1 Introduction: choices in designs of pension schemes

We examine optimal investment performance sharing rules between future pensioners and sponsors/managersFootnote ¹ of private pension schemes. Ideally, a performance sharing rule should align the interests of the sponsors and those of the pensioners, so that a sponsor seeking an investment strategy that maximizes his own utility also maximizes the utility of the pensioners. In reality, there are a variety of performance sharing rules currently in use that leave some conflicts of interests between the two parties and result in equilibrium investment strategies that depend on information available to the two parties and their relative bargaining power.

In many European countries, the UK for instance, private pension schemes fall into two main categories, defined benefits (DB) and defined contributions (DC). A pensioner in a DC scheme bears the full performance risk of the fund into which he and his employer make regular contributions; the fund manager collects a fixed fee. By contrast, a pensioner in a DB company scheme bears no investment performance risk but takes a credit risk on the scheme's sponsoring firm that guarantees the defined benefits. Typically, the defined benefits are a fraction of final salary multiplied by the number of years in employment. Such DB schemes are referred to more specifically as ‘final salary schemes’. The benefits from other DB schemes may be based on average salary or other more stable references. The DB scheme sponsor must make up any investment performance deficit with additional contributions but retains any performance surplus (usually by reducing or deferring contributions).Footnote ² This performance risk is all the more visible in the UK and countries using IAS accounting standards now that new regulationsFootnote ³ aimed at giving credit risk protection to DB pensionersFootnote ⁴ require that a DB fund surplus or deficit be reported by the sponsoring firm as a special item on its balance sheet. These regulations also require that the surplus or deficit be evaluated on a fair value basis as the mark-to-market value of the fund's assets less the present value of the fund's liabilities discounted at ‘AA’ Libor rates for matching maturities. Thus, the sponsoring firm's financial standing is affected by the volatility of both assets and liabilities in its pension scheme.Footnote ⁵

An unintended consequence of these new regulations is that many firms found that their DB schemes were becoming too risky and therefore stopped offering them to new employees, closed existing schemes to new contributions, and induced existing members to transfer to alternative schemes.Footnote ⁶ The remaining DB schemes – and they will still represent the majority of company schemes in the UK and other countries for some years to come – are now managed more conservatively. Asset allocations have been tilted away from equities towards long-term bonds to match liabilities more closely and thus reduce surplus volatility. But long-term expected returns are thereby reduced. Thus, the new regulations designed to provide greater security (lesser credit risk) to DB pensioners resulted in fewer DB schemes being offered and an increase in the cost of the remaining schemes for the sponsoring firms. They may have also contributed to the decline of the equity markets in the few years leading to their implementation. So, pension regulations not only affect the balance of risks between pensioners and sponsors but also affect investment strategies and, ultimately, global market performance.

DB and DC schemes are but two instances in a wide spectrum of possible risk sharing rules, and neither may be ideal from the point of view of pensioners. Some intermediate performance sharing rules may be more attractive to pensioners because they would align better their interests with those of the sponsor. We analyze a particular class of performance sharing rules that guarantee a minimum return to the pensioners. Some governments allow this type of performance sharing rule and regulate them to ensure that the sponsors are able to meet their commitments. An illustration is provided by PensionskassenFootnote ⁷ in Germany, which typically are life insurance companies. Pensionskassen guarantee at retirement date the contributions plus interest compounded at a fixed rate, currently set by law to at least 2.25% per year. Every year pensioners accumulate either this guaranteed minimum return on previous contributions or 90% of the fund's annual return, if higher.Footnote ⁸ In future, the guaranteed minimum may be linked to inflation, but for the moment it is simply raised by a flat 1% to 3.25% to compensate for inflation.Footnote ⁹

The German Pensionskassen scheme is an instance of a generic type of an intermediate performance sharing rule between DB and DC rules, namely a DC scheme with a minimum guarantee for the pensioners and a share of the surplus for the sponsor. We shall not attempt to reflect the detailed workings of Pensionskassen, or limit ourselves to their current investment strategies, which currently use fixed income instruments almost exclusively. Rather, we shall examine how the choice of investment strategy by the fund manager depends on structure and choice of parameters of the performance sharing rule and, consequently, affects the benefits of pensioners. The onus is on the pension fund industry to propose, and on regulators to approve, performance sharing rules that benefit pensioners.

Performance sharing rules for pension schemes and related dynamic investment strategies have been studied in a variety of contexts. In one group of studies, the authors investigate regulatory schemes with defined minimum guarantees. They evaluate the advantages of introducing such guaranteed returns and examine the corresponding optimal investment strategies. Deelstra et al. (Reference Deelstra, Grasselli and Koehl2003) search for optimal investment strategies for a DC scheme with a fixed minimum guarantee in a continuous time framework. They develop an analytic approach in the context of a complete financial market and illustrate their results with a numerical analysis. Boulier et al. (Reference Boulier, Huang and Taillard2001) consider a DC scheme with a minimum guaranteed stochastic interest rate. They develop a quantitative method to determine an optimal dynamic allocation among three different asset classes: cash, long-term bonds, and stocks. Doskeland and Nordahl (Reference Doskeland and Nordahl2008) show that annual guarantees have a negative effect on the final wealth of the investor and that a lifetime guarantee is preferable, if a guarantee is required at all. We come to the same conclusion in this paper. Several other studies, such as Brennan (Reference Brennan1993), analyze the effect of a minimum guarantee on investor's welfare. He analyzes, inter alia, bonus policies with reversionary bonuses of life insurance companies and shows them to be inefficient. Jensen and Sorensen (Reference Jensen and Sorensen2001) demonstrate that a guaranteed minimum return may lead to a significant expected utility loss measured with a CRRA utility function. Consiglio et al. (Reference Consiglio, Saunders and Zenios2006) present a model to compare portfolio performance under different policy structures for with-profits funds. Hansen and Miltersen (Reference Hansen and Miltersen2002) examine funds offering a guaranteed minimum rate of return and introduce a complex smooth surplus-sharing rule between the investor and the fund that is fair to both.

A second group of studies concentrates on the optimal design of pension fund schemes leading to optimal return profiles for investors. Some studies investigate how a minimum guarantee should be designed optimally. Deelstra et al. (Reference Deelstra, Grasselli and Koehl2004) study the optimal design of a guarantee in a DC framework. By means of optimal control theory, they maximize the expected utility of the fund manager under the assumption of a power utility function but they do not provide numerical results. Doskeland and Nordahl (Reference Doskeland and Nordahl2008) also analyze how to optimally design traditional funds and pension funds with a guaranteed minimum rate of return. To find how to increase investors' welfare (measured with a certainty equivalent) they use numerical methods and find that, with a CRRA utility, they cannot justify the existence of a minimum guarantee. Pézier (Reference Pézier2008) also finds that, with exponential utility functions, optimal portfolio returns are linear in asset returns; however, he finds that with power utilities, optimal portfolio returns may be convex, linear, or concave in the risky asset return depending on the sensitivity of the local risk aversion coefficient to wealth and the risk/return characteristics of the risky asset.

A third group of studies focuses on the optimal management of pension funds with dynamic investment strategies. A stochastic pension fund model is used by Cairns (Reference Cairns1996) who considers a generalized constant proportion portfolio insurance to manage a pension fund in continuous time and gives numerical examples to show that a very good approximation to discrete time models is reached. For the funding level, he also derives the stationary distribution. Martinelli and Milhau (Reference Martellini and Milhau2009) examine dynamic allocation strategies for pension funds to find an integrated model for asset-liability management. They find that the cost of short-term funding constraints is unexpectedly low but the lack of dynamic risk management can be costly. Other studies compare the performance of static and dynamic investment strategies. Bertrand and Prigent (Reference Bertrand and Prigent2005) analyze and compare two very common portfolio insurance strategies: an option based portfolio insurance (OBPI) and a constant proportion portfolio insurance (CPPI). They consider various criteria for their comparison and conclude that there is no dominant strategy either state-wisely or stochastically to the first order. Zagst and Kraus (Reference Zagst and Kraus2008) compare the same two portfolio insurance methods but consider their stochastic dominance up to third order criteria. They derive parameter conditions that lead to a second or third order dominance of the CPPI strategy.

In a recent study, Amenc et al. (Reference Amenc, Martellini and Sender2009) examine the impact of regulatory and institutional frameworks on pension fund management and show the challenges European regulatory developments pose to pension funds. Their two main findings are that if regulators would tolerate short-term risk, the pension system would be more stable and pension funds should further develop internal models to analyze investment strategies. This study reveals growing concerns about the influence of regulations on the management of pension funds, but, as seen in our literature overview, there are still few studies on what could be considered as optimal design of pension plans. Our paper addresses this issue.

The next two sections describe our problem setting. In Section 2, we define four notional assets available for investment, denoted Cash, Bond, Equity, and Market (a constant mix of Bond and Equity) and specify their price dynamics. In Section 3, we describe the yearly and cumulative performance sharing rules we want to compare. We also describe two types of investment strategies: constant value mix (CM) investment strategies and constant proportionality portfolio insurance (CPPI) style strategies. To complete the setting, we specify the expected utility criterion used to assess investment performance from the points of view of pensioners and sponsors. Section 4 presents the results of CM strategies under a yearly performance sharing rule; they reveal a sharp conflict of interest between pensioners and sponsors and poor results for both. Section 5 shows that a cumulative performance sharing rule alleviates this conflict and improves the results for both parties. Sections 6 and 7 revisit the sharing rules in Sections 4 and 5 but with CPPI style investment strategies. The main advantage of CPPI style strategies is to reduce the downside risk for both pensioners and fund managers. Under the cumulative performance sharing rule, there is no conflict of interest between these two parties and CPPI style strategies do not improve on the performance of the optimal CM strategy. But by reducing the risk of a loss for the fund manager, they reduce the credit exposure of the pensioners. We conclude in Section 8 on the advantages of introducing new pension schemes with profit sharing rules intermediate between the traditional DB and DC schemes, provided they are suitably designed and supervised, and on the relative advantages of CPPI strategies over CM strategies in the presence of a liability constraint.

2 Assets and asset price dynamics

We define four notional assets for investing pension contributions. We call them Cash, Bond, Equity, and Market. In the following, we give brief introductions into the theory of the modeling of those assets and present our approaches to it. The overarching assumption in this section is that Brownian motions can be used to model the uncertainty in the dynamics of financial assets. We use Brownian motions in the subsequent subsections for modeling interest rates and equity prices.

Interest rate modeling

Vasicek (Reference Vasicek1977) proposes a time-homogeneous interest rate model. He assumes that a spot rate, r, follows an Ornstein–Uhlenbeck (O–U) process under a real-world probability measure

(1)

where r ₀, a, b, and σ_r are positive constants and W _r is a Brownian process.

We calibrate the Vasicek model on the yields of the 30-year German government Bund, the 5-year BoBL,Footnote ¹⁰ and the 6-month Treasury Bill with daily data from January 2000 until June 2008.Footnote ^11,Footnote ¹² The corresponding calibrated parameters a, b, and σ_r are associated with the modified durations (MD) of these instruments which are, approximately, 15 years, 4.5 years, and 6 months respectively.Footnote ¹³ Matching parameters for intermediate MDs are obtained by interpolation. We also find a correlation of 0.5 between the yields of the long-term bond and the 6-month Bill. These parameters are plotted in Figure 1. Although the O–U process allows negative yields, the probability of a negative yield is negligible with these parameters.Footnote ¹⁴

Figure 1. Interpolated Vasicek model parameters (Right scale for parameter a, left scale for parameters b and σ_r)

An O–U process for an instantaneous rate could be used as a single factor model for the entire yield curve. However, such modeling would lack realism and flexibility. In our implementation, it would grossly underestimate the volatility of long-term bonds and imply perfect correlation between long-term and short-term yields.

We prefer a more empirical approach to the modeling of returns on a portfolio of bonds. We assume that the fund manager always takes positions in notional par bonds with yield r and modified duration MD so that over a short period dt the return on a bond position to first order terms is

(2)

where P _t is the bond price at t.

We use this formula to define Cash returns with MD=1 and Bond returns with MD=15 when the fund's remaining life is greater or equal to 15 years and MD equal to the remaining fund's life during the last 15 years. In other words, the Cash and Bond assets are notional instruments with defined durations such that Cash corresponds to a one-year government Bill and Bond approximates first a 30-year par Bund and then, during the last 15 years of the fund, approximates shorter-term par Bunds or BoBLs. Adjusting the MD of the Bond asset with the time to maturity of the fund reduces the uncertainty in the cumulative performance of the Bond asset.

Equity market modeling

For the Equity asset, we assume that the price S _t follows a geometric Brownian motion

(3)

where S ₀, μ, and σ_s are positive constants and W _s is a Brownian process. Over a time interval dt, the Equity return is

(4)

We calibrate the Equity model parameters with daily data for the German DAX equity index from June 1980 until June 2008.Footnote ^15,Footnote ¹⁶ The estimated yearly parameters are μ=0.0904 and σ_s=0.2084.

Finally, we define a single risky asset as a constant value mix of 60% Bond and 40% Equity. We call it the ‘Market’ asset. Indeed, the mix so defined has about the highest Sharpe ratio achievable with the Bond and Equity assets.

Equity was not significantly correlated with Cash and Bond during the period January 2000 to June 2008; we therefore take the Brownians W _s and W _r to be independent. A correlation between Bond and Equity performance could easily be introduced to reflect different circumstances.

Although we use historical data to calibrate our chosen price processes for our chosen assets, it does not necessarily mean that our choice of representative assets is ideal and that the corresponding processes and parameters lead to good forecasts of future returns. However, we found that these assets and processes are sufficiently representative to validate our general conclusions. Sensitivity analyses to the choice of parameters show that realistic variations would not change our conclusions about optimal performance sharing rules and the relative merits of CM and CPPI strategies. We evaluate the performance of a fund with alternative performance sharing rules and CM and CPPI strategies by simulating 10,000 paths of monthly Cash, Bond, Equity, and Market returns. This number of simulations proved to be sufficient to yield stable results.

3 Performance sharing rules, investment strategies, and preference criteria

We assume that 40 equal yearly contributions are made into a Fund (the ‘Fund’).Footnote ¹⁷ It does not matter for this study whether the contributions are made by a cohort of members of the scheme (the ‘Pensioner’),Footnote ¹⁸ their employer, the state, or jointly by several contributors. These contributions are immediately and fully invested in the Fund. The Fund manager (the ‘Manager’) guarantees a minimum return on the contributions, or Floor, plus participation to the performance of the Fund if that participation exceeds the guaranteed minimum return. In reality, some management fees would probably be charged either directly on the contributions or on the current value of assets under management, or a combination of both.Footnote ¹⁹ For the sake of simplicity, we do not model any specific fee arrangement but recognize that the Manager will require a satisfactory return for sponsoring a scheme.

The performance sharing rule is characterized by two parameters: α, the guaranteed minimum return on contributions, and β, the percentage participation in the performance of the Fund. We understand that German Pensionskassen apply the sharing rule to yearly returns in the following way. Denote by V _t the value of the Fund at the beginning of year t before the yearly contribution c _t is received, F _t the guaranteed minimum value, or Floor, for the Pensioner, and R _t the share of the Fund value already attributed to the Pensioner. Then, at the beginning of year t+1, before a new contribution is made, the value of the Fund attributed to the Pensioner is

(5)

the value attributed to the Manager, M _t, is

(6)

and the new value of the Floor is

(7)

with, at inception, R ₁=V ₁=F ₁=0.

The value of the Floor at any time can be calculated from the contributions up to that point. For example, with uniform contributions c per year from t=1 to t=T, the Floor at T+1 is

(8)

On the other hand, the value of the Fund and the shares attributed to the Pensioner and the Manager for t>1 are stochastic variables depending on investment performance.

According to this yearly performance sharing rule (YPSR), the Manager provides the Pensioner with a yearly in-the-money put option and a fraction β of a yearly call of same strike as the put on the performance of the Fund in return for a fraction (1−β) of that call. This amounts to a series of cliquet optionsFootnote ²⁰ with variable known strikes on the value of the Fund itself influenced by the choice of investment strategy. It could be unprofitable for the Manager unless he finds a suitably low risk and high yielding asset to invest in.

Alternatively, the same Floor could be guaranteed to the Pensioner with the excess cumulative (rather than yearly) performance of the Fund above the Floor being shared between the Pensioner and the Manager in the proportions β: (1−β). That is, one could define the value of the Fund attributed to the Pensioner at the beginning of year t+1, before a new contribution is made as

(9)

with F _t+1 defined as before.

Thus, the Manager would provide a single 40-year American style put option with the monotonically increasing Floor defined by (7) rather than a more costly series of yearly put options.Footnote ²¹ This might induce the Manager to seek riskier investments with correspondingly higher expected returns and the pension scheme could, on balance, become more attractive to the Pensioner. We call this sharing rule the cumulative performance sharing rule (CPSR) as opposed to the previous yearly performance sharing rule (YPSR).

Among all imaginable investment strategies using Cash, Bond, Equity, and Market assets, we focus on CM and CPPI style strategies.Footnote ²². Each strategy of either type is characterized by the parameter ω defining the allocation to the risky asset, the rest being allocated to Cash. With CM strategies, ω is the proportion of the total Fund value allocated to the risky asset. The Fund is rebalanced every month to maintain ω constant. Many traditional pension Fund management strategies are approximately CM strategies because pension Fund trustees often stipulate narrow ranges for allocations to major asset classes. CM strategies are contrarian strategies. Every month some of the risky asset is bought (sold) if it has underperformed (outperformed) the return on Cash. We use alternatively Bond, Equity, and Market for the risky asset.

But for the Manager the risk of providing a minimum guaranteed return to the pensioners could be excessive if he were to invest a fixed fraction of the Fund in a risky asset according to a CM strategy. If the risky asset performs badly, the Fund value could approach zero and the Manager would have to pay any minimum guaranteed return out of his own resources. To reduce this risk, the Manager could find it preferable to manage dynamically his allocation to the risky asset so as to reduce it to zero if the Fund value falls to the minimum guaranteed return. In general, the performance of a dynamic strategy depends on the ability of the Manager to forecast the volatility and other dynamics of the risky asset he chooses. Modeling such capabilities to decide on the best choice of dynamic strategy would be complex and would reflect the timing and asset selection skills of the Manager. Instead, we consider a systematic CPPI style strategy, independent of the forecasting skills of the Manager. A CPPI strategy ensures that a minimum performance, or Floor – typically the minimum guaranteed return – is achieved by allocating to a risky asset a constant proportion (or multiplier) of the excess of the Fund value above the Floor, the excess is called Buffer. By analogy with the CM strategy, we denote the CPPI multiplier ω. We use exclusively Market for the risky asset and rebalance the Fund at monthly intervals. Should the Buffer become nil or negative, the Fund would be entirely invested in the relevant risk-free asset.Footnote ²³ We implement the CM and the CPPI strategies both for a yearly and a cumulative performance sharing rule.

On a yearly basis (YPSR), the Buffer B_s for the CPPI strategy at time s, t⩽s<t+1, is

(10)

where, using previous notations, V _s is the value of the Fund at time s and D _s is the discounted value of the minimum due to the Pensioner at year t+1 before a new contribution is made, that is

(11)

where r _t is the Cash rate at time s. The term in square brackets on the right-hand side of (11) is the new Floor value at year t+1 (before a new contribution c _t+1 is made) plus the excess above the Floor already attributed to the Pensioner at year t.Footnote ²⁴ At the end of the year, a new Buffer is calculated and the CPPI process is repeated. The Manager must make up any negative Buffer value to deliver the minimum benefits guaranteed to the Pensioner. If the leverage ω is not large, it is improbable that the Buffer would ever become negative.

On a cumulative basis (CPSR), the Buffer is defined as in (26) but the discount value D _s at time s, t⩽s<t+1, is defined as

(12)

which is the discounted value of the minimum guaranteed to investors at maturity T from contributions already made at time t. The risk-free rate r is chosen as the relevant Bond yield (as defined before). Should the Buffer become nil or negative, the Fund would be fully invested at that yield until maturity.Footnote ²⁵

For the evaluation of the investment strategies, we rely on utility analysis and calculate the certainty equivalents (CE) of the CM and CPPI strategies for both the Manager and the Pensioner. Each CE should be interpreted as the minimum amount for sure that the beneficiary would be willing to receive at maturity instead of facing an uncertain terminal value. A CE is a primitive concept that encapsulates the risk attitude of the beneficiary. To evaluate CEs systematically we assume the risk attitudes of the Pensioner and of the Manager can be characterized by exponential utility functionsFootnote ²⁶

(13)

The single parameter λ, a local coefficient of risk tolerance, is usually in a range from 10% to 25% of the net worth of the beneficiary. For illustration, we assume λ_P=40 for the Pensioner and λ_M=15 for the Manager.Footnote ²⁷

Thus, the CEs are functions of the sharing rule type, its parameters α and β, the coefficients of risk tolerance of the beneficiaries, and the relevant investment strategy parameter ω. We assume that the Manager always adopts the investment strategy, ω*(α, β) that maximizes his CE. It should be of interest to the pension industry and to regulators to find out which choice of sharing rule parameters α and β maximizes the CE of the Pensioner in these circumstances. We shall say that a triplet (α, β, ω) defines a plan, a triplet (α, β, ω*) a Manager's optimal plan, and a triplet (α*, β*, ω*) a Pensioner's optimal plan.

4 Performance with constant mix strategies and yearly performance sharing rule

Figure 2 shows probability densities of the Fund value a year after the last of 40 units of yearly contributions is made and the Manager has implemented a CM strategy either all in Cash, all in Bond, or with ω=30% in Market (and 70% in Cash). With the chosen asset price dynamics, these densities do not have simple analytical forms; they are obtained by simulating 10,000 asset return scenarios. Their main statistics and those of an All Equity investment are reported in the second column of Table 1 under the heading ‘Fund’.

Figure 2. Final fund value distributions under various CM strategies and YPSR

Table 1. Final performance with alternative CM strategies under YPSR

(The final value of the Fund is shared between Pensioner and Manager according to a yearly performance sharing rule with guaranteed return α as shown and β=90%)

The All Cash investment leads to an approximately normal distribution with low expected return (86.18) and low standard deviation (6.38). Both All Bond and 30%-Market investment strategies yield also approximately normal distributions but with higher expected values (108.22 and 107.38, respectively); the All Bond investment is also less risky than the All Cash investment, whereas the 30%-Market investment is more risky (standard deviation of 4.72, 6.38, and 13.44, respectively). The All Equity investment, on the other hand, yields a highly positively skewed final value distribution with much higher expected value (406.88) and standard deviation (622.37).

How these final Fund values are shared between the Manager and the Pensioner under the YPSR is critical. The key observation is that all of these investment strategies are unattractive to the Manager. There are only three combinations with positive yet insufficient expected returns:Footnote ²⁸ the 30%-Market investment strategy with α=2.25%, the All Cash investment strategy with α=2.25%, and, marginally positive, the All Cash investment strategy with α=3.00%. Moreover, these combinations are the least attractive to the Pensioner. It may surprise at first that the All Bond investment strategy, which leads to the least risky final Fund value distribution, is less attractive to Managers than the All Cash and the 30%-Market strategies. The reason is that the yearly returns of the Bond strategy are highly volatile and therefore less attractive for the Manager. Clearly, the interests of the Pensioner and the Manager are in conflict and Managers cannot extract a sufficient return to sponsor these schemes.

For a more comprehensive comparison of the attractiveness of alternative sharing rules and investment strategies, we calculate their respective CEs for the Pensioner and the Manager. The CEs of the Pensioner and the Manager with the All Cash, All Bond, and 30%Market CM strategies are shown in Table 2. For each pension plan we calculate the average utility over n=10,000 scenarios and calculate the corresponding CEs using the exponential utility function

(14)

As expected, all CEs are lower than the means reported in Table 1 for the corresponding plans because of uncertainties in terminal values. The CEs of the Manager are negative except for the All Cash strategy and the 30%-Market strategy when α=2.25%. They are most negative with the All Bond strategy because of the relatively high volatility of Bond on a yearly basis. The CEs of the Pensioner are always above the Floor but unattractive with the All Cash strategy compared to the 30%-Market strategy. They are highest for the All Bond strategy that is the least attractive from the Manager's perspective.

Table 2. Certainty equivalents of CM strategies All Cash, All Bond, and 30%-Market under YPSR

(Yearly performance sharing rule with Pensioner participation β=90% in all cases)

We therefore focus our attention on CM strategies with Cash and Market assets and explore the continuum of CE outcomes with Market allocations from 0% to 40% and guaranteed minimum rates of return from 2.25% to 3.75%. In all cases, the Pensioner participation is 90%. The results are plotted in Figure 3 for the Pensioner and Figure 4 for the Manager.

Figure 3. Certainty equivalent of the Pensioner with Cash/Market CM strategies under YPSR

Figure 4. Certainty equivalent of the Manager with Cash/Market CM strategies under YPSR

The CE of the Pensioner is always well above the Floor. It increases moderately with the increase in the guaranteed minimum and it increases markedly with increasing Market allocation, especially if the guaranteed minimum is high. On the other hand, the CE of the Manager is positive only for very low Market allocations. It decreases moderately with an increasing guaranteed minimum. It increases slightly with small market allocations, reaches a maximum with about 5–10% allocated to Market, and then decreases more and more rapidly with higher Market allocations. This confirms the obvious conflict of interest between the investment strategy preferences of the Pensioner and the Manager. The former, being protected by the Floor, prefers the higher expected return brought by higher Market allocations, whereas the latter, providing the downside protection, prefers only a small Market allocation.

This conflict of interest could be reduced by modifying the sharing rule in favor of the Manager. This would induce him to adopt an investment strategy with a larger allocation to Market, thus benefiting the Pensioner as well. Figures 5 and 6 show the CEs of the Pensioner and the Manager as a function of the Pensioner participation, β, when the guaranteed minimum rate of return is α=2.25%. The Pensioner always prefers both a greater participation, and a greater Market allocation as shown by the constant Market allocation curves in Figure 5. The Manager, on the other hand, prefers a lower participation going to the Pensioner, but his optimal level of Market allocation varies as a function of Pensioner participation. We see in Figure 6 that the Manager's optimal Market allocation is near zero when Pensioner participation is near 100%, but increases to near 40% when Pensioner participation decreases from 100% to 55%. In an oligopoly where the Manager would be able to optimize his choice of investment strategy for his own benefit and with little regards for the Pensioner, the Pensioner's CE would evolve as shown by the Manager's optimal allocation curve in Figure 5. It remains very flat with highs around 88 for β=100% and β=55% and a low around 85 for β=80%. An adequate performance for the Manager would be 10 or above and would be attainable only if Pensioner participation were no more than 75%. On the other hand, if the Manager had to compete with others for the custom of the Pensioner, he would be able to reduce their upside participation down to 75%, necessary for a viable business, but Pensioners could force the Manager to allocate 30% to the Market asset because it would cost Manager little to do so. Thus, the CEs would be at least 10 for the Manager and close to 95 for the Pensioner. That would be markedly better than under the current rule for Pensionskassen with a Pensioner participation of 90%, which makes the business unviable for the Manager and still poor value (CE around 87) for the Pensioner. It remains that a Pensioner's CE of about 95 under the YPSR (α=2.25%, β=75%, ω=30%) is still poor compared to what can be achieved with a CPSR rule, as we shall see next.

Figure 5. Certainty equivalent of the Pensioner as a function of the Pensioner's participation and the Market asset allocation under YPSR

Figure 6. Certainty equivalent of the Manager as a function of the Pensioner's participation and the market asset participation under YPSR

5 Performance with constant mix strategies and cumulative performance sharing rule

We found in Section 4 that the YPSR produces final value distributions for the Pensioner that are well above the guaranteed minimum value, or Floor. In fact, due to frequent participations in the yearly Fund performance, these distributions for All Cash, All Bond, and 30%-Market CM strategies are close to normality. If instead, one applies a CPSR with same Floor and same Pensioner participation, the corresponding final value distributions for the Pensioner are truncated distributions easily obtained from the final Fund value distributions in Figure 2. The Pensioner obtains 90% (or generally the participation β) of the final value above the Floor, if any, or the Floor when the final value of the Fund does not reach the Floor. The Manager obtains 10% (or generally (1−β)) of the excess of the final value of the Fund above the Floor, if any, or pays out to the Pensioner the shortfall when the final value of the Fund does not reach the Floor. These distributions are generally more attractive to the Manager and less so to the Pensioner than the distributions with matching investment strategies under the YPSR as we can see from Table 3, which should be compared with Table 2 for YPSR. We also add in Table 3 an All Market investment strategy, which is now attractive for the Manager.

Table 3. Certainty equivalents of CM strategies All Cash, All Bond, and 30%-Market under CPSR

(Cumulative performance sharing rule with Pensioner participation β=90% in all cases)

We observe only small differences between YPSR and CPSR for an All Cash investment strategy, especially with low levels of α, because it is almost guaranteed to produce low volatility results greater than the guaranteed minimum return. On the other hand, the All Bond and the 30%-Market strategies become more attractive to the Manager and less so to the Pensioner. That is because the Bond asset has low final uncertainty (although it has relatively high yearly volatility). But the best CM strategy among the four displayed in Table 4 is clearly the All Market strategy. It is best both for the Pensioner and the Manager. It is also the only strategy that may appear sufficiently attractive for a Manager to sponsor such a scheme. The final Fund value distribution is plotted in Figure 7.

Figure 7. Final Fund value distribution with all market strategy under CPSR

Table 4. Final performance with CPPI strategy (ω=2) under YPSR

(The final value of the Fund is shared between Pensioners and Manager according to a yearly performance sharing rule with guaranteed return α as shown and β=90%)

The only drawback from the Pensioner's perspective is that under the All Market strategy there is a risk that the final Fund value does not exceed the Floor and therefore the Pensioner takes a credit risk on the Manager.

Figures 8 and 9 show the CEs of the Pensioner and the Manager for CM strategies with Cash and Market assets with Market allocations from 0% to 100% and guaranteed minimum rate of return on contributions from 2.25% to 3.75%. The Pensioner participation is still 90%. These figures with the CPSR correspond to Figures 3 and 4 with the YPSR.

Figure 8. Certainty equivalent of the Pensioner with Cash/Market CM strategies under CPSR

Figure 9. Certainty equivalent of the Manager with Cash/Market CM strategies under CPSR

As before, the CE of the Pensioner increases moderately with the increase in the guaranteed minimum and it increases markedly with increasing Market allocation. But now the CE of the Manager also increases with increasing market allocations in the range ω=0% to 100%. It decreases moderately with an increasing guaranteed minimum, but remains mostly positive. Therefore, there is still some conflict of interest between the Pensioner and the Manager on the guaranteed minimum return, but both prefer higher Market allocations. Maximum CEs would be reached for allocations to Market beyond 100%, if the Manager were allowed to cash-leverage his investments. We conclude that under CPSR the All Market investment strategy, if cash-leverage is not allowed, offers the best achievable CEs for both the Pensioner and the Manager. The CEs of the Pensioner (above 136) are greatly superior to those achievable with a YPSR. The CE of the Manager is above 10 when α=2.25% and could be kept above 10 for larger minimum guarantees if the upside participation were reduced slightly below 90%.

6 Performance with CPPI strategies and yearly performance sharing rule

We now consider CPPI style investment strategies using the Cash and Market assets only. Specifically, the amount invested in the Market asset every month is a constant multiplier, ω, of the Buffer, the difference between the value of the Fund and the discounted minimum value of the benefits already promised to the Pensioner. The rest of the Fund is invested in Cash. This CPPI strategy is first used with the YPSR. The distributions of final Fund values are plotted in Figure 10 for a leverage ω=2. There is one distribution for each of the three guaranteed minimum returns on contribution, α=2.25%, 3.00%, and 3.75%, because the investment strategy now depends on the level of the Floor. The expected final values are seen to decrease slightly with increasing Floors and so do the standard deviations. This is confirmed by the statistics in Table 4 in the columns headed ‘Fund’. Clearly, the higher the Floor, the smaller the Buffer, and the lower the average investment in the Market asset. But the Buffer remains always very small so that all three distributions have approximately the same shape and statistics as the All Cash distribution examined in Section 4 (Figure 2 and Table 1); the expected values and standard deviations are only marginally greater than with the All Cash strategy.

Figure 10. Final fund value distributions with CPPI strategy (ω=2) under YPSR

The same conclusions can be drawn for the shares of the final Fund value attributed to the Pensioner and the Manager when the Manager's participation to the upside is β=90%. The corresponding statistics are also reported in Table 4 (columns headed ‘Pens.’ and ‘Man.’, respectively). Naturally, with increasing Floor levels the expected values for the Pensioner increase and those for the Manager decrease. The standard deviations of these distributions are small so that the corresponding CEs, shown in Table 5, are barely below the relevant expected values. For ease of comparison, Table 5 repeats the results of the All Cash investment strategy of Section 4, which corresponds to a leverage ω=0, and adds the cases ω=5 and ω=10. As ω increases, both Pensioner's and Manager's CEs increase marginally; there is no conflict of interest. However, there is conflict of interest on the choice of minimum guarantee and, in all cases, the CEs of the Manager are too low to make the business viable under the YPSR.

Table 5. Certainty equivalents of CPPI strategies under YPSR

(Pensioners yearly surplus participation β=90% in all cases)

A more complete picture of CEs for both Pensioner and Manager as a function of the minimum guaranteed rate α (from 2.25% to 3.75%) and the leverage ω (from 0 to 10) is given in Figures 11 and 12. The Pensioners surplus participation is 90% in all cases. These figures should be compared to Figures 3 and 4 with CM strategies.

Figure 11. Certainty equivalent of the Pensioner with CPPI strategies under YPSR

Figure 12. Certainty equivalent of the Manager with CPPI strategies under YPSR

Because there is little if any conflict of interest between Manager and Pensioner about seeking a high investment leverage, there is little room for improving the Pensioner's CE by modifying the performance sharing rule in favor of the Manager so that the Manager would adopt a more profitable investment strategy. Figures 13 and 14 show the CEs of the Pensioner and the Manager under a CPPI strategy as a function of the Pensioner's participation, β, when the guaranteed minimum rate of return is α=2.25%. In an oligopoly, the Manager would maximize his own CE by choosing the optimal ω. That optimal leverage is in the region of ω=10 when β is less than 85% but then decreases to reach 5 when β reaches 97% and drops below 5 when β reaches 100%. Correspondingly, the Pensioner's CE reaches a maximum of about 89 when β=96%. But again, to make the plan viable for the Manager, say with a CE greater than 10, pension authorities should reduce the Pensioner's upside participation in the Fund performance to β=75% or less, in which case the CE of the Pensioner under a CPPI strategy becomes inferior to the CE under a CM strategy with the Market asset. In a competitive market, the Pensioner might push the Manager to increase the multiplier ω to increase his CE. We conclude that if pension authorities want to design viable pension schemes with guaranteed minimum return on contributions and a YPSR, they should keep the guaranteed minimum return low (well below risk-free rates), and, assuming our parameters, offer an upside performance participation to the Pensioner of no more than 75%, then the Manager is likely to adopt a CM investment strategy with around 30% invested in the Market asset (the risky asset mix with maximum Sharpe ratio) and 70% in Cash as we found in Section 4. CPPI strategies are not preferable under the YPSR.

Figure 13. Certainty equivalent of the Pensioner with CPPI strategies as a function of the Pensioner's participation and the leverage under YPSR

Figure 14. Certainty equivalent of the Manager with CPPI strategies as a function of the Pensioner's participation and the leverage under YPSR

7 Performance with CPPI strategies and cumulative performance sharing rule

We now investigate the use of CPPI style investment strategies with the CPSR using exclusively the Cash and Market assets as before. The CEs of the Pensioner and the Manager are shown in Table 6 for a selection of leverage factors from ω=0 to ω=10. In all cases, the Pensioner participation is β=90%. Table 6 should be compared to Table 3 for CM strategies with CPSR. When ω=0, we have an All Cash strategy showing the same results as in Table 3 for the corresponding All Cash investment. As we observed before, this plan is unattractive for the Manager, especially when the guaranteed minimum return is high.

Table 6. Certainty equivalents of CPPI strategies under CPSR

(CPSR with Pensioner participation β=90% in all cases)

As ω increases, the average investment in the Market asset increases towards a 100% self-imposed no-cash-leverage ceiling; both the Manager's and the Pensioner's CEs increase rapidly at first, especially when the guaranteed minimum return is low, and then reach a plateau. The reason is simple. At first, the size of the Buffer is limited by the value of the Fund; then it is limited by the no-cash leveraging constraint. When ω increases, the average amount invested in the Market asset increases. When ω=1, the average investment in the Market asset is about 43% of the Fund value, whereas when ω=2, it reaches 87%. Since we found in Section 5 that the All Market investment strategy is superior to all other CM strategies with lesser allocations to the Market asset, we can therefore expect the CPPI strategies with high ω to approach but never exceed the performance of the All Market strategy. With α=2.25% and ω=10, the CEs of the Pensioner and the Manager are 135.48 and 10.17 respectively, which comes close to the corresponding CEs of 136.64 and 10.23 with the All Market investment strategy (see bottom row of Table 3). With larger guaranteed minimum returns on contribution, the Buffers are smaller and the CPPI strategies run more distant seconds to the All Market strategy. The CEs of the Manager drop below 10 but could be restored at that level by reducing the Pensioner's upside participation below 90%.

A fuller picture of CE variations as a function of the leverage ω from 0 to 20 and α from 2.25% to 3.75% is given in Figure 15 for the Pensioner and Figure 16 for the Manager. There is little gain to be found by increasing ω beyond 5. At that leverage level, both the Pensioner and the Manager lose if the guaranteed minimum return on contributions is increased because it reduces the Buffer. Only at values of ω well below 1 does the raising of the guaranteed minimum return increase slightly the Pensioner's CE.

Figure 15. Certainty equivalent of the Pensioner with CPPI strategies under CPSR

Figure 16. Certainty equivalent of the Manager with CPPI strategies under CPSR

So, under CPSR, a CPPI strategy with a leverage ω=5 performs almost as well as the All Market strategy. The main advantage of the CPPI strategy is that the probability of the Buffer becoming negative is negligible; therefore, the Pensioner takes no material credit risk on the Manager compared to the small credit risk he takes under the All Market investment strategy.