How To Value Life Insurance In A Stochastic Financial Environment

FAIR VALUAION OF VARIOUS PARICIPAION SCHEMES IN LIFE INSURANCE BY PIERRE DEVOLDER AND INMACULADA DOMÍNGUEZ-FABIÁN ABSRAC Fair valuation is becomin a maor concern for actuaries, especially in the perspective of IAS norms. One of the key aspects in this context is the simultaneous analysis of assets and liabilities in any sound actuarial valuation. he aim of this paper is to illustrate these concepts, by comparin three common ways of ivin bonus in life insurance with profit: reversionary, cash or terminal. For each participation scheme, we compute the fair value of the contract takin into account liability parameters (uaranteed interest rate and participation level) as well as asset parameters (market conditions and investment stratey). We find some equilibrium conditions between all those coefficients and compare, from an analytical and numerical point of view, the systems of bonus. Developments are made first in the classical binomial model and then extended in a Black and Scholes economy. KEYWORDS Fair value, participation scheme, asset and liability manaement. INRODUCION If for a lon time life insurance could have been considered as a sleepin beauty, thins have chaned dramatically as well from a theoretical point of view as from industrial concerns. Nowadays, the financial risks involved in life insurance products are surely amonst the most important challenes for actuaries. he need to update our actuarial backround takin into account the real financial world has been recently emphasized by Hans Bühlman in a recent editorial in ASIN BULLEIN (Bühlman ()). he classical way of handlin financial revenues in life insurance was characterized by two assumptions: stationarity of the market (no term structure of interest rate) and absence of uncertainty (deterministic approach); all this leadin to the famous actuarial paradm of the technical uaranteed rate: all the future was summarized in one maic number. Clearly thins are not so simple and life insurance is a perfect example of stochastic process (even more than non life); the two dimensions of time and ASIN BULLEIN, Vol. 35, No., 5, pp. 75-97

76 P. DEVOLDER AND I. DOMÍNGUEZ-FABIÁN uncertainty are completely involved in the products: lon term aspect and financial risk (Norber ()). he purpose of this paper is to focus on the stochastic aspect of the return and to show how to calibrate technical conditions of a life insurance product, usin classical financial models. In this context fair valuation is the central topic. he comin introduction of accountin standards for insurance products will undoubtedly increase the importance of fair valuation of life continencies. Even if mortality risk can deeply influence the profitability of a life insurer (for instance in the annuity market), it seems clear that the financial risk is the heart of the matter. Pricin and valuation of life insurance products in a stochastic financial environment started first for equity linked policies with a maturity uarantee (Brennan and Schwartz (976), Delbaen (986), Aase and Persson (994), Nielsen and Sandmann (995)). he newness of these products explains probably why historically they were the first studied with stochastic financial models. But more classical products, like insurance with profit, are at least as important for the insurance industry and induce also clearly maor concerns in terms of financial risk. Different models of valuation of these contracts have been proposed based on the classical neutral approach in finance (Briys and De Varenne (997), Bacinello (, 3b)). he importance to develop, in this field, ood models and to analyse all the embedded options, will surely increase with the IAS world (Grosen and Jorensen ()). Even if for competitive or leal reasons the pricin of these products has very often limited deree of freedom, their valuation requires a complete understandin of various risks involved in each product. Amonst them, the relation between the uaranteed rate and the participation level is of first importance, in close connection with the asset side. he aim of this paper is to focus on this aspect of participation, for the one hand by introducin some links between the liability conditions and the asset stratey and, for the other hand, by comparin three classical bonus systems. wo problems are presented: first if all the parameters of the contract are fixed (actuarial valuation of an existin product), formulas of fair value are iven. Secondly, in order to desn new parameters of a product, equilibrium conditions are developed. he paper is oranised as follows. Section presents the main assumptions used in terms of assets and liabilities in a financial binomial environment and introduces three different participation systems: reversionary, cash and terminal bonus. In section 3, a first model on a sinle period is developed, showin how to compute fair values and equilibrium conditions in function of the chosen investment stratey. In this case, it appears clearly that the three participation schemes are identical. hen section 4 extends the fair valuation computation in a multiple period model; each participation system havin then different valuation. Section 5 is devoted to the equilibrium condition in the multiple period; in particular, we show that the conditions are identical for the reversionary and the cash bonus even if in eneral fair values are different as seen in section 3. his motivates to take a deeper look on the relation between fair values in these two systems. Section 6 shows that cash systems have a ber valuation when the contract is unbalanced in favour of the insurer and vice versa. Section 7 illustrates numerically the results and section 8 extends

FAIR VALUAION OF VARIOUS PARICIPAION SCHEMES IN LIFE INSURANCE 77 the results in a continuous time framework, usin the classical Black and Scholes economy. In this case, we show that explicit formulas for the fair values and the equilibrium value of the participation rate are still available; but implicit relations are only possible for the equilibrium value of the uaranteed rate.. NOAIONS AND DEFINIIONS We consider a life insurance contract with profit: in return for the initial payment of a premium, the policyholder obtains, at maturity, a uaranteed benefit plus a participation bonus, based on the eventual financial surplus enerated by the underlyin investments. Insurer is supposed to be risk neutral with respect to mortality. Furthermore, we assume independence between mortality and financial elements. he asset and the liability sides of the product can be characterized as follows:.. Liability side We consider a pure endowment policy with sinle premium p issued at time t and maturatin at time. he benefit is paid at maturity at time t if and only if the insured is still alive. No surrender option exists before maturity. We denote by i the uaranteed technical interest rate and by x the initial ae. he survival probability will be denoted by p x. If we assume without loss of enerality that the initial sinle premium is equal to p x, the uaranteed benefit paid at maturity (disreardin loadins and taxes) is simply iven by: G ] ] i p i p ] x We introduce then the participation liability; we denote by B the participation rate on the financial surplus ( < B ). hree different participation schemes will be compared: Reversionary bonus : a bonus is computed yearly and used as a premium in order to buy additional insurance (pure endowment insurance increasin benefit only at maturity). Cash bonus : the bonus is also computed yearly but is not interated in the contract. It can be paid back directly to the policyholder or transferred to another contract without uarantee. erminal bonus : the bonus is only computed at the end of the contract, takin into account the final surplus. We denote by B, B C and B the correspondin participation rates.

78 P. DEVOLDER AND I. DOMÍNGUEZ-FABIÁN.. Asset side We assume perfectly competitive and frictionless markets, in a discrete time framework with one risky asset and one risk less asset. he annual compounded risk free rate is supposed to be constant and will be denoted by r. he risky asset is supposed to follow a binomial evolution (Cox-Ross-Rubinstein (979)); two returns are possible at each period: a ood one denoted by u and a bad one denoted by d. In order to avoid any arbitrae opportunity, we suppose as usual: d <r ) m: volatility (m > l) u r l m d r l m () On this market, the insurer is supposed to invest a part of the premium in the risky asset and the other part in the risk less asset. A stratey is then defined by a coefficient (with constraint < ) ivin the risky part of the investment. he return enerated by a stratey is a random variable, takin one of two possible values denoted by u and d and defined as follows: u u ( )(r) r (l m) () d d ( )(r) r (l m) (3) he case will not be taken into account. 3. A FIRS ONE PERIOD MODEL We start with the very special case. hen by definition the three ways of ivin bonus, as defined in section., are identical. A contract is characterized by its vector of technical and financial parameters: v (i,b,) (4) he other parameters (r,u,d) can be seen as constraints of the market.

FAIR VALUAION OF VARIOUS PARICIPAION SCHEMES IN LIFE INSURANCE 79 3.. Fair valuation Usin the standard risk neutral approach in finance, the fair value of the contract can be expressed as the discounted expectation of its future cash flows, under the risk neutral measure, and takin into account the survival probability. he risk neutral probabilities, associated respectively with the ood return and the bad return, are iven by: p r d m l - - u - d u - ] r p - p u - d he correspondin liabilities are respectively: m l (5) L i B[u (i)] i B[r i (l m)] (6) L i B[d (i)] i B[r i (l m)] he fair value of a contract v is then defined by: (v) p x (v) with (v) defined as the financial fair value of the benefit and iven by: v ] r! pl m- l m l < i B r i l m i B r i l m r 8 6 - ^ h@ B 8 6 - ^ - h@ BF m- l m l i B r- i ^l mh r- i ^l- mh r < < 6 @ 6 @ FF (7) he financial fair value consists of two parts: (v) G (v)p (v) with G (v): fair value of the uarantee r i (8) P (v): fair value of the participation correspondin to a call option and iven by: P (v) B m - l m l P P r < F (9)

8 P. DEVOLDER AND I. DOMÍNGUEZ-FABIÁN with P [r i (l m)] () P [r i (l m)] o o further, we have to make some assumptions on the values of P and P. Of course, we have P P. he followin situations can happen for a iven contract: Case : P P Case : P r () he contract becomes purely deterministic, and the fair value is then: (v) r i () Case : P <P his case can be considered as the realistic assumption: If the risky asset is up, there is a surplus and participation. But if the risky asset is down, the uarantee is playin and there is no participation. he condition can be written as follows: In this situation, the fair value becomes: r (l m) i < r (l m) (3) m l (v) i B r i l m - r < - ^ ^ hhf (4) Case 3: <P < P In this case, the technical uaranteed rate is so low that even in the down situation of the market, there is a surplus: i < r (m l) (5)

FAIR VALUAION OF VARIOUS PARICIPAION SCHEMES IN LIFE INSURANCE 8 he fair value becomes then: v m- l m l ] i B r i l m r i l m r < < ^ - ^ hh ^ - ^ - hhff (6) i B r-i r 6 ] @ he fair value of the participation is ust based on the difference between the risk free rate and the uaranteed rate. 3.. Equilibrium A vector v of parameters is said to be equilibrated if the correspondin initial fair value is equal to the sinle premium paid at time t : (v) p p x (7) that is equivalent to (v). Since relation (7) implies: (v) r i we have a first eneral equilibrium condition iven by: i r (8) So, in order to obtain equilibrium, the technical uaranteed rate must always be lower or equal to the risk free rate. We try now to look at equilibrium situations in the three cases presented in section 3.. Case : P P In this case, relation () shows that i > r, and no equilibrium is possible. Case : P <P akin into account simultaneously relation (8) and (3), we must have: r (l m) i < r (9) he relation can be written in terms of equilibrum values of each of the parameters of the contract: Guaranteed rate (function of the participation rate and the stratey coefficient): m - l i r B -B ^m -lh ()

8 P. DEVOLDER AND I. DOMÍNGUEZ-FABIÁN with conditions: < B and <. he condition (9) is well respected, takin into account these two conditions. Participation rate (function of the uaranteed rate and the stratey coefficient): B with conditions: < and r (m l) i < r. r i m - l < - r - i ^l mh F () Stratey coefficient (function of the uaranteed rate and the participation rate): he purpose here is to see if, for a couple of coherent values of the technical parameters i and B, there exists an asset stratey eneratin an equilibrium situation: r i B m l B - - ^ - h m - l () with conditions: for i < r and B < m - (i.e. for all B since l m - l >) B _ m - l i b) if i r -B ^m -lh Case 3: <P < P he equilibrium condition on the fair value becomes then, by (6): i B r i - r 6 ] @ or B(r i) r i Because in this case i < r, this implies B All this development shows that the real interestin situation with a non-trivial solution is the case. 4. FAIR VALUAION IN MULIPLE PERIOD MODELS We extend here the computation of fair values for a eneral maturity. he three participation schemes defined in section.. must now be studied separately. 4.. Reversionary bonus akin into account the binomial structure of the returns, the total benefit to be paid at maturity is a random variable iven by B() L L, where is

FAIR VALUAION OF VARIOUS PARICIPAION SCHEMES IN LIFE INSURANCE 83 the number of up cases of the risky asset amonst the years and L and L are the correspondin total returns. he fair value is then iven by: C p p L L p pl pl - - ] r xb x l! b r l 6 v p p 6 ] v@ x @ (3) where C denotes the binomial coefficient and (v) is the financial fair value on one year (cf. (7)). 4.. Cash bonus In this case, each year, the rate of bonus is applied only to the reserve accumulated at rate i and takin into account the survival probabilities. For an initial sinle premium equal to p p x, the reserve V(t) to use at time t is iven by: V(t) p( i) t / t p x p x ( i) t / t p x he part of the liabilities to be paid at time t ( t,,...,) in case of survival as cash bonus is a random variable iven by: in up case: C (t) B c P ( p x ( i) t / t p x ) in down case: C (t) B c P ( p x ( i) t / t p x ) he fair value can be expressed as discounted expected value of all futures cash flows under the risk neutral measure and takin into account the survival probabilities: x! t t ] v p b r i p pc t p C t l b t x^ r l ] ] h p r i i B p p P p P x C x ] b l! ^ h t ] r p ] v x t- t where (v) is the financial fair value and is iven by: v r i r i B p P p P C ] - ] ] b l ^ h ] r r- i (4) p, p and P, P bein defined in (5) and () In the particular case P ;P >, this ives: v ] ] r ^m-lh^ r- i ^l mhh ] r -] > ] BC H r- i (5)

84 P. DEVOLDER AND I. DOMÍNGUEZ-FABIÁN he correspondin value for the reversionary bonus is (cf. (3)): ] v ] r ^m -lh^r- i ^l mhh i B G (6) As expected, for these two values are indentical. A deeper comparison between (5) and (6) will be developed in section 6, after calculation of equilibrium values in section 5. 4.3. erminal bonus he bonus is only computed at the end of the contract, comparin the final technical liability ( i) with the asset value at maturity. If we denote by A () the terminal assets at time t, usin stratey, the fair value can be written as follows: ] v ] r p x px i B EQ A - i ] v 9] 7 ] ] A C where Q is the risk neutral measure and E Q denotes expectation under Q. Alternatively, we can express this fair value in terms of option price: ] ] v B C_ A; ; ] i ] r i (7) where C(A ; ;(i) ) is the price of a call option on the asset A, with maturity and strike price ( i) akin into account the binomial structure of the model, the eneral form of this price is iven by: C_ A - ; ; i - ] i! 8ud - ] C pp r he fair value is then iven explicitly by: ] ] ] B 9 B C - - ] v! ud - ] C pp r r ] 8 B (8) As example, let us look at the model on two periods of time ( ). If we denote by a the minimal number of umps in order to ive participation: a inf{ N :(r (l m)) ( r (l m)) >(i) }, the followin situations can happen:

FAIR VALUAION OF VARIOUS PARICIPAION SCHEMES IN LIFE INSURANCE 85 First situation: a > he technical uaranteed rate is so b that no participation can be iven 8] $ u B. he fair value is ust: ] v r i : D Second situation: a he only case where bonus can be iven is the situation of two up umps of the risky asset: u d ( i) ] B d n 8u i - ] BH hird situation: a As soon as there is one up ump on the two periods, bonus is iven: d ( i) ] B > d n u i r _ - ] i ] V m - l _ ud - ] ih W ^ mh W X (3) Fourth situation: a A bonus is iven each year whatever are the returns of the asset: he fair value is simply: ] v d >(i) (3) ] ] r B < - b r i l F (33) he first and the last situations can be considered as deenerate. If we want to avoid these limit situations, we must have: d ( i) < u (34)

86 P. DEVOLDER AND I. DOMÍNGUEZ-FABIÁN We are then in situation or situation 3. Remark: It is easy to see that we will be automatically in situation 3 (a ) if: m < l l( r) (35) and i < r i. he first condition (35) is independent of the characteristics of the contract; it ust means that the volatility has not to be too important. ii. he second condition [i < r], related to the contract, seems to be quite reasonable (cf. (8) on one period). 5. EQUILIBRIUM RELAION IN MULIPLE PERIOD MODELS he aim of this section is to eneralize on periods the equilibrium relations obtained in section 3.. for one period, usin the explicit formulas of fair value obtained in section 4. 5.. Reversionary bonus Formula (3) shows clearly that: (v) (v) Equilibrium results obtained in section 3.. are unchaned in a multiple period model with reversionary bonus. 5.. Cash bonus Formulas (5) and (6), for instance, show that the fair values are normally different usin the reversionary bonus or the cash bonus. Nevertheless, we will see that the equilibrium values of the parameters of the contract are the same. Usin (4), the contract will be equilibrated in the cash bonus scheme if: 7] r - ] A ( r) (i) B C ( p P p P ) ] r- (36) Like in section 3.., we can consider different cases, dependin on the values of P and P.

FAIR VALUAION OF VARIOUS PARICIPAION SCHEMES IN LIFE INSURANCE 87 Case : P P No equilibrium is possible like in the reversionary scheme. Case : P <P (central assumption) Usin the values (5) and () of P and P, formula (36) becomes: ( r) (i) B m- l 7] r - ] A C ( r i (l m)) ] r- or r i B m - l C [r i (l m)] m - l or i r B C m - B ^ m - l h c (37) identical to relation () obtained for the reversionary bonus. Case 3: <P < P Formula (36) becomes then: ( r) (i) B C 7] r - ] A which implies like in the reversionary case: B C Conclusion: he equilibrium conditions on the parameters of the contract are the same for the reversionary bonus and for the cash bonus. 5.3. erminal bonus Formula (8) ives as equilibrium condition: ] ] r B ] r! - - - ] 8ud i B C pp (38) From this relation, it is already possible to obtain an equilibrium value for the participation rate, as a function of the uaranteed rate and the stratey coefficient (to be compared with formula () for the reversionary bonus): B ] r - ]! 8 ] B - - ud - C pp (39) On the other hand, if the participation rate is known, it is possible to express the equilibrium value of the uaranteed rate as follows:

88 P. DEVOLDER AND I. DOMÍNGUEZ-FABIÁN ] i - - r B! u d C p p a - - B! C p p a ] - m - l m l ] - d d - - r B! C u d n n a - m - l m l - B! C d n d n a (4) where a is defined as usual by: a inf{k N : u k d k >(i) } Unfortunately, this relation is not explicit because the coefficient a depends on the level of i. he relation can be computed, assumin a certain value for a and then check afterwards if the condition on a is fulfilled. As example, let us see aain what happens on two periods. We first concentrate on the standard situation a ; this means that constraint (3) has to be fulfilled. Startin from (4), we will et an equilibrium candidate for i and check afterwards this constraint. For and a, we et: ] m - l m - l ] r - B > d n u u d H ^ mh m - l m - l - B > d n H ^ mh (4) with the followin condition to check: d ( i) < u d Similary, we can try to find a candidate for the situation a ; now constraint (9) has to be checked. Usin the same approach, the equilibrium value is iven by: ] m - l ] r -B d n u m - l - B d n (4) with the followin condition to check: u d ( i) < u

FAIR VALUAION OF VARIOUS PARICIPAION SCHEMES IN LIFE INSURANCE 89 6. ANALYICAL COMPARISON BEWEEN REVERSIONARY AND CASH SYSEMS Reversionary and cash systems seem not so different; the only difference is the interation of the bonus inside the contract. Moreover, we saw in section 5 that the equilibrium conditions are the same, even if the fair value formulas look quite different. From now on, and without loss of enerality, we will work without mortality effect. he aim of this section is to prove the followin relation: (reversionary) > (cash) if and only if the participation rate is reater than its equilibrium value. In order to et this relation, we have to compare the valuations in the two participation schemes usin the same parameters: For the reversionary bonus (cf. (3)): with K [p P p P ] pl @ ] v b pl i BK r l 6 b r l 6] @ For the cash bonus (cf. (4)): v r i r i B K ] - ] ] b l C ] r r- i Developin the reversionary formula ives: - ] v ]! C ] BK ] r ] > H he condition to have a ber fair value for the reversionary system becomes then: ] ] r ] r! - C BK i > i ] ] b l r BK ] r -] ] r r- i Or: ] r - ] - -! C ] BK ] > r- i Or assumin K! (i.e. P > ): -! C ] BK ] - ] BK ] BK - ] BK ] r - ] r- i > (43)

9 P. DEVOLDER AND I. DOMÍNGUEZ-FABIÁN Let us consider the function: x i ] - ] G(x) x with ] r - ] For x r i, we have: G(r i) r- i On the other hand, it is easy to show that for x >, the function G is strictly increasin. So when condition BK > r i is fulfilled, the fair value for the reversionary bonus is ber than the fair value for the cash bonus and vice versa. his last condition can be written as follows: B[p P p P ] > r i (44) akin into account the different cases studied in section 3., condition (44) becomes: Case : P P : not relevant here (K! ). Case : P m- l Bp P B m [r i (l m)] > r i r i m - l < - r- i ^l mh F which means that the participation rate is ber than its equilibrium value (cf. ()). Case 3: r i Or B > that is the equilibrium value in that case. 7. NUMERICAL ILLUSRAION In this section, we will compare the three participation schemes in terms of fair values and equilibrium values of the parameters in a two periods model and without mortality effect. In terms of financial market, we will use a central scenario based on the followin values: r riskfree rate.3 l risk premium. m volatility.6

FAIR VALUAION OF VARIOUS PARICIPAION SCHEMES IN LIFE INSURANCE 9 FIGURE In terms of investment stratey underlyin the product, we will mainly compare two choices:.: conservative stratey.6: aressive stratey 7.. Fair values for different uarantees and participation levels Fure shows for each chosen stratey (conservative or aressive) the relation between the uaranteed rate, the participation level and the fair value of the contract. he fair value is clearly an increasin function of the participation level, whatever is the participation scheme. 7.. Equilibrium values Like seen in section 5., the equilibrium conditions on the parameters of the contract are the same for the reversionary bonus and for the cash bonus. Fure compares, for the aressive investment stratey, the equilibrium values of the participation rate and of the uaranteed rate between reversionary and terminal bonus. 7.3. Fair value and equilibrium value able compares for various values of the technical parameters, chosen in relation with their equilibrium values, the fair values in the three participation schemes for the aressive investment stratey (6% in risky asset). Fures in bold and italic correspond to situations where the initial fair value is ber than the paid premium.

9 P. DEVOLDER AND I. DOMÍNGUEZ-FABIÁN FIGURE ABLE 6% i.5 i.5 i.6 Equilibrium values of B ERMINAL BONUS interest rate,76 %,995485,978396,96357,45 4%,6553,9856995,973645,4 6%,5856,99363,985763,47 8%,9958,33,99788,6 %,666,7699 REVERSIONARY BONUS,657 %,99754,97939,9654,6 4%,469,98755,97858,8 6%,933,995345,986767,54 8%,783,3498,99348,6 %,47467,6844 CASH BONUS,657 %,99759,97988,966363,6 4%,43,987777,977,8 6%,8473,9953737,9888,54 8%,7695,34696,9949,6 %,45357,5656 Clearly, the difference between the different bonus schemes would be much more pronounced in models on more than two periods. 8. GENERALIZAION IN CONINUOUS IME MARKE he principles of comparison between the different participation schemes can be easily adapted in a continuous time financial market. We develop here the

FAIR VALUAION OF VARIOUS PARICIPAION SCHEMES IN LIFE INSURANCE 93 model, usin the Black and Scholes environment. he classical assumptions on the market are supposed to be fulfilled. wo kinds of assets are supposed to exist: the risk less asset X, linked to the risk free rate: dx (t) rx (t)dt where r ln(r) is the instantaneous risk free rate. the risky fund X, modelled by a eometric Brownian motion: where w is a Wiener process. dx (t) X (t)dt sx (t)dw(t) Once aain, the risk neutral probability measure will be denoted by Q. he reference portfolio of the insurer consists of a constant proportion invested in the risky fund and a proportion ( ) invested in the risk free asset. he evolution equation of this portfolio denoted by X is then iven by: dx (t) ( ( )r)x (t)dt sx (t)dw(t) (45) 8.. he one period model We extend here the results of section 3 obtained in a binomial environment. On one period, the three kinds of participation schemes are identical. We compute the fair values for a iven contract v (i,b,) and obtain equilibrium conditions on the coefficients in order to have a fair contract. he fair value is iven now by: v p,, ] r i Bc i x b ^ hl (46) where c(i,,) represents the price of a call option on the reference portfolio X for one period and for a strike price equal to the uarantee ( i). In the Black and Scholes environment this price is iven by: where c^ i,, h r E X i Q `^ ] - ] h F d i,,,, r i F d i - ^ ^ hh ^ ^ hh d i,, ln r s / s i ^ h b b l l d ^i,, h d ^i,, h - s F is the cumulative distribution function of a standard normal variable.

94 P. DEVOLDER AND I. DOMÍNGUEZ-FABIÁN Finally the fair value can be written as follows: v p,,,, r i B F d i r i F d i x - ] b b ^ ^ hh ^ ^ hhll (47) he equilibrium condition iven by (7) can be expressed, in this model, as an explicit equilibrium value B of the participation rate, for a iven uaranteed rate i and a iven stratey coefficient (equivalent of formula () in the binomial model): B r- i (48) ] rf^ d ^i,, hh - ] F^d ^i,, hh Implicit relations can only be obtained in this model if we want to solve it for the two other parameters (equilibrium value respectively for the uaranteed rate and for the stratey coefficient). 8.. Fair value in multiple period models 8... Reversionary bonus Exactly as in the discrete case and takin into account the structure of the return process, the fair value for a contract of periods usin a reversionary bonus is iven by: ] v p,,,, r i B F d i r i F d i x - b b ^ ^ hh ^ ^ hhll (49) 8... Cash Bonus he fair value is expressed as the discounted expected value of all future cash flows under the risk neutral measure Q and the survival probabilities: v p p t Q ] xb t x l! t r where CB(t) is the cash bonus paid at time t: r i E ^CB] th ] t- px EQ^ CB] th Bcd ] n r c i,, p ] ^ h Finally, the fair value is iven by: r p,, xe C v ] b r i B r c i l ] ^ h ] t x ] - ] r r- i o (5) 8..3. erminal Bonus he fair value will have the same structure as in the one period model:

FAIR VALUAION OF VARIOUS PARICIPAION SCHEMES IN LIFE INSURANCE 95 ] v p,, r i B c i xd b l ^ hn (5) with: where Q a_ cî,, h b E X - i r l ] ] i k F^d i,,,, r i F d i ^ hh - b l ^ ^ hh d i,, ln r s / s i ^ h b b l l d î,, h d î,, h - s 8.3. Equilibrium relation in multiple period models 8.3.. Reversionary Bonus Accordin to formula (49) and like in the binomial model, the equilibrium condition is the same as in the one period model. 8.3.. Cash Bonus Usin formula (5), the equilibrium condition becomes for the cash bonus: r i r i B r c i,, C ] - ] b l ] ^ h ] r ] r- or: B C r- i r- i ] rc^ i,, h ] rf^ d î,, hh - ] F^d î,, hh which is aain equal to the equilibrium value on one period (cf. (48)). 8.3.3. erminal Bonus Usin formula (5), the equilibrium condition for the terminal bonus becomes: B ] r - ] ] r F^ d î,, hh - ] F^d î,, hh (5) 8.4. Comparison between reversionary and cash systems A same methodoloy as in section 6 can be used in order to obtain a rankin between fair values for reversionary and cash bonus when the parameters are

96 P. DEVOLDER AND I. DOMÍNGUEZ-FABIÁN not in equilibrium. Indeed usin respectively formulas (49) and (5), the fair values can be written as follows: in the reversionary case: in the cash case : ] v v p i BK* ] x ] ] r with: K* ] rc^i,, h p * r i r i B K ] - ] eb l ] r ] r- i o x C which have exactly the same form as in the binomial case. So the same conclusion can be drawn. 9. CONCLUSION In this paper, we have developed various formulations in order to compare the fair value for life insurance products based on three participation schemes: reversionary, cash and terminal bonus, takin into account simultaneously the asset side and the liability side in a multiple period model. We have shown that the fair value depends on the investment stratey (and on the associated risk), on the participation level and on the uaranteed rate but also on the bonus system chosen. We have found some explicit equilibrium conditions between all these parameters. A deep comparison has been made between the three participation schemes, as well in terms of computation of the fair value as in the equilibrium conditions. Usin first a binomial model, we have obtained closed forms and iven clear interpretations on the link between the market conditions, the volatility of the assets and the parameters of the product. A same approach, leadin to similar conclusions, has been proposed in a time continuous model. he model could be also extended in order to take into account other aspects like surrender options, periodical premiums or the lonevity risk. REFERENCES AASE, K.K. and PERSSON, S.A. (994) Pricin of unit-linked life insurance policies, Scandinavian Actuarial Journal,, 6-5. BACINELLO, A.R. () Fair pricin of Life Insurance participatin policies with a minimum interest rate uaranteed, ASIN Bulletin, 3(), 75-98. BACINELLO, A.R. (3a) Fair valuation of a uaranteed life insurance participatin contract embeddin a surrender option, Journal of Risk and Insurance, 7(3), 46-487. BACINELLO, A.R. (3b) Pricin uaranteed life insurance participatin policies with annual premiums and surrender option, North American Actuarial Journal, 7(3), -7. BRIYS, E. and DE VARENNE, F. (997) On the risk of insurance liabilities: debunkin some common pitfalls, Journal of Risk an Insurance, 64(4), 673-694.