# A cohort-based Partial Internal Model for demographic risk

Gian Paolo Clemente<EMAIL_ADDRESS>Department of Mathematics
for Economic, Financial and Actuarial Sciences
Università Cattolica del Sacro Cuore, Milano  Francesco Della Corte
<EMAIL_ADDRESS>Nino Savelli<EMAIL_ADDRESS>
###### Abstract

We investigate the quantification of demographic risk in a framework
consistent with the market-consistent valuation imposed by Solvency II. We
provide compact formulas for evaluating inflows and outflows of a portfolio of
insurance policies based on a cohort approach. In this context, we maintain
the highest level of generality in order to consider both traditional policies
and equity-linked policies: therefore, we propose a market-consistent
valuation of the liabilities. In the second step we evaluate the Solvency
Capital Requirement of the idiosyncratic risk, linked to accidental mortality,
and the systematic risk one, also known as trend risk, proposing a formal
closed formula for the former and an algorithm for the latter. We show that
accidental volatility depends on the intrinsic characteristics of the policies
of the cohort (Sums-at-Risk), on the age of the policyholders and on the
variability of the sums insured; trend risk depends both on accidental
volatility and on the longevity forecasting model used.

###### keywords:

Demographic risk, Solvency II, Market-consistent valuation, Risk Theory,
Cohort approach

[figure]capposition=top [table]capposition=top

##  1 Introduction

In the recent years, there has been a trend towards market consistent
valuation of assets and liabilities in the insurance field. Both Solvency II
Directive [16] and IFRS accountings standard [19] defined a market-consistent
valuation of technical liabilities. Market-consistent valuation indeed plays a
crucial role in regulatory compliance and financial solvency assessment for
insurance companies. Regulators require insurers to value their liabilities
using market-consistent methods to ensure accurate and reliable financial
reporting.
In particular, assessing demographic risk is an important aspect of market-
consistent valuation for life insurance company. Demographic risk refers to
the uncertainty and variability associated with demographic factors such as
mortality rates, longevity, and morbidity. It is a crucial component in
valuing insurance liabilities accurately, as these factors have a significant
impact on the timing and amount of future policy benefits and claims
obligations. The assessment of demographic risk involves incorporating
appropriate demographic assumptions and models into the valuation process.
These assumptions are typically based on historical data, expert judgment, and
relevant population statistics. The aim is to capture the inherent variability
and uncertainty in demographic factors to produce more realistic and reliable
valuation outcomes.
In the literature, a common approach to assess capital requirement for
demographic risk is through stochastic modelling and several contributions
have been provided (see, e.g., [4, 18, 25, 27, 30]).
Several papers focused also on market-valuation in life insurance. In
particular, Dhaene et al. (see [15]) introduce a fair valuation of liabilities
related to a portfolio in a single period framework, such that it is both
mark-to-market for any hedgeable part of a claim, and mark-to-model for any
claim that is independent of financial market evolutions. [14] and [13] focus
instead on the market-consistent evaluation in continuous time. A valuation of
some life insurance contracts in a stochastic interest rate environment taking
into account the default risk of the underlying insurance company is given in
[3]. [17] consider typical participating policy showing that they can be
decomposed into a risk free bond element, a bonus option, and a surrender
option. A dynamic model is then constructed in which these elements can be
valued separately using contingent claims analysis. [23] propose a valuation
framework based on replication over multiple 1-year time periods by a
periodically updated portfolio of assets.
The mathematical framework that leads to market-consistent values for
insurance liabilities is presented in detail in [31] and aspects related to
Solvency topics are treated in [33]. Approaches based on two and three steps
procedure have been also provided for market-consistent valuation of
liabilities. [2] combine quadratic hedging with application of a risk measure
on the residual liability, to obtain a cost-of-capital margin. A three-step
hedge-based valuation for the valuation of hybrid claims is given in [21].
However few attention has been paid to combine market-consistent valuation and
the assessment of capital requirement for demographic risk. By assessing
demographic risk within the framework of market-consistent valuation,
insurance companies can better understand and manage their exposure to
longevity risk, mortality risk, and other demographic uncertainties. This
enables insurers to make informed decisions regarding pricing, reserving, and
risk management strategies.
Therefore, we propose here a novel methodology for assessing capital
requirement for mortality and longevity risk of linked policies. To this end,
starting from the framework defined in [31] and [33], we consider a portfolio
of homogenous policies and we provide a method for the evaluation of
idiosyncratic and trend risks considering also financial effects. We show that
accidental volatility is mainly related to the sum-at-risk of the considered
contracts, the age of the policyholders and the variability of the insured
sums. Trend risk is instead related to the accidental volatility and to the
volatility of the model used for forecasting future longevity behaviour.
Furthermore, we use a matrix approach that allows us to simulate a large
number of scenarios in an extremely short time, in order to obtain a punctual
and robust estimate of the risk-based capital.
A numerical application is developed in order to provide additional insights
about main drivers that affect capital requirement.
The paper is organized as follows. In Section 2 we introduce some preliminary
aspects of the model. In Section 3, we outline the mathematical framework of
the model and in Section 4, we present the Cohort Valuation Portfolio using a
matrix approach. Moving on to Section 5, we introduce the stochastic model
designed to identify and quantify idiosyncratic and trend risks. Lastly, in
Section 6, we provide a numerical example with a specific emphasis on the
assessment of the Solvency Capital Requirement in line with Solvency II rules.

##  2 Preliminaries

We indicate with $\Omega$ a given set, then with $\mathcal{F}$ a family of
subsets of $\Omega$ with usual desired properties (see e.g., [24]), hence
$\mathcal{F}$ is a $\sigma$-algebra and $(\Omega,\mathcal{F})$ is a measurable
space.
Considering a generic fixed positive number $n$, we assume that for each
$t\in[0,n]$ there is a $\sigma$-algebra; the collection of $\sigma$-algebras
$\mathcal{F}_{t}$ is a filtration, denoted with $\mathbb{F}$. We define with P
the probability measure that, to every set $\emph{A}\in\mathcal{F}$ assigns a
number in $[0,1]$ (i.e., $\emph{P}(\emph{A})$) coherently with real-word
evidence.
The starting point coincides with the definition of the vector of the
cashflows, generally indicated with X 111Random variables are indicated with
capital letters, while deterministic values with small letters

$\textbf{X}=(\emph{X}_{0},\emph{X}_{1},...,\emph{X}_{n})\in\emph{L}^{2}_{n+1}(\emph{P},\mathbb{F})$
(1)

where $\emph{L}^{2}_{n+1}(\emph{P},\mathbb{F})$ is a Hilbert space with scalar
product $\langle\cdot,\cdot\rangle$:

$\displaystyle\emph{E}\left[\sum_{t=0}^{n}X^{2}_{t}\right]<\infty$ for all
$\textbf{X}\in\emph{L}^{2}_{n+1}(\emph{P},\mathbb{F})$ (2)
$\displaystyle\emph{E}\left[\langle\textbf{X},\textbf{Y}\rangle\right]=\emph{E}\left[\sum_{t=0}^{n}X_{t}\cdot
Y_{t}\right]<\infty$ for all
$\textbf{X},\textbf{Y}\in\emph{L}^{2}_{n+1}(\emph{P},\mathbb{F})$ (3)
$\displaystyle||\textbf{X}||={\langle\textbf{X},\textbf{X}\rangle}^{1/2}<\infty$
for all $\textbf{X}\in\emph{L}^{2}_{n+1}(\emph{P},\mathbb{F})$ (4)

For several purposes, contracts are grouped in cohorts in the actuarial
practice (see, e.g., market consistent embedded value, IFRS, solvency capital
requirement). Therefore, we consider here the following definition of cohort.

######  Definition 2.1 (Cohort).

We define a cohort a set of policyholders that have underwritten the same
contract and that have the same characteristics. The only difference inside a
single cohort regards the sums insured. It is therefore assumed that the
survival (and death) of individual policyholders is independent from each
other and that all those who belong to the same cohort have the same survival
probability.

At the inception date (i.e., $t=0$), we consider a number of $l_{0}$
policyholders in the cohort and we denote with $\textbf{s}_{0}=[s_{k,0}]$
($k=1,2,...,l_{0}$) the vector of the sums insured of the policyholders. At
the inception, each policyholder buys $s_{k,0}$ shares of the portfolio by
paying the necessary premium to build the replicating portfolio. At each time
period $t$ (with $t>0$), the information available is represented by
$\mathcal{F}_{t}$. Therefore, the stochastic vector of the sums insured at
time $t$ is defined as follows:

$\textbf{S}_{t}=\begin{pmatrix}s_{1,0}\\\ s_{2,0}\\\ ...\\\
s_{l_{0},0}\end{pmatrix}\circ\begin{pmatrix}\mathbb{I}_{1,0}^{L}\\\
\mathbb{I}_{2,0}^{L}\\\ ...\\\
\mathbb{I}_{l_{0},0}^{L}\end{pmatrix}\circ...\circ\begin{pmatrix}\mathbb{I}_{1,t-1}^{L}\\\
\mathbb{I}_{2,t-1}^{L}\\\ ...\\\ \mathbb{I}_{l_{0},t-1}^{L}\end{pmatrix}$ (5)

where with “$\circ$” we indicate the Hadamard product (also known as element-
wise product) and the generic $\mathbb{I}_{k,t-1}^{L}$ is a dichotomic
$\mathcal{F}_{t}$-measurable random variable. It assumes the value 1 if the
$k$-th policyholder survived in $(t-1,t]$, 0 otherwise.

######  Assumption 2.1 (Valuation functional and deflators).

We assume that $\mathcal{Q}$ :
$\emph{L}^{2}_{n+1}(\emph{P},\mathbb{F})\rightarrow\mathbb{R}$ is a linear,
positive, continuous and normalized functional on
$\emph{L}^{2}_{n+1}(\emph{P},\mathbb{F})$ (hence, $\mathcal{Q}$ is a valuation
functional).
For more information on the valuation functional $\mathcal{Q}$, we refer to
[7] and [8]. However, we point out that $\mathcal{Q}_{t}(\textbf{X})$ is a
monetary value, $\mathcal{F}_{t}$-measurable and therefore stochastic at time
$0$.
Taking advantage of the Riesz–Fréchet representation theorem (see Theorem 2.5
in [31]), there exists a
$\bm{\varphi}\in\emph{L}^{2}_{n+1}(\emph{P},\mathbb{F})$ such that for all
$\textbf{X}\in\emph{L}^{2}_{n+1}(\emph{P},\mathbb{F})$ we have

$\mathcal{Q}_{0}\left(\textbf{X}_{(0)}\right)=\dfrac{1}{\varphi_{0}}\emph{E}\left[\sum_{t=0}^{n}\varphi_{t}X_{t}\right]=\emph{E}\left[\langle\left(\bm{\varphi}_{(0)}\right)^{\intercal},\bm{X}_{(0)}\rangle\right]$
(6)

Notice that we use the notation $\textbf{X}_{(t)}$ to indicate a generic
vector X which elements are $\left(X_{t}\right)_{t,t+1,...,n}$222This notation
is taken from [31].

The vector $\bm{\varphi}$ is the state price deflator, with the usual
properties to be $\mathbb{F}$-adapted and $(\emph{P},\mathbb{F})$-martingale,
it has square integrable components, it is normalized ($\varphi_{0}=1$), its
components are strictly greater than 0 and, whereas the market is complete,
it’s unique.
Since the deflators assign market-consistent values to the cashflows $X_{h}$,
with $h\in[1,n]$, it is possible to represent the reserves at time $t$, with
$t\in[0,h-1]$ for outstanding cashflows as

$\mathcal{R}_{t}^{(h)}=\mathcal{Q}_{t}\left(\textbf{X}_{(h)}\right)=\dfrac{1}{\varphi_{t}}\emph{E}\left[\sum_{s=h}^{n}\varphi_{s}X_{s}\bigg{|}\mathcal{F}_{t}\right]$
(7)

######  Definition 2.2 (Fair value principle).

Article 76 of Directive 2009/138/EC (usually called Solvency II, see [16])
requires that the insurance and reinsurance undertaking hold technical
provisions measured at fair value, i.e., “the current amount insurance and
reinsurance undertakings would have to pay if they were to transfer their
insurance and reinsurance obligations immediately to another insurance or
reinsurance undertaking”.
In literature, many authors have dealt with the topic in life insurance. Ante
litteram, Brennan & Schwartz (see [5]) evaluated the equilibrium pricing of
equity-linked life insurance policies with an asset value guarantee. Dhaene
et. al. (see [15]), investigated the fair value of liabilities intended as
both a market-consistent valuation for hedgeable claims (or parts of them),
and actuarial for non-hedgeable claims. Delong et. al. (see [14] and [13]),
extended the market consistent and actuarial valuation to the continuous time.
We assume here that $\mathcal{Q}$ is a valuation functional (for further
details see Definition 2.1 or [7]) hence, for a given state price deflator
$\bm{\varphi}$, the process
$\left\\{\mathcal{Q}_{t}(\textbf{X})\right\\}_{t=0,...,n}$ is consistent with
an arbitrage-free valuation scheme.
In conclusion, we highlight that $\bm{\varphi}$-consistency implies that
$\left(\varphi_{t}\mathcal{Q}_{t}\left(\textbf{X}_{h}\right)\right)_{t=0,...,n}$
is a $\left(\emph{P},\mathbb{F}\right)$-martingale.

######  Assumption 2.2 (Independent split of filtrations).

As in [31] and [33], we factorize the filtered probability space
$(\Omega,\mathcal{F},\emph{P},\mathbb{F})$ into a product space to obtain an
independent decoupling:

$\mathbb{T}=\left(\mathcal{T}_{t}\right)_{t=0,...,n}$ (8)
$\mathbb{G}=\left(\mathcal{G}_{t}\right)_{t=0,...,n}$ (9)

where formula (8) refers to the filtration of insurance technical events and
formula (9) refers to the filtration of financial events. We assume that
$\mathbb{T}$ and $\mathbb{G}$ are independent w.r.t the probability measure P
and that, for every $t$, $\mathcal{F}_{t}$ is generated by $\mathcal{T}_{t}$
and $\mathcal{G}_{t}$, hence
$\mathcal{T}_{t},\mathcal{G}_{t}\subset\mathcal{F}_{t}$ for every $t$.
Therefore, the state price deflator $\bm{\varphi}$ has a product structure

$\varphi_{t}=\varphi_{t}^{\mathbb{T}}\cdot\varphi_{t}^{\mathbb{G}}$ (10)

##  3 The model framework

######  Assumption 3.1 (Model assumptions).

We consider now a equity-linked endowment policy with constant annual premiums
that guarantee the beneficiary a stochastic amount if the insured dies and a
capital if the policyholder survives up to maturity $t=n$. For the sake of
simplicity, we assume the same structure of guarantee for both benefits, but
the model can be easily extended to different cases.
We build a model framework coherent with the market-consistent valuation of
Solvency II and we assume :

  1. 1.

In every $t$, with $t\in[0,n-1]$, the undertaking collects the premiums at the
beginning of the year according to the cohort who survived at time $t$ and to
the sums insured;

  2. 2.

Claims that occur in the period $t\in(t,t+1]$ depend on the realizations of
i.i.d. Bernoulli random variables and are paid at the end of the one-year
period.

Considering a generic valuation instant $t$, we present a definition of the
future in and out cashflows that is consistent with the aforementioned model
assumptions, in particular that premiums are paid at the beginning of each
time span and benefits are paid at the end. We denote this quantity with
$\textbf{X}_{(t)}=\left[X_{\tau}\right]_{\tau=t,...,n}$ where each element
$X_{\tau}$ is defined as follows:

$X_{\tau}=\begin{cases}-X_{\tau}^{in}&\text{if $\tau=t$,}\\\
X_{\tau}^{out}-X_{\tau}^{in}&\text{if $\tau>t$,}\end{cases}$ (11)

where

$X_{\tau}^{out}=\langle\left(\textbf{S}_{\tau-1}\right)^{\intercal},\bm{\Lambda}^{B}_{\tau-1},U^{out,\;\tau}_{\tau}\rangle$
(12)

and

$X_{\tau}^{in}=\langle\left(\textbf{S}_{\tau}\right)^{\intercal},\bm{\mathds{1}},U^{in,\;\tau}_{\tau}\rangle$
(13)

In formula (12), $\textbf{S}_{\tau-1}$ is a $l_{0}\times 1$ vector of insured
sums as specified in Definition 2.1, $\bm{\Lambda}_{\tau-1}^{B}$ is a
$l_{0}\times 1$ $\mathbb{T}$-adapted matrix of insurance technical variables
which elements are $\mathbb{I}_{k,\tau-1}^{D}$ with $k\in[1,l_{0}]$. Because
of Definition 2.1, they are i.i.d. Bernoulli’s random variables with parameter
equals to $q_{x+\tau-1}$, i.e, the (realistic) death probability of the
policyholders.
We assume that $\left(U_{\tau}^{out,\;t}\right)_{t\in[0,n]}$ is a
$\mathbb{G}$-adapted stochastic process and represents the process of the
financial portfolio $\mathcal{U}^{out}_{\tau}$ used to replicate the outflow
at time $\tau$; moreover, we specify that the superscript $t$ denotes that it
is the price of $\mathcal{U}^{out}_{\tau}$ at time $t$ and, lastly, that the
assumption is completely identical for
$\left(U_{\tau}^{in,\;t}\right)_{t\in[0,n]}$.
In case $\textbf{S}_{\tau-1}\in\left\\{0,1\right\\}$, the term
$\langle\left(\textbf{S}_{\tau-1}\right)^{\intercal},\bm{\Lambda}^{B}_{\tau-1}\rangle$
represents the r.v. number of deaths in the period $(\tau-1,\tau]$ and it is
consistent with the formulation provided in [31].
As regards to formula (13), $\bm{\mathds{1}}$ is the $l_{0}\times 1$ unitary
vector and $U_{\tau}^{in,\;\tau}$ is a $\mathbb{G}$-adapted stochastic process
related to the process of the financial portfolio $\mathcal{U}^{in}_{\tau}$
used to replicate inflows collected at time $\tau$.
As in [31] and [33], we are assuming a product structure of insurance cash
flows X, splitting them into a headgeable $\mathcal{G}_{t}$-measurable part
and a non-hedgleable $\mathcal{T}_{t}$-measurable part.
It is therefore possible to extend formula (7) to the cohort context to
calculate the reserve of outstanding premiums and liabilities at a generic
time $t$.
First of all, let us denote with $\mathbb{I}_{k,t}^{L}$ a Bernoulli random
variable with parameter $p_{x+t}$, equal to the second order annual survival
probability of a policyholder aged $x$ at the inception. The r.v. assumes the
value 1 if the policyholder $k$ survives; moreover we assume policyholders are
i.i.d.. Consequently we have the r.v.
$\mathbb{I}_{k,t}^{D}=1-\mathbb{I}_{k,t}^{L}$ for describing the death of the
policyholder.

######  Definition 3.1 (The mathematical reserve for outstanding premiums and
liabilities).

Therefore, we define the mathematical reserves for outstanding premiums and
liabilities at time $t$ as

$\displaystyle\mathcal{R}_{t}$
$\displaystyle=\mathcal{Q}_{t}\left(\textbf{X}_{(t)}\right)$ (14)
$\displaystyle=\langle\emph{E}\left(\textbf{S}_{t}\big{|}\mathcal{T}_{t}\right)^{\intercal},\dfrac{1}{\varphi_{t}^{\mathbb{T}}}\emph{E}\left[\left(\bm{\varphi}_{(t+1)}^{\mathbb{T}}\circ\bm{\Lambda}_{(t)}^{L}\circ\bm{\Lambda}_{(t)}^{B}\right)\big{|}\mathcal{T}_{t}\right],\dfrac{1}{\varphi_{t}^{\mathbb{G}}}\emph{E}\left[\left(\bm{\varphi}_{(t+1)}^{\mathbb{G}}\circ\textbf{U}_{(t+1)}^{out}\right)\big{|}\mathcal{G}_{t}\right]\rangle$
$\displaystyle-\langle\emph{E}\left(\textbf{S}_{t}\big{|}\mathcal{T}_{t}\right)^{\intercal},\dfrac{1}{\varphi_{t}^{\mathbb{T}}}\emph{E}\left[\left(\bm{\varphi}_{(t)}^{\mathbb{T}}\circ\bm{\Lambda}_{(t)}^{L}\right)\big{|}\mathcal{T}_{t}\right],\dfrac{1}{\varphi_{t}^{\mathbb{G}}}\emph{E}\left[\left(\bm{\varphi}_{(t)}^{\mathbb{G}}\circ\textbf{U}_{(t)}^{in}\right)\big{|}\mathcal{G}_{t}\right]\rangle$

where $\bm{\Lambda}_{(t)}^{L}=\left[\lambda^{L}_{(t),k,\tau}\right]$,
$\bm{\Lambda}_{(t)}^{B}=\left[\lambda^{B}_{(t),k,\tau}\right]$ with
$k=1,...,l_{0}$, $\tau=1,...,n-t$ are $l_{0}\times(n-t)$ matrices that
consider the survival of the policyholder and the unitary benefit in a
endowment contract333It is possible to rewrite $\bm{\Lambda}_{(t)}^{B}$ for
other types of policies. For instance, for a Term Insurance the elements of
the last column (i.e. $\tau=n-t$) would be $\mathbb{I}_{k,n-1}^{D}$. For a
Pure Endowment, all the elements of the matrix would be equal to $0$, except
those of the last column equal to $\mathbb{I}_{k,n-t}^{L}$., respectively. We
define indeed

$\lambda^{L}_{(t),k,\tau}=\begin{cases}1&\text{if $\tau=1$,}\\\
\prod_{h=0}^{\tau-2}\mathbb{I}_{k,t+h}^{L}&\text{if $\tau\geq 1$,}\end{cases}$
(15)

and

$\lambda^{B}_{(t),k,\tau}=\begin{cases}\mathbb{I}_{k,t+\tau-1}^{D}&\text{if
$\tau<n-t$,}\\\ 1&\text{if $\tau=n-t$,}\end{cases}$ (16)

Therefore, notice that the Hadamard product
$\left(\bm{\Lambda}_{(t)}^{L}\circ\bm{\Lambda}_{(t)}^{B}\right)$ results in a
$l_{0}\times(n-t)$ indicator matrix, which (stochastic) elements assume the
value $1$ if the generic policyholder is entitled to the benefit at the time
specified on the column of the matrix. With a slight abuse of notation, we
specify that $\bm{\varphi}^{\mathbb{T}}_{t+1}$ is also a $l_{0}\times(n-t)$
matrix, which columns are composed of the same deflators.

##  4 The Cohort Valuation Portfolio

The purpose of this section is to show the construction of a valuation
portfolio (VaPo) and to obtain its value. Given a certain series of future
cash flows, a Valuation Portfolio is a portfolio of financial instruments that
replicate the specific cash flow. Therefore, we present the steps for building
a VaPo which final result is consistent with the cohort approach.

  1. Step 1:

Choice of financial instruments: The first step concerns the choice of
financial basis: with reference to formula (11), we are interested in two
groups of financial instruments: those for replicating outflows and those for
inflows. As regards the former, the choice depends exclusively on the value of
the outflows: in the most generic way, here we define with
$\mathcal{U}^{out}_{t}$ the portfolio of financial instruments to replicate
the outflow $X_{t}^{out}$. The $\mathcal{U}^{out}_{t}$ portfolio can be
understood as a linear combination of weights $y^{out}_{i,t}$ of financial
instruments available on the market $\mathcal{M}$:

$\mathcal{U}^{out}_{t}=\sum_{i\in\mathcal{M}}y_{i,t}^{out}\cdot\mathcal{U}^{out}_{i,t}$
(17)

In an analogous way, with reference to the inflows, we have:

$\mathcal{U}^{in}_{t}=\sum_{i\in\mathcal{M}}y_{i,t}^{in}\cdot\mathcal{U}^{in}_{i,t}$
(18)

  2. Step 2:

Determination of the number of portfolio shares to be held to replicate the
cash flows After defining the financial instruments to replicate outflows and
premiums, it is necessary to identify the best estimate of the number of such
instruments that the insurance company must hold to replicate the general
future cash flows $\textbf{X}_{(t)}$ In formulas:

$\displaystyle\textbf{X}_{(t)}\rightarrowtail$ $\displaystyle
VaPo_{t}\left(\textbf{X}_{(t)}\right)=$ (19)
$\displaystyle=\langle\emph{E}\left(\textbf{S}_{t}\big{|}\mathcal{T}_{t}\right)^{\intercal},\emph{E}\left[\left(\bm{\Lambda}_{(t)}^{L}\circ\bm{\Lambda}_{(t)}^{B}\right)\big{|}\mathcal{T}_{t}\right],\bm{\mathcal{U}}_{(t+1)}^{out}\rangle$
$\displaystyle-\langle\emph{E}\left(\textbf{S}_{t}\big{|}\mathcal{T}_{t}\right)^{\intercal},\emph{E}\left[\left(\bm{\Lambda}_{(t)}^{L}\right)\big{|}\mathcal{T}_{t}\right],\bm{\mathcal{U}}_{(t)}^{in}\rangle$

  3. Step 3:

The account principle to evaluate financial instruments This last step
presents an accounting principle which associates a monetary value to each
financial instrument. Of all the possible accounting principles, we choose to
use a fair value valuation consistent with the Solvency II framework, where
each financial instrument is valued at the traded market value. We indicate
this accounting principles with $\mathcal{E}_{t}$ with $t\in[0,n]$ and we
define the accounting value of the cash-flows as follows:

$\displaystyle\textbf{X}_{(t)}\rightarrowtail$
$\displaystyle\;\mathcal{Q}_{t}\left(\textbf{X}_{(t)}\right)=\mathcal{E}_{t}\left(VaPo_{t}\left(\textbf{X}_{(t)}\right)\right)$
(20)
$\displaystyle=\langle\emph{E}\left(\textbf{S}_{t}\big{|}\mathcal{T}_{t}\right)^{\intercal},\emph{E}\left[\left(\bm{\Lambda}_{(t)}^{L}\circ\bm{\Lambda}_{(t)}^{B}\right)\big{|}\mathcal{T}_{t}\right],\mathcal{E}_{t}\left(\bm{\mathcal{U}}_{(t+1)}^{out}\right)\rangle$
$\displaystyle-\langle\emph{E}\left(\textbf{S}_{t}\big{|}\mathcal{T}_{t}\right)^{\intercal},\emph{E}\left[\left(\bm{\Lambda}_{(t)}^{L}\right)\big{|}\mathcal{T}_{t}\right],\mathcal{E}_{t}\left(\bm{\mathcal{U}}_{(t)}^{in}\right)\rangle$

Introducing the values of financial instruments, we have:

$\mathcal{Q}_{t}\left(\textbf{X}_{(t)}\right)=\langle\emph{E}\left(\textbf{S}_{t}\big{|}\mathcal{T}_{t}\right)^{\intercal},\emph{E}\left[\left(\bm{\Lambda}_{(t)}^{L}\circ\bm{\Lambda}_{(t)}^{B}\right)\big{|}\mathcal{T}_{t}\right],\textbf{U}_{(t+1)}^{out,\;t}\rangle-\langle\emph{E}\left(\textbf{S}_{t}\big{|}\mathcal{T}_{t}\right)^{\intercal},\emph{E}\left[\left(\bm{\Lambda}_{(t)}^{L}\right)\big{|}\mathcal{T}_{t}\right],\textbf{U}_{(t)}^{in,\;t}\rangle$
(21)

In insurance practice it is very common to use an ad hoc demographic table for
pricing. This table, called the first order demographic base, leads to a
distorted probability (far from any best estimate) whose use allows the
creation of an expected demographic profit. Considering a generic indicator
$\mathbb{I}_{k,0}^{L}$ of the matrix $\bm{\Lambda}_{(0)}^{L}$, we define

$(\varphi_{1}^{\mathbb{T}}/\varphi_{0}^{\mathbb{T}})=\tilde{\varphi}_{1}^{\mathbb{T}}$
(22)

and

$\tilde{\varphi}_{1}^{\mathbb{T}}(1)=\dfrac{{}_{1}p^{*}_{x}}{{}_{1}p_{x}}$
(23)
$\tilde{\varphi}_{1}^{\mathbb{T}}(0)=\dfrac{1-_{1}p^{*}_{x}}{1-_{1}p_{x}}$
(24)

As known in the literature (see, e.g, [26]), premiums of a policy are
calculated as the expected present value of the benefits, computed with
prudent technical bases (also called first-order bases).
In this paper we assume that the insurance company prices the policies of the
cohort using a safety loading only with reference to the technical risk, i.e.,
the demographic one. In other words, the value of financial instruments in the
Cohort VaPo protected are calculated throughout the market-consistent
accounting principle $\mathcal{E}$. Typically, in the traditional insurance
market, insurance companies ensure the non-certainty of default (ruin
probability theorem) by inserting implicit safety loading in the pricing
phase. For instance, considering a without profit traditional policy it is
possible to use deflators of the type

$\varphi_{t}^{\mathbb{G}}=(1+i^{*})^{-t}$ (25)

where $i^{*}$ is a return rate (first order financial basis) that the company
is likely to be able to earn on the capital market through the investment of
premiums and mathematical reserves. In this paper, we propose a premium
calculation structure where the undertaking sells the collateral at exactly
the market price. The purpose is to assess the Solvency Capital Requirement
linked to demographic risk and the use of implicit safety loading influences
the risk profile of undertaking. Our goal is to evaluate only the demographic
risk (and therefore the profit), eliminating an additional source of profit of
a purely financial nature.

######  Definition 4.1 (Cohort VaPo protected and premiums calculus).

The cohort VaPo protected, obtained by replacing the insurance technical risk
by their probability distorted conditional expectations at time $0$, is
defined as follows:

$\displaystyle\textbf{X}_{(0)}\rightarrowtail$ $\displaystyle
VaPo_{0}^{prot}\left(\textbf{X}_{(0)}\right)=$ (26)
$\displaystyle=\langle\left(\textbf{s}_{0}\right)^{\intercal},\emph{E}\left[\left(\bm{\varphi}_{(1)}^{\mathbb{T}}\circ\bm{\Lambda}_{(0)}^{L}\circ\bm{\Lambda}_{(0)}^{B}\right)\big{|}\mathcal{T}_{0}\right],\bm{\mathcal{U}}_{(1)}^{out}\rangle$
$\displaystyle-\langle\left(\textbf{s}_{0}\right)^{\intercal},\emph{E}\left[\left(\bm{\varphi}_{(0)}^{\mathbb{T}}\circ\bm{\Lambda}_{(0)}^{L}\right)\big{|}\mathcal{T}_{0}\right],\bm{\mathcal{U}}_{(0)}^{in}\rangle$

Hence, by formula (18), weights $y_{i,t}^{in}$ are calculated as a solution of
the equation

$\mathcal{E}_{0}\left(VaPo_{0}^{prot}\left(\textbf{X}_{(0)}\right)\right)=0$
(27)

or, analogously

$\langle\left(\textbf{s}_{0}\right)^{\intercal},\emph{E}\left[\left(\bm{\varphi}_{(1)}^{\mathbb{T}}\circ\bm{\Lambda}_{(0)}^{L}\circ\bm{\Lambda}_{(0)}^{B}\right)\big{|}\mathcal{T}_{0}\right],\textbf{{U}}_{(1)}^{out,\;0}\rangle=\langle\left(\textbf{s}_{0}\right)^{\intercal},\emph{E}\left[\left(\bm{\varphi}_{(0)}^{\mathbb{T}}\circ\bm{\Lambda}_{(0)}^{L}\right)\big{|}\mathcal{T}_{0}\right],\textbf{U}_{(0)}^{in,\;0}\rangle$
(28)

We now present a definition of best estimate liabilities that is consistent
with both the market-consistent valuation of Solvency II and with Assumptions
3.1.

######  Definition 4.2 (Best estimate liabilities of a cohort of endowment
contracts).

We define the best estimate of liabilities of a cohort of endowment contracts
with an annual premium as follows:

$\displaystyle\mathcal{R}_{t}=\mathcal{Q}_{t}\left(\textbf{X}_{(t)}\right)=$
$\displaystyle\mathcal{E}_{t}\left(VaPo_{t}\left(\textbf{X}_{(t)}\right)\right)$
(29)
$\displaystyle=\langle\emph{E}\left(\textbf{S}_{t}\big{|}\mathcal{T}_{t}\right)^{\intercal},\emph{E}\left[\left(\bm{\Lambda}_{(t)}^{L}\circ\bm{\Lambda}_{(t)}^{B}\right)\big{|}\mathcal{T}_{t}\right],\mathcal{E}_{t}\left(\bm{\mathcal{U}}_{(t+1)}^{out}\right)\rangle$
$\displaystyle-\langle\emph{E}\left(\textbf{S}_{t}\big{|}\mathcal{T}_{t}\right)^{\intercal},\emph{E}\left[\left(\bm{\Lambda}_{(t)}^{L}\right)\big{|}\mathcal{T}_{t}\right],\mathcal{E}_{t}\left(\bm{\mathcal{U}}_{(t)}^{in}\right)\rangle$

The case of single premium can be obtained as a specific case of formula (29)
where second term should be neglected.

Comparing formulas (28) and (29),net of the difference relating to the instant
of valuation, it is noticeable the absence of
$\bm{\varphi}_{(0)}^{\mathbb{T}}$ in formula (29). The best estimate reserve
is indeed defined as the expected present value of the cash flows, using the
most up-to-date demographic information and market prices (under the non-
arbitrage assumption).

##  5 Demographic risk and Solvency

In order to introduce the proposal for the quantification of the capital
requirement related to demographic risk, some distinctive features of the
legislation (Solvency II) are presented below, highlighting the aspects that
influence the structural characteristics of the proposed model.
In particular, within the several sources of risks considered by the Solvency
II regulation, we provide here a stochastic model aimed at assessing the
capital requirement related to both longevity risk and mortality risk. These
risks are defined by evaluating both the risk of loss or of adverse change in
the value of insurance liabilities, resulting from changes in the level, trend
or volatility of mortality rates (see Art. 105 of [16]). Consistently with the
legislation, a model is therefore presented that focuses on the two risks of
losses and unfavourable variations of insurance liabilities and that considers
both the effects of longevity and mortality.

###  5.1 The model framework

Examining a generic period $(t,t+1]$, as shown in Figure 1, we consider to
have available, at the beginning of the year, the best estimate
$\mathcal{R}_{t}$ (see (29)) and the deterministic premiums $X_{t}^{in}$.
These amounts, properly invested, should cover the r.v. claims $X_{t+1}^{out}$
and the new reserves $\mathcal{R}_{t+1}$.

Figure 1: A graphical representation of the book entries in a generic year $t$
and considering an annual time horizon

The starting point therefore coincides considering the quantity $V_{t+1}$
defined as

$\displaystyle V_{t+1}=$
$\displaystyle\langle\emph{E}\left(\textbf{S}_{t}\big{|}\mathcal{T}_{t}\right)^{\intercal},\emph{E}\left[\left(\bm{\Lambda}_{(t)}^{L}\circ\bm{\Lambda}_{(t)}^{B}\right)\big{|}\mathcal{T}_{t}\right],\textbf{U}_{(t+1)}^{out,\;t+1}\rangle$
(30)
$\displaystyle-\langle\emph{E}\left(\textbf{S}_{t}\big{|}\mathcal{T}_{t}\right)^{\intercal},\emph{E}\left[\left(\bm{\Lambda}_{(t+1)}^{L}\right)\big{|}\mathcal{T}_{t}\right],\textbf{U}_{(t+1)}^{in,\;t+1}\rangle$

Therefore, we assume to buy $VaPo_{t}\left(\textbf{X}_{(t)}\right)$ at time
$t$ at its market price $\mathcal{R}_{t}$. In addition to this quantity, we
collect the inflow $X_{t}^{in}$ from the policyholders. At the end of the time
span, i.e. in $t+1$, the market price total value is $V_{t+1}$.
We define the Claims Development Result (CDR) $CDR_{t+1}$ as

$CDR_{t+1}=V_{t+1}-X_{t+1}^{out}-\mathcal{R}_{t+1}$ (31)

We observe that in formula (31) the available financial information is
described by $\mathcal{G}_{t+1}$, while the insurance technical one is
represented by $\mathcal{T}_{t+1}$. Moreover, within $V_{t+1}$ the expected
value is calculated on the basis of $\mathcal{T}_{t}$, while in
$\mathcal{R}_{t+1}$ the expected value of the technical risk is conditioned on
the information contained in $\mathcal{T}_{t+1}$.
We then introduce the quantity $\hat{\mathcal{R}}_{t+1}$

$\displaystyle\hat{\mathcal{R}}_{t+1}=$
$\displaystyle\langle\emph{E}\left(\textbf{S}_{t+1}\big{|}\mathcal{T}_{t+1}\right)^{\intercal},\emph{E}\left[\left(\bm{\Lambda}_{(t+1)}^{L}\circ\bm{\Lambda}_{(t+1)}^{B}\right)\big{|}\mathcal{T}_{t}\right],\mathcal{E}_{t+1}\left(\bm{\mathcal{U}}_{(t+2)}^{out}\right)\rangle$
(32)
$\displaystyle-\langle\emph{E}\left(\textbf{S}_{t+1}\big{|}\mathcal{T}_{t+1}\right)^{\intercal},\emph{E}\left[\left(\bm{\Lambda}_{(t+1)}^{L}\right)\big{|}\mathcal{T}_{t}\right],\mathcal{E}_{t+1}\left(\bm{\mathcal{U}}_{(t+1)}^{in}\right)\rangle$

that represents the best estimate in $t+1$ under the assumption that the
technical risk is conditioned on the information contained in
$\mathcal{T}_{t}$.
Hence, we split the CDR in two components as follows:

$CDR_{t+1}^{Idios}=V_{t+1}-X_{t+1}^{out}-\hat{\mathcal{R}}_{t+1}$ (33)

and

$CDR_{t+1}^{Trend}=\hat{\mathcal{R}}_{t+1}-\mathcal{R}_{t+1}$ (34)

It could be noticed that two sources of uncertainties of demographic nature
are present. The first is given by the accidental mortality of the cohort,
related the volatility due to the mortality/longevity probabilities (i.e, the
idiosyncratic risk). The second source is related to the fact that new
information available during the year could lead to a modification of the best
estimate at the end of the year. For instance, this aspect occurs when a trend
in mortality or longevity rates is observed. This risk will be referred to as
trend risk.

###  5.2 The idiosyncratic risk

We focus here on the idiosyncratic risk defined as the risk of monetary loss
resulting from changes in level or volatility of mortality rates. It concerns
both polices whose benefit is linked to the death of the policyholder and
policies whose benefit is linked to the survival of the policyholder.

######  Definition 5.1.

In a one-year time horizon, we define the idiosyncratic profit and loss
extending formula (33):

$\displaystyle CDR_{t+1}^{Idios}=$
$\displaystyle\langle\emph{E}\left(\textbf{S}_{t}\big{|}\mathcal{T}_{t}\right)^{\intercal},\emph{E}\left[\left(\bm{\Lambda}_{(t)}^{L}\circ\bm{\Lambda}_{(t)}^{B}\right)\big{|}\mathcal{T}_{t}\right],\textbf{U}_{(t+1)}^{out,\;t+1}\rangle$
(35)
$\displaystyle-\langle\emph{E}\left(\textbf{S}_{t}\big{|}\mathcal{T}_{t}\right)^{\intercal},\emph{E}\left[\left(\bm{\Lambda}_{(t+1)}^{L}\right)\big{|}\mathcal{T}_{t}\right],\textbf{U}_{(t+1)}^{in,\;t+1}\rangle$
$\displaystyle-\langle\left(\textbf{S}_{t}\right)^{\intercal},\bm{\Lambda}^{B}_{t},U^{out,\;t+1}_{t+1}\rangle$
$\displaystyle-\langle\emph{E}\left(\textbf{S}_{t+1}\big{|}\mathcal{T}_{t+1}\right)^{\intercal},\emph{E}\left[\left(\bm{\Lambda}_{(t+1)}^{L}\circ\bm{\Lambda}_{(t+1)}^{B}\right)\big{|}\mathcal{T}_{t}\right],\textbf{U}_{(t+2)}^{out,\;t+1}\rangle$
$\displaystyle+\langle\emph{E}\left(\textbf{S}_{t+1}\big{|}\mathcal{T}_{t+1}\right)^{\intercal},\emph{E}\left[\left(\bm{\Lambda}_{(t+1)}^{L}\right)\big{|}\mathcal{T}_{t}\right],\textbf{U}_{(t+1)}^{in,\;t+1}\rangle$

As usual in a life insurance balance-sheet (see, e.g., [31] and [33]) the
profit is here defined as the sufficiency of initial best estimate and
premiums collected to cover the payment of benefits, due to the claims
occurred during the year, and the assessment of the final best estimate.
Notice that in the idiosyncratic component we want to catch only the risk due
to the volatility of deaths in the year $(t,t+1]$, that affects both the
payments in $t+1$ and the sums insured used in the new best estimates
calculation. Therefore, we assume that the insurance company does not change
its assumptions on future mortality, considering any change in mortality
during the year to be accidental. Indeed, the effect on the final best
estimate due to a revision of demographic assumptions is devoted to the trend
risk.
We provide in the following Theorem a compact formulation of the r.v.
idiosyncratic risk. Proof is reported in A.

######  Theorem 1 (Compact formulation of Idiosyncratic Risk).

Considering a generic cohort of policyholders consistent with Definition 2.1,
if the premiums are paid in advance, the claims are paid at the end of the
occurrence year and the best estimates at the end of the year are calculated
on the same demographic and financial bases used at the beginning year, it is
possible to define the idiosyncratic risk as

$CDR_{t+1}^{Idios}=\langle\left[\emph{E}\left(\textbf{S}_{t}\circ\bm{\Lambda}_{t}^{B}\big{|}\mathcal{T}_{t}\right)-\emph{E}\left(\textbf{S}_{t}\circ\bm{\Lambda}_{t}^{B}\big{|}\mathcal{T}_{t+1}\right)\right]^{\intercal},\mathds{1}\cdot\eta_{t+1}\rangle$
(36)

where $\eta_{t+1}=\left(U_{t+1}^{out,\;t+1}-\beta_{t+1}\right)$ with
$\beta_{t+1}$ best estimate rate of a policyholder of the cohort calculated in
$t+1$ with technical bases estimated in $t$; or, analogously, $\beta_{t+1}$ is
the generic element of
$\bm{\beta}_{t+1}=\langle\emph{E}\left(\bm{\Lambda}_{(t+1)}^{L}\circ\bm{\Lambda}_{(t+1)}^{B}\big{|}\mathcal{T}_{t}\right),\textbf{U}_{(t+2)}^{out,\;t+1}\rangle-\langle\emph{E}\left(\bm{\Lambda}_{(t+1)}^{L}\big{|}\mathcal{T}_{t}\right),\textbf{U}_{(t+1)}^{in,\;t+1}\rangle$.

According to formula (36), if the accidental (idiosyncratic) mortality of the
year is higher than expected, the insurer will have to cope with a greater
amount of claims $U_{t+1}^{out,\;t+1}$. In this scenario, the firm will
release best estimate liabilities which unit rate for each policyholder is
$\beta_{t+1}$.
It is noteworthy that we can interpret the term $\eta_{t+1}$ as the expected
value at time $t$ of the Sum-at-Risk rate at time $t+1$ of a policyholder.
Notice that this result generalizes the simplified case of non equity-linked
contracts where $U_{t+1}^{out,\;t+1}$ would be equal to one (see, e.g. [29],
[10] and [11]).
We can focus now on the main cumulants of $CDR_{t+1}^{Idios}$. In particular,
using the tower property:

$\emph{E}\left(CDR_{t+1}^{Idios}\big{|}\mathcal{F}_{t}\right)=0$ (37)

Therefore, on average the best estimate at time $t$ added to the premiums, is
sufficient to meet the payments of the expected benefits and the provision of
new best estimates calculated on the same demographic basis. Figure 3 shows
that, if the accidental mortality of the year is different444The sign of the
variation in mortality that produces a loss depends on the type of contract
from the expected one, a profit or loss is created.

Figure 2: Claims development result representation in $[t,t+1]$

In this way, the idiosyncratic profit definition presented in formula (35) is
perfectly consistent with the concept of Claims Development Result, and the
result is consistent with [33] and [32].
Considering the random variable defined in formula (35), we prove (see A for
details) that the standard deviation of the idiosyncratic profit is equal to

$Var\left(CDR_{t+1}^{Idios}\big{|}\mathcal{F}_{t}\right)=\left(l_{t}\cdot\
q_{x+t}\cdot(1-q_{x+t})\cdot\bar{\textbf{S}}_{t}^{2}\right)\cdot\emph{E}\left(\eta^{2}_{t+1}\big{|}\mathcal{G}_{t}\right)$
(38)

where $\bar{\textbf{S}}_{t}^{j}$ denotes the $j$-raw moment of the
$\mathcal{F}_{t}$-adaptable vector $\textbf{S}_{t}$. We therefore observe that
the volatility of the idiosyncratic risk increases with the ageing of the
cohort555Only for cohorts with $q_{x+t}<0.5$, therefore all ages except for
the extreme ages, the size of the cohort $l_{t}$ and a higher variability
within the sums insured. Additionally, it depends on the path of the expected
Sum-at-Risk rate, that, as well-known, is related to the the type of policies
considered. For instance, it has a decreasing behaviour for a endowment
contract and a convex one for a term insurance with annual premiums.

######  Definition 5.2 (Risk measure).

We define the generic risk measure (see [1]) as a function $\varsigma$

$\varsigma:\emph{L}^{2}(\emph{P})\rightarrow\mathbb{R}^{+};\qquad
X\rightarrow\varsigma(X)$ (39)

General pleasant properties for a risk measure are normalization
($\varsigma(0)=0$), translation invariance ($\varsigma(c+X)=c+\varsigma(X)$),
positive homogeneity(if $c>0$, then $\varsigma(c\cdot X)=c\cdot\varsigma(X)$),
subadditivity ($\varsigma(X+Y)\leq\varsigma(X)+\varsigma(Y)$), monotonicity
(if $X\leq Y$ P-almost surely, then $\varsigma(X)\leq\varsigma(Y)$),
expectation boundedness ($\varsigma(X)>\emph{E}^{\emph{P}}(-X)$).

Although Value at Risk does not satisfy all the properties in Definition 5.2,
following Solvency II regulation (see Art. 102), the Solvency Capital
Requirement can be computed with a partial internal model as follows:

$\varsigma\left(CDR_{t+1}^{Idios}\right)=-VaR_{0.5\%}\left(CDR_{t+1}^{Idios}\right)$
(40)

where $VaR_{0.5\%}$ stands for Value at Risk calculated with a confidence
level equal to $99.5\%$.
Our aim is to also provide a possible undertaking specific approach for
quantifying the capital requirement due to idiosyncratic risk. To this end,
adapting the formula provided for non-life premium and reserve risk, we define
a factor-based structure for capital assessment. Knowing the characteristics
of the distribution of $CDR_{t+1}^{Idios}$, it is therefore possible to write

$SCR_{L,Idios}=f\left[CDR_{t+1}^{Idios}\right]\cdot\sigma\left(CDR_{t+1}^{Idios}\right)$
(41)

where $f\left[CDR_{t+1}^{Idios}\right]$ is a multiplier, appropriately
calibrated, of the standard deviation, calculated as a function of the
distribution characteristics of the r.v. idiosyncratic profit.

###  5.3 The trend risk

Comparing the definitions of the whole demographic profit (see Figure 1 and
the idyosincratic profit (see formula (35)), we have to consider the
possibility that the best estimate a time $t+1$, given by $\mathcal{R}_{t+1}$,
can be could differ from the reserve accounted in the idiosyncratic profit.

######  Definition 5.3.

We define the random variable demographic profit (or loss) due to trend risk
$CDR_{t+1}^{Trend}$ as

$\displaystyle CDR_{t+1}^{Trend}=$
$\displaystyle\langle\emph{E}\left(\textbf{S}_{t+1}\big{|}\mathcal{T}_{t+1}\right)^{\intercal},\langle\emph{E}\left(\bm{\Lambda}_{(t+1)}^{L}\circ\bm{\Lambda}_{(t+1)}^{B}\big{|}\mathcal{T}_{t+1}\right)-\emph{E}\left(\bm{\Lambda}_{(t+1)}^{L}\circ\bm{\Lambda}_{(t+1)}^{B}\big{|}\mathcal{T}_{t}\right),\textbf{U}_{(t+2)}^{out,\;t+1}\rangle$
(42)
$\displaystyle-\langle\emph{E}\left(\bm{\Lambda}_{(t+1)}^{L}\big{|}\mathcal{T}_{t+1}\right)-\emph{E}\left(\bm{\Lambda}_{(t+1)}^{L}\big{|}\mathcal{T}_{t}\right),\textbf{U}_{(t+1)}^{in,\;t+1}\rangle\rangle$

It is noteworthy that, the Claim Development Result linked to the trend risk
depends on two sources of uncertainty: since in $t+1$ the technical
information $\mathcal{T}_{t+1}$ is different from that of the previous instant
$\mathcal{T}_{t}$, the insurance company can update (albeit to a limited
extent) future expectations. Secondly, it is specified that, as in
idiosyncratic risk context, the adjustment of technical expectations implies
purchases or sales of shares in replicating portfolios: for this reason, trend
risk also incorporates a share of financial risk.
In this context, the distribution of $CDR_{t+1}^{Trend}$ is therefore
influenced by both the accidental mortality of the year (given by
$\textbf{S}_{t+1}$) and the revision of the demographic basis. In conclusion,
we highlight that since $\mathcal{F}_{t}\subset\mathcal{F}_{t+1}$, the tower
property implies that

$\emph{E}\left(CDR_{t+1}^{Trend}\big{|}\mathcal{F}_{t}\right)=0$ (43)

######  Definition 5.4.

(SCR for Trend risk) We define the Solvency Capital requirement for Trend risk
as

$SCR_{L,Trend}=\varsigma\left(CDR_{t+1}^{Trend}\right)$ (44)

Quantile of $CDR_{t+1}^{Trend}$ can be carried out through different
methodologies. We propose a simple approach based on Monte Carlo, capable of
capturing the aforementioned dependencies between accidental mortality and
trend risk.
We propose the following algorithm:

  1. 1.

Fit a projection model (see, e.g, [20], [6], [9], [28], [27], etc.) to
forecast expected mortality rates using train data available at time $t$;

  2. 2.

Calculate $\textbf{U}_{(t+2)}^{out,\;t+1}$ and $\textbf{U}_{(t+1)}^{in,\;t+1}$
simulating the underlying financial variables $\textbf{Z}_{t+1}$;

  3. 3.

For each policyholder $k$, simulate $H$ times the r.v. $\mathbb{I}_{k,t}^{L}$
from i.i.d. Bernoulli distributions with parameter $p_{x+t}$ given by the
expected mortality rates obtained at step 1;

  4. 4.

Compute for each simulation the stochastic vector $\textbf{S}_{t+1}$ of the
sums insured at time $t+1$;

  5. 5.

For each simulation, fit the same mortality model used at step 3 on a train
set enriched with additional information simulated under real-word
probabilities;

  6. 6.

Using values at steps 2, 4 and 5, compute for each simulation
$CDR_{t+1}^{h,Trend}$ with $h\in[1,H]$;

  7. 7.

Calculate formula (44) as

$SCR_{L,Trend}=-inf\left[CDR_{t+1}^{h,Trend}:\emph{F}_{{CDR}_{t+1}^{Trend}}\left(CDR_{t+1}^{h,Trend}\right)>0.5\%\right]$
(45)

Therefore, having obtained the distribution of $CDR_{t+1}^{Trend}$, through
the $H$ simulations, we calculate the quantity
$-VaR_{p=0.5\%}\left(CDR_{t+1}^{n,Trend}\right)$ as the opposite of the lower
extreme of the obtained vector, weighted by the probabilities of both
$\textbf{S}_{t+1}$ and $\textbf{Z}_{t+1}$.

##  6 The case study

We develop here a case study in order to investigate the behaviour of the
model in describing both idiosyncratic and trend risk. The characteristics of
the cohort and of the contracts and some preliminary aspects are outlined in
Section 6.1. Main results regarding idiosyncratic and trend risks are provides
in Section 6.2 and 6.3, respectively.

###  6.1 Preliminaries

We summarize in Table 1 main characteristics of the cohort analysed. In
particular, we consider a cohort of 10,000 policyholders who have subscribed
equity-linked endowment policies, each with its own specific sum insured. If
the generic $k$-th policyholder dies in the time span $(t-1;t]$ the
beneficiary is entitled to receive a benefit equal to $S_{k,t}\cdot
max\left(\dfrac{U_{t}}{U_{0}},(1+i_{gar})^{t}\right)$ where $U_{t}$ is the
market value of the underlying asset $\mathcal{U}$ (the equity) at time $t$
and $i_{gar}$ is the minimum guaranteed rate.

Table 1: Model parameters Characteristics | Value
---|---
Number of policyholders | 10,000
Cohort age | 50
Policies duration | 10 years
2nd order demographic base | Lee-Carter applied on 1852-2019 Italy data
1st order demographic base | 2nd order $q_{x}$ stressed of 20%
Risk-free rate | Costant and equals to 2%
Guaranteed rate $i_{gar}$ | 1%
Average sum insured | 100,000.00
CoV of $s_{0}$ | 2

Assuming $U_{0}=1$, the general payoff
$max\left(\dfrac{U_{t}}{U_{0}},(1+i_{gar})^{t}\right)$ can be hedged through
the ownership of the equity $\mathcal{U}$ and a put option
$\mathcal{P}\left(\mathcal{U},(1+i_{gar})^{t}\right)$ written on
$\mathcal{U}$, with a strike price and a maturity equals to $(1+i_{gar})^{t}$
and $t$, respectively.
According to the number of equities that the company must hold to form the
replicating portfolio, we can define the following relation:

$\sum_{t=1}^{n-1}\emph{E}\left(D_{x+t}\big{|}\mathcal{T}_{0}\right)+\emph{E}\left(L_{x+n}\big{|}\mathcal{T}_{0}\right)=l_{0}$
(46)

where $D_{x+t}$ denotes the r.v. number of deaths within the cohort between
$t-1$ and $t$ and $L_{x+n}$ the r.v. number of policyholders survived at
maturity.
Therefore, $s_{k,0}$ indicates the number of replicating portfolio shares due
to the $k$-th policyholder. Hence, the insurance company, by buying exactly
$s_{0}=\sum_{k=1}^{l_{0}}s_{k,0}$ equities, will certainly be able to meet
part of the liabilities; the risk of incurring losses for the company concerns
only the part of outflows covered by Put options. We represent the whole
Hedging Portfolio in Table 2.

Table 2: The Hedging Portfolio bought in $t=0$ Financial Instrument | Amount in the hedging portfolio
---|---
$\mathcal{U}$ | $s_{0}$
$\mathcal{P}\left(\mathcal{U},(1+i_{gar})^{1}\right)$ |  $s_{0}\cdot$_E_ $\left(D_{50+1}\big{|}\mathcal{T}_{0}\right)$
… | …
$\mathcal{P}\left(\mathcal{U},(1+i_{gar})^{10}\right)$ | $s_{0}\cdot\left[\emph{E}\left(D_{50+10}\big{|}\mathcal{T}_{0}\right)+\emph{E}\left(L_{50+10}\big{|}\mathcal{T}_{0}\right)\right]$

By assuming a constant risk-free flat rate over time, the prices of Zero
Coupon Bonds can be easily determined. However, it is important to note that
while the model presented in this paper is applicable to a wide range of
inflows, equity-linked policies in insurance practice are typically financed
either through a Single Premium or Recurrent Premiums. The utilization of
Regular Annual Premiums would necessitate spreading the purchase cost of
equities throughout the entire premium duration. In such a scenario, the
insurance company would incur a substantial initial outflow before recovering
the costs, which is unsustainable given the negative best estimate of the
mathematical reserve. This situation could create an incentive for the
policyholder to terminate the contract.
Furthermore, we assume that we are in an efficient market context, all
relevant information is reflected in the prices of the underlying assets,
there are neither dividends nor transition costs and prices are described by
Geometric Brownian Motions (GBM). In particular

$dU_{t}=\mu\;U_{t}\;dt+\sigma\;U_{t}\;dW_{t}$ (47)

where $W_{t}$ is a Wiener process. We calibrated the parameters of the GBM in
order to obtain

$\begin{cases}\emph{E}\left(U_{t}\right)=(1+15\%)^{t}&\\\
CoV\left(U_{t}\right)=1\\\ \end{cases}$ (48)

it was therefore possible to calculate the prices of put options at time $t$
with maturity $m$ with the well-known Black & Scholes formula.

###  6.2 Idiosyncratic risk results

In Table 3 we present the results of the simulation model aimed at obtaining
the distribution of $CDR_{t+1}^{Idios}$ at different time instants (displayed
in Figure 3). First of all, we specify that we performed 1 million simulations
for each instant of time, obtaining consistent results. The number of
simulations has been indeed validated by verifying that the simulated
characteristics of the distribution converge to the exact ones.

Table 3: Numerical results of $CDR_{t+1}^{Idios}$ $\bm{CDR_{t+1}^{Idios}}$ | t=1 | t=3 | t=5 | t=7
---|---|---|---|---
Simulated mean | -1,106 | 700 | -983 | -591
Simulated standard deviation | 1,029,540 | 1,197,463 | 1,394,818 | 1,604,043
Simulated skewness | 12.63 | 11,36 | 11,36 | 11,47
Solvency Capital Requirement | 3,526,052 | 4,085,160 | 4,845,989 | 5,638,829

As shown before, the average of the distributions is centered on zero. With
reference to the standard deviation, we notice two contrasting effects. On the
one hand, a higher volatility due to an increased death probability over time.
On the other hand, the expected sum at risk rate decreases over time. However
results show a prevalence of the mortality rates leading to an increase of the
standard deviation over time.
It is also observed a relevant positive skewness quite stable over time. As a
consequence we have an increased capital that is equal to more than 3 times
the volatility.

Figure 3: Idiosyncratic Claims Development Result representation in $[t,t+1]$

It is noteworthy that formula (41) shows some common features with the factor-
based approach provided for the assessment of idiosyncratic component of
longevity risk in Quantitative Impact Study n.2 (see [12]). The SCR for
idiosyncratic mortality risk was indeed defined in QIS2 as follows:

$SCR^{Id,QIS2}=2.58\cdot\sqrt{\dfrac{q\cdot(1-q)}{l}}\cdot Sum-at-Risk.$ (49)

The simulation framework based on the cohort approach therefore highlights two
innovative aspects:

  * •

The variability of the vector of the insured sums might be considered;

  * •

The fact that the distributions of the Claims Development Results for
Idiosyncratic Risk have positive skewness and are all leptokurtic (see Figure
3), imposes the use of a multiplier greater than the 2.58 obtainable under the
assumptions of Normal distribution. A better proxy could be the multiplier
equal to 3 used for the Premium & Reserve Risk of the Non-Life Underwriting
Risk macro-module.

###  6.3 Trend Risk results

Figure 4: Trend claims development result representation in $[t,t+1]$

Figure 4 and Table 4 show the results of the simulation model aimed at
quantifying the Claims Development Result for Trend Risk. For th sake of
simplicity, future mortality rates have been obtained using the Lee-Carter
model. This choice is related to the fact that at each simulation it is
necessary to re-fit the mortality projection model and, therefore, the
computation times are particularly more consistent, despite the matrix
representation of the components. However, the approach can be tested with
alternative models for forecasting mortality.
The results of the model highlight two phenomena: on the one hand, the
volatility linked to the new reserve estimates is significantly lower than the
idiosyncratic volatility. This phenomenon was foreseeable, since the sample
used to estimate the initial table were based on data from 1872 to 2019; an
additional unit number/information content can only have a limited effect on
updating expectations. On the second hand, we observe that the standard
deviation of the CDR for Trend Risk decreases over time. Indeed, any revision
of expectations has an increasingly limited effect as the expiry of the
policies approaches because a change of expectations affects only a few cash
flows.
In conclusion, we observe that also for the CDR Trend, 3 times the standard
deviation is a good proxy of the Solvency Capital Requirement.

Table 4: Numerical results of $CDR_{t+1}^{Trend}$ $\bm{CDR_{t+1}^{Trend}}$ | t=1 | t=3 | t=5 | t=7
---|---|---|---|---
Simulated mean | -2,269 | -3,374 | -2,948 | 1,772
Simulated standard deviation | 158,451 | 149,561 | 125,348 | 41,175
Simulated skewness | -0.31 | -0.32 | -0.33 | -0.34
Solvency Capital Requirement | 546,330 | 521,549 | 424,445 | 137,818

###  6.4 Conclusions

In conclusion, the proposed model incorporates the market-consistent actuarial
valuation proposed by Solvency II and effectively computes the key
characteristics of both components of demographic risk: idiosyncratic and
trend. The model shows an excellent performance when applied to a single
cohort of policyholders. Additionally, it can be extended to assess the
riskiness of portfolios composed of multiple cohorts, serving as a valuable
model point.
One notable advantage of the model is its efficient computational framework,
leveraging matrix notation to significantly reduce computation times. By
utilizing this approach, actuarial calculations can be performed more swiftly,
facilitating timely analysis and decision-making processes.
Furthermore, we emphasize the importance of considering the volatility of sums
insured, element that is neglected by approaches based on classical models for
forecasting mortality. This volatility fluctuations has indeed a substantial
impact on both the demographic and trend distributions and the inclusions of
this aspect in the model allows a more accurate assessment of the associated
risks.
The numerical analyses show the applicability and effectiveness of the model
within the context of Solvency II. It offers valuable insights and practical
tools for actuarial professionals to enhance their risk assessment and
management capabilities.

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##  Appendix A Proofs

We report here the proof of relation (36) in Theorem 1.

###### Proof.

Recalling formula (30), it is possible to rewrite it by extracting the first
terms from the cross products

$\displaystyle V_{t+1}=$
$\displaystyle\langle\emph{E}\left(\textbf{S}_{t}\big{|}\mathcal{T}_{t}\right)^{\intercal},\emph{E}\left(\bm{\Lambda}_{t}^{B}\big{|}\mathcal{T}_{t}\right),U_{t+1}^{out,\;t+1}\rangle$
(50)
$\displaystyle+\langle\emph{E}\left(\textbf{S}_{t}\big{|}\mathcal{T}_{t}\right)^{\intercal},\emph{E}\left[\left(\bm{\Lambda}_{(t+1)}^{L}\circ\bm{\Lambda}_{(t+1)}^{B}\right)\big{|}\mathcal{T}_{t}\right],\textbf{U}_{(t+2)}^{out,\;t+1}\rangle$
$\displaystyle-\langle\emph{E}\left(\textbf{S}_{t}\big{|}\mathcal{T}_{t}\right)^{\intercal},\emph{E}\left[\left(\bm{\Lambda}_{t+1}^{L}\right)\big{|}\mathcal{T}_{t}\right],U_{t+1}^{in,\;t+1}\rangle$
$\displaystyle-\langle\emph{E}\left(\textbf{S}_{t}\big{|}\mathcal{T}_{t}\right)^{\intercal},\emph{E}\left[\left(\bm{\Lambda}_{(t+2)}^{L}\right)\big{|}\mathcal{T}_{t}\right],\textbf{U}_{(t+2)}^{in,\;t+1}\rangle$

By definition of $\textbf{S}_{t+1}$ (see Definition 5.1) and exploiting cross
product algebra, $\hat{\mathcal{R}}_{t+1}$ can be re-written as

$\displaystyle\hat{\mathcal{R}}_{t+1}=$
$\displaystyle\langle\emph{E}\left(\textbf{S}_{t+1}\big{|}\mathcal{T}_{t+1}\right)^{\intercal},\langle\emph{E}\left(\bm{\Lambda}_{(t+1)}^{L}\circ\bm{\Lambda}_{(t+1)}^{B}\big{|}\mathcal{T}_{t}\right),\textbf{U}_{(t+2)}^{out,\;t+1}\rangle$
(51)
$\displaystyle-\langle\emph{E}\left(\bm{\Lambda}_{(t+1)}^{L}\big{|}\mathcal{T}_{t}\right),\textbf{U}_{(t+1)}^{in,\;t+1}\rangle\rangle$
$\displaystyle=\langle\emph{E}\left(\textbf{S}_{t}\circ\left(\bm{\mathds{1}}-\bm{\Lambda}_{t}^{B}\right)\big{|}\mathcal{T}_{t+1}\right)^{\intercal},\langle\emph{E}\left(\bm{\Lambda}_{(t+1)}^{L}\circ\bm{\Lambda}_{(t+1)}^{B}\big{|}\mathcal{T}_{t}\right),\textbf{U}_{(t+2)}^{out,\;t+1}\rangle$
$\displaystyle-\langle\emph{E}\left(\bm{\Lambda}_{(t+1)}^{L}\big{|}\mathcal{T}_{t}\right),\textbf{U}_{(t+1)}^{in,\;t+1}\rangle\rangle$

Implementing formulas (51) and (50) in formula (35), it is possible to write
$CDR_{t+1}^{Idios}$ as

$\displaystyle CDR_{t+1}^{Idios}=$
$\displaystyle\langle\left(\emph{E}\left(\textbf{S}_{t}\circ\bm{\Lambda}_{t}^{B}\big{|}\mathcal{T}_{t}\right)-\emph{E}\left(\textbf{S}_{t}\circ\bm{\Lambda}_{t}^{B}\big{|}\mathcal{T}_{t+1}\right)\right)^{\intercal},\left(\mathds{1}\cdot
U_{t+1}^{out,\;t+1}\right)\rangle$ (52)
$\displaystyle+\langle\left(\emph{E}\left(\textbf{S}_{t}\circ\left(\bm{\mathds{1}}-\bm{\Lambda}_{t}^{B}\right)\big{|}\mathcal{T}_{t}\right)-\emph{E}\left(\textbf{S}_{t}\circ\left(\bm{\mathds{1}}-\bm{\Lambda}_{t}^{B}\right)\big{|}\mathcal{T}_{t+1}\right)\right)^{\intercal},\left(\mathds{1}\cdot\beta_{t+1}\right)\rangle$

where $\beta_{t+1}$ is the generic element of the vector $\bm{\beta}_{t+1}$,
defined as

$\bm{\beta}_{t+1}=\langle\emph{E}\left(\bm{\Lambda}_{(t+1)}^{L}\circ\bm{\Lambda}_{(t+1)}^{B}\big{|}\mathcal{T}_{t}\right),\textbf{U}_{(t+2)}^{out,\;t+1}\rangle-\langle\emph{E}\left(\bm{\Lambda}_{(t+1)}^{L}\big{|}\mathcal{T}_{t}\right),\textbf{U}_{(t+1)}^{in,\;t+1}\rangle$
(53)

$\beta_{t+1}$ represents the best-estimate rate of the policyholder $k$ (we
specify that all elements of the vector are equal due to assumptions of i.i.d.
on survival of policyholders).
With simple algebra we obtain:

$CDR_{t+1}^{Idios}=\langle\left(\emph{E}\left(\textbf{S}_{t}\circ\bm{\Lambda}_{t}^{B}\big{|}\mathcal{T}_{t}\right)-\emph{E}\left(\textbf{S}_{t}\circ\bm{\Lambda}_{t}^{B}\big{|}\mathcal{T}_{t+1}\right)\right)^{\intercal},\left(\mathds{1}\cdot
U_{t+1}^{out,\;t+1}-\mathds{1}\cdot\beta_{t+1}\right)\rangle$ (54)

Considering linearity of the expected value formula (36) is proved. ∎

###### Proof.

Here we prove formulas (38).
We introduce the hedgeable filtration (see [22])
$\mathbb{H}=\left(\mathcal{H}_{t}\right)_{t}=0,...,n-1$ and

$\mathcal{H}_{t+1}=\sigma\left\\{\mathcal{T}_{t},\mathcal{G}_{t+1}\right\\}$
(55)

We observe

$\displaystyle Var\left(CDR_{t+1}^{Idios}\big{|}\mathcal{F}_{t}\right)$ (56)
$\displaystyle=\emph{E}\left[Var\left(CDR_{t+1}^{Idios}\big{|}\mathcal{H}_{t+1}\right)\big{|}\mathcal{F}_{t}\right]+Var\left[\emph{E}\left(CDR_{t+1}^{Idios}\big{|}\mathcal{H}_{t+1}\right)\big{|}\mathcal{F}_{t}\right]$
$\displaystyle=\emph{E}\left[Var\left(CDR_{t+1}^{Idios}\big{|}\mathcal{H}_{t+1}\right)\big{|}\mathcal{F}_{t}\right]\geq
0$

Switching from a matrix representation to an extended one, formula (36) can be
represented as

$CDR_{t+1}^{Idios}=\sum_{k=0}^{l_{0}}s_{k,t}\cdot\left[\emph{E}\left(\Lambda_{k,t}^{B}\big{|}\mathcal{T}_{t}\right)-\emph{E}\left(\Lambda_{k,t}^{B}\big{|}\mathcal{T}_{t+1}\right)\right]\cdot\eta_{t+1}$
(57)

Let define as
$Z_{t}=\langle\left(\textbf{S}_{t}\right)^{\intercal},\bm{\Lambda}_{t}^{B}\rangle$
the total sums insured of occurred deaths at time $t$ and $Z_{k,t}$ the sum
insured of occurred death of the single $k$-th policyholder. Since the generic
element of $\bm{\Lambda}_{t}^{B}$ is distributed as a Bernoulli r.v. with
parameter $q_{x+t}$, the moment generating function (mgf)
$M_{\lambda_{k,t}^{B}}(m)$ is

$M_{\lambda_{k,t}^{B}}(m)=q_{x+t}\cdot e^{m}+(1-q_{x+t})$ (58)

Considering the policyholders’ sums insured, we have:

$M_{Z_{k,t}}(m)=q_{x+t}\cdot e^{m\cdot s_{k,t}}+(1-q_{x+t})$ (59)

Considering that elements of $\bm{\Lambda}_{t}^{B}$ are identical distributed,
we have:

$M_{Z_{t}}(m)=\left(q_{x+t}\cdot e^{m\cdot
s_{k,t}}+(1-q_{x+t})\right)^{l_{t}}$ (60)

Therefore, the cumulant generating function is defined as:

$\Psi_{Z_{t}}(m)={l_{t}}\cdot ln\left(q_{x+t}\cdot e^{m\cdot
s_{k,t}}+(1-q_{x+t})\right)$ (61)

Hence, the variance is defined as:

$Var(Z_{t})=l_{t}\cdot\ q_{x+t}\cdot(1-q_{x+t})\cdot\bar{\textbf{S}}_{t}^{2}$
(62)

where $\bar{\textbf{S}}_{t}^{j}$ denotes the $j$-raw moment of the
$\mathcal{F}_{t}$-adaptable vector $\textbf{S}_{t}$. ∎