Financial Valuation of a New Generation Participating Life-Insurance Contract

Financial Valuation of a New Generation
Participating Life-Insurance Contract∗
Claudio Pacati∗∗
Abstract
In this paper we analyze a typical “new generation” partici-
pating life-insurance contract: the single-premium whole life par-
ticipating policy with zero technical rate and low-level surrender
penalty. In such a contract there is no “traditional” mortality risk
and the financial content is prevailing, since the benefits are yearly
readjusted according to the return of the fund where the reserve is
invested, with a yearly minimum guarantee. This kind of policy is
widely sold by Italian companies in these years and can be seen as
an open-term investment in the fund, protected by the minimum
guarantee.
The aim of this paper is to perform mark-to-market valuation
of this contract in order to obtain the stochastic reserve, the fair
value of the embedded option and the value of business in-force,
the “technical” component of the embedded value.
Keywords and phrases: participating life-insurance policy, whole-life in-
surance, financial valuation, mark-to-market reserve, value of business
in-force, embedded value.
Introduction
This paper deals with a particular participating policy type: the single-
premium participating whole life, with zero technical rate and low-level
∗Invited conference at the 6th Spanish-Italian Meeting on Financial Mathematics
(Trieste, 3–5 July 2003).
∗∗Universit`a di Siena, Dipartimento di Economia Politica, p.zza S. Francesco, 7,
I–53100 Siena SI; email: pacati@unisi.it.

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surrender penalty. This kind of contract is a typical example of the
“new generation” participating policies sold in these years by Italian
companies, designed to emphasize their financial content and to be sold
as an alternative to purely financial instruments, by traditional insurance
agents as well through bancassurance channels. In fact, this policy has
little actuarial content: at any time the statutory reserve is always equal
to the benefit for the case of death at the same time. Thus the policy is
sold mainly for its financial features: it is very similar to an open-term
investment in the a fund managed by the company, protected by a yearly
minimum guarantee.
Participating life-insurance policies are very popular in Italy. They
are sold since the early 80’s and where originally introduced to protect
the insured benefits from inflation. The basic idea of the participating
rule is the following: the insurance company invests the mathematical
reserve of the policy in a fund, the segregated fund, whose yearly return
I is shared between the company and the insured. A readjustment rate
ρ is contractually defined as a function of I and applied to raise for the
same year the insured capital, according to a rule that depends on the
policy type. A quite general form of the contractual definition of ρ is
J − i
ρ = max
, ρ
1 + i
min
,
(1)
where i is the technical rate of the policy, ρmin ≥ 0 is the yearly minimum
guaranteed and J is the assigned return, the part of I assigned to the
insured. A typical contract defines J according to the following rules:
• the insured gains a fraction β ∈ (0, 1] of I, the participation coef-
ficient,
• but the company has to retain at least iret ≥ 0,
• however, in any case, the insured has the right to at least a fraction
βmin ∈ (0, β] of I.
Hence:
J = max [min (βI , I − iret) , βminI] .1
(2)
1Notice that, when the return of the fund is negative, J cannot be properly seen as
a “part” of it. In fact, if I < 0, since β, iret and βmin are non negative, J = βminI > I.
However, the term assigned return has become standard in the Italian insurance
practice.
2

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Due to the participating rule, the benefits of the policy are random
variables with regard to both actuarial and financial uncertainty. The
former concerns the benefit’s type and payment date, whereas the latter
affects the benefit amount. From a financial point of view, the policy
is a derivative contract, with underlying the return of the segregated
fund: the various yearly minima in (1) and (2), and mainly the yearly
minimum ρmin, embed into the policy a quite complex financial option.
Furthermore, the option is of cliquet type: the indexation rule applies
every year, consolidating the benefit level reached by the revaluation
occurred in previous years.
Since the participating rule links the benefit’s amount to the capital
market, the valuation method to be used to price the policy has to be
consistent with the valuation methods used in capital markets. This re-
quirement is well understood both in the theory (see e.g. [5], [7] and, for
the Italian case, [13], [10], [1], [2] and [3]) and in the insurance practice
(see e.g. [11]). Accordingly, in section 3 we determine the mark-to-
market reserve of the policy, also called stochastic reserve to emphasize
the fact that the valuation is done in a mark-to-market setting, hence
considering a stochastic evolution of interest rates, in contrast to “tradi-
tional” constant rate valuations. Also the embedded value of the policy
has to be computed in a mark-to-market sense. Following [10] we derive
its the “technical” component, the value of business in-force in section 4,
together with its decomposition in investment gain, mortality gain and
surrender gain, by comparing the statutory and the mark-to-market re-
serves, computed under different actuarial assumptions.
The last section of the paper is devoted to some valuation results,
obtained through Monte Carlo simulations.
1
The contract
Let us consider the case of a whole life participating policy with zero
technical rate (i = 0). Consider a policy sold at time zero to an insured
of age x. The benefit, the insured capital, is paid at death of the insured,
whenever it occurs. Assume for simplicity that death can occur only at
integral times, “just after” the revaluation. Let the insured capital be
initially defined to be C0; its level at time T = 1, 2, . . . , is given by the
recurrent relation
CT = CT−1(1 + ρT ) ,
(3)
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T
I  
T
ρ
 
T
&
 
&
&
 
&
&
 
 
 
 
 
 
 
 
 
 
 
ρmin
¨¨
¨
 
 
E
0
iret
iret
IT
 
1−βmin
1−β
JT
 
······························
 
Figure 1: Readjustment rate ρT as a function of the fund’s return IT
(i = 0).
where ρT is the readjustment rate for year T . For this policy i = 0
and hence by (2) and (1) the assigned return JT for year T and the
readjustment rate ρT become
JT = max [min (βIT , IT − iret) , βminIT ] ,
(4)
ρT = max [min (βIT , IT − iret) , βminIT , ρmin] ,
(5)
where IT is the return of the segregated fund in the same year. Figure 1
shows the graph of ρT as a function of IT in a non-degenerate case,
where the constants β, βmin, iret and ρmin are set in such a way that
all the minima can occur. In the Italian practice, typical values for
these constants are β ≥ 90%, βmin ≥ 75%, 0% ≤ iret ≤ 1.25% and
2% ≤ ρmin ≤ 4%. Of course, if iret = 0%, the presence of βmin has no
effect and the rule becomes simpler.
The insured capital at time T can be written in the closed form
T
CT = C0
(1 + ρh) ,
(6)
h=1
that emphasizes its path-dependent nature. In the sequel we will use
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the notation
T −t
Φt,T =
(1 + ρt+h) ,
(7)
h=1
so that CT = C0Φ0,T or, starting from any integral time 0 ≤ t < T ,
CT = CtΦt,T .
The contract provides also the insured with a surrender option: at
any time T , he or she can surrender the policy and receive the redemption
value
ΣT = CT γT ;
(8)
the redemption coefficient γT ≤ 1 depends in a deterministic and con-
tractually defined way on surrender time T . In these contracts sur-
renders are penalized at a very low level; a typical example could be
γT = 1 − (3 − T )+/100, allowing for a very little surrender penalty only
in the first two years. For simplicity we will assume that surrender can
occur only at integral dates, “just after” the revaluation of the insured
capital.
In fact, the surrender option is the most important feature of the
contract. The insurer does not expect the insured to persist until death
and the contract is typically sold as an open-term investment in the
segregated fund.
From a financial point of view, the surrender option is an american
put option embedded in the contract: at any time T the insured has
the right to sell back the contract to the insurer, at the strike price
ΣT . This option could be priced by standard no-arbitrage techniques,
assuming rational exercise by the insured. However, there are two major
drawbacks to this assumptions:
• The insured is not a financial institution and surrender may be
driven more frequently by the evolution of his or her consumption
plan than by the market evolution. For financial instruments, this
assumption was empirically proven to be correct in [6], referring to
early redemption of Canadian savings bonds, and in [15], referring
to mortgage-backed securities.
• The information on the value and on the asset-allocation of the
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segregated fund is not public2, the yearly return is known with a
lag of one or two months and no benchmark is contractually nor
indicatively defined.3 Even a rational insured does not have all the
information needed to test the rational exercise condition, since
he or she cannot compare the redemption value to the prosecution
value and is hence not able to rationally exercise the surrender
option.
These considerations lend us to apply the more traditional idea of mod-
elling surrender uncertainty through experience-based elimination ta-
bles. To this end, we will consider an actuarial probabilistic framework
with two elimination causes: death and surrender; we will furthermore
assume that elimination events are independent of market events.
2
Traditional valuation
The statutory technical reserve of the policy, that is to say the level
of funding the company has to maintain by law, is the net premium
mathematical reserve and is defined in a traditional actuarial setting.
Let us denote by probI the first order probability distribution of the
future lifetime Tx of the insured and let qI be the first order probability
t,k
of death in year t + k, conditional to survivor at age x + t:
qIt,k = probI(x + t + k − 1 ≤ Tx < x + t + k | Tx > x + t) .
(9)
The traditional reserve R(t) at time t is defined considering the sum
insured at time t, not considering future readjustments nor the surrender
option, using the first order probability distribution and discounting at
the technical rate:
∞
∞
R(t) = Ct
qIt,k(1 + i)−k = Ct
qIt,k = Ct .
(10)
k=1
k=1
Since at any time the traditional reserve equals the death benefit at that
time, from a traditional point of view the policy has no mortality risk.
2In fact Italian companies do publish a brief quarterly report on the segregated
fund, containing the (book) value of the fund and an asset-allocation summary, but
with a significant lag with respect to the reference date.
3Notice that this lack of information does not regards the reference funds of Italian
unit-linked policies. For these funds the unit value is published at least weekly and
an (indicative) asset allocation and a benchmark are contractually defined.
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The main drawback of the traditional approach is that the valuation
is performed as if the policy where non-participating. In this way, the
embedded minima are ignored and the method is unable to price the
minimum guarantee option embedded in the readjustment mechanism.
Also the surrender option is not considered, but this is a minor prob-
lem. Due to the former “approximation”, the redemption value Σt+k at
future time t + k has coherently to be “approximated” by Ctγt+k ≤ Ct.
Ignoring the surrender option gives hence a “prudential” valuation.
3
Mark-to-market valuation
In contrast to the traditional valuation framework, the mark-to-market
approach is able to consider the financial uncertainty affecting the bene-
fits. This uncertainty comes from the market where the fund’s manager
invests the policy reserve. Managers of Italian segregated funds typically
invest the main part (at least 80%) of the reserve in bonds and the rest
in stocks. The amount invested in corporate bonds is negligible, so we
can ignore default risk. We will therefore consider a market model with
two sources of uncertainty: interest rate risk and stock-market risk.
We will model interest rate risk through the one-factor Cox, Ingersoll
and Ross (CIR) model ([8], see also [9] and [14]): if rt is the market spot
rate at time t, we will assume it follows a square-root mean-reverting
diffusion process
√
drt = α(γ − rt)dt + ρ rtdW rt ,
(11)
where W rt is a standard Brownian motion and α, γ and ρ are positive
constant parameters, with 2αγ/ρ2 ≥ 1. We furthermore assume that
market price of interest rate risk is of the form
√r
q(t, r
t
t) = π
,
(12)
ρ
with π a constant parameter.
Stock-market uncertainty is considered by modelling the stock-index
St as a Black and Scholes [4] log-normal process, with constant drift and
volatility parameters µ and σ:
dSt = µStdt + σStdW S
t
,
(13)
7

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where W S
t is a standard Brownian motion.
We assume finally that the two Brownian motions driving rt and St
are correlated:
cov dW rt, dW St = ρr,Sdt ;
(14)
the instantaneous correlation coefficient ρr,S is assumed to be constant.
It is well known that this model is complete and arbitrage-free and
that the risk-neutral dynamics of the state variables is
√
drt = α(γ − rt)dt + ρ rtdW rt ,
(15)
dSt = rtStdt + σStdW S
t
,
(16)
where W rt and W St are the risk-neutral Girsanov transformations of the
two Brownian motions W rt and W St, and α = α − π and γ = αγ/α are
the drift parameters of the risk-neutral spot rate dynamics.
In this market-model, the stochastic evolution of the fund’s return
I is completely specified once specified the trading strategy the fund’s
manager follows. According to what said before, we will assume that,
for each integer T ,
S
B
I
T − ST −1
T − BT −1
T = Q
+ (1 − Q)
,
(17)
ST−1
BT−1
where Q is the fraction of the fund held in stocks and B is the market
value of a self-financing bond portfolio.
A standard no-arbitrage argument shows that the market price at
time t of a random payment YT at time T > t, subject only to financial
uncertainty, is given by
V (t, Y
ru du
T ) = Et
YT e− Tt
,
(18)
where Et is the risk-neutral expectation implied by the risk-neutral ver-
sion of the model and conditional to the market information at time
t. Notice that, by standard no-arbitrage arguments, we have for every
integer T > t that
T −t
V
t,
(1 + IT ) = 1 .
(19)
k=1
8

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Now consider at time t the payment provided at time T by our policy.
It is of the form
YT = 1E α C
T
ET
T
(20)
= 1E α C
,
(21)
T
ET
tΦtT
where ET is the elimination event originating the payment, 1E its indi-
T
cator function and αE is 1 or γ
T
T , depending on the type of elimination
(death or surrender). The random variable YT is affected by both finan-
cial and actuarial uncertainty. However, by the independence assump-
tion made at the end of section 1, the randomness of the first factor of
(21) is only of actuarial type, whereas the last factor is affected only by
financial uncertainty and αE and C
T
t are known at valuation time. Thus
the market value of YT becomes
V (t, Y
ru du
T ) = prob∗
t (ET )αE C
e− T
t
(22)
T
tEt
ΦtT
= prob∗(ET )αE C
) ,
(23)
T
tV (t, ΦtT
where prob∗ is the risk-adjusted probability measure of actuarial events
(death and surrender), conditional to life and persistency of the insured
at time t, and V (t, Φt ) is the stochastic valuation factor at time t for
T
time T : it is the value of one unit of cash invested in the fund and
revaluated up to T at the stochastic yearly readjustment rate ρ.
If we denote the valuation factor by
φ(t, T ) = V (t, Φt,T ) ,
(24)
the value of the policy at time t, that is to say the mark-to-market
reserve, is
∞
∞
V (t) = Ct
q∗t,kφ(t, t + k) + Ct
s∗t,kγt+kφ(t, t + k) ,
(25)
k=1
k=1
where q∗ (s∗ ) is the prob∗
t,k
t,k
t -probability of life and persistency at time
t + k − 1 and death (surrender) at time t + k. Mark-to-market re-
serve is therefore computed knowing the following quantities at time
t: the insured capital in-force at time t, the term structure of valu-
ation factors, the term-structure of death- and surrender-elimination
risk-adjusted probabilities.
9

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Standard measures of financial instantaneous risk can be obtained in
the same way, computing the derivatives of the valuation factors with re-
spect to both the state variables. The stock-market delta of the mark-to-
market reserve is the elasticity of V (t) with respect to the state variable
St,
S
∂
S
∞
∂
∆
t
tCt
S (t) =
V (t) =
(q∗
φ(t, t + k) .
V (t) ∂S
t,k + s∗
t,kγt+k)
t
V (t)
∂St
k=1
(26)
The interest rate sensitivity of the mark-to-market reserve is the semi-
elasticity of V (t) with respect to the state variable rt,
1
∂
C
∞
∂
∆
t
r(t) =
V (t) =
(q∗
φ(t, t + k) . (27)
V (t) ∂r
t,k + s∗
t,kγt+k)
t
V (t)
∂rt
k=1
It is worth to obtain a separate value for the minimum guarantee.
Since ρmin is the most relevant of the three minima, we consider the pol-
icy as a derivative contract with underlying the yearly assigned returns
Jt+k. For every k, define the base readjustment rate to be
ρbt+k = Jt+k ,
(28)
k
Φbt,t+k =
(1 + ρbt+k) ,
(29)
h=1
the base valuation factor to be
φb(t, t + k) = V (t, Φbt,t+k)
(30)
and the base stochastic reserve to be
∞
V b(t) = Ct
q∗t,k + s∗t,kγt+k φb(t, t + k) .
(31)
k=1
This quantity is the reserve of a policy version with ρmin = −1, i.e.
without the minimum guarantee. The difference
V p(t) = V (t) − V b(t)
(32)
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