Mean-variance inefficiency of CRRA and CARA utility functions for portfolio selection in defined contribution pension schemes

We consider the portfolio selection problem in the accumulation phase of a defined contribution pension scheme in continuous time, and compare the mean-variance and the expected utility maximization approaches. Using the embedding technique pioneered by Zhou and Li (2000) we first find the efficient frontier of portfolios in the Black-Scholes financial market. Then, using standard stochastic optimal control we find the optimal portfolios derived via expected utility for popular utility functions. As a main result, we prove that the optimal portfolios derived with the CARA and CRRA utility functions are not mean-variance efficient. As a corollary, we prove that this holds also in the standard portfolio selection problem. We provide a natural measure of inefficiency based on the difference between optimal portfolio variance and minimal variance, and we show its dependence on risk aversion, Sharpe ratio of the risky asset, time horizon, initial wealth and contribution rate. Numerical examples illustrate the extent of inefficiency of CARA and CRRA utility functions in defined contribution pension schemes.Mean-variance approach, efficient frontier, expected utility maximization, defined contribution pension scheme, portfolio selection, risk aversion, Sharpe ratio

On efficiency of mean-variance based portfolio selection in DC pension schemes

We consider the portfolio selection problem in the accumulation phase of a defined contribution (DC) pension scheme. We solve the mean-variance portfolio selection problem using the embedding technique pioneered by Zhou and Li (2000) and show that it is equivalent to a target-based optimization problem, consisting in the minimization of a quadratic loss function. We support the use of the target-based approach in DC pension funds for three reasons. Firstly, it transforms the difficult problem of selecting the individual's risk aversion coefficient into the easiest task of choosing an appropriate target. Secondly, it is intuitive, flexible and adaptable to the member's needs and preferences. Thirdly, it produces final portfolios that are efficient in the mean-variance setting. We address the issue of comparison between an efficient portfolio and a portfolio that is optimal according to the more general criterion of maximization of expected utility (EU). The two natural notions of Variance Inefficiency and Mean Inefficiency are introduced, which measure the distance of an optimal inefficient portfolio from an efficient one, focusing on their variance and on their expected value, respectively. As a particular case, we investigate the quite popular classes of CARA and CRRA utility functions. In these cases, we prove the intuitive but not trivial results that the mean-variance inefficiency decreases with the risk aversion of the individual and increases with the time horizon and the Sharpe ratio of the risky asset. Numerical investigations stress the impact of the time horizon on the extent of mean-variance inefficiency of CARA and CRRA utility functions. While at instantaneous level EU-optimality and efficiency coincide (see Merton (1971)), we find that for short durations they do not differ significantly. However, for longer durations - that are typical in pension funds - the extent of inefficiency turns out to be remarkable and should be taken into account by pension fund investment managers seeking appropriate rules for portfolio selection. Indeed, this result is a further element that supports the use of the target-based approach in DC pension schemes.Mean-variance approach; efficient frontier; expected utility maximization; defined contribution pension scheme; portfolio selection; risk aversion; Sharpe ratio

Optimal composition of hybrid/blended real estate portfolios

Purpose: The purpose of this paper is to establish an optimum mix of liquid, publicly traded assets that may be added to a real estate portfolio, such as those held by open-ended funds, to provide the liquidity required by institutional investors such as UK defined contribution pension funds. This is with the objective of securing liquidity while not unduly compromising the risk-return characteristics of the underlying asset class. This paper considers the best mix of liquid assets at different thresholds for a liquid asset allocation, with the performance then evaluated against that of a direct real estate benchmark index. Design/Methodology/Approach: The authors employ a mean-tracking error optimisation approach in determining the optimal combination of liquid assets that can be added to a real estate fund portfolio. The returns of the optimised portfolios are compared to the returns for portfolios that employ the use of either cash or listed real estate alone as a liquidity buffer. Multivariate Generalised Autoregressive models are used along with rolling correlations and tracking errors to gauge the effectiveness of the various portfolios in tracking the performance of the benchmark index. Findings: The results indicate that applying formal optimisation techniques leads to a considerable improvement in the ability of the returns from blended real estate portfolios to track the underlying real estate market. This is the case at a number of different thresholds for the liquid asset allocation and in cases where a minimum return requirement is imposed. Practical Implications: The results suggest that real estate fund managers can realise the liquidity benefits of incorporating publicly traded assets into their portfolios without sacrificing the ability to deliver real estate-like returns. However, in order to do so, a wider range of liquid assets must be considered, not just cash. Originality/value: Despite their importance in the real estate investment industry, comparatively few studies have examined the structure and operation of open-ended real estate funds. To the authors’ knowledge, this is the first study to analyse the optimal composition of liquid assets within blended or hybrid real estate portfolios

Joined-Up Pensions Policy in the UK: An Asset-Libility Model for Simultaneously Determining the Asset Allocation and Contribution Rate

The trustees of funded defined benefit pension schemes must make two vital and inter-related decisions - setting the asset allocation and the contribution rate. While these decisions are usually taken separately, it is argued that they are intimately related and should be taken jointly. The objective of funded pension schemes is taken to be the minimization of both the mean and the variance of the contribution rate, where the asset allocation decision is designed to achieve this objective. This is done by splitting the problem into two main steps. First, the Markowitz mean-variance model is generalised to include three types of pension scheme liabilities (actives, deferreds and pensioners), and this model is used to generate the efficient set of asset allocations. Second, for each point on the risk-return efficient set of the asset-liability portfolio model, the mathematical model of Haberman (1992) is used to compute the corresponding mean and variance of the contribution rate and funding ratio. Since the Haberman model assumes that the discount rate for computing the present value of liabilities equals the investment return, it is generalised to avoid this restriction. This generalisation removes the trade-off between contribution rate risk and funding ratio risk for a fixed spread period. Pension schemes need to choose a spread period, and it is shown how this can be set to minimise the variance of the contribution rate. Finally, using the result that the funding ratio follows an inverted gamma distribution, shortfall risk and expected tail loss are computed for funding below the minimum funding requirement, and funding above the taxation limit. This model is then applied to one of the largest UK pension schemes - the Universities Superannuation Scheme

Optimal funding and investment strategies in defined contribution pension plans under Epstein-Zin utility

A defined contribution pension plan allows consumption to be redistributed from the plan member’s working life to retirement in a manner that is consistent with the member’s personal preferences. The plan’s optimal funding and investment strategies therefore depend on the desired pattern of consumption over the lifetime of the member. We investigate these strategies under the assumption that the member has an Epstein-Zin utility function, which allows a separation between risk aversion and the elasticity of intertemporal substitution, and we also take into account the member’s human capital. We show that a stochastic lifestyling approach, with an initial high weight in equity-type investments and a gradual switch into bond-type investments as the retirement date approaches is an optimal investment strategy. In addition, the optimal contribution rate each year is not constant over the life of the plan but reflects trade-offs between the desire for current consumption, bequest and retirement savings motives at different stages in the life cycle, changes in human capital over the life cycle, and attitude to risk

Longevity risk and capital markets: The 2009-2010 update

This Special Issue of the North American Actuarial Journal contains ten contributions to the academic literature all dealing with longevity risk and capital markets. Draft versions of the papers were presented at Longevity Five: the Fifth International Longevity Risk and Capital Markets Solutions Conference that was held in New York on 25-26 September 2009. It was hosted by J. P. Morgan and St John’s University and organized by the Pensions Institute at Cass Business School, London, and the Edmondson-Miller Chair at Illinois State University.Longevity Risk; Capital Market