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QUANTIFYING THE TRADE-OFF BETWEEN INCOME STABILITY AND THE NUMBER OF MEMBERS IN A POOLED ANNUITY FUND
Abstract
The number of people who receive a stable income for life from a closed pooled annuity fund is studied. Income stability is defined as keeping the income within a specified tolerance of the initial income in a fixed proportion of future scenarios. The focus is on quantifying the effect of the number of members, which drives the level of idiosyncratic longevity risk in the fund, on the income stability. To do this, investment returns are held constant, and systematic longevity risk is omitted. An analytical expression that closely approximates the number of fund members who receive a stable income is derived and is seen to be independent of the mortality model. An application of the result is to calculate the length of time for which the pooled annuity fund can provide the desired level of income stability.
... It is nevertheless possible to generate lifelong incomes by investing in survivor funds over successive time intervals. This offers an alternative to group self-annuitization schemes of Piggott et al. (2005) as well as to modern tontines or pooled annuity funds developed for example by Bernhardt and Donnelly (2021), Chen et al. (2021), and Hieber and Lucas (2022). ...
... Survivor funds offer a single terminal payout to participants. This is in contrast to modern tontines or pooled annuity funds considered for instance by Bernhardt and Donnelly (2021) in the homogeneous case, by Bernhardt and Qu (2021) with heterogeneous contributions and by Chen et al. (2021) with multiple cohorts. Since survivor funds correspond to pure endowment insurance contracts without the life table guarantee and since life annuities can be obtained as a sequence of pure endowment contracts with increasing maturities, lifelong incomes can be generated by investing in survivor funds with different time horizons. ...
Survivor funds are financial arrangements where participants agree to share the proceeds of a collective investment pool in a predescribed way depending on their survival. This offers investors a way to benefit from mortality credits, boosting financial returns. Following Denuit (2019, ASTIN Bulletin , 49 , 591–617), participants are assumed to adopt the conditional mean risk sharing rule introduced in Denuit and Dhaene (2012, Insurance: Mathematics and Economics , 51 , 265–270) to assess their respective shares in mortality credits. This paper looks at pools of individuals that are heterogeneous in terms of their survival probability and their contributions. Imposing mild conditions, we show that individual risk can be fully diversified if the size of the group tends to infinity. For large groups, we derive simple, hierarchical approximations of the conditional mean risk sharing rule.
The question how individuals should (optimally) annuitize their wealth remains of high relevance in light of longevity risk and volatile capital markets. In this article, we first present traditional and innovative products and strategies for the decumulation of wealth during retirement, based on a review of 72 selected academic articles in peer-reviewed journals. We further identify relevant factors that generally influence the conception of these products from the retirees’ perspectives, and derive implications for product developers, before concluding with avenues of future research. Our results indicate that innovative suggestions often comprise tontine-like structures, exploit actuarial and accounting smoothing in various ways, defer annuitization to higher ages, or combine it with long-term care options, for instance. Key areas of future research in this field include the consideration of both insurer and retiree perspectives in the analysis of products, using behavioral considerations when evaluating the retirees’ perspective, and taking into account the impact of costs or expenses. While recent articles increasingly consider these aspects, manifold opportunities for future research remain.
In this chapter I provide some background on the reasons (I think) a traditional fund company might want to introduce a modern tontine as it relates to the unique challenges that people face managing their financial affairs towards the end of the human lifecycle.
For centuries, mathematicians and, later, statisticians, have found natural research and employment opportunities in the realm of insurance. By definition, insurance offers financial cover against unforeseen events that involve an important component of randomness, and consequently, probability theory and mathematical statistics enter insurance modeling in a fundamental way. In recent years, a data deluge, coupled with ever-advancing information technology and the birth of data science, has revolutionized or is about to revolutionize most areas of actuarial science as well as insurance practice. We discuss parts of this evolution and, in the case of non-life insurance, show how a combination of classical tools from statistics, such as generalized linear models and, e.g., neural networks contribute to better understanding and analysis of actuarial data. We further review areas of actuarial science where the cross fertilization between stochastics and insurance holds promise for both sides. Of course, the vastness of the field of insurance limits our choice of topics; we mainly focus on topics closer to our main areas of research.
Expected final online publication date for the Annual Review of Statistics, Volume 9 is March 2022. Please see http://www.annualreviews.org/page/journal/pubdates for revised estimates.
For insurance companies in Europe, the introduction of Solvency II leads to a tightening of rules for solvency capital provision. In life insurance, this especially affects retirement products that contain a significant portion of longevity risk (e.g., conventional annuities). Insurance companies might react by price increases for those products, and, at the same time, might think of alternatives that shift longevity risk (at least partially) to policyholders. In the extreme case, this leads to so-called tontine products where the insurance company’s role is merely administrative and longevity risk is shared within a pool of policyholders. From the policyholder’s viewpoint, such products are, however, not desirable as they lead to a high uncertainty of retirement income at old ages. In this article, we alternatively suggest a so-called tonuity that combines the appealing features of tontine and conventional annuity. Until some fixed age (the switching time), a tonuity’s payoff is tontine-like, afterwards the policyholder receives a secure payment of a (deferred) annuity. A tonuity is attractive for both the retiree (who benefits from a secure income at old ages) and the insurance company (whose capital requirements are reduced compared to conventional annuities). The tonuity is a possibility to offer tailor-made retirement products: using risk capital charges linked to Solvency II, we show that retirees with very low or very high risk aversion prefer a tontine or conventional annuity, respectively. Retirees with medium risk aversion, however, prefer a tonuity. In a utility-based framework, we therefore determine the optimal tonuity characterized by the critical switching time that maximizes the policyholder’s lifetime utility.
A new stream of research proposes how people can increase their income in retirement by pooling their mortality risk. How one of these mortality risk-sharing rules could be implemented in practice, as part of a retirement income scheme, is considered. A potential advantage of the scheme is that a retiree’s housing wealth can be monetised to provide an income stream. This would mean that retirees can continue living in their home, without needing to downsize. It may be most attractive to the millions of single pensioners, particularly those who are “asset-rich and cash-poor”. Other types of assets that could be included and how to mitigate selection risks are assessed. A way of smoothing the raw mortality credits in order to make the scheme more appealing to potential members is proposed. An illustrative premium calculation suggests that the cost of the smoothing is very small compared to the potential attractiveness of an enhanced, smoothed income.
There is growing interest in the design of pension annuities that insure against
idiosyncratic
longevity risk while pooling and sharing
systematic
risk. This is partially motivated by the desire to reduce capital and reserve requirements while retaining the value of mortality credits; see for example, Piggott
et al.
(2005) or Donnelly
et al.
(2014). In this paper, we generalize the
natural retirement income tontine
introduced by Milevsky and Salisbury (2015) by combining heterogeneous cohorts into one pool. We engineer this scheme by allocating tontine shares at either a premium or a discount to par based on both the age of the investor and the amount they invest. For example, a 55-year old allocating $10,000 to the tontine might be told to pay $200 per share and receive 50 shares, while a 75-year old allocating $8,000 might pay $40 per share and receive 200 shares. They would all be mixed together into the same tontine pool and each tontine share would have equal income rights. The current paper addresses existence and uniqueness issues and discusses the conditions under which this scheme can be constructed
equitably
— which is distinct from
fairly
— even though it isn't
optimal
for any cohort. As such, this also gives us the opportunity to compare and contrast various pooling schemes that have been proposed in the literature and to differentiate between arrangements that are socially equitable, vs. actuarially fair vs. economically optimal.
Tontines were once a popular type of mortality-linked investment pool. They promised enormous rewards to the last survivors at the expense of those died early. And, while this design appealed to the gambling instinct, it is a suboptimal way to generate retirement income. Indeed, actuarially-fair life annuities making constant payments–where the insurance company is exposed to longevity risk–induce greater lifetime utility. However, tontines do not have to be structured the historical way, i.e. with a constant cash flow shared amongst a shrinking group of survivors. Moreover, insurance companies do not sell actuarially-fair life annuities, in part due to aggregate longevity risk.
A simple closed-form approximation for the median of the beta distribution
Beta(a, b) is introduced: (a-1/3)/(a+b-2/3) for (a,b) both larger than 1 has a
relative error of less than 4%, rapidly decreasing to zero as both shape
parameters increase.
We explicitly derive and explore the optimal consumption and portfolio policies of a loss-averse individual who endogenously updates his or her reference level over time. We find that the individual protects current consumption by delaying painful reductions in consumption after a drop in wealth, and increasingly so with higher degrees of endogeneity. The incentive to protect current consumption is stronger with a medium wealth level than with a high or low wealth level. Furthermore, this individual adopts a conservative investment strategy in normal states and typically a more aggressive strategy in good and bad states. Endogeneity of the reference level increases overall risk-taking and generates an incentive to reduce risk exposure with age even without human capital. The welfare loss that this individual would suffer under the conventional constant relative risk aversion (CRRA) consumption and portfolio policies easily exceeds 10%.
This paper was accepted by Tyler Shumway, finance.
Increases in the life expectancy, the low interest rate environment and the tightening solvency regulation have led to the rebirth of tontines. Compared to annuities, where insurers bear all the longevity risk, policyholders bear most of the longevity risk in a tontine. Following Donnelly and Young (2017), we come up with an innovative retirement product which contains the annuity and the tontine as special cases: a tontine with a minimum guaranteed payment. The payoff of this product consists of a guaranteed payoff and a call option written on a tontine. Extending Donnelly and Young (2017), we consider the tontine design described in Milevsky and Salisbury (2015) for designing the new product and find that it is able to achieve a better risk sharing between policyholders and insurers than annuities and tontines. For the majority of risk-averse policyholders, the new product can generate a higher expected lifetime utility than annuities and tontines. For the insurer, the new product is able to reduce the (conditional) expected loss drastically compared to an annuity, while the loss probability remains fairly the same. In addition, by varying the guaranteed payments, the insurer is able to provide a variety of products to policyholders with different degrees of risk aversion and liquidity needs.
Testing whether a random variable has a specific distribution is often done by
Kolmogorov-Smirnov test; the CDF of the corresponding test statistic approaches,
asymptotically (with increasing sample size) a Jacobi theta function. Unfortunately,
the convergence is rather slow (for a sample of size n = 100, the maximum error is
still about 2.6%); in this article I derive corrections to this approximation to make it substantially more accurate, even when used with a sample size as small as 10.
This paper analyses the consumption-investment problem of a loss averse investor with an s-shaped utility over consumption relative to a time-varying reference level. Optimal consumption exceeds the reference level in good times and descends to the subsistence level in bad times. Accordingly, the optimal portfolio is dominated by a mean-variance component in good times and rebalanced more aggressively toward stocks in bad times. This consumption-investment strategy contrasts with customary portfolio theory and is consistent with several recent stylized facts about investor’ behavior. I also analyze the joint effect of loss aversion and persistence of the reference level on optimal choices. Finally, the strategy of the loss-averse investor outperforms the conventional Merton-style strategies in bad times, but tends to be dominated by the conventional strategies in good times.
This paper explicitly derives the optimal dynamic consumption and portfolio choice of a loss averse agent who endogenously updates his reference level. His optimal choice seeks protection against consumption losses due to downside financial shocks. This induces a (soft) guarantee on consumption and is due to loss aversion. Furthermore, his optimal consumption choice gradually adjusts to financial shocks. This resembles the payout streams of financial plans that respond sluggishly, smoothing investment returns to reduce payout volatility, and is due to endogenous updating. The welfare losses associated with various suboptimal consumption and portfolio strategies are also evaluated. They can be substantial.
This study explores the issue of what is an appropriate default equity glide-path for client portfolios during the retirement phase of the life cycle. We find, surprisingly, that rising equity glide-paths in retirement – where the portfolio starts out conservative and becomes more aggressive through the retirement time horizon – have the potential to actually reduce both the probability of failure and the magnitude of failure for client portfolios. This result may appear counter-intuitive from the traditional perspective, which is that equity exposure should decrease throughout retirement as the retiree’s time horizon (and life expectancy) shrinks and mortality looms. Yet the conclusion is actually entirely logical when viewed from the perspective of what scenarios cause a client’s retirement to “fail” in the first place. In scenarios that threaten retirement sustainability – e.g., an extended period of poor returns in the first half of retirement – a declining equity exposure over time will lead the retiree to have the least in stocks if/when the good returns finally show up in the second half of retirement (assuming the entire retirement period does not experience continuing poor returns). With a rising equity glide-path, the retiree is less exposed to losses when most vulnerable in early retirement and the equity exposure is greater by the time subsequent good returns finally show up. In turn, this helps to sustain greater retirement income over the entire time period. Conversely, using a rising equity glide-path in scenarios where equity returns are good early on, the retiree is so far ahead that their subsequent asset allocation choices do not impact the chances to achieve the original retirement goal.
The study of the empirical process and the empirical distribution function is one of the major continuing themes in the historical development of mathematical statistics. The applications are manifold, especially since many statistical procedures can be viewed as functional on the empirical process and the behavior of such procedures can be inferred from that of the empirical process itself. We consider the empirical process per se, as well as applications to tests of fit, bootstrapping, linear combinations of order statistics, rank tests, spacings, censored data, and so on.
Many of the classical results for sums of iid rv's have analogs for empirical processes, and many of these analogs are now available in best possible form. Thus we concern ourselves with empirical process versions of laws of large numbers (LLN), central limit theorems (CLT), laws of the iterated logarithm (LIL), upper-class characterizations, large deviations, exponential bounds, rates of convergence and orthogonal decompositions with techniques based on martingales, special constructions of random processes, conditional Poisson processes, and combinatorial methods.
Good inequalities are a key to strong theorems. In Appendix A we review many of the classic inequalities of probability theory. Great care has been taken in the development of inequalities for the empirical process throughout the text; these are regarded as highly interesting in their own right. Exponential bounds and maximal inequalities appear at several points.
Because of strong parallels between the empirical process and the partial sum process, many results for partial sums are also included. Chapter 2 contains most of these.
Our main concern is with the empirical process for iid rv's, though we also consider the weighted empirical process of independent rv's in some detail. We ignore the large literature on mixing rv's, and confine our presentation for k-dimensions and general spaces to an introduction in the final chapter.
We emphasize the special Skorokhod construction of various processes, as opposed to classic weak convergence, wherever possible. We feel this makes for simpler and more intuitive proofs. The Hungarian construction is also considered. It is usually more cumbersome for weak convergence results, since there is no single limiting Brownian bridge. However, it can be used to provide strong limit theorems even though the Skorokhod construction cannot.
The financial industry has recently seen a push away from structured products and towards transparency. The trend is to decompose products, such that customers understand each component as well as its price. Yet the enormous annuity market combining investment and longevity has been almost untouched by this development.
We suggest a simple decomposed annuity structure that enables cost transparency and could be linked to any investment fund. It has several attractive features: (i) it works for any heterogeneous group; (ii) participants can leave before death without financial penalty; and (iii) participants have complete freedom over their own investment strategy.
Group self-annuitization (GSA) schemes are designed to share uncertain future mortality experience including systematic improvements. Challenges for designing group pooled schemes include decreasing average payments when mortality improves significantly, decreasing numbers in the pool at older ages, and the impact of dependence from systematic mortality improvements across different ages of members in the pool. This article uses a multiple-factor stochastic mortality model in a simulation study to show how pooling can be made more effective and to quantify the limitations of these pooling schemes arising from the impact of systematic longevity risk.
Various types of structures that enable a group of individuals to pool their mortality risk have been proposed in the literature. Collectively, the structures are called pooled annuity funds. Since the pooled annuity funds propose different methods of poolingmortality risk, we investigate the connections between them and find that they are genuinely different for a finite heterogeneous membership profile. We discuss the importance of actuarial fairness, defined as the expected benefits equalling the contributions for each member, in the context of pooling mortality risk and comment on whether actuarial unfairness can be seen as solidarity between members. We show that, with a finite number of members in the fund, the group self-annuitization scheme is not actuarially fair: some members subsidize the other members. The implication is that the members who are subsidizing the others may obtain a higher expected benefit by joining a fund with a more favorable membership profile. However, we find that the subsidies are financially significant only for very small or highly heterogeneous membership profiles.
We propose an optimization criterion that yields extraordinary consumption smoothing compared to the well known results of the life-cycle model. Under this criterion we solve the related consumption and investment optimization problem faced by individuals with preferences for intertemporal stability in consumption. We find that the consumption and investment patterns demanded under the optimization criterion is in general offered as annuity benefits from products in the class of ‘Formula Based Smoothed Investment-Linked Annuities’.
In this paper, we study the optimal dynamic asset allocation strategy for the ELA scheme of DC pension plan during the distribution phase. In an ELA scheme of DC pension plan, the assets are invested in equities and bonds, and are distributed to the plan participants by an actuarial method. The survived participant can also obtain a survival credit from the mortality risk-sharing implicit in the pension plan. The goal of the scheme is to maintain the stable purchasing power of the plan participants, i.e., to minimize the square deviations of the distribution and a predetermined level by choosing the optimal dynamic asset allocation proportions. We formalize the problem into a continuous-time stochastic optimal control problem and establish the optimal dynamic asset allocation strategy by stochastic dynamic programming method. We obtain the optimal dynamic asset allocation proportions invested in the equities and bonds, and give an economical explanation of the key factors influencing the strategy.
Executive Summary This paper establishes new guidelines for determining the maximum "safe" initial withdrawal rate, defined as (1) never requiring a reduction in withdrawals from any previous year, (2) allowing for systematic increases to offset inflation, and (3) maintaining the portfolio for at least 40 years. It evaluates the maximum safe initial withdrawal rate during the extreme period from 1973 to 2003 that included two severe bear markets and a prolonged early period of abnormally high inflation. It tests the performance of balanced multi-asset class portfolios that utilize six distinct equity categories: U.S. Large Value, U.S. Large Growth, U.S. Small Value, U.S. Small Growth, International Stocks, and Real Estate. Two portfolios (65 percent equity and 80 percent equity) are evaluated in conjunction with systematic Decision Rules that govern portfolio management, sources of annual income withdrawals, impact of years with investment losses and withdrawal increases to offset ongoing inflation. This paper finds that applying these Decision Rules produces a maximum "safe" initial withdrawal rate as high as 5.8 percent to 6.2 percent depending on the percentage of the portfolio that is allocated to equities.. "How much income can I safely take from my investment portfolio?" is one of the most critical and complex questions on which a financial planner must advise a client.
An annuity is an arrangement in which an individual makes a one-time payment (the premium) to an insurer in exchange for a lifetime payment stream. Ideally, the expected present value of the payment stream matches the premium, making it a fair annuity. In practice, the payment stream is smaller than that, providing the insurer with a profit margin. This paper proposes a fair tontine annuity (FTA), an arrangement that provides a lifetime payment stream whose expected present value matches that of a fair annuity. The FTA is based on a fair tontine. In a fair tontine, a group of members contribute to a pool, and each time a member dies, her contribution is divided among surviving members. The distribution to surviving members is made in unequal portions, according to a plan that provides each member with a fair bet, meaning a bet whose expected gain is zero. We show that, under broad conditions, such a plan exists and is readily constructed. Members may be of any age and gender and may contribute any desired amount. New members may join at any time, allowing the fair tontine to operate in perpetuity.The FTA is formed by adding a few enhancements to the fair tontine. The result is something that closely resembles an annuity. The member makes a one-time contribution and receives payments on a fixed schedule (e.g., monthly) that last for his lifetime. The value of each payment is a random amount, with an expected value that is identical to the payment that would be made by a fair annuity. Thus, the FTA offers a higher expected payout than an insurer-provided annuity, since no profit is being extracted. Simulations show that the FTA outperforms a typical insurer-provided FTA not just on the average, but for virtually every member who lives more than just a few years, even with a pool as small as a few hundred members. Since the FTA imposes no risk on the provider, it can be offered by vendors other than insurers, such as mutual fund houses, retail brokers, etc.
In this paper, we construct a model for examining the demand for annuities together with the possible implications of adverse selection when an individual consumer has access to both a private annuity market and a market with a pooled annuity fund. An earlier paper by Piggott et al. [Piggott, J., Valdez, E.A., Detzel, B., 2005. The simple analytics of a pooled annuity fund. J. Risk Insur. 72, 497–520] provides a formal analysis of the payout adjustments from a longevity risk-pooling fund, an arrangement referred to in the paper as Group Self Annuitization (GSA). In such a pooled arrangement, the annuitants will bear their own cohorts’ systematic risk, but the cohort will share the idiosyncratic risk. The resulting return on the pooled annuity fund can be expressed as the product of a return from an ordinary annuity multiplied by a random variable that accounts for the adjustment that is due to deviations from expectation of mortality and investments. As demonstrated in this paper, a simple analysis of economic choice provides that it is possible to reduce the implications of adverse selection in a pooled annuity fund. It is well-documented that empirically, individuals do not find private annuity funds an attractive form of investment despite the potential welfare benefits that can be drawn from annuitization. A pooled annuity fund is an alternative to the conventional private annuity fund that may be considered more cost-effective.
This paper investigates the dynamic consumption and portfolio choice of an investor with habit formation in preferences and access to a complete financial market with time-varying investment opportunities. An exact and simple characterization of the optimal behavior under general, possibly non-Markov, dynamics of market prices is derived. Relative to the benchmark case of time-additive power utility, habit formation affects the hedge component of the optimal portfolio differently than the speculative component. The quantitative effects of habit formation are studied in three concrete settings. Firstly, a closed-form solution of the optimal consumption and portfolio choice with mean-reverting stock returns is derived. Secondly, with Cox–Ingersoll–Ross interest rate dynamics the optimal strategies are expressed in terms of the solution to a partial differential equation, which has an explicit solution for time-additive utility, but not with habit formation. Thirdly, a new model with both mean-reverting stock returns and stochastic interest rates is studied. Overall, the numerical examples show that, while hedging demands for various assets are affected differently by habit persistence, the main effect on relative asset allocations stems from the fact that some assets (bonds and cash) are better investment objects than others (stocks) when it comes to ensuring that future consumption will not fall below the habit level.
This paper presents the optimal continuous time dynamic consumption and portfolio choice for pooled annuity funds. A pooled annuity fund constitutes an alternative way to protect against mortality risk compared to purchasing a life annuity. The crucial difference between the pooled annuity fund and purchase of a life annuity offered by an insurance company is that participants of a pooled annuity fund still have to bear some mortality risk while insured annuitants bear no mortality risk at all. The population of the pool is modelled by employing a Poisson process with time-dependent hazard-rate. It follows that the pool member’s optimization problem has to account for the stochastic investment horizon and for jumps in wealth which occur if another pool member dies. In case the number of pool members goes to infinity analytical solutions are provided. For finite pool sizes the solution of the optimization problem is reduced to the numerical solution of a set of ODEs. A simulation and welfare analysis show that pooled annuity funds insure very effectively against longevity risk even if their pool size is rather small. Only very risk averse investors or those without access to small pools are more inclined to pay a risk premium to access private life annuity markets in order to lay off mortality risk completely. As even families constitute such small pools the model provides theoretical justification for the low empirical annuity demand.
Doob [1] has given heuristically an appealing methodology for deriving asymptotic theorems on the difference between the empirical distribution function calculated from a sample and the actual distribution function of the population being sampled. In particular he has applied these methods to deriving the well known theorems of Kolmogorov [2] and Smirnov [3]. In this paper we give a justification of Doob's approach to these theorems and show that the method can be extended to a wide class of such asymptotic theorems.
The equity premium puzzle, identified by Rajnish Mehra and Edward C. Prescott, states that, for plausible values of the risk aversion coefficient, the difference of the expected rate of return on the stock market and the riskless rate of interest is too large, given the observed small variance of the growth rate in per capita consumption. The puzzle is resolved in the context of an economy with rational expectations once the time separability of von Neumann-Morgenstern preferences is relaxed to allow for adjacent complementarity in consumption, a property known as habit persistence. Essentially, habit persistence drives a wedge between the relative risk aversion of the representative agent and the intertemporal elasticity of substitution in consumption. Copyright 1990 by University of Chicago Press.
This article provides a formal analysis of payout adjustments from a longevity risk-pooling fund, an arrangement we refer to as group self-annuitization (GSA). The distinguishing risk diffusion characteristic of GSAs in the family of longevity insurance instruments is that the annuitants bear their systematic risk, but the pool shares idiosyncratic risk. This obviates the need for an insurance company, although such instruments could be sold through a corporate insurer. We begin by deriving the payout adjustment for a single entry group with a single annuity factor and constant expectations. We then show that under weak requirements a unique solution to payout paths exists when multiple cohorts combine into a single pool. This relies on the harmonic mean of the ratio of realized to expected survivorship rates across cohorts. The case of evolving expectations is also analyzed. In all cases, we demonstrate that the periodic-benefit payment in a pooled annuity fund is determined based on the previous payment adjusted for any deviations in mortality and interest from expectations. GSA may have considerable appeal in countries which have adopted national defined contribution schemes and/or in which the life insurance industry is noncompetitive or poorly developed. Copyright The Journal of Risk and Insurance.
Optimal portfolio choice with loss aversion over consumption. The Quarterly Review of Economics and Finance
- G Curatola
Conserving client portfolios during retirement, Part IV
- Bengen