AN INSTITUTIONAL EVALUATION OF PENSION FUNDS AND LIFE INSURANCE COMPANIES


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1 AN INSTITUTIONAL EVALUATION OF PENSION FUNDS AND LIFE INSURANCE COMPANIES DIRK BROEDERS, AN CHEN, AND BIRGIT KOOS Abstract. This paper analyzes two different types of annuity providers. In the first case, the annuity is provided by a collective non profit) defined benefit pension scheme with no external shareholders. The residual risk in this pension scheme is collectively born by the beneficiaries in the fund. In the second case, the annuity is provided by a forprofit) life insurance company with external shareholders. The residual risk is then born by the shareholders of the insurance company who require a positive risk adjusted return. In both cases, we allow the nominal annuity to be discretionarily increased with the performance of the institution. In the case of a pension fund this is known as contingent indexation and the jargon for an insurance company is with profit. The paper employs a contingent claim approach to evaluate the risk return tradeoff for consumers requiring a defined benefit taking into account the difference in regulatory regimes between pension funds and insurance companies. Meanvariance analysis is conducted to evaluate under which circumstances which annuity provider is more preferable. Keywords: Pension plans, barrier options, contingent claim approach, meanvariance analysis JEL: G11, G23 1. Introduction Typically, pension funds and life insurance companies are both key annuity providers. Next to the government, they are important institutions in the world to arrange old age provisions in an efficient manner. From Figure 1, the heterogeneity of financing vehicles is observed in funded pension arrangements across OECD countries. Table 1 shows the importance of pension funds and life insurers by the size of assets under management for North America, Europe and the UK. Date. May 3, De Nederlandsche Bank DNB), Supervisory Policy Division, Strategy Department, PO Box 98, 1000 AB Amsterdam, The Netherlands, Tel: , Fax: E Mail: Netspar and Department of Business Administration III, Faculty of Economics and Law, University of Bonn, Adenauerallee 2442, Bonn, Germany. Tel: , Fax: , E Mail: Department of Business Administration III, Faculty of Economics and Law, University of Bonn, Adenauerallee 2442, Bonn, Germany. Tel: , Fax: , E Mail: 1
2 Denmark Netherlands 2) Iceland Swit zerland United States Canada Australia United Kingdom 2) Finland Sweden Ireland Japan Germany 3) Portugal Belgium 2) Spain 2) New Zealand Mexico Poland Hungary Austria 4) Korea France Norway Czech Republic Italy Slovak Republic Turkey Greece Pension funds 1,021) 512) 30) 59) 155) 13) 98) 38) 11) 28) 68) 155) 23) 6) 62) 2) 4) 0) Book reserves 218) 110) Pension insurance contracts Other types 1) 385) 914) 22) 462) 15,886) 1,306) 687) 2,117) 163) % GDP Figure 1. Heterogeneity of financing vehicles used in funded pension arrangements across OECD countries, 2006; Source: Region Pension funds Life insurers North America 10, % 3, % Europe 2, % 4, % UK 1, % 2, % Total 14, , 835 Table 1. Total Size of funds under management by region; Source: OECD totals in 1,00 bn USD for There are however distinct institutional differences between these two providers, as illustrated in Table 2. Pension funds usually have the form of a nonprofit organisation or a trust. As such, the collective beneficiaries are both the liability holder as well as the equity holder. The residual risk is therefore born by the beneficiaries. In a continuum of overlapping generations this may also include future generations. New entrants to a pension fund might be confronted with losses that occurred in the previous period. Also, in many jurisdictions also the company behind the pension scheme is either explicitly or implicitly involved in risk sharing. E.g. in case of underfunding or in case of significant overfunding) contributions to the fund can be increased or decreased) relative to the cost effective level. If there is a pension guarantee fund, like the Pension Benefit Guarantee Corporation in the US or the Pension Protection Fund in the UK, the residual risk is also covered by these institutions. Specifically in the case the corporation defaults and the pension rights have to be protected. In order to satisfy the non discretionary or defined benefit pension promise, pension funds are generally subject to a full funding requirement. This requires that at all times the market value of the assets should be at least equal to the market value of the non discretionary liabilities, i.e. the funding ratio should always be in excess of 100%. Some countries, like the Netherlands explicitly prescribe that pension 2
3 Pension fund Insurance company Institutional structure Trust Incorporation Who bears residual risk? Future) beneficiaries are Beneficiaries are bond holders residual claim holders, risk external stock holders sharing with sponsor bare residual risk Contract specifications  Guaranteed benefit Defined benefit pension annuity  Indexation policy Contingent indexation related with profit to inflation or wage growth Investment policy Typically large mismatch Typically matching of guaranteed liabilities Regulation parameters  Confidence level Typically low, e.g. 97.5% Typically high 99.5%  Recovery period Typically high, e.g. 3 years Typically short, e.g. 3 months Policy variables Contribution rate Asset allocation Indexation policy Initial surplus Premium Asset Allocation With profit policy Debt/Equity ratio Table 2. Difference between pension funds and insurance companies funds hold additional assets. This surplus is necessary to absorb short term deviations in the funding ratio. The required surplus is usually a function of the mismatch risk between assets and liabilities. A high mismatch risk requires a large surplus, and in case of assetliability matching the surplus can be kept to a minimum. The available surplus assets minus liabilities) can be regarded as the equity of the pension fund in the sense that it offers unlimited risk absorption. On the other hand it cannot be considered as true equity as it does not offer voting rights. The board of a pension fund typically consists of representatives of both employers and employees resembling the collective risk sharing on hand. This fussiness creates the peculiarity that a pension scheme can be analysed in different ways. It can be regarded a risk management problem as well as an investment problem. Risk management s primary focus is to secure the defined benefit liabilities. From this perspective, the pension scheme can decide to limit mismatch risks, for example by hedging interest rate risks or inflation risks using bonds and derivatives. However, if pension provisioning is regarded as an investment problem, then other considerations play a role. From the perspective of an integrated life cycle model it can be more attractive to take investment risk in the pension scheme asset portfolio. This may hold especially true for younger participants who have substantial human capital with limited risk and who can offer flexible labor supply to offset financial risk, as in Bodie, Merton and Samuelson 1992). The ambiguity above implies that pension scheme trustees face a complex problem 3
4 involving weighing the interests of all stakeholders against each other. A forprofit insurance company, usually in the form of an Incorporation Inc), can also provide annuities. 1 Now the beneficiaries act as the bond holders while the shareholders provide equity. The shareholders therefore accept the residual risk. E.g. if the return on the assets of the insurance company is negative or if there is underwriting loss, this will be absorbed by the shareholders. On the other hand, if the performance is above average the shareholders will yield from this. However, the shareholders have a limited liability. In case of a default the shareholders will not lose more than their initial investment. As the shareholders are the residual claimants and the owner, they also decide on the policy and risk management. The beneficiaries or consumers have no say in this. Pension regulation is less strict. Some countries allow substantial recovery periods for a pension fund to restore sufficient funding, see Broeders and Chen 2008). Insurance supervision on the other hand is stricter. If the solvency ratio available over required solvency level) is inadequate the supervisory authorities will react promptly and the company will be liquidated if there is no resurrection in the short run. This way the consumers are relatively certain that they do not lose significant value at liquidation. Interestingly, a confidence level of 97.5% for pension funds is significantly lower than the 99.9% confidence level in the future Solvency II framework for insurers. In particular, this questions the levelplaying field between pension funds and insurers both writing comparable long term obligations. However, the difference may be explained by the additional policy instruments pension funds possess like being able to raise future contributions and cutting back on indexation of pensions when necessary. As a ruleofthumb, this greater flexibility should therefore reflect more or less the difference in confidence levels. Furthermore, there are distinct differences in the investment policy between pension funds and life insurance companies. On average pension funds run a larger mismatch risk when compared to life insurance companies. This is shown in table 1. Pension funds in the US, Europe and in the UK invest more heavily on equity investments. Investments in listed and non listed equities, real estate and alternatives are dominant for pension funds. Insurance companies are more focussed on asset and liability matching and prefer fixed income assets. To our knowledge, there are still no papers dealing with the institutional differences between pension funds and insurance companies. The present paper aims to fill this gap and makes some comparative analysis of these two institutions. First, we specify the contracts fund. 1 An insurance company can also take the form of a mutual. This is much comparable to a pension 4
5 Region Pension funds Life insurers North America Equities 62% 4% Bonds 33% 78% Other 5% 18% Europe Equities 40% 26% Bonds 50% 59% Other 10% 16% UK Equities 66% 43% Bonds 25% 39% Other 9% 18% Table 3. Asset allocation by region; Source: IMF, OECD provided by these two institutions. Hereby we are already able to notice the difference between these two institutions by looking at the fair combinations of contract parameters, particularly the fair participation rate with which contract holders are allowed to participate in the surpluses of the institutions. This part of the analysis is based on the valuation of the issued contracts. Second, we address the difference issue of the two institutions from the perspective of the contract holders, i.e. we mainly aim to answer the question, in which institution a potential annuity buyer will prefer to invest? In order to answer this question, we look at the expected value and the variance of the terminal contract payoff combined with the contract holder s preferences 2. The remainder of the paper is organized as follows. Section 2 introduces the contracts provided both by the pension fund and the insurance company. Section 3 investigates the valuation of the contracts which serves for the fair contract analysis and some numerical results are illustrated. Section 4 conducts a meanvariance analysis to find out which institutions are more preferable for a certain unnecessarily riskneutral) potential annuity buyer. Section 5 concludes the paper. 2. Contract specification This section aims to introduce the general contract payoffs, i.e. the payoff when there is no underfunding at the maturity date and the rebate payment when premature default occurs. When there is no premature default, we distinguish between the pension fund and 2 Since Chen and Suchanecki 2007) and Broeders and Chen 2008) have provided a very detailed analysis on the recovery period, we ignore the effects of the recovery period in this paper. 5
6 the life insurance company. There, we are already able to see the first difference between these two institutions Payoff in case of no premature default. We consider the pension plan of a single representative participant who has to work another T years. Let us assume the pension plan was issued at time t 0 = Collective pension fund. At time 0, the pension fund issues a conditionally indexed defined benefit pension plan to a representative group of beneficiaries who provides an upfront contribution P 0. The pension fund also receives an amount of initial contributions from the sponsor S 0 at time 0. Consequently, the initial asset value of the pension fund is given by the sum of the contributions from both the beneficiary and the sponsor, i.e. = P 0 + S 0. From now on, we shall denote S 0 = 1 α) with α [0, 1]. The pension fund invests the proceeds in a diversified portfolio of risky and nonrisky assets. At retirement T, the beneficiaries receive a lump sum nominal pension of L. In addition, the pension plan has the objective to increase pension rights by i% per annum, where i% might be related to, say, the average expected CPI or wage growth. Since the determination of this parameter should take into consideration many factors, in reality this procedure is fairly complicated. Here, for simplicity we assume i is deterministic and constant and a fully indexed pension is then L = Le it. However, it should be noted that the actual outcome of the pension plan is contingent on the funding ratio at maturity T, which is defined as the ratio of the pension fund s assets A T ) to its liability. At maturity T, given that the assets are sufficiently high A T > L), the beneficiary not only receives an indexed pension of L, but is allowed to participate in the surplus of pension funds A T L) with a participation rate δ, where δ [0, 1] is the surplus distribution parameter. For instance, δ = 0.75 means that the pension beneficiary receive 3/4 of the surplus. When the assets of the pension fund do not perform well, we distinguish between two scenarios: A T < L and L A T < L. In the latter case, the assets value A T is assigned to the beneficiary, whereas in the former case the guaranteed amount L is paid out to the beneficiary. Since a pension fund does not have external shareholders, there is instead the corporate pension plan sponsor, a pension guarantee fund or the government bearing the residual risk, when the assets are insufficient to cover the guaranteed benefits L. To sum up, at the maturity date T the payoff to the beneficiary is assuming no early termination): L, if A T < L ψ B A T ) = A T, if L A T L L + δa T L), if A T > L. More compactly, we can split the above payoff into two parts: ψ B A T ) = min{max{a T, L}, L} + δ max{a T L, 0}. 6
7 ψ BA T ), ψ SA T ) L L δ L L L AT Figure 2. The payoffs ψ B A T ), ψ S A T ) to the beneficiaries and the sponsor given no premature closure collective pension fund). The first component on the righthand side is capped by L, i.e. it corresponds to the payoff of a traditional pension plan where the beneficiary sells off any payoffs above L and is not entitled to sharing in the surplus of the pension fund. The second component corresponds to the surplus participation which allows the beneficiary to share in the pension fund s surplus with a participation rate δ. Rephrasing the first component, we can rewrite this payoff to: ψ B A T ) =L + [A T L] + 1 δ)[a T L] +, 1) where we have used [x] + := max{x, 0}. This payoff consists of three parts: a promised amount L, a long call option on the assets with strike equal to the promised payment L, and a short call option with strike equal to L multiplied by 1 δ). The latter represents the money returned to the pension plan sponsor by the beneficiaries to cover the shortfall risk. Including this compensation, the total payoff to the pension plan sponsor at maturity ψ S A T ), is given by: or more compactly, A T L, if A T < L ψ S A T ) = 0, 1 δ)a T L), if L A T L if A T > L ψ S A T ) = 1 δ)[a T L] + [L A T ] +. The payoff can be decomposed into two terms: a long call option which corresponds to the bonus received by the sponsor and a short put option reflecting the deficit which he 7
8 Assets Liabilities Equity E 0 = P 0 = 1 α) ) Liabilities P 0 = α ) Table 4. Initial capital structure of a forprofit insurance company covers in case of underperformance of the assets. The payoffs of the beneficiary and the sponsor are illustrated in figure Forprofit life insurance company. A forprofit insurance company, usually in the form of an incorporation Inc.), can also provide annuities. The beneficiaries act as the bond holders in this situation and the shareholders provide equity. A simplified) initial capital structure of a forprofit insurance company is described in Table 4. That is, for simplicity, we suppose that the representative beneficiary also policyholder or liability holder) whose premium payment at the beginning of the contract constitutes the liability of the insurance company, denoted by P 0 = α, α [0, 1] the same as in collective pension fund framework), and the representative equity holder whose equity is accordingly denoted by E 0 = 1 α) form a mutual company, the life insurance company. Now at the maturity date T, the outstanding liability that the insurance company should provide the beneficiary who survives time T is given by A T if A T < L ψ L A T ) = L if L A T L α L + δ l αa T L) if A T > L α or more compactly [ ψ L A T ) = L + δ l α A T L + [L A T ] α] + 2) with A t denoting now the insurance company s assets value at time t [0, T ]. As illustrated in Figures 2 and 3, the payoff of the beneficiary in a collective pension fund and in a forprofit insurance company framework shows quite different patterns. More specifically, when the final asset s value is not sufficiently high A T < L), in a withprofit life insurance contract, the contract holder will obtain A T due to the limited liability of the equity holder, whereas in a pension plan, a floor here L, provided by the sponsor) is ensured to the beneficiary. The equity holders accept the residual risk but will not lose more than their initial investment. Therefore, when the firm s assets are underperforming, only the existing assets A T will be assigned to the surviving beneficiary. It implies that the forprofit insurance company in fact does not provide the beneficiary with a guaranteed amount L under all circumstance. When the assets perform moderately L A T L), 8
9 ψ LA T ) L L L α δ l A T Figure 3. The payoffs ψ L A T ), ψ E A T ) to the beneficiary and the liability and equity holder given no premature closure life insurance company). a withprofit life insurance provides its contract holder with the guaranteed amount L, whereas in a pension plan, the entire assets value is assigned to the beneficiary. Finally, if the assets perform well A T > L), both the collective and withprofit pension plan allow their contract holder to participate in their surplus. The differences are twofold. One lies in the triggering condition for the surplus sharing. In a collective pension fund framework, the surplus is distributed to the beneficiary when the assets value exceeds the fully indexed pension L = Le it, whereas in the case of a forprofit insurance company, the triggering bound is L/α. The second difference concerns the participation rate. It corresponds to δ in a collective pension plan and δ l α in a forprofit insurance company framework. It indicates that the bonus payment depends on the initial) contribution rate of the liability holder in the case of a forprofit life insurance company. When the performance of the firm s assets is above average, the equity holder will benefit from this. The payoff of the equity holder is residually given by [ ψ E A T ) = [A T L] + δ l α A T L +. 3) α] The payoff of the equity holder is also given in Figure Premature default formulation. We now turn to the case when a liquidation is enforced by the regulator. We suppose that the regulator monitors the firm s assets value A t continuously because a company has to be solvent at any time 3. Default and 3 In particular, in the case of a continuous monitoring by the regulator, when policies also include surrender options, the company should be able to give back the promised amount at any time. 9
10 liquidation 4 are carried out by the regulator when the insurer s firm assets A t become too low, mathematically when they hit some deterministic timedependent barrier B t : B t = η L e rt t) 4) for t [0, T ], where r is continuously compounded year on oneperiod bond. We assume η 1, this parameter η may be regarded as a regulation parameter controlling the strictness of the regulation rule. The liquidation time τ is given by τ = inf{ t [0, T ] A t B t }. 5) If τ T, a premature liquidation results. The liquidation date is constructed as the first time of the firm s asset hitting the barrier from above. Upon premature liquidation, a rebate payment Θ B τ) = Θ L τ) = B τ 6) is provided to the beneficiary for both the collective pension fund and the forprofit insurance company case). The pension fund or the equity holder in case of a life insurance company) receive Θ S τ) = 0 Θ E τ) = 0). Hereby we have assumed to ignore liquidation costs and other types of costs. 3. Fair Contract Analysis The first question which comes to mind is how different the contract parameters particularly the participation rate in the surpluses) these two contracts offer Theoretical analysis. The faircontractprinciple can be addressed to answer this question. Similar study as in Broeders and Chen 2008) and Grosen and Jørgensen 2002), a fair contract is determined when the market value of each party coincide with its initial contribution to the pension fund or life insurance company, i.e. V k ) = E [ e rt ψ k A T )1 {τ>t } ] + E [ e r τ Θ k τ)1 {τ T } ], k = B, S, L, E, where B, S, L, E stands for beneficiary, sponsor, liability holder, and equity holder respectively. The market value corresponds to the expected discounted payment under the unique equivalent martingale measure P and E is used to denote the expectation taken under this measure. For the valuation, the firm s assets are assumed to evolve according to da t = A t r dt + σ dw t ) where σ is the instantaneous rate of return of the asset under the riskneutral probability measure P and Wt is a standard Brownian motion under this measure. 4 For instance, in the US Bankruptcy Code in the Chapter 7 Bankruptcy Procedure, default leads to an immediate liquidation. 10
11 The value V k ) k = B, L, S, E) consists of time 0 value of several components. Determining the market values of the benefits for each party boils down to deriving several values: the value of fixed payment given that there is no default during [0, T ] τ > T ), a downandout call option, a downandout put option and the expected discounted rebate payment if there is a default τ T ). More specifically, we have V B ) =F P L) + DOCL) 1 δ)doc L) + RP B V S ) =1 δ)doc L) DOP L) + RP S V L ) =F P L) + δ l αdocl/α) DOP L) + RP B V E ) =DOCL) δ l αdocl/α) + RP S. The value of the fixed payment F P L) is equal to E [e r T L1 {τ>t } ] = Le rt Nd, B 0 )) A0 B 0 ) ) Nd B 0, )) with d ± S, K) = ln S K )± 1 2 σ2 T σ and B T 0 = L e r T. The value of the downandout call option with strike K, K = L, L/α, L is given by E [e rt [A T K] + 1 {τ>t } ] = Nd +, K e rt )) K e rt N d, Ke rt } )) ) )) A0 B 2 0 B N d + 2 0, max{k e rt, B 0 } K e rt N B 0 The value of the downandout put option DOP L) can be calculated by ))) B d 2 0, max{k e rt, B 0 } E [e rt [L A T ] + 1 {τ>t } ] [ =e rt L N d, Le rt )) Nd + B 0, )) Nd B 0, ) + A ] 0 Nd B 2 B 0 B 0, Le rt )) 0 [ N d +, Le rt )) N d +, B 0 )) B 0 Nd + B 0, )) + B ] 0 Nd + B 2 A 0, Le rt )) 0 And the value of the rebate payment RP B is determined by E [e rτ ηle rt τ) 1 {τ T } ] =ηle rt N d, B 0 )) + The value of RP S can be calculated analogously. A0 B 0 ) ) Nd B 0, )) Numerical illustration. The valuation formula obtained in the last subsection are implemented here in order to conduct a fair contract analysis. The parameters are set as follows: r = 5%, L = 120, i = 1%, = 100, T = 15, α = 0.9 and η = Figures 4 and 5 illustrate the fair combinations between the risk level σ and the participation rate δ pension fund case) or δ L life insurance company case) for different η 11
12 L Η 0.9 Η 0.95 Η Η 0.9 Η 0.95 Η Σ Figure 4. Fair combinations of σ and δ for different values of η pension) Σ Figure 5. Fair combinations of σ and δ l for different values of η insurance). levels. Since we assume that both the beneficiary and the liability holder obtain a rebate payment corresponding to the barrier level at the default time, they own the same value when we assign the same value to η or barrier level). In other words, the market value of the benefits of the beneficiary and that of the life insurance company differ in the contract payoff when there is no premature default. First of all, overall it is observed that δ is much lower than δ L. It is just due to the fact that the sponsor indeed ensures the beneficiary with a guaranteed amount L at the maturity date if no premature default occurs. Whereas the insurance company only has limited liability, which implies that only the remaining asset value A T is provided to the liability holder when underfunding A T < L) occurs at the maturity T. As a compensation, δ l should be larger than δ in order to make the contract fair. In other words, fairness implies that the insurance company offers a higher participation rate to its clients than the pension fund. Second, let us take a look at the effect of the risk level on the contract value, and consequently on the fair participation rate. As the volatility goes up, the value of the downandout call increases, while the value of the downandout put increases with the volatility at first and then decreases humpshaped). The value of the fixed payment goes down and the rebate term behaves similarly to the downandout put, i.e., goes up at first then goes down after a certain level of volatility is reached. It implies that the total effect of σ on the contract value is indeed ambiguous. Here, in the framework of pension fund, for the given parameters, the contract value decreases in σ, hence a positive relation between δ and σ. In contrast, the contract value of the liability holder life insurance case) goes up in σ, which leads to a negative relation between δ l and σ. Third, the effect of η the regulation parameter) can be observed from the figures too. It is noted that different ηvalues lead to different values of the barrier B t = ηle rt t) ). As η the barrier) is set higher, the values of the downandout call and put decrease, so does the value of the fixed payment. In contrast, the expected value of the rebate increases with the barrier. In all, the contract value might decrease or increase in η when the barrier is set higher. In the pension fund case, before a critical δvalue is reached, the contract value decreases in η high δ values combined with low η values to achieve the fair contract). After a critical δvalue is reached, the contract increases in δ high δ values shall be combined with high η values to achieve the 12
13 L L 100 L 120 L L 100 L 120 L Σ Figure 6. Fair combinations of sigma and delta for different values of L pension) Σ Figure 7. Fair combinations of sigma and delta for different values of L insurance). F P DOC DOP RP σ η L Table 5. Effects of η, σ and L for different payoff components. fair contract). Whereas in the life insurance case, the contract value increases in η, which leads to a negative relation between δ l and η. Similar figures are plotted in order to analyze the effect of L. Since L appears almost in each component of the contract value, it plays a quite manifold role. To start with, the lower the L, the lower the barrier level, the less likely the barrier will be hit from the underlying asset from above. Furthermore, the magnitude of the rebate payment is discounted Lvalue multiplied by the regulation parameter). Consequently, the expected rebate payment is reduced as L goes down. Second, the same arguments make the effect of L on the fixed payment component becomes ambiguous. For instance, if the probability that the barrier is not breached during the contract life time dominates, the fixed component increases in L. Third, the value of the call option with strike L rises as a smaller value of the guarantee L shall be provided. The reduction in the barrier level through L intensifies the positive effect of L on the call value. Finally, the value of the put option decreases in the strike L, whereas the survival probability increases in L, which together makes the entire impact of L on the put value unclear. The effects of η, σ and L on different contract components are summarized in Table Mean variance analysis Through the analysis in Section 3, we notice the difference of the contract parameters particulary the participation rate) a fair contract provides. A naive annuity purchaser might already make his pension plan by choosing the contract which provides him a higher 13
14 participation rate. Note that we have determined the fair participation rate by using the same risk level volatility) for both contracts. However, as already mentioned in the introduction, pension funds and life insurances are very differently exposed to the risky assets. Hence, in what follows we want to make a comparative analysis to find out what financial institution is more preferred by a potential annuity buyer, provided that some differences, like regulatory constraints and the volatility level are taken into consideration. Since there is a onetoone correspondence between the volatility and the default probability c.f. Bernard and Chen 2008)), we assume that the pension fund and the life insurer will follow a risk management strategy with the maximal volatility level that keeps the institutions satisfy the regulatory constraints confidence level 97.5% for pension fund and 99.9% for the life insurer for oneyear time horizon). In addition to the form of the contract, the annuity buyer cares about the performance of the contract and the related risks, i.e. the expected contract payoffs and the corresponding variances. Therefore, we assume that he will choose the contract which leads to a higher value of his expected utility of the terminal wealth: E[Uterminal payment)] =UP 0 ) + U P 0 ) E[terminal payment P 0 ] + 1/2U P 0 ) E[terminal payment P 0 ) 2 ] where P 0 is the initial investment of the annuity buyer. The terminal payment shall be understood as the contract payoff at T if there is no premature default or the accumulated rebate payment if premature default occurs. U represents the utility function of the potential annuity buyer. Hereby we have assumed that the expected utility is merely dependent on the mean and variance of the terminal wealth. A risk averse annuity buyer so that U < 0) will prefer lower variance of terminal wealth to higher for fixed expected wealth. It should be noted that the third or higher moments of the terminal wealths are usually relevant for the analysis except for the quadratic utility where the third or higher moments are equal to zero). However, the analysis is valid if the rate of returns of the asset is normally distributed which is the case in our context). Furthermore, the expected E is now taken under the real world measure P rather than the equivalent martingale measure P. This is due to the fact that we are interested here in the performance of the contract. Under the real world measure P, the firm s assets are assumed to evolve according to da t = A t µdt + σdw t ), where µ, σ are positive constants and W t a martingale under this measure. Since the initial investment and rebate payment are identical for both contracts, we obtain the following 14
15 expected utility difference between these two contracts: E[Upension)] E[Ulife)] =U P 0 ) + U P 0 )) E[ [A T L] + 1 δ)[a T L] + δ l α ] 1 {τ>t } + 1 [ L 2 U P 0 ) E + [AT L] + 1 δ)[a T L] +) ] 2 1{τ>T } [ A T L ] ) + + [L A T ] + α [ ) E L + δ l α A T L + 2 ) [L A T ] α] + 1 {τ>t }. 7) When this difference is positive, the contract provided by the pension fund is chosen over that provided by the life insurer and otherwise the life insurer is more preferable. To follow the above reasonings and to proceed our analysis, we shall determine the default probabilities to determine to critical volatility level) and the first and second moments of the terminal contract payoff Default Probability. Provided that the assets of the pension fund and of the insurance company are assumed to follow the same stochastic evolution, the resulting default/liquidation probability shall be equal for the same initial value, barrier level and investment policy). The default/liquidation probability is characterized by the probability that the firm s assets have hit or fallen below the barrier before the maturity date, i.e. P τ T ) = P inf {A t B t } T t [0,T ] ). 8) In this framework, this probability can be computed explicitly. c.f. Bernard and Chen 2008)) P τ T ) =N d + B 0, e µ r)t ) ) ) 2ˆµ A0 σ + 2 N d B 0, e µ r)t ) ) 9) with ˆµ = µ r σ2. The critical risk level σ is determined numerically by solving 2 P τ T ) =N d + B 0, e µ r)t ) ) ) 2ˆµ A0 σ + 2 N d B 0, e µ r)t ) ) = ε 10) B 0 where ε is the default probability constraint. For a oneyear time horizon, the default probability constraint stipulated by Solvency II is 2.5% and 0.5% for the pension fund and life insurance company respectively. Table 6 illustrates several critical volatility level for diverse scenarios. First of all, hereby the default probability has been adjusted to T = 15 timehorizon. The higher the ηvalue, the higher the barrier level and the more probable occurs. In order to make the default probability remain equal to ε, the critical value for the volatility shall be reduced. Therefore a negative relation between η and ε are observed. 15 B 0
16 η Default probability constraint ε = 5% ε = 6% ε = 7% ε = 8% ε = 9% η = η = η = Table 6. Critical volatility levels for different values of η and ε with parameters: = 100; L = 100; T = 15; r = 0.05; i = 0.01; α = 0.9; µ = First and second moments of terminal payoff. The determination of the expected utility difference between these two contracts c.f. 7)) boils down to deriving the the first and second moments of terminal contract payoffs under the real world measure P. Since the expected final payment can be derived quite similarly as for the valuation part, we will jump to the results straightforwardly. For K = L, L α, LeiT, it holds E[[A T K] + 1 {τ>t } ] = e µt N d +, X e µ T ) ) KN d, X e µ T ) ) ) 2µ r+ 2 1 σ2 ) e µt B0 σ 2 N d + B 0 ) 2, X e µt ) ) ) 2µ r 1 2 σ2 ) B0 σ + K 2 N d B 0 ) 2, X e µt ) ) with X := max{b T, K}. The fixed payment is determined by E[L 1 {τ>t } ] =LN d, B 0 e µ r)t ) ) L A0 B 0 ) 2µ r 1 2 σ2 ) σ 2 N d B 0, e µ r)t ) ). The down and out put option is then determined by E[L A T ] + 1 {τ>t } ] [ =L N d, Le µ T )) Nd + B 0, e µ r)t )) + B0 ) 2 σ 2 µ r 1 2 σ2 ) Nd B 2 0, Le µt )) ) 2µ r)+σ 2 B0 σ 2 Nd + B 0, e r µ)t )) + B0 B0 ) 2 σ 2 µ r 1 2 σ2 ) Nd B 0, e µ r)t ) ] [ e µt N d +, Le µt )) N d + e µ r)t, B 0 )) 16 ) 2µ r)+σ 2 σ 2 Nd + B 2 0, Le µt ))]
17 In order to compute the second moments of the terminal payments, we shall calculate several expectations which take the form of E [ [A T K] + 1 {τ>t } ) 2] =E[A 2 T 1 {AT >K,τ>T }] 2K E[A T 1 {AT >K,τ>T }] + K 2 E[1 {AT >K,τ>T }] = E[A 2 T 1 {AT >K}] E[A 2 T 1 {AT >K,τ T }] ) 2K E[A T 1 {AT >K}] 2K E[A T 1 {AT >K,τ T }] ) + K 2 E[1 {AT >K}] K 2 E[1 {AT >K,τ T }] ). 11) Using the following notations: b := 1 σ ln B 0 ; m = µ r 1/2σ2, k = 1 σ σ all the six terms in 11) are explicitly given by ln K ) rt, ln E[A 2 T 1 {AT >K}] =A 2 0e 2µ+σ2)T K N + µ + ) 3 2 σ2 )T σ T ) E[A 2 T 1 {AT >K,τ T }] =A 2 0e 2rT e 2bm+2σ) e 2mσT +2σ2 T 2b k + m + 2σ)T ) N T ln 2K E[A T 1 {AT >K}] =2K e µt K N + µ + ) 1 2 σ2 )T σ T 2K E[A T 1 {AT >K,τ T }] =2K e rt e 2bm+σ) e mσt e 1 2 σ2t N ln K 2 E[1 {AT >K}] =K 2 K N + µ ) 1 2 σ2 )T σ T ) 2b k + mt K 2 E[1 {AT >K,τ T }] =K 2 e 2bm N. T ) 2b k + m + σ)t T When coming to compute the second moment of the downandout put, we rely on the following relations: E[[L A T ] + ) 2 1 {τ>t } ] =E[L A T ) 2 1 {τ>t,at <L}] =E[L 2 1 {τ>t,at <L}] 2 LE[A T 1 {τ>t,at <L}] + E[A 2 T 1 {τ>t,at <L}] ) = E[L 2 1 {AT <L}] E[L 2 1 {τ T } ] + E[L 2 1 {τ T,AT >L}] ) 2L E[A T 1 {AT <L}] E[A T 1 {τ T,AT >B T }] + E[A T 1 {AT B T }]) + E[A T 1 {τ T,AT >L}] ) + E[A 2 T 1 {AT <L}] E[A 2 T 1 {τ T,AT >B T }] + E[A 2 T 1 {AT B T }])) + E[A 2 T 1 {τ T,AT >L}]. 17
18 Furthermore, we shall calculate some terms which have the following forms: E [ [A T K 1 ] + [A T K 2 ] + 1 {τ>t } ] =E[A T K 1 )A T K 2 )1 {AT >max{k 1,K 2 },τ>t }] =E[A 2 T K 1 + K 2 )A T + K 1 K 2 )1 {AT >max{k 1,K 2 },τ>t }] =E[A 2 T 1 {AT >max{k 1,K 2 }}] E[A 2 T 1 {AT >max{k 1,K 2 },τ T }]) K 1 + K 2 )E[A T 1 {AT >max{k 1,K 2 }}] E[A T 1 {AT >max{k 1,K 2 },τ T }]) + K 1 K 2 E[1 {AT >K}] E[1 {AT >K,τ T }]) Finally, it should be noted that [ [ E A T L ] + [L A T ] + 1 {τ>t }] = 0. α Table 7 illustrates some numerical results concerning the expected utility difference resulting from the two contracts. We take linear risk neutrality), quadratic, exponential, log and power utility as examples. For all the utilities, we first demonstrate the results given that the pension fund and life insurance should satisfy the same regulatory constraints ε, according to which the maximal allowed volatility level is determined. For the resulting volatility levels, the fair participation rates are computed. It is observed that a riskneutral annuity buyer prefers life insurance company to pension fund. In case of risk neutrality the expected utility difference is reduced to only one term because the second derivative of a linear function is equal to zero. In other words, only the expected value of the terminal wealth is crucial. In this case, the annuity buyer does not care about the guaranteed payment as much as risk averse potential buyers and goes for high risk and high return. Therefore, his choice is a life insurance company rather than a collective pension fund. An annuity buyer with power and quadratic utility will buy annuity at a life insurance company too. The other utility functions lead to a decision for pension fund. Second, we also take a look at the case when different default probability constraints are used ε = 5% for life insurer and ε = 9% for pension fund), which implies that the insurance company and pension fund can use risk management strategies with different volatilities. For the chosen parameters, different confidence levels for pension fund and life insurance do not change our result. Overall, the expected utility differences show extremely small values, it is due to the fact that the weights, i.e. the first and second derivatives of utility are rather small. This paper is still in working process. More numerical results and interpretations are to be added. 5. Conclusion The present paper analyzes the institutional differences between pension funds and insurance companies, which are both important annuity providers. In case of the pension 18
19 utility function institution ε σ crit δ, δ L EU decision difference linear ux) = x pension 5% 11.69% 73.06% insurance insurance 5% 11.69% 96.24% pension 9% 13.06% 73.59% insurance insurance 9% 13.06% 95.34% pension 9% 13.06% 73.59% insurance insurance 5% 11.69% 96.24% quadratic ux) = x 0.001x 2 pension 5% 11.69% 73.06% 18,20 insurance insurance 5% 11.69% 96.24% pension 9% 13.06% 73.59% insurance insurance 9% 13.06% 95.34% pension 9% 13.06% 73.59% insurance insurance 5% 11.69% 96.24% exponential ux) = 1 e 0.8x pension 5% 11.69% 73.06% pension insurance 5% 11.69% 96.24% pension 9% 13.06% 73.59% pension insurance 9% 13.06% 95.34% pension 9% 13.06% 73.59% pension insurance 5% 11.69% 96.24% log ux) = logx) pension 5% 11.69% 73.06% 0.34 pension insurance 5% 11.69% 96.24% pension 9% 13.06% 73.59% 0.32 pension insurance 9% 13.06% 95.34% pension 9% 13.06% 73.59% pension insurance 5% 11.69% 96.24% power ux) = x pension 5% 11.69% 73.06% insurance insurance 5% 11.69% 96.24% pension 9% 13.06% 73.59% insurance insurance 9% 13.06% 95.34% pension 9% 13.06% 73.59% insurance insurance 5% 11.69% 96.24% Table 7. Expected utility differences with parameters: = 100; L = 120; T = 15; r = 0.05; i = 0.01; α = 0.9; µ =
20 fund the annuity is provided by a collective defined pension scheme with no external shareholders while the life insurance company is with external shareholders. First, we determine the fair combinations of contract parameters by applying a contingent claim approach and compare the fair participation rates of both contracts. We find out that the fair participation rate in case of the pension fund is much lower than the participation rate in case of the life insurance. Second, by using meanvariance analysis we answer the question, what contract a annuity buyer would prefer if he chooses the contract which leads to a higher value of his expected utility of the terminal wealth. This part of the analysis is done by incorporating the regulatory requirements, which determine the maximal allowed volatility of the risk management strategies the life insurer or pension fund can trade in. We observe that the annuitybuying choice of a potential annuity purchaser depends on his preferences crucially. 20
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