Optimal life insurance with no-borrowing constraints: duality approach and example

We solve an optimal portfolio choice problem under a no-borrowing assumption. A duality approach is applied to study a family’s optimal consumption, optimal portfolio choice, and optimal life insurance purchase when the family receives labor income that may be terminated due to the wage earner’s premature death or retirement. We establish the existence of an optimal solution to the optimization problem theoretically by the duality approach and we provide an explicitly solved example with numerical illustration. Our results illustrate that the no-borrowing constraints do not always impact the family’s optimal decisions on consumption, portfolio choice, and life insurance. When the constraints are binding, there must exist a wealth depletion time (WDT) prior to the retirement date, and the constraints indeed reduce the optimal consumption and the life insurance purchase at the beginning of time. However, the optimal consumption under the constraints will become larger than that without the constraints at some time later than the WDT.     

Household Life Cycle Protection: Life Insurance Holdings, Financial Vulnerability, and Portfolio Implications

Using the Survey of Consumer Finances, we examine the life cycle demand for different types of life insurance. Specifically, we test for the consumer's aversion to income volatility resulting from the death of a household's wage‐earner through the purchase of life insurance. We first develop a financial vulnerability index to control for the risk to the household. We then examine the life cycle demand for life insurance using several definitions of life insurance. We find, in contrast to previous research, that there is a relationship between financial vulnerability and the amount of term life or total life insurance purchased. In addition, we find older consumers use less life insurance to protect a certain level of financial vulnerability than younger consumers. Secondly, our study provides evidence that life insurance demand is jointly determined as part of a household's portfolio. Finally, we consider the impact of family members' nonmonetary contribution on the household's life cycle protection decision. Our results provide some evidence that households take into account the value of nonmonetary contribution in their insurance purchase.     

Portfolio Choice and Life Insurance: The CRRA Case

We solve a portfolio choice problem that includes life insurance and labor income under constant relative risk aversion (CRRA) preferences. We focus on the correlation between the dynamics of human capital and financial capital and model the utility of the family as opposed to separating consumption and bequest. We simplify the underlying Hamilton–Jacobi–Bellman equation using a similarity reduction technique that leads to an efficient numerical solution. Households for whom shocks to human capital are negatively correlated with shocks to financial capital should own more life insurance with greater equity/stock exposure. Life insurance hedges human capital and is insensitive to the family's risk aversion, consistent with practitioner guidance.     

Optimal life insurance purchase and consumption/investment under uncertain lifetime

In this paper, we consider optimal insurance and consumption rules for a wage earner whose lifetime is random. The wage earner is endowed with an initial wealth, and he also receives an income continuously, but this may be terminated by the wage earner’s premature death. We use dynamic programming to analyze this problem and derive the optimal insurance and consumption rules. Explicit solutions are found for the family of CRRA utilities, and the demand for life insurance is studied by examining our solutions and doing numerical experiments.     

Reinsurance decisions in life insurance: An empirical test of the risk–return criterion

Recent studies have analyzed optimal reinsurance contracts within the framework of profit maximization and/or risk minimization. This type of framework, however, does not consider reinsurance as a tool for capital management and financing. In the present paper, we consider different proportional reinsurance contracts used in life insurance (viz., quota-share, surplus, and combinations of quota-share and surplus) while taking into account the insurer's capital constraints. The objective is to determine how different reinsurance transactions affect the risk/reward profile of the insurer and whether factors, such as claims severity, premiums, and insurer's risk appetite, influence the choice of a proportional reinsurance coverage. We compare each reinsurance structure based on actual insurance company data, using the risk–return criterion. This criterion determines the type of reinsurance that enables insurer to retain the largest underwriting profits and/or minimize the risk of the retained claims while keeping the insurer's risk appetite constant, assuming a given capital constraint. The results of this study confirm that the choice of reinsurance arrangement depends on many factors, including risk retention levels, premiums, and the variance of the sum insured values (and therefore claims). As such, under heterogeneous insurance portfolio single type of reinsurance arrangement cannot maximize insurer's returns and/or minimize the risk, therefore a combination of different reinsurance coverages should be employed. Hence, future research on optimal risk management choices should consider heterogeneous portfolios while determining the effects of different financial and risk management tools on companies' risk–return profiles.     

Health and Mortality Delta: Assessing the Welfare Cost of Household Insurance Choice

We develop a pair of risk measures, health and mortality delta, for the universe of life and health insurance products. A life‐cycle model of insurance choice simplifies to replicating the optimal health and mortality delta through a portfolio of insurance products. We estimate the model to explain the observed variation in health and mortality delta implied by the ownership of life insurance, annuities including private pensions, and long‐term care insurance in the Health and Retirement Study. For the median household aged 51 to 57, the lifetime welfare cost of market incompleteness and suboptimal choice is 3.2% of total wealth.     

The Impact of Disability Insurance on a Portfolio of Life Insurances

The aim of this article is to study the impact of disability insurance on an insurer's risk situation for a portfolio that also consists of annuity and term life contracts. We provide a model framework using discrete time nonhomogeneous bivariate Markov renewal processes and in a simulation study focus on diversification benefits as well as potential natural hedging effects (risk-minimizing or risk-immunizing portfolio compositions) that may arise within the portfolio because of the different types of biometric risks. Our analyses emphasize that disability insurances are a less efficient tool to hedge shocks to mortality and that their high sensitivity toward shocks to disability risks cannot be easily counterbalanced by other life insurance products. However, the addition of disability insurance can still considerably lower the overall company risk.     

Human capital investment and optimal portfolio choice

We analyze how an individual should optimally invest in human capital when he also has financial wealth. We treat the individual's possibilities to take more education as expansion options and apply real option analysis. In addition, we characterize the individual's optimal consumption strategy and portfolio weights. The individual has a demand for hedging financial risk, labor income risk, and also wage level risk.     

Pension Fund Design under Long-term Fairness Constraints

We consider optimal portfolio insurance for a mutually owned with-profits pension fund. First, intergenerational fairness is imposed by requiring that the pension fund is driven towards a steady state. Subject to this condition the optimal asset allocation is identified among the class of constant proportion portfolio insurance strategies by maximising expected power utility of the benefit. For a simple contract approximate analytical results are available and accurate, whereas for a more involved contract Monte Carlo methods must be applied to pick out the best design. The main insights are (i) aggressive investment strategies are disastrous for the clients; (ii) in most cases it is optimal to gear the bonus reserve; and (iii) the results are far less sensitive to the agent's risk aversion than in the classical case of Merton (1969), and as opposed to Merton horizon matters even with constant investment opportunities (because of the serial dependence between bonuses).     

On systematic mortality risk and risk-minimization with survivor swaps

A new market for so-called mortality derivatives is now appearing with survivor swaps (also called mortality swaps), longevity bonds and other specialized solutions. The development of these new financial instruments is triggered by the increased focus on the systematic mortality risk inherent in life insurance contracts, and their main focus is thus to allow the life insurance companies to hedge their systematic mortality risk. At the same time, this new class of financial contract is interesting from an investor's point of view, since it increases the possibility for an investor to diversify the investment portfolio. The systematic mortality risk stems from the uncertainty related to the future development of the mortality intensities. Mathematically, this uncertainty is described by modeling the underlying mortality intensities via stochastic processes. We consider two different portfolios of insured lives, where the underlying mortality intensities are correlated, and study the combined financial and mortality risk inherent in a portfolio of general life insurance contracts. In order to hedge this risk, we allow for investments in survivor swaps and derive risk-minimizing strategies in markets where such contracts are available. The strategies are evaluated numerically.     

Central bank haircut policy

We present a model of central bank collateralized lending to study the optimal choice of the haircut policy. We show that a lending facility provides a bundle of two types of insurance: insurance against liquidity risk as well as insurance against downside risk of the collateral. Setting a haircut therefore involves balancing the trade-off between relaxing the liquidity constraints of agents on one hand, and increasing potential inflation risk and distorting the portfolio choices of agents on the other. We argue that the optimal haircut is higher when the central bank is unable to lend exclusively to agents who actually need liquidity. Finally, for a temporary surprise drop in the haircut, the central bank can be more aggressive than when setting a permanent level of the haircut.     

Optimal consumption,portfolio and life insurance rules for an uncertain lived individual in a continuous time model

A continuous time model for optimal consumption, portfolio and life insurance rules, for an investor with an arbitrary but known distribution of lifetime, is derived as a generalization of the model by Merton (1971). The classic Tobin-Markowitz separation theorem obtains with the mutual funds being identical to those obtained under the assumption of certain lifetime. The investor is found to have a ‘human capital’ component of wealth, which is independent of his preferences and risky market opportunities and represents the certainty equivalent of his future net (wage) earnings. Explicit solutions, which are linear in wealth, are found for the investor with constant relative risk aversion.     

Does operating risk affect portfolio risk? Evidence from insurers' securities holding

This study empirically examines, in the setting of insurance companies, the hypothesis that investors facing more operating risk may behave as if they were more risk averse in investment decisions. Specifically, we study how operating risk from underwriting insurance policies affects insurers' risk taking behavior in their portfolio investments. We find that insurers with higher volatilities in underwriting incomes and cash flows are more conservative in their financial investment risk taking – they have lower credit risk exposure in their bond investments, as well as lower portfolio weights on risky bonds and equities. Further, insurers' portfolio risk exposure is sensitive to the risk of permanent underwriting income shocks but insensitive to the risk of transitory shocks. Transitory operating risk, however, is significantly related to portfolio risk when insurers face tight financing constraints. Our findings suggest a substitutive effect of operating risk on investment decisions by financial institutions.     

Saving and investing for early retirement: A theoretical analysis

We study optimal consumption and portfolio choice in a framework where investors adjust their labor supply through an irreversible choice of their retirement time. We show that investing for early retirement tends to increase savings and reduce an agent's effective relative risk aversion, thus increasing her stock market exposure. Contrary to common intuition, an investor might find it optimal to increase the proportion of financial wealth held in stocks as she ages and accumulates assets, even when her income and the investment opportunity set are constant. The model predicts a decrease in risk aversion following strong market gains like those observed in the nineties.     

The effect of limited information and estimation risk on optimal portfolio diversification

This paper analyzes the optimal portfolio choice problem when security returns have a joint multivariate normal distribution with unknown parameters. For the case of limited, but sufficient (sample plus prior) information, we show that for a general family of conjugate priors, the optimal portfolio choice is obtained by the use of a mean-variance analysis that differs from traditional mean-variance analysis due to estimation risk. We also consider two illustrative cases of insufficient sample information and minimal prior information and show that in these cases it is asymptotically optimal for an investor to limit diversification to a subset of the securities. These theoretical results corroborate observed investor behavior in capital markets.     

Strategic technology adoption and hedging under incomplete markets

We investigate the implications of technological innovation and non-diversifiable risk on entrepreneurial entry and optimal portfolio choice. In a real options model where two risk-averse individuals strategically decide on technology adoption, we show that the impact of non-diversifiable risk on the option timing decision is ambiguous and depends on the frequency of technological change. Compared to the complete market case, non-diversifiable risk may accelerate or delay the optimal investment decision. Moreover, strategic considerations regarding technology adoption play a central role for the entrepreneur’s optimal portfolio choice in the presence of non-diversifiable risk.     

Optimal hedge fund portfolios under liquidation risk

We use an expected utility framework to integrate the liquidation risk of hedge funds into portfolio allocation problems. The introduction of realistic investment constraints complicates the determination of the optimal solution, which is solved using a genetic algorithm that mimics the mechanism of natural evolution. We analyse the impact of the liquidation risk, of the investment constraints and of the agent's degree of risk aversion on the optimal allocation and on the optimal certainty equivalent of hedge fund portfolios. We observe, in particular, that the portfolio weights and their performance are significantly affected by liquidation risk. Finally, tight portfolio constraints can only provide limited protection against liquidation risk. This approach is of special interest to fund of hedge fund managers who wish to include the hedge fund liquidation risk in their portfolio optimization scheme.     

Asset Allocation and the Liquidity Premium for Illiquid Annuities

Academics and practitioners alike have developed numerous techniques for benchmarking investment returns to properly adjust seemingly high numbers for excessive levels of risk. The same, however, cannot be said for liquidity, or the lack thereof. This article develops a model for analyzing the ex ante liquidity premium demanded by the holder of an illiquid annuity. The annuity is an insurance product that is akin to a pension savings account with both an accumulation and decumulation phase. We compute the yield (spread) needed to compensate for the utility welfare loss, which is induced by the inability to rebalance and maintain an optimal portfolio when holding an annuity. Our analysis goes beyond the current literature, by focusing on the interaction between time horizon (both deterministic and stochastic), risk aversion, and preexisting portfolio holdings. More specifically, we derive a negative relationship between a greater level of individual risk aversion and the demanded liquidity premium. We also confirm that, ceteris paribus, the required liquidity premium is an increasing function of the holding period restriction, the subjective return from the market, and is quite sensitive to the individual's endowed (preexisting) portfolio.     

An analysis of VaR-based capital requirements

We study the behavior of a financial institution subject to capital requirements based on self-reported VaR measures, as in the Basel Committee's Internal Models Approach. We view these capital requirements and the associated backtesting procedure as a mechanism designed to induce financial institutions to reveal the risk of their investments and to support this risk with adequate levels of capital. Accordingly, we consider the simultaneous choice of an optimal dynamic reporting and investment strategy. Overall, we find that VaR-based capital requirements can be very effective not only in curbing portfolio risk but also in inducing revelation of this risk.     

Commitment and Lapse Behavior in Long‐Term Insurance: A Case Study

This article presents a case study of a portfolio of individual long‐term insurance contracts sold by a Spanish mutual company. We describe the risk levels, the rating structure, and the implied cross‐subsidies on a portfolio of policies providing health, life, and long‐term care insurance. We show evidence of reclassification risk through the history of disability spells. We also analyze the lapse behavior and seek to provide a rationale for the portfolio’s dynamics. We discuss the lack of commitment from the policyholders (lapses) and from the mutual company (which took a run‐off decision). Finally, we draw conclusions regarding the design of such contracts.