Cost of capital and surrender options for guaranteed return life insurance contracts

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 q S0e gT , λ  ST − S0egT r + S0e gT , (3) or LT = S0e gT  ~} € Minim ium Garanteed Term inal b enefi t + λMax q 0, ST − S0egT r  ~} € Call option (4) 4 1 3 5 7 9 λ = 1.0 λ = 0.6 Payoff at Maturity for a participating contract 6 The price of a contract having such a payoff is calculated in the following way: V0 = EQ k e−rTLT l , (5) which leads to V0 = S0e (g−r)T + λCBS0 (S0, S0e gT , r, σ, T ) (6) where CBS0 (S0, S0e gT , r, σ, T ) represents the Black Scholes price of a European call option with initial price of the underlying S0, a strike value S0egT , 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 TC0erT 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 + TC0. At maturity T, the value of the assets of the company is defined in the following way, AT = (TC0 + (V0 − S0)) erT + ST . (7) The benefits to be paid to the policyholder are given by LT = S0e gT + λMax q 0, ST − S0egT r . (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 = TC0] 6 α. (9) Assuming that the initial value of the shareholder’s capital is TC0, we can write AT < LT ⇔ (TC0 + (V0 − S0)) erT + ST < S0egT + λMax q 0, ST − S0egT r but we also have AT < LT ⇔ AT < LgT = S0e gT , . because insolvency can occur if and only if the value of assets is lower than the value of the guaranteed benefits. Therefore {AT < LT } = q (TC0 + (V0 − S0)) erT + ST < S0egT r and hence P [AT < LT ] = P ⎡ ⎢⎢⎣ST <  S0e gT − (TC0 + (V0 − S0)) erT   ~} € β ⎤ ⎥⎥⎦ = P [ST < β] . (10) The relation P [ST < β] = α will be satisfied if and only if β = S0e (µ−σ 2 2 )T+σ √ TwT (11) where wT = Φ−1(α) is the inverse of the standard normal distribution. 7 Therefore the initial target capital is given by TC0 =  S0e gT − 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 =  S0egT + λMax 0, ST − S0egT if ST ≥ β AT = (TC0 + (V0 − S0)) erT + ST if ST < β , (13) where β defines the threshold value of the stock price indicating insolvency, {ST < β}⇔ q AT < L g T = S0e gT r . We can also write the payoff in the following way, LT = S0e gT + λMax q 0, ST − S0egT r · 1{ST≥β} +  (TC0 + (V0 − S0)) erT + ST  · 1{ST<β}, (14) or LT = S0e gT · 1{ST≥β} + λ(ST − S0egT ) · 1{ST≥Max{S0egT ,β}} +  (TC0 + (V0 − S0)) erT + ST  · 1{ST<β}. (15) 0.5 1 1.5 2 0.6 0.8 1.2 1.4 1.6 1.8 Payoff at Maturity as a function of ST considering the default of the insurer 8 Comparing to the true guarantee case, the contract includes an additional put option in favour of the company. Ignoring default implies an overpricing of the contract and results therefore in an additional value creation for the shareholders. The price of such contracts are calculated in the following way V0 = EQ k e−rTLT l (16) which leads to V0 · erT = S0egT ·Q [ST ≥ β]− λS0egT ·Q k ST ≥Max q S0e gT , β rl +λEQ k ST · 1{ST≥Max{S0egT ,β}} l +Q [ST < β] · (TC0 + (V0 − S0)) erT +EQ  ST · 1{ST<β}  , or V0 = S0e (g−r)T ·N (d2)− λS0e(g−r)T ·N (h2) + λS0 ·N (h1) + (TC0 + (V0 − S0)) ·N (−d2) + S0 ·N (−d1) , (17) with d1 = ln  S0 β  +  r + σ 2 2  T σ √ T , d2 = d1 − σ √ T (18) and h1 = ln S0 K  +  r + σ 2 2  T σ √ T , h2 = h1 − σ √ T , K =Max q S0e gT , β r . (19) The target capital is defined in a similar way as in the case of a "true" guarantee: TC0 =  S0e gT − β  e−rT − (V0 − S0) . (20) Replacing in the previous formula gives the following expression for the price of the contract: V0 = S0e (g−r)T ·N (d2)− λe(g−r)TS0 ·N (h2) + λS0 ·N (h1) +  S0e gT − β  e−rT ·N (−d2) + S0 ·N (−d1) . (21) When default is considered, the price of the contact depends directly on the value of the target capital. Therefore, the pricing and the valuation of the target capital have to be solved simultane- ously. The difference in price compared with the "true" guarantee case is a deduction of an amount corresponding to the value of the right of the shareholder to not inject additional capital in the company. 9 The following example shows the variation of the price of the contract and the target capital with a change in the level of solvency α. 0.5 0.6 0.7 0.8 0.9 1− α 0.2 0.4 0.6 0.8 1 8__Price , −−TC< Price and Target capital as function of 1-α, S0 = 1, r = 0.15, µ = 0.17, g = 0.08, σ = 0.3, λ = 0.95, T = 1 The price of the contract and the target capital are increasing functions of the solvency level α. We have lim α→0 TC0 (α) = +∞, but this convergence is slow due to the short-tail distribution of the financial uncertainty W which is assumed to be normally distributed. The target capital is more sensitive to the solvency level than the price is. The level of the solvency α will have an impact on the payoff only when the stock price ST is in the critical region below β. The probability to be in this critical region decreases with lower levels of α, this explains why the price is less sensitive than the target capital. We note also that the price of the contract is increasing with the level of the target capital. In order to charge the policyholder with a maximum price, the company could be tempted to increase indefinitely the level of the target capital. But as we will see in the next section, capital has a cost, and the company can only have a limited resort to shareholders’ capital. 10 3 Cost of Capital 3.1 Incomplete markets and cost of capital Insurance companies are required to maintain a minimum level of capital in order to satisfy the solvency level. In the previous section this capital was represented by the target capital TC0. A rational client will usually prefer to buy his contract from the more secure company. A simple way for an insurance company to reduce its ruin probability is to increase the target capital to relatively high levels. In practice we note that the access to equity capital has a cost. This cost is represented by an extra return requirement by the shareholders on their invested assets. Part of this extra return covers the risk that in case of default shareholders may not receive the full value of their investment. Covering the default risk is equivalent to going back to the situation where pricing was performed under the assumption of a "true" guarantee. In which case the value of the limited liability put option was not deducted from the price of the contract. The practice shows that in addition to the assumption of a "true" guarantee, the shareholders will still require an extra compensation. This extra compensation is justified not only by capital costs resulting from market imperfections such as frictional costs, double taxation or agency costs but mainly from the fact that the policyholder agrees to pay more than the cost of the contract for the shareholder. We will summarize all these extra compensations to the shareholder under the term of "cost of capital". And we consider the cost of capital to be entirely explained by the risk aversion of the policyholder who can not access or replicate the payoff of the contract outside the offer of the company. In such incomplete markets there is usually no uniqueness of prices satisfying no-arbitrage conditions. The final prices will be chosen by the insurance company according to the level of the competition in the market. For a new contract, the insurer will fix the price at the highest level, and afterwards the price will be adjusted with the inflow of new competitors. Let us consider the following model to illustrate the incomplete market situation. Let us assume that the insurance company wants to sell a contract with a term of one year (T=1). We assume that this contract covers entirely a contingent claim C faced by the policyholder. The preferences of the policyholders and the shareholders at time T are described by exponential utilities with different absolute risk aversions αP and αS respectively. Let uP be the utility function of the policyholder and uS the utility function of the shareholder. uP (x) = 1− e−αP x uS(x) = 1− e−αSx And let WP and WS be the initial wealth of the policyholder and shareholder respectively. We will define the price of the contract by the indifference price. We assume that if the economic agents are not buying or selling the contract, then they will invest their initial capital in the riskless asset and will wait to face the contingent claim at time t=T. The indifference price pS for the shareholder is defined in the following way: uS (WS(1 + r)) = E [uS ((WS + pS) (1 + r)− C)] (22) this gives, pS = 1 αS(1 + r) ln  E k eαSC l (23) pS is the minimal price at which the shareholder will accept to sell the contract. The indifference price of the policyholder is defined as follows: E [uP (WP (1 + r)−C)] = E [uP ((WP − pP ) (1 + r) + C − C)] = uP ((WP − pP ) (1 + r)) (24) The first term can also be written as E [uP (WP (1 + r)− C)] = E k 1− e−αP (WP (1+r)−C) l = 1−E k e−αP (WP (1+r)−C) l = 1− e−αP (WP (1+r))E k eαPC l . 11 The second term in (24) is given by uP ((WP − pP ) (1 + r)) = 1− e−αP ((WP−pP )(1+r)). Plugging in the equation (24) gives e−αP (WP (1+r))E k eαPC l = e−αP ((WP−pP )(1+r)), −αP (WP (1 + r)) + ln  E k eαPC l = −αP ((WP − pP ) (1 + r)) , ln  E k eαPC l = αP pP (1 + r) from which it follows that pP = 1 αP (1 + r) ln  E k eαPC l . pP is the maximal price that the policyholder is willing to pay for the insurance contract. The assumption of a higher risk aversion for the policyholder comparing to the shareholder implies pP ≥ pS . The policyholder is ready to buy the contract at a price pP higher than the price offered by the shareholder pS . We can define the difference of prices pp − pS as a source of an excess return for the shareholder. Even if the shareholder assets are invested riskless, he will receive a higher return than the risk free rate due to the risk aversion of the policyholders. In such a situation, we will say that the capital held in the company has a cost; this cost is represented by the price differential pp − pS . Consequently, the level of the target capital can not be increased indefinitely; an optimal level has to be defined. This will be the purpose of the next sections. The following example gives an illustration of the relation between the indifference price and the level of risk aversion. Example : We assume that the policyholder is facing a contingent claim C given by C =  Cu = 1.5 with probability pu = 0.5 Cd = 0.5 with probability pd = 0.5 The company proposes an insurance contract to cover this claim. The risk aversion of the policy- holder and the shareholder are represented respectively by αP and αS . The computation of the indifference prices according to the methodology described above gives the following result, 12 Indifference prices A transaction between the policyholder and the company can occur only if pS ≤ pP . In general, a sufficient condition for the occurrence of a transaction will be a sufficiently large price differential pP − pS to cover the default risk on the shareholder’s assets but also any frictional costs. 3.2 Pricing under cost of capital In this chapter we consider the same models as defined earlier, and we will look at the additional impact of the cost of capital. The company fixes at the beginning of the contract the maximum level of the insolvency risk α that they can tolerate. The target capital corresponds to the minimum capital injected initially in the company such that the ruin probability at maturity is lower than the level α. We assume that the cost of the capital is defined by an additional return γ on the target capital TC0 injected by the shareholders. This additional return is in excess of the invested returns and is assumed to take in account the risk aversion of the policyholders. For instance if TC0 is invested in the riskless asset returning r then the return allocated to the shareholder will be γ + r. In such a situation the company can not increase indefinitely the level of the target capital without altering the performance of their business. An inappropriately high level of the target capital may discourage certain policyholders from buying their contract from the company, because they may consider the cost of the capital to be too high. 3.2.1 The case of a ’true’ guarantee The payoff to the policyholder is defined in the same way as in the previous sections: LT =Max q S0e gT , λ(ST − S0egT ) + S0egT r (25) We assume that in addition to the fair price of the contract, the policyholder is required to pay initially an amount B to cover the cost of the shareholders’ capital. Instead of applying such an upfront charge, an other possibility could be to charge at maturity a portion of the surplus of the policyholder. The required value by the shareholder at time T is given by TC0e(r+γ)T . 13 If we assume that B is invested in the riskless asset, we should have the following relation (TC0 +B) e rT = TC0e (r+γ)T , and hence B = TC0  eγT − 1  . (26) The total premium paid by the policyholder should be P = V0 +B (27) where V0 is defined in a similar as in the section 2.2: V0 = S0e (g−r)T + λCBS0 (S0, S0e gT , r, σ, T ). (28) This additional premium B will also have an impact on the target capital. At maturity the value of the assets is given by AT = (TC0 + (V0 − S0) +B) erT + ST (29) The target capital TC0 has been defined such that the following condition is satisfied: P [AT < LT | Initial shareholder capital = TC0] 6 α. The insolvency event is defined as follows: {AT < LT } = (TC0 + (V0 − S0) +B) erT + ST < S0egT + λMax q 0, St − S0egT r . When the company is in an insolvency situation, there is no surplus distributed to the policyholder, therefore {AT < LT }⇔ q AT < L g T = S0e gT r or {AT < LT } = q (TC0 + (V0 − S0) +B) erT + ST < S0egT r , which implies P {AT < LT } = P q ST <  S0e gT − (TC0 + (V0 − S0) +B) erT r = P {ST < β} . The relation P {ST < β} = α will be satisfied if and only if β = S0e(µ−σ 2 2 )T+σ √ TwT with wT = Φ−1(α) the inverse of the standard normal distribution. Therefore the target capital is given by TC0 =  S0e gT − β  e−rT − (V0 − S0)−B. (30) Replacing B by its value defined above gives TC0 =  S0e gT − β  e−rT − (V0 − S0)− TC0  eγT − 1  . (31) Finally, TC0 = e −γT  S0e gT − β  e−rT − (V0 − S0)  . (32) The target capital is a decreasing function of the cost of capital γ. A shareholder expecting a higher excess return γ has to inject a lower amount of capital in the company because the target capital is partly financed by the policyholder. Comparing to the case studied in the section 2.2, we notice that the target capital becomes now relevant for the policyholder’s decision through the cost of capital B that the policyholder has to bear. In addition, for a similar value of the solvency level α, the required shareholder’s capital injection has reduced by an amount corresponding to the cost of capital B. 14 The following example illustrates the impact of the capital cost γ on the price of the contract and on the target capital. 0.2 0.4 0.6 0.8 1 γ 0.2 0.4 0.6 0.8 1 1.2 8__ P0, _ _B, −−TC0< Total price P0, Cost of Capital B and target capital TC0 as function of γ. S0=1, r=0.05, µ = 0.07 , g=0.04, σ=0.3, λ=0.95, T=1, α=0.01. With the assumption of a "true" guarantee, the total price P0 is changing in the same proportions as the cost of capital B for an in increase in the capital cost γ. The initial price of the contract V0 which excludes the cost of capital remains indifferent to the level of the target capital TC0. 3.2.2 The case of a conditional guarantee The payoff to the policyholder is now given by LT =  S0egT + λMax 0, ST − S0egT if ST ≥ β AT = (TC0 + (V0 − S0) +B) erT + ST if ST < β . (33) Here, β defines the level of the stock price ST at which the value of the assets is lower than the value of the liabilities, i.e. β indicates the solvency region {ST < β}⇔ AT < L g T = S0e gT . The payoff can also be written in the following way, LT =  S0e gT + λMax q 0, ST − S0egT r · 1{ST≥β} +  (TC0 + (V0 − S0) +B) erT + ST  · 1{ST<β}. (34) LT = S0e gT · 1{ST≥β} + λ  ST − S0egT  · 1{ST≥Max{S0egT ,β}} +  (TC0 + (V0 − S0) +B) erT + ST  · 1{ST<β}. (35) As previously, the required value by the shareholder at time T is TC0e(r+γ)T , the additional return γ is financed by an upfront payment by the policyholder of an amount B. If we assume that B is invested riskless, (TC0 +B) e rT = TC0e (r+γ)T ⇒ B = TC0  eγT − 1  . The total premium paid by the policyholder is P0 = V0 +B. (36) This additional premium B has an impact on the target capital. At maturity t=T, the value of the assets is given by AT = (TC0 + (V0 − S0) +B) erT + ST . (37) 15 The target capital TC0 is defined such that the following condition is satisfied: P [AT < LT ] 6 α. By the definition of β, P [AT < LT ] 6 α ⇐⇒ P [ST < β] 6 α, where β = S0e(µ− σ2 2 )T+ √ TσwT with wT = Φ−1(α). The target capital is given by TC0 =  S0e gT − β  e−rT − (V0 − S0)−B. (38) The price of such a contract is calculated exactly in a similar way as in the case without cost of capital and gives: V0 = S0e (g−r)T ·N (d2)− λS0e(g−r)T ·N (h2) + λS0 ·N (h1) + (TC0 + (V0 − S0) +B) ·N (−d2) + S0 ·N (−d1) , (39) or V0 = S0e (g−r)T ·N (d2)− λS0e(g−r)T ·N (h2) + λS0 ·N (h1) (40) +  S0e gT − β  e−rT ·N (−d2) + S0 ·N (−d1) (41) with d1 = ln  S0 β  +  r + σ 2 2  T σ √ T , d2 = d1 − σ √ T , (42) h1 = ln S0 K  +  r + σ 2 2  T σ √ T , h2 = h1 − σ √ T , K =Max q S0e gT , β r . The final price that the policyholder has to pay is P0 = V0 +B. A reduction in the solvency level α will lead to a lower premium paid by the policyholder. This results from the following two effects: - a reduction in the target capital which lead to a lower associated cost of capital, - an increase in the probability of default which leads to a higher deduction in the premium for the value of the limited liability put option. The following example illustrates this relation between the solvency level α, the price of the contract before cost of capital V0, the target capital TC0 and the cost of capita B. 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−α 0.2 0.4 0.6 0.8 1 1.2 __V 0, −− TC0, __ __ B Total price V0,target capital TC0 and cost of capital B as a function of 1-α. S0=1, r=0.05, µ = 0.07, g=0.04, σ=0.3, λ=0.95, γ = 0.2. 16 4 Multi-period extension 4.1 General framework In this section we will try to extend the one period model of the previous sections to a multi-period situation. Let At and Lt be respectively the value of the assets and the liabilities of the company at time t. We assume that the contract is still of a European type in case of non default and at default the policyholder will receive the remaining value of the assets. In the multi-period case, our first task will be to define an appropriate risk measure. The guaranteed cash flow at time T is given by LgT = S0e gT . We will be interested in the first passage time τ = inf q t > 0 : At < e−r(T−t)E [L g T | Ft] r where the company is in an insolvency situation. The value of the assets is given by At = (TC0 + (V0 − S0) +B) ert + St (43) We can rewrite the default time as follows: τ = inf q t > 0 : (TC0 + (V0 − S0) +B) ert + St < e−r(T−t)S0egT r with B = TC0  eγT − 1  τ = inf q t > 0 : St < e−r(T−t)S0egT −  TC0 + (V0 − S0) + TC0  eγT − 1  ert r = inf q t > 0 : St < S0egT−r(T−t) −  V0 − S0 + TC0eγT  ert r Using the expression of the stock price under the physical probability, we can rewrite, τ = inf  t > 0 : S0e(µ− σ2 2 )T+σWt <  S0egT−r(T−t) −  V0 − S0 + TC0eγT  ert  = inf  t > 0 :Wt < 1σ  ln  S0e gT−r(T−t)−(V0−S0+TC0eγT )ert S0  − (µ− σ2 2 )T  = inf {t > 0 :Wt < κ(t)} Given the continuity of the Brownian motionWt, we have at insolvency time τ, Aτ = e−r(T−τ)S0egT . 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 TC0 at the level α is the minimum capital satisfying the following condition P [τ ≤ T | Initial shareholder capital = TC0] 6 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 =  S0egT + λMax 0, ST − S0egT if τ > T Aτ = (TC0 + (V0 − S0) +B) erτ + Sτ if τ ≤ T where Aτ = (TC0 + (V0 − S0) +B) erτ + Sτ , B = TC0  eγT − 1  , Sτ = e −r(T−t)S0e gT −  TC0 + (V0 − S0) + TC0  eγT − 1  erτ . Replacing the values of B and Sτ in the expression of Aτ gives back Aτ = e−r(T−τ)S0egτ and LT∧τ =  S0egT + λMax 0, ST − S0egT if τ > T Aτ = e−r(T−τ)S0egT if τ ≤ T . (45) 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. 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 = TC0  eγT − 1  . 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. TC0 +B ~} € shareholders’ fund invested in the risk free asset + V0 − S0 ~} € shareholders’ part paid out to shareholders as a div idend + S0~}€ policyholder’s fund Invested in the risky asset The new structure of the capital of the company after the payment of the dividend is as follows: TC0 +B ~} € shareholders’ fund ~} € Invested in the risk less asset + S0~}€ policyholder’s fund ~} € Invested in the risky asset 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 = (TC0 +B) e rt + St. (46) The default time will be given by τ = inf q t ≥ 0 : St < e−r(T−t)S0egT − (TC0 +B) ert r . (47) 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 = S0e gτ−r(T−τ) − (TC0 +B) erτ . (48) The payout to the policyholder is now given by LT∧τ =  S0egT + λMax 0, ST − S0egT if τ > T Aτ = (TC0 +B) erτ + Sτ = e−r(T−τ)S0egT if τ ≤ T (49) In the following section we will treat the two periods case. 18 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 q t ∈ {t1, T} : At < e−r(T−t)EQ [LgT | Ft] r , with t1 ≤ T. We can also write τ = inf q t ∈ {t1, T} : (TC0 +B) ert + St < e−r(T−t)S0egT r . (50) The target capital TC0 at the level α is the minimum capital satisfying the condition P [τ ∈ {t1, T}] 6 α. In our discrete case, the ruin probability before the maturity T is given by P [τ ∈ {t1, T}] = P [τ = t1] + P [{τ = T} ∩ {τ 6= t1}] . (51) The ruin events can also be expressed as follows: {τ = T}⇔ q (TC0 +B) e rT + ST ≤ S0egT r ⇔ {WT ≤ wT } , {τ 6= t1}⇔ (TC0 +B) e rt1 + St1 > S0e gt1 ⇔ {Wt1 > wt1} . with wt = ln Kt S0 −  µ−σ 2 2  t σ and Kt = S0e gt − (TC0 +B) ert for t ∈ {t1, T} . Moreover we have the following relationship wt1 = wT +  µ− σ2 2  (T − t1) σ . (52) Replacing in the above equations gives, P [τ ∈ {t1, T}] = P [Wt1 wt1}] . (53) It can easily be shown that, P [WT ≤ wT ,Wt1 > wt1 ] = E  Φ  wT −Wt1√ T − t1  · 1{Wt1>wt1}  . (54) where Φ is the cumulative standard normal distribution. Finally, P [τ ∈ {t1, T}] = P [Wt1 < wt1 ] +E ⎡ ⎢⎢⎣Φ ⎛ ⎜⎜⎝ wt1 −  µ−σ 2 2  (T−t1) σ −Wt1√ T − t1 ⎞ ⎟⎟⎠ · 1{Wt1>wt1} ⎤ ⎥⎥⎦ (55) The equation P [τ ∈ {t1, T}] 6 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 =  S0egT + λMax 0, ST − S0egT if τ > T Aτ = (TC0 +B) erτ + Sτ = e−r(T−τ)S0egT 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 = S0e gT · 1{τ>T} + λ  ST − S0egT  · 1{ST≥S0egT} · 1{τ>T} +((TC0 +B) e rτ + Sτ ) · 1{τ≤T}. (57) The price is given by V0 = EQ k e−r(T∧τ)LT∧τ l , (58) which leads to V0 = S0e (g−r)T · EQ  1{τ>T}  +λe−rTEQ k ST · 1{ST≥S0egT} · 1{τ>T} l −λS0e(g−r)TEQ k 1{ST≥S0egT} · 1{τ>T} l +EQ k e−rTS0e gT · 1{τ≤T} l 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 5 Surrender options Life insurance contracts are usually long term contracts, they require important amount of invest- ments 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 guaran- teed 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 = TC0e rt + 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 =  S0egt + λ Max 0, St − S0egt if τ > t Aτ = e−r(T−τ)S0egT if τ ≤ t , t ≤ T. (59) 21 The default time τ is given by τ = inf q t > 0 : TC0e rt + St < e −r(T−t)S0e gT r and corresponds to the first time where the value of the assets is lower than the minimum guar- anteed 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 ∂t + rS ∂V ∂S + 1 2 σ2S2 ∂2V ∂S2 − rV = 0 for St > h(t), V (St = h(t)) = e−r(T−t)S0egT , (60) h(t) = e−r(T−t)S0e gT − TC0ert. (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) ≥ S0egt + λ Max 0, St − S0egt for 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)  = h0 h(t) dSt − St h0 h(t)2 ∂h ∂t dt, dZt = Zt  r − 1 h(t) ∂h ∂t  dt+ σdWt  . (64) We define now the value of the contract by the process U (t, Z) = V  t, h (t) h0 Z  (65) ⇔ V (t, S) = U  t, h0 h (t) S  Rewriting the equation above using the expression of Z gives ∂U ∂t + Z  r − 1 h(t) ∂h ∂t  ∂U ∂Z + 1 2 σ2Z2 ∂2U ∂Z2 − rU = 0, Zt > h0, (66) U(Zt = h0) = e −r(T−t)S0e gT . (67) or ∂U ∂t + F 1(Z, t) ∂U ∂Z + F 2(t) ∂2U ∂Z2 − rU = 0, Zt > h0, (68) U(Zt = h0) = e −r(T−t)S0e gT . (69) where F 1(Z, t) = Z  r − 1 ht ∂h ∂t  , 22 F 2(Z, t) = 1 2 Z2σ2. In addition we impose the following condition: U(t, Zt) ≥ S0egt + λ Max  0, h (t) h0 Zt − S0egt  for Zt > h0. (70) 5.1.1 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 ∂U ∂Z = Ui,j+1 − Ui,j−1 2∆Z , (71) - Forward approximation for ∂U∂t ∂U ∂t = Ui+1,j − Ui,j ∆t , (72) - Approximation for ∂ 2U ∂Z2 ∂2U ∂Z2 =  Ui,j+1 − Ui,j ∆Z − Ui,j − Ui,j−1 ∆Z  1 ∆Z = Ui,j+1 + Ui,j−1 − 2Ui,j ∆Z2 . (73) Substituting the above equations in the differential equation (66) gives Ui+1,j − Ui,j ∆t +F 1(h0+j∆Z, i∆t) Ui,j+1 − Ui,j−1 2∆Z +F 2(h0+j∆Z, i∆t) Ui,j+1 + Ui,j−1 − 2Ui,j ∆Z2 = rUi,j . (74) By rearranging the terms we have,  − 1 2∆ZF 1(h0 + j∆Z, i∆t) + 1∆z2F 2(h0 + j∆Z, i∆t)  Ui,j−1 +  − 1 ∆t − 2 ∆Z2F 2(h0 + j∆Z, i∆t)  Ui,j +  1 2∆ZF 1(h0 + j∆Z, i∆t) + 1∆Z2F 2(h0 + j∆Z, i∆t)  Ui,j+1 + Ui+1,j ∆t = rUi,j . We can also rewrite ∆t  1 2∆ZF 1(h0 + j∆Z, i∆t)− 1∆Z2F 2(h0 + j∆Z, i∆t)  Ui,j−1 +∆t  1 ∆t + 2 ∆Z2F 2(h0 + j∆Z, i∆t) + r  Ui,j −∆t  1 2∆ZF 1(h0 + j∆Z, i∆t) + 1∆Z2F 2(h0 + j∆Z, i∆t)  Ui,j+1 = Ui+1,j or aijU i i,j−1 + b i jUi,j + c i jUi,j+1 = Ui+1,j for i = 1 : N − 1 and j = 1 :M − 1 (75) 23 where aij = ∆t  1 2∆Z F 1(h0 + j∆Z, i∆t)− 1 ∆Z2 F 2(h0 + j∆Z, i∆t)  , (76) bij = ∆t  1 ∆t + 2 ∆Z2 F 2(h0 + j∆Z, i∆t) + r  , (77) cij = −∆t  1 2∆Z F 1(h0 + j∆Z, i∆t) + 1 ∆Z2 F 2(h0 + j∆Z, i∆t)  . (78) Now consider the boundary condition of the contract (1) UN,j = S0egT + λMax 0, ST,j − S0egT , UN,j = S0egT + λMax q 0, h(T )h0 (h0 + j∆Z)− S0e gT r . (2) Ui,0 = S0egi∆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 = S0egi∆t + λMax q 0, h(i∆t)h0 (h0 +M∆Z)− S0 e gi∆t r . We solve for i=N-1 to 0 aijUi,j−1 + b i jUi,j + c i jUi,j+1 = Ui+1,j for i = 1 : N − 1 and j = 1 :M − 1. The matrix representation is as follows: ⎡ ⎢⎢⎣ bi1 c i 1 ai2 b i 2 c i 2 · · · aiM−1 b i M−1 ⎤ ⎥⎥⎦ ⎡ ⎢⎢⎢⎣ Ui,1 Ui,2 ... Ui,M−1 ⎤ ⎥⎥⎥⎦ = ⎡ ⎢⎢⎢⎣ Ui+1,1 − ai1Ui,0 Ui+1,2 ... Ui+1,M − ciM−1Ui,M ⎤ ⎥⎥⎥⎦ The following graphic shows the price of the contract as a function of Z and t: Z t S00 =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 s r(t)dWt (79) 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 (80) with B (t, T ) = 2(eγ(T−t)−1) (γ+a)(eγ(T−t)−1)+2γ and A (t, T ) =  2(e(a+γ)(T−t)/2−1) (γ+a)(eγ(T−t)−1)+2γ 2ab/σ2 with γ = √ a2 + 2σ2. The guaranteed rate corresponding to the yield of the bond of maturity T is defined by g = − ln (P (0, T )) T (81) We assume that the premium π paid by the policyholder corresponds exactly to the price of one unit of the bond P(0,T), 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 + δr. 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 = − ln (P (t, T )) T − t (82) Let τ = inf {t ∈ [0, T ] : yt = g + δr} The payoff to the policyholder is defined by Lτ∧T =  πegT if τ ≥ T πegτ if τ < T (83) 25 Lτ∧T = πe gT 1{τ≥T} + πe gτ1{τ<T} (84) This contract corresponds to the combination of a zero coupon bond of maturity T and a surrender option SOδr The cost of the surrender option is defined as follows: - If there is no surrender, then at maturity the company receives an amount of πegT from the investment in the bond and pays πegT to the policyholder. This lead to a net cash flow of zero amounts. - If the policyholder surrenders at time τ before maturity, then the company receives πegTP (τ , T ) by selling the bond and pays an amount πegτ to the policyholder. The cost at time τ of the surrender option is given by: SO0(δr) =  0 if τ > T π  egTP (τ , T )− egτ  if τ ≤ T (85) The value of the surrender option at time t=0 is calculated in the following way PSO0(δr) = EQ k e− UT∧τ 0 r(s)dsπ  egTP (τ , T )− egτ  1{τ<T} l (86) We can apply the Monte Carlo method to calculate numerically the value of the surrender option SO0(δr). For that, we divide the interval [0,T] into K equal intervals of length h, and we approximate the diffusion of r by rˆk+1 − rˆk+1 = a (rˆk+1, tk) + b (rˆk, tk) zk+1 √ h (87) k=0,1,..(K-1); t0 = t, ..., tK = (t + T ); rˆ0 = r(t) and the z’s are independent standard normal variates. Repeating the calculation for different values of the threshold δr gives the following graphic, 0.00 0.01 0.02 0.03 0.04 0.05 Surrender threshold 0. 00 0. 01 0. 02 0. 03 0. 04 0. 05 P ric e of th e S ur re nd er O pt io n T=10, r0 =0.07, a=0.1, b=0.13 , σ =0.015, π = 1 26 For small values of the threshold δr, the value of the surrender option is close to zero because for small interest rate movements the bond price remains close to the price of the guaranteed bond. We notice that in a first stage, the price of the option increases with δr. This results from the fact that the option is more and more in the money at exercise time. In a second stage, the price of the option decreases to zero because the probability of reaching high levels of the threshold becomes smaller. Let’s define δ∗ = argmax δr∈[0,+∞] {PSO0(δr)} If we assume that at time t=0, the policyholder chooses a target threshold δ and does not change his mind until the maturity by surrendering his contract whenever yt = g + δr then δ∗ will be the threshold that the policyholder should choose in order to maximise the value of the surrender option. But policyholders are not always behaving rationnaly, therefore some of them could choose a different threshold than δ∗. Using the pattern of the surrender option prices (PSO0(δr), δr)δr∈[0,+∞] , we can model the behaviour of the policyholders as follows: Let Ψ be the random variable representing the threshold at which the policyholder will exercise the surrender option. We can assume that Ψ is distributed as follows: P [Ψ ≤ x] = ] +∞ o PSO0(δr)dδr −1 ] x o PSO0(δr)dδr (88) In practice such a calculation will be tedious, and this distribution will always remain an approx- imation of the true behaviour of the policyholders. A computationally more efficient method will be to derive a parametric distribution which captures certain proprieties of the surrender option prices. For instance we can require the distribution to be right skewed in order to reflect the fact that most policyholders will react with a delayed time. An other requirement could be to set the mean of the distribution such that in average the policyholders are initially choosing δ∗ as a target threshold. 27 6 Conclusion In this thesis we tried to build a unifying valuation framework for the pricing and the capital require- ment for an insurance contract. A high level of capital reflects the soundness of a company but the capital is costly and therefore an optimal level has to be defined. The final purpose of an insurance company being to provide protection for risk averse policyholders, the cost of capital must be born by the policyholders. For that reason capital valuation can not be dissociated from the pricing issue. For one-period situation, the problem could easily be solved. However for multi- period cases, the complexity of the problem becomes considerable. The main sources of the difficulty are due to the choice of the risk measure but also to the choice of the investment strategy. To overcome this difficulty we assumed that an amount corresponding to the value of the limited liability put option was initially paid out as dividend to the shareholder. We could then after defining an appropriate risk measure, analyse the case of two periods. In practice, the target capital is usually defined at portfolio level and includes diversification benefits. A further extension of our work could be the incorporation of the diversification effects and the analyse of the optimal asset allocation strategy. In the second part of this thesis we analysed two different situations of surrender options. 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