example, the paid-up valuation principle described in Section 11 will provide another, where $\tau_{t}^{L}=0$.

8.6 Surplus With Valuation Premium Rate $\tau_{t}^{L}$

We omit the proof of the following proposition, which follows similar lines to the proof of Proposition 4 (see Corollary 5.1 in Haçarız et al. (2024)).

Proposition 5

If the valuation premium rate $\tau_{t}^{L}$ is substituted for the contractual premium rate $P_{t}$ in calculating policy values $V_{t}^{L}$ under technical basis ${\cal B}^{L}$, then:

[TABLE]

$\Box$

That is, $(\tau_{t}^{L}-\pi_{t}^{L})$ is a loading which is capitalized at outset. Referring again to Corollary 5.1 in Haçarız et al. (2024), we have the following.

Corollary 4

The pure premium loading $P_{t}-\pi_{t}^{L}$ may be written as:

[TABLE]

in which ‘Loading 2’ is capitalized at once as in Proposition 5 and ‘Loading 1’ falls into surplus when it is received, as we shall see in Proposition 6. $\Box$

Before proceeding to Proposition 6 we need to formalize the use of non-contractual cashflows in calculating policy values and other quantities.

9 Non-contractual Cashflows and Actuarial Bases

9.1 Non-contractual Cashflows

Valuation premium rates $\tau_{t}^{L}\not=P_{t}$ introduce non-contractual cashflows. Non-contractual cashflows also arise often other calculations.

• (a)

Surrender Values: Premium and valuation bases may commonly omit lapses, equivalent to assuming null lapse rates in a model that otherwise equates the value of benefits on lapsing to the policy value at that time111111Cantelli’s Theorem (Cantelli 1914, Hoem 1988, Christiansen et al. 2014). (Hoem 1988, Section 8) noted that second-order calculations might use cashflows (such as surrender values) absent from the corresponding first-order calculations. However lapseing benefits will certainly appear as an accounting fact in the experience basis and the surplus.

• (b)

Bonus: Any distribution of surplus to with-profit policies involves non-contractual cashflows. Bonus is considered in Section 13.

Haçarız et al. (2024) defined the ‘enhanced valuation basis’ $\tilde{\cal B}^{L}$ to be the pair $({\cal B}^{L},\tau_{t}^{L})$121212We have changed the notation of Haçarız et al. (2024) to agree with ours; the valuation premium rate we call $\tau_{t}^{L}$ they called $\pi_{t}^{*}$.. In the following, we make this more general, by parcelling up technical basis and cashflows into a single entity, which we call an actuarial basis.

9.2 Actuarial Bases

Suppose we are given a generic technical basis denoted by ${\cal B}^{Z}$, and, associated with it, some (stochastic) cumulative cashflows denoted by $B^{Z}(t)$ defining an insurance contract. Define the pair $({\cal B}^{Z},B^{Z}(t))$ to be an actuarial basis, denoted by ${\cal A}^{Z}$. Actuarial basis ${\cal A}^{Z}$ parametrizes the Thiele equation:

[TABLE]

where $\delta_{t}^{Z}$ and $\mu_{t}^{Z}$ are from the technical basis ${\cal B}^{Z}$ and $\tau_{t}^{Z}$ and $S_{t}^{Z}$ are from the cashflows $B^{Z}(t)$. If $g(n)=\bar{S}$ the solution is a policy value, ${\cal A}^{Z}$ is a valuation basis and we may write ${\cal A}^{Z}={\cal A}^{L}$ and $g(t)=V_{t}^{L}$, while if $g(0)=0$ the solution is an accumulation and we may write ${\cal A}^{Z}={\cal A}^{A}$ and $g(t)=W_{t}^{A}$.

Table 4 shows ‘standard’ definitions. For convenience we denote contractual cashflows equivalently by $B(t)=B^{P}(t)=B^{M}(t)$. Note that the valuation actuarial basis ${\cal A}^{L}$ generalizes and replaces the ‘enhanced valuation basis’ $\tilde{\cal B}^{L}$ defined in Haçarız et al. (2024).

10 Surplus with Non-contractual Cashflows

10.1 Cashflow Values $X^{L,M}(t)$ and Surplus

Given the valuation actuarial basis ${\cal A}^{L}=({\cal B}^{L},B^{L}(t))$ and the experience actuarial basis ${\cal A}^{M}=({\cal B}^{M},B^{M}(t))$, define the ${\cal F}_{n}$-measureable $X^{L,M}(t)$ for $0\leq t\leq n$ as:

[TABLE]

The valuer looking back from time $t$ sees known cashflows $dB^{M}(r)$ $(r\leq t)$, and looking ahead sees the ${\cal B}^{L}$-expectation of future cashflows $dB^{L}(r)$ $(r>t)$. Then the stochastic surplus at time $t$ is $\mbox{E}_{L}[X^{L,M}(t)\mid{\cal F}_{t}]$:

[TABLE]

Forming the modeled surpluses by taking ${\cal F}_{0}$-expectations under ${\cal B}^{M}$, we get (showing only the discounted case as an example):

[TABLE]

From now on unless stated otherwise, policy values $V_{t}^{L}$ and accumulations $W_{t}^{A}$, surplus $\Theta^{L,M}(t)$ and $\tilde{\Theta}^{L,M}(t)$, and so on, are based on the referenced actuarial bases ${\cal A}^{L}$ and ${\cal A}^{A}$ and so on, including their cashflow parts.

10.2 Surplus

Proposition 6

Let ${\cal A}^{M}$ and ${\cal A}^{L}$ be the experience basis and a valuation basis, respectively. Let $R^{L}(t)=S_{t}^{L}-V^{L}_{t}$ be the sum at risk at time $t$ under ${\cal A}^{L}$. Then for $t\in(0,n)$:

[TABLE]

Proof: Follow the same steps as in the proof of Proposition 3. At equation (59), $S_{t}\,dN(t)$ becomes $S_{t}^{M}\,dN(t)=S_{t}^{L}\,dN(t)+(S_{t}^{M}-S_{t}^{L})\,dN(t)$; and at equation (60), $P_{t}$ becomes $\tau_{t}^{L}$; from which after some rearrangement the additional term in d\big{(}(B^{M}(t)-B^{L}(t)\big{)} in (a) and (b) arises. Then (c) and (d) follow directly by taking ${\cal F}_{0}$-conditional expectations of d\big{(}(B^{M}(t)-B^{L}(t)\big{)}. $\Box$

Define an augmented systematic part of the surplus131313The contribution formula from (Berger 1939, p.141) referred to in Ramlau-Hansen (1988a) includes the ‘technical basis component’ from equation (76) and a ‘cashflow component’ derived from premium-related and per-policy expenses. The latter component is distinct from the cashflow component in equation (76), which excludes expenses, and in particular it is not the same as the valuation premium term $(\tau_{t}^{M}-\tau_{t}^{L})$. Likewise, see the contribution formula including expenses in (Norberg 2001, equation (7.5)). as the rate:

[TABLE]

noting that $P_{t}=\tau_{t}^{M}$. Then a simple rearrangement yields the following.

Corollary 5

Under the same assumptions, (a) and (b) of Proposition 6 can be written in terms of systematic parts and martingale residuals as:

[TABLE]

$\Box$**

Note the following.

• (a)

The augmented systematic component of surplus now includes the loading $(\tau_{t}^{M}-\tau_{t}^{L})=(P_{t}-\tau_{t}^{L})$, see Corollary 4. Proposition 5 showed valuation premium loadings $(\tau^{L}-\pi^{L})$ to have been capitalized into surplus at outset.

• (b)

The new term $\mu^{M}_{t}\,(S_{t}^{M}-S_{t}^{L})$ is often zero, which simplifies the results above.

• (c)

These new terms appear to fall outside the decomposition of surplus into orthogonal martingale-driven components in Jetses & Christiansen (2022).

• (d)

Proposition 6 and Corollary 5 express the analysis of surplus well-known in many valuation regimes that admit premium loadings, including inter alia the treatment of certain expenses, which we have ignored (Berger 1939), US-style contribution bonus (Homans 1863) and approaches that supplanted the Northampton method in the UK to control the emergence of premium loadings into surplus (McCandlish 1876, Turnbull 2017). They may also be extended to profit-testing techniques that focus entirely on profit loadings (Anderson 1959).

Noting the initial condition $\tilde{\Theta}^{L,\omega}(0)=\tilde{\Theta}^{L,L}(0)=-V_{0}^{L}=0$, and the terminal net cashflow \Delta\big{(}B^{M}(n)-B^{L}(n)\big{)}=-1_{\{N(n)=0\}}\,(\bar{S}^{M}-\bar{S}^{L}) we have the following.

Corollary 6

Either by forming complete differentials in (a) and (c) of Proposition 6, or by integrating (b) and (d) directly, we have for $0\leq t<n$:

[TABLE]

and for $t=n$:

[TABLE]

$\Box$**

Corollary 6 shows that choosing the valuation basis amounts to choosing a subdivision of the surplus into surplus capitalized at outset as $-V_{0}^{L}$, and surplus that emerges later. Proposition 5 (and equation (67)) shows how this also subdivides the premium loadings. The next section completes the relationships shown in Figure 4 by showing that the discounted modeled surplus $\tilde{\Theta}^{L,M}(n)$ is independent of the valuation basis.

10.3 Invariance Under Change of Valuation Basis

(Haçarız et al. 2024, Proposition 5.3 (iii)) showed that the EPV of total surplus under the experience basis did not depend on the choice of valuation basis141414This result was well-known in jurisdictions where the valuation basis was not mandatory (for example, see Fisher & Young (1965)) but had not, to our knowledge, been shown before in the modern setting..

Proposition 7

Under the assumptions of Proposition 6, the EPV $\tilde{\Theta}^{L,M}(n)$ of the total surplus under the experience basis ${\cal A}^{M}$ does not depend on the valuation basis ${\cal A}^{L}$.

Proof: Consider:

[TABLE]

The first term on the right-hand side gives:

[TABLE]

and the second term is $-v^{M}(t)\,V_{t}^{L}\,dN(t)$, see equation (8). Add and subtract:

[TABLE]

noting that $P_{t}=\tau_{t}^{M}$ and $S_{t}^{M}\,dN(t)=S_{t}^{L}\,dN(t)+(S_{t}^{M}-S_{t}^{L})\,dN(t)$. Gathering terms, the right-hand side of (83) is in total:

[TABLE]

Then since:

[TABLE]

we have, on integrating expression (86) and equating it to (87):

[TABLE]

Add the discounted terminal net cashflow $-v^{M}(n)\,1_{\{N(t)=0\}}\,(\bar{S}^{M}-\bar{S}^{L})$ to both sides, noting that $V_{n}^{L}=\bar{S}^{L}$, then taking conditional expectations $\mbox{E}_{M}[\,\bullet\mid{\cal F}_{0}\,]$ we have finally:

[TABLE]

in which the left-hand side is the EPV under ${\cal B}^{M}$ of the total surplus with respect to ${\cal A}^{L}$ (equation (82)) and the right-hand side is the EPV of the total cashflow and is independent of ${\cal A}^{L}$. $\Box$

11 Quasi-contractual Cashflows

11.1 Quasi-contractual Cashflows

No actuarial basis can include all possible events or cashflows. Events beyond the model can change the cashflows that the actuary has to value; we call cashflows subject to such contingencies quasi-contractual. The problem they create, in the context of a continuous-time model, is the following. At time $t$, the actuary faces one set of facts and cashflows. By time $t+dt$, the facts and cashflows may have changed. For example:

• (a)

lapsing a policy triggers payment of any cash value and cancels future premiums;

• (b)

making a policy paid-up reduces future benefits and cancels future premiums; or

• (c)

declaring a reversionary bonus (an option of the insurer) increases future benefits and (we suppose) leaves future premiums unchanged.

Examples (a) and (b) above could be handled by transitions to a new state in a multiple-state model. That would not be so easy in example (c). We conclude that there are occasions when it will not do to use the same actuarial basis at time $t+dt$ as was used at time $t$. Facts and not only assumptions may have changed.

11.2 Continuous Change of Valuation Actuarial Basis

We consider the case, which includes the examples above, where the technical basis is fixed but the cashflow component of the valuation actuarial basis changes continuously. We denote this by indexing ${\cal A}^{L_{t}}$ by time $t$. Heuristically, this means the following.

• (a)

At time $t$ policy values on $[t,n]$ are calculated using ${\cal A}^{L_{t}}$. In particular, the policy value at time $t+dt$ is ‘projected’ to be $V_{t+dt}^{L_{t}}$.

• (b)

By time $t+dt$ the valuation basis has changed to ${\cal A}^{L_{t+dt}}$, and the policy value at time $t+dt$ is actually $V_{t+dt}^{L_{t+dt}}$.

• (c)

Neglecting terms of second order and above, the total increment $dV_{t}^{L_{t}}$ is:

[TABLE]

More formally, where Thiele’s equation is invoked replace $dV_{t}^{L}$ by the total derivative:

[TABLE]

noting that both $r$ and $t$ are annual time units so $dt/dr=1$.

In the remainder of this section we give two examples involving quasi-contractual cashflows and dynamic actuarial bases: (a) the paid-up valuation principle suggested by Linnemann (2002, 2003) (Section 12); and (b) some aspects of bonus on participating contracts (Section 13). It will be helpful in both to follow Linnemann (2002) and define the passivum $k^{Z}(t)$ at time $t$ under generic actuarial basis ${\cal A}^{Z}$ as:

[TABLE]

to be the EPV of future benefit cashflows, and the activum $a^{Z}(t)$ as:

[TABLE]

to be the EPV of future premium cashflows. So, for example, the equation of value under ${\cal B}^{Z}$ may be expressed as $k^{Z}(0)=a^{Z}(0)$, with $\tau_{t}^{Z}$ the unknown premium.

12 Example 1: The Paid-up Valuation Principle

12.1 The Paid-up Valuation Principle

Motivated by Danish regulation, Linnemann (2002) introduced a paid-up valuation principle which recognized the policyholder’s right at any time to cease paying premiums but leave a reduced sum insured in force. (To be clear, the method values premium-paying policies, but based upon with this right.) The principle assumed that on being made paid-up at time $t$, the prospective policy value $V_{t}^{L}$ on the first-order technical basis ${\cal B}^{L}$ funded reduced benefits for $r\in(t,n]$, which we denote by $S_{r}^{L_{t}}$ (death) and $\bar{S}^{L_{t}}$ (maturity), all pro rata the original benefits so the ‘shape’ of future benefits was fixed.

Under this principle, in determining the reserve to be held at time $t$ it is assumed that a premium-paying policyholder makes their policy paid-up at time $t$ and their contractual benefits are reduced by a factor $\kappa^{L}(t)$ given by:

[TABLE]

The cumulative cashflow $B^{L_{t}}(r)$ under ${\cal A}^{L_{t}}$ for $t\in(0,n]$ and $r\in(t,n]$ is then:

[TABLE]

and future policy values on $r\in(t,n]$ are given by:

[TABLE]

Figure 5 shows an example of the policy values $V_{t}^{L}$ under an endowment assurance (with level unit benefits and level premium rate) compared with the paid-up policy values $V_{r}^{L_{t}}$ on $r\in(t,n]$ with the policy being made paid-up at three selected times $t$.

12.2 Surplus Under the Paid-up Valuation Principle

Noting that $V_{r}^{L_{t}}$ satisfies the Thiele equation:

[TABLE]

on $r\in(t,n]$ (see Figure 5) and plugging this into equation (93) gives, after some algebra:

[TABLE]

Therefore, valuation on the paid-up principle results in the same policy values and surpluses as before, and all the machinery of Section 10.2 holds.

Replacing ${\cal A}^{M}$ by ${\cal A}^{L}$ in Section 10.2, equation (76) gives the systematic surplus:

[TABLE]

Then by setting $r=t$ in equation (98) and differentiating, using (99), we see that:

[TABLE]

This is the premium decomposition in (Linnemann 2003, Section 3.2): if the policyholder chooses to pay the premium $\pi_{t}^{L}\,dt$ at time $t$, it purchases the increment in future benefits and meets the excess mortality cost (the first and second terms respectively in equation (102)) leaving net surplus of zero (all under ${\cal A}^{L}$)

12.3 Paid-up Policy Values Under Alternative Bases

Linnemann (2003) suggested applying the paid-up valuation principle on the second-order technical basis. The actual cashflows are of course unchanged, so the actuarial valuation basis at time $t$, which we may denote by ${\cal A}^{M_{t}}$, is the pair $({\cal B}^{M},B^{L_{t}}(r))$. Figure 6 shows, as is easily checked, that policy values under ${\cal A}^{M_{t}}$ are simply those under ${\cal A}^{L_{t}}$ scaled by a factor $k^{M}(t)/k^{L}(t)$. Note that while $V_{t}^{L_{t}}=V_{t}^{L}$ we have $V_{t}^{M_{t}}\not=V_{t}^{M}$ (the last quantity as in equation (50)), in fact if ${\cal B}^{L}$ is a ‘safe-side’ technical basis then:

[TABLE]

(Linnemann 2003, equation (6.1.5)), but we do have $V_{0}^{M_{0}}=0$. Therefore, this approach allows for some realism of both technical basis and cashflows, reducing capital needs, but without the undesirable feature of capitalizing future surplus at outset.

Following the same steps as in equation (102) we see that at time $t$, a premium of:

[TABLE]

will purchase the increment in paid-up benefit and fund the cost of death benefits above $S_{t}^{L_{t}}$, so the balance of premium, $\pi_{t}^{L}-\pi_{t}^{M_{t}}$ emerges as (expected) surplus at time $t$. Linnemann (2004) suggested using the approach where: (a) realism is desired in the liability valuation because assets must be taken at market values; and (b) the loadings $\pi_{t}^{L}-\pi_{t}^{M_{t}}$ in future premiums are bonus loadings which it is desired not to capitalize in advance.

13 Example 2: Bonus

13.1 Surplus, Dividends and Bonus

Under participating or with-profit policies, surplus is converted into various forms of bonus and such an allocation, once declared, is irreversible. Ramlau-Hansen (1991) considered three kinds of bonus, as follows:

• (a)

cash or contribution bonus under which surplus is distributed as cash as it is earned;

• (b)

reversionary bonus under which surplus is used to buy additional future benefits; and

• (c)

terminal bonus under which surplus is returned to policyholders when they exit.

We do not have space to consider the merits of bonus systems here, except to note that surplus is an asset, until converted into bonus, at which point it becomes a liability, and the consequent incease in policy values may be called the cost of bonus. The literature is exemplified by (Asmussen & Steffensen 2020, p.184) who say: “The surplus $Y$ generates dividends that are paid into the account $X$ from which the payments, including bonus payments, are paid to the policy holders”, a process which is illustrated below.

[TABLE]

Mathematical representations must simplify this, see Ramlau-Hansen (1991), Linnemann (1994), Norberg (1999, 2001). In the example below we omit dividends and model reversionary bonus funded directly out of surplus, which includes methods from UK practice.

13.2 An Example: Compound Reversionary Bonus

Bonus provides our second example of quasi-contractual cashflows, and for demonstration we choose reversionary bonus declared continuously, because it leads to transparent procedures. Let ${\cal A}^{Z}$ be a generic actuarial basis, and associate with it a non-decreasing ${\cal F}_{t^{-}}$-measureable bonus accumulation function $\tilde{\beta}^{Z}(t)$ with the following properties.

• (a)

The death benefit payable at time $t$ is $S_{t}^{Z}\,\tilde{\beta}^{Z}(t)$, and similarly the maturity value payable at time $n$ is $\bar{S}^{Z}\,\tilde{\beta}^{Z}(n)$.

• (b)

There exists an ${\cal F}_{t^{-}}$-measureable ‘force of bonus’ $\beta^{Z}_{t}$ such that \tilde{\beta}^{Z}(t)=\exp\big{(}\int_{0}^{t}\beta^{Z}_{r}\,dr\big{)} (therefore $\tilde{\beta}^{Z}(0)=1$). We call $\beta^{Z}_{t}$ a bonus rate and emphasise that it modifies the cashflows and does not form part of the technical basis ${\cal B}^{Z}$.

The ‘null’ assumption $\beta^{Z}_{t}=0$ ($t\geq 0$) means non-participation, and $\beta^{Z}_{t}=b/(1+bt)$ with $b>0$ gives simple reversionary bonus, so the definition is quite general, but clearly it is aligned with assuming $\beta^{Z}_{t}$ is constant, which gives level compound reversionary bonus.

A valuation actuarial basis at time $t$ may assume future bonus rates of $\beta^{L}_{r}$ $(r\geq t)$, but past bonus declarations at rates $\beta^{M}_{r}$ $(r<t)$ are factual. So, we assume that there is a family of valuation actuarial bases $\{{\cal A}^{L_{t}}\}_{0\leq t\leq n}$, where $B^{L_{t}}(r)$ is given by:

[TABLE]

As a simple example, suppose $\beta^{L}_{t}=0.02$ and $\beta^{M}_{t}=0.03$, so the valuation actuarial basis anticipates bonus at 2% per annum but the experience is that 3% per annum is declared. If the basic death benefit is $S$ then policy values calculated at time $t$ anticipate a death benefit of $S\,\exp(0.03t)\,\exp(0.02(r-t))$ at time $r$ ($t\leq r\leq n$).

• (a)

At time $t$, past cashflows $dB^{L_{t}}(r)$ for $r\in[0,t]$ recognize factual bonus declarations.

• (b)

At time $t$, future cashflows $B^{L_{t}}(r)$ for $r\in(t,n]$ include assumed bonus declarations after time, part of ${\cal A}^{L_{t}}$.

• (c)

Technical bases ${\cal B}^{L_{t}}={\cal B}^{L}$ under all ${\cal A}^{L_{t}}$, hence also expectations $\mbox{E}_{L_{t}}=\mbox{E}_{L}$.

• (d)

The analogue of Scandinavian-style regulation would be assume as the premium basis ${\cal A}^{P}={\cal A}^{L_{0}}$, so that $V_{0}^{L_{0}}=0$. In the UK this would be called a bonus reserve valuation with premiums calculated on the same basis.

Proceeding to the calculation of surplus, the cashflow value function receives its final embellishment as follows:

[TABLE]

and notation for surplus equations allowing for past and future bonus declarations is shown in Table 5 for convenience. Policy values $V_{t}^{L_{t}}$ under ${\cal A}^{L_{t}}$ are the solution on $(t,n]$ of the Thiele equation:

[TABLE]

with boundary condition $V_{n}^{L_{t}}=\bar{S}^{L}\,\tilde{\beta}^{L_{t}}(n)\,\tilde{\beta}^{M}(t)\,(\tilde{\beta}^{L_{t}}(t))^{-1}$. For brevity we may write $V^{L}_{t}$ for $V^{L_{0}}_{t}$. We also need the following definition of the policy value of the benefits alone, where we use the same notation as the passivum of equation (94):

[TABLE]

The main result of this section is as follows. Define the sums at risk as:

[TABLE]

and the systematic component of surplus as the rate:

[TABLE]

Proposition 8

Suppose actuarial bases ${\cal A}^{M}$ (experience) and $\{{\cal A}^{L_{t}}\}_{0\leq t\leq n}$ (valuation) include in their respective cashflow components reversionary bonus at rates $\beta^{M}_{r}$ and $\beta^{L}_{r}$ respectively. Then the rate of surplus emerging at time $t$ is:

[TABLE]

Proof: (a) Follow the same steps as in Proposition 6 and Corollary 5. Where $dV_{t}^{L}$ appears at equation (60) (here $dV_{t}^{L_{t}}$) differentiate directly as in equation (93):

[TABLE]

Both $r$ and $t$ are annual time units so $dt/dr=1$ and:

[TABLE]

and the rest of the proof follows the same steps as before.

(b), (c) and (d) follow the same steps as in Proposition 6, adapted as in (a) above. $\Box$

13.3 Comments

• (a)

The multiplicative structure of compound bonus additions leads to the appearance of the linear term in $(\beta_{t}^{M}-\beta_{t}^{L})$ in the systematic component of the surplus.

• (b)

If $\beta_{t}^{P}>0$, so that future bonus is anticipated in the premium basis, we say the premiums have been explicitly loaded for bonus.

• (c)

If $\beta_{t}^{L}>0$, so that future bonus is anticipated in the valuation basis, we have a bonus reserve valuation.

• (d)

If $\beta_{t}^{M}\not=\beta_{t}^{L}$ (including a net premium valuation, $\beta_{t}^{L}=0$) the systematic component of the surplus includes the additional cost of bonus shown in equation (109).

• (e)

Premiums (resp. policy values) may be computed conveniently by calculating the passivum ‘as if’ for a contract without bonus at interest rate $\delta_{t}^{P}-\beta_{t}^{P}$ (resp. $\delta_{t}^{L}-\beta_{t}^{L}$). This arithmetical sleight of hand has no deeper implications. From there it is a short step to simply using a safe-side technical basis — Scandinavian-style regulation.

14 Conclusions

We started by noting the concordance of the Northampton method of policy valuation — discarded in the UK by about 1850 — and the Scandinavian-style regulation underlying recent probabilistic models. Norberg (2004) summarized the key features, including: (a) the duality between policy values deterministic as solutions of Thiele’s equation; and (b) policy values as conditional expectations in a Markov model. A single first-order technical basis specifying interest and mortality suffices for both. A single technical basis then defines martingale random noise associated with a life insurance contract, see Section 5 (Bühlmann 1976, Ramlau-Hansen 1988b). Two technical bases suffice to describe surplus under Scandinavian-style regulation, see Section 6 (Ramlau-Hansen 1988a, Norberg 1991).

Scandinavian-style regulation is characterized by restricting safe-side margins to the elements of the technical basis. Other valuation methodologies admit safe-side loadings also in premiums, and their definition involves the valuation of non-contractual premiums. Haçarız et al. (2024) adjusted the valuation technical basis in an ad hoc manner for this purpose; in Section 9 we extended this more generally to an actuarial basis, in which the cashflows as well as the parameters in the associated Thiele equation are specified as part of the basis. This introduces extra elements into the surplus (Section 10). The key outcomes for risk management (Haçarız et al. 2024) are a split of surplus into loadings capitalized at outset and loadings falling into surplus as premiums are paid (Corollary 4) and the fact that the conditional expectation $\mbox{E}_{M}[\,\bullet\mid{\cal F}_{0}\,]$ of surpluses is independent of the valuation actuarial basis (Proposition 7).

This analysis can usefully be extended to cases where actual cashflows are dynamic, by introducing a time-indexed family of actuarial bases into the valuation, with a fixed technical basis and variable (quasi-contractual) cashflows in the associated Thiele equations (Section 11). As examples we applied this construction to surplus arising under the paid-up valuation principle of Linnemann (2003) in Section 12, and reversionary bonus in a simple participating setup (bonuses purchased directly out of surplus) in Section 13.

Acknowledgements

This study is part of the research programme at the Research Centre for Longevity Risk — a joint initiative of NN Group and the University of Amsterdam, with additional funding from the Dutch Government’s Public Private Partnership programme. We are grateful to Dr Per Linnemann and Prof Dr Marcus Christiansen for comments which helped us to improve the paper.

Competing Interests

None.