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Optimal asset allocation of pension funds in regret theoretic framework
The major contribution of this paper lies in the use of regret theory to analyse the optimal asset allocation of a pension fund that maximises the expected modified utility of its final wealth. Unlike the standard expected utility framework in which a pension fund manager independently considers only the investment choice that he makes and performs utility maximisation without any recourse to other investment choices that could have been made, regret theory gives room for the fund manager to account for his investment choice as well as other feasible investment choices that could have made. In essence, the fund manager experiences regret if the outcome of his investment choice is worse than the outcome of at least one of his forgone alternatives, and he rejoices if otherwise. Because of the anticipation of future regret, we set up the objective function—the expected modified utility of the fund’s final wealth— in such a way as to incorporate regret function. The presence of this regret function distinguishes regret theory from the traditional expected utility framework. In this light, therefore, we develop a set-up aimed at examining the extent to which the anticipation of future regret influences the choice and optimal asset allocation of a pension fund.
Accessible literature on asset allocation problems and optimal financial portfolios for pension funds almost completely neglects regret theory and widely favors expected utility maximisation. In addition to other limitations and violations of the traditional expected utility theory, so elegantly demonstrated and documented in the behavioral economics literature, our major discontent with the theory is that it assumes individuals consider each possible outcome independently of other outcomes. This can be interpreted to mean that fund investment managers care only about their investment choices. However, as has been demonstrated in the behavioral finance literature, fund managers do experience regret whenever alternative investment choices yield better returns ex-post. Even though they ex-ante felt convinced that their investment decisions were optimal, fund managers still harbor a feeling of regret for not having made the right investment decisions whenever their ex-post returns on investments turn out to be worse off. As a simple illustration, consider a fund manager who can receive a $5 return on investment for each dollar invested in the debt capital market and either a $7.5 or $3.5 return on investment for each dollar invested in the equity capital market. If he takes a huge position in the equity capital market and finally receives a $3.5 for each dollar invested in equity, he may almost surely experience a feeling of regret for getting less than he would have gotten if he had taken little or no position in the equity capital market. Next time, this experience will shape his investment decisions and therefore make him averse to regret and this aversion will in turn force him to incorporate regret into his decision making process. This idea is well-documented in the so called behavioral decision theory under uncertainty. Unfortunately, however, nowhere has it been used to analyse the optimal asset allocation of a pension fund.
The concept of regret theory is intuitively straightforward. Regret is a cognitively mediated emotion of pain and anger when people observe that they took a bad decision ex-ante and could have taken an alternative decision with better outcome. In capital markets, people experience regret when their investments give a worse performance than an alternative investment they could have easily chosen. This, for instance, is in contrast with disappointment, which is experienced when a negative outcome happens relative to prior expectation. Regret is strongly associated with a feeling of responsibility for the choice that has been made and is known to influence decision-making under uncertainty. Regret is a powerful negative emotion; the anticipation of future regret is so strong that it forces even Harry Markowitz to turn against his very own Nobel winning asset allocation theory when confronted with a financial decision on his pension plan. His quote: ‘I should have computed the historical covariance of the asset classes and drawn an efficient frontier. Instead I visualized my grief if the stock market went way up and I wasn’t in it—or if it went way down and I was completely in it. My intention was to minimize future regret, so I split my pension scheme contributions 50-50 between bonds and equities.’ ‘Harry Markowitz. As quoted in Zweig, 1998, ‘America’s top pension fund’, Money, 27, page 114’ . This gives support and adds credibility to the claim that regret does influence optimal investment decision of a pension fund. Anticipation of future experience of regret may lead individuals to make decisions that contrast the expected utility paradigm. This assertion will be investigated in the context of the optimal asset allocation of a pension fund in the course of this research.
Regret theory, due to Bell (1982) and Loomes and Sugden (1982), proposes a normative theory of choice under uncertainty that explains many observed violations of the axioms that the traditional expected utility theory is built upon. Regret theory involves the regret or rejoice that a person can feel when he gets outcome instead of outcome. The theory assumes that people are rational but base their decisions not only on expected payoffs or utility but also on expected regret, so that they try to anticipate future regret and consistently incorporate it into their investment decisions. The incorporation of regret yields a modified utility and people reach their investment decisions by maximizing the expected value of this modified utility. This makes the theory suitable for analysing the optimal asset allocation of a pension fund.
Unlike other institutional investors, the case of pension funds requires the introduction of two new characteristics: (i) the different behaviors of the fund wealth during the accumulation Ac and decumulation Dc phases, and (ii) the mortality risk. Also, regret risk is required because we are working in a regret theoretic framework. So, this thesis considers three dimensions of risk: traditional risk (volatility of final wealth), regret risk and mortality risk. To the best of our knowledge, no work on optimal asset allocation has ever simultaneously taken these risks into account to study the optimal asset allocation of a pension fund. The only work, at least to our knowledge, which considers these risks in pension fund research and asset allocation theory, does not consider them all at once. For instance, Bajeux-Besnainou and Jordon  consider only volatility risk, Michenaud and Solnik  consider volatility risk and regret risk and Battocchio, Menoncin and Scaillet  consider volatility risk and mortality risk.
Michenaud and Solnik  study the currency hedging techniques for foreign assets in a regret theoretic framework and derive some interesting implications for long and short hedging positions when a foreign currency appreciates or depreciates ex-post. In contrast, our methodology allows the derivation of approximate closed-form solutions for the optimal investment choices available to a pension fund. While the intuition of applying regret theory to asset allocation is not new, this is the first time/one of the few times that a formal regret theoretic approach is applied to a pension fund with mortality risk.
As we motivated above, regret is a major factor when making investment choices because institutional investors, more often than not, care about their choices relative to other strategies they could have employed. Although there has been observed evidence in favor of the influence of regret on decision-making under uncertainty as well as the axiomatic and normative appeal of regret theory for investment choices, it is surprising that the theory has caught only little attention in the field of finance, Michenaud and Solnik . For instance, Braun, Mitchell and Volkman  apply regret theory to asset allocation in defined contribution pension schemes. They find that an investor who takes regret into account will hold more risky assets (stocks) when the equity premium is low but less risky assets when the equity premium is high. Mitchell and Muermann  apply regret theory to demand for insurance. Dodonova and Khoroshilov  apply a pseudo regret theory to asset pricing. Michenaud and Solnik  apply regret theory to portfolio optimisation. All these models offer comparative statics or approximate explicit solutions for investment rules outside the case of a pension fund.
In this paper, instead, we provide approximate explicit optimal solutions for investment rules within the context of regret theory in the case of a pension fund which manages employees’ contributions towards retirement. In particular, during the active years of the employees, the fund wealth increases because of the contributions that the employees make towards retirement while, after retirement, the fund wealth decreases because of the pension payments that the pension fund makes to the retired employees. Following Battocchio, Menoncin and Scaillet , we suppose that a representative employee has no other choice at the retirement date than to receive a pension until the death time , which we assume to be stochastic. The pension fund then maximises the expected modified utility of its final wealth, in anticipation of future regret.
In our model the contribution and pension rates are constant and linked by a feasibility condition that guarantees the convenience of both the pension fund and the representative employee to amicably enter the pension contract. We argue why this feasibility condition must hold and derive its approximate closed-form expression under the assumption that the death time follows a log-logit distribution. We emphasise that our result is quite different from the closed form expression obtained under the assumption of a Gompertz-Makeham and Weibull distributed death time in Battocchio, Menoncin and Scaillet , and remark that our motivation for this choice of distribution for the death time stems from the fact that death-survival analyses for a random death time are best done under the assumption of a log-logit distribution .
To summarise, in addition to other important results, our major contribution in this paper is systematic. We integrate regret into a well-defined objective function and this allows us to derive optimal investment strategies that reflect the risk and regret aversion of a pension fund.
To this end, the paper flows as follows. Chapter 2 presents the financial model for the pension fund and explains some very important concepts that will aid the understanding of other ideas presented in subsequent chapters. Chapter 3 discusses the importance of regret theory for pension fund decisions and describes our modeling framework as well as the computation of the feasibility condition on the contribution and pension rates when the death time follows a Log-logit distribution. Chapter 4 presents the objective function for the pension fund and the computation of the optimal allocation rule. Chapter 5 discusses the main practical implications of our results for the effective management of a pension fund and concludes with direction to future research.
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