Until recently traditional valuation methods in the life insurance industry were based on deterministic projections of risk factors. As long as life insurers were primarily underwriting diversifiable risks, the deterministic approach although not sufficiently appropriate could provide good approximations. The delay in the recent realization by the insurers for the need for more efficient valuation standards was mainly due to the opacity of traditional accounting systems.
Cost of Capital and Surrender Options for Guaranteed Return Life Insurance Contracts Alexis Bailly February 27, 2005 Master of Advanced Studies in Finance ETH - University of Zürich Abstract The opacity of traditional accounting systems for insurance companies is well known This was confirmed recently by unexpected repercussions of stock market and interest rates movements on the financial strength of many insurance companies To improve transparency, new valuation standards are initiated by regulators or by professional bodies such as actuaries or accountants Whether the purpose is pricing or risk management, the new standards are all based on a market consistent framework where assets and liabilities are valued at market value Traditionally the pricing and the risk capital assessment are treated separately In this thesis we build a unifying valuation framework where these two components can not be dissociated To reflect the incompleteness of insurance markets and the limited access to equity capital, we introduce the notion of cost of capital We analyse the impact of the cost of capital on the valuation of life insurance contracts with guarantees In the last part of this thesis we focus on surrender options that represent today one of the most obscure risk for life insurers We present models where the pricing can be precisely performed, but we also discuss certain aspects of policyholder’s behaviour Acknowledgement I would like to thank the supervisors of my master’s thesis, Prof Damir Filipovi´ for his valuable c comments and for his availability, Prof Paul Embrechts for useful discussions and Jon Bardola who supported me with the necessary time of work and also encouraged me in the selection of this practical topic Contents Introduction Pricing and capital requirement for life insurance policies 2.1 The structure of the contract 2.1.1 The model 5 2.2 2.3 The case of a ’true’ guarantee The case of a conditional guarantee Cost of Capital 3.1 Incomplete markets and cost of capital 3.2 Pricing under cost of capital 3.2.1 The case of a ’true’ guarantee 3.2.2 The case of a conditional guarantee 11 11 13 13 15 Multi-period extension 17 4.1 General framework 17 4.2 Simplified framework 18 4.2.1 The two-period case 19 Surrender options 21 5.1 Surrender options in the case of conditional guarantee 21 5.1.1 Finite difference approach 23 5.2 Surrender triggered by interest rates movements 25 Conclusion 28 Introduction Until recently traditional valuation methods in the life insurance industry were based on deterministic projections of risk factors As long as life insurers were primarily underwriting diversifiable risks, the deterministic approach although not sufficiently appropriate could provide good approximations The delay in the recent realization by the insurers for the need for more efficient valuation standards was mainly due to the opacity of traditional accounting systems The statutory accounting system is a summary of the cash flows and reserve movements during the year This accounting system could be suitable for industrial companies but not for life insurers where cash flows from one individual contract would be spread over long term periods Considering the usual cash flow pattern of insurance contracts described by an initial loss followed by a sequence of small profits, the statutory accounting is unable to give any indication on the profitability of a new policy To overcome this deficiency new standards such as US GAAP have been introduced to allow a smoothing of expenses and revenues over the term of the contract Unfortunately these latest standards will be even more opaque and will not help an active management due to the attenuation by smoothing of the volatility risk During the 90’s, actuaries introduced the Embedded Value standards Basically, the Embedded Value is the discounted value of future statutory profits, where all economic and operational assumptions are projected in a deterministic way Embedded Value as a nominal value may not be very significant but when comparing with the previous standards, the change in Embedded Value could give an important insight into the sources of value creation (for instance investments, mortality, lapses ) The main drawback remains the inability to value the costs of options and guarantees The use of deterministic methods, allowed life insurance companies to ignore the sources of volatility, and in particular the occurrence of extreme events In the last 10 years, insurance companies started to record extreme losses resulting from unanticipated changes in interest rates, stock markets or longevity The pressure from the investment community is becoming more and more intense The modern standards currently in development are either called Fair Value, Market Consistent Value or Stochastic Embedded Value; the objectives are the same, valuing assets and liabilities at purchase value by taking in account all sources of variability Traded assets and liabilities will be valued at market value and the non traded ones will be valued at the price of the replicating portfolio The main challenge for insurance companies is the implementation of stochastic models in order to value the costs of embedded options and guarantees In this paper, our focus will be on market consistent valuation at the level of an individual policy Traditionally the issue of the pricing of an insurance contract was completely separated from the issue of risk capital assessment In this thesis, we propose a unifying valuation framework, where the two issues are treated simultaneously The thesis is organised as follows In Section we describe the structure of the contract and we introduce the general framework for pricing participating policies with guaranteed return In Section we first introduce a definition for the cost of capital by considering the incompleteness of the insurance market Then we analyse the impact of the cost of capital on the valuation of insurance contracts In Section we extend our framework to a multiperiod situation, we discuss some practical issues and analyse a simplified two-period case In Section 5, we focus on the pricing of surrender options and discuss the issue of the policyholder behaviour We end with some conclusions Pricing and capital requirement for life insurance policies 2.1 The structure of the contract In this section we develop a framework to analyse a single premium participating insurance contract with minimum guarantee Under this contract the policyholder pays a single premium P0 at time t=0 and the insurer is obliged to pay at maturity t=T a specified amount depending on the performance of a reference fund The participating contract consists of two elements - a fixed guaranteed payment and a call option which gives the policyholder a fraction of the surplus We define the surplus as the policyholder’s fund value less the guaranteed amount Such contracts provide generally a guaranteed benefit in case of death, but in our analysis we will ignore mortality and will only focus on financial risks Moreover, the default possibility of an insurance company is never zero and in case of such an event the company will not be able to fulfil its obligations towards the policyholder Therefore the default of the insurance company must be considered in a market consistent framework In practice this default is often ignored by the insurance companies concerning their retail business It is difficult for the policyholder to estimate the impact of the solvability of the insurer on the price of their contract However in an insurer-reinsurer relationship, this risk is not ignored and the financial soundness of the reinsurer will have an important impact on the price of a treaty We will treat both situations - true guarantee under the assumption of a non defaultable company and - conditional guarantee under the assumption of a defaultable company 2.1.1 The model We consider an economy with two traded assets - a risk free bank account with price process B and a risky asset with price process S We assume that the financial market is frictionless and can be represented by a probability space (Ω, F, {Ft } , P ) {Ft }t∈[0,T ] is the filtration generated by a one-dimentional Brownian motion, W, which represents the financial uncertainty in the economy We assume that there exists a constant risk free interest rate r such that the dynamic of the bank account is given by dBt = rBt dt, B0 = (1) The dynamic of the risky asset is described by the following stochastic differential equation, dSt = µSt dt + σSt dWt (2) The insurance contract offers a guaranteed minimum return g We assume that a fraction λ ∈ (0, 1] of the surplus is given to the policyholder at maturity, λ is defined at the beginning of the contract At time t=0 the total premium received by the insurance company is P0 In addition, we assume that the shareholders of the company have to inject a certain level of target capital TC0 in order to guarantee the solvency of the company We define the ruin probability ψ as the probability that at maturity t=T, the value of the assets of the company AT is lower than the value of the liabilities LT , ψ (u) = P [AT < LT | Initial shareholder capital = u] We distinguish the physical probability P used here to express the ruin probability from the risk neutral probability Q that we will use later in the valuation of the prices of insurance contracts The management of the company fixes a target α ∈ (0, 1) for the maximum value of the ruin probability The target capital TC0 is the minimum capital satisfying this constraint on the ruin probability and is given by T C0 = inf {u : ψ (u) ≤ α} Moreover, we assume that immediately after receiving the premium, the company decides the following investment strategy: - F0 = S0 is invested in the risky asset, - P0 − F0 = P0 − S0 is invested in the riskless asset, - TC0 is invested in the riskless asset It is appropriate to assume that the target capital is entirely invested in the riskless asset given that its main purpose is a role of buffer against adverse movements in the financial market Concerning the split of the initial premium between risky and riskless asset, we assume that it is a pure decision of the management of the company Our choice here is justified by the fact that we will consider F0 = S0 to be the reference fund for the attribution of the policyholder’s benefits 2.2 The case of a ’true’ guarantee In this section we assume that the insurance company will fulfil in any situation its obligations towards the policyholder This is equivalent to assume that the shareholder will inject at maturity additional capital in the company if required For instance this could be observed in a situation where a holding company decides to inject additional capital in an affiliate in order to avoid reputational issues We can also assume that the regulator is monitoring closely the solvency of the company and by constraining the investment strategy prevents the company from becoming insolvent For an initial fund of the policyholder L0 = S0 , the payoff at maturity to the policyholder can be described as LT = Max S0 egT , λ ST − S0 egT + S0 egT , (3) or S0 egT LT = + λMax 0, ST − S0 egT M in im iu m G a nte ed Term in a l b e n e fi t C a ll o p tio n λ = 1.0 λ = 0.6 Payoff at Maturity for a participating contract (4) The price of a contract having such a payoff is calculated in the following way: V0 = EQ e−rT LT , (5) BS V0 = S0 e(g−r)T + λC0 (S0 , S0 egT , r, σ, T ) (6) which leads to BS where C0 (S0 , S0 egT , r, σ, T ) represents the Black Scholes price of a European call option with initial price of the underlying S0 , a strike value S0 egT , interest rate r, volatility σ and maturity T At time t=0 the total premium received by the insurance company is P0 = V0 In addition the shareholder injects a certain level of target capital TC0 in order to guarantee the solvency The assumption of "true" guarantee implies that if at maturity TC0 erT is not enough then an additional injection will be made The initial investment strategy is as follows: - F0 = S0 is invested in the risky asset - V0 − F0 = V0 − S0 is invested in the riskless asset - TC0 is invested in the riskless asset The total value of the assets of the company at date t=0 is given by A0 = V0 + T C0 At maturity T, the value of the assets of the company is defined in the following way, AT = (T C0 + (V0 − S0 )) erT + ST (7) The benefits to be paid to the policyholder are given by LT = S0 egT + λMax 0, ST − S0 egT (8) We assume that the target level of the ruin probability α, is defined by the shareholder at time t=0 The target capital TC0 is the minimum capital satisfying the following relation P [AT < LT | Initial shareholder capital = T C0 ] α (9) Assuming that the initial value of the shareholder’s capital is T C0 , we can write AT < LT ⇔ (T C0 + (V0 − S0 )) erT + ST < S0 egT + λMax 0, ST − S0 egT but we also have AT < LT ⇔ AT < Lg = S0 egT , T because insolvency can occur if and only if the value of assets is lower than the value of the guaranteed benefits Therefore {AT < LT } = (T C0 + (V0 − S0 )) erT + ST < S0 egT and hence ⎡ ⎤ ⎢ ⎥ P [AT < LT ] = P ⎢ST < S0 egT − (T C0 + (V0 − S0 )) erT ⎥ = P [ST < β] ⎣ ⎦ (10) β The relation P [ST < β] = α will be satisfied if and only if β = S0 e(µ− σ )T +σ √ T wT where wT = Φ−1 (α) is the inverse of the standard normal distribution (11) Therefore the initial target capital is given by T C0 = S0 egT − β e−rT − (V0 − S0 ) (12) In this section we assumed that at maturity the shareholder will honour its obligations in any situation, therefore the target capital has no importance from the policyholder’s point of view The contract is clearly defined; the only information important for the policyholder is the price Here, the target capital is only a requirement from the regulator and there is no additional cost associated The price of the contract is independent of the level of the target capital and therefore when the pricing is performed there is no necessity to consider simultaneously the question of the minimum capital requirement The non consideration of the default risk of the insurer within the pricing of a contract fairly represents the common practice in the market This practice results from the information asymmetry existing between policyholders and shareholders 2.3 The case of a conditional guarantee Many insurance companies are set in the form of a limited liability company For these companies, in case of insolvency there is no obligation for the shareholders to inject additional capital By considering the default of the company, the payoff to the policyholder becomes LT = S0 egT + λMax 0, ST − S0 egT AT = (T C0 + (V0 − S0 )) erT + ST if ST ≥ β , if ST < β (13) where β defines the threshold value of the stock price indicating insolvency, {ST < β} ⇔ AT < Lg = S0 egT T We can also write the payoff in the following way, LT = S0 egT + λMax 0, ST − S0 egT · 1{ST ≥β} + (T C0 + (V0 − S0 )) erT + ST · 1{ST : S0 e(µ− t > : Wt < σ2 )T +σW t σ ln < S0 egT −r(T −t) − V0 − S0 + T C0 eγT ert S0 egT −r(T −t) −(V0 −S0 +T C0 eγT )ert S0 − (µ − σ2 )T = inf {t > : Wt < κ(t)} Given the continuity of the Brownian motion Wt , we have at insolvency time τ , Aτ = e−r(T −τ ) S0 egT The ruin time τ is depending simultaneously on the target capital TC0 and the value of the contract V0 Assuming that V0 is given, the target capital T C0 at the level α is the minimum capital satisfying the following condition P [τ ≤ T | Initial shareholder capital = T C0 ] α (44) If surrender occurs before the maturity T, the payoff to the policyholder takes place at surrender time τ , otherwise the payoff takes place at maturity T The payoff is defined as follows: LT ∧τ = where S0 egT + λMax 0, ST − S0 egT Aτ = (T C0 + (V0 − S0 ) + B) erτ + Sτ if τ > T if τ ≤ T Aτ = (T C0 + (V0 − S0 ) + B) erτ + Sτ , B = T C0 eγT − , Sτ = e−r(T −t) S0 egT − T C0 + (V0 − S0 ) + T C0 eγT − erτ Replacing the values of B and Sτ in the expression of Aτ gives back Aτ = e−r(T −τ ) S0 egτ and if τ > T S0 egT + λMax 0, ST − S0 egT (45) Aτ = e−r(T −τ ) S0 egT if τ ≤ T As we mentioned before the stopping time indicating the insolvency τ depends on both the level of the target capital and the price of the contract Moreover the values of TC0 and V0 are depending on each other This recursive dependence, in addition to the path dependent property of the contract makes the pricing particularly difficult LT ∧τ = 17 4.2 Simplified framework To solve the problem of the recursivity, we make the assumption that the company has the possibility to segregate the initial fund V0 between the policyholder and the shareholders in the following way S0 is initially attributed to the policyholder V0 − S0 + B is initially attributed to the shareholders One of the main difficulties in our valuation is resulting from the fact that the price of the option is part of the capital at risk The shareholder’s part of the initial premium (V0 − S0 + B) was kept within the company and invested in the riskless asset In such a situation the target capital was depending on the value of the options and inversely the cost of options was depending on the level of the target capital The cost of capital is paid by the policyholder and is given by B = T C0 eγT − To simplify the problem and avoid partially the recurrence situation, we will assume that once the premium is paid an amount corresponding to (V0 − S0 ) is paid out immediately as a dividend to the shareholders T C0 + B + sh a re h o ld e rs’ fu n d inve ste d in th e risk free a sse t V0 − S0 + sh a reh o ld ers’ p a rt p a id o u t to sh a re h o ld e rs a s a d iv id en d S0 p o lic y h o ld e r’s fu n d Invested in th e risk y a sse t The new structure of the capital of the company after the payment of the dividend is as follows: T C0 + B + S0 sh a reh o ld ers’ fu n d p o licy h o ld er’s fu n d Invested in th e risk less a sse t Inve ste d in th e risk y a sset The evolution of the assets of the company is no more depending on the value of the option and the recurrence effect is cancelled The value of the assets at time t is given by At = (T C0 + B) ert + St (46) τ = inf t ≥ : St < e−r(T −t) S0 egT − (T C0 + B) ert (47) The default time will be given by We are now in a situation where the stopping time is no more depending on the value of the contract V0 V0 is still depending on TC0 , but TC0 is no longer depending on V0 , and the difficulty of the problem has considerably reduced Because of the continuity of the Brownian motion, at insolvency time we will have Sτ = S0 egτ −r(T −τ ) − (T C0 + B) erτ (48) The payout to the policyholder is now given by LT ∧τ = S0 egT + λMax 0, ST − S0 egT Aτ = (T C0 + B) erτ + Sτ = e−r(T −τ ) S0 egT In the following section we will treat the two periods case 18 if τ > T if τ ≤ T (49) 4.2.1 The two-period case We assume that at an intermediary time t1 before maturity T, we check the solvency position of the company This means that we are deriving the solvency position of the company from the balance sheet situation at year end Of course, such a static method may not seem effective however by increasing the number of intermediary time steps we can still increase considerably the efficiency of the risk management process We define the insolvency time by τ = inf t ∈ {t1 , T } : At < e−r(T −t) EQ [Lg | Ft ] , with t1 ≤ T T We can also write τ = inf t ∈ {t1 , T } : (T C0 + B) ert + St < e−r(T −t) S0 egT (50) The target capital T C0 at the level α is the minimum capital satisfying the condition P [τ ∈ {t1 , T }] α In our discrete case, the ruin probability before the maturity T is given by (51) P [τ ∈ {t1 , T }] = P [τ = t1 ] + P [{τ = T } ∩ {τ 6= t1 }] The ruin events can also be expressed as follows: {τ = T } ⇔ (T C0 + B) erT + ST ≤ S0 egT ⇔ {WT ≤ wT } , {τ 6= t1 } ⇔ (T C0 + B) ert1 + St1 > S0 egt1 ⇔ {Wt1 > wt1 } K ln S t − µ− σ σ t and Kt = S0 egt − (T C0 + B) ert for t ∈ {t1 , T } with wt = Moreover we have the following relationship wt1 = wT + µ− σ2 (T − t1 ) σ (52) Replacing in the above equations gives, (53) P [τ ∈ {t1 , T }] = P [Wt1 < wt1 ] + P [{WT ≤ wT } ∩ {Wt1 > wt1 }] It can easily be shown that, wT − Wt1 √ T − t1 where Φ is the cumulative standard normal distribution Finally, ⎡ ⎛ P [WT ≤ wT , Wt1 > wt1 ] = E Φ ⎢ ⎜ wt − P [τ ∈ {t1 , T }] = P [Wt1 < wt1 ] + E ⎢Φ ⎜ ⎣ ⎝ µ− σ · 1{Wt (T −t1 ) √ σ T − t1 >wt1 (54) } ⎞ ⎤ ⎥ − Wt1 ⎟ ⎟·1 ⎥ ⎠ {Wt1 >wt1 } ⎦ (55) The equation P [τ ∈ {t1 , T }] α , can be solved by numerical methods, we first obtain the value of wt1 and then the value of the target capital TC0 The payout to the policyholder is given by: LT ∧τ = S0 egT + λMax 0, ST − S0 egT Aτ = (T C0 + B) erτ + Sτ = e−r(T −τ ) S0 egT if τ > T if τ ≤ T (56) The payoff to the policyholder takes place at surrender time τ if the surrender occurs before the maturity T, otherwise the payoff takes place at maturity T 19 We can also write the payoff in the following format LT ∧τ = S0 egT · 1{τ >T } + λ ST − S0 egT · 1{ST ≥S0 egT } · 1{τ >T } + ((T C0 + B) erτ + Sτ ) · 1{τ ≤T } (57) The price is given by V0 = EQ e−r(T ∧τ ) LT ∧τ , (58) which leads to V0 = S0 e(g−r)T · EQ 1{τ >T } +λe−rT EQ ST · 1{ST ≥S0 egT } · 1{τ >T } −λS0 e(g−r)T EQ 1{ST ≥S0 egT } · 1{τ >T } +EQ e−rT S0 egT · 1{τ ≤T } The price V0 can now easily be calculated using numerical methods such as Monte Carlo The impact of the cost of capital γ is measured only under the events {τ ≤ T } Increasing the target capital TC0 will increase the price of the contract but this increase will be partially offset by the lower probability of the event {τ ≤ T } 20 Surrender options Life insurance contracts are usually long term contracts, they require important amount of investments from the policyholders Many customers will be reluctant to the idea of locking high amounts of money for such long periods Therefore for marketing reasons, the insurers may offer the possibility to the policyholder to terminate the contract prior to maturity Different reasons may explain the decision of the policyholder to surrender: - The policyholder behaves rationally and notes that it is the optimal time to surrender his contract - The policyholder may surrenders for personal needs of the invested amounts at this time or may simply behave in a non rational way In our analysis we will assume that the policyholder is perfectly informed and takes rational decisions The policyholder will take the decision to surrender in order to maximise the value of his wealth In practice, insurance companies will partially cover the cost of surrender options by applying surrender penalties In general the penalties are predefined as a percentage of the guaranteed benefit This percentage could be a decreasing function of the in force period of the contract Usually higher surrender penalties will be applied in the first period of the contract This is due to the higher sensitivity of bonds of longer duration but also to the fact that the insurer may not have received enough periodical charges to cover initial expenses In this chapter we cover two different situations In a first section we consider the situation of surrender options for guaranteed return insurance contracts where the policyholder’s fund is entirely invested in a risky asset with a price following a geometric Brownian motion This study builds on Grosen and Jorgensen (2001), where they analyse surrender options for participating contracts with minimum guaranteed return They develop a finite difference algorithm for contracts where the participating bonus is attributed yearly to the policyholder In their study the default of the company has not been considered In our case we develop an algorithm for contracts where the participating bonus is allocated only at the end of the contract but in addition we consider the possibility of the default of the company In the second section we analyse a case that has not been treated previously in the literature We consider a situation where the policyholder’s fund is entirely invested in bonds After assuming a particular stochastic interest rate model, we value the surrender options under the assumption that the policyholder’s decision is based on the level of the interest rates 5.1 Surrender options in the case of conditional guarantee The contract considered is a participating contract with minimum guaranteed return g The guaranteed return is conditional to the solvency of the company In addition we assume that the policyholder can terminate the contract before the maturity T In this section we will not focus on the calculation of the value of the target capital We assume that the shareholders have initially injected in the company an amount TC0 and our problem will be to determine the price of the contract conditional to the shareholders’ capital TC0 We consider the following investment strategy at time t=0: - TC0 is invested in the bank account, - an amount corresponding to S0 is invested in the risky asset The value of the assets at time t is given by At = T C0 ert + St In case of insolvency the regulator informs the policyholder who receives the remaining part of the assets of the company and afterwards the company is shut down If surrender occurs before default then the payoff takes place at surrender time t≤ T , otherwise the payoff takes place at default time τ The payoff is given as follows: Lt∧τ = S0 egt + λ Max 0, St − S0 egt Aτ = e−r(T −τ ) S0 egT 21 if τ > t if τ ≤ t , t ≤ T (59) The default time τ is given by τ = inf t > : T C0 ert + St < e−r(T −t) S0 egT and corresponds to the first time where the value of the assets is lower than the minimum guaranteed return Let V(t,St ) be the value process of the payoff Lt defined above In accordance with the standard framework of Black and Scholes (1973), the value of the contract V satisfies the following partial differential equation ∂V ∂2V ∂V + rS + σ2 S − rV = ∂t ∂S ∂S f or St > h(t), V (St = h(t)) = e−r(T −t) S0 egT , h(t) = e−r(T −t) S0 egT − T C0 ert (60) (61) The function h represents the value of the assets of the company at the insolvency point τ In addition, the existence of the surrender option before maturity imposes the following condition: V (t, St ) ≥ S0 egt + λ Max 0, St − S0 egt f or St > h(t) (62) We are in a similar situation as in the case of an American barrier option, but here the barrier h(t) is time dependent We can transform the problem to a constant barrier situation using the transformation Zt = h0 St h(t) (63) The dynamic of the new process is then given by dZt = d h0 St h(t) dZt = Zt = r− h0 h0 ∂h dSt − St dt, h(t) h(t)2 ∂t ∂h h(t) ∂t (64) dt + σdWt We define now the value of the contract by the process U (t, Z) = V ⇔ V (t, S) = U t, h (t) Z h0 t, (65) h0 S h (t) Rewriting the equation above using the expression of Z gives ∂h ∂U +Z r− ∂t h(t) ∂t ∂2U ∂U + σ2 Z − rU = 0, ∂Z ∂Z Zt > h0 , U (Zt = h0 ) = e−r(T −t) S0 egT (66) (67) or ∂U ∂2U ∂U + F (Z, t) + F (t) − rU = 0, ∂t ∂Z ∂Z U (Zt = h0 ) = e−r(T −t) S0 egT where F (Z, t) = Z r − 22 ∂h ht ∂t , Zt > h0 , (68) (69) F (Z, t) = 2 Z σ In addition we impose the following condition: U(t, Zt ) ≥ S0 egt + λ Max 0, 5.1.1 h (t) Zt − S0 egt h0 for Zt > h0 (70) Finite difference approach We describe below an algorithm to value to price of the contact using a finite difference scheme Let us define Zmax as the arbitrary upper boundary of process Zt on the interval [0, T ] Let M and N be the number of the subdivisions of respectively the space Z and the time T, M · ∆Z + h0 =Zmax and N · ∆T = T Ui,j denotes the value of the contract at (i∆T, h0 + j∆Z) For an interior point (i,j), using a fully implicit method, the discretization gives the following expressions: - Symmetric approximation for ∂U ∂Z Ui,j+1 − Ui,j−1 ∂U = , ∂Z 2∆Z - Forward approximation for (71) ∂U ∂t Ui+1,j − Ui,j ∂U = , ∂t ∆t ∂ U ∂Z - Approximation for ∂2U = ∂Z (72) Ui,j − Ui,j−1 Ui,j+1 − Ui,j − ∆Z ∆Z Ui,j+1 + Ui,j−1 − 2Ui,j = ∆Z ∆Z (73) Substituting the above equations in the differential equation (66) gives Ui+1,j − Ui,j Ui,j+1 − Ui,j−1 Ui,j+1 + Ui,j−1 − 2Ui,j +F (h0 +j∆Z, i∆t) +F (h0 +j∆Z, i∆t) = rUi,j ∆t 2∆Z ∆Z (74) By rearranging the terms we have, − 2∆Z F (h0 + j∆Z, i∆t) + + − ∆t − + F (h0 ∆Z F (h0 2∆Z F (h0 ∆z + j∆Z, i∆t) Ui,j + j∆Z, i∆t) + F (h0 ∆Z We can also rewrite ∆t 2∆Z F (h0 + j∆Z, i∆t) − F (h0 ∆Z +∆t ∆t −∆t F (h0 2∆Z + + j∆Z, i∆t) Ui,j−1 + j∆Z, i∆t) Ui,j+1 + F (h0 ∆Z Ui+1,j ∆t = rUi,j + j∆Z, i∆t) Ui,j−1 + j∆Z, i∆t) + r Ui,j + j∆Z, i∆t) + F (h0 ∆Z + j∆Z, i∆t) Ui,j+1 = Ui+1,j or i Ui,j−1 + bi Ui,j + ci Ui,j+1 = Ui+1,j f or i = : N − and j = : M − j j j 23 (75) where 1 F (h0 + j∆Z, i∆t) − F (h0 + j∆Z, i∆t) , 2∆Z ∆Z = ∆t j (76) + F (h0 + j∆Z, i∆t) + r , ∆t ∆Z (77) bi = ∆t j ci = −∆t j 1 F (h0 + j∆Z, i∆t) + F (h0 + j∆Z, i∆t) 2∆Z ∆Z Now consider the boundary condition of the contract (1) UN,j = S0 egT + λMax 0, ST,j − S0 egT , UN,j = S0 egT + λMax 0, h(T ) (h0 + j∆Z) − S0 egT h0 (78) (2) Ui,0 = S0 egi∆t (3) If Zmax is chosen large enough, we will be beyond the optimal exercise boundary and therefore we can assume that the value of the contract at Z=Zmax is given by Ui,M = S0 egi∆t + λMax 0, h(i∆t) (h0 + M∆Z) − S0 egi∆t h0 We solve for i=N-1 to Ui,j−1 + bi Ui,j + ci Ui,j+1 = Ui+1,j for i = : N − and j = : M − j j j The matrix representation is as follows: ⎡ bi ⎢ ⎢ ⎣ ci bi ci ··· −1 M bi M−1 ⎤⎡ ⎥⎢ ⎥⎢ ⎦⎢ ⎣ Ui,1 Ui,2 Ui,M −1 ⎤ ⎡ ⎥ ⎢ ⎥ ⎢ ⎥=⎢ ⎦ ⎣ Ui+1,1 − Ui,0 Ui+1,2 Ui+1,M − ci −1 Ui,M M The following graphic shows the price of the contract as a function of Z and t: Z ⎤ ⎥ ⎥ ⎥ ⎦ t S0 =100 , Zmax = 15, dZ=0.1, T= 1, dT=0.1 , r=0.05, σ=0.2, g=0.19, λ=90%, TC0 =90 Considering the default of the company results in a lower price of the contract when the stock price is close to the insolvency point This is due to the fact that at insolvency the policyholder is obliged to terminate the contract although continuing the contract would be optimal The impact of the limited liability put option is reducing with a shorter time to maturity 24 5.2 Surrender triggered by interest rates movements After the stock market crisis of the end of 90’s, the insurance companies have conducted a de-risking of their balance sheet The main action was to sell equities and invest in fixed income instruments in order to match durations of the assets and liabilities Today the assets of insurance companies are predominantly invested in bonds and the level of the interest rates are very low The major concern for the life insurance industry is the risk of a substantial increase in interest rates and a reaction of policyholders to surrender their contracts Insurance companies are not able to evaluate precisely the costs of this pending risk The difficulty is resulting from the fact that we don’t know accurately when the policyholder will take the decision to surrender One possibility could be to assume that the surrender will happen at the optimal time, but the practical experience has shown that the behaviour of the policyholder is not always rational We consider a contract which proposes a guaranteed return to the policyholder In addition we assume that the policyholder has the possibility to surrender at any time and is entitled to receive the guaranteed rate The premium paid by the policyholder is entirely invested in bonds with maturities similar to the contract maturity T, in such a way that the assets and liabilities are perfectly matching The initial guaranteed interest rate corresponds to the yields of the bonds available at time t=0 Under these assumptions the contract does not include any risks, unless the policyholder decides to surrender We assume that the short rates process follows the Cox Ingersoll Ross model dr(t) = (a − br(t))dt + b (79) r(t)dWt Under this model the expression of the zero coupon bond prices of maturity T are given as follows: P (t, T ) = A(t, T )e−B(t,T )r 2(eγ(T −t) −1) B (t, T ) = (γ+a) eγ(T −t) −1 +2γ ( ) √ with γ = a2 + 2σ2 with and A (t, T ) = (80) 2(e(a+γ)(T −t)/2 −1) 2ab/σ (γ+a)(eγ(T −t) −1)+2γ The guaranteed rate corresponding to the yield of the bond of maturity T is defined by ln (P (0, T )) (81) T We assume that the premium π paid by the policyholder corresponds exactly to the price of one unit of the bond P(0,T), g=− π = P (0, T ) If the surrender option was excluded from the contract, P(0,T) would be the market price of the contract Of course when a surrender option is included the price of the contract will be higher than the price of the bond The main problem that we face when we try to price the surrender options is that we don’t know in advance when the policyholder will take the decision to surrender For that, in a first step we fix initially a threshold δr , and we assume that the policyholder will surrenders whenever the yields are reaching g + δ r In fact we are looking at different scenarios where the policyholder surrenders at predefined shifts in interest rates The yield process is given by yt = − Let τ = inf {t ∈ [0, T ] : yt = g + δr } ln (P (t, T )) T −t (82) πegT if τ ≥ T πegτ if τ < T (83) The payoff to the policyholder is defined by Lτ ∧T = 25 Lτ ∧T = πegT 1{τ ≥T } + πegτ 1{τ T if τ ≤ T (85) The value of the surrender option at time t=0 is calculated in the following way P SO0 (δ r ) = EQ e− T ∧τ r(s)ds π egT P (τ , T ) − egτ 1{τ