Financial Valuation of a New Generation
Participating Life-Insurance Contract∗
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
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.
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: email@example.com.
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
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
1 + i
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-
• 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.
J = max [min (βI , I − iret) , βminI] .1
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
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. ,  and, for
the Italian case, , , ,  and ) and in the insurance practice
(see e.g. ). 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  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.
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
CT = CT−1(1 + ρT ) ,
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 ] ,
ρT = max [min (βIT , IT − iret) , βminIT , ρmin] ,
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
CT = C0
(1 + ρh) ,
that emphasizes its path-dependent nature. In the sequel we will use
(1 + ρt+h) ,
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
ΣT = CT γT ;
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
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
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 , referring to
early redemption of Canadian savings bonds, and in , referring
to mortgage-backed securities.
• The information on the value and on the asset-allocation of the
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
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.
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
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) .
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 .
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.
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.
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 (, see also  and ): if rt is the market spot
rate at time t, we will assume it follows a square-root mean-reverting
drt = α(γ − rt)dt + ρ rtdW rt ,
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
t) = π
with π a constant parameter.
Stock-market uncertainty is considered by modelling the stock-index
St as a Black and Scholes  log-normal process, with constant drift and
volatility parameters µ and σ:
dSt = µStdt + σStdW S
where W S
t is a standard Brownian motion.
We assume ﬁnally that the two Brownian motions driving rt and St
cov dW rt, dW St = ρr,Sdt ;
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 ,
dSt = rtStdt + σStdW S
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 ,
T − ST −1
T − BT −1
T = Q
+ (1 − Q)
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
T ) = Et
YT e− Tt
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
(1 + IT ) = 1 .
Now consider at time t the payment provided at time T by our policy.
It is of the form
YT = 1E α C
= 1E α C
where ET is the elimination event originating the payment, 1E its indi-
cator function and αE is 1 or γ
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 are known at valuation time. Thus
the market value of YT becomes
V (t, Y
T ) = prob∗
t (ET )αE C
= prob∗(ET )αE C
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
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 ) ,
the value of the policy at time t, that is to say the mark-to-market
V (t) = Ct
q∗t,kφ(t, t + k) + Ct
s∗t,kγt+kφ(t, t + k) ,
where q∗ (s∗ ) is the prob∗
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
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
S (t) =
V (t) =
φ(t, t + k) .
V (t) ∂S
t,k + s∗
The interest rate sensitivity of the mark-to-market reserve is the semi-
elasticity of V (t) with respect to the state variable rt,
V (t) =
φ(t, t + k) . (27)
V (t) ∂r
t,k + s∗
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 ,
(1 + ρbt+k) ,
the base valuation factor to be
φb(t, t + k) = V (t, Φbt,t+k)
and the base stochastic reserve to be
V b(t) = Ct
q∗t,k + s∗t,kγt+k φb(t, t + k) .
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)