Financial Valuation of a New Generation Participating Life-Insurance Contract

Transcription

1 Financial Valuation of a New Generation Participating Life-Insurance Contract Claudio Pacati Abstract In this paper we analyze a typical new generation participating life-insurance contract: the single-premium whole life participating policy with zero technical rate and low-level surrender penalty. In such a contract there is no traditional mortality risk and the financial content is prevailing, since the benefits are yearly readjusted according to the return of the fund where the reserve is invested, with a yearly minimum guarantee. This kind of policy is widely sold by Italian companies in these years and can be seen as an open-term investment in the fund, protected by the minimum guarantee. The aim of this paper is to perform mark-to-market valuation of this contract in order to obtain the stochastic reserve, the fair value of the embedded option and the value of business in-force, the technical component of the embedded value. Keywords and phrases: participating life-insurance policy, whole-life insurance, financial valuation, mark-to-market reserve, value of business in-force, embedded value. Introduction This paper deals with a particular participating policy type: the singlepremium participating whole life, with zero technical rate and low-level Invited conference at the 6 th Spanish-Italian Meeting on Financial Mathematics (Trieste, 3 5 July 2003). Università di Siena, Dipartimento di Economia Politica, p.zza S. Francesco, 7, I Siena SI; pacati@unisi.it.

2 surrender penalty. This kind of contract is a typical example of the new generation participating policies sold in these years by Italian companies, designed to emphasize their financial content and to be sold as an alternative to purely financial instruments, by traditional insurance agents as well through bancassurance channels. In fact, this policy has little actuarial content: at any time the statutory reserve is always equal to the benefit for the case of death at the same time. Thus the policy is sold mainly for its financial features: it is very similar to an open-term investment in the a fund managed by the company, protected by a yearly minimum guarantee. Participating life-insurance policies are very popular in Italy. They are sold since the early 80 s and where originally introduced to protect the insured benefits from inflation. The basic idea of the participating rule is the following: the insurance company invests the mathematical reserve of the policy in a fund, the segregated fund, whose yearly return I is shared between the company and the insured. A readjustment rate ρ is contractually defined as a function of I and applied to raise for the same year the insured capital, according to a rule that depends on the policy type. A quite general form of the contractual definition of ρ is ( ) J i ρ = max 1 + i, ρ min, (1) where i is the technical rate of the policy, ρ min 0 is the yearly minimum guaranteed and J is the assigned return, the part of I assigned to the insured. A typical contract defines J according to the following rules: the insured gains a fraction β (0, 1] of I, the participation coefficient, but the company has to retain at least i ret 0, however, in any case, the insured has the right to at least a fraction β min (0, β] of I. Hence: J = max [min (βi, I i ret ), β min I]. 1 (2) 1 Notice that, when the return of the fund is negative, J cannot be properly seen as a part of it. In fact, if I < 0, since β, i ret and β min are non negative, J = β min I > I. However, the term assigned return has become standard in the Italian insurance practice. 2

3 Due to the participating rule, the benefits of the policy are random variables with regard to both actuarial and financial uncertainty. The former concerns the benefit s type and payment date, whereas the latter affects the benefit amount. From a financial point of view, the policy is a derivative contract, with underlying the return of the segregated fund: the various yearly minima in (1) and (2), and mainly the yearly minimum ρ min, embed into the policy a quite complex financial option. Furthermore, the option is of cliquet type: the indexation rule applies every year, consolidating the benefit level reached by the revaluation occurred in previous years. Since the participating rule links the benefit s amount to the capital market, the valuation method to be used to price the policy has to be consistent with the valuation methods used in capital markets. This requirement is well understood both in the theory (see e.g. [5], [7] and, for the Italian case, [13], [10], [1], [2] and [3]) and in the insurance practice (see e.g. [11]). Accordingly, in section 3 we determine the mark-tomarket reserve of the policy, also called stochastic reserve to emphasize the fact that the valuation is done in a mark-to-market setting, hence considering a stochastic evolution of interest rates, in contrast to traditional constant rate valuations. Also the embedded value of the policy has to be computed in a mark-to-market sense. Following [10] we derive its the technical component, the value of business in-force in section 4, together with its decomposition in investment gain, mortality gain and surrender gain, by comparing the statutory and the mark-to-market reserves, computed under different actuarial assumptions. The last section of the paper is devoted to some valuation results, obtained through Monte Carlo simulations. 1 The contract Let us consider the case of a whole life participating policy with zero technical rate (i = 0). Consider a policy sold at time zero to an insured of age x. The benefit, the insured capital, is paid at death of the insured, whenever it occurs. Assume for simplicity that death can occur only at integral times, just after the revaluation. Let the insured capital be initially defined to be C 0 ; its level at time T = 1, 2,..., is given by the recurrent relation C T = C T 1 (1 + ρ T ), (3) 3

4 ρ min J T 0 i ret 1 β min I T ρ T i ret 1 β I T Figure 1: Readjustment rate ρ T (i = 0). as a function of the fund s return I T where ρ T is the readjustment rate for year T. For this policy i = 0 and hence by (2) and (1) the assigned return J T for year T and the readjustment rate ρ T become J T = max [min (βi T, I T i ret ), β min I T ], (4) ρ T = max [min (βi T, I T i ret ), β min I T, ρ min ], (5) where I T is the return of the segregated fund in the same year. Figure 1 shows the graph of ρ T as a function of I T in a non-degenerate case, where the constants β, β min, i ret and ρ min are set in such a way that all the minima can occur. In the Italian practice, typical values for these constants are β 90%, β min 75%, 0% i ret 1.25% and 2% ρ min 4%. Of course, if i ret = 0%, the presence of β min has no effect and the rule becomes simpler. The insured capital at time T can be written in the closed form C T = C 0 T h=1 (1 + ρ h ), (6) that emphasizes its path-dependent nature. In the sequel we will use 4

5 the notation T t Φ t,t = (1 + ρ t+h ), (7) h=1 so that C T = C 0 Φ 0,T or, starting from any integral time 0 t < T, C T = C t Φ t,t. The contract provides also the insured with a surrender option: at any time T, he or she can surrender the policy and receive the redemption value Σ T = C T γ T ; (8) the redemption coefficient γ T 1 depends in a deterministic and contractually defined way on surrender time T. In these contracts surrenders are penalized at a very low level; a typical example could be γ T = 1 (3 T ) + /100, allowing for a very little surrender penalty only in the first two years. For simplicity we will assume that surrender can occur only at integral dates, just after the revaluation of the insured capital. In fact, the surrender option is the most important feature of the contract. The insurer does not expect the insured to persist until death and the contract is typically sold as an open-term investment in the segregated fund. From a financial point of view, the surrender option is an american put option embedded in the contract: at any time T the insured has the right to sell back the contract to the insurer, at the strike price Σ T. This option could be priced by standard no-arbitrage techniques, assuming rational exercise by the insured. However, there are two major drawbacks to this assumptions: The insured is not a financial institution and surrender may be driven more frequently by the evolution of his or her consumption plan than by the market evolution. For financial instruments, this assumption was empirically proven to be correct in [6], referring to early redemption of Canadian savings bonds, and in [15], referring to mortgage-backed securities. The information on the value and on the asset-allocation of the 5

6 segregated fund is not public 2, the yearly return is known with a lag of one or two months and no benchmark is contractually nor indicatively defined. 3 Even a rational insured does not have all the information needed to test the rational exercise condition, since he or she cannot compare the redemption value to the prosecution value and is hence not able to rationally exercise the surrender option. These considerations lend us to apply the more traditional idea of modelling surrender uncertainty through experience-based elimination tables. To this end, we will consider an actuarial probabilistic framework with two elimination causes: death and surrender; we will furthermore assume that elimination events are independent of market events. 2 Traditional valuation The statutory technical reserve of the policy, that is to say the level of funding the company has to maintain by law, is the net premium mathematical reserve and is defined in a traditional actuarial setting. Let us denote by prob I the first order probability distribution of the future lifetime T x of the insured and let qt,k I be the first order probability of death in year t + k, conditional to survivor at age x + t: q I t,k = probi (x + t + k 1 T x < x + t + k T x > x + t). (9) The traditional reserve R(t) at time t is defined considering the sum insured at time t, not considering future readjustments nor the surrender option, using the first order probability distribution and discounting at the technical rate: R(t) = C t qt,k I (1 + i) k = C t qt,k I = C t. (10) Since at any time the traditional reserve equals the death benefit at that time, from a traditional point of view the policy has no mortality risk. 2 In fact Italian companies do publish a brief quarterly report on the segregated fund, containing the (book) value of the fund and an asset-allocation summary, but with a significant lag with respect to the reference date. 3 Notice that this lack of information does not regards the reference funds of Italian unit-linked policies. For these funds the unit value is published at least weekly and an (indicative) asset allocation and a benchmark are contractually defined. 6

7 The main drawback of the traditional approach is that the valuation is performed as if the policy where non-participating. In this way, the embedded minima are ignored and the method is unable to price the minimum guarantee option embedded in the readjustment mechanism. Also the surrender option is not considered, but this is a minor problem. Due to the former approximation, the redemption value Σ t+k at future time t + k has coherently to be approximated by C t γ t+k C t. Ignoring the surrender option gives hence a prudential valuation. 3 Mark-to-market valuation In contrast to the traditional valuation framework, the mark-to-market approach is able to consider the financial uncertainty affecting the benefits. This uncertainty comes from the market where the fund s manager invests the policy reserve. Managers of Italian segregated funds typically invest the main part (at least 80%) of the reserve in bonds and the rest in stocks. The amount invested in corporate bonds is negligible, so we can ignore default risk. We will therefore consider a market model with two sources of uncertainty: interest rate risk and stock-market risk. We will model interest rate risk through the one-factor Cox, Ingersoll and Ross (CIR) model ([8], see also [9] and [14]): if r t is the market spot rate at time t, we will assume it follows a square-root mean-reverting diffusion process dr t = α(γ r t )dt + ρ r t dw r t, (11) where Wt r is a standard Brownian motion and α, γ and ρ are positive constant parameters, with 2αγ/ρ 2 1. We furthermore assume that market price of interest rate risk is of the form q(t, r t ) = π rt ρ, (12) with π a constant parameter. Stock-market uncertainty is considered by modelling the stock-index S t as a Black and Scholes [4] log-normal process, with constant drift and volatility parameters µ and σ: ds t = µs t dt + σs t dw S t, (13) 7

8 where Wt S is a standard Brownian motion. We assume finally that the two Brownian motions driving r t and S t are correlated: cov ( dw r t, dw S t ) = ρ r,s dt ; (14) the instantaneous correlation coefficient ρ r,s is assumed to be constant. It is well known that this model is complete and arbitrage-free and that the risk-neutral dynamics of the state variables is dr t = α( γ r t )dt + ρ r t d W r t, (15) ds t = r t S t dt + σs t d W S t, (16) where W t r and W t S are the risk-neutral Girsanov transformations of the two Brownian motions Wt r and Wt S, and α = α π and γ = αγ/ α are the drift parameters of the risk-neutral spot rate dynamics. In this market-model, the stochastic evolution of the fund s return I is completely specified once specified the trading strategy the fund s manager follows. According to what said before, we will assume that, for each integer T, I T = Q S T S T 1 S T 1 + (1 Q) B T B T 1 B T 1, (17) where Q is the fraction of the fund held in stocks and B is the market value of a self-financing bond portfolio. A standard no-arbitrage argument shows that the market price at time t of a random payment Y T at time T > t, subject only to financial uncertainty, is given by V (t, Y T ) = (Y Ẽt T e ) T t r u du, (18) where Ẽt is the risk-neutral expectation implied by the risk-neutral version of the model and conditional to the market information at time t. Notice that, by standard no-arbitrage arguments, we have for every integer T > t that ( ) T t V t, (1 + I T ) = 1. (19) 8

9 Now consider at time t the payment provided at time T by our policy. It is of the form Y T = 1 ET α ET C T (20) = 1 ET α ET C t Φ tt, (21) where E T is the elimination event originating the payment, 1 ET its indicator function and α ET is 1 or γ T, depending on the type of elimination (death or surrender). The random variable Y T is affected by both financial and actuarial uncertainty. However, by the independence assumption made at the end of section 1, the randomness of the first factor of (21) is only of actuarial type, whereas the last factor is affected only by financial uncertainty and α ET and C t are known at valuation time. Thus the market value of Y T becomes ( V (t, Y T ) = prob t (E T )α ET C t Ẽ t Φ tt e ) T t r u du (22) = prob (E T )α ET C t V (t, Φ tt ), (23) where prob is the risk-adjusted probability measure of actuarial events (death and surrender), conditional to life and persistency of the insured at time t, and V (t, Φ tt ) is the stochastic valuation factor at time t for time T : it is the value of one unit of cash invested in the fund and revaluated up to T at the stochastic yearly readjustment rate ρ. If we denote the valuation factor by φ(t, T ) = V (t, Φ t,t ), (24) the value of the policy at time t, that is to say the mark-to-market reserve, is V (t) = C t qt,k φ(t, t + k) + C t s t,k γ t+kφ(t, t + k), (25) where qt,k (s t,k ) is the prob t -probability of life and persistency at time t + k 1 and death (surrender) at time t + k. Mark-to-market reserve is therefore computed knowing the following quantities at time t: the insured capital in-force at time t, the term structure of valuation factors, the term-structure of death- and surrender-elimination risk-adjusted probabilities. 9

10 Standard measures of financial instantaneous risk can be obtained in the same way, computing the derivatives of the valuation factors with respect to both the state variables. The stock-market delta of the mark-tomarket reserve is the elasticity of V (t) with respect to the state variable S t, S (t) = S t V (t) = S tc t V (t) S t V (t) (qt,k + s t,k γ t+k) φ(t, t + k). S t (26) The interest rate sensitivity of the mark-to-market reserve is the semielasticity of V (t) with respect to the state variable r t, r (t) = 1 V (t) = V (t) r t C t V (t) (qt,k + s t,k γ t+k) φ(t, t + k). (27) r t It is worth to obtain a separate value for the minimum guarantee. Since ρ min is the most relevant of the three minima, we consider the policy as a derivative contract with underlying the yearly assigned returns J t+k. For every k, define the base readjustment rate to be the base valuation factor to be ρ b t+k = J t+k, (28) k Φ b t,t+k = (1 + ρ b t+k ), (29) h=1 and the base stochastic reserve to be V b (t) = C t φ b (t, t + k) = V (t, Φ b t,t+k ) (30) ( q t,k + s t,k γ ) t+k φ b (t, t + k). (31) This quantity is the reserve of a policy version with ρ min = 1, i.e. without the minimum guarantee. The difference V p (t) = V (t) V b (t) (32) 10

11 is the mark-to-mark value of the cliquet minimum guarantee put embedded in the contract. 4 Notice that, for every integer k, 1 T t {I k+k>0} Φb t,t+k < 1 T t {I k+k>0} h=1 k (1 + I t+h ). (33) In normal situations, the event T t {I k+k > 0} has a high risk-neutral probability and by (19) typically one has for each k and hence φ b (t, t + k) < 1. (34) V b (t) < R(t), (35) whereas the same is not true for the mark-to-market reserve, that can be significantly greater than the statutory reserve, due to the value of the embedded put options. 4 Value of business in-force In this section we apply the methodology proposed in [10] in order to obtain the contract s mark-to-market value of business in-force (VBIF). To this end we repeat the mark-to-market valuation with different actuarial probabilities: first-order: we use first-order survivor probability distribution and assume that surrenders are disallowed; second-order: we use a second-order survivor probability distribution (the part of prob t concerning survivor only) and assume that surrenders are disallowed. We denote by a superscript I ( II ) the valuation results obtained using first-order (second-order) assumptions. Notice that the real markto-market results are the ones obtained in the previous section using the 4 Of course, one could alternatively consider the policy as a derivative contract with underlying the fund s return. From this point of view, defining ρ b t+k = I t+k and repeating the same arguments one obtains the value of the global put embedded in the contract. 11

12 risk-adjusted actuarial probability; these variations are introduced only to decompose the VBIF. Mark-to-market VBIF E(t) at time t is obtained by comparing the statutory and the mark-to-market reserve at the same time: E(t) = R(t) V (t). (36) It measures the mark-to-market value of the profits the company will get during the lifetime of the policy. We will denote by e(t) = E(t) R(t) (37) the ratio between VBIF and statutory reserve, i.e. the percentage of the present reserve that will not be paid as benefits and hence belongs to the insurer. VBIF can be decomposed in where E(t) = F (t) + M(t) + L(t), (38) F (t) = R(t) V I (t) is the financial gain, (39) M(t) = V I (t) V II (t) is the mortality gain, (40) L(t) = V II (t) V (t) is the surrender gain. (41) (42) We will also consider the base VBIF and the base versions of the VBIF s components, defined using the base mark-to-market-reserve, at the various orders and denoted by a superscript b. Financial gain F (t) is the value of purely financial future profits. By (10) and the first-order version of (25) and recalling that C t = R(t) it can be written as F (t) = R(t) [1 φ(t, t + k)]. (43) q I t,k Recalling that, for every k, 1 = (1 + i) k, the ratio f(t) = F (t) R(t) = q I t,k [1 φ(t, t + k)] (44) 12

13 is hence the first-order weighted mean of the term-structure of differences between technical discount factors and mark-to-market valuation factors. Assuming (34) for each k, as it is usual the case, it holds f b (t) = F b (t) R(t) 0. (45) The quantity f b (t) measures the mark-to-market value of the retained part of future fund s returns, as a fraction of the present fund s level. In contrast, due to the minimum guarantee, f(t) can be negative. Notice also that, in contrast to R(t), the first order mark-to-market reserve and hence F (t) depends on the first-order survivor probability measure. Mortality gain is the mark-to-market value of future profits given by differences between first-order and second-order mortality. Denoting by q t,k = qt,k I qii t,k, we have M(t) = R(t) q t,k φ(t, t + k) (46) and m(t) = M(t) R(t) = q t,k φ(t, t + k). (47) Notice that, from a traditional point of view, the policy has no mortality risk and hence there cannot be future mortality profits. However, since mark-to-market reserve depends on the term structure of mortality rates, different probability distributions produce different values. In the Italian practice, survivor probabilities are originated by mortality tables and first-order year-by-year mortality rates are typically higher than the corresponding second-order ones. Hence, for non-extreme ages, the first terms of the term-structure of q s are positive and the last are negative. Since the term structure of valuation factors tends to be decreasing, in typical cases we have a positive mortality gain. In other words, the second order probability measure increases the persistence of the reserve in the fund, allowing to Company for more financial profits. Surrender gain measures the mark-to-market value of the effects of surrenders. Denoting by q t,k = qt,k II q t,k, we have L(t) = R(t) [ q t,k s t,k γ ] t+k φ(t, t + k) (48) 13

14 and l(t) = L(t) R(t) = [ q t,k s t,k γ ] t+k φ(t, t + k). (49) Notice that, in contrast to m(t), l(t) tends to be negative. The fact that surrenders are now allowed typically decreases the persistence of the reserve in the fund. If the consequent loss of financial profits is not compensated by the redemption penalties, the global effect on profits is negative. Finally, notice that we can decompose VBIF in an alternative way, in order to show at VBIF level the effects of the minimum guarantee. Since we have that V p (t) = V (t) R(t) + R(t) V b (t) = E b (t) E(t) (50) and, by using the notation E(t) = E b (t) V p (t) (51) p(t) = V p (t) R(t), (52) we have that e(t) = e b (t) p(t). (53) This decomposition presents VBIF as his base component minus the put component. The base VBIF is hence the VBIF of the policy without the minimum guarantee, whose negative effect on VBIF is measured by the put component. 5 Numerical computations In this section we present some numerical calculation examples. consider the following contractual parameters: We β = 90%, i ret = 1%, β min = 75% (54) 14

15 Figure 2: Term structures of market interest rates 31/12/2002. and the four cases ρ min = 2%, 2.5%, 3% and 4%. The redemption coefficient will be assumed to be defined by the contractual rule γ T = 1 (T 3) + /100. Assume x = 40 years and t = 0. As seen in section 3, the valuation problem can be split in two separate parts: the financial part (calculation of the valuation factors) and the actuarial part (calculation of the elimination probabilities). For what about the financial part, we consider the market at date 31/12/2002. Risk-neutral parameters for the interest rate part of the model has been estimated on euro interest rate swap market data (end of day mid quotes), using the methodology proposed in [12]: r t = , α = , γ = , ρ = Figure 2 shows the term structure of interest rates implied by the CIR model. We model the bond portfolio W, following [10], by a quarterly 3-year zero coupon bond trading strategy. For the stock index S we take the MIB30; at the valuation date the 1-year historical volatility of the index was σ = 28%; we set also 15

16 Figure 3: Term structures of valuation rates 31/12/2002. ρ r,s = 0.1 and take Q = 10%. The valuation factors and their derivatives are computed through Monte Carlo simulation. To this end, the risk-neutral version of the model has been approximated by a stochastic Euler scheme, with time step 1/48 of year. Figure 3 shows the graph of the term structures of valuation rates, defined as [ i val (t, T ) = 1 φ(t, T ) ] 1 T t 1, (55) for time to maturities till 30 years. The figure contains one curve for each considered guarantee level and one curve of base valuation rates. Notice that the base rates are always positive, and this means that base valuation factors are between 0 and 1, whereas the valuation rates can be negative and hence valuation factors can be greater than 1. This means that for ρ min = 4% we have a negative financial gain, with positive base component. Table 1 reports the valuation factors for some time to maturities, with their estimation standard errors. For what about the actuarial part of the valuation, we take as firstorder probability measure the one originated from the Italian SIM92 16

17 Table 1: Valuation factors 31/12/2002. T t φ(t, T ) (standard error) φ b (t, T ) ρ min = 2% ρ min = 2.5% ρ min = 3% ρ min = 4% (.0000) (.0000) (.0000) (.0000) (.0002) (.0002) (.0002) (.0001) (.0003) (.0003) (.0003) (.0002) (.0004) (.0004) (.0004) (.0003) (.0005) (.0005) (.0005) (.0004) (.0006) (.0006) (.0006) (.0005) (.0007) (.0007) (.0007) (.0006) (.0008) (.0007) (.0007) (.0007) (.0008) (.0008) (.0008) (.0008) (.0009) (.0009) (.0009) (.0009) (.0009) (.0009) (.0009) (.0009) (.0012) (.0012) (.0012) (.0012) (.0014) (.0014) (.0014) (.0015) (.0015) (.0016) (.0016) (.0017) (.0016) (.0017) (.0018) (.0020) (.0018) (.0019) (.0020) (.0024) (.0020) (.0021) (.0022) (.0027) (.0020) (.0022) (.0024) (.0030) (.0021) (.0023) (.0026) (.0034) (.0022) (.0024) (.0027) (.0037) table. The second-order probability will be the one obtained from the same table by considering only 60% of the year-by-year mortality rate. Finally we assume a constant yearly redemption rate of 4%; the riskadjusted prob -probability is constructed by joining the second-order survivor probability with the redemption rates in the standard way. Figure 4 shows the term structures of the elimination probabilities qt,k I, qii t,k, qt,k, s t,k and of the sum q t,k + s t,k. Valuation results for an insured capital of C t = R(t) = 100 are shown in table 2. Table 3 presents the VBIF of the policy and its decomposition. 17

18 Figure 4: Term structures of elimination probabilities. Table 2: Mark-to-market reserve ρ min R(t) V (t) V b (t) V p (t) 2% % % % Table 3: VBIF and its decomposition ρ min E(t) F (t) M(t) L(t) 2% % % %

19 References [1] Bacinello, A.R., Fair Pricing of Life Insurance Participating Policies with a Minimum Interest Rate Guaranteed, Astin Bulletin 31(2001), [2] Bacinello, A.R., Fair Valuation of the Surrender Option Embedded in a Guaranteed Life Insurance Participating Policy, Quaderni del Dipartimento di Matematica Applicata alle Scienze Economiche Statistiche e Attuariali Bruno de Finetti, Università a degli Studi di Trieste, 8(2001). [3] Bacinello, A.R., Pricing Guaranteed Life Insurance Participating Policies with Periodical Premiums and Surrender Option, Quaderni del Dipartimento di Matematica Applicata alle Scienze Economiche Statistiche e Attuariali Bruno de Finetti, Università a degli Studi di Trieste, 1(2002). [4] Black, F., Scholes, M., The Pricing of Options and Corporate Liabilities, Journal of Political Economy, 81(1973), 3. [5] Boyle, P.P., Recent Research on the Risk Return Relationship in Financial Economics, in Goovaerts, M., de Vylder, F., Haezendonck, J. (eds.) Insurance and Risk Theory, Dordrecht, Reidel, [6] Brennan, M.J., Schwartz, E.S., Savings Bonds: Theory and Empirical Evidence, New York, Monograph Series in Finance and Economics, 4, [7] Bühlmann, H., New MATH for Life Actuaries, Working paper, May [8] Cox, J.C., Ingersoll, J.E., Ross, S.A., A Theory of the Term Structure of Interest Rates, Econometrica, 53(1985), 2. [9] De Felice, M., Moriconi, F., La teoria dell immunizzazione finanziaria. Modelli e strategie, Bologna, Il Mulino, [10] De Felice, M., Moriconi, F., Finanza dell assicurazione sulla vita. Principî per l asset-liability management e per la misurazione dell embedded value, Gruppo di ricerca su Modelli per la finanza matematica, Working paper n. 40, giugno

20 [11] Morgan Stanley, Revolution, Equity Research, January 31, [12] Pacati, C., Estimating the Euro Term Structure of Interest Rates, Roma, Research Group on Models for Mathematical Finance, Working paper 32, March [13] Pacati, C., Valutazione di portafogli di polizze vita con rivalutazione agli ennesimi, Roma, Gruppo di ricerca su Modelli per la finanza matematica, Working paper 38, aprile [14] Rogers, L.C.G., Which Model for Term-Structure of Interest Rates Should One Use?, Mathematical Finance 65(1995), [15] Schwartz, E.S., Torous, W.N., Prepayment and the valuation of mortgage-backed securities, Journal of Finance, 44(1989), 2. 20