# Life insurance policies with cash flows subject to random interest rate
changes

David Baños  D. Baños: Department of Mathematics, University of Oslo, Moltke
Moes vei 35, P.O. Box 1053 Blindern, 0316 Oslo, Norway<EMAIL_ADDRESS>
###### Abstract.

The main purpose of this work is to derive a partial differential equation for
the reserves of life insurance liabilities subject to stochastic interest
rates where the benefits and premiums depend directly on changes in the
interest rate curve. In particular, we allow the payment streams to depend on
the performance of an overnight technical interest rate, making them
stochastic as well. This opens up for considering new types of contracts based
on the performance of the insurer’s returns on their own investments. We
provide explicit solutions for the reserves when the premiums and benefits
vary according to interest rate levels or averages under the Vasicek model and
conduct some simulations computing reserve surfaces numerically. We also give
an example of a reinsurance treaty taking over pension payments when the
insurer’s average returns fall under some specified threshold.

Key words and phrases: Reserve, stochastic reserve, interest rate, stochastic
interest rate, Thiele’s equation, Thiele’s PDE, regime switching interest
rate.

MSC2010: 60H30, 91G20, 91G30, 91G60, 35Q91.

##  1\. Introduction

In actuarial science, a reserve is a liability equal to the present value of
future cash flows that the insurance company promises to pay out to the
insured under certain conditions. In easier terms, a reserve is an estimate of
_how much_ the insurance company should charge today in order to meet future
payments owed to the insured. This quantity is the basis to ensure solvency of
the company and the standard way of computing premiums. The literature
sometimes distinguishes between the terms _present value_ of future
obligations and _actuarial reserve_. The former is the cost of the insurance
itself, while the latter is obtained by subtracting the premiums provided by
the insured, which are used to pay back to the insured. In this sense, the
paid-in premiums should match, in average, the future obligations in such a
way that the reserve at the entry of the contract is null.

In life insurance, the main sources of risk of a policy are the state of the
insured which triggers the payments, and the future development of the return
on the financial investments or technical interest rate. In this note, we
focus on the latter and model its risk via a continuous time stochastic
process of Itô-diffusion type and Markovian. We suggest a way to make cash
flows interest rate dependent and provide some new examples to show how this
may relax the burden of low interest rate regimes.

Life insurance claims in the context of stochastic interest rates has been
studied before. We mention [1, 2] for some specific interest rate models and
[3, 7, 9] for more general models in the framework of Heath-Jarrow-Morton. A
model for the interest rate of diffusion type was considered by Norberg and
Møller in [10] and by the authors in [4] where they look at unit-linked
insurances with variance risk, as well.

A classical way of computing reserves in the continuous time setting is by
solving the so-called Thiele’s differential equation, which, in the case of
deterministic interest rate is an ordinary differential equation and, in the
case of stochastic interest rate, a partial differential equation (PDE). The
corresponding Thiele’s equation for the case of stochastic rates was derived
by Norberg and Møller in [10] and later risk adjusted by Persson in [11]. In
the present manuscript we use the _no arbitrage_ approach as in [11] to price
insurance claims. A justification of why this is the right pricing approach
can be found there. In addition, a complete and thourough discussion of the no
arbitrage approach to Thiele’s equation is creditted to Steffensen in [12].
While interest rate there was taken to be deterministic, the reader may
benefit from a precise and excellent discussion on the topic.

Although the above-mentioned works do consider stochastic rates, their models
assume that payments are deterministic given the state of the policyholders
stipulated by contract. In this note we want to generalize the type of
contracts to those looking at interest rate behaviour both at punctual times
and path-dependent past averages. In the spirit of [11] we generalize Thiele’s
partial differential equation (Thiele’s PDE) to incorporate interest rate
dependent cash flows. The motivation to do so comes from the fact that
insurance policies have historically considered fixed returns for their
policyholders. This has proven to carry the risk of low returns in the bond
market and as a consequence, limitations in order to cover the agreed future
liabilities. In this regard, we propose new policies that take into account
the behaviour of the returns and how to price them. We also propose a
reinsurance treaty which covers a percentage of the pension policies in case
of low returns.

The paper is organized as follows: In Section 2 we introduce our modelling
framekwork for the insurance and the interest rate models in two independent
probability spaces coupled together. In Section 3 we review the classical
pricing of financial derivatives and adapt the Feynman-Kac formula to both
European and Asian-type options, where the latter are options whose contingent
claim is a function of the integral of the underlying. Then we derive a Thiele
partial differential equation for insurance contracts whose cash flows depend
on the performance of the interest rate. We look at specific regime switching
contracts in Section 4. The examples are given in general terms, but the
simulations and analysis of concrete results are provided under the Vasicek
interest rate model using a rather simplified model for Norwegian mortality.
More specifically we look at: a pure endowment with reduction on premiums
under high interest rate levels, pension plans with pension rise under high
interest rate regimes, interest rate caps and floors insurances, a binary
endowment associated to two average interest rate levels and finally a
reinsurance treaty taking over part of the pensions if the average returns
during the premium phase are below some threshold.

##  2\. Framework

Our modelling framework will consist of two independent probability spaces. On
the one hand, the states of the insured at any given time, which will be
modelled by a Markov chain and on the other hand, the value of an overnight
technical interest rate.

###  2.1. The insurance model

Let $(\Omega_{1},\mathcal{A}_{1},\mathbb{P}_{1})$ be a complete probability
space where $\Omega_{1}$ is the set of outcomes, $\mathcal{A}_{1}$ is a
suitable $\sigma$-algebra and $\mathbb{P}_{1}$ is a probability measure on
$(\Omega_{1},\mathcal{A}_{1})$ and we denote by $\mathbb{E}_{1}$ the
expectation under $\mathbb{P}_{1}$. This space carries a stochastic process
$X=\\{X_{t},t\in[0,T]\\}$ where $T>0$ is a time horizon, possibly infinite. We
will consider the ($\mathbb{P}_{1}$-augmented) filtration generated by $X$ and
denote it by $\mathcal{F}^{X}=\\{\mathcal{F}_{t}^{X}\\}_{t\in[0,T]}$. Since we
consider no further information on this space, it is natural to take
$\mathcal{A}_{1}=\mathcal{F}_{T}^{X}$. We assume that $X$ is a regular
continuous time Markov chain taking values in a finite state space
$\mathscr{S}$ with transition probabilities

$p_{ij}(s,t)\triangleq\mathbb{P}_{1}\left[X_{t}=j|X_{s}=i\right],\quad
i,j\in\mathscr{S}.$

By regular here, we mean that the transition rates

$\mu_{ij}(t)\triangleq\lim_{\begin{subarray}{c}h\to 0\\\
h>0\end{subarray}}\frac{p_{ij}(t,t+h)}{h},\quad i,j\in\mathscr{S},\quad i\neq
j$

and

$\mu_{i}(t)\triangleq\lim_{\begin{subarray}{c}h\to 0\\\
h>0\end{subarray}}\frac{1-p_{ii}(t,t+h)}{h},\quad i\in\mathscr{S}$

exist for every $t\in[0,T]$, are finite and the functions $\mu_{ij}$,
$i,j\in\mathscr{S}$ are continuous. Hence, we can recover
$\\{p_{ij}\\}_{i,j\in\mathscr{S}}$ from $\\{\mu_{ij}\\}_{i,j\in\mathscr{S}}$
via Kolmogorov’s equations.

On this space we inherently have the following processes

$I_{i}^{X}(t)=\begin{cases}1,\mbox{ if }X_{t}=i,\\\ 0,\mbox{ if }X_{t}\neq
i\end{cases},$ $N_{ij}^{X}(t)=\\#\\{s\in(0,t):X_{s^{-}}=i,X_{s}=j\\}.$

Here, $\\#$ denotes the counting measure and
$X_{t^{-}}:=\lim_{\begin{subarray}{c}u\to t\\\ u<t\end{subarray}}X_{u}$ the
left a.s.-limit of $X$ at the point $t$. The random variable $I_{i}^{X}(t)$
tells us whether the insured is in state $i$ at time $t$ and $N_{ij}^{X}(t)$
tells us the number of transitions from $i$ to $j$ in the whole period
$(0,t)$.

######  Definition 2.1 (Stochastic cash flow).

A stochastic cash flow is a stochastic process $A=\\{A_{t}\\}_{t\geq 0}$ with
almost all sample paths with bounded variation.

More concretely, we will consider cash flows described by an insurance policy
entirely determined by its policy functions. We denote by $a_{i}(t)$,
$i\in\mathscr{S}$, the sum of payments from the insurer to the insured up to
time $t$, given that we know that the insured has always been in state $i$.
Moreover, we will denote by $a_{ij}(t)$, $i,j\in\mathscr{S}$, $i\neq j$,
denotes the payments which are due when the insured switches state from $i$ to
$j$ at time $t$. We always assume that these functions are of bounded
variation (almost surely in the case of random payments). The cash flows we
will consider are entirely described by the policy functions defined by an
insurance policy.

######  Definition 2.2 (Policy cash flow).

We consider payout functions $a_{i}(t)$, $i\in S$ and $a_{ij}(t)$, $i,j\in S$,
$i\neq j$ for $t\geq 0$ of bounded variation. The (stochastic) cash flow
associated to this insurance is defined by

$A(t)=\sum_{i\in S}A_{i}(t)+\sum_{\begin{subarray}{c}i,j\in S\\\ i\neq
j\end{subarray}}A_{ij}(t),$

where

$A_{i}(t)=\int_{0}^{t}I_{i}^{X}(s)da_{i}(s),\quad
A_{ij}(t)=\int_{0}^{t}a_{ij}(s)dN_{ij}^{X}(s).$

The quantity $A_{i}$ corresponds to the accumulated liabilities while the
insured is in state $i$ and $A_{ij}$ for the case when the insured switches
from $i$ to $j$.

The value of a stochastic cash flow $A$ at time $t$ will be denoted by
$V(t,A)$ and is defined as

$V(t,A)\triangleq\frac{1}{v(t)}\int_{t}^{\infty}v(s)dA(s),$

where $v$ is a suitable discount factor (e.g. $v(t)$ can be today’s value of
one monetary unit at time $t$ with respect to a technical interest rate, see
Definition 2.4).

The stochastic integral is a well-defined pathwise Riemann-Stieltjes integral
since $A$ is of bounded variation with probability one. Since we wish to have
the best forecast for the reserve at a given time $t$, it is natural to
project on to the available information of the insurance company which, at
this point, is given by $\mathcal{F}_{t}^{X}$. Hence, the present value of the
cash flow $A$ given information $\mathcal{F}_{t}^{X}$ is given by

$\displaystyle
V_{\mathcal{F}^{X}}^{+}(t,A)\triangleq\mathbb{E}_{1}[V(t,A)|\mathcal{F}_{t}^{X}].$
(2.1)

In the above expression one can use any type of information in the conditional
expectation to obtain proper reserves. In this case however, we may use the
Markovianity of $X$ in order to compute $V_{\mathcal{F}^{X}}^{+}(t,A)$ as an
actual value, that is by the Markov property of $X$,

$V_{\mathcal{F}^{X}}^{+}(t,A)=\mathbb{E}_{1}[V(t,A)|\mathcal{F}_{t}^{X}]=\mathbb{E}_{1}[V(t,A)|\sigma(X_{t})]=H(t,X_{t})$

for some Borel-measurable function $H$, and hence

$V_{i}^{+}(t,A)\triangleq H(t,i)$

which corresponds to (2.1) and denotes the present value of $A$ given that we
know that the insured is in state $i$ at time $t$. An often abuse of notation
is to simply write

$V_{i}^{+}(t,A)=\mathbb{E}_{1}[V(t,A)|X_{t}=i].$

Having established our insurance model, we will next introduce our market
model with stochastic interest rate in an independent probability space.

###  2.2. The technical interest rate model

Let $(\Omega_{2},\mathcal{A}_{2},\mathbb{P}_{2})$ be a complete probability
space where $\Omega_{2}$ is the set of outcomes, $\mathcal{A}_{2}$ is a
suitable $\sigma$-algebra and $\mathbb{P}_{2}$ is a probability measure on
$(\Omega_{2},\mathcal{A}_{2})$ and we denote by $\mathbb{E}_{2}$ the
expectation under $\mathbb{P}_{2}$. This space carries a stochastic process:
$r=\\{r_{t},t\in[0,T]\\}$ modelling the value of an overnight technical
interest rate. On this space we consider the ($\mathbb{P}_{2}$-augmented)
filtration generated by this process, denoted by
$\mathcal{F}^{r}=\\{\mathcal{F}_{t}^{r}\\}$ and since no further information
is considered we take $\mathcal{A}_{2}=\mathcal{F}_{T}^{r}$. A natural
modeling assumption at this point is to assume that $\mathcal{F}^{X}$ and
$\mathcal{F}^{r}$ are independent.

More concretely, $r$ will be governed by the following stochastic differential
equation of Itô type.

$\displaystyle dr_{t}=\lambda(t,r_{t})dt+\tau(t,r_{t})dW_{t},\quad
r_{0}=x\in\mathbb{R},\quad t\in[0,T].$ (2.2)

Here, $W$ is a standard Wiener process. Moreover,
$\lambda,\tau:[0,T]\times\mathbb{R}\rightarrow\mathbb{R}$ are measurable
functions such that a pathwise unique global strong solution exists, and all
moments are finite. For instance, if they satisfy the Lipschitz property
uniformly in time, have at most linear growth and $\tau$ is away from $0$, we
have such result.

Moreover, $r$ from (2.2) is a strong Markov process admitting a stochastic
flow of homeomorphisms satisfying the flow property, i.e
$r_{t}^{s,r_{s}^{0,x}}=r_{t}^{0,x}$, where

$\displaystyle
r_{t}^{s,x}=x+\int_{s}^{t}\lambda(u,r_{u}^{s,x})du+\int_{s}^{t}\tau(u,r_{u}^{s,x})dW_{u},\quad
0\leq s\leq t\leq T,\quad x\in\mathbb{R}.$ (2.3)

######  Remark 2.3.

Conditions for the existence of unique global strong solutions admitting a
stochastic flow of homeomorphisms can be relaxed considerably. Low regularity
can be considered if one wishes to, but since this is not the purpose of this
work, we impose the classical Itô assumptions.

######  Definition 2.4 (Discount factor).

We define the (stochastic) discount factor as the process

$v(t)=e^{-\int_{0}^{t}r_{s}ds},\quad t\geq 0,$

which models the current value of one monetary unit at time $t$.

Hence, a liability $Z$ which is due at time $s>0$ has a present value today of
$v(s)Z$, and value $\frac{v(s)}{v(t)}Z$ at time $t$, $0\leq t\leq s$.

###  2.3. The model

From now on we work on the probability space $(\Omega,\mathcal{A},\mathbb{P})$
where $\Omega=\Omega_{1}\times\Omega_{2}$,
$\mathcal{A}=\mathcal{A}_{1}\otimes\mathcal{A}_{2}$ and
$\mathbb{P}=\mathbb{P}_{1}\times\mathbb{P}_{2}$ and denote by $\mathbb{E}$ the
expectation under $\mathbb{P}$. The overall information considered by the
insurer is $\mathcal{F}_{t}=\mathcal{F}_{t}^{X}\otimes\mathcal{F}_{t}^{r}$,
$t\in[0,T]$ and we have for an $\mathcal{F}_{T}$-measurable random variable
$Y$ on $(\Omega,\mathcal{A},\mathbb{P})$ that

$\mathbb{E}[Y|\mathcal{F}_{t}]=\mathbb{E}_{1}[\mathbb{E}_{2}[Y|\mathcal{F}_{t}^{r}]|\mathcal{F}_{t}^{X}]=\mathbb{E}_{2}[\mathbb{E}_{1}[Y|\mathcal{F}_{t}^{X}]|\mathcal{F}_{t}^{r}]$

for all $t\in[0,T]$.

We will simultaneously work on $\mathbb{P}$ or restricted to $\mathbb{P}_{i}$,
$i=1,2$ when necessary, without really distinguishing notations whenever the
context is clear. For example, if $Y\in L^{1}(\Omega,\mathcal{A},\mathbb{P})$
and $Y$ only depends on $\omega_{1}$ then $\mathbb{E}[Y]$ is implicitly both
under $\mathbb{P}$ and $\mathbb{P}_{2}$ since
$\mathbb{E}[Y]=\int_{\Omega}Y(\omega)\mathbb{P}(d\omega)=\int_{\Omega_{1}}Y(\omega_{1})\mathbb{P}_{1}(d\omega_{1})\int_{\Omega_{2}}\mathbb{P}_{2}(d\omega_{2})=\mathbb{E}_{2}[Y]$
since we are working with probability measures.

###  2.4. Present value of a policy when $r$, $a_{i}$ and $a_{ij}$ are
deterministic

We assume for a moment that $v(t)$ is known and $a_{i}$, $a_{ij}$,
$i,j\in\mathscr{S}$ are deterministic. In particular, the only information
available for the insurer is the state of the insured. Then [6, Theorem
4.6.10] states that the present value of a cash flow $A$ given by an insurance
policy entirely described by the functions $a_{i}$ and $a_{ij}$ is given by

$\displaystyle
V_{i}^{+}(t,A)=\sum_{j\in\mathscr{S}}\left[\int_{t}^{\infty}\frac{v(s)}{v(t)}p_{ij}(t,s)da_{j}(s)+\sum_{\begin{subarray}{c}k\in\mathscr{S}\\\
k\neq
j\end{subarray}}\int_{t}^{\infty}\frac{v(s)}{v(t)}p_{ij}(t,s)\mu_{jk}(s)a_{jk}(s)ds\right].$
(2.4)

The proof of the above formula is a consequence of [6, Theorem 4.6.3] which
corresponds to (2.1) using the Markov property on the process $X$. The formula
itself is quite intuitive. The (future) value of the policy is the sum over
all states of discounted accumulated payment streams coming from the insured
being or changing state.

If we assume that $a_{i}$ are a.e. differentiable with derivative
$\dot{a}_{i}$ and with possibly countable discontinuities at say, points
$t_{1},\dots,t_{n}\in[0,\infty)$, $n\geq 1$ then we can recast (2.4) as

$\displaystyle\begin{split}V_{i}^{+}(t,A)=&\,\sum_{j\in\mathscr{S}}\bigg{[}\sum_{l=1}^{n}\frac{v(t_{l})}{v(t)}p_{ij}(t,t_{l})\Delta
a_{j}(t_{l})\mathbb{I}_{\\{t<t_{l}\\}}+\int_{t}^{\infty}\frac{v(s)}{v(t)}p_{ij}(t,s)\dot{a}_{j}(s)ds\\\
&+\sum_{\begin{subarray}{c}k\in\mathscr{S}\\\ k\neq
j\end{subarray}}\int_{t}^{\infty}\frac{v(s)}{v(t)}p_{ij}(t,s)\mu_{jk}(s)a_{jk}(s)ds\bigg{]},\end{split}$
(2.5)

where $\Delta a_{j}(t)\triangleq a_{j}(t)-a_{j}(t^{-})$ for all $t\geq 0$ and
$j\in\mathscr{S}$.

This is like saying that we assume that the functions $a_{i}$ admit a density
(in the generalised sense) of the form

$\displaystyle da_{j}(t)=\sum_{l=1}^{n}\Delta
a_{j}(t_{l})\delta_{t_{l}}(t)dt+\dot{a}_{j}(t)dt,\quad t\in[0,\infty),\quad
j\in\mathscr{S},$ (2.6)

where $y\mapsto\delta_{x}(y)$ is a (generalised) function called the Dirac
delta function which has the property that

$\int_{-\infty}^{\infty}\delta_{x}(y)f(y)dy=f(x),\quad x\in\mathbb{R}$

for all locally integrable functions $f$. It can, for instance, be shown that

$\varphi_{\varepsilon}(y)=\frac{1}{\sqrt{2\pi\varepsilon}}e^{-\frac{1}{2\varepsilon}(y-x)^{2}},\quad\varepsilon>0$

converges in $L^{1}(\mathbb{R})$ as $\varepsilon\to 0$ to $\delta_{x}$.

Formula (2.5) in the case of stochastic interest rates has been studied by
[11] and the corresponding Thiele’s partial defferential equation was derived.
Here, in addition, we are interested in the case where $a_{i}$ and $a_{ij}$
also are stochastic and subject to the performance of $r$. Hence, the measure
in (2.6) (or rather its density) is also stochastic.

##  3\. Pricing and reserving life insurance subject to cash flows with
stochastic interest rate

As we can see in (2.5), if we assume that $a_{i}$, $a_{ij}$ are adapted to
$\mathcal{F}^{r}$, then we need to price claims of the form

$\Delta a_{j}(s)=\Delta a_{j}(r_{\leq
s}),\quad,\dot{a}_{j}(s)=\dot{a}_{j}(s,r_{\leq s}),\quad
a_{jk}(s)=a_{jk}(s,r_{\leq s}),$

for possibly varying maturities $s\geq t$. Here, $r_{\leq t}$ denotes the
whole trajectory of $r$ up to time $t$, and $\Delta a_{j},\dot{a}_{j},a_{jk}$
are suitable functionals on the space of continuous functions.

We will focus on claims that are functions of European and Asian type options
on $r$. For convenience introduce the following notation

$\displaystyle\overline{r}_{s,t}\triangleq\int_{s}^{t}r_{u}du,\quad
s,t\in[0,T],\quad s\leq t,$ (3.1)

and $\overline{r}_{t}\triangleq\overline{r}_{0,t}$.

We will consider payoff functions of the form

$\theta\left(r_{s},\overline{r}_{s}\right),$

for fixed $s\in[0,T]$ and some suitable function
$\theta:\mathbb{R}^{2}\rightarrow\mathbb{R}$. The risk neutral pricing
approach tells us that if the market is free of arbitrage, there will be a
martingale measure $\mathbb{Q}$ such that the price at time $t$ of an option
with payoff $\theta\left(r_{T},\overline{r}_{T}\right)$ is given by

$\displaystyle\pi_{t}(T)\triangleq\mathbb{E}_{\mathbb{Q}}\left[\frac{v(T)}{v(t)}\theta\left(r_{T},\overline{r}_{T}\right)\Big{|}\mathcal{F}_{t}\right].$
(3.2)

By the Markov property of $r$ and the fact that
$\overline{r}_{T}=\overline{r}_{t}+\overline{r}_{t,T}$ we may express
$\pi_{t}(T)$ as a function $u$ of the states of $r_{t}$ and $\overline{r}_{t}$
at time $t$. That is,

$\displaystyle\pi_{t}(T)=u\left(t,r_{t},\overline{r}_{t}\right)$ (3.3)

where

$\displaystyle
u(t,x,y)\triangleq\mathbb{E}_{\mathbb{Q}}\left[e^{-\int_{t}^{T}r_{s}^{t,x}ds}\theta\left(r_{T}^{t,x},y+\int_{t}^{T}r_{s}^{t,x}ds\right)\right],$

and $r_{\cdot}^{t,x}$ denotes the process given in (2.3).

The pricing measure $\mathbb{Q}$ is given by

$\displaystyle\frac{d\mathbb{Q}}{d\mathbb{P}}\Big{|}_{\mathcal{F}_{t}}=Z_{t},\quad
t\in[0,T],$

where

$Z_{t}\triangleq\mathcal{E}\left(\int_{0}^{\cdot}\gamma_{s}dW_{s}\right)_{t},\quad
t\in[0,T],$

and where $\mathcal{E}(Z)_{t}\triangleq\exp(Z_{t}-\frac{1}{2}[Z,Z]_{t})$ for
an Itô process $Z$ and $\gamma$ is an adapted process in
$L^{2}([0,T]\times\Omega)$.

Girsanov’s theorem implies that the process $W^{\mathbb{Q}}$ defined by

$W_{t}^{\mathbb{Q}}\triangleq W_{t}-\int_{0}^{t}\gamma_{s}ds,\quad t\in[0,T],$

is a $\mathbb{Q}$-Wiener processes.

More concretely, assuming that $\gamma_{t}=\gamma(t,r_{t})$ we have that the
$\mathbb{Q}$-dynamics of $r$ are given by

$\displaystyle
dr_{t}=(\lambda(t,r_{t})+\tau(t,r_{t})\gamma(t,r_{t}))dt+\tau(t,r_{t})dW_{t}^{\mathbb{Q}},\quad
r_{0}=x,\quad t\in[0,T].$

The following result is the celebrated Feynman-Kac formula which can be used
to find the value of (3.3) using the theory of partial differential equations.
The classical formula is often offered for European type options. Here, we
provide a version for path-dependent options adapted to our purposes.

######  Theorem 3.1 (Feynman-Kac formula).

Let $Y_{t}$ be an Itô diffusion given by

$dY_{t}=b(t,Y_{t})dt+a(t,Y_{t})dB_{t},\quad Y_{0}\in\mathbb{R},\quad
t\in[0,T],$

where $b:[0,T]\times\mathbb{R}\rightarrow\mathbb{R}$,
$\sigma:[0,T]\times\mathbb{R}\rightarrow\mathbb{R}$ satisfy classical Itô
assumptions for existence and uniqueness and $B$ is a standard Wiener process.

Let $R,S:[0,T]\times\mathbb{R}\rightarrow\mathbb{R}$ be two continuous
functions. Consider the function $(t,x,y)\mapsto U(t,x,y)$ solving the
following PDE

$\displaystyle\partial_{t}U+b(t,x)\partial_{x}U+x\partial_{y}U+\frac{1}{2}a^{2}(t,x)\partial_{x}^{2}U-R(t,x)U=0,$
(3.4)

with terminal condition $U(T,x,y)=\psi(t,x,y)$.

Then

$U(t,x,y)=\mathbb{E}\left[e^{-\int_{t}^{T}R(s,Y_{s})ds}\psi\left(Y_{T},\int_{0}^{T}Y_{s}ds\right)\Big{|}Y_{t}=x,\int_{0}^{t}Y_{s}ds=y\right]$

for all $(t,x)\in[0,T]\times\mathbb{R}$.

###### Proof.

Define the process

$\displaystyle Z_{s}\triangleq
e^{-\int_{t}^{s}R(v,Y_{v})dv}U\left(s,Y_{s},\int_{0}^{s}Y_{v}dv\right),\quad
t\leq s\leq T.$ (3.5)

We denote by $\partial_{s}U$, $\partial_{x}U$ and $\partial_{y}U$ the partial
derivaties of $(s,x,y)\mapsto U(s,x,y)$ with respect to $s$, $x$ and $y$,
respectively. Itô’s formula yields

$\displaystyle dZ_{s}=$
$\displaystyle\,-R(s,Y_{s})e^{-\int_{t}^{s}R(v,Y_{v})dv}Uds+e^{-\int_{t}^{s}R(v,Y_{v})dv}dU$
$\displaystyle=e^{-\int_{t}^{s}R(v,Y_{v})dv}\left[-R(s,Y_{s})U+\partial_{s}U+b(s,Y_{s})\partial_{x}U+Y_{s}\partial_{y}U+\frac{1}{2}a(s,Y_{s})^{2}\partial_{x}^{2}U\right]ds$
$\displaystyle+e^{-\int_{t}^{s}R(v,Y_{v})du}a(s,Y_{s})\partial_{x}UdB_{s}$

The finite variation part is $0$ since $U$ satisfies PDE (3.4). Hence,

$Z_{T}-Z_{t}=\int_{t}^{T}e^{-\int_{t}^{s}R(v,Y_{v})dv}a(s,Y_{s})\partial_{x}U\left(s,Y_{s},\int_{0}^{s}Y_{v}dv\right)dB_{s}.$

Taking expectations and using the martingale property of the Itô integral we
have

$\displaystyle\mathbb{E}\left[Z_{T}\Big{|}Y_{t}=x,\int_{0}^{t}Y_{s}ds=y\right]=\mathbb{E}\left[Z_{t}\Big{|}Y_{t}=x,\int_{0}^{t}Y_{s}ds=y\right].$
(3.6)

Due to (3.5) we have

$\mathbb{E}\left[Z_{t}\Big{|}Y_{t}=x,\int_{0}^{t}Y_{s}ds=y\right]=U(t,x,y).$

Hence, by (3.6) and the fact that $U(T,x,y)=\psi(x,y)$ we conclude that

$U(t,x,y)=\mathbb{E}\left[e^{-\int_{t}^{T}R(v,Y_{v})dv}\psi\left(Y_{T},\int_{0}^{T}Y_{s}ds\right)\Big{|}Y_{t}=x,\int_{0}^{t}Y_{s}ds=y\right].$

∎

Applying the above result to the case $Y_{t}=r_{t}$, $R(t,x)=x$ we have

$b(t,x)=\lambda(t,x)+\gamma(t,x)\tau(t,x),\quad a(t,x)=\tau(t,x).$

Hence, $u$ from (3.3) solves the following PDE

$\displaystyle\partial_{t}u+(\lambda(t,x)+\gamma(t,x)\tau(t,x))\partial_{x}u+x\partial_{y}u+\frac{1}{2}\tau(t,x)^{2}\partial_{x}^{2}u-xu=0$
(3.7)

with final condition $u(T,x,y)=\theta(x,y)$.

Denote the differential operator

$\displaystyle\begin{split}L=&\,(\lambda(t,x)+\gamma(t,x)\tau(t,x))\partial_{x}+x\partial_{y}+\frac{1}{2}\tau(t,x)^{2}\partial_{x}^{2}.\end{split}$
(3.8)

We will denote by $u_{T}^{\theta}$ the solution to the PDE (3.7) with payoff
function $\theta$ and maturity time $T$. That is

$\displaystyle\partial_{t}u_{T}^{\theta}(t,x,y)+Lu_{T}^{\theta}(t,x,y)=xu_{T}^{\theta}(t,x,y),\quad
u_{T}^{\theta}(T,x,y)=\theta(x,y).$ (3.9)

From now on, we will assume that the policy functions $a_{j}$ are
$\mathbb{P}$-a.s. a.e. differentiable with (stochastic) Lebesgue-Stieltjes
measure $da_{j}$ given by

$\displaystyle da_{j}(t)=\sum_{l=1}^{n}\Delta
a_{j}(t_{l})\delta_{t_{l}}(t)dt+\dot{a}_{j}(t)dt,\quad t\in[0,\infty),\quad
j\in\mathscr{S},$ (3.10)

with

$\displaystyle\Delta
a_{j}(t_{l})=f_{j}\left(r_{t_{l}},\overline{r}_{t_{l}}\right),\quad\dot{a}_{j}(t)=g_{j}\left(t,r_{t},\overline{r}_{t}\right),\quad
t\in[0,\infty),\quad j\in\mathscr{S},$ (3.11)

and that

$\displaystyle a_{jk}(t)=h_{jk}\left(t,r_{t},\overline{r}_{t}\right),\quad
t\in[0,\infty),\quad j,k\in\mathscr{S},$ (3.12)

for some functions $f_{j}:\mathbb{R}^{2}\rightarrow\mathbb{R}$,
$g_{j},h_{jk}:[0,T]\times\mathbb{R}^{2}\rightarrow\mathbb{R}$,
$j\in\mathscr{S}$, $k\neq j$ where we recall that $\overline{r}_{t}$ is the
notation in (3.1). These functions describe the stream of payments for each
state of the insured and for the transitions between states subject to
interest rate instantaneous and average changes.

In general, the cash flows coming from a policy are stochastic due to the fact
that the states of the insured are stochastic, while the payments are
deterministic, given that the state is known. Here, the payments are also
stochastic _per se_ , since they also depend on the interest rate curve.

Using the notation from (3.10) and (3.12) in connection with (3.3) and (2.5)
we have that the prospective reserve is given by,

$\displaystyle\begin{split}V_{i,\mathcal{F}}^{+}(t,A)=&\,\sum_{j\in\mathscr{S}}\sum_{l=1}^{n}p_{ij}(t,t_{l})u_{t_{l}}^{f_{j}}(t,r_{t},\overline{r}_{t})\mathbb{I}_{\\{t<t_{l}\\}}+\sum_{j\in\mathscr{S}}\int_{t}^{\infty}p_{ij}(t,s)u_{s}^{g_{j}(s,\cdot)}(t,r_{t},\overline{r}_{t})ds\\\
&+\sum_{\begin{subarray}{c}j,k\in\mathscr{S}\\\ k\neq
j\end{subarray}}\int_{t}^{\infty}p_{ij}(t,s)\mu_{jk}(s)u_{s}^{h_{jk}(s,\cdot)}(t,r_{t},\overline{r}_{t})ds.\end{split}$

’ In particular, we see that the prospective reserve of interest-rate-linked
policies with stochastic interest rate can be expressed as a function
$V(t,r_{t},\overline{r}_{t})$ of $t$, $r_{t}$ and $\overline{r}_{t}$. This is
due to the fact that the process $\overline{r}$ is $\mathcal{F}$-adapted and
$r$ is Markovian. We can characterize the function $V_{i}(t,x,y)$,
$(t,x,y)\in[0,T]\times\mathbb{R}^{2}$ by deriving the so-called Thiele’s
(partial) differential equation.

From now on, and without loss of generality, we assume $n=1$ and $t_{1}=T>0$.
Thus

$\displaystyle\begin{split}V_{i,\mathcal{F}}^{+}(t,A)=&\,\sum_{j\in\mathscr{S}}p_{ij}(t,T)u_{T}^{f_{j}}(t,r_{t},\overline{r}_{t})+\sum_{j\in\mathscr{S}}\int_{t}^{T}p_{ij}(t,s)u_{s}^{g_{j}(s,\cdot)}(t,r_{t},\overline{r}_{t})ds\\\
&+\sum_{\begin{subarray}{c}j,k\in\mathscr{S}\\\ k\neq
j\end{subarray}}\int_{t}^{T}p_{ij}(t,s)\mu_{jk}(s)u_{s}^{h_{jk}(s,\cdot)}(t,r_{t},\overline{r}_{t})ds.\end{split}$
(3.13)

where $u$ above satisfies the PDE (3.7) with the corresponding maturity times
and terminal conditions.

In the above expression the first term corresponds to the benefits associated
to being in state $j$ at the end of the contract, the second term corresponds
to inflow and outflow of benefits and premiums and the last term are benefits
from transitions between $j$ to $k$. We see that at the end of the contract we
have indeed $V_{i,\mathcal{F}}^{+}(T,A)=f_{i}(r_{T},\overline{r}_{T})$. The
following theorem is the corresponding Thiele’s parial differential equation
for (3.13), that is policies with payment streams subject to both sudden and
average changes in the interest rate curve.

######  Theorem 3.2 (Thiele’s partial differential equation).

Let $A$ be the payout function determined by the policy functions $f_{i}$,
$g_{i}$, $i\in\mathscr{S}$ and $h_{ij}$, $i,j\in\mathscr{S}$, $j\neq i$ as
defined in (3.11) and (3.12). Denote by $V_{i,\mathcal{F}}^{+}(t,A)$,
$t\in[0,T]$ the value of an insurance contract at time $t$ given that the
insured is in state $i\in\mathscr{S}$ at time $t$ and given the information
$\mathcal{F}_{t}$. Then

$V_{i,\mathcal{F}}^{+}(t,A)=V_{i}(t,r_{t},\overline{r}_{t}),$

where the function $V_{i}:[0,T]\times\mathbb{R}^{2}\rightarrow\mathbb{R}$ is
the solution to the following partial differential equation

$\displaystyle\begin{split}\partial_{t}V_{i}=&\,xV_{i}-g_{i}(t,x,y)-\sum_{j\neq
i}\mu_{ij}(t)(h_{ij}(t,x,y)+V_{j}-V_{i})-LV_{i}\end{split}$ (3.14)

with $L$ being the differential operator defined as

$\displaystyle L=$
$\displaystyle\,(\lambda(t,x)+\gamma(t,x)\tau(t,x))\partial_{x}+x\partial_{y}+\frac{1}{2}\tau^{2}(t,x)\partial_{x}^{2}.$
(3.15)

The boundary condition is given by $V_{i}(T,x,y)=f_{i}(x,y)$.

###### Proof.

The PDE given in (3.14) is a second order parabolic linear PDE. It is
therefore well-posed and admits a unique solution under the assumption that
the coefficients are continuous. See e.g. [5].

Observe that if $\theta_{T}^{1}$ and $\theta_{T}^{2}$ are two final conditions
for the PDE given in Theorem 3.1 and $u^{\theta_{T}^{1}}$ and
$u^{\theta_{T}^{2}}$ denote their solutions then
$u^{\theta_{T}^{1}+\theta_{T}^{2}}$ solves the PDE corresponding to the final
condition $\theta_{T}^{1}+\theta_{T}^{2}$. This is a trivial consequence of
the uniqueness of the PDE and the fact that $u^{\theta}$ appearing in (3.2) is
linear in $\theta$.

Define the function

$V_{i}(t,x,y)=G_{i}^{T}(t,x,y)+\int_{t}^{T}F_{i}^{s}(t,x,y)ds,\quad
t\in[0,T],\quad i\in\mathscr{S},$

where

$\displaystyle
G_{i}^{T}(t,x,y)\triangleq\sum_{j\in\mathscr{S}}p_{ij}(t,T)u_{T}^{f_{j}}(t,x,y),\quad
t\in[0,T],\quad i\in\mathscr{S},$

and

$\displaystyle
F_{i}^{s}(t,x,y)\triangleq\sum_{j\in\mathscr{S}}p_{ij}(t,s)u_{s}^{\theta_{j}^{s}}(t,x,y),\quad
t\in[0,T],\quad s\in[t,T],\quad i\in\mathscr{S},$ (3.16)

where

$\theta_{j}^{s}(x,y)\triangleq
g_{j}(s,x,y)+\sum_{\begin{subarray}{c}j,k\in\mathscr{S}\\\ k\neq
j\end{subarray}}\mu_{jk}(s)h_{jk}(s,x,y).$

Then the (stochastic) reserve $V_{i,\mathcal{F}}(t,A)$ with payout function
$A$ determined by the policy functions $f_{i}$, $g_{i}$ and $h_{ij}$ as
defined in (3.11) and (3.12) is given by

$V_{i,\mathcal{F}}^{+}(t,A)=V_{i}(t,x,y)\Big{|}_{(x,y)=(r_{t},\overline{r}_{t})}=G_{i}^{T}(t,r_{t},\overline{r}_{t})+\int_{t}^{T}F_{i}^{s}(t,r_{t},\overline{r}_{t})ds.$

Our arguments will be for fixed $T$ and $s\geq t$ being maturity times. Hence
$u_{T}^{f_{j}}$ and $u_{s}^{\theta_{j}^{s}}$ are well-defined.

First, observe that by Kolmogorov’s backward equation we have

$\displaystyle\partial_{t}p_{ij}(t,s)=\sum_{\begin{subarray}{c}k\in\mathscr{S}\\\
k\neq i\end{subarray}}\mu_{ik}(t)(p_{ij}(t,s)-p_{kj}(t,s)).$ (3.17)

Further, we can find a straightforward relation between the derivatives of
$F_{i}^{s}$ and those of $u_{s}^{\theta_{j}^{s}}$,

$\displaystyle\partial_{t}F_{i}^{s}$ $\displaystyle=\sum_{j}\sum_{k\neq
i}\mu_{ik}(t)(p_{ij}(t,s)-p_{kj}(t,s))u^{\theta_{j}^{s}}+\sum_{j}p_{ij}(t,s)\partial_{t}u^{\theta_{j}^{s}}$
$\displaystyle=\sum_{k\neq
i}\mu_{ik}(t)(F_{i}^{s}-F_{k}^{s})+\sum_{j}p_{ij}(t,s)\partial_{t}u^{\theta_{j}^{s}},$

where we used relation (3.17) first and (3.16) thereafter. Moreover, if $L$ is
the differential operator of (3.15), then

$LF_{i}^{s}=\sum_{j}p_{ij}(t,s)Lu_{s}^{\theta_{j}^{s}},$

since $L$ is linear.

For easier readability we drop the point $(t,r_{t},\overline{r}_{t})$ in the
notation. Now, we compute the Itô differential of
$F_{i}^{s}(t,r_{t},\overline{r}_{t})$ under the risk neutral measure
$\mathbb{Q}$ in two ways: first, by direct defition, i.e.

$\displaystyle
dF_{i}^{s}=(\partial_{t}F_{i}^{s}+LF_{i}^{s})dt+\tau(t,r_{t})\partial_{x}F_{i}^{s}dW_{t}^{\mathbb{Q}},$
(3.18)

and now we compute $dF_{i}^{s}(t,r_{t},\overline{r}_{t})$ using the relation
(3.16) in connection with the identity (3.17),

$\displaystyle\begin{split}dF_{i}^{s}=&\,\left(\sum_{k\neq
i}\mu_{ik}(t)(F_{i}^{s}-F_{k}^{s})+\sum_{j}p_{ij}(t,s)\left(\partial_{t}u_{s}^{\theta_{j}^{s}}+Lu_{s}^{\theta_{j}^{s}}\right)\right)dt+\tau(t,r_{t})\partial_{x}F_{i}^{s}dW_{t}^{\mathbb{Q}}.\end{split}$

Now, we use the fact that $u_{s}^{\theta_{j}^{s}}$ is a solution to
$\partial_{t}u_{s}^{g_{j}^{s}}+Lu_{s}^{g_{j}^{s}}=xu_{s}^{g_{j}^{s}}$. Hence,

$\displaystyle dF_{i}^{s}=\left(\sum_{k\neq
i}\mu_{ik}(t)(F_{i}^{s}-F_{k}^{s})+r_{t}F_{i}^{s}\right)dt+\tau(t,r_{t})\partial_{x}F_{i}^{s}dW_{t}^{\mathbb{Q}}.$
(3.19)

Equating (3.18) and (3.19) we obtain the following PDE in time and space for
the function $(t,x,y)\mapsto F_{i}^{s}(t,x,y)$ for fixed $s$:

$\displaystyle\partial_{t}F_{i}^{s}+LF_{i}^{s}=\sum_{k\neq
i}\mu_{ik}(t)(F_{i}^{s}-F_{k}^{s})+xF_{i}^{s}.$ (3.20)

On the other hand, recall that
$V_{i}(t,r_{t},\overline{r}_{t})=V_{i}(t,x,y)\Big{|}_{(x,y)=(r_{t},\overline{r}_{t})}$
and

$V_{i}(t,x,y)=G_{i}^{T}(t,x,y)+\int_{t}^{T}F_{i}^{s}(t,x,y)ds.$

Therefore,

$\partial_{t}V_{i}(t,x.y)=\partial_{t}G_{i}^{T}(t,x,y)+\int_{t}^{T}\partial_{t}F_{i}^{s}(t,x,y)ds-\lim_{\begin{subarray}{c}s\to
t\\\ s>t\end{subarray}}F_{i}^{s}(t,x,y),$

where we used Lebesgue’s dominated convergence theorem and the fundamental
theorem of calculus. Observe that

$\displaystyle\lim_{\begin{subarray}{c}s\to t\\\
s>t\end{subarray}}F_{i}^{s}(t,x,y)$
$\displaystyle=\sum_{j}p_{ij}(t,t)u_{t}^{\theta_{j}^{t}}(t,x,y)=u_{t}^{\theta_{i}^{t}}(t,x,y)=\theta_{i}^{t}(x,y)$
$\displaystyle=g_{i}(t,x,y)+\sum_{k\neq i}\mu_{ik}(t)h_{ik}(t,x,y).$

Altogether,

$\int_{t}^{T}\partial_{t}F_{i}^{s}(t,x,y)ds=\partial_{t}V_{i}(t,x,y)-\partial_{t}G_{i}^{T}(t,x,y)+g_{i}(t,x,y)+\sum_{k\neq
i}\mu_{ik}(t)h_{ik}(t,x,y).$

In particular, evaluating at $(t,x,y)=(t,r_{t},\overline{r}_{t})$ we have

$\displaystyle\begin{split}\int_{t}^{T}\partial_{t}F_{i}^{s}(t,r_{t},\overline{r}_{t})ds\\\
&=\partial_{t}V_{i}(t,r_{t},\overline{r}_{t})-\partial_{t}G_{i}^{T}(t,r_{t},\overline{r}_{t})+g_{i}(t,r_{t},\overline{r}_{t})+\sum_{k\neq
i}\mu_{ik}(t)h_{ik}(t,r_{t},\overline{r}_{t}).\end{split}$ (3.21)

Integrating (3.20) with respect to $s$ on the region $[t,T]$, interchanging
integration and differentiability, using (3.21) and
$V_{i}(t,x,y)=G_{i}^{T}(t,x,y)+\int_{t}^{T}F_{i}^{s}(t,x,y)ds$ we obtain

$\partial_{t}(V_{i}-G_{i}^{T})+g_{i}+\sum_{k\neq
i}\mu_{ik}(t)h_{ik}+L(V_{i}-G_{i}^{T})=\sum_{k\neq
i}\mu_{ik}(t)\left(V_{i}-V_{k}+G_{k}^{T}-G_{i}^{T}\right)+x(V_{i}-G_{i}^{T}).$

All the terms involving $G_{i}^{T}$ will cancel each other. Indeed, observe
that

$\displaystyle\partial_{t}G_{i}^{T}$
$\displaystyle=\sum_{j}\partial_{t}p_{ij}(t,T)u_{T}^{f_{j}}+\sum_{j}p_{ij}(t,T)\partial_{t}u_{T}^{f_{j}}$
$\displaystyle=\sum_{k\neq
i}\mu_{ik}(t)(G_{i}^{T}-G_{k}^{T})+\sum_{j}p_{ij}(t,T)\partial_{t}u_{T}^{f_{j}}.$

Also,

$\displaystyle\partial_{t}G_{i}^{T}+LG_{i}^{T}$ $\displaystyle=\sum_{k\neq
i}\mu_{ik}(t)(G_{i}^{T}-G_{k}^{T})+\sum_{j}p_{ij}(t,T)(\partial_{t}u_{T}^{f_{j}}+Lu_{T}^{f_{j}})$
$\displaystyle=\sum_{k\neq i}\mu_{ik}(t)(G_{i}^{T}-G_{k}^{T})+xG_{i}^{T},$

where we used that $\partial_{t}u_{T}^{f_{j}}+Lu_{T}^{f_{j}}=xu_{T}^{f_{j}}$.

As a result, we obtain $V_{i}(T,x,y)=f_{i}(x,y)$ and

$\partial_{t}V_{i}=xV_{i}-g_{i}(t,x,y)-\sum_{k\neq
i}\mu_{ik}(t)h_{ik}(t,x,y)+\sum_{k\neq i}\mu_{ik}(t)(V_{i}-V_{k})-LV_{i}$

and the result follows by grouping together the sums over $k\in\mathscr{S}$,
$k\neq i$. ∎

The following result is a direct consequence of Thiele’s PDE obtained in
Theorem 3.2 showing that one can aggregate reserves from policies of equal
ages and expiration dates.

######  Corollary 3.3.

Assume we have $n\geq 1$ policies with the same maturities $T$ and all
policyholders enter the contract at the same age. Then the present value of
all reserves

$\overline{V}_{i}(t,x,y)\triangleq\sum_{k=1}^{n}V_{i}^{k}(t,x,y),\quad
i\in\mathscr{S},$

where $V_{i}^{k}(t,x,y)$ corresponds to the reserve of policy $k$ at time $t$
and interest rate state $x$, given that the policy holder is in state
$i\in\mathscr{S}$, satisfies the PDE

$\partial_{t}\overline{V}_{i}=x\overline{V}_{i}-\overline{g}_{i}-\sum_{j\neq
i}\mu_{ij}(t)(\overline{h}_{ij}+\overline{V}_{j}-\overline{V}_{i})-L\overline{V}_{i},$

with terminal condition $\overline{V}_{i}(T,x,y)=\overline{f}_{i}(x,y)$ where

$\overline{f}_{i}(x,y)=\sum_{k=1}^{n}f_{i}^{k}(x,y),\quad\overline{g}_{i}(x,y)=\sum_{k=1}^{n}g_{i}^{k}(x,y),\quad\overline{h}_{ij}(x,y)=\sum_{k=1}^{n}h_{ij}^{k}(x,y).$

Here, $f_{i}^{k}$, $g_{i}^{k}$ and $h_{ik}^{k}$ denote the policy functions
for the $k$th policyholder.

##  4\. Life insurance policies with stochastic policy functions

Henceforward, we assume that $\mathscr{S}=\\{\ast,{\dagger}\\}$. Since there
are only two states and one of them is absorbing, we drop the notations $\ast$
and ${\dagger}$. That is to say, we simply write $\mu$ to denote the force of
mortality, $f$, $g$ and $h$ the policy functions and $V$ the mathematical
reserve given that the insured is alive, since otherwise $V_{\dagger}\equiv
0$.

For simulations purposes, we will assume the Gompertz-Makeham law of mortality
on $\mu$ given by

$\mu(t)=\alpha_{0}+\alpha_{1}\exp(\alpha_{2}t),\quad t\geq
0,\quad\alpha_{0},\alpha_{1},\alpha_{2}\in\mathbb{R}.$

This law of mortality describes the age dynamics of human mortality rather
accurately in the age window from about $30$ to $80$ years of age, which is
good enough for our analysis. For this reason, we excluded the very first and
last age groups from the data. We obtain $\hat{\alpha_{0}}=0.00127529$,
$\hat{\alpha_{1}}=2.51137\cdot 10^{-6}$ and $\hat{\alpha_{2}}=0.1271853$ which
are the least squares estimates obtained by fitting Norwegian mortality from
2019 (both genders together). See Statistics Norway, table: 05381 for the
employed data. Our examples will assume that the age of the insured at the
beginning of the contract is fixed to $30$ years old.

We will introduce the following notations

$\displaystyle
U_{s}^{K}(t,r_{t})\triangleq\mathbb{E}_{\mathbb{Q}}\left[e^{-\int_{t}^{s}r_{u}du}\mathbb{I}_{\\{r_{s}\geq
K\\}}|\sigma(r_{t})\right],$ (4.1)

and

$\displaystyle\overline{U}_{s}^{K}(t,r_{t},\overline{r}_{t})\triangleq\mathbb{E}_{\mathbb{Q}}\left[e^{-\int_{t}^{s}r_{u}du}\mathbb{I}_{\\{\overline{r}_{s}\geq
Ks\\}}|\sigma(r_{t})\otimes\sigma(\overline{r}_{t})\right],$ (4.2)

which appear naturally in many common policy specifications with regimes, such
as endowment, term insurance, pensions, etc.

For simulation purposes we consider the dynamics of the Vasicek short-rate
model, which are given by

$\displaystyle dr_{t}=a(b-r_{t})dt+\sigma dW_{t},\quad
r_{0}\in\mathbb{R},\quad t\in[0,T],$ (4.3)

where $a,b,\sigma\in\mathbb{R}$, $\sigma>0$ are model parameters. In this case
the market price of risk $\gamma(t,x)\equiv\gamma\in\mathbb{R}$ is an
additional constant parameter. Since the Vasicek model has the property that
it is invariant under change of measure, we simply take $\gamma=0$ and the
reader may adjust the interest market price of risk by a modification of $a$
and $b$.

Under the model in (4.3), one can find fairly explicit expressions. For
example, PDE (3.9) can be solved in closed form for some specific terminal
conditions $\theta$.

Introduce the notations

$\mu_{r_{s}|r_{t}}(h|x)\triangleq
xe^{-ah}+b(1-e^{-ah}),\quad\mu_{\overline{r}_{t,s}|r_{t}}(h,x)\triangleq(x-b)\frac{1}{a}(1-e^{-ah})+bh,$
$\sigma_{r_{s}|r_{t}}^{2}(h)\triangleq\frac{\sigma^{2}}{2a}\left(1-e^{-2ah}\right),\quad\sigma_{\overline{r}_{t,s}|r_{t}}^{2}(h)\triangleq\frac{\sigma^{2}}{a^{2}}\left[h-2\frac{1}{a}(1-e^{-ah})+\frac{1}{2a}\left(1-e^{-2ah}\right)\right]$

and the covariance

$\sigma_{r_{s}|r_{t},\overline{r}_{t,s}|r_{t}}(h)\triangleq\frac{\sigma^{2}}{a}\left[\frac{1}{a}(1-e^{-ah})-\frac{1}{2a}\left(1-e^{-2ah}\right)\right],$

which gives the correlation function

$\rho_{r_{s}|r_{t},\overline{r}_{t,s}|r_{t}}(h)\triangleq\frac{\sigma_{r_{s}|r_{t},\overline{r}_{t,s}|r_{t}}(h)}{\sigma_{r_{s}|r_{t}}(h)\sigma_{\overline{r}_{t,s}|r_{t}}(h)}.$

It holds that

$r_{s}|r_{t}\sim
N(\mu_{r_{s}|r_{t}}(s-t|r_{t}),\sigma_{r_{s}|r_{t}}^{2}(s-t)),\quad\overline{r}_{t,s}|r_{t}\sim
N(\mu_{\overline{r}_{t,s}|r_{t}}(s-t,r_{t}),\sigma_{\overline{r}_{t,s}|r_{t}}^{2}(s-t)).$

From the above, one can deduce that

$\displaystyle\begin{split}U_{s}^{K}(t,r_{t})&=\\\
&=e^{-\mu_{\overline{r}_{s}|r_{t}}(h|x)+\frac{1}{2}\sigma_{\overline{r}_{s}|r_{t}}^{2}(h)}\Phi\left(-\rho_{r_{s}|r_{t},\overline{r}_{t,s}|r_{t}}(h)\sigma_{\overline{r}_{s}|r_{t}}(h)+\frac{\mu_{r_{t,s}|r_{t}}(h,x)-K}{\sigma_{r_{t,s}|r_{t}}(h)}\right)\Bigg{|}_{(h,x)=(s-t,r_{t})},\end{split}$

where $\Phi$ denotes the distribution function of a standard normally
distributed random variable. Similarly, one can show that

$\displaystyle\begin{split}\overline{U}_{s}^{K}(t,r_{t},\overline{r}_{t})&=\\\
\\\
&=e^{-\mu_{\overline{r}_{s}|r_{t}}(h|x)+\frac{1}{2}\sigma_{\overline{r}_{s}|r_{t}}^{2}(h)}\Phi\left(-\sigma_{\overline{r}_{s}|r_{t}}(h)+\frac{\mu_{\overline{r}_{t,s}|r_{t}}(h,x)-Ks+y}{\sigma_{\overline{r}_{t,s}|r_{t}}(h)}\right)\Bigg{|}_{(h,x,y)=(s-t,r_{t},\overline{r}_{t})}.\end{split}$

Observe that when $K=-\infty$ then we obtain the classical price of a zero-
coupon bond at time $t$ with maturity $s$ and when $K=\infty$ both
expectations are null.

###  4.1. Pure endowment with premium reduction on high interest rate levels

Let $E>0$ be the guaranteed endowment to be paid at the end of the contract
$T$ upon survival. Let $\rho\in[0,1]$ be a reduction factor and $K>0$ an
interest rate level above which premiums are reduced by a factor of $1-\rho$.
Then the value of this contract, premiums taken into account is given by

$\displaystyle V^{\rho}(t,r_{t})=$
$\displaystyle\,-\pi^{\rho}\int_{t}^{T}p_{\ast\ast}(t,s)\left(U_{s}^{-\infty}(t,r_{t})-\rho
U_{s}^{K}(t,r_{t})\right)ds+Ep_{\ast\ast}(t,T)U_{T}^{-\infty}(t,r_{t}),$

where $U_{s}^{-\infty}$ is the function given in (4.1).

We choose $\pi^{\rho}$ in such a way that $V^{\rho}(0,r_{0})=0$. In this case,
the expected difference between reserves is given by

$\mathbb{E}[V^{\rho}(t,r_{t})-V^{0}(t,r_{t})]=(\pi^{0}-\pi^{\rho})\int_{t}^{T}p_{\ast\ast}(t,s)\mathbb{E}[U_{s}^{-\infty}(t,r_{t})]ds+\pi^{\rho}\rho\int_{t}^{T}\mathbb{E}[U_{s}^{K}(t,r_{t})]ds.$

In Figure 1 we show an example of two random interest rate curves and the
corresponding reserves for a contract ($\times 1\,000$) with and without
reduction and the difference between such reserves. The parameters for the
interest rate model are $r_{0}=0.03$, $a=0.1$, $b=0.2$, $\gamma=0$,
$\sigma=0.01$. The contract pays an endowment of $\$100\,000$ in $T=10$ years
for a person who is $30$ years old today. For the reduction case we take a
threshold of $K=0.04$ and a premium reduction of $\rho=20\%$.

Figure 1. On top: Reserves for an endowment of $\$100\,000$ in $\times 1\,000$
units with no reduction (in blue) and with a reduction of $20\%$ on premiums
for interest rate above $K=4\%$ (in red). In the middle: difference between
reserves. On the bottom: the corresponding interest rate (random) outcomes.
The premiums obtained are $\pi^{0}=\$8\,770.28$ and $\pi^{\rho}=\$9\,092.40$
when $r_{0}=3\%$.

One may argue that $\pi^{\rho}=\$9\,092.40$ is a rather high premium for an
endowment of $\$100\,000$, but the insured could have potentially profited
from long high interest regimes as, for instance, the outcome on the right.
There, the insured has to pay approximately $(1-\rho)\pi^{\rho}=7\,273.92$ for
eight years of high interest rate and $\pi^{\rho}=\$9\,092.40$ for the two
first years of the contract. Thus, a total amount of $\$76\,376.16$. Under the
same outcome, an insured with no reduction would have paid
$10\pi^{0}=\$87\,702.87$. In any case, to overcome the potential issue of
paying too high premiums, one can add a premium refund at the end of the
contract if the rates are low, or as we will see later, define a policy based
on average rates rather than current rates.

We can see in Figure 1 the reserves and their differences for two (random)
interest rate regimes; one with regime mostly under $4\%$ (right), and another
with regime crossing $K=4\%$ (left). Both outcomes shows that the reserves for
the case of a policy with premium reduction requires slightly higher reserves.
In Figure 2, we show the mean difference in the long-run.

(a) $1\,000$ simulations ($\sim 16$ min)

(b) $10\,000$ simulations ($\sim 2$ h $36$ min)

(c) $100\,000$ simulations ($\sim 1$ d $1$ h $40$ min)

(d) Multiples of $10\,000$ simulations

Figure 2. Mean difference reserve between an endowment with reduction
$\rho=20\%$ above $K=4\%$ and no reduction. We can see that the difference is
more prominent and significant at the beginning of the contract and that the
approximation is more reliable after $10\ 000$ simulations.

To finish this example we show the reserve surface, i.e. the function
$(t,x)\mapsto V^{\rho}(t,x)$ for $\rho=20\%$ and the surface of the difference
between the reserves with reduction and without.

(a) Reserve surface, $\rho=20\%$

(b) Surface of the difference

Figure 3.

The surfaces in Figure 3 gives us complete information of the behaviour of the
reserves and their differences for each plausible interest rate value $x$. One
can think that all outcomes from Figure 1 are paths along the surface starting
at interest $0.03$ and ending at $\$100\,000$.

###  4.2. Pension insurance with pension bonus during high interest rate
regimes

In the case of a pension insurance, the present value $V^{\rho}$ of a policy
with periodical premiums $\pi$ paying a periodical pension of $P$ if interest
is low and $(1+\rho)P$ if interest is high, from time $\hat{T}$ until end of
life is given by

$\displaystyle V^{\rho}(t,r_{t})=$
$\displaystyle\,-\pi\int_{\min\\{t,\hat{T}\\}}^{\hat{T}}p_{\ast\ast}(t,s)U_{s}^{-\infty}(t,r_{t})ds$
$\displaystyle+P\int_{{\max\\{t,\hat{T}\\}}}^{\infty}p_{\ast\ast}(t,s)\left(U_{s}^{-\infty}(t,r_{t})+\rho
U_{s}^{K}(t,r_{t})\right)ds.$

The PDE associated to $V^{\rho}$ in this case is given by

$\partial_{t}V=xV+\pi\mathbb{I}_{\\{x<K,0\leq
t<\hat{T}\\}}-P\left(1+\rho\mathbb{I}_{\\{x\geq
K\\}}\right)\mathbb{I}_{\\{\hat{T}\leq t<\infty\\}}+\mu(t)V-LV,$

where

$LV=\left[a(b-x)+\gamma\tau\right]\partial_{x}V-\frac{1}{2}\tau^{2}\partial_{x}^{2}V$

with boundary condition $\displaystyle\lim_{t\to\infty}V(t,x)=0$.

Insurance companies usually set the end of the contract at an age of $120$ or
similar. Hence, a computationally more friendly boundary condition for the
case that the insured is $30$ years old would be $V(90,x)=0$ being the maximum
length of the contract $T=90$ years.

We still consider a person who is $30$ years old today and will retire at the
age of $70$, that is in $\hat{T}=40$ years from now. In Figure 4 we show an
example of two random interest rate curves and the corresponding reserves for
a contract ($\times 1\,000$) with and without bonus on pensions and the
difference between such reserves. The parameters for the interest rate model
are $r_{0}=0.03$, $a=0.1$, $b=0.2$, $\gamma=0$, $\sigma=0.01$. The contract
pays a pension $P$ of $\$20\,000$ yearly if $r_{t}<4\%$ and $(1+\rho)P$ if
$r_{t}\geq 4\%$. We take a pension bonus of $\rho=20\%$.

Figure 4. On top: Reserves for a pension insurance of $\$20\,000$ yearly in
$\times 1\,000$ units with no bonus (in blue) and with a bonus of $20\%$ on
pensions for interest rate regimes above $K=4\%$ (in red). In the middle:
difference between reserves. On the bottom: the corresponding interest rate
(random) outcomes. The premiums obtained are $\pi^{0}=\$8\,611.31$ and
$\pi^{\rho}=\$8\,910.87$

(a) $50$ simulations ($\sim 53$ minutes)

(b) $500$ simulations ($\sim 9$h $31$ minutes)

(c) Multiples of $100$ simulations. Dotted line corresponds to $10$ and thick
line to $500$.

Figure 5. Mean difference reserve between a pension insurance with bonus of
$\rho=20\%$ above $K=4\%$ and no bonus. We can see that the difference is more
prominent and significant around the retirement age.

(a) Reserve surface, $\rho=20\%$

(b) Surface of the difference

Figure 6. Both surfaces have been computed using the direct formula for the
reserve ($\sim 31$ minutes)

###  4.3. Interest rate caps and floors insurance

One can also look at four classical options on $r_{T}$. Figure 7 shows the
surfaces for the present value of an interest rate cap, an interest rate
floor, a call and a put option with strike rate $4\%$ at the end of the
contract. We obtain these present values by solving the PDE from Theorem 3.2
which in this case is given by

$\partial_{t}V=xV+\mu(t)V-LV,$

with terminal conditions $V_{\mbox{cap}}(T,x)=E\mathbb{I}_{\\{x\geq K\\}}$,
$V_{\mbox{floor}}(T,x)=E\mathbb{I}_{\\{x\leq K\\}}$,
$V_{\mbox{call}}(T,x)=E\max\\{x-K,0\\}$ and
$V_{\mbox{put}}(T,x)=E\max\\{K-x,0\\}$. We use an explicit finite difference
method to obtain the solutions. The curve we see at $t=0$ corresponds to the
fair premium of the contract.

(a) Reserve surface for a cap

(b) Reserve surface for a floor

(c) Reserve surface for a caplet

(d) Reserve surface for a floorlet

Figure 7. European options on interest rate with strike $K=4\%$. Surface is
obtained solving Thiele’s PDE (3.14) with an explicit finite difference method
with step sizes $h=0.1$ for time and $l=1/12$ for space. Execution time around
$0.2$ seconds.

###  4.4. Binary endowment based on average interest rate

Let $E_{1}$ and $E_{2}$ be two endowments. The policy pays $E_{1}$ upon
survival at expiry time $T$ if the average interest rate during the contract
time is above $K$ and $E_{2}$ otherwise. Mathematically, the present value of
this policy is given by

$V(t,r_{t},\overline{r}_{t})=p_{\ast\ast}(t,T)\left[E_{1}\overline{U}_{T}^{K}(t,r_{t},\overline{r}_{t})+E_{2}\left(\overline{U}_{T}^{-\infty}(t,r_{t},\overline{r}_{t})-\overline{U}_{T}^{K}(t,r_{t},\overline{r}_{t})\right)\right],$

where here $\overline{U}_{s}^{K}$ is the function defined in (4.2).

In Figure 8 we show the reserves for a binary endowment of $E_{1}=\$150\,000$
if the average is above $K=4\%$ at the end of the contract and
$E_{2}=\$100\,000$ if the average is below under the same model as in the
previous examples. The reserve is indeed stochastic.

Figure 8. On top: Reserves for a binary endowment of $E_{1}=\$150\,000$ in
$\times 1\,000$ units if average rate is above $K=4\%$ and $E_{2}=\$100\,000$
otherwise. On the bottom: the corresponding interest rate (random) outcomes
(dashed line) and the running average during the contract (solid line). The
premium obtained was $\pi=\$9\,516.71$ when $r_{0}=3\%$. Vasicek parameters:
$a=0.1$, $b=0.2$, $\sigma=0.01$.

###  4.5. A reinsurance treaty on pensions when insurer’s average return is
low

Using the new model introduced in Section 3 we can consider reinsurance
agreements between the insurer and the reinsurer. If we assume that $r$ models
the return on investments for the insurer, then their liabilities depend on
$r$. If returns are low, then the pension liabilities increase, making it
difficult to meet the requirements with their customers. One possibility to
relax such risk could be to cede some of the risk (of low returns) to the
reinsurer. In this example, we consider two polices; one for the insured and
another one for the insurer (with the reinsurer). For a pension policy paying
a yearly pension $P$ from $\hat{T}$ upon death (or high enough $T$) we
consider

$\dot{a}_{\ast}^{ins}(s)=g_{\ast}^{ins}(s,r_{s},\overline{r}_{s})=P,\quad
s\in[\hat{T},T),$

giving rise to

$\displaystyle V^{ins}(t,r_{t})\triangleq
P\int_{\max\\{t,\hat{T}\\}}^{T}p_{\ast\ast}(t,s)U_{s}^{-\infty}(t,r_{t})ds,\quad
t\in[\hat{T},T]$

the value of the policy at each time. On the other hand, we can look at the
performance of $\overline{r}_{t}$ from the contract start to the time pension
payments start $\hat{T}$. If average returns are too low we can cede some of
the liabilities to the reinsurer by purchasing a (stochastic) endowment
(re)insurance in a similar fashion as in Section 4.4, which pays $100\rho\%$
of the value of the pension at time $t=\hat{T}$, that is the following
(stochastic) endowment

$\rho
V^{ins}(\hat{T},r_{\hat{T}})\mathbb{I}_{\\{\overline{r}_{\hat{T}}<K\hat{T}\\}}.$

The value of this endowment (re)insurance is thus given by

$\displaystyle
V^{re}(t,r_{t},\overline{r}_{t})=p_{\ast\ast}(t,\hat{T})\mathbb{E}_{\mathbb{Q}}\left[e^{-\int_{t}^{\hat{T}}r_{u}du}\rho
V^{ins}(\hat{T},r_{\hat{T}})\mathbb{I}_{\\{\overline{r}_{\hat{T}}<K\hat{T}\\}}\Big{|}\sigma(r_{t})\otimes\sigma(\overline{r}_{t})\right].$

The above conditional expectation is rather involved. This example shows why
the PDE derived in Theorem 3.2 may be useful when looking at policies with
stochastic payments where explicit expressions cannot directly be obtained.
The reserve $V^{re}(t,x,y)$ is a function of three variables: time $t$, level
of return $x$ and level of the average return $y$. It satisfies the following
PDE

$\partial_{t}V^{re}=xV^{re}+\mu(t)V^{re}-LV^{re},\quad t\in[0,\hat{T}],$

where $L$ is the differential operator in (3.15). The terminal condition is
given by

$V^{re}(\hat{T},x,y)=\rho V^{ins}(\hat{T},x)\mathbb{I}_{\\{y<K\hat{T}\\}}.$

Here, $V^{ins}(t,x)$ is a function of two variables where again, $t$ is time,
$x$ is the level of the return and it satisfies the following PDE

$\partial_{t}V^{ins}=xV^{ins}-P+\mu(t)V^{ins}-LV^{ins},\quad t\in[\hat{T},T],$

with terminal condition $V^{ins}(T,x)=0$.

## References

  * [1] Bacinello, A.R., Ortu, F. (1993). Pricing Guaranteed Securities-linked Life Insurance under Interest Rate Risk. In proceedings from AFIR (Actuarial Approach for Financial Risks) 3rd International Colloquium, Rome, Italy, volume 1, pages 35–55.
  * [2] Bacinello, A.R., Ortu, F. (1993). Single an Periodic Premiums for Guaranteed Equity-Linked Life Insurance under Interst Rate Risk: The "Lognormal + Vacisek" Case. In Peccati L. and M. Viren editors, Financial Modelling, pages 1–25. Physica-Verlag, Heidelberg, Germany.
  * [3] Bacinello, A. R., Persson, S. A. (2000) Design and Pricing of Equity-Linked Life Insurance Under Stochastic Interest Rates. Available at SSRN: https://ssrn.com/abstract=276701 or http://dx.doi.org/10.2139/ssrn.276701
  * [4] Baños, D. R., Lagunas-Merino, M., Ortiz-Latorre, S. (2020) Variance and Interest Rate Risk in Unit-Linked Insurance Policies. Risks 8(3), 84.
  * [5] Evans, L. Partial Differential Equations. American Mathematical Society. 2nd edition. 2010.
  * [6] Koller, M. Stochastic Models in Life Insurance. Springer-Verlag Berlin Heidelberg. Ed. 1. 2012.
  * [7] Kurtz, A. (1996). Pricing of Equity-linked Life Insurance Policies with an Asset Value Guarantee and Periodic Premiums. In proceedings from AFIR (Actuarial Approach for Financial Risks) 6th International Colloquium, Nürnberg, Germany, volume 2, pages 1485–1496.
  * [8] Nielsen, J.A., Sandmann, K., (1995). Equity-linked life insurance: a model with stochastic interest rates. Insurance: Mathematics and Economics 16, 225–253.
  * [9] Nielsen, J.A., Sandmann, K., (1996). Uniqueness of the Fair Premium for Equity-Linked Life Insurance Contracts. The Geneva Risk and Insurance Review 21(1), 65–102.
  * [10] Norberg, R., Møller, C. M. (1996). Thiele’s Differential Equation with Stochastic Interest Rate of Diffusion Type. Scand. Actuarial J. 1, 37–49.
  * [11] Persson, S. A. (1998). Stochastic Interest Rate in Life Insurance: The Principle of Equivalence Revisited. Scand. Actuarial J. 2, 97–112.
  * [12] Steffensen, M. (2000). A no arbitrage approach to Thiele’s differential equation. Insurance: Mathematics and Economics 27, 201–214.