On Surrender and Default Risks∗ Olivier Le Courtois †

Hidetoshi Nakagawa ‡

The rst version: July 29, 2010 This version: October 8, 2010 Abstract This article examines certain types of saving institutions or insurance companies that are subject to surrender and default risks, in a stochastic interest rate context. In the setting under study, investors are endowed with an option to surrender. The goal of the article is to study how this option impacts the default risk of the issuing company and the value of the contracts it issues. Surrender risk has been extensively studied in arbitrated markets, using trees or least-squares Monte-Carlo methods for valuations, although practitioners often rely on econometric methods. We deal with surrender risk in a third way, assuming policyholders have sets of information and preferences that dier from those of nancial market agents, but without relying on econometric methods. In particular, policyholders are supposed to be only partially rational (at least in the nancial sense). This is done by modeling surrender risk through a Cox process correlated to the assets and interest rate dynamics. The article provides formulas for the dynamics of the assets of the issuing rm (these dynamics drive the default time of the company), and for the valuation of liabilities and equity. A numerical illustration is provided.

Keywords Surrender risk, default risk, interest rate risk, Cox process, participating policies.

∗ This research was supported by Grant-in-Aid for Scientic Research (A) No. 20241038 from Japan Society for the Promotion of Science (JSPS). † O. Le Courtois is at the Center for Financial Risks Analysis (CEFRA), EM Lyon Business School, France. ‡ H. Nakagawa is at the Graduate School of International Corporate Strategy (ICS), Hitotsubashi University, Japan.

1

Introduction This article deals with the joint surrender and default risk of certain types of saving institutions and insurance companies issuing participating contracts. Examples of such contracts were considered and analyzed by Briys and de Varenne (1994, 1997, 2001) in the nineties; they guarantee a bonus proportional to the performance of the assets of the company. Since Briys and de Varenne, the literature expanded a lot. See, for instance, the works of Bacinello (2001), Miltersen and Persson (2003), Bernard, Le Courtois and Quittard-Pinon (2005), Tanskanen and Lukkarinen (2003), or Ballotta (2005), to cite only a few. In nance and insurance, studying surrender risk mainly means being able to price an option to pay back a credit in anticipation. This important issue arises in a number of situations where products (issued by both nance and insurance companies) are subject to a so-called repayment or surrender risk. This risk that the investor pays back in advance has been scrutinized within many academic publications, and a number of dierent methods have been employed in the last two decades. However, no unitary framework, or unanimous solution, has emerged until now. Among the various situations in which surrender risk appears, two frequent ones are the surrender of insurance (in fact savings) contracts and the prepayment of mortgage-backed securities. The nance literature addressing the second situation is by far the largest one. In contrast, the insurance literature developing on surrender risk of participating contracts is more restricted, though not less interesting. Surrender risk is dicult to assess because of its economic nature. People can surrender because of rational arbitrage possibilities (e.g. falling interest rates and therefore advantageous capacities to renance), apparently non rational behaviors, like seasonal surrenders, or due to personal considerations. Surrender risk is consequently not an easy risk to manage, occurring as it does from a number of causes that dier in nature. In particular, the traditional nance literature, since it is based on arbitrage-free valuation of options, can not be readily applied. Although surrender options are particular types of American options, they can not be priced with option-based methodologies. A seminal article from the nance literature is the one by Schwartz and Torous (1989). These authors developed one of the rst frameworks to price mortgage-backed securities, under a reasonable set of assumptions including the randomness of interest rates. Schwartz and Torous model their prepayment function as a typical proportional-hazard function where a log-logistic hazard function multiplies the exponential of several critical variables. Among these variables can be found the long-term interest rate, a simple indicator of seasonality, and an indicator of outstanding principal due. Their interest rate model lets the short term and the long term interest rates evolve. A number of articles modeled prepayment risk based on, or extending, the contribution of Schwartz and Torous. Another interesting article, that of d'Andria, Elie and Boulier (1991), models surrender risk in full generality. The analysis therein can nd applications in nance as well as in insurance. These authors model the speed of surrenders directly, taking into account the dierent causative factors. They do their computations based on a Vasicek model. The article by Albizzati and Geman (1994) 2

is one of the rst valuing surrender options in insurance contracts; it relies on predened functionals of prepayment rate. See also Pras (1998) for a use of regressions on simple analytical forms with respect to the main problem drivers. The works of Bacinello (2003), as well as Andreatta and Corradin (2003), are more recent, important studies in the actuarial literature on surrender risk. The latter papers develop on the pure market valuation of surrender options in insurance contracts. For instance, Bacinello relies on trees, while Andreatta and Corradin apply the now well-known Longsta and Schwartz (2001) least-squares Monte-Carlo method, to price these options as simple American or Bermudan options. In any of these studies, surrender risk is viewed as a purely nancial and rational risk, without any component (such as seasonality) exogenous to the market. As already mentioned, this is an interesting, but incomplete approach. In this article, we assume that investors surrender along a Cox-type model. That is, we suppose that surrenders are driven by an intensity, that may be deterministic or stochastic. The correlation linking this intensity to interest rate and stock market is a measure of how investors are rational in the nancial sense, and of how their behavior responds to personal, non-nancial motivations. We study the impact of adopting such a framework on the default risk of the issuer and on the prices of the contracts it issues. We dene the general setting in section 1 and derive relationships for the assets' dynamics in section 2. In section 3, we concentrate on the pricing of liabilities and equity. Section 4 illustrates the results obtained in sections 2 and 3, while section 5 concludes the study.

1 Framework We start by dening the general framework that we use for the modeled companies and surrenders. 1.1

Qualitative description of the problem

Let us consider an asset management entity (for example, an investment trust fund or an insurance company that issues participating contracts) that is supposed to satisfy the following qualitative assumptions. 1. Initial fund structure: The entity collects the resources at time 0 equally from a number of individuals to make a fund pool. In return for the fund, the entity gives some contingent claim to the participants. The entity supposes that all the participants are common in terms of the contract conditions, anticipations, economic information and attitude toward risk. Additional capital can be levied in the form of equity at time zero. 2. Fund management: The entity can trade a non-defaultable risky asset in the nancial market. The entity starts to invest its assets at time 0 and liquidates them at maturity or when a default condition is satised. Specically, the entity liquidates the fund pool before the maturity if the value of its assets falls down below a given threshold. Moreover, the entity admits no additional funds on the way (we make this assumption to simplify the exposition: it is trivial to remove this restriction), while 3

early-withdrawals, or surrenders (hereafter we use surrenders for abuse of usage) in units of one share of the fund pool are allowed. 3. Surrenders: At each surrender, the entity cashes out some amount from the fund pool, and then pays o a part of the cash to the participant who surrenders while the remaining cash is withheld as a management cost. The payo rule to participants at surrender is predetermined. The entity supposes that surrenders by the participants are regarded as sudden events rather than as the result of value maximization through the option of surrender. If all the participants are rational, they should surrender at the same time when it is the most optimal to exercise the surrender option. However such coincidence is hardly observed as a matter of fact, and it is not unnatural to suppose that the fund entity regards surrenders as sudden events. 4. Default: Upon default, the company is liquidated and the proceeds are equally distributed according to the contract terms to the remaining participants. 5. Liquidation at maturity: The fund pool is naturally liquidated at the maturity and the proceeds are equally distributed according to the contract terms to the remaining participants  unless an early liquidation occurred or there are no remaining participants. Moreover, the entity guarantees some prexed amount of payo to the remaining participants at the maturity even if the asset has not performed so well. The remainder after paying all the participants is passed on to the equity holder. The main problem is to achieve the quantitative relationship between the value of the assets and each contingent claim value in consideration of the possibility of default as well as surrender. 1.2

Mathematical description of the problem

Let (Ω, A, P ) denote a complete probability space. We will also consider another probability measure Q equivalent to P . We will regard Q as a chosen  though not unique  risk-neutral probability measure (its nature will be discussed afterwards). We denote by W and W Q one-dimensional Brownian motions under the probability measures P and Q, respectively, whereas r is the stochastic process that stands for the default-free instantaneous interest rate. We also make use of the ltration F = {Ft , t ∈ R+ }, where

Ft := σ{rs , Ws | s ≤ t}.

(1)

This ltration (and all the ltrations dened below) is supposed to be rightcontinuous and complete. F stands for the market information generated by the prices of tradable assets, interest rate and so on. We traditionally assume for any process Y , Yt− := lim Ys , if this limit exists. Now we present addis↑t

tional notations to mathematically describe the problem, based on the previous subsection. 4

1. Assets I: Let T ∈ (0, ∞) be the maturity of the investment. We denote by A the dynamics of the assets of the company. In the case of no liquidation and no surrenders, we suppose that in the real world
dAt = At µt dt + σt dWt . where µ and σ are F -adapted processes that satisfy some technical conditions so as to ensure the existence of a solution. In this situation, A is F -adapted (it will not be the case in the forthcoming developments). At time 0, when the company is constituted, the amount A0 is levied from both policyholders and stockholders. 2. Threshold process I: Denote by L the threshold process that triggers the liquidation of the company before the maturity (see below for the specication of this event). 3. Surrenders: We denote by τ i (i = 1, · · · , I0 ) the surrender time of the ith participant, where I0 is the initial number of policyholders. Throughout this paper, we assume that simultaneous surrenders almost surely do not occur; that is, P (τ i = τ j ) = 0 for any i, j (i 6= j). We assume that the entity views surrender events as sudden. More specically, we do not assume that the τ i 's are F -stopping times, which amounts to relying on reduced-form credit risk valuation. Let also Nti = 1{τ i ≤t} be the indicator function of the ith participant's surrender. Accompanying this, we dene the number of cumulative surI0 X Nti . renders up to time t by Nt = i=1

From the assumption of (almost surely) no simultaneous surrenders, it follows that N can jump with size 1 almost surely. N is by nature a counting process. Then, we dene the ltrations H = {Ht , t ∈ R} and G = {Gt , t ∈ R} by

Ht = σ{Ns | s ≤ t},

Gt = Ft ∨ Ht .

We can see H as the surrender information and G as the full information. We suppose that W , A, and L are G -adapted, similarly to the τ i 's. We suppose that some amount is withdrawn from the assets at each surrender. Denote by ϕ a G -adapted càdlàg positive process that stands for the withdrawal from the assets upon surrender. Let A¯ be the process that stands for the value of one share of the assets for each investor who has not yet surrendered. In formal terms, we write at any time t1 At , (2) A¯t := I0 − Nt where I0 − Nt is the number of remaining policyholders. 1 By convention, viewing At = 00 as zero after the time when Nt comes to I0 implies I0 −Nt

¯ t dened that (2) is actually well dened at any time t. The same argument holds for L afterward.

5

We also dene the threshold process for a single participant who has not surrendered until time t by

¯ t := L

Lt . I0 − Nt

(3)

¯ t can be viewed as the discount value of the minimum The quantity L amount guaranteed to the participant who has not surrendered until maturity T . Finally, the quantity ϕt can be decomposed into the payo to the partic¯ t− ), where F¯S (., .) ipant who surrenders at time t, denoted by F¯S (A¯t− , L is a positive measurable function, and into a remaining positive amount that corresponds to management costs. 4. Assets II: On the basis of the above framework for surrenders, we can now present our model for the dynamics of the assets A of the company

dAt = At− (µt dt + σt dWt ) − ϕt− dNt   ϕt− dNt . = At− µt dt + σt dWt − At−

(4)

Quantities are taken at time t−, for a surrender occurring at time t, because time t refers to the dynamics after the rebalancing of the assets. 5. Default: The investment vehicle can be liquidated in anticipation due to default, which can occur at time τd dened by

τd = inf{t ∈ (0, T ] | At ≤ Lt }.

(5)

In other words, the company enters default when the value of the assets drops below the threshold process. Note that it is not apparent under these conditions whether the liquidation time τd is an F -stopping time. Note also that because of (2) and (3), τd can be expressed as

¯ t }. τd = inf{t ∈ (0, T ] | A¯t ≤ L

(6)

The contingent payo at the liquidation time is given by

F¯ D (A¯τd ), where F¯ D (.) is a measurable positive function. 6. Liquidation at maturity: At the maturity, the rm is naturally liquidated. ¯ T a constant that represents the amount guaranteed at the Denote by L maturity T for each participant. The contingent payo at the maturity ¯ T ), where F¯ (., .) is a measurable positive can be generally given by F¯ (A¯T , L function of two variables (or one variable in some instances). 7. Threshold Process II: Let us now present our model for the threshold process L that guides the potential default of the rm   dNt dLt = Lt− ρg dt − . (7) I0 − Nt− 6

This equation implies that Lt increases deterministically with time if no dNt surrender happens while it jumps downwards by IL0t− −Nt− if one surrender occurs at time t. Intuitively, Lt can be considered as the entity's total discounted critical liability that is implied from the minimum guarantee at maturity. One surrender implies a reduction of the remaining participants, and therefore, as with the assets, leads to a proportional reduction of the default threshold.

Remark 1.1.

We discuss here the range of the payo functions F¯ S (a, `), F¯ D (a), ¯ t− ) ≤ ϕt− . What is given and F¯ (a, `). First, we have by denition F¯ S (A¯t− , L ¯ t− ) to the participant who surrenders at time t is inferior as a payo F¯ S (A¯t− , L to ϕt− , which is the total cost of this surrender to the company. The dier¯ t− ) constitutes the management costs that are incurred ence ϕt− − F¯ S (A¯t− , L upon surrender. Second, it is clear that the contingent payo F¯ D (A¯τd ) at the liquidation time τd satises the inequality F¯ D (A¯τd ) ≤ A¯τd due to the tacit principles of equal sharing and limited liability. Looking at the total assets, this means that, upon default, not more can be redistributed to policyholders than what is remaining in the rm's possession. At the scale of policyholders, this means that, upon default, no policyholders are advantaged at the expense of ¯ T ) at the maturity should satisfy others. Finally, the contingent payo F¯ (A¯T , L ¯ ¯ ¯ ¯ ¯ LT ≤ F (AT , LT ) ≤ AT due to the minimum guarantee and the tacit principle of equal sharing. The rst inequality is necessarily achieved if the rm could survive until T without defaulting: by denition at that time the rm has enough cash to fulll its duties. The second inequality simply means that the rm cannot distribute more than what remains in its possession, and that distributions are done among policyholders on a fair basis.

2 Mathematical results on the assets and threshold values In this section, we use the semimartingale framework of linear stochastic differential equations to obtain explicit representations in the real world of the solutions to the stochastic dierential equations (4) and (7) given in the previous section. 2.1

On the assets dynamics

A

As a preparation for the discussion below, dene the continuous (P, F)-local martingale E(σ · W ) by   Z t 2 Z t σu du + σu dWu . (8) E(σ · W )t := exp − 0 0 2

2.1.1 General case The next lemma gives a direct representation of the general equation (4) describing the evolution of the assets process.

7

Lemma 2.1. The unique solution to

(4) is given by  Z t µu du E(σ · W )t At = A0 exp 0  Z t Z t E(σ · W )t µu du exp − ϕs− dNs . E(σ · W )s s 0

(9)

Proof. See Appendix A.1. Dividing formula (9) by the number of remaining participants, we obtain on {I0 > Nt }  Z t A0 ¯ µu du E(σ · W )t At = exp I0 − Nt 0  Z t Z t E(σ · W )t 1 µu du exp − ϕs− dNs . (10) I0 − Nt 0 E(σ · W )s s Also, we can easily express the stochastic dierential equation that A¯ satises.

Lemma 2.2.

A¯

satises the next stochastic dierential equation on {I0 > Nt } dA¯t = A¯t− (µt dt + σt dWt ) +

A¯t− − ϕt− dNt . I0 − Nt

(11)

Proof. See Appendix A.2. It is important to remark that in this framework, if a surrender happens at time τ , A¯ jumps as follows

A¯τ − − ϕτ − A¯τ = A¯τ − + , I0 − Nτ which expresses that when a surrender occurs, the assets that were attached to the surrendering policyholder are not necessarily completely consumed if A¯τ − > ϕτ − . In that situation, the dierence A¯τ − − ϕτ − is distributed to the remaining I0 − Nτ policyholders. On the contrary, when the management costs incurred upon surrender are big enough for A¯τ − < ϕτ − , then the cashow A¯τ − − ϕτ − aects all remaining policyholders. The latter situation is the one where surrender depletes progressively the assets until a critical level is reached and, thus, where surrender induces default. Remember from (6) that

¯ t }. τd = inf{t > 0 | A¯t ≤ L

(12)

It is important to remark that in full generality τd is not an F -stopping time because A¯t depends on the history of N up to time t, as can be seen from (10). This is one of the most essential characteristics of this framework.

8

2.1.2 Case

ϕt = A¯t

Next we consider the case where the entity cashes out one participant's share of the assets at each surrender. Equivalently, we suppose in the rest of this subsection that the withdrawal ϕt at surrender is given by ϕt = A¯t , that is, the dynamics of At is specied by   dNt . (13) dAt = At− µt dt + σt dWt − I0 − Nt− In this special case, the solution of equation (13) has some good properties, and is useful to achieve an explicit representation of the solution Lt to the equation (7), as we will see later. The following lemma gives the solutions of (13).

Lemma 2.3. The solution to

(13) is represented by Z t  I 0 − Nt . At = A0 exp µu du E(σ · W )t I0 0

(14)

Proof. See Appendix A.3. It is interesting to observe in this case that the value of the assets become a simple product of the value of the assets of an otherwise equivalent rm allowing no surrenders and of the remaining number of participants. To understand this feature more precisely, we give the dynamics of the assets per remaining participant on the basis of formula (14) Z t  ¯ ¯ At = A0 exp µu du E(σ · W )t , (15) 0

where obviously A¯0 = Thus, A¯t satises2

A0 I0 .


dA¯t = A¯t µt dt + σt dWt .

(16)

which can also be obtained by replacing directly ϕt = A¯t in equation (11). Equations (15) and (16) show us that, under the assumption ϕt = A¯t , A¯ remains unchanged upon a surrender event and is a continuous process (as long as µ and σ are continuous). In this setup, policyholders do not see the eect of the surrenders of other policyholders. This is of course an over-simplied setup, especially due to the natural features of the contracts sold in practice. 2.2

On the liquidation threshold

L

In order to achieve an explicit representation of the solution Lt to equation (7) of the threshold process, it is useful to consider the case of ϕt = A¯t discussed in the latter part of the previous subsection. The reason for this is that the jump dNt part of (13) coincides with that of (7). I0 − Nt− The following result can be obtained as a corollary of Lemma 2.3; it is valid whatever the specication of ϕ. 2 A distinct proof can be achieved by applying It o's lemma for semimartingales to Formulas (13) and (2). See Theorem 4.57 in Jacod and Shiryaev (2003).

9

Corollary 2.4. The solution to

(7)

is explicitly represented by

Lt = L0 eρg t

I0 − Nt . I0

(17)

Proof. Notice that equation (7) is equivalent to equation (13) of At under the

assumption of ϕt = A¯t if we replace µu and σu in (13) with ρg and 0 respectively. Therefore the explicit representation in the middle term of (17) is straightforward from Lemma 2.3. From this result, it follows that

¯ t = L0 eρg t , L I0 or equivalently that

(18)

¯t = L ¯ t ρg dt dL

¯ is deterministic, continuous, and not impacted by surrender meaning that L events. This property is independent of the specication of ϕt and of the number of surrenders. From a nancial point of view also, this makes sense: the critical liability (threshold) to individual policyholders should be independent of the size of the pool to which they pertain, whatever the type of contracts that are sold.

Remark 2.5. Now, we can look at τd when ϕt = A¯t . It follows from (12), (15), and the above discussion that Z t  n o ¯ ¯ T e−ρg (T −t) , τd = inf t > 0 | A0 exp µu du E(σ · W )t ≤ L (19) 0

and thus we nd that τd is an F -predictable stopping time in this situation. In full generality, its law follows from simulations3 . 3 If both the expected return and the asset volatility are constant, that is, µ ≡ µ and t

σt ≡ σ , then we see the following   2 ¯0 exp µ − σ t + σWt ≤ L ¯ T e−ρg (T −t) A 2  ¯T  1 σ2 L 1 µ− − ρg t + ρg T − log ¯ ≤ 0. ⇐⇒Wt + σ 2 σ A0

We can use a general result on the reection principle of Brownian motion (see, for example, Lemma 3.1.2 in Bielecki and Rutkowski (2002)) to obtain   σ2 2  P (τd ≤ s) = Φ(f1 (s)) + exp − 2 µ − − ρg ρg T − log σ 2

¯ T  L Φ(f2 (s)), ¯0 A

where Φ(·) is the standard normal distribution function and  ¯T   L 1 ρg T − log ¯ + µ − f1 (s) = − √ σ s A0  ¯T   1 L ρg T − log ¯ − µ − f2 (s) = − √ σ s A0

10

  σ2 − ρg s , 2   σ2 − ρg s . 2

3 Valuation and capital structure of the rm We set ourselves in a valuation context, namely, in a context where A, L, N ... are considered in the risk-neutral world where the drift of A is the risk-free rate process r. The indicator process of early liquidation M , dened ∀t by Mt = 1{τd ≤t} , is also viewed in the risk-neutral universe, similar to τd . The main question that arises is, how can the number of surrenders N , viewed typically as a Cox process of intensity λ, be passed from the universe P to a universe Q. Consider the Girsanov theorem for marked point processes (with a marker equal to 1): the change of measure guided by the local martingale ξ dened by dξt = ξt− (ζ − 1) (dNt − λt dt), where ζ is a positive predictable process, allows N to be viewed as a Q-Cox process of intensity ζλ. ζ , assumed constant in the rest of this article, can be understood as a risk premium that arises in equilibrium between rms and investors, which is translated into an additional loading for surrender risk  a technical risk to use an actuarial term. It is important to note that the Q measure can not be identied univocally (surrender risk not being hedgeable using traded assets). We indeed choose a Q measure, among others, by specifying the parameter ζ . Note also that in full generality incompleteness can arise from the stochasticity of the volatility σ . 3.1

Market value of liabilities w.r.t. policyholders

We start with a denition.

Denition 3.1 (Liabilities w.r.t. policyholders). We dene the market value of the total liability of the rm with respect to policyholders by the following expression V0 := V01 + V02 + V03 ,

where h i RT ¯T ) , V01 := EQ (I0 − NT )(1 − MT )e− 0 ru du F¯ (A¯T , L "Z # T R 2 − 0s ru du ¯ D ¯ V0 := EQ (I0 − Ns− )e F (As− )dMs ,

(20) (21)

0

V03 := EQ

"Z

T

(1 − Ms )e−

Rs 0

# ru du

¯ s− )dNs . F¯ S (A¯s− , L

(22)

0

These three terms admit the following meaning. V01 stands for the value of the total payo distributed at maturity T to the participants who have not surrendered. Clearly, 1 − MT is the indicator of the event no liquidation until time T , and I0 − NT is the number of the remaining participants at T . V02 stands for the value of the total payo granted upon liquidation before maturity. This expectation incorporates an integral over all the possible liquidation times: if the entity liquidates the assets at time τd , then dMτd = 1, and the amount F¯ D (A¯τd ) is distributed to each and any of the I − Nτd − participants still with the company at that time. Finally, V03 is the value of the cumulative payo distributed (before maturity or early liquidation time) to the participants surrendering their contracts. Again, an integral appears, but now it is performed along surrender times. Each time τ i a surrender occurs, dNτ i is equal to 1, 11

¯ τ i ) is distributed to the participant, provided liquidation has not and F¯ S (A¯τ i , L occurred yet (equivalently 1 − Mτi − = 1). Note also that not (1 − Ms− ) but (1 − Ms ) is used in the integrand because we suppose that liquidation occurs prior to surrender. Using an actuarial term, V0 could be described as the market-consistent value (according to the rm) of the future payos to participants, meaning that V0 consists of the risk-neutral expectation of the future payos to participants discounted at the risk-free rate. 3.2

Equity

We start with a useful lemma. Z t

Lemma 3.2. Assume that

A¯s− σs dWsQ

0

h

A0 = EQ (I0 − NT )(1 − MT )e

−

RT 0

ru du

is a (Q, G)-true martingale. We have i

A¯T + EQ

"Z

T

(I0 − Ns− )e

−

Rs 0

# ru du

A¯s− dMs

0

"Z

T −

(1 − Ms )e

+ EQ

Rs 0

# ru du

ϕs− dNs .

(23)

0

Proof. From Lemma 2.2, it follows that    Rt  Rt A¯t− − ϕt− d e− 0 ru du A¯t = e− 0 ru du A¯t− σt dWtQ + dNt . I0 − Nt In addition, by the integral-by-part formula (in dierentiation form) and the above equation, we have   Rt d (I0 − Nt ) e− 0 ru du A¯t  Rt  h i Rt R· = (I0 − Nt− ) d e− 0 ru du A¯t − e− 0 ru du A¯t− dNt − d N, e− 0 ru du A¯ s   R Rt ¯t− − ϕt− A Q − 0t ru du − r du = (I0 − Nt− ) e A¯t− σt dWt + dNt − e 0 u A¯t− dNt I 0 − Nt Rt A¯t− − ϕt− d [N, N ]s − e− 0 ru du I0 − Nt Rt = (I0 − Nt− ) e− 0 ru du A¯t− σt dWtQ

− Nt− )(A¯t− − ϕt− ) − (I0 − Nt )A¯t− − (A¯t− − ϕt− ) dNt I0 − Nt Rt Rt = (I0 − Nt− ) e− 0 ru du A¯t− σt dWtQ − e− 0 ru du ϕt− dNt . (24) + e−

Rt 0

ru du (I0

The third equality follows from the property of [N, N ]t ≡ Nt , and the last equality follows from the relation I0 − Ns− − 1 = I0 − Ns if dNs = 1. Note that the rst expectation term ofR the right-hand side in (23) contains the product of T (1 − MT ) and (I0 − NT )e− 0 ru du A¯T . Using equation (24) and the integral-by-

12

part formula again, since N0 = M0 = 0, we have R

T (I0 − NT )(1 − MT )e− 0 ru du A¯T Z T   Rt = I0 A¯0 + (1 − Ms− )d (I0 − Nt ) e− 0 ru du A¯t

0 T

Z

(I0 − Ns− )e−

−

Rs 0

ru du

h i R· A¯s− dMs − M, (I0 − N )e− 0 ru du A¯

0 T

Z

(I0 − Ns− )(1 − Ms− )e−

= A0 + Z

0 T

(1 − Ms− )e−

−

Rs 0

ru du

Rs 0

ru du

T

A¯s− σs dWsQ

ϕs− dNs

0

Z −

T

(I0 − Ns− )e−

Rs 0

ru du

A¯s− dMs −

Z

T

e−

Rs 0

ru du

ϕs− d[M, N ]s .

0

0

The last term of this equation corresponds to a surrender payment made upon default. This is a situation that we do not want to allow: when a surrender induces default, namely when N and M occur simultaneously, we assume the rm is liquidated and the surrendering participant receives a payment F¯ D but not F¯ S . We thus equate the last term to zero and Ms− can be substituted for Ms in the integrand of the third term. Finally, we obtain R

T (I0 − NT )(1 − MT )e− 0 ru du A¯T Z T Rs = A0 + (I0 − Ns− )(1 − Ms− )e− 0 ru du A¯s− σs dWsQ

Z −

0 T

(I0 − Ns− )e

−

Rs 0

ru du

A¯s− dMs −

0

Z

T

(1 − Ms− )e−

Rs 0

ru du

ϕs− dNs .

0

Moreover, Z tthe Q-expectation of the Rsecond term of the right-hand side vans ishes since (I0 − Ns− )(1 − Ms− )e− 0 ru du A¯s− σs dWsQ can be considered as 0

a (Q, G)-true martingale due to the boundedness of the rst three terms in the integrand and to the hypothesis of the lemma. This yields the result of the lemma. Z t Remark 3.3. It can be checked that in most situations X = A¯s− σs dWsQ is 0

a (Q, G)-true martingale. It is naturally a local martingale, and Corollary 4 to Theorem 26 of Chapter II in Protter (2005) tells us that if EQ ([X, X]+∞ ) < +∞, then we obtain a (in fact square-integrable) martingale. It can be checked, for instance, that the Heston specication of volatility (CIR process) satises this constraint. The above lemma rst allows us to check a coherence principle.

Proposition 3.4. Assume that the conditions of Lemma 3.2 hold. Moreover if ¯ t− ) ≤ ϕt− for any t ∈ (0, T ], and if for any a > 0 F¯ S (A¯t− , L F¯ (a, .) ≤ a, F¯ D (a) ≤ a,

(25)

then V0 ≤ A0 . In particular, if all the equalities in the above inequality conditions hold, we can easily see that V0 = A0 is satised. 13

Proof. Since V0 = V01 + V02 + V03 , and applying the conditions of the proposition to equations (20), (21) and (22), we can write "Z # T i h R R − 0T ru du ¯ − 0s ru du ¯ AT + EQ (I0 − Ns− )e As− dMs V0 ≤ EQ (I0 − NT )(1 − MT )e 0

"Z

T

+ EQ

(1 − Ms )e−

Rs 0

# ru du

ϕs− dNs

0

= A0 , where the last equality follows from Lemma 3.2. We can then approach the denition of equity.

Proposition 3.5.

A0 − V 0

can be decomposed as follows

A0 − V0 := E01 + E02 + D0 ,

where h RT  i ¯T ) , E01 := EQ (I0 − NT )(1 − MT )e− 0 ru du A¯T − F¯ (A¯T , L "Z # T Rs  E02 := EQ (I0 − Ns− )e− 0 ru du A¯s− − F¯ D (A¯s− ) dMs ,

(26) (27)

0

"Z

T

(1 − Ms )e

D0 := EQ

−

Rs 0

ru du

#  S ¯ s ) dNs . ϕs− − F¯ (A¯s− , L

(28)

0

Proof. Directly, based on the denition of V0 and Lemma 3.2.

Remark 3.6 (Present value of management costs).

The quantity D0 dened by equation (28) admits a natural interpretation. It corresponds to the discounted value of the management costs that are incurred between time 0 and time T . From there, we obtain the following denition of equity.

Denition 3.7 (Equity). We dene the market value of equity as the dierence between the value of the assets and the sum of the market value of the liabilities with respect to policyholders and of the discounted future management costs, namely as E0 := A0 − V0 − D0 = E01 + E02 .

(29)

Equity (positive, as the sum of two positive quantities) is logically the sum of what remains to stockholders at the maturity (if it is reached) and of what goes to stockholders (admittedly due to deviations from the absolute priority rule) upon default. We have thus constructed a coherent framework for valuing the various liabilities of a nancial company issuing contracts that can be surrendered. Using actuarial terms, E0 can be viewed as the market-consistent value of the prots provided by the business to the stockholders of the company, or as an embedded value, or as a value of business in force (beyond this, allowing for the arrival of new contractors would lead to the creation of a value of future business). 14

3.3

Case study

This subsection further explains equations (20), (21), and (22) in the case where ϕt = A¯t . We replace A¯t− with A¯t in the expressions because here A¯t is continuous due to (15), as long as µ and σ are continuous. Furthermore, we suppose that N is independent of F , that is, independent of A¯, r, and hence M , throughout the rest of this subsection, so that policyholders are supposed to surrender on personal considerations alone, and independent of the markets. Besides, we specify the payo functions of the contingent claim as follows

F¯ (a, `) = γT · a, F¯ D (a) = a, F¯ S (a, `) = γS · a, where both γT (∈ (0, 1]) and γS (∈ (0, 1]) are constants. Rt For subsequent discussions, remember that e− 0 ru du A¯t = A¯0 E(σ · W Q )t due to (15), where E(σ · W Q )t , given in (8), is assumed a (Q, G)-true martingale. At rst, thanks to the additional assumption of independence, we can write as follows h i RT V01 = (I0 − EQ [NT ])EQ (1 − MT )e− 0 ru du γT A¯T   = (I0 − EQ [NT ])γT A¯0 EQ (1 − MT )E(σ · W Q )T . (30) It is possible to compute the expectation EQ [NT ] if the intensity model of NT is specied, while the second expectation can be developed in a closed or ¯ ). Next, semi-closed form (see Bernard et al. (2006) for simple expressions of L we note that the following equality can be obtained easily R

T (1 − MT )e− 0 ru du A¯T = (1 − MT )A¯0 E(σ · W Q )T (Z ) Z T T
Q Q = A¯0 (1 − Ms− )E(σ · W )s dNs + Ns− d (1 − Ms )E(σ · W )s .

0

0

Therefore, we can express V03 as follows "Z # T V 3 = γS A¯0 EQ (1 − Ms− − ∆Ms )E(σ · W Q )s dNs 0

0



  = γS A¯0 EQ (1 − MT )E(σ · W Q )T EQ [NT ] "Z # "Z T
Q − EQ Ns− d (1 − Ms )E(σ · W )s − EQ 0

Q

#

E(σ · W )s d[M, N ]s

0

( = γS A¯0

T

  EQ (1 − MT )E(σ · W Q )T EQ [NT ] − EQ

"Z

#)

T

Ns− d (1 − Ms )E(σ · W Q )s

0

The last equality follows from [M, N ] ≡ 0 because of the independence be-

15




.

tween Mt and Nt . Moreover, we have "Z # T
Q EQ Ns− d (1 − Ms )E(σ · W )s 0

"Z

#

T Q

= −EQ

"Z

Q

Ns− E(σ · W )s dMs + EQ 0

"Z = −EQ

#

T

Ns− (1 − Ms− )dE(σ · W )s 0

#

T Q

Ns− E(σ · W )s dMs , 0

where the last equality follows from the fact that E(σ · W Q )s is not only a (Q, F)-martingale, but also a (Q, G)-martingale. Finally, we obtain ( "Z #) T   V 3 = γS A¯0 EQ (1 − MT )E(σ · W Q )T EQ [NT ] + EQ Ns− E(σ · W Q )s dMs . 0

0

Hence, from (21) and (31), it follows that   V02 + V03 = γS A¯0 EQ (1 − MT )E(σ · W Q )T EQ [NT ] "Z # T + A¯0 EQ (I0 − (1 − γS )Ns− ) E(σ · W Q )s dMs .

(31)

(32)

0

¯τ = L ¯ T e−ρg (T −τd ) follows from As a matter of fact, the equality A¯τd = L d ¯ the a.s. path continuity of At . This implies, formally, A¯0 E(σ · W Q )s dMs = e−

Rs 0

ru du

¯ T e− A¯s dMs = L

Rs 0

ru du−ρg (T −s)

dMs .

Thus, the second term of (32) can be represented as follows "Z # T

¯ T EQ L

(I0 − (1 − γS )Ns− ) e−

Rs 0

ru du−ρg (T −s)

dMs

0

¯T =L

Z

¯T =L

Z

T

i h Rs EQ (I0 − (1 − γS )EQ [Ns− ]) e− 0 ru du−ρg (T −s) τd = s Q(τd ∈ ds)

T

h Rs i (I0 − (1 − γS )EQ [Ns− ]) e−ρg (T −s) EQ e− 0 ru du τd = s Q(τd ∈ ds).

0

0

(33)

We can replace EQ [Ns− ] with EQ [Ns ] if Nt has a continuous compensator. Assume, hereafter, that Nt has a continuous compensator. Grouping (30), (32), and (33), we nally obtain   V0 = A¯0 {γT (I0 − EQ [NT ]) + γS EQ [NT ]} EQ (1 − MT )E(σ · W Q )T Z T h Rs i ¯T +L (I0 − (1 − γS )EQ [Ns ]) e−ρg (T −s) EQ e− 0 ru du τd = s Q(τd ∈ ds). 0

16

Finally, we note that

¯t) > 0 MT = 0 ⇐⇒ τd > T ⇐⇒ min (A¯t − L 0≤t≤T  Z t  Z t o n σu2  Q ¯ T e−ρg (T −t) > 0 ¯ ru − σu dWu − L ⇐⇒ min A0 exp du + 0≤t≤T 2 0 0 Z t nZ t ¯T L σ2  o ⇐⇒ min σu dWuQ − log ¯ + ρg (T − t) + ru − u du > 0. 0≤t≤T 2 A0 0 0   Thus, we expect that EQ (1 − MT )E(σ · W Q )T can be computed by a joint distribution of a Brownian motion and its R t minimum up to time T through time change techniques so as to transform 0 σu dWuQ into a Brownian motion, as long as µ and σ are deterministic. 4

4 Numerical illustration In this section, we consider as an illustration a type of participating contract that some insurance companies issue and that is dened by the payo functions

F¯ (a, `) = ` + max{δ(α · a − `), 0}, F¯ D (a) = a, F¯ S (a, `) = `, where both α and δ are constants contained in (0, 1]. Such a specication of F¯ (a, `) is similar to the payo of a participating contract discussed in Briys and de Varenne (1994, 1997, 2001) and Bernard, Le Courtois and Quittard-Pinon (2005). We assume ϕt = β A¯t , where β is a positive constant. The specication of S ¯ F (a, `) = ` implies that it is relatively inconvenient for the participants to 4 If r ≡ r and σ ≡ σ , where r and σ are constants, let t t

Yt := WtQ +

1 σ

 r−

σ2 − ρg 2

 t+

1 σ



¯T  L ρg T − log ¯ . A0

We can use Cor. 3.1.2 in Bielecki and Rutkowski (2002) to achieve for y ≥ 0    ¯ T  L σ2 2 ρg T − log ¯ Q(YT ≥ y, τd ≥ T ) = Φ(g1 (T )) − exp − 2 r − − ρg Φ(g2 (T )), σ 2 A0

where Φ(·) is the standard normal distribution function and 



¯T   L ρg T − log ¯ + r− A0 σ T     ¯T 1 L g2 (T ) = √ −σy − ρg T − log ¯ + r− A0 σ T

g1 (T ) =

1 √

−σy +

σ2 − ρg 2 σ2 − ρg 2

  T ,   T .

So, we have    h i σ2 T EQ (1 − MT )E(σ · W Q )T = EQ (1 − MT ) exp − + σWTQ 2   ¯ T  L = EQ (1 − MT ) exp σYT − rT + log ¯ A0 h i ¯T L −rT σYT = ¯ e EQ (1 − MT )e A0 Z ∞ ¯T L = ¯ e−rT eσy Q(YT ∈ dy, τd > T ). A0 0

17

¯ t even if the surrender since they receive only at most the single threshold L fund performance is quite strong. First, we explain the specication and implementation of the model and give the employed parameter sets. After that, we study sensitivities of the default probability P (τd ≤ T ) up to maturity T with respect to the initial surrender intensity as well as the common correlation parameter of a Gaussian copula that expresses the dependence among surrender times. Finally, we examine the corresponding sensitivities of the initial values of the various liabilities, that is of V01 , V02 , V03 , V0 , E0 , and D0 . 4.1

Valuation in practice

It is possible to specify the model further by making the following assumptions. We suppose rst that the interest rate dynamics under P is given by

drt = ar (br − rt )dt + σr dzt , where ar , br and σr are positive constants, and z is a standard P -Brownian motion. The surrender intensity, which drives the process N is modeled under P accordingly dλt = aλ (bλ − λt )dt + σλ dxt , where aλ , bλ , and σλ are positive constants and x is a P -standard Brownian motion. To compute each surrender time, one performs i

Zt

τ = inf{t|

λs ds ≥ i },

0

where each i (i = 1, · · · , I0 ) is a reduced exponential random variable. Then, the τ i 's are sorted in increasing order. We assume the following correlations between the Brownian motion W , which drives the asset value, and the Brownian motions z and x

dWt dzt = ρAr dt, dWt dxt = ρAλ dt, dzt dxt = ρrλ dt, and we model the dependence between surrender times through a Gaussian copula (with a unique common correlation coecient ρ) that is applied to the i 's. We employ this copula for the sake of simplicity. The Gaussian copula admits the advantage of its being easy to calibrate and implement. Here, instead of specifying it with a general matrix of correlation, we used a unique parameter describing the correlation between any two pairs of underlying variables. The reason of this choice is to exhibit how copulas can be introduced in this framework, without adding complexity in the results by the introduction of too many stylized features and parameters. We leave to the reader the trivial implementation of other copulas such as the Student t. Note also that this is a standard method in CDO pricing for implementing dependent default times generated by a Cox process. Finally note that the intensities are simulated in such a way that they never become negative (although the probability is very small) by making them rebound at zero, if necessary. The simulations are performed in the real 18

world for the computation of default probabilities and in the risk-neutral world for the valuation of liabilities. Note that the number of trials for each simulation is set at 20,000 times.

A0 100

L0 80

ρg 0.0375

δ 0.9

I0 1000

T 15

α 0.8

β 1.05

Table 1: Case study parameters on the initial conditions of the fund and the contingent payo First, we set the parameters on the initial conditions of the fund and the contingent payo as seen in Table 1. Note that T is the maturity of the contracts, given in years. Because we choose to discretize over all open days, we have 15 × 252 = 3780 time steps used in the simulations.

r0 0.04

br 0.055

ar 35

λ0 0.05

bλ 0.05

aλ 30

ζ 0.1

µ 0.07

Table 2: Case study parameters on the other parameters In Table 2, we read rst the intrinsic risk-free rate and surrender intensity parameters. To simplify matters, we assume that z is also a Q-Brownian motions, that is, we assume a null interest rate premium (the reader aiming at introducing this quantity can calibrate it from historical drifts and volatilities of government bonds or Libor rates). As a result, the interest rate parameters are naturally the same under P and Q. Next, following the discussion at the beginning of section 3, we assume the surrender intensity under the risk-neutral probability Q is given by multiplying the surrender intensity process λ under P by ζ , which is the premium for the technical risk that allows the simulation of the surrender counting process N in the risk-neutral world (the parameter ζ can be calibrated implicitly by reversing the model on policies of known quoted prices and underlying pool composition). Coherently, we presume that x is also a Q-Brownian motion. Then, the parameters of the surrender intensity under Q can be specied via multiplying λ0 , aλ , and σλ by ζ . In the sensitivity analysis of the default probability and the valuations below, we change the value of λ0 from 0 to 0.1 under the assumption λ0 = bλ . In addition, µ is the historical drift of A: it is used for the computation of real-world default probabilities.

σA 0.08

σr 0.05

σλ 0.05

ρAr -0.5

ρAλ -0.5

ρrλ 0.5

ρ 0.1

Table 3: Case study parameters on the volatilities and the correlations The rst six parameters of Table 3 present the parameters of the variancecovariance matrix of the assets A, risk-free rate r, and surrender intensity λ. The last parameter ρ stands for the common correlation used in the Gaussian copula that links surrender times and is set at 0.1 for the basic case. For the sensitivity analysis below, we vary the value of the common correlation ρ between 0.1 and 0.9. 19

4.2

Sensitivities of the default probability

In Figure 1, we plot the historical default probability, as dened in (12), with respect to the intensity. By this, we mean that we set the initial value of the intensity λ0 equal to the mean reversion parameter bλ , and that we vary this common parameter between 0 and 0.1. All other parameters admit the values dened in the above tables. We observe that the default probability is strictly non-null for λ0 = bλ null or small. This non-null value gives the probability of a default due to bad management of the assets, in the case of contracts that are nearly (because σλ is non-null) never surrendered. Then, and not surprisingly, we observe that the probability of default increases when the surrender intensity is more important. This also conrms the intuition of this article: surrender risk can indeed be a cause of default risk. 0.155

0.2 0.15

0.19

0.14

0.18

Default Probability

Default Probability

0.145

0.135 0.13 0.125 0.12

0.17

0.16

0.15

0.115

0.14 0.11 0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

l

0.13 0.1

0

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

ρ

Figure 1: Comparison of the default probability P (τd ≤ 15) for changing the initial surrender intensity λ0 (under the assumption λ0 = bλ ) between 0 and 0.1

Figure 2: Comparison of P (τd ≤ 15) for changing the common correlation parameter ρ (of a Gaussian copula) between 0.1 and 0.9

Figure 2 graphs the real-world default probability of the issuing entity with respect to the Gaussian copula parameter ρ. Recall that this is the parameter that denes the degree of dependence between surrender times. We let ρ vary between 0.1 and 0.9. All other parameters are unchanged. We observe that the more the surrender times are correlated (independently of their overall arrival rate), the more default is likely to occur. To put it dierently, clustering of surrender times is more amenable to inducing default than their uniform repartition in time. 4.3

Sensitivities of the liabilities

We start by examining how the liabilities toward investors, dened by equations (20), (21), and (22), respond to changes in the surrender intensity and surrender clustering. Then, we concentrate on the sensitivity of equity and of the present value of management costs, as dened by equations (29) and (28) respectively.

20

90

80 80

70

70 V10 V2 0 3 V0

50

V10

60

2

V0

V0

Liabilities

Liabilities

60

40 30

V30

50

V0 40

20

30 10

20 0 0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

λ0=bλ

10 0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

ρ

Figure 3: Comparison of the liabilities V01 , V02 , V03 , and V0 for various λ0 values (under the assumption λ0 = bλ ) between 0 and 0.1

4.3.1 Sensitivities of

V01,2,3

and

Figure 4: Comparison of V01 , V02 , V03 , and V0 for various ρ values between 0.1 and 0.9

V0

Figure 3 shows that the studied investment is reasonably fair with respect to policyholders in the sense that the overall liability of the rm V0 toward them is little sensitive to the rate of investors surrendering their contract. However, looking at individual liabilities, we observe huge discrepancies. For instance, V01 , corresponding to the value of the payos paid at maturity, falls dramatically with surrender intensity: an average intensity of 5% reduces this value by half compared with a null starting level of surrender intensity. On the contrary, V03 , corresponding to the value of the payos paid to investors when they surrender, increases remarkably with surrender intensity. Only V02 , corresponding to the value of the payos paid upon default, is rather insensitive to surrender intensity: when this intensity increases, more defaults take place, as shown above, but when these defaults occur, the compensating payment is by denition distributed to fewer policyholders. As we see, these two eects oset each other approximately. Finally, Figure 4 shows that all these liabilities are nearly insensitive to the clustering parameter ρ. If this parameter can lead to an increase of the default probability, as shown above, the osetting eects explain why V01,2,3 and V0 are insensitive to ρ.

4.3.2 Sensitivities of

E0

and

D0

Figure 5 shows rst, and not surprisingly, that the present value of the management costs incurred when policyholders surrender their contracts is an increasing function of the surrender intensity. More interesting is the behavior of equity (in the current context, and from the denition of F¯ D , E0 = E01 ), which is clearly a decreasing function of the surrender intensity. It is certainly not in the interest of the stockholders constituting this company to err on the level of surrender intensity that is going to prevail within the pool of policyholders. For the sake of brevity, we do not show the plot of E0 and D0 with respect to ρ; as in (4), it shows a marked insensitivity toward the clustering parameter.

21

25

Equity and Management Costs’PV

E0 D0

20

15

10

5

0

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

λ =b 0

λ

Figure 5: Comparison of the equity E0 and the expected discount total management cost D0 for various λ0 values (under the assumption λ0 = bλ ) between 0 and 0.1

5 Final comments The present article exhibits how surrender risk can induce default risk for certain types of saving institutions or insurance companies, and develops the pricing of their liabilities in such a setting. In particular, it builds a framework where agents surrender not only because of nancially rational reasons but also due to other factors (assuming an intensity of surrenders that is correlated, though not perfectly, to market variables). The framework presented here can be readily expanded to take into account additional real-world features that are not taken into account in the core of the text in order to simplify the exposition. To start with, consider the modeling of the assets of the company. Implicitly, and for the sake of simplicity, we have viewed the assets as a black box. In practice, assets are not just bought and held until maturity, but they are managed. A classical way of managing assets is the constant proportion portfolio insurance method, presented by Perold (1986) and Black and Perold (1992) and widely used by practitioners. According to this method, the assets are grouped into two subfunds, a risky one and a risk-free one. A oor that should not be crossed is dened (in our context, this would typically be the threshold process L), and the dierence between the total value of the assets and the oor is the cushion. Then, the amount of risky asset is proportional to the cushion, the proportionality coecient being called a multiplier. In mutual funds, this coefcient is typically about 3, while for insurance companies managing their assets, it is rather close to 1. The dynamics of the cushion, and thus of the total portfolio, can be easily obtained in a framework such as ours, where the risky sub portfolio is modeled by a lognormal-type (with potentially stochastic volatility) process with compound Poisson process jumps. See, for instance, Prigent (2007) for more details on this method and on other portfolio management methods. We have assumed that policyholders could surrender their contracts but that 22

the rm functioned with a pool of policyholders constituted at time 0. Indeed, it is possible to take into account the arrival of new contractors by modeling the assets as follows   ϕt− ψt− dAt = At− µt dt + σt dWt − dNt + dχt , At− At− where ψ is the amount that is paid at the inception of the contract and χ is the counting process that drives the arrival of new contractors. The model for the threshold process, as given by (7), can clearly be enlarged, though this is extremely dependent on the type of monitoring that is incurred by the rm. In particular, the regulations and supervision of saving institutions and of insurance companies can be very dierent. The threshold can be modeled as a simple constant number or as the discounted value (typically at the rate r) of the future critical liabilities (such as a potential minimum guarantee). It can also be modeled as an implicit process that is chosen by the managers of the fund in order to maximize equity. See, for instance, Leland (1994) for more details on implicit default barriers. Two important features should be taken into account when modeling asset dynamics: stochastic volatility and the presence of jumps. The above framework naturally allows for the presence of stochastic volatility, but we have not added, for the sake of simplicity, jumps in the dynamics (other than those that stem from surrenders, which form the core of this paper). To add a compound Poisson process component J for the market jumps in the assets dynamics is indeed a straightforward procedure   ϕt− dNt . dAt = At− µt dt + σt dWt + dJt − At− See, for instance, Le Courtois and Quittard-Pinon (2006) for a structural model of default with assets modeled by a jump diusion. Finally, it is easy to incorporate mortality risk in the framework dened in this article. Indeed, adding mortality risk simply corresponds to multiplying terms by life/death probabilities (assuming that mortality risk is independent of market risk and that it is diversiable by working with suciently big pools of investors). For instance, with T px the probability that an individual aged x at time 0 has survived until time T , we can rewrite V01 as follows h i RT ¯T ) , V01 = T px EQ (I0 − NT )(1 − MT )e− 0 ru du F¯ (A¯T , L provided all policyholders are aged x at time 0. When this is not the case, a simple summation over x extends the formula.

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26

A Proofs A.1

Proof of Lemma 2.1

Proof. Let Z

t

Z

0

Z

t

σu dWu , Ht := −

µu du +

Zt :=

t

ϕu− dNu , 0+

0

where both Zt and Ht are semimartingales. Equipped with this notation, we can rewrite equation (4) as follows

dAt = At− dZt + dHt ,

H 0 = A0 .

(34)

This is a linear stochastic dierential equation, whose solution is known. See Protter (2005) for a general account on such equations and Jaschke (2003) for the complete solution. In our particular case, we can write    Z t  X ∆Hs ∆Zs  , At = E D (Z)t H0 + dH − d[H, Z]cs − E D (Z)−1 s−  s 1 + ∆Zs  0 0
where

 Y  1 c (1 + ∆Zt )e−∆Zt E (Z)t = exp Zt − [Z, Z]t 2 D

(35)

0
is the Doléans-Dade exponential, solution of the equation dBt = Bt− dZt . We readily have Z t    Z t  Z t σ2 E D (Z)t = exp µu − u du + σu dWu := exp µu du E(σ · W )t . 2 0 0 0 Then, we observe that the quadratic covariation of the continuous parts of H and Z , namely [H, Z]c , is null because H is a pure P jump process. Furthermore, ∆Hu ∆Zu we note that because Z presents no jumps, then 1+∆Zu is also null, so 0
that

dHs − d[H, Z]cs −

X ∆Hu ∆Zu = −ϕs− dNs . 1 + ∆Zu

0
The result of the lemma follows immediately.

A.2

Proof of Lemma 2.2

Proof. Let us remember that A satises (4) dAt = At− (µt dt + σt dWt ) − ϕt− dNt . In addition, we have, on the set {I0 > Nt },     1 1 1 d = − dNt I0 − Nt I0 − Nt I0 − Nt− 1 = dNt , (I0 − Nt )(I0 − Nt− ) 27

since Nt − Nt− = 1 if dNt = 1 and Nt − Nt− h = 0 otherwise. i 1 Furthermore, the quadratic covariation A, I0 −N is calculated as follows t

 A,

1 I0 − N

 = t

X A0 + ∆As ∆ I0 − N0



0
1 I0 − Ns

 =

A0 X −ϕs− + ∆Ns . I0 (I0 − Ns )(I0 − Ns− ) 0
The rst equality is obtained from Theorem 28 in Chapter II of Protter (2005). Using the integral by parts formula (see Corollay 2 of Theorem 22 in Chapter II of Protter (2005) ), we have       1 1 1 At = dAt + At− d + d A, dA¯t = d I0 − Nt I0 − Nt− I0 − Nt I0 − N t At− ϕt− At− = (µt dt + σt dWt ) − dNt + dNt I0 − Nt− I0 − Nt− (I0 − Nt )(I0 − Nt− ) ϕt− − dNt (I0 − Nt )(I0 − Nt− )   ϕt− 1 A¯t− dNt − 1+ dNt = A¯t− (µt dt + σt dWt ) + I0 − Nt I0 − Nt− I0 − Nt A¯t− ϕt− = A¯t− (µt dt + σt dWt ) + dNt − dNt . I0 − Nt I0 − Nt We used again the fact Nt − Nt− = 1 if dNt = 1 to obtain the last equality.

A.3

Proof of Lemma 2.3

Proof. Let Z Zt =

t

Z

0

t

Z

t

σu dWu −

µu du + 0

0+

dNu , I0 − Nu−

where Z is a semimartingale with jumps. Then, equation (13) can be represented as (36)

dAt = At− dZt .

Equation (36) is a linear stochastic dierential equation. The generic solution of this type of equation is a well known result (see for instance Theorem 4.61 of Jacod and Shiryaev (2003)), and can be written as  Y  1 D c (1 + ∆Zs ) e−∆Zs . (37) At = A0 E (Z)t = A0 exp Zt − [Z, Z]t 2 0
A straightforward application of the It o isometry allows to compute the quadratic variation of the continuous part of Z Z t [Z, Z]ct = σu2 du. 0

28

As far as jump contributions are concerned, one readily has, with s denoting jump times   Z t  Y X dNu −∆Zs   , e = exp − ∆Zs = exp 0+ I0 − Nu− 0
0
and

Y

(1 + ∆Zs ) =

0
Y  0
1−

∆Ns I0 − Ns−



Y  I0 − Ns− − ∆Ns  = I0 − Ns− 0
As N remains constant between jump times, thus jumps at time s, we can write

Y

(1 + ∆Zs ) =

0
I0 −Ns I0 −Ns−

= 1 unless Ns

I0 − Nt I0 − Nt I0 − 1 I0 − 2 · ··· = . I0 I0 − 1 I0 − (Nt − 1) I0

Inputing the above results in formula (37) yields the explicit representation (14) of A as the solution to (13).

29