a course on financevol 2

8/12/2019 A Course on Financevol 2

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Groupe Consultatif Actuariel EuropeenSummer School 2002

Organized by Istituto Italiano degli Attuari

Massimo De Felice Franco Moriconi

A Course on

Finance of Insurancevol. 2

Milano, 10-12 July 2002Universit` a Cattolica Milano


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Mailing Address

Massimo De FeliceDipartimento di Scienze Attuariali e FinanziarieUniversit` a di Roma La SapienzaVia Nomentana, 41 00161 Romae-mail: [email protected]

Franco MoriconiDipartimento di EconomiaFacolt` a di EconomiaUniversit` a di PerugiaVia A. Pascoli, 1 06100 Perugiae-mail: [email protected]

Copyright c 2002 by M. De Felice, F. Moriconi. All rights reserved.


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Contents

Keywords, phrases. Methods, techniqes. *.1*.2

Introduction Towards a nance of insurance 1Valuation of the life insurance business: a general framework. Asset-liability approach. Fair valuation.A new toolkit for corporate governance. Selected bibliography.

Chapter 1 The standard actuarial valuation framework 9Endowement policy with constant sum insured. The actuarial uncertainty. Technical means. The stan-dard actuarial valuation framework. The reserve as a budget constraint. Participating endowment policy.The standard actuarial valuation of a participating policy. The readjustment as an interest rate crediting.General participating mechanism.

Chapter 2 Valuation principles 23The general asset-liability framework. Asset-liability management. The valuation principle. Some prop-erties of the value funcion. Technical means and valuation factors. Valuation of payment streams.

Stochastic reserve. VBIF: the stochastic reserve approach. Decomposition of VBIF.

Chapter 3 Basic nance of life insurance contracts 36The nancial structure of a policy. The embedded options. Put decomposition. Call decomposition. Theminimum guarantees in a multiperiod contract. The return of the reference fund.

Chapter 4 The valuation model. Value and risk measures 43The valuation model. Interest rate uncertainty. Stock price uncertainty. Hedging argument and valuationequation. The risk-neutral probability. The endogenous term structure. Integral expression of prices.Measures of basis risk. Value-at-Risk. Risk capital.

Chapter 5 Applying the valuation model 54Calibration of the valuation model. Computation of the valuation factors. Term structures of valuation

rates. The return of the reference fund. Valuation during the life of the policy. Cost of the embeddedput and expected funds return. More details about market effects.

Chapter 6 Other embedded options 76Surrender options. Guaranteed annuity conversion options.

Chapter 7 Valuation of outstanding policy portfolios 80Partecipating policies with constant annul premiums. Different benets in case of death. Portfoliovaluation. Controlling the balance of assets and liabilities.

Chapter 8 Alternative valuation methods 92VBIF: the annual prots approach. Equivalence with the stochastic reserve approach. Actuarial expecta-tion of future annual prots. Mortality gain. Investment gain. Valuation of the annual prots. Alternative

valuation methods. Risk-neutral probabilities. Risk-adjusted discounting. RAD under scenario.Chapter 9 Unit-linked and index-linked policies 107Unit-linked endowment policy. Similarities with participating policies. The standard valuation frame-work. Reserve and sum insured. Unit-linked policies with minimum guarantee. Put decomposition.Stochastic reserve and VBIF. Prots from management fees. Surrenders. Index-linked endowment pol-icy. Similarities with participating (and u-l) policies. The standard valuation framework. Financial risk.Stochastic reserve and VBIF.

Appendix An elementary model for arbitrage pricing 124The derivative contract. Single period binomial model. The hedging (or replication) argument. Therisk-neutral valuation. Valuing a life insurance liability. Valuing the investment gain. Example.

References 138


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Chapter 7 Valuation of outstanding policy portfolios

Partecipating policies with constant annul premiumsDifferent benets in case of death

Portfolio valuation

Controlling the balance of assets and liabilities

c MDF-FM Finance of Insurance vol. 2, p. 80


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Participating policies with constant annual premiums

If the annual premiums are not readjusted (i.e. Ak A0 ), the read-

justment of benets is different from the full readjustment rule:

C k = C k 1 (1 + k ) .

Typically, the increment C k is determined as the benet of anadditional single premium endowment over the residual life of the

principal policy. The additional policy is nanced by the excessreturn on the investment of the savings premium Ask .

The intensity of the readjustment of benets will depend on x, n, k .Ceteris paribus :

for policies with equal values of x and n , the readjustment willbe increasing w.r. to k;

for policies with equal values of n and k, the readjustment willbe decreasing w.r. to x.

An approximating rule

(independent of x):

C k = C k 1 (1 + k ) C 0 1 kn

k .



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Different benets in case of death

In many policies benets payable in case of death ( C Dk ) are different

from benets payable if the insured is alive ( C L

n ).=

computation of separated streams of technical means C D

t,k , C L

t,n ; computation of separated valuation factors uD (t, k ), uL (t, n ).

Technical means of premiums and benets

valutation date 31/12/1998

c a s h

o w s

( m i l l i o n

E u r o

)

maturity year



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Portfolio valuation

Since the calculation of the valuation factors u(t, k ) involves Monte

Carlo procedures, the valuation of a portfolio of outstanding poli-cies can be highly time consuming if the contracts are not properlyaggregated

For single premium policies or for policies readjusting both premi-ums and benets the valuation factors only depend on k t :

u (t, k ) = u(k t )

= a single structure of valuation factors is needed for each classof policies.

In the general case, for each class of policies a different stream of valuation factors is required for different values of x , n and n t .



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Reserves(million Euro)

policy traditional stochastic diff. %(a) (b) (a-b) (a-b)/a

CAP 3 +1.5 1,062 1,054 9 0.83CAP 3 +1 739 755 -16 -2.22CAP 3 +1 3,114 3,174 -60 -1.94CAP 4 +0 1,450 1,453 -2 -0.17CAP 3 +0 629 492 137 21.76

CAP 2.5+0 69 36 33 47.34SP 3 +1 28 29 -1 -3.05SP 4 +0 69 71 -2 -2.94SP 3 +0 136 137 -1 -1.05SP 2.5+0 14 14 -0 -0.04NP 3 +0 174 161 13 7.68PORTFOLIO 7,485 7,376 108 1.45

Legend

CAP: Constant Annual Premiums (indexed benefits)SP: Single Premium (indexed benefits)NP: Non Participating (constant premiums and benefits)3+1.5: technical rate 3%, minimum guaranteed 4.5%



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Components of stochastic reserves(million Euro)

policy benefits premiums diff.(a) (b) (a-b)

CAP 3 +1.5 1,306 253 1,054CAP 3 +1 1,076 321 755CAP 3 +1 5,789 2,615 3,174CAP 4 +0 3,981 2,528 1,453CAP 3 +0 2,799 2,307 492CAP 2.5+0 435 398 36SP 3 +1 29 0 29SP 4 +0 71 0 71SP 3 +0 137 0 137SP 2.5+0 14 0 14NP 3 +0 166 5 161PORTFOLIO 15,802 8,426 7,376

policy benefits survival death(a+b) (a) (b)

CAP 3 +1.5 1,306 1,242 64CAP 3 +1 1,076 1,005 71CAP 3 +1 5,789 5,479 310CAP 4 +0 3,981 3,723 257CAP 3 +0 2,799 2,587 211CAP 2.5+0 435 397 38SP 3 +1 29 28 1SP 4 +0 71 69 3SP 3 +0 137 129 8SP 2.5+0 14 13 1

NP 3 +0 166 160 5PORTFOLIO 15,802 14,833 969



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Basis Risk

Stochastic duration

policy premiums benefits survival death

CAP 3 +1.5 2.15 2.57 2.59 2.23

CAP 3 +1 2.69 2.91 2.94 2.45

CAP 3 +1 3.24 3.33 3.35 2.87

CAP 4 +0 3.55 3.91 3.95 3.33

CAP 3 +0 3.70 4.14 4.19 3.55

CAP 2.5+0 3.87 4.34 4.41 3.74

SP 3 +1 . 1.94 1.94 2.03

SP 4 +0 . 2.07 2.07 1.87

SP 3 +0 . 2.18 2.20 1.95

SP 2.5+0 . 2.23 2.25 1.95

NP 3 +0 1.51 2.94 2.93 3.14

PORTFOLIO 3.42 3.49 3.52 3.07

Delta

policy premiums benefits survival death

CAP 3 +1.5 0.00 0.00 0.00 0.00

CAP 3 +1 0.00 0.07 0.07 0.06

CAP 3 +1 0.00 0.05 0.05 0.05

CAP 4 +0 0.00 0.03 0.03 0.03

CAP 3 +0 0.00 0.02 0.02 0.02

CAP 2.5+0 0.00 0.02 0.02 0.01

SP 3 +1 . 0.10 0.10 0.11

SP 4 +0 . 0.11 0.11 0.11

SP 3 +0 . 0.13 0.13 0.13

SP 2.5+0 . 0.14 0.14 0.14

NP 3 +0 0.00 0.00 0.00 0.00

PORTFOLIO 0.00 0.04 0.04 0.03



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Embedded options(million Euro)

Put decomposition of benefits

policy benefits base put %(a+b) (a) (b) b/(a+b)

CAP 3 +1.5 1,264 1,190 73 5.80CAP 3 +1 1,073 975 97 9.09CAP 3 +1 5,776 5,152 624 10.81CAP 4 +0 3,978 3,526 453 11.37CAP 3 +0 2,798 2,558 240 8.58CAP 2.5+0 435 402 33 7.62SP 3 +1 29 26 2 7.89SP 4 +0 71 67 5 6.67SP 3 +0 137 129 8 5.86SP 2.5+0 14 13 1 5.32NP 3 +0 0 0 0 .PORTFOLIO 15,574 14,038 1,537 9.87

Call decomposition of benefits

policy benefits guaranteed call %(a+b) (a) (b) b/(a+b)

CAP 3 +1.5 1,264 1,207 57 4.47CAP 3 +1 1,073 941 132 12.30CAP 3 +1 5,776 4,817 959 16.61



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Value of Business In Force(million Euro)

VBIF without minimum guarantees (a) 2,841Value of minimum guarantees (b) 982

VBIF (a-b) 1,858Investment gain 108Mortality gain 157Surrender gain 269Value of loadings 1,324

The value of minimum guarantees (b) is computed on third order basis.

Value of Business In Force by policy type (million Euro)

Inv. Mort. Sur. Load. VBIF Value VBIF withoutgain gain gain of m.g. min. guar.

CAP 3 +1.5 9 5 15 65 94 66 161CAP 3 +1 -16 10 14 82 89 82 171CAP 3 +1 -60 43 109 517 608 437 1,046CAP 4 +0 -2 49 56 345 447 254 702CAP 3 +0 137 44 55 267 504 111 615CAP 2.5+0 33 8 6 46 93 14 107SP 3 +1 -1 -0 1 0 -0 2 2SP 4 +0 -2 -0 2 0 -0 5 4SP 3 +0 -1 -0 8 0 6 8 13SP 2.5+0 -0 -0 1 0 1 1 2NP 3 +0 13 -0 2 1 16 2 18PORTFOLIO 108 157 269 1,324 1,858 982 2,841

The value of minimum guarantees is computed on third order basis.



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Controlling the balance of assets and liabilities

The corresponding asset portfolio is evaluated using the same pricing

model used for the policy portfolio

Same pricing model, same valuation date, same calibration

the values A t and V t (and their sensitivities) can be coherentlycompared.



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ALM analysis

Policy portfolio(million Euro)

Price Duration Delta

Premiums 8,426 3.42 0.00Benefits 15,802 3.49 0.04Reserve/Gap 7,376 0.07 0.04

Investment portfolio(million Euro)

Price % Duration Delta

Bond 6,176 71.7 1.41 0.00Stock 2,441 28.3 0.00 0.28Total 8,617 100.0 1.41 0.28

VaR of the investment portfolio(99%, 10 days)

Price Amm VaR %

Bond 6,176 95.06 bp 61 0.99Stock 2,441 -8.30 % 203 8.30Total 8,617 . 264 3.06



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ALM analysis

Asset-liability portfolio

Price Duration Delta

Investments (a) 8,617 1.41 0.28Premiums (b) 8,426 3.42 0.00Asset (a+b) 17,043 2.41 0.14Liabilities (c) 15,802 3.49 0.04

A/L Portfolio (a+b-c) 1,241 -1.07 0.11

VaR of the asset-liability portfolio(99%, 10 days)

Price Interest % Stock % VaR (pb) VaR (%)

Investments 8,617 -50 -0.6 203 2.4

Premiums 8,426 -121 -1.4 0 0.0Asset 17,043 -170 -1.0 203 1.2Liabilities 15,802 227 1.4 -49 -0.3

A/L Portfolio 1,241 56 4.5 153 12.3

The interest rate VaR of the A/L portfolio corresponds to Amm=-76.35 bp.For an interest rate movement of +95.06 bp the VaR is negative.

Netting the VaR of the investments

Amm VaR % Amm VaR %(Inv.) (Inv.) (A/L) (A/L)

Bond 95.06 pb 61 0.99 -76.35 pb 56 0.91Stock -8.30 % 203 8.30 -8.30 % 153 6.27Total 264 3.06 209 2.43



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Chapter 8 Alternative valuation methods

VBIF: the annual prots approachEquivalence with the stochastic reserve approach

Actuarial expectation of future annual prots

Mortality gain

Investment gain

Valuation of the annual prots

Alternative valuation methods

Risk-neutral probabilities

Risk-adjusted discounting

RAD under scenario



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VBIF: the annual prots approach

The standard approach for determining VBIF is based on an in-

vestment argument.

We refer for simplicity to a single premium endowment.

During the life of the policy the company must maintain a capital atthe level of the reserve process:

R k := 1{T x >k 1 } R k ,

However, at time k 1 the reserve R k 1 can be invested in the refer-ence fund, providing an annual rate of return I k .The amount R k 1 (1 + I k ) realized at time k, net of the new reservelevel R k and of the liability Y k represents the technical gain in yeark .

The VBIF at time 0 can be obtained as the present value of thesequence of the annual gains.

At time 0 the annual prots emerging from the policy can berepresented by the cash ow stream:

G = {G k , k = 1 , 2, . . . , n } ,where:

G k = R k 1 (1 + I k ) R k Y k .Then the VBIF at time 0 is given by:

E 0 = V (0; G ) =n

k =1V 0; G k .



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Equivalence with the stochastic reserve approach

Under the no arbitrage assumption in perfect market the annual

prots approach is equivalent to the stochastic reserve approach.The previous expression can be explicitely written as:

E 0 =n

k =1

V 0; R k 1 (1 + I k ) n 1

k =1

V 0; R k n

k =1

V 0; Y k . ( )

By the reinvestment security theorem:

V t ; R k 1 (1 + I k ) = V t ; R k 1 .Thus the rst sum in (*) can be expressed as:

n

k =1

V 0;

R k 1 (1 + I k ) =

n

k =1

V 0;

R k 1 = R 0 +

n 1

k =1

V 0;

R k ;

thus expression (*) reduces to:

E 0 = R 0 n

k =1

V 0 = R 0 V 0 .



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Actuarial expectation of future annual prots

Taking the expectation of the actuarial random variables, the ex-

pected future gains G are dened by: G k =

k 1 px [R k 1 (1+ I k ) (1 q x + k 1 )R k q x + k 1 C k ] , k < n ,

n 1 px [R n 1 (1+ I n ) C n ] , k = n .

Subtracting the quantity:

R k 1 (1 + k )(1 + i) (1 q x + k 1 ) R k q x + k 1 C k ,

which is equal to zero by the equilibrium constraint, we get theHomans formula:

Gk =

k 1 px R k 1 (I k m k )

+ ( C k

Rk

) (q x + k 1 q x

+k

1 ) , k < n ,n 1 px R n 1 (I n m n ) , k = n ,

where m k := (1 + k )(1 + i) 1, that is:

m k = max { I k , i} .

Under rst order basis, i.e. if:

I k i and q x + k 1 q x + k 1 ,

then all the expected prots are zero:

G k = 0 , k = 1 , 2, . . . , n .c MDF-FM Finance of Insurance vol. 2, p. 95


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To better characterize the minimum guarantee embedded in thepolicy, let us consider the investment gain of an analogous policywithout minimum guarantee; this is the base payoff , dened as:

B k = K k 1k 1

j =1

(1 + I j ) (1 ) I k .

Of course: B k G I k .The guarantee payoff , or the put payoff , is the difference:

P k = B k G I k 0 .

For k = 1 we have:

G I 1 = R

0 (I 1 m 1 ) = R 0 (1 ) I 1 max{i I 1 , 0} .

That is

G I 1 = B 1 P 1 ,where:

B1 = R

0 (1 ) I 1 ,

P 1 = R 0 max{i I 1 , 0} .



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Valuation of the annual prots

The VBIF at time 0 can be obtained as the value of the future annual

prots G

; under our assumptions:V (0; G k ) = V (0; G k ) ;

hence:

V (0; G ) =n

k =1

V (0; G k ) .We are mainly interested in the investment component of the annualprots; we have:

V (0; G I k ) = V (0; G I k ) ,which can be written as:

V (0; G I k ) = V (0; B k ) V (0; P k ) .Under the fair valuation approach, the sum of this values over thelife of the policy must be equal to the investment component givenby the stochastic reserve approach.

Dening:

k := ( I k m k )k 1

j =1

(1 + m j ) ,

the value of the investment gain can be expressed as:

G I k = K k 1 k ,where:

the technical mean K k 1 is determined by actuarial assumptions

on the probability measure P (1) ; the factors k are determined by capital market uncertainty.



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Alternative valuation methods

Risk-neutral probabilities (RNP)

The risk-neutral probability (RNP) approach is natural when thevaluation problem is set up in the framework of contingent claimspricing. Under the arbitrage principle in a perfect market:

V 0; k = E Q0 [k (0, k )] , (RNP)

where:E Q0 is the expectation operator taken with respect to the risk-neutral probability Q , conditional on the information at time0;

(0, k ) is a stochastic discount factor on the time interval [0 , k ].

The discount factor (0, k ) and the risk-neutral probability Q mustbe specied under an appropriate stochastic model.

Once the sources of market uncertainty are specied in the model, and Q are the same for all the securities which depend on these riskfactors= if the model is calibrated in order to match the observed price of traded securities, it can be applied to non-traded securities, providingcoherent pricing.

Remark. The valuation of the options embedded in life insurancepolicies with the RNP method can be considered a problem in Real

Option Analysis .[Copeland, Antikarov, 2001]



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Risk-adjusted discounting (RAD)

The standard approach to calculating VBIF consists in taking the

natural expectation of the random payoff k and then discountingit at an appropriate risk-adjusted force of interest r a ; that is:

V 0; k = e r a k E 0 k . (RAD)

The RAD method is widely used in capital budgeting applications,where it is also referred to as the Net Present Value method.

The risk premium r a r is usually determined by the observationof past returns on assets of similar insurance rms, using popularmodels as the Capital Asset Pricing Model or the Dividend DiscountModel.

Pros: RAD method is easier to communicate to practitioners is the most intuitive in a single-period setting

Cons:

RAD method becomes very complicated when the problem is

inherently intertemporal high degree of subjectivity is involved in the practical assess-ment of both the expected payoff and the risk-adjusted rate;this problem is even more important when option-like payoffsare considered it can be argued that just this difficulty gave an impetus to

the development of the option pricing theory and of the RNPmethod.



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RAD under scenario

Scenario methods are typically used in practical applications of the

RAD approach, in order to derive an estimate of the natural expec-tation E 0 k .

Since the technical mean k is a function of the realized return I kof the reference fund, a best estimate I k of this random variableis taken and the expectation of k is derived correspondingly.For illustration purposes, let us assume:

I k = E 0 I k .

A problem obviously arises since if k is a non linear function of I kthe property:

E 0 k (I k ) = k I

k

in general does not hold.

The embedded options are far-out-of-the-money at the policy is-suance and in typical market conditions they remain out-of-the-money during the life of the policy; i.e. normally the assumed sce-nario is such that, for each future year k:

I k > i/ ,

which corresponds to:mk = I

k ;

hence:

K k 1 E 0 k = K k 1k 1

j =1

(1 + I j ) (1 ) I

k = E 0 Bk . the embedded options are not captured under the scenariomethod.c MDF-FM Finance of Insurance vol. 2, p. 102


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l u e

v a

l u e

( a

)

( b )

( b

- a

)

C A P

_ 1

4 %

1 ,

6 2 0

, 0 0 0

1 6 4

, 4 0 8

1 1 7

, 5 3 6

1 7 0

, 8 7 3

5 3

, 3 3 7

C A P

_ 2

2 .

5 -

3 %

9 3

, 9 4 8

2 5

, 5 0 6

2 7

, 2 5 6

3 2

, 0 1 8

4 ,

7 6 3

C A P

_ 2

4 %

5 0 2

, 7 5 6

6 9

, 1 1 8

4 7

, 7 2 2

8 0

, 2 3 1

3 2

, 5 0 9

I A P

_ 1

4 %

9 3 9

, 2 9 7

1 0 2

, 3 1 4

5 6

, 0 8 0

1 0 6

, 4 7 2

5 0

, 3 9 2

I A P

_ 2

2 ,

5 -

3 %

3 5

, 4 2 7

1 2

, 7 5 4

1 3

, 8 7 7

1 6

, 2 2 4

2 ,

3 4 7

I A P

_ 2

4 %

1 4 8

, 7 4 3

2 7

, 6 5 4

1 9

, 6 9 2

3 1

, 8 5 8

1 2

, 1 6 4

S P

2 ,

5 -

3 %

9 7

, 3 1 5

3 ,

2 2 4

1 ,

2 5 7

3 ,

2 4 0

1 ,

9 8 2

S P

4 %

4 4

, 3 1 9

1 ,

4 8 0

- 4 2 0

1 ,

2 2 2

1 ,

6 4 0

R S P

2 ,

5 -

3 %

3 2 2

, 6 3 2

5 5

, 8 8 1

6 2

, 9 5 0

8 0

, 0 7 9

1 7

, 1 2 8

R S P

4 %

7 9

, 8 1 1

5 ,

9 1 7

6 4 4

5 ,

8 6 8

5 ,

2 2 3

T I

4 %

6 ,

4 0 5

5 ,

8 3 2

6 ,

3 5 8

6 ,

3 5 8

0

g r o u p

2 ,

5 -

3 %

2 8 4

, 5 5 0

7 ,

1 9 8

1 ,

9 5 7

9 ,

6 4 3

7 ,

6 8 5

g r o u p

4 %

8 2 4

, 7 9 4

3 0

, 4 6 2

- 3 7 8

6 5

, 1 9 4

6 5

, 5 7 2

p o r t

f o l

i o

5 ,

0 0 0

, 0 0 0

5 1 1

, 7 4 9

3 5 4

, 5 3 3

6 0 9

, 2 7 8

2 5 4

, 7 4 6

l e g e n

d

-

C A P :

C o n s t a n t

A n n u a

l

P r e m

i u m s

( i

n d

e x e

d

b e n e

f i

t s

)

-

I A P :

I n

d e x e

d

A n n u a

l

P r e m

i u m s

( i

n d

e x e

d

p r e m

i u m s

a n

d

b e n e

f i

t s

)

-

S P :

S i

n g

l e

P r e m

i u m

( i

n d

e x e

d

b e n e

f i

t s

)

-

R S P :

R e c u r r

i n g

S i

n g

l e

P r e m

i u m

( i

n d

e x e

d

b e n e

f i

t s

)

-

T I :

T e r m

I n s u r a n c e



31/64

Chapter 9 Unit-linked and index-linked policies

Unit-linked endowment policySimilarities with participating policiesThe standard valuation frameworkReserve and sum insuredUnit-linked policies with minimum guaranteePut decompositionStochastic reserve and VBIFProts from management feesSurrendersIndex-linked endowment policySimilarities with participating (and u-l) policiesThe standard valuation frameworkFinancial riskStochastic reserve and VBIF



32/64

Unit-linked endowment policy

Given an investment fund, let F t be the market value at time t of

one unit of the fund.A unit-linked endowment policy with term n years for a life aged x

provides for payment of

a number N D of units at the end of the year of death if thisoccurs within the rst n years ( term insurance ),

otherwise

a number N L of units at the end of the n th year ( pure endow-ment ).

If the policy is single premium , the insured pays a lump sum U attime 0.

The insured benet in case of death at time k is:

C Dk = N D F k ;

if the insured is alive at time n the benet is:

C Ln =

N L F n

the insured sums are contractually dened in stochastic units .

Typically the management of the reference fund is under the insurercontrol.



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Similarities with participating policies

Let:

C k : benet (eventually) paid at time k; F t : market value of the reference fund;

I k := F k /F k 1 1: annual rate of return of the fund at time k.

The benets at time k are given by:

C 0 = N F 0 ,C k = C k 1 (1 + I k ) , k = 1 , 2, . . . , n .

For 0 h k n, we can dene the readjustment factors :

(h, k ) :=k

j = h +1(1 + I j ) = F kF h

(being ( k, k) = 1).

Hence:C k = C 0 (0, k) .



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The standard valuation framework

At time 0 we have the liability stream:

C = C k ; k = 1 , 2, . . . , n ;

where:C k =

C k , with prob. P 0(C k ; k)0 , with prob. 1 P 0(C k ; k)

The net single premium is given by:

U = C D0n

k =1

P (1)0 (C D

0 ; k) + C L

0 P(1)0 (C

L

0 ; n ) ,

or:U = F 0 N 0 ,

where:

N 0 := N D

n

k =1

P (1)0 (C D

0 ; k) + N L P (1)0 (C

L

0 ; n ) .

rst order basis: probability P (1) and technical rate i = 0.

Similarly, the net premium reserve at time k = 0 , 1, . . . , n is denedby:

R k = F k N k ,

where:

N k =:

N Dn

j = k +1

P (1)

k (C D

j ; j

) + N L P (1)

k (C L

n ;n

).



35/64

=

The policy can be hedged by the insurer by purchasing N 0

units at time 0; the hedging strategy is a replicating strategy, because the port-

folio purchased at time 0 replicates (on the average) the futureliabilities;

the hedging strategy is a static strategy;

the hedging strategy is not completely riskless, because of mor-tality uncertainty;

If N D = N L = N the hedging strategy is a riskless strategy.Both nancial and actuarial risk are eliminated from the policy.



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Reserve and sum insured

The reserve at time k can be expressed as:

Rk = C D

k

n

j = k +1

P k (C D

j ; j ) + C L

n P k (C L

n ; n)

= ( C Dk C L

k )n

j = k +1

P k (C D

j ; j )

+ C L

k

n

j = k +1P k (C

D

j ; j ) + C L

k P k (C L

n ; n) .

Since:

n

j = k +1

P k (C D

j ; j ) + P k (C L

n ; n) =n

j = k +1j 1 /1 q x + n px = 1 ,

one has:

Rk = C L

k + ( C D

k C L

k )n

j = k +1

P k (C D

j ; j ) .

If N D = N L = N , then: C Dk = C L

k = C k = N F k , k ; hence:

Rk = C k

the reserve at time k is equal to the current value C k of theN = U/F 0 units purchased at time 0 the contract is not exposed to mortality risk.

If N D > N L the reserve is greater than C Lk .



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Unit-linked policies with minimum guaratee

Assume that the insured benets C k cannot be lower than a oor

value NM k xed at time 0:

C k = max {NF k , NM k } .

e.g.:M k := F 0 (1 + g)k ,

with g: a minimum guaranteed annual return maturity guarantee .

Remark. Tipically the reference fund F has a substantial equitycomponent. Thus also negative values of g can be of interest.

The insured sum can also be expressed as:

C n = NF 0 maxF nF 0

, (1 + g)n ,

hence:

C n = C 0 (0, n ) ,

where:C 0 = NF 0 ,

and:(0, n ) = max

F n

F 0

, (1 + g)n .



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Put decomposition

Since C n can be written as:

C n = N F n + N max{M n F n , 0} ,

the policy is equivalent to a u-l policy with sum insured N F n andwithout minimum guarantee, plus a contract providing at time n thepayoff:

P n := N max{M n F n , 0} .

This is the payoff of a portfolio of N european put options on theprice of the unit, with exercise date n and strike price M n .

= In principle, the insurer can eliminate nancial risk by purchas-ing the put options.

Remark. The expression:

C n = C 0 maxF nF 0

, (1 + g)n ,

can be written as:

C n = C 0 max

n

k =1

F kF k 1 ,

n

k =1(1 + g) .

The payoff of a policy with annual guarantees can be expressed in-stead as:

C n = C 0n

k =1

max F kF k 1

, (1 + g) .

Of course in a multiple period contract there is a signicant differencebetween the two guarantees.c MDF-FM Finance of Insurance vol. 2, p. 114


39/64


40/64

Prots from management fees

Assume that at each year end the fund pays to the insurer manage-

ment fees determined as a fraction f of the current NAV; at time kthe value F k of the fund is now:

F k = F k (1 f )k ,

where F k is the value of an analogous fund without managementfees.

The sum insured is now C k = NF k ; hence:

V (0; C k ) = N (1 f )k V (0; F k ) = NF 0 (1 f )k = R 0 (1 f )k .

Therefore the stochastic reserve at time 0 is:

V 0 =n

k =1

P 0 (C k ; k) V (0; C k )

= R 0n

k =1

P 0 (C k ; k) (1 f )k < R 0 .

The VBIF is given by:

E 0 = R 0 V 0 = R 0 1

n

k =1 P0 (C k ; k) (1 f )

k

.

Remark. For a policy without embedded options the VBIF is inde-pendent of the fund investment strategy.

Remark. If the policy provides minimum return guarantees the value

of the embedded put option is subtracted from the VBIF.The put price is generally depending on the investment strategy.c MDF-FM Finance of Insurance vol. 2, p. 116


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43/64

Similarities with participating (and u-l) policies

Let:

C k : benet (eventually) paid at time k; F t : market value of the reference index;

I k := F k /F k 1 1: annual rate of return of the index at time k.

Given the initial sum insured C 0 , the benets at time k are given by:

C k = C 0 (0, k) , k = 1 , 2, . . . , n .

where the function:

(0, k) = ( F 1 , F 2 , . . . , F k ) ,

is contractually xed at time 0.

Remark. It is relevant to observe that in the i-l policies the referenceindex is observed on the market and cannot be inuenced by theinsurer.



44/64

The standard valuation framework

At time 0 we have the liability stream:

C = C k ; k = 1 , 2, . . . , n ;

where:C k =

C k , with prob. P 0(C k ; k)0 , with prob. 1 P 0(C k ; k)

The net single premium is given by:

U = C D0n

k =1

(1 + i) k P (1)0 (C D

0 ; k) + C L

0 (1 + i) n P (1)0 (C

L

0 ; n ) ,

rst order basis: probability P (1) and technical rate i.

If the policy is fully indexed the technical interest rate is set equalto 0.

The net premium reserve at time k = 0 , 1, . . . , n is dened by:

R k =: C D

k

n

j = k +1

(1 + i) ( j k ) P (1)k (C D

j ; j )

+ C Ln (1 + i) (n k ) P (1)

k (C L

n ; n ) .



45/64

Financial risk

To meet solvency requirements the insurer purchases a portfolio of

assets backing the contract. At each date t [0, n ] the market valueA t of the asset portfolio cannot be lower than the technical reserve:

A t R t .

Classical scheme. The insurer is involved in a replicating invest-

ment strategy providing the result At R t for any t.[Brennan, Schwartz, 1976]

Remark. If the functions include minimum guarantees the repli-cating strategy is a dynamic hedging strategy, as prescribed by theoption pricing theory.

Scheme with underlying security. At time 0 let us consider astochastic zcb with maturity n and terminal payoff:

Y n := (0 , n ) .

Assume that the zcb is traded on the market at the price Q t

anassume that:

Q 0 = 1 , at time 0

Q t = (0 , t ), for each t < n

the equality A t = R t is guaranteed if at time 0 the insurer pur-

chases A0 = C 0 units of this zcb.



46/64

In actual contracts the equality Q t = (0 , t ) is obtained bydenition since the price Q t is used as the reference index; that isthe function is dened as:

(0, t ) := Q tQ 0

, t .

In a policy written on an underlying security the insurer is notfaced with investment risk.

If the issuer of the underlying security is defaultable the policyinvolves counter-party risk. This default risk can be faced bythe insurer or by the policyholder, depending on the speciccontractual clauses.

If the price of the underlying security is determined on a non

efficient market, the insurer can incur in losses in case of re-demption if the price Q t is greater than the fair value of thesecurity. This surrender risk can be reduced by stipulating abuy-back ageement with the bond issuer.

Typically the underlying security of the i-l policy is a structured bond which includes minimum return guarantees. If the priceQ t is not efficiently determined the insurer needs an appropriatepricing model in order to control possible deviations of Q t fromits fair value.



47/64


48/64


49/64


50/64

Single period binomial model

Let t = 0 and T = 1 and assume (slightly changing notations) the

following binomial evolution of the undrlying price.

0 1

uF with prob. pF

dF with prob. 1 p

stochastic growth factor , with possible values u or d.Let u > d .

Assume there exists a riskless investment opportunity (the riskless

bond) with interest rate r in [0, 1] deterministic growth factor: m := 1 + r .

We suppose that F pays no dividends and we make the usual perfectmarket assumptions; that is:

no transaction costs, no taxes;

short sales are allowed; the agents are price taker and prefer more to less;

the securities are innitely divisible;

riskless arbitrage opportunities are precluded.

A rst consequence: to prevent arbitrage the following inequalities

must hold:u > m > d .



51/64

The hedging (or replication) argument

Correspondingly to the evolution of F , we have the derivative price

evolution:

0 1

F u = uF with prob. pF

F d = dF with prob. 1 p

D u = g(uF ) with prob. pD D d = g(dF ) with prob. 1 p

Let us consider a portfolio containing units of F and the amountB in riskless bond.The price evolution of this portfolio is given by:

uF + mB with prob. pF + B

dF + mB with prob. 1 p

In order that the portfolio replicates the contingent claim payoff thefollowing equalities must hold:

uF + mB = G udF + mB = G d



52/64

Solving these equations, we obtain:

= D u D d

(u d) F ,

and:B =

u D d d D u(u d) m

.

For these values of anf B the portfolio exactly replicates the ter-minal value of D (the equivalent portfolio , or replicating portfolio ).

To avoid arbitrage the price of this ptf must be equal to the price of the derivative (the low of one price); that is:

D = F + B =

= D u D d

u d +

u D d d D u(u b) m

=

= 1m

m uu d

D u + u mu d

D d .

This equation can be rewritten as:

D = 1m

q D u + (1 q ) D d .

where:q :=

m du d

The value of the derivative D is independent on the natural prob-ability p.

The value of the derivative does not depend on investors attitudetoward risk.c MDF-FM Finance of Insurance vol. 2, p. 128


53/64

The risk-neutral valuation

The contingent claim price can be expressed as discounted the ex-

pectation:D 0 =

11 + r

E Q0 D 1

where E Qt is the expectation operator with respect to the probabilityq , which is referred to as risk-neutral probability.

The expected return of F (with respect to the natural probability p)is given by:

E F :=E 0 [F 1 ]

F 0 1

=( u 1) p + ( d 1)(1 p) .

If the expectation is taken with respect to q one has:

E Q0 [F 1 ]F 0

1 = ( u 1) q + ( d 1)(1 q )

= ( u 1) m du d

+ ( d 1) u mu d

= m 1 = r .



54/64

Valuing a life insurance liability

Single premium pure endowment maturing at time 1, with current

sum insured C 0 an technical rate i.The policy is participating, with reference return I 1 := F 1 /F 0 1and participation coefficient : hence the benet a time 1 is:

Y 1 = C 01 + max { I 1 , i}

1 + i ,

or:

Y 1 = R 1 + max { I 1 , i} ,where R := C 0 / (1 + i).

We have:

0 1

F u = uF with prob. pF

F d = dF with prob. 1 p

I u = u 1 with prob. pI

I d = d 1 with prob. 1 p

Y u

= R [1 + max { (u 1), i}]Y Y d = R [1 + max { (d 1), i}]



55/64

Using the expression:

Y 1 = R 1 + I 1 + R max{i I 1 , 0} ,

we can value separately the linear component (the base compo-nent):

L 1 = R 1 + I 1 ,

and the put component:

P 1 = R max{i I 1 , 0} .

For the base component we have:L u = R [1 + (u 1)] = R [(1 ) + u ] ,

L d = R [1 + (d 1)] = R [(1 ) + d ] .Hence we nd:

L = 1m

q L u + (1 q ) L d

= R 1m

(1 ) + [q u + (1 q ) d] ,

and: L =

L u L d(u d) F

= RF

.

For the put component we have:

P = 1m

q P u + (1 q ) P d ,

and: P =

P u P d(u d) F

,

where:P u = R max{i (u 1), 0} ,

P d = R max{i (d 1), 0} .Since P u P d , then P 0.c MDF-FM Finance of Insurance vol. 2, p. 131


56/64

Valuing the investment gain

Referring to the same policy, we consider the investment gain of the

insurer at time 1, given by:

G 1 = R I 1 max{ I 1 , i} ,

which can be written as:

G 1 = R (1 ) I 1 R max{i I 1 , 0} .

Thus G 1 can be written as the difference between the linear compo-nent:

H 1 = R (1 ) I 1 ,

and the put component:

P 1 = R max{i I 1 , 0} .

The linear component is now given by:

H u = R [(1 ) (u 1)] ,

H d = R [(1 ) (d 1)] .

Hence we nd:

H = 1m

q H u + (1 q ) H d

= R 1m

(1 ) q u + (1 q ) d 1 ,

and: H = H u H d(u d) F = (1

) RF .



57/64

Example

Let:

u = 1 .1 , d = 1 /u, r = 5% , F = 10 (market parameters) ;C 0 = 102, i = 2% (hence R = 100) , = 0 .8 (policy features) .

Under the binomial scheme:

F u = 11F = 10

F d = 9 .09091

I u = 1 .1 1 = 0 .1I

I d = 0 .909091 1 = 0.091

Y u = 100 (1 + max {0.8 0.1, 0.02}) = 108Y

Y d = 100 (1 + max {0.8 0.091, 0.02}) = 102

The risk-neutral probability is:

q = m du d =

1.05 0.9090911.1 0.909091 = 0 .7381 ,

and the value of the insurance liability is:

V (0; Y 1 ) = E Q0 (Y 1 )

1 + r =

1m

q Y u + (1 q ) Y d

= 11.05

0.7381 108 + (1 0.7381) 102

= 11.05

106.4286 = 101.361 .



58/64


59/64

Put component

P u = R max{i I u , 0}

= 100 max{0.05 0.8 0.1, 0} = 0 ,P d = R max{i I d , 0}

= 100 max{0.05 0.8 0.091, 0} = 9 .27273 .

Thus the put value is:

P = 1m

q P u + (1 q ) P d

= 11.05

0.7381 0 + (1 0.7381) 9.27273

= 2 .31293 .

with delta:

P = P u

P d

(u d) F = 0 2.3129311 9.09091 = 4.8571 .

In fact one can obtain:

V = L + P = 99 .0476 + 2 .31293 = 101 .361 .

= L + P = 8 4.8571 = 3 .1429 .



60/64

The investment gain

The investment gain generated by the policy at time 1 can have the

values,G u = R I u max{ I u , i}

= 100 0.1 max{0.8 0.1, 0.05} = 2 ,

G d = R I d max{ I d , i}

= 100 0.091 max{0.8 0.091, 0.05} = 11.0909 .

Therefore the value of the investment gain is negative:

G = 1m

q G u + (1 q ) G d

= 11.05

0.7381 2 + (1 0.7381) 11.0909

= 1.36054 .

with delta:

G = G u G d(u d) F

= 2 + 11 .090911 9.09091

= 6 .8571 .

For the linear component we have:

H u = R [(1 ) I u ] = R [(1 0.8) 0.1] = 2 ,H d = R [(1 ) I d ] = R [(1 0.8) 0.091] = 1.818218.

Hence one obtains:

H = 1m

q H u + (1 q ) H d

= 11.05 0.7381 2 + (1 0.7381) 1.818218

= 0 .95238 .c MDF-FM Finance of Insurance vol. 2, p. 136


61/64

with: H = (1 )

RF

= 0 .2 10 = 2 .

Performing the valuation by G = H P we get:

G = H P = 0 .95238 2.31293 = 1.36054 .

The retained interest H is not sufficient to offset the cost of theminimum guarantee.

Remark. The difference:

E := R V = 100 101.361 = 1.361 ,

is the (investment component of) the VBIF generated by the con-tract.

Remark. For a participation coefficient = 0 .6 one would obtain:

V = 99 .9546 , L = 98 .0952 , P = 1 .8594 , H = 1 .90476 ,

E = R V = 100 99.9546 = 0 .0454 .



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63/64

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De Felice, M., Moriconi, F., Un modello per la progettazione e la valutazione di polizze indicizzate e rivalutabili , Rapporto per lINA, Roma, giugno 1994.

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De Felice, M., Moriconi, F., Finanza dellassicurazione sulla vita. Princip per lasset-liability management e per la misurazione dellembedded value , Gruppo diricerca su Modelli per la nanza matematica, Working paper n. 40, giugno2001a.De Felice, M., Moriconi, F., Lassicurazione tra probabilismo e corporate gover-nance , Assicurazioni, 68(2001b), 3-4.

De Felice, M., Moriconi, F., Salvemini, M.T., Italian trasury credit certicates (CCTs): theory, practice, and quirks , BNL Quarterly Review, 185(1993), 127-168.

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de Finetti, B., Teoria delle probabilit`a , Torino, Einaudi, 1970; english version:

Theory of Probability , Chichester, Wiley, 1974.Gerber, H.U., Life Insurance Mathematics , Berlin, Springer, 1997 3 .

Grosen, A., Jrgensen, P.L., Fair valuation of life insurance liabilities: The im-pact of interest rate guarantees, surrender options, and bonus policies , Insurance:Mathematics and Economics, 26(2000), 1.

Kaufmann, R., Gadmer, A., Klett, R., Introduction to Dynamic Financial Analysis ,Astin Bulletin 31(2001), 1.

Merton, R.C., On the application of the continuous-time theory of nance to nan-cial intermediation and insurance , Geneva Papers on Risk and Insurance, 14(1989),225-261.

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