Pricing Formulae for Financial Options and Guarantees Embedded in Profit Sharing Life Insurance Policies

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1 Pricing Formulae for Financial Options and Guarantees Embedded in Profit Sharing Life Insurance Policies Gilberto Castellani Università di Roma La Sapienza Franco Moriconi Università di Perugia Massimo De Felice Università di Roma La Sapienza Claudio Pacati Università di Siena Version 1.1 February 10, 2007 Summary In the QIS3 Pre-Test Technical Specifications, CEIOPS is providing examples of valuation models for small life-insurance firms, proposing Black-Scholes type pricing formulae for financial options and guarantees FOG. If this kind of approach is considered viable by CEIOPS, a large set of closed form solutions is readily available, allowing small firms to easily compute price and risk of a variety of life insurance FOG. We present in this paper a number of such pricing formulae, mainly referring to FOG embedded in Italian profit sharing PS policies. While the pricing expressions in QIS3 document are strictly developed in a Black-Scholes setting, the formulae provided here are based on an extension of the Black-Scholes model, obtained via an implied volatility function which captures the actual volatility of the bond investment strategy chosen by the insurer. We shaw in the paper that this extension provides a quite good approximation of a completely consistent valuation model including interest rate risk. This point is illustrated by extended numerical examples. 1 Introduction In the recent actuarial literature a growing attention has been devoted to the market consistent arbitragefree valuation of financial options and guarantees FOG embedded in life insurance policies. An important case of FOG are the financial options embedded in some kind of profit-sharing PS policies, where the PS mechanism is based on an explicit participating rule contractually specified at the policy inception. In these policies the payoff representing the benefits to policyholders can be expressed as a minimum guaranteed benefit plus a call option representing the extra benefits or, equivalently, as a variable risky benefit plus a put option, which represents the minimum guarantee. Whichever the representation chosen, the financial component of these PS policies can be modelled as a derivative contract having the return on investments as the underlying. Of course the insurer has some degree of discretion in managing the investment fund; hence any valuation model should take into account the possible management actions. In any case measuring price and risk of this class of PS policies is a complex problem of asset-liability management in a stochastic setting. Italian companies experience on profit sharing policies. PS policies with embedded FOG represent a large component of the life insurance business on the Italian market. As a consequence in the last two decades some leading Italian insurance companies have gained a wide experience in developing and applying arbitrage-free models appropriate for the valuation of portfolios of such contracts. Early approaches to the problem were based on models using limited information on the outstanding portfolios. In these models, which will be referred to as type a models, the investment strategy followed by the insurer was highly stylized and little room was leaved to additional management actions. Though being based on simplified assumptions, these models were consistent arbitrage-free models, well suited for pricing and hedging long term interest 1

2 rate derivatives 1. Recently more sophisticated models have been introduced for the valuation of Italian PS policy. In these models, which we shall denote type b models, all the details of the outstanding portfolio of assets and liabilities are considered and the management actions are chosen among the possible ones in order that price and risk of the embedded options are optimally reduced. In particular the accounting rules under which the reference fund return is determined are fully taken into account and the eventual unrealized gains in the segregated fund are optimally used in the investment strategy 2. Given the high complexity of the payoffs to be priced, type b models are typically based on Monte Carlo procedures. However in an appendix in [CDFMP-05] closed form expressions for the price of the embedded options are also derived in the framework of a type a model. These pricing formulae were provided there in order to give a simple illustration of the price decomposition as intrinsic value plus time value, as proposed by the CFO Forum in 2004 [CFO-04]. The profit sharing policies in the Solvency II framework. The problem of PS policies has become a central one in the Solvency II project, since it has been recognised that the PS mechanism should allow a risk mitigation, hence a reduction of capital requirements to the insurers. The intuition behind risk mitigation is that transferring part of the profits to the policyholder typically implies that also investment risk is partially transferred. The problem of determining the correct risk reduction for profit sharing RPS is currently referred to as the K-factor issue. Obviously the RPS problem for the Italian policies can be consistently tackled using the stochastic models just developed for the asset-liability management applications. In the Solvency II framework, type b models can be classified as internal models, while type a models are natural candidates for the class of the simplified and standardised models well suited for small firms. In a recent paper [DFM-07] a standardised model for determining the RPS in the Italian policies has been provided, based on a type a two-factor model largely experimented on outstanding portfolios. At the same time, in the QIS3 Pre-Test Technical Specifications [QIS3], CEIOPS is providing examples of valuation models for small firms, proposing Black-Scholes type pricing formulae. This formulae are essentially the same proposed in the appendix in [CDFMP-05]. We point out that if this kind of approach is considered viable by CEIOPS, a large set of closed form solutions is readily available, allowing small firms to easily compute price and risk of a variety of PS policies. We present in this paper a number of such pricing formulae. It is important to observe that while the pricing expressions in [QIS3, pp , 55-60] are strictly developed in a Black-Scholes setting, the formulae provided here are based on an extension of the Black-Scholes model, obtained via an implied volatility function which captures the actual volatility of the bond investment strategy chosen by the insurer. We shaw in the paper that this extension provides a quite good approximation of the completely consistent valuation model of type a introduced in [DFM-07]. This point is illustrated by extended numerical examples. 2 Option-like representation of participating benefits We refer to pure endowment profit-sharing policies with single or constant annual premium. 3 In order to focus our discussion on financial risks, we simplify the valuation problem by assuming zero probability of early termination death or surrender. Therefore the insured benefit will be paid for certainty at the policy s term and all the premiums if any will be paid for certainty at the contractually specified dates. We will consider the standard profit-sharing rules used by Italian companies. 2.1 Single premium policies With respect to a policy with term n, let S 0 be the initial sum insured and, for each policy year t = 1, 2,..., n, let S t be the sum insured at time t, increased by the profit sharing rule in period [0, t]. The profit sharing rule for single premium policies is defined by the recurrent equation: S t = S t ρ t, t = 1,..., n, 1 1 An illustration of this kind of approach can be found in [DFM-02a], [DFM-02b] and [DFM-05]. In [DFM-05, p. 81] some applications of a type a model by Italian life insurance companies are described. 2 Models of type b are illustrated in [DFM-04] and [CDFMP-05]. 3 All the results can be extended to any other class of profit-sharing policies endowments, whole life insurance, annuities,... and to any premium type revaluable annual premium or single recurrent premium. 2

3 where ρ t is the so-called revaluation rate, defined as: where: ρ t := max{ min{β I t, I t h} i, δ } I t is the rate of return of the investment portfolio in policy year t, i is the technical interest rate, δ 0 is a minimum guaranteed spread over the technical rate, h 0 is a minimum return retained by the insurer, β 0, 1] is the participation coefficient., 2 Equations 1 and 2 define the standard profit sharing rule used in Italian policies, which is called the full revaluation rule. All the parameters other than I t are contractually fixed. The statutory reserve R t is currently defined in the traditional way, i.e. as if the policy would be non-participating with sum insured S t ; under these assumptions: R t = S t n t. 3 In 2 the parameter δ provides the minimum ρ t value, hence the minimum annual return credited to the sum insured. Combining equations 2 and 3 the statutory reserve increases at the rate ρ t := i + ρ t, with annual minimum in := i + δ. Typical values of the parameters for Italian outstanding policies are h 1%, β 80% and in varying between 1% and 4.5%, in combinations i, δ with i > 0 and δ > 0, i = 0 and δ = 0, and also i > 0 and δ > 0. The annual revaluation rate embodies in general two optional components. To identify them we have to distinguish between tree cases, all of them relevant in the Italian practice. a If β = 1 then I t I t h for each I t and the revaluation rate can be written as: ρ t = δ + 1 max {I t h +, 0}. 4 Equation 4 is the call decomposition of the revaluation rate, which is expressed as the sum of the minimum value δ and 1/ call options having the return I t as the underlying and h + as the strike value. A similar decomposition holds for the statutory reserve s revaluation rate: Case a is quite typical in Italian bancassurance. ρ t = + max {I t h +, 0}. 5 b If /h+ β < 1, using well known properties of the max and min operators the revaluation rate can be written as: { { max I t h + 1 βmax I t h } } 1 β, 0, 0 ρ t = δ +. 6 In this case we have: h + h 1 β ; hence, by straightforward calculations, the double max in 6 can be written as the difference between a call option with strike h + and 1 β call options with strike h/1 β, both having the same underlying I t. Therefore the call decomposition of the revaluation rate corresponds to the expressions: and: ρ t = δ + 1 max {I t h +, 0} 1 β max { I t h 1 β, 0 { ρ t = + max {I t h +, 0} 1 βmax I t h } 1 β, 0. 7 }, 3

4 c If β < /h + then: and the call decomposition is: ρ t = δ + h 1 β < h + < β, β { max I t i } m β, 0, where the revaluation rate is given by the sum of the minimum rate δ and β/ call options on I t with strike /β. The corresponding call decomposition for ρ t is: { ρ t = + β max I t i } m β, 0. 8 In figure 1 we show the graphs of ρ t as a function of I t in the three cases considered. Since the transformation ρ t ρ t is a downward translation by i and a scaling by 1/ > 0, the corresponding graphs for ρ t have the same form. I t I t I t ρ t h ρ t h h ρ t 0 h+ It 0 β h+ h 1 β It 0 h+ β It 1.a: β = 1 1.b: h + β < 1 1.c: β < im h + Figure 1: Graphs of ρ t in the three cases discussed. Remark. Using the put-call parity relation the put decompositions of ρ t and ρ t can be immediately obtained from the call decompositions previously considered. Let us now consider the valuation problem for the single premium policy at time t. The benefit to be paid at the policy term n can be written as: where: Φ t,n := S n = S t Φ t,n, n 1 + ρ k, is the full revaluation factor for the interval [t, n]. The fair value or stochastic reserve of the single premium policy at time t is: V t = S t V t, Φ t,n, 9 where V t, X T is the market value at time t of a purely financial contract with payoff X T at time T. The options embedded in the contract are a complex combination of the 1-year options embedded in the annual revaluation rates. The terminal payoff can be decomposed as: S n = S t Φ min t,n + S t Φt,n Φ min t,n, 10 4

5 where: Φ min t,n := min Φ t,n = 1 + δ n t, is the contractual minimum value of the full revaluation factor. Equation 10 is the call decomposition of the benefit into the sum of the guaranteed benefit the first term and the non-guaranteed benefit the second term. From a financial point of view the former is a deterministic zero-coupon-bond ZCB, the latter is a cliquet european call option having the annual return of the investment portfolio as the underlying. The call decomposition 10 of the benefit induces the call decomposition of the fair value: The fair value of the guaranteed benefit is given by: V t = G t + C t. G t := S t V t, Φ min t,n = S t 1 + δ n t v t,n, where v t,t := V t, 1 T is the market value at time t of a unit ZCB maturing at time T the market discount factor at time t for the maturity T. The fair value of the non-guaranteed benefit is: C t = S t V t, Φ t,n Φ min t,n. The fair value of the guaranteed benefit can be immediately derived observing the value of market discount factor at the valuation date, whereas the computation of the fair value of the call component requires a more sophisticated approach, based on suitable stochastic models typically solved by appropriate numerical procedures e.g. see [DFM-02b], [DFM-05], [CDFMP-05]. Another useful expression is the put decomposition. If one defines: the terminal payoff can be decomposed as: ρ b k := min{β I k, I k h} i, 11 n Φ b t,n := 1 + ρ b k, S n = S t Φ b t,n + S t Φt,n Φ b t,n. 12 The first term is the base benefit, i.e. the benefit that would be paid at the term n if the policy would not provide any minimum guarantee. 4 The second term in 12 is the cliquet european put option providing the minimum guarantee the protective put. Considering the the fair value, one has the put decomposition of the stochastic reserve: V t := B t + P t := S t V t, Φ b t,n + S t V t, Φ t,n Φ b t,n. Usually, both the put component P t and the base component B t have to be evaluated using stochastic market models and numerical methods Policies with constant annual premium Let us now consider a policy with constant annual pure premium Π, where the sum insured is readjusted in each year t using the so-called t/n revaluation rule 6 : S t = S t ρ t S 0 n t n ρ t, 13 4 Assuming, as in practice occurs, that the investment portfolio is limited liability, then min I t = 1 and 11 can be obtained by posing δ = 1 h/ in 2. 5 There is an exception, sometimes occurring in the Italian practice. If β = 1, i = 0 and h = 0, then ρ b k = I k for all k and since: V¼t, n 1 + I k ½ =1 one has B t = S t. Even in this easy case there is no analogous easy formula both for the put component P t and for the call component C t. 6 This rule is a contractual standard in the Italian market and is also used for other classes of constant annual premium policies. 5

6 where ρ t is the revaluation rate defined by 2. The complete statutory reserve R t at time t has the traditional definition: n t 1 R t = S t n t Π n k S = t n t 1 n t 1 Π, if i > 0, i k=1 S t n t 1Π, if i = 0. Under the t/n profit sharing rule the statutory reserve is readjusted in each year at a rate approximatively equal to ρ t 7. Let us consider the valutation problem at policy year t. In [P] it was shown that the recurrent equation 13 has the closed form solution: S n = n t n S 0 + S t n t n S 0 Φ t,n + 1 n S 0 Θ t,n, 14 where Φ t,n is the full revaluation factor and: Θ t,n := n k tρ k Φ k,n = Φ k,n n t. 15 Using equation 14 we obtain the call decomposition for the terminal payoff: S n = n t n S 0 + S t n t n S 0 Φ min t,n + 1 n S 0 Θ min t,n + S t n t 16 n S 0 Φt,n Φ min 1 t,n + n S 0 Θt,n Θ min t,n, where: Θ min t,n := Φ min k,n 1 + δ 1 + δn t 1 n t, if δ > 0, n t δ 0, if δ = 0. is the contractual minimum value for the factor Θ t,n. Equation 16 is the call decomposition for the benefit of the annual premium policy with t/n revaluation rule. The second line is the optional component of the decomposition is a cliquet european option obtained as a complex combination of the 1-year optional components of the revaluation rates ρ t+1,..., ρ n ; the first line is the guaranteed component, which is a zcb. As for single premium policies, decomposition 16 induces a similar decomposition on the stochastic reserve: where: and: G t = C t = V t = V t, S n Π v t,k = G t + C t, [ n t n S 0 + S t n t n S 0 Φ min t,n + 1 ] n S 0 Θ min t,n v t,n Π S t n t n S 0 V t, Φ t,n Φ min t,n + 1 n S 0 V t, Θ t,n Θ min t,n. As for single premium policies, the fair value of the guaranteed benefit G t can be easily computed using the term structure of market discount factors observed at the valuation date, whereas the fair valuation of the guaranteed benefit is a more complex task, requiring the use of stochastic models. The same considerations apply to the obvious put decomposition. 7 The t/n revaluation rule was born as an approximation of the exact rule, for which the traditional statutory reserve increases each year at rate ρ t. Notice that under the assumption made, if i = 0 the statutory reserve increases exactly by ρ t each year. v t,k, 6

7 3 The representation of the investment strategy As in [DFM-05] we consider a constant-mix bond-equity strategy. The rate of return of the reference fund for year [t 1, t] is modelled as: I t := F t F t 1 F t := 1 q W t W t 1 W t 1 + q E t E t 1 E t 1, where: q is the equity proportion of the reference fund, W t is the market value of the portfolio bond component, E t is the market value of the equity component. The trading strategy of the bond component is modelled as a buy-and-sell strategy of a zcb with Macaulay duration D, with trading period t: every t years the zcb is sold and the resulting market price is completely reinvested in a new zcb with duration D. The unitary total return from this investment strategy after k trading periods is: k vr j t ; D t Wk t = vr j 1 t ; D. j=1 Different choices of D and t provide different volatility structures for Wt. To give the intuition of these effects we illustrate in figure 2 some simulated paths of the total returns generated by two alterntive strategies. Figure 2: Sample paths of investment return under alternative strategies Coc-Ingersoll-Ross model. 7

8 4 Closed form expressions In this section we derive closed form solutions for the stochastic reserve and for the call and put components, extendig the results of part IV of [CDFMP05], under the same simplified assumptions. Both single premium and constant annual premium policies are considered. As shown in section 2, to solve the valuation problem at time t for a policy with term n one has only to compute the valuation factors V t, Φ t,n and V t, Θ t,n. 4.1 The extended Black-Scholes model We assume that the market value of the investment fund F t has the following log-normal risk-neutral dynamics: df t = r t F t dt + σ F t dz t, 17 where r t is the risk free spot rate and Z t a standard Brownian motion. As it is well-known, the stochastic differential equation 17 has the closed form solution: where: F T = F t e r F t,t 1 2 σ2 T t+σz T Z t, for all t T, r F t,t = 1 T t T t r u du, is the continuously compounded forward rate in [t, T] on annual basis. We further assume that the risk free spot rate is deterministic; hence for any T s the forward rate r F T,s is known at the valuation date. In particular, the discount factor v T,s = exp r F T,s s T is known at time t. Finally, we further extend the model by assuming a time dependent implied volatilty structure: for every T s we denote by σ T,s the implied volatility of the reference fund for the period [T, s] on annual basis. 4.2 Derivation of the pricing formulae To simplify notations we shall denote by a single subscript the 1-year forward rate and volatility: r F T := rf T 1,T, σ T := σ T 1,T, for each T t + 1. Consider first the 1-year valuation problem for the unit factor Φ. Since: Φ t,t+1 = 1 + ρ t+1, we can apply to the call decompositions 5, 7 and 8 of section 2.1 the Black-Scholes pricing formula with risk free rate rt+1 F and volatility σ t+1. After some simplifications we obtain: [ Nd δ ] m + h Nd 2 e rf t+1, if β = 1, Nd 1 1 βnd 3 + V t, Φ t,t+1 = [ δ m + h Nd β + h ] if Nd 4 e rft+1, h + β < 1, 18 βnd 5 [ δ β + i ] m Nd i 6 e rf t+1 m, if β <, h + 8

9 where: d 1 = rf t+1 log m + h σ2 σ t+1, d 2 = d 1 σ t+1 = rf t+1 log m + h 1 2 σ2 t+1 σ t+1, 19 d 3 = rf t+1 log[1 + h/1 β] σ2 t+1 σ t+1, d 4 = d 3 σ t+1 = rf t+1 log[1 + h/1 β] 1 2 σ2 t+1 σ t+1, 20 d 5 = y t 1,t log[ m /β] σ2 t+1 σ t+1, d 6 = d 5 σ t+1 = rf t+1 log[ m/β] 1 2 σ2 t+1 σ t Remark. The value V t, Φ t,t+1 depends on the valuation date only trough r F t+1 and σ t+1, which are deterministic. Under our assumptions this implies that the 1-year value V T, Φ T,T+1 made at a future time T > t is known at the valuation date t. We are now able to tackle the general valuation problem. For each T t + 1 let us denote by u T = V T 1, Φ T 1,T the 1-year valuation factor. Using the 1-year valuation result 18 and the martingality and linearity of market prices we obtain: V t, Φ t,n = V t, Φ t, V n 1, Φ,n = V t, Φ t, u n Since: = u n V t, Φ t,n 2 V n 2, Φ n 2, = u u n V t, Φ t,n 3 V n 3, Φ n 3,n n = u k. Turning to the valuation of Θ t,n, which is needed for constant annual premium policies, we have by 15: one has: V t, Θ t,n = V t, Φ k,n n tv t,n. V t, Φ k,n = u k+1 u k+2 u n v t,k, V t, Θ t,n = v t,k n h=k+1 22 u k n tv t,n. 23 Remark. The valuation formulae 22 and 23 depend only on the term structure of the forward rates: { r F t+1, rt+2 F,..., rn F }, 24 and on the term structure of the unit valuation factors: {u t+1, u t+2,..., u n }. The u k factors in turn depend only on the term structure of forward rates and on the term structure of volatilities: {σ t+1, σ t+2,..., σ n }. 25 Hence the valuation factors are only determined by the term structures 24 and 25. Now we can substitute the closed form expression for the valuation factors in the equations of the stochastic reserve and of their components of section 2. For the single premium policy, after some simplifications we get: V t = S t n u k, 26 G t = S t 1 + δv t,n, 27 [ n ] C t = S t u k 1 + δv t,n. 28 9

10 For the constant annual premium policy one has: V t = S t n t n S n 0 u k + 1 n S 0 v t,k n h=k+1 u h Π v t,k ; 29 G t = S t n t n S 0 S t n t n S 0 = S t v t,n Π C t = 1 + δ n t v t,n + 1 n S [ 0 Θ min t,n + n t] v t,n Π 1 + δ n t v t,n + 1 n S δ 1 + δn t 1 δ v t,n Π v t,k ; v t,k, if δ > 0, v t,k, if δ = 0; S t n t [ ] n n S 0 u k 1 + δ n t v t,n n S 0 [ n v t,k S t n t n S n 0 h=k+1 ] u h 1 + δ 1 + δn t 1 v t,n, δ u k + 1 n S 0 v t,k n h=k+1 if δ > 0, u h S t v t,n, if δ = Remark. Equations are closed form expressions for the stochastic reserves and their call decompositions. In a similar way one can obtain closed form solutions for the put decompositions. Remark. The fair values derived depend only on the term structures of interest rates and volatilities of the investment portfolio at the valuation date. 4.3 After-shock valuations Shock on the fund s value In the valuation fomulae the time t value of the investment portfolio F t does not appears explicitely. However there is a hidden dependence that has to be considered in calculating derivatives or variations. Let us consider again the revaluation factor Φ t,t+1 referring for simplicity sake to the case β = 1 and h = 0. Writing equation 2 for t + 1 and expressing 1 + I t+1 as F t+1 /F t we have: 1 + ρ t+1 = 1 + ρ t+1 = max{1 + I t+1, m } = 1 { max F t+1, } m. F t F t Hence a contract with payoff 1 + ρ t+1 at time t + 1 is equivalent to a 1-year investment of one unit in the reference fund, with a minimum guaranteed return. More explicitely, the investor buys N = 1/F t shares of the fund and receives after one year the market value of the shares with a floor per share M = 1+ /F t ; that is the payoff: 1 + ρ t+1 = N max {F t+1, M}. If we assume that immediately after time t say at time t + a shock on the market occurs modifying the fund value but not the interest rate term structure, then the constants N and M, which are fixed at time t, remains unchanged, whereas the fund after-shock value Ft := F t + embodies the shock effects on the portfolio value. By the standard Black-Scholes formula the before-shock and after-shock values of the contract are: V t, 1 + ρ t+1 = N [M v t,t+1 + F t Nd 1 M v t,t+1 Nd 2 ], V t +, 1 + ρ t+1 = N [M v t,t+1 + F t Nd 1 M v t,t+1 Nd 2 ], 10

11 where d 1 and d 2 are defined in 19 here with h = 0 and d 1 and d 2 are: d 1 = log F t M rf t σ t+1, d 2 σ = d 1 σ t+1. t+1 Let η := F t /F t. Then by back-substituting M = m /F t, N = 1/F t and Φ t,t+1 = 1 + ρ t+1 / and denoting by V t, := V t +, the after-shock values, we get: V t, Φ t,t+1 = η N d σ log η [ δ m d N 2 + 1σ ] log η e rf t+1. Applying the same argument to the general case and posing u t+1 := V t, Φ t,t+1, after some calculations we obtain: N d 1 + log η σ η [ δ m + h N d 2 + log η ] e rf t+1, if β = 1, σ u t+1 = N η [ + d 1 + log η σ βn d 5 + log η σ η 1 βn 1+δ 1++h N d 3 + log η σ d 2 + log η σ β+h ] if N d 4 + log η σ e rf t+1, h + β < 1, [ δ β + i m N d 6 + log η ] i e rf t+1 m, if β <. σ h + 32 Therefore the effect of the shock the 1-year unit fair value is, also in the general case, the translation of d 1 and d 2 by log η/σ and the scaling of the first component by the η factor. Remark. For k > t + 1 the 1-year valuation factors u k are not affected by the value of the shock at time t. The after-shock valuation factors for single and annual premium policies are: V t, Φ t,n = u t+1 n u k = u t+1 V t, Φ t,n, u t+1 +2 V t, Θ t,n = V t, Φ k,n n tv t,n = V t, Φ t,n + u = t+1 1 V t, Φ t,n + V t, Θ t,n. u t+1 V t, Φ k,n n tv t,n Using these valuation factors in equations 26 and 28 we obtain the after-shock fair value and its nonguaranteed component for single premium policies. The guaranteed component G t does not changes. After some calculations we obtain: V t = u t+1 u t+1 V t and V t := V t V t = u t+1 1 V t, u t+1 G t = G t and G t := G t G t = 0, u Ct = C t + t+1 u 1 V t and C t := Ct C t = t+1 1 V t. u t+1 u t+1 11

12 The corresponding results for costant annual premium policies are: Vt = V t + S t n t 1 u S t V t, Φ t,n, n u t+1 G t = G t, Ct = C t + S t n t 1 u S t V t, Φ t,n, n u t+1 and: V t = S t n t 1 u S t V t, Φ t,n, n u t+1 G t = 0, C t = S t n t 1 u S t V t, Φ t,n. n u t+1 Remark. The shock type considered in this subsection is related to the equity shock in [QIS3]. If the investment portfolio is an all-equity portfolio the formulae given here can be used to obtain after-equityshock fair values in the sense of [QIS3] by posing η = 1 + ξ, where ξ is the combined volatility factor for the equity position in the investment portfolio. If the fund has an equity proportion q 0, 1, then we can also use these formulae posing η = q1 + ξ + 1 q = 1 + qξ Interest rate shock Another type of shock to be considered is the shock on the term structure of interest rates. An interest rate shock at time t + has two effects: i the value of the investment portofolio changes by a factor η; 8 ii the 1-year rates used in the valuation are shifted. For each k, let us denote by r k F the after-shock version of rf k. By the previous analysis it is clear that the 1-year unit value u t+1 has to be modified in order to reflect both the effects; that is: i the same modification by η factor as in 32 has to be applied; ii the after-shock term structure has to be used in the valuation formula 32 including the expressions of the d functions; Let ũ t+1 denote the after-shock 1-year unit value, obtained by modifying u t+1 as described in i and ii. As in the previous case, the shock effect on the value of investment portfolio does not affect the valuation factors u k for k > t + 1. However, because of the shift of all interest rates, the change in ii has to be applied. Let us denote by ũ k the 1-year unit factors obtained by replacing the interest rate rk F by rf k in the u k formula including the d functions. Futhermore, let us denote by Ṽ t, := V t +, the after-shock market value operator and by Ṽ t, the value computed as in section 4.2 but with the forward rates rf k instead of the rk F. The after-shock valuation factors for single and annual premium policies are: Ṽ t, Φ t,n = ũ t+1 n ũ k = ũ t+1 Ṽ t, Φ t,n, ũ t+1 +2 Ṽ t, Θ t,n = Ṽ t, Φ k,n n tṽ t,n = Ṽ t, Φ t,n + ũ = t+1 1 Ṽ t, Φ t,n + ũ Ṽ t, Θ t,n. t+1 Ṽ t, Φ k,n n tv t,n By using these valuation factors in equations 26 and 28 we obtain the after-shock fair value and its non-guaranteed component for single premium policies. Also the guaranteed component G t changes, because 8 Of course η 1 if and only if the fund has an interest rate sensitive component. 12

13 of the change in the discount factor v t,n. After some calculations we obtain: Ṽt = ũ t+1 Ṽ t ũ t+1 and V t := Ṽ t V t = ũ t+1 Ṽ t V t = ũ t+1 G t = G t and G t := G t G t = G, C t = ũ t+1 ũ t+1 1 Ṽ t + C t and C t := C t C t = ũ t+1 ũ t+1 Ct C t = ũ t+1 1 Ṽ t + ũ V t, t+1 ũ t+1 1 Ṽ t + ũ C t, t+1 where the operator applied to a value X t denotes the difference X t := X t X t. Of course: G t = S t vt,n. The analogous expressions for the constant annual premium policy can be obtained by simple but boring calculations. Remark. The shock considered in this subsection is similar to the interest rate shock defined in [QIS3]. The factor η has to be computed by measuring the effects of the shock on the interest rate sensitive component of the investment portfolio: if the equity proportion is q [0, 1] and if ζ is the percentage change in the interest rate sensitive component of the investment portfolio, then one has η = q + 1 q1 + ζ = qζ. 5 Numerical examples For illustration, we refer to the same numerical examples of [DFM-07, sec. 4.4]. Both single premium and constant annual premium policies are considered, with = 2.5%, 3% and 4% and with time-to-maturity τ = 10 years. All the policies have by construction statutory reserve R t = 100. We performed the valuation with respect to 8 possible trading strategies for the investment portfolio, given by all the combinations of the equity proportion q = 0%, 5%, 10% and 30% and the duration of the bond component D = 3, 4 years. We fixed the valuation date t = 31/12/2005. The Cox-Ingersoll-Ross CIR one-factor model has been calibrated on the interest rates both central and shocked specified in the technical document [QIS2]. In figure 3 the term structures of interest rates corresponding to the QIS2 scenarios dots joined by dashed lines and the CIR yield curves calibrated on these data solid lines are illustrated. Figure 3: Calibration of the CIR term structure on the QIS2 interest rates scenarios The volatility of the stock index was set at the level σ E = 25%. For the investment strategy a trading period t = 1 month has been assumed. Given these parameters, we calibrated the implied volatility function σ t considering the total return under the specified values of q, D and t. The form of the volatility functions for t 40 years is illustrated in figures 4 and 5. In tables 1 4 we present a comparison between the Monte Carlo and the closed form approach. The tables report figures for the key quantities of the valuation: technical provision TP, fair value of guaranteed 13

14 Figure 4: Term structures of implied volatility of returns for different portfolios and strategies with D = 3. Figure 5: Term structures of implied volatility of returns for different portfolios and strategies with D = 4. benefits TP G and of non-guaranteed benefits TP N, solvency capital requirement and reduction for profit sharing both for interest rate risk SCR int and RPS int and for equity risk SCR eq and RPS eq, and the K-factors for the two risk drivers K int and K eq. In table 1 we reproduce table 1 of [DFM-07], containing the Monte Carlo valuation results for single premium policies. Table 2 on the same page reports the same quantities computed with the closed form approximations. Tables 3 taken from [DFM-07, table 2] and 4 present the same comparison for constant annual premium policies. The results, though preliminar, looks promising: the differences are very small and close to the Monte Carlo error affecting figures of tables 1 and 3. 14

15 q% D % TP TP G TP N SCR int RPS int K int% SCR eq RPS eq K eq% Table 1: Single premium policies with τ = 10 years Monte Carlo valuation. q% D % TP TP G TP N SCR int RPS int K int% SCR eq RPS eq K eq% Table 2: Single premium policies with τ = 10 years closed form approximation. 15

16 q% D % TP TP G TP N SCR int RPS int K int% SCR eq RPS eq K eq% Table 3: Constant annual premium policies with τ = 10 and n = 20 years Monte Carlo valuation. q% D % TP TP G TP N SCR int RPS int K int% SCR eq RPS eq K eq% Table 4: Constant annual premium policies with τ = 10 and n = 20 years closed form approximation. 16

17 References [BS] Black F., Scholes M., The Pricing of Options and Corporate Liabilities, Journal of Political Economy, , 3. [CFO-04] CFO Forum, European Embedded Value Principles, [CDFMP-05] Castellani, G., De Felice, M., Moriconi, F., Pacati C., Embedded Value in Life Insurance, Working paper, March [CIR] Cox J.C., Ingersoll J.E. jr., Ross S.A., A Theory of the Term Structure of Interest Rates, Econometrica , [DFM-02a] De Felice, M., Moriconi, F., Finanza dell assicurazione sulla vita. Principî per l asset-liability management e per la misurazione dell embedded value, Giornale dell Istituto Italiano degli Attuari, LXV2002, 1-2. [DFM-02b] De Felice, M., Moriconi, F., A course on Finance of insurance, Groupe Consultatif Actuariel Europeen Summer School 2002, Milan. [DFM-04] De Felice, M., Moriconi, F., Market consistent valuation in life insurance. Measuring fair value and embedded options, Giornale dell Istituto Italiano degli Attuari, LXVII2004, 1-2. [DFM-05] De Felice, M., Moriconi, F., Market based tools for managing the life insurance company, Astin Bulletin, , 1. [DFM-07] De Felice, M., Moriconi, F., K-factor parameters. The determinants of risk reduction in the Italian with-profit life insurance policies, Working Paper, January [P] Pacati, C., The t/n Profit Sharing Rule, Research Group on Insurance Companies and Pension Funds. Valuation Models and Pension Funds, Working Paper 12, May [QIS2] CEIOPS, QIS2 Technical Specifications, CEIOPS-PI-08/06, [QIS3] CEIOPS, QIS3 Pre-Test Technical Specifications, February