DEVELOPMEN AND PRICING OF A NEW PARICIPAING CONRAC Carole L. Bernard,* Olivier A. Le Courtois, and François M. Quittard-Pinon ABSRAC his article desins and prices a new type of participatin life insurance contract. Participatin contracts are popular in the United States and European countries. hey present many different covenants and depend on national reulations. In the present article we desin a new type of participatin contract very similar to the one considered in other studies, but with the uaranteed rate matchin the return of a overnment bond. We prove that this new type of contract can be valued in closed form when interest rates are stochastic and when the company can default.. INRODUCION Life insurance contracts, especially participatin contracts, are the subject of a hue literature. Because these contracts bear many covenants (e.., surrender options, bonus options) and are subject to many potential risks (e.., interest rates, default, leal and mortality risks), buildin valuation tools and methods to price and manae them is important. he rise of new accountin standards, with the implementation by the International Accountin Standard Board (IASB) of International Accountin Standards (IAS) 3 and 39 and of International Financial Reportin Standard (IFRS) 4, suests that fair value has become the key concept of corporate finance and insurance theory. In that sense this evolution calls for a stroner interation of financial and actuarial methods. Briys and de Varenne (994, 997a) built a simple framework for the pricin of life insurance contracts that has been used often in the literature (see, e.., Grosen and Jørensen 00). hey value life insurance contracts in a stochastic interest rate environment and take into account the default risk of the issuin company. Nevertheless, as in the Merton model of a firm s capital structure, default here can occur only at the contract maturity. his is an obvious limitation of their framework. Briys and de Varenne (997b) in a different manner valued risky debt in a stochastic interest rate environment where the default barrier is stochastic, say, proportional to a zero-coupon bond. his model, which is an extension of the Black and Cox model where default can occur at any time, is particularly interestin from a financial viewpoint. o the best of our knowlede, this approach has not yet been used in the insurance domain. Since the articles of Briys and de Varenne, many papers have been written on participatin contracts. Amon these contributions Miltersen and Persson (003) should be noted, who provide closed-form formulas for uaranteed investment contracts with participatin covenants. Bacinello (00) values life insurance contracts with binomial trees. his method proves to be very useful in dealin with various actuarial features: for example, mortality, surrender options, minimum uarantees, annual participatin bonuses, and periodic premia. anskanen and Lukkarinen (003) studied participatin life insur- * Carole L. Bernard obtained her PhD at the University of Lyon ; address: 55 chemin des Peyrières, 0700 Boulieu les Annonay, France; e-mail: carole.bernard@netcourrier.com. Olivier A. Le Courtois, PhD and Series 7, is an Associate Professor of Finance at the EM Lyon Graduate School of Manaement; address: 3 Avenue Guy de Collonue, 6934 Ecully Cedex, France; e-mail: lecourtois@em lyon.com. François M. Quittard-Pinon, PhD, is a Professor of Finance at the University of Lyon ; address: 50 Avenue ony Garnier, 69007 Lyon, France; e-mail: fquittard@wanadoo.fr. 79
80 NORH AMERICAN ACUARIAL JOURNAL, VOLUME 0, NUMBER 4 ance contracts in a Black and Scholes framework. In that case the insured has the riht to surrender at some prespecified dates and is able to chane the reference portfolio. he pricin is done via numerical schemes. One can also note Ballotta (005), who evaluated with-profit contracts based on an asset modeled by a jump diffusion process. In conclusion, a hue diversity of products, models, and numerical methods exist with which to tackle the challene of valuin fair value life insurance contracts. In Bernard, Le Courtois, and Quittard-Pinon (005), some participatin contracts are priced under both default risk and a stochastic term structure of interest rates. hese authors represent the firm assets by a standard eometric Brownian motion correlated with the drivin factor of the interest rates. heir interest rate framework is the Vasicek one, within which an exponential volatility structure is used. o solve their problem and avoid heavy numerical schemes, they rely on the extended Fortet (943) method first developed in finance by Collin-Dufresne and Goldstein (00), which is based on the implementation of recurrence equations. It should be noted that althouh use of the extended Fortet s equations allows the latter authors to price participatin contracts quite quickly (up to a few minutes of computation time) compared to Monte Carlo methods, their numerical scheme is not instantaneous yet. In this article we construct a new type of contract close to the one presented in Bernard, Le Courtois, and Quittard-Pinon (005) but where the minimum uaranteed rate is of a different nature. We assume that, under this new contract, the life insurance company uarantees a overnment rate to the insured, in other words, a stochastic interest rate: if and when the company defaults, the covenant uarantees a sum proportional to the value of a overnment bond, whose value is unknown at that time by definition. We shall denote these new contracts as Government Rate Guarantee Participatin Life Insurance Contracts (GP-LICs). In a similar fashion, the more classical contracts studied by Bernard, Le Courtois, and Quittard-Pinon will be denoted as Constant Rate Guarantee Participatin Life Insurance Contracts (CP-LICs). he contribution of this article is to show that the introduced new contracts, or GP-LICs, can be priced in closed form under a Vasicek term structure of interest rates. We also do an empirical analysis of this contract s prices and sensitivities.. DESIGN OF A NEW CONRAC We shall study in this article a particular type of contract (GP-LIC) that is an extension of the participatin contract (CP-LIC) studied in Bernard, Le Courtois, and Quittard-Pinon (005). he oriinality of this new contract lies in its minimum uarantee, which is proportional to the value of a overnment bond and hence stochastic.. Contract Payoff We classically assume that the funds raised by the insurance company constitute its assets and are modeled by a lonormal diffusion. From the proceeds made by the company, a part is distributed as a minimum rate and another part as a bonus or participatin interest on the financial success of the firm. he oriinality of a GP-LIC lies in its minimum uaranteed rate. We assume that the company uarantees a rate proportional to that of a set of overnment zero-coupon bonds maturin, as a first step, at the same time as the contract. Indeed, if the company defaults, the insured recover the amount they would have obtained by investin initially in overnment bonds, times a proportionality coefficient. We suppose that the insured invest the initial capital L 0 in the participatin policy. Leavin aside the participatin bonus temporarily, the minimum uarantee is equivalent to buyin L 0 /P(0, ) overnment zero-coupon bonds havin an initial value equal to P(0, ) at time 0. Here is chosen inferior to one, so as to allow the company to uarantee a minimum rate inferior to the overnment rate. Indeed, at maturity this position is worth L 0 /P(0, ) contracts, whose value is equal to P(, ) ; in other words, it is worth l L 0 /P(0, ). Provided P(0, ), the insured have a minimum amount uaranteed at time that is superior to their initial investment L 0. Now, if the company defaults at time t, a minimum amount L 0 P(t, )/P(0, ) is uaranteed (value at time t of L 0 /P(0, ) l t
DEVELOPMEN AND PRICING OF A NEW PARICIPAING CONRAC 8 zero-coupon bonds maturin at time ) by the covenant. his means in particular that we are nelectin for the time bein and at this level the additional bankruptcy costs that may occur at time t. Let A be the assets of the firm, modeled as a lonormal process. he initial contribution of the insured, L 0, satisfies the followin relationship: L 0 A 0, where A 0 is the initial value of the assets. For the initial amount of equity, it readily verifies E 0 ( )A 0. he insured receive at maturity of their contracts, provided the company did not bankrupt in the meantime, A if A l () l if l L A l l (A l ) if A, l where, in the first situation, the company has defaulted at maturity, and the assets are returned to the insured; whereas in the second situation, the company performs correctly, and the insured receive the uaranteed rate defined above. Finally, in the third situation, an additional participatin rate is redistributed. Note that in the discriminatin value l /, where the participatin rate starts bein distrib- uted, the coefficient appears, puttin forward the equitable distribution of benefits between the insured and the equityholders. Initially, the insured possess the amount A 0,orL 0. At maturity, one simply has to compare A with l, to decide whether a participation rate is added to the uaranteed rate. o sum up, the final payoff to the insured can be written as L() l (A l ) (l A ), (.) where one adds to the sum promised to the insured a bonus option correspondin to a participation in the benefits of the company and a put option directly related to the terminal default risk of the issuin company. A first idea would be to value these contracts by takin the risk-neutral expectation of the above payoff. his would essentially mean computin a linear combination of standard European options whose closed-form formulas could be easily obtained. In the next pararaph, we shall extend this contract to a settin where default can happen at any time (put differently, we shall move from a Mertontype model to a Black and Cox-type model).. A More Refined Default Model A life insurance company must be able to honor its commitments to the insured. It should be solvent at any time, and not only at the maturity of the issued contracts (this corresponds to assumin classically that a covenant forces the company to be solvent at any time even if it pays back the insured only at a fixed date in the future). In the case we consider, the insured has a minimum amount uaranteed l t at any time t if the company defaults early. Let us recall that this is the value at time t of L 0 /P(0, ) zero-coupon bonds maturin at time. Of course, in the real world, this is a basis for the calculation of what is really oin to be distributed to the insured upon default, because bankruptcy costs will have to be taken into account. Default occurs when the level of the assets is not sufficient to reimburse the insured. Let be the company s early default time. It may be written as inf{s /As l s}, (.) where is a proportionality coefficient whose meanin is that the manaers cannot fully anticipate bankruptcy and declare it when the assets attain a level ls (which is inferior to ls at time s). Now, let be the bankruptcy costs parameter: upon default, a fraction of the assets are wasted to cover various costs such as court fees. he amount of assets remainin at default l has to be diminished
8 NORH AMERICAN ACUARIAL JOURNAL, VOLUME 0, NUMBER 4 usin to obtain the residual amount l that will be redistributed to the insured. In full enerality, and should be estimated from past records on the assets values and recovery rates of defaultin life insurance companies. he payoff to the insured at default time, assumin and, may be written as L() l. (.3) he eneral pricin formula for our contract then can be established usin the above expressions L () and L (), iven in equations (.) and (.3). Denotin by r the risk-free interest rate process, one readily has under the risk-neutral measure Q the formula allowin one to compute V, the value of a GP-LIC: his equation can be written explicitly as 0rs ds 0rs ds Q L L V (0) [(e ()) (e ()) ]. 0 rs ds 0 rs ds Q V (0) [e (l (A l ) (l A ) ) e l ]. (.4) o do the valuation of this uarantee, one has to postulate some dynamics for the interest rates and assets, which we will do in the followin subsection..3 Assets and Interest Rate Dynamics We set ourselves in a eneral framework where we need to know for our study the forward-neutral dynamics of the assets A t and the zero-coupon bonds P(t, ). We assume that the assets follow a lonormal dynamics correlated to the interest rates, which themselves possess an exponential volatility structure P. he interest rate model considered here is driven by a unique factor, correlated to the one of the assets, as mentioned before. Settin 0 and a 0, the volatility structure is expressed simply as a(t) P(t, ) ( e ). a Under the risk-neutral measure Q, the dynamics of the assets A t and the zero-coupon bond P(t, ) are expressed as and da t A t Q rt dt dz (t) (.5) dp(t, ) Q rt dt P(t, ) dz (t), P(t, ) where Z Q Q (t) and Z (t) are standard Q-Brownian motions with a correlation coefficient equal to. Q Q Q Q Let us now construct a Brownian motion Z independent from Z, that is, such that dz dz 0. It is possible to split up Z Q into the two followin components: Q Q Q dz (t) dz (t) dz (t). We have therefore decorrelated the pure interest rate risk from the other sources of risk. he dynamics of the assets iven in equation (.5) now can be reexpressed as da t Q Q t A t r dt (dz (t) dz (t)). Recall that the Radon-Nikodym density allowin one to build the forward-neutral measure Q is defined by
DEVELOPMEN AND PRICING OF A NEW PARICIPAING CONRAC 83 dq Q 0 P (s, ) dz (s)(/) 0 P(s, ) ds e. dq In this case the short-term interest rate dynamics obey the relationship Q dr a( r )dt dz (t), t t t where t /a ( e a(t) ) and we have defined a new Brownian motion satisfyin under Q Q Q the relationship: dz dz P (t, ) dt. Q Q Q We also define Z such that Z and Z be noncorrelated Q -Brownian motions. he dynamics of A t and P(t, ) under Q are written as Q Z and da t Q Q t P A t (r (t, )) dt (dz dz ) dp(t, ) Q (r (t, )) dt (t, ) dz t P P. P(t, ) After interatin these two dynamics, one obtains and t t A 0 Q Q t P 0 0 A exp ( (u, t) ) dz (u) dz (u) P(0, t) (u, t) (u, )( (u, t) ) t P 0 P P du t t P(0, ) Q P P P P 0 0 P(t, ) exp ( (u, ) (u, t)) dz ( (u, ) (u, t)) du. P(0, t) Finally, note that the followin dynamics will also be useful: t t A A P P(t, ) P(0, ) 0 0 exp ( (u, ) ) dz (u) dz (u) t 0 Q Q t P 0 (( (u, ) ) ( )) du (.6) Let us now see the exhibition of the main formulas ivin the price of a GP-LIC in the settin just defined..4 Main Formulas We are now able to ive a eneral valuation formula for our uarantee. Startin from equation (.4), and movin toward the forward-neutral world, one obtains (see the Appendix for more details) V (0) P(0, ) Q [(l (A l ) (l A ) ) l ]. (.7) We wish to write the above formula in a simplified form: and we define for this purpose: V (0) P(0, )[GF BO PO LR], (.8) GF l ( E ) 6 7 3 PO l (E8 E 4) E9 E 5 LR l E, BO (E E ) l (E E ) where the fundamental contributions to the contract s value express as follows:
84 NORH AMERICAN ACUARIAL JOURNAL, VOLUME 0, NUMBER 4 E [A Q A (l /) ] l E 3 Q A, E4 Q [A l, ] E [A 5 Q A l ] E Q [ ] E [A 6 Q A (l /)] E Q 7 l A E8 Q [A l ] E [A 9 Q A l ]. he next section will compute explicitly the nine contributions defined above. 3. CONRAC VALUAION We start by detailin the mechanics allowin to compute the first subcontract term, E. 3. Computation of E E is the probability that bankruptcy occurs before. Usin equation (.), this is the probability that A u crosses L 0 /P(0, ) P(u, ), in other words, that the process A u /P(u, ) crosses the barrier L 0 /P(0, ) before, or written more explicitly: A u P(u, ) u[0, [ E Q inf l. Start by notin that equation (.6) is written as Au A0 Nu(/)(u) e, P(u, ) P(0, ) where the differential of N is defined by and the quadratic variation of N is Q Q dns ( P(s, ) ) dz (s) dz (s), u (u) N u [( P(s, ) ) ( )] ds. (3.) 0 he key of the computation of E is the Dubins-Schwarz theorem (see, e.., Karatzas and Shreve 99), which states that there exists a unique Q -Brownian motion B such that u [0, ], Nu N0 B (u). Usin this representation theorem, the searched probability becomes A u P(u, ) u[0, [ A 0 Nu(/)(u) Q min e l P(0, ) u[0, ] P(0, )l B(u) (/)(u) Q min (e ) A u[0, ] 0 E Q inf l Q min Bs s ln(). s[0,( )] It appears from this formula that we need to know only the law of the minimum of an arithmetic
DEVELOPMEN AND PRICING OF A NEW PARICIPAING CONRAC 85 Brownian motion to compute E (it can be found, e.., in Jeanblanc, Yor, and Chesney 006). One then obtains ln() () ln() () () () E. o simplify notation, we use the auxiliary functions and defined by ln(x) () ln(x) () (x), (x), () () where denotes the cumulative standard normal distribution function. In this settin one finally obtains for E : E () (). We shall now see that the computation of E, thouh different from the one of E, follows readily, based on the same tools and principles. 3. Computation of E Let us first recall the expression of E : E [A Q {A l /} ] [A Q {A l /} inf ]. u[0, ](A u/p(u, ))l o compute this formula, two steps are in order. First, one has to apply Girsanov s theorem to move from the probability Q to a new probability Q allowin us to simplify reatly the above expression (by ettin rid of A ). hen, another application of the Dubins-Schwarz theorem under the new probability Q will help us reach E. Define the followin Radon-Nikodym measure: dq Q Q exp ( P(u, ) ) dz (u) dz (), dq 0 0 where () is defined as in equation (3.). hanks to Girsanov s theorem, it is possible to construct under Q the two standard Brownian motions Z and Z defined by and dz (s) dz Q 0 P s (s) ( (u, ) ) du s Q 0 dz (s) dz (s) du. Let us express A under Q: A0 P P(0, ) 0 0 A exp ( (u, ) ) dz (u) dz (). hen the expression of E becomes Define the martinale H as l E Q A ; inf l. A0 Au P(0, ) u[0, [ P(u, )
86 NORH AMERICAN ACUARIAL JOURNAL, VOLUME 0, NUMBER 4 s s H ( (u, ) ) dz (u) dz s P. 0 0 Note that the quadratic variation of H is equal to the one of N; therefore, we denote it by. Due to the Dubins-Schwarz theorem, there exists a Q -Brownian motion B that satisfies his allows simplifyin E as H B. s A lp(0, ) lp(0, ) 0 ( ) s P(0, ) A0 s[0,( )[ A0 E Q B () ln, inf B s ln. (s) Finally, usin the joint law of an arithmetic Brownian motion and its infimum, one obtains A0 E (). P(0, ) Applyin the same method, we are able to compute the expressions of E 3, E 4, and E 5. 3.3 Final Results As far as the five first terms are concerned, they are obtained followin the above methodoloy, and write as E () () A0 E () P(0, ) 3 E ( ) (3.) E4 () ( () ()) A 0 5 E [ ( ) ( ( ) ( ))]. P(0, ) he four last terms, where does not appear explicitly, are expressed readily as simple Gaussian functions: l E6 M;V ;, E7 M ln l ln(l ) M 8 9 E, E (M ; V ;l ), V V (3.3) where A follows a lonormal law with moments M and V, and where and are defined by and m ln(a) X (m; ; a) [e exa] expm
DEVELOPMEN AND PRICING OF A NEW PARICIPAING CONRAC 87 ln(a) m X (m; ; a) [e exa] expm when X is a random variable distributed as (m, ). Formulas (3.) and (3.3) are truly closed-form formulas of the market value of our contract and can be computed instantaneously once all the parameters are specified. In the last section, we shall provide some interestin numerical results obtained usin these formulas. 4. NUMERICAL ANALYSIS In the first part of our numerical analysis, we ive some numerical results related to GP-LICs and explain how to set the different parameters to obtain fair-priced contracts. A second part is devoted to the comparison of GP-LICs and CP-LICs (whose characteristics are recalled in subsection 4.3). he only difference between these contracts is their uaranteed part. Indeed, the uaranteed interest rate of a GP-LIC is proportional to the yield of a zero-coupon bond P(0, ), while a CP-LIC uarantees a constant interest rate r. We will first describe this former contract, before comparin it to GP-LICs. 4. Data We ive below our chosen parameter values. Some of them will chane durin the numerical study, and we shall state whether we take the followin values or not: A 0 00 0.85 a 0.4 0.008 0. 0. 0 P(0, ) 0.6703 0.6 0.4 Let us first recall the meanin of the above coefficients. A 0 refers to the initial assets value of the company, and is a coefficient yieldin the part invested by the insured (i.e., L 0 A 0 85). he two parameters a and define the volatility P of the instantaneous interest rate process. he correlation coefficient between the assets process A and the instantaneous interest rate process r is. he volatility of the assets is set at 0%, which is quite low and is due to the presence of investment rade bonds in the portfolio of the insurance company. stands for the maturity of the contract; we suppose it is equal to 0 years. P(0, ) is a overnment zero-coupon bond maturin at. Finally, is the scale factor on the threshold trierin bankruptcy and is set to 60%. Note that some parameters have not been iven yet: the participatin bonus and the proportional coefficient that determines the minimum uarantee. hey will be specified when needed. Let us now present some numerical results on the fair valuation of GP-LICs and their components. 4. he New Contract First recall how the fairness of a contract can be assessed. One has set the contract parameters in such a way as to establish the equality between the initial policyholders investment L 0 and the initial market value V (0) of their contracts. hat is, formally: L0 V (0). V (0) can be computed instantaneously thanks to formulas (.8), (3.), and (3.3). Assinin a value to all the parameters except one, we obtain the remainin parameter s value by means of a simple root-
88 NORH AMERICAN ACUARIAL JOURNAL, VOLUME 0, NUMBER 4 search alorithm. Note that the participation coefficient is extremely easy to obtain in terms of the other parameters (for this, use eq. [.8]): L 0 GF PO LR P(0, ). (4.) (E E ) l (E E ) 6 7 3 In such contracts two parameters play a key role: the uaranteed rate and the participation coefficient. We first concentrate on the uaranteed rate, which is proportional to the yield of the zerocoupon bond P(0, ). Indeed, our contract uarantees the amount L 0 /P(0, ) at time. Now, let us define y 0 accordin as y Le 0 0 L 0 ; P(0, ) this is the yield that can be anticipated at time 0 by the insured, assumin subsequent bankruptcy will not occur. Of course, this is only an indication at time 0 of the true yield of a GP-LIC; note also that in this case the yield at time t is unknown because of the stochastic nature of the uarantee. Keepin the values of our list above (except for the volatility that is allowed to chane), we compute and raph in Fiure the fair participation coefficient with respect to y 0, at some iven fixed values of the volatility ( 6%, 0%, 3%, and 5%). he interpretation of the two first curves ( 6% and 0%) is very standard: they are neatively sloped because in a eneral context a hiher-yield y 0 should be compensated by a lower participation coefficient. When the volatility of the underlyin assets portfolio is hiher ( 3% or 5%), one can observe different curves where tends to be an increasin function of y 0. he interpretation is as follows: when is hih, the insured are facin a quite important default risk; because of this and the fact that extractin more value from the assets (by distributin y 0 ) increases the ruin probability, policyholders will require hiher participatin rates under hiher yields y 0. he horizontal dashed line ives a limit beyond which fair contracts cannot be built without seriously harmin shareholders. Clearly, not every choice of parameters will yield acceptable fair contracts. In particular, some parameters obey reulatory constraints and cannot be fixed arbitrarily. For instance, in France participation coefficients should be hiher than 85%, uaranteed interest rates (correspondin here to the Fiure Participation Coefficient with Respect to y 0. 0.9 0.8 δ 0.7 0.6 0.5 σ =6% σ =0% σ=3% σ =5% 0.4 0 0.005 0.0 0.05 0.0 0.05 0.03 0.035 0.04 y 0
DEVELOPMEN AND PRICING OF A NEW PARICIPAING CONRAC 89 yield y 0 ) should be less than 75% of the averae overnement yield, and should necessarily be inferior to 96% (existence of minimum solvency marin of 4%). Parameters should rane between realistic values: the participation coefficient should preferably be less than 00%, and uaranteed interest rates must of course be positive. hese constraints are illustrated in Fiure, where plots of the participation coefficient with respect to can be observed. In this raph the first coordinate ranes between 80% and 00%, and y 0 is set to 0.5%, %, and 3%. Under the existin constraints some contracts cannot exist; for instance, it is impossible to set 84%, y 0 3% and ive a participation rate of 80%. he rane of possibilities is located in the rey zone (raphin the constraints 0.85 and 0.96). An important remark is that when solvency is in daner (when equity decreases or increases), then for the contract to be fair, its associated participation coefficient should increase. he above numerical study reveals that the contracts we introduced (GP-LICs) display very standard features. Our next oal is to show in what respects they defer from other existin contracts (CP-LICs). 4.3 Comparison with Existin Contracts We first recall briefly the desin of CP-LICs, based on the description made in Bernard, Le Courtois, and Quittard-Pinon (005). hen we study the dependence of GP-LICs and CP-LICs on the uaranteed interest rate, the interest rate volatility, and the correlation between the interest rate and the assets processes. 4.3. Description A CP-LIC is a participatin contract with minimum and constant (instantaneously compounded) interest rate uaranteed r, and a participation coefficient equal to. Note that in the case of CP-LICs, r holds between 0 and and is contractual (whereas in the case of GP-LICs, y 0 is just an equivalent yield representin the rate uaranteed from time 0, only, and up to time ). he initial investment of the insured is L 0 ; at maturity, in the case of no prior default, he or she r will receive the investment put up by the uaranteed rate, that is, L L 0e. At that time, he or she should also et the participatin part of the contract, (A l ), provided the company performed well. Default risk is taken into account by introducin a reulatory barrier in the valuation model of rt the contract. he level of the assets of the company has to be above Le at any iven time t; is 0 Fiure Participation Coefficient with Respect to Participation Coefficient δ. 0.8 y 0 =0.5% y 0 =% y 0 =3% 0.8 0.84 0.88 0.9 0.96 α
90 NORH AMERICAN ACUARIAL JOURNAL, VOLUME 0, NUMBER 4 rt the default or the first passae time of the assets process at the barrier B t Le 0 L t. A CP- LIC therefore admits, under the risk-neutral probability Q, the valuation formula rs ds rs ds 0 0 Q V (0) [e (L (A L ) (L A ) ) e L ]. Under a stochastic interest rate environment, this formula cannot be developed in closed form because the minimum uaranteed interest rate (and hence the default barrier) is deterministic and not proportional to risk-free zero-coupon bonds as is the case for the new contracts introduced in this article. Instead, this formula can be developed in semiclosed form, as shown by Bernard, Le Courtois, and Quittard-Pinon (005), and based on the Collin-Dufresne and Goldstein (00) approach. We use their methodoloy to price CP-LICs, with a sliht adaptation because a clear distinction between and had not been made in their paper. In fact, in a simple model where interest rates are constant and not stochastic, this standard contract can be priced in closed form as shown by Grosen and Jørensen (00). We will use the notation V 3 for the value of a CP-LIC evaluated in a constant interest rate model. We recall the notation V (respectively V ) for the price of a GP-LIC (respectively a CP-LIC) evaluated under a stochastic interest rate assumption. Recall also that the yield of a GP-LIC is proportional to the yield of a overnment zero-coupon bond maturin at time. Before comparin the two contracts, we want them to uarantee approximately the same yield. In fact, a GP-LIC s yield can be known only at time 0 (because the contractual uarantee at time t is proportional to P(t, ), which is stochastic). Denotin by r the minimum uaranteed rate of a CP-LIC, we set for the GP-LIC in such a way as to satisfy the followin equality: r Le 0 L0, P(0, ) where it is meant that both contracts should start offerin the same initial yield (put differently, r y 0 should hold). 4.3. Participation Coefficient of a Fair Contract We choose the parameters iven earlier and raph in Fiure 3 the value of the participation coefficient with respect to r (where r ranes between % and 3% and is equal to y 0 ) for both contracts. Recall that the participation coefficient admits a closed-form expression. Indeed, equation (4.) ives the Fiure 3 as a Function of r 0.96 0.94 0.9 V V V 3 : r=3.9% 0.9 δ 0.88 0.86 0.84 0.8 0.8 0.0 0.05 0.0 0.05 0.03 r
DEVELOPMEN AND PRICING OF A NEW PARICIPAING CONRAC 9 correspondin formula for the case of GP-LICs (a similar expression can be found in Bernard, Le Courtois, and Quittard-Pinon (005) for the case of CP-LICs). We need to affect a value to the constant risk-free interest rate to compute V 3, that is, the value of a CP-LIC in a constant interest rate framework. We chose r 3.9% for the pricin of V 3 : this is this particular value that makes the third curve in Fiure 3 close enouh to the two first ones. Fiure 3 is very typical, and its shape can be explained simply: to compensate for a low uaranteed rate, the insurance company has to provide a hih level of participation on the assets performance. In the remainder of this study, we shall assume r %, and keep the fair value of for each of the existin three situations. his means that we will assume 89.70% as far as GP-LICs will be concerned, 90.5% for CP-LICs in a stochastic interest rate context, and 3 89.63% for CP-LICs in a constant interest rate context. 4.3.3 Default Probability We denote by E the default probability, and display its dependence with reards to the minimum uaranteed rate r in Fiure 4. It is interestin to interpret how the default probability varies. First of all, and this is common sense, it increases with r. Indeed, when all the other parameters are kept constant, an increase of r means an increase of the payout rate withdrawn from the assets, and hence a hiher default probability. Second and more interestin, for a very analoous desin and similar parameters, a GP-LIC is less likely to induce bankruptcy of the issuin company than a CP-LIC. Let us explain this feature. Note first that the CP-LIC s default barrier is defined by r (t) l e, which is the discounted value at r of the terminal uaranteed amount. On the other hand, the GP-LIC s default barrier is constructed as l P(t, ), which is the terminal uaranteed amount discounted by means of a risk-free zero-coupon bond. Note that the stochastic interest rate settin constructed above imposes a smaller default barrier than the constant interest rate settin. his is because r is usually much smaller than a risk-free zerocoupon bond rate; in other words, e P(t, ). Also note that even thouh P(t, ) is stochastic, r (t) r (t) in eneral it will never rise to the level of e, because of the very small value of r that is postulated. Fiure 4 E with Respect to r 0.08 V V V 3 : r=3.9% 0.04 E 0.0 0.006 0.00 0.0 0.05 0.0 0.05 0.03 r
9 NORH AMERICAN ACUARIAL JOURNAL, VOLUME 0, NUMBER 4 he conclusion is that GP-LICs, althouh built with floatin barriers (movin in standard market conditions in the opposite way of the assets), bear smaller default probabilities than CP-LICs. his is because the barrier of GP-LICs is always smaller than that of CP-LICs, because of the reulatory low value of r. Note also that E is small for CP-LICs computed with V 3, in other words, with a constant interest rate model. Of course, these values (that would be obtained as simplifications by actuaries and would lead them to underestimate the risk of CP-LICs) are wron. A constant interest rate model is not sufficient to price efficiently such contracts (for instance, and obviously, it cannot take into account the correlation between the assets and interest rates), and this is another conclusion of our study. 4.3.4 Sensitivity to the Assets Volatility and Interest Rate Parameter Let us recall that the parameter values iven earlier. We assumed r % and computed the fair value of for each of the existin three situations ( 89.70%, 90.5%, 3 89.63%). here the assets volatility is set to 0%. So the situation 0% represents a fair contract. We raph in Fiure 5 the contract value as a function of the underlyin assets volatility. he first element that can be noticed is that all contracts have values that start increasin with the level of volatility and then decrease (after an optimum at 0%) when the volatility continues increasin. Actually, the optimum corresponds to the fair coefficients computed based on our parameters. All parameters bein equal, we observe the impact of a volatility chane on the contract market value. Decreasin the volatility below 0% corresponds to decreasin the appeal of the product to the investors. It necessarily decreases its market value. Increasin it induces an increase of the default probability and therefore a decrease of the policy value. his effect is indeed related to the fact that we started from a fair contract. Comparin both contracts, it appears that V always remains hiher than V and has a smaller tendency to decrease with respect to the volatility. Aain, this is an advantae of GP-LICs. Less sensitive to the volatility of the assets, it should be of reater interest to the investors. his is, of course, a consequence of the way we construct the floatin-rate uarantee (reducin in particular the default probability of the issuin company), which is the only distinction between the two contracts. his type of contract, which is less likely to induce default, is compatible with a hiher level of, and this is a sufficient reason for it to be worth more than a CP-LIC. Finally, Fiure 6 clearly exhibits the hiher sensitivity to interest rates of a CP-LIC compared to a GP-LIC. Here is the parameter drivin the size of the volatility of the interest rates, and could be Fiure 5 V with Respect to 86 84 8 V V V : r=3.9% 3 Contract s Value 80 78 76 74 7 70 0 0.05 0. 0.5 0. 0.5 0.3 0.35 σ
DEVELOPMEN AND PRICING OF A NEW PARICIPAING CONRAC 93 Fiure 6 V with Respect to Contract s Value 85. 85. 85 84.9 V V V 3 : r=3.9% 84.8 0.00 0.006 0.0 0.04 0.08 ν=0.008 ν interpreted as a proxy of this volatility itself. Our conclusion at this stae is that a GP-LIC is more akin to resist environment chanes than a standard participatin contract. 5. CONCLUSION In a eneral framework takin into account actual features such as stochastic interest rates and default probability, we suest studyin a new contract bearin many characteristics of usual participatin life insurance contracts. his new contract is desined in a way that leads to an easy understandin of its behavior. Indeed, it reacts to financial and economic parameters qualitatively in a similar way as classical participatin contracts and seems to have very interestin manaement properties. he main technical arument for buildin this contract is that we value it in closed form. he practical construction of GP-LIC contracts, based on the use of overnment bonds, which are extremely liquid instruments, seems easy to achieve. 6. ACKNOWLEDGMENS he authors wish to thank Loïc Belze, Monique Jeanblanc, Yannick Maleverne, Honorine N Dri, Rivo Randrianarivony, Philippe Spieser, an anonymous referee at the NAAJ, and the participants of the AFIR and IME 005 conferences and of the New Mathematical Methods in Risk heory workshop in the honor of Hans Bühlmann for their many useful and insihtful comments. APPENDIX GOING FORWARD-NEURAL We explain briefly how to o from formula (.4) to formula (.7) rs ds rs ds 0 0 Q V(0) [e (l (A l ) (l A ) ) e l ] V(0) P(0, ) Q [(l (A l ) (l A ) ) l ]. he main difficulty here is to show that
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DEVELOPMEN AND PRICING OF A NEW PARICIPAING CONRAC 95 ANSKANEN, ANI J., AND JANI LUKKARINEN. 003. Fair Valuation of Path-Dependent Participatin Life Insurance Contracts. Insurance: Mathematics and Economics 33, 595 609. Additional discussions on this paper can be submitted until April, 007. he authors reserve the riht to reply to any discussion. Please see the Submission Guidelines for Authors on the inside back cover for instructions on the submission of discussions.