ON A FAIR VALUE MODEL FOR PARTICIPATING LIFE INSURANCE POLICIES

Investment Management and Financial Innovations, Volume 3, Issue 2, 2006
105
ON A FAIR VALUE MODEL
FOR PARTICIPATING LIFE INSURANCE POLICIES
Fabio Baione, Paolo De Angelis, Andrea Fortunati
Abstract
The aim of this paper is to analyze both the term structure of interest and mortality rates
role for evaluating a fair value of a life insurance business. In particular, a fair value accounting
impact on reserve evaluations is discussed comparing a traditional deterministic model based on
local rules for an Italian balance sheet calculation and a stochastic one based on a diffusion process
for both mortality and financial risks. As proposed by IAS Board we will separate the embedded
derivatives from their host contracts, so the fair value of a traditional life insurance contract would
be expressed as the value of four components: the basic contract, the participation option, the op-
tion to annuitise and the surrender option. A numerical application to a traditional Italian life in-
surance policy is discussed.

Key words: Fair pricing, participation option, surrender option, guaranteed annuity op-
tion, Black&Scholes-CIR framework, Longstaff-Schwartz Least-Squares Approach.
JEL Classification: C15, G13, G22.
1. Introduction
Literature on International Accounting Standards in the last three years has been im-
proved, not only in financial area, by authoritative scientific and professional contributions. The
expectation for a final version of an accounting standards guide line for insurance companies by
IASB organisation has given rise to an intense methodological debate focused on technical solu-
tions for accounting rules application within insurance sector.
The meaning of “Fair Value” extended to insurance contracts involves, by a side, an
adoption of new strategic choices for managing resources and structures and, on the other side,
represents a way in which actuarial theory could assume a primarily role in methodological paths
definition.
More in general, when a liquid market of insurance contracts is available, Fair Value of
an insurance contract is equal to its market value; otherwise it is necessary to obtain an estimate of
the market value using a theoretical model consistent with economic operators’ behaviour in a
particular risk condition. The forthcoming IAS guide line proposes that insurance liabilities should
be valued as if they are traded among well-informed, independent investors in a liquid market-
place. IAS guide line allows using of stochastic methods to estimate future cash flows (liabilities
including embedded options) arising from the contract.
In this context, insurance companies should adequate their internal procedures to charac-
terize and estimate all the policies’ components at fair value. As an example, traditional Italian
participating life insurance contracts, also known as with-profits, entitle the policyholder of a cer-
tain part of the profits generated by the assets associated with the contract; these profits are cred-
ited to the mathematical reserve increasing the insurer’s liabilities. According to the recent litera-
ture (Grosen and Jørgensen (2000), Bacinello (2001)), the value of the previous contract could be
split into two components: a basic contract and a participation or bonus option; a bonus option is a
participating European-style option where the benefit is annually adjusted according to the per-
formance of a reference fund. Furthermore, if a contract provides that the policyholder can surren-
der the contract before maturity, this component is called surrender option. According to the recent
insurance literature on this argument, as in the regular option literature, a contract with a surrender
option is called American, a contract without a surrender option is called European (Bacinello
(2003), Andreatta and Corradin (2003)). At last, some policies enable the policyholder to convert
cash benefit at maturity into a guaranteed annuity payable throughout the remaining lifetime, cal-
culated at a guaranteed rate. A guaranteed annuity option is a contract that provides the policy-

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106
Investment Management and Financial Innovations, Volume 3, Issue 2, 2006
holder with the right to receive at maturity either a cash payment or an annuity, depending on
which has the greater value (Milevsky and Promislow (2001), Ballotta and Haberman (2003)).
In reference to embedded derivatives, IFRS 4 clarifies that “an insurer needs not account
for an embedded derivative separately at fair value if embedded derivative meets the definition of
an insurance contract”. Moreover “insurers will not need to separate surrender options within dis-
cretionary participatory features (DPF) contracts, irrespective of whether they transfer significant
insurance risk or not”. Then, the above directions establish that insurance companies have to
evaluate the fair value of the embedded derivatives but not necessarily account them separately
from the host contract.
References on the application of accounting rules based on Fair Value are easily founded
in literature; De Angelis (2001) reviews international guide line for Fair Valuation of insurance
companies; De Felice and Moriconi (2001) introduce a mark to market model for a fair pricing of
life insurance participating policies with a minimum interest rate guaranteed.
In actuarial literature most papers deal with models for Fair Value of life insurance liabili-
ties with embedded options; in particular Milevsky and Promislow (2001) propose a stochastic
approach to model the future mortality hazard rate in insurance contract with option to annuitise;
Bacinello (2003) deals with the problem of pricing a guaranteed life insurance participating policy
which embeds a surrender option; Andreatta and Corradin (2003) propose a similar approach to
price the embedded options via Monte Carlo simulation; Ballotta and Haberman (2003) propose a
theoretical model for evaluating guaranteed annuity conversion options.
The aim of this paper is to analyze an actuarial model to compare reserves evaluated on
the basis of local rules with reserves calculated on a Fair Value basis, considering a stochastic ap-
proach for both mortality and financial risk. We focus, in particular, on the fair valuation of a sur-
rendable participating contract with minimum return guaranteed and option to annuitise via Monte
Carlo simulation. We implement the recent contributions of Bacinello (2003), Andreatta and Cor-
radin (2003), considering the case of a guaranteed annuity option as in Ballotta and Haberman
(2003). Moreover, our approach describes expected cash flows of the fair value liabilities and em-
bedded derivates between inception and term.
Section 2 describes theoretical model used for the comparison described above. Section 3
presents an application of the model discussed in Section 2 for an evaluation of a surrendable par-
ticipating policy with minimum return guaranteed and option to annuitise.
2. A Fair Value of the embedded options in a guaranteed life insurance par-
ticipating policy
2.1. It does not exist a unique definition of Fair Value for whole insurance contracts.
However definition proposed by Financial Accounting Standards Board (FASB) for financial
transaction declares “Fair Value is an estimate of the price an entity would have realized if it had
sold an asset or paid if it had been relieved of a liability on the reporting date in an arm’s-length
exchange motivated by normal business considerations. That is, it is an estimate of an exit price
determined by market interactions”.
A similar definition of Fair Value is proposed by International Accounting Standards
Committee (IASC): “The amount for which an asset could be exchanged or liability settled, be-
tween knowledgeable, willing parties in an arm’s length transaction”; therefore, in the traditional
conditions of market efficiency, Fair Value of an insurance policy could be equalized to its equi-
librium-price. In absence of an efficient market, Fair Value could be estimated through a consis-
tent theoretical bid/ask model joined with similar assets and liabilities.
Valuation techniques include using recent arm’s length market transactions between
knowledgeable, willing parties, if available, referring to the current fair value of another instru-
ment that is substantially the same, discounted cash flow analysis and option pricing models.
The chosen valuation technique makes maximum use of market inputs and relies as little
as possible on entity-specific inputs. It incorporates all factors that market participants would con-
sider in setting a price and is consistent with accepted economic methodologies for pricing finan-
cial instruments. The fair value is based on:

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Investment Management and Financial Innovations, Volume 3, Issue 2, 2006
107
• observable current market transactions in the same instrument;
• a valuation technique whose variables include primarily observable market data and
that is calibrated periodically to observable current market transactions in the same in-
strument or to other observable current market data;
• a valuation technique that is commonly used by market participants to price the in-
strument and has been demonstrated to provide realistic estimates of prices obtained in
actual market transactions (see IASB (2004)).
In order to present an actuarial model for a fair value estimation of a traditional life insur-
ance contract, we suppose to operate in a traditional efficient market. We assume, in fact, that fi-
nancial and insurance markets are perfectly competitive, frictionless, and free of arbitrage oppor-
tunities. Moreover, all the agents are supposed to be rational and non-satiated, and to share the
same information.
Consider {r ;t = ,
1
µ
t
}
,....
2
and {
; = ,
1
x+ t
t
}
,....
2
as two diffusion processes driven the
instantaneous interest rate and the intensity of mortality (referred to an insured aged x at issue), by
the filtrations
r
F and
µ
F respectively; with reference to a generic insurance contract pay-out,
the two stochastic processes are defined on a probability space (Ω, r,µ
F
, P) such that
r,µ
r
µ
F
= F ∪ F .
Fair Value of a generic life insurance contract in t ∈ [ ,
0 s] is expressed as follows:
⎡⎛
⎞
⎤

FV ( V
v t
CFL
v t
CFA F
, (2.1)
t
x ) = E
ˆ ⎢ ∑ ( ,τ )
−
,τ
r ,
τ
∑ ( )
µ
τ
t
⎥
⎢⎜⎜
⎟
⎟
⎝τ (∈t,s)
τ (
∈ t,s)
⎣
⎠
⎥
⎦
where Εˆ denotes the usual expectation under the risk-neutral probability measure;
v(t,τ ) is the stochastic discount factor dependent on the spot-rate dynamic between t and τ ;
CFLτ and CFAτ , are the annual random cash flows of insurance company and in-
sured/policyholder respectively, jointly dependent on the spot rate and intensity of mortality dy-
µ
r,µ
namics;
r
F , F and F
are the σ-algebras associated with the above defined filtrations.
t
t
t
2.2. As stated in Section 1, to compute the fair value of a surrendable participating en-
dowment policy with option to annuitise, we separate the whole contract in its components consid-
ering a basic contract, a participation option, an option to annuitise and a surrender option.
The basic contract is a standard endowment policy with benefit C , net constant annual
0
premium1 P , technical rate i . The Fair Value is expressed as

(2.2)
FV (
B
V
C
v t t
q
v t s
p
P
v t t
p
F
t
x )
⎪
⎧⎛ ⎡ s−t
⎤
s t
⎞ r µ ⎪⎫
= Eˆ⎨
∑
−
+τ
+
−
τ
τ −
x+t
s −t
x+t
∑
+ −
0
( ,
)
⎢
1 / 1
( , )
( ,
)1
,
.
⎥
τ 1
⎜
⎜
−
x +t ⎟⎟ t
⎬
⎪⎝
⎣τ 1=
⎦
τ 1
=
⎠
⎪
⎩
⎭
In order to value the participation option it is necessary to compute the fair value of the
non-surrendable participating contract. In accordance with this contract the insurer pays a benefit
τ +
τ
C if the insured dies within τ and
1; otherwise C if the insured is alive at maturity s .
s
Then, the fair value of the non-surrendable participating contract is given by:
⎧⎛
P
⎡ s−t
⎤
s t
⎞
⎫

FV ( V
C
v t t
q
C v t s
p
P
v t t
p
F
(2.3)
t
x )
⎪
r µ ⎪
= Eˆ⎨ ∑
−
+τ
+
−
τ
t +τ −
τ −
x +t
s
s −t
x +t
∑
+ −
⎢
1 ( ,
) 1/1
( , )
( ,
)1
,
,
⎥
τ 1
⎜
⎜
−
x+t ⎟⎟ t
⎬
⎪⎝ ⎣τ 1=
⎦
τ 1
=
⎠
⎪
⎩
⎭
where the participation rule is

1 The annual premium is computed using first order technical basis.

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108
Investment Management and Financial Innovations, Volume 3, Issue 2, 2006
⎛
τ ⎞

C = C
1+ φ − C 1
−
(2.4)
1
0 ⎜ −
φ
τ
τ
(
τ )
⎟ τ
⎝
s ⎠
and
⎛ηI − i
τ
⎞

φ = max
τ
⎜
,i
. (2.5)
min ⎟
⎝ 1− i
⎠
η ∈[ ]
1
,
0 is the participation coefficient, I represents the rate of return of the reference
t
portfolio during t-th anniversary of policy (see Section 3), i
is the minimum readjustment
min
measure.
The fair value of the participation option could be easily computed as the difference be-
tween FV ( P
V and FV ( B
V .
t
x )
t
x )
Some traditional Italian policies enable the policyholder to convert cash benefit at matur-
ity into a guaranteed annuity payable throughout the remaining lifetime, calculated at a guaranteed
rate G .
A guaranteed annuity option is a contract that provides the policyholder with the right to
receive at maturity either a cash payment or an annuity, depending on which has the greater value.
The guaranteed annuity option pay-out at maturity is expressed as (Ballotta and Haber-
man (2003))

max(G C a ,C = C + G C max ,
0 a
− K = C + OTA , (2.6)
s
x+s
s )
s
s
( x+s
)
s
s
where K = 1/ G , OTA = G C max ,
0
−
and
s
s
( a
K
x+s
)
⎛ ω−(s+x)
µ ⎞

a
=
r ,
Eˆ
v s, s τ
p
F
. (2.7)
x+ s
∑ ( + )
⎜
⎜
τ
x+ s
s
⎟
⎟
⎝ τ =1
⎠
In that case we have to express the fair value of a non-surrendable participating contract
with option to annuitise as

FV ( OTA
V
= FV V + OTA v t s
p
F
. (2.8)
t
x
)
( P
t
x )
Eˆ[
s
( , )
r,µ
s−t
x+t
t
]
The fair value of OTA could be easily computed as the difference between FV ( OTA
V

t
x
)
and FV ( P
V .
t
x )
At last, our aim is the computation of the surrender option’s fair value as the difference
between the fair value of the whole contract and the fair value of the non-surrendable participating
contract with option to annuitise.
According to Grosen and Jørgensen (2000) and Bacinello (2003) the whole contract is an
American-style contract that embeds a surrender option. A surrender option is an American-style
option that enables the policyholder to surrender the policy and receive the so called surrender
value.
Typically, the surrender value of a constant periodical premiums policy is given by
t
− −

R = C
+ C − C 1+ i
, (2.9)
t
0
( t
0 )(
sur ) (s t )
s
where i
is an annually compounded discount rate.
sur
The mechanism underlying a surrender option is the following: at any time t=1,2,..,s-1,
the policyholder compares the surrender value with the expected payoff from the continuation
value, and exercises the option if the surrender value is higher.

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Investment Management and Financial Innovations, Volume 3, Issue 2, 2006
109
In order to price the surrender option, Grosen and Jørgensen (2000) and Bacinello (2003)
propose a binomial tree model à la Cox, Ross and Rubinstein (1979): the fair price of the whole
contract and the continuation price can be computed by means of a backward recursive procedure
operating from time s-1 to time 0. Instead of a binomial tree model Andreatta and Corradin (2003)
use the Least Squares Monte Carlo Approach following Longstaff and Schwartz (2001).
In these articles authors value the surrender option for a surrendable participating contract
without option to annuitise. So the continuation value at time s-1 for a given path j is expressed as
(see Andreatta and Corradin (2003) p. 18):
( j)
( j)

W
= v
− ,
1
− . (2.10)
s−1
(s
s) ( j)
C
P
s
To distinguish our model from those described above, we have introduced in the con-
tinuation value of a surrendable participating policy a guaranteed annuity option, remembering
Italian policy features. If we suppose to be at time s-1, we have to compute a different continuation
value. To continue means receive, at time s, the benefit C
, if the insured dies within s-1 and s,
s 1
−
or to be entitled of a contract whose total random value equals max(G C a
,C
s
x+
, if the in-
s
s )
sured is alive. Therefore, the continuation value at time s-1 for a given path j is expressed by:
( j)
( j)

W
= v
− ,
1
+ max
,
− . (2.11)
s−1
(s
s) ( j)
( j) ( j)
( j)
[C q
s−1
x+s−1
(GC a C
s
x+s
s
)px+s−1] P
The fair value of the whole contract FV (
T
V
= F is therefore the maximum be-
s 1
−
x )
( j)
s 1
−
( j)
tween the continuation value and the surrender value Rs 1
− :
( j)
( j)
( j)

F
= max W , R
. (2.12)
s 1
−
( s 1− s 1−)
Assume now to be at time t < s – 1. As in the Fackler-Fourer’s recursive formula, to con-
tinue means to immediately pay the premium P and to receive, at time t+1, the benefit C , if the
t
insured dies within one year, or to be entitled of a contract whose total random value (including
the option of surrendering it in the future), equals F if the insured is alive. Therefore the con-
t 1
+
tinuation value at time t is given by the following expression:

W =
+ +
+
−
t
[C q v
t
x+
(2.13)
t (t, t
)
ˆ
1
p E
x+t
(v(t,t )1F
P
t 1
+ )]
.
Longstaff and Schwartz (2001) propose that the conditional expected value of the future
option value ˆ
E(v(t,t + )
1 F
can be estimated from the cross-sectional information in the simu-
t 1
+ )
lation by using least squares, that is by regressing the discounted realized payoffs from continua-
tion on functions of the values of the state variables1.
Finally, the fair value of the surrender option could be evaluated as the difference be-
tween FV ( T
V and FV ( OTA
V
.
t
x
)
t
x )
3. An application of the fair value actuarial model
3.1. The demographic stochastic process {µ
;t = ,
1
is described by a Mean-
x+t
}
,....
2
Reverting Brownian Gompertz (MRBG) model; in particular, we take into account a traditional
actuarial approach, where T is a random variable representing the remaining lifetime of a policy-
x

1 Dealing with Andreatta and Corradin (2003) we choose to base the regression on two state variables, the cash benefit and
the reference fund and regress according to a third-order polynomial model
ˆE(v(t,t + )
1
j
F
≈ a + a C + a C + a C + a S + a S + a S + a C S + a C S + a C S .
t 1
+ )
2
3
2
3
2
2
1
2
t
3
t
4
t
5 t
6 t
7
t
8
t
t
9
t
t
10
t
t

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110
Investment Management and Financial Innovations, Volume 3, Issue 2, 2006
holder aged x ; as a consequence the probability of survival to time s , conditional on being alive
at age x , is equal to
p = Prob T > s F . (3.1)
s
x
(
µ
x
t )

We define µ x+t t: to be the hazard rate for an individual aged x + t , at calendar year t , it
follows that 0 can be arranged as
⎡ s
−
ˆ
∫ µ dt
x t t
:
⎤

p =
+
E e 0
F
. (3.2)
s
x
⎢
µ
t ⎥
⎣
⎦
The time evolution of the hazard rate µ x+t t: is expressed by an exponential form as fol-
+
lows
gx,s t
x t
Y
µ
=
e
σ
µ
,
with
g ,σ , µ
> 0 , where g resumes on time s the
x+t t:
x 0
:
x,s
x
x 0
:
x,s
deterministic correction due to age x and the effect of longevity risk; { t
Y } is a stochastic process
introduced to model random variations in the forecast trends; σ represents the standard deviation
x
of the process {µ
;t = ,
1
; in particular the stochastic process { t
Y } is described by a
x+t:t
}
,....
2
mean reverting diffusion process

dY = −bY dt + dW , Y = 0, b ≥ 0 , (3.3)
t
t
t
0
where b is the mean reversion coefficient and {W is a standard Brownian motion.
t }
The time dynamic of the instantaneous interest rate (spot rate) {r ;t = ,
1
is mod-
t
}
,....
2
elled by a mean reverting square root diffusion equation as in Cox, Ingersoll and Ross model
(CIR); therefore, we assume the following stochastic equation:

dr = k θ − r dt + σ
r dZ , (3.4)
t
(
t )
r
r
t
t
where k is the mean reversion coefficient, θ is the long term rate, σ r is the volatility
parameter and { r
Z is a standard Brownian motion.
t }
Fair pricing of an insurance participating policy also depends on reference portfolio’s dy-
namic; in particular we assume to work in a Black-Scholes economy where the reference portfolio
is compounded mainly by a bond index and a minority by a stock index. The two components are
described by the following equation
(
⎧1:stock index
i )
(i)
(i) (i)
(i)


dS
= r S dt + σ S dZ ,
i =
(3.5)
t
t
t
t
t
⎨
⎩2 :

bond index
(i)
(i
(i)
where S
σ
t ,
) e {Zt }, are, for each reference portfolio’s component, market price,
volatility parameter and a Wiener process. At last, the three sources of financial uncertainty are
correlated:
(k )
( j)



dZ dZ
= ρ dt k, j = ,
1 ,
2 r , (3.6)
t
t
k , j
hence, reference portfolio could be expressed as a combination of the random variables
introduced above

S = −α S + S
α
. (3.7)
t
(1 ) (1)
(2)
t
t

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Investment Management and Financial Innovations, Volume 3, Issue 2, 2006
111
As a result, the annual rate of return of the reference fund at time t is defined as:
S

I =
t
−1. (3.8)
t
St 1−
3.2. Since a closed form solution of expression 0 is not available, a policy’s Fair Value
can be obtained via Monte Carlo simulation. 0 reports some contract features used for numerical
analysis.
Table 1
Contract Features
Sex
Male
Age
55
Duration
10
Technical rate
1.00%
Mortality Table
SIM 92
Sum Assured
100 €
Participation coefficient
87.5%
Minimum Adjustment Measure
0.00%
Guaranteed rate for annuity
5.78%
Annually compounded surrender discount rate
3.00%
Reference portfolio participation coefficient:

Stock index
10%
Bond Index
90%

0 shows parameters estimated on market data and used in mortality and financial risk dif-
fusion processes. In particular, with reference to CIR model, risk-adjusted parameters are cali-
brated on market value of euro swap interest rates through Brown and Dybvig (1986) framework.
Parameters for MRGB model are calibrated on mortality hazard rates derived from an Italian pro-
jected life table called “RG48”. At last, reference fund parameters are estimated on daily market
value of Emu-Bond Index and MSCI World Index observed between 2001 and 2003.
Table 2
Estimated Parameters set
µ
S
t
r

x+t
t
(1)
0
r
0.015268
b
0.5
S0
123.57
k
(2)

0.245439
σ
0.18
S0
1240.22
θ
0.058359
g
0.10
( )
1
σ
0.01
σ


(2)
r
0.053524
σ

0.11

With reference to an insured aged 55 and a contractual term of ten years, 0 reports, the
annual expected cash flow of mathematical reserve computed with local rules, fair value of basic
contract, participating option, option to annuitise, surrender option and whole contract respec-
tively; we simulate 10,000 paths for the mortality hazard rate, the spot rate and the reference fund,
described in 0, 0 and 0.

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112
Investment Management and Financial Innovations, Volume 3, Issue 2, 2006
Table 3
Local Reserve and Fair Value of a contract with embedded options
Participating
Local Re-
Basic Con-
Participation
Option to
Surrender
Whole Con-
Year
contract / Lo-
serve
tract
Option
annuities
Option
tract
cal Reserve
0 -
-17.37
16.01 - 0.70
-0.66
1 9.19 -7.89 16.44 92.99% - 0.72
9.27
2 18.97 1.81 17.00 99.18% - 0.75 19.57
3 29.40 11.87 17.69 100.56% - 0.79 30.35
4 40.54 22.39 18.49 100.83% - 0.84 41.72
5 52.46 33.46 19.40 100.78% - 0.89 53.76
6 65.17 45.17 20.42 100.64% - 0.96 66.55
7 78.78 57.59 21.55 100.45% - 1.03 80.16
8 93.37 70.81 22.78 100.23% - 1.10 94.68
9 108.93 84.91 24.10 100.07% - 1.17 110.18
10 125.51 100.00 25.51 100.00% -
- 125.51

The introduction of a fair value accounting system produces a reduction of insurance li-
abilities and so a different profits distribution over time until the policy’s maturity. In particular,
for a 10-year contract analyzed, 0 shows that:
• the liability increase is an average about of 1.91%;
• the participating option is the most relevant embedded contract’s component, perform-
ing in average about of 34.76% over the whole contract value;
• a high performance of the surrender option component, in average about of 1.44% over
the whole contract value, is explained by the implicit option annuity component’s
value equals zero, in reference to a guaranteed annuity coefficient used in the tradi-
tional Italian life insurance policies.
0 shows second moments, variation coefficient and skewness of contract value compo-
nents. These parameters offer a measure of riskness in reference to each contracts component, use-
ful to calculate adequate margins for risk under a Fair Value account system.
Table 4
Second Moments, Variation Coefficient and Skewness
Fair Value Whole contract
Fair Value Surrender Option
Year
σ
µ
σ
µ
σ

3

3
µ
3
σ
σ
µ
3
σ
0 5.04 1.07 2.81 7.57 12.67
-1.03
1 5.20 0.36 2.76 7.74 12.55
-1.02
2 5.43 0.22 2.64 7.92 12.29
-1.00
3 5.76 0.16 2.45 8.07 11.87
-0.97
4 6.20 0.13 2.20 8.17 11.30
-0.96
5 6.80 0.12 1.92 8.15 10.54
-0.93
6 7.59 0.11 1.64 7.99 9.60
-0.91
7 8.62 0.10 1.42 7.58 8.48
-0.89
8 9.89 0.10 1.27 6.75 7.03
-0.87
9 11.49 0.10 1.16 1.59 1.55
1.79
10 12.81 0.10 1.14
-
-
-

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Investment Management and Financial Innovations, Volume 3, Issue 2, 2006
113
To analyse value’s sensitivity of each component, we have derived regions of Fair Value in
correspondence to the most relevant parameters: K , i
and t .
sur
In particular in t = 0 :
• Figure 1 presents the whole contract Fair Value behaviour in relation to the two pa-
rameters K and i
, showing a non increasing monotone trend, concave in increas-
sur
ing the two parameters;
• Figure 2 shows the surrender option Fair Value behaviour in relation to the two pa-
rameters K and i
; it highlights a non decreasing monotone trend, convex in in-
sur
creasing the K parameter, while it is concave in increasing the i
parameter.
sur

Fig. 1. Fair Value of the Whole contract

Fig. 2. Fair Value of the Surrender Option

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114
Investment Management and Financial Innovations, Volume 3, Issue 2, 2006
At last, Figure 3 presents the annuity option Fair Value behaviour in relation to the two
coordinates t and K ; it can be observed that results found in Ballotta and Haberman (2003) are
confirmed, as the annuity option Fair Value shows a non increasing monotone trend, concave in
increasing the two parameters.

Fig. 3. Fair Value of the Option to Annuitise

References
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lio of Italian Life Guaranteed Participating Policies: a Least Squares Monte Carlo Approach”,
Real option theory meets practice, 8th Annual International Conference, Montrèal Canada
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2. BACINELLO A.R. (2001), “Fair Pricing of Life Insurance Participating Policies with a Mini-
mum Interest Rate Guaranteed”, Astin Bulletin, 31, 275-297.
3. BACINELLO A.R. (2003), “Fair Valuation of a the Surrender Option Embedded in a Guaran-
teed Life Insurance Participating Policy”, The Journal of Risk and Insurance 70 (3), 461-487.
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tion”, www.casact.org.
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Econometrica, 53:-408.
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compagnie di Assicurazione: implicazioni attuariali”, Seminario tenuto presso l’Istituto
Italiano degli Attuari, Roma, 29 Marzo, 2001.

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