Constant Relative Risk Aversion Preferences Research Articles

ALM for insurers with multiple underwriting lines and portfolio constraints: a Lagrangian duality approach

We investigate how an insurance firm can optimally allocate its assets to back up liabilities from multiple non-life business lines. The insurance risks are modeled by a multidimensional jump-diffusion process that accounts for simultaneous claims in different insurance lines with policy limits. We use Lagrangian convex duality techniques to derive optimal investment-underwriting strategies that maximize the expected utility from dividends and final wealth over a finite horizon. We examine how risk aversion, prudence, portfolio constraints, and multivariate insurance risk affect the firm’s earnings retention. We obtain explicit solutions for optimal strategies under constant relative risk aversion preferences. Finally, we illustrate our results with numerical examples and show the impact of co-integration for asset-liability management with multiple sources of insurance risk.

Optimal Asset Allocation for CRRA and CARA Insurers under the Vasicek Interest Rate Model

This paper considers the reinsurance-investment problem with interest rate risks under constant relative risk aversion and constant absolute risk aversion preferences, respectively. Stochastic control theory and dynamic programming principle are applied to investigate the optimal proportional reinsurance-investment strategy for an insurer under the Vasicek stochastic interest rate model. Solving the corresponding Hamilton-Jacobi-Bellman equation via the Legendre transform approach, the optimal premium allocation strategies maximizing the expected utilities of terminal wealth are derived. In addition, several sensitivity analyses and numerical illustrations are given to analyze the impacts of different risk preferences and interest rate fluctuation on the optimal strategies. We find that the asset allocation and reinsurance ratio of the insurer are correlated with risk preference coefficient and interest rate fluctuation, and the insurance company may adjust the reinsurance-investment strategy to deal with interest rate risk.

The value and cost of more stages in stochastic programing: a statistical analysis on a set of portfolio choice problems

Sequential decision problems under uncertainty are commonly studied with stochastic programing. An important modeling choice is the number of stages. More stages allow additional information to be captured, but is associated with a coarser representation of uncertainty may worsen solution quality. In this paper, we study this trade-off, with the objective to advance the understanding of how the number of stages affect solution quality in stochastic programing. We show: (i) how the optimistic bounds from stochastic programing gradually suggest improved performance with more stages, while the real solution quality simultaneously deteriorates; and (ii) that real performance can be improved by adding stages, but only up to some point, after which more stages are detrimental. Further, we highlight the importance of understanding what creates the value of more stages in the problem of interest, and particularly if this can be captured in models with few stages. The numerical experiments are based on the classic portfolio choice problem of an investor with constant relative risk aversion preferences, maximizing the expected utility of terminal wealth. We study instances with proportional transaction costs and predictability in returns, which takes this problem into an inherently multi-stage nature.

Mean-variance versus utility maximization revisited: The case of constant relative risk aversion

We examine if mean-variance (M-V) is a good proxy for portfolios based on the Constant Relative Risk Aversion (CRRA) utility function. M-V portfolios are considered good proxies for portfolios from several utility functions which is why they are routinely used in the portfolio theory literature as the benchmark. Our results clearly show this is not the case. Low risk CRRA portfolios are in many cases very different to M-V portfolios, especially with respect to downside risk. If a risk-free asset is available, in many cases, no M-V portfolio is an adequate proxy for CRRA portfolios. The implications of our findings are that: i) M-V portfolios are a poor proxy for investors with CRRA preferences, ii) CRRA portfolios are more suited to investors who care about downside risk than M-V portfolios, and iii) the efficacy of M-V to proxy for utility maximization should be examined more thoroughly.

Hedging Demand in Long-Term Asset Allocation with an Application to Carry Trade Strategies

Abstract We derive a closed-form expression for the mean and marginal hedging demand on risky assets in long-term asset allocation problems for individuals with constant relative risk aversion preferences. Our parametric portfolio policy rule accommodates an arbitrarily large number of state variables for predicting the state of nature and number of assets in the portfolio. The closed-form expression for the hedging demand is exact under polynomial specifications of the portfolio policy rule and a suitable approximation for unknown smooth parametric portfolio policy rules using Taylor expansions. The hedging demand on risky assets depends positively on the predictability of the risky asset and the persistence of the predictors, and negatively on the degree of investor’s relative risk aversion. We illustrate these insights empirically for a basket of currencies by showing the outperformance of rebalancing carry trade strategies over different investment horizons against a short-term (myopic) portfolio.

THE EFFECT OF THE ASSUMED INTEREST RATE AND SMOOTHING ON VARIABLE ANNUITIES

AbstractIn this paper, we consider the risk–return trade-off for variable annuities in a Black–Scholes setting. Our analysis is based on a novel explicit allocation of initial wealth over the payments at various horizons. We investigate the relationship between the optimal consumption problem and the design of variable annuities by deriving the optimal so-called assumed interest rate for an investor with constant relative risk aversion preferences. We investigate the utility loss due to deviations from this. Finally, we show analytically how habit-formation-type smoothing of financial market shocks over the remaining lifetime leads to smaller year-to-year volatility in pension payouts, but to increases in the longer-term volatility.

Stocks for the log-run and constant relative risk aversion preferences

Almost Stochastic Dominance implies that stocks are preferred over bonds in the long-run by all preferences with bounded marginal utility. Unfortunately, this analysis excludes the very central Constant Relative Risk Aversion (CRRA) preferences, which have unbounded marginal utility. We derive a simple closed-form solution for the portfolio-choice problem of CRRA investors which, with the empirical parameters, implies that stocks are preferred over bonds for all CRRA investors with relative risk aversion smaller than 3.9. As a by-product, the analysis offers an alternative derivation of the continuous-time CAPM, that does not rely on dynamic programming, and allows for ambiguous investment horizons.

Importance sampling in stochastic optimization: An application to intertemporal portfolio choice

In this paper, we propose an approach to construct an analytical approximation of the zero-variance importance sampling distribution. We show specifically how this can be designed for the classic intertemporal portfolio choice problem with proportional transaction costs and constant relative risk aversion preferences. We compare the method to standard variance reduction techniques in single-period optimization and multi-stage stochastic programming formulations of the problem. The numerical experiments show that the method produces significant improvements in solution quality. In the single-period setting, the number of scenarios can be reduced by a factor of 400 with maintained solution quality compared to the best standard method; Latin hypercube sampling. Using importance sampling in multi-stage formulations, the gaps between lower and upper bound estimates are reduced by a factor of 26-500 with maintained scenario tree size. On a higher level, we consider analytical approximations of the zero-variance importance sampling distribution to be a promising method to improve solution quality in stochastic optimization.

On the Gains of Using High Frequency Data in Portfolio Selection

Abstract This paper analyzes empirically the performance gains of using high frequency data in portfolio selection. Assuming Constant Relative Risk Aversion (CRRA) preferences, with different relative risk aversion levels, we compare low and high frequency portfolios within mean-variance, mean-variance-skewness and mean-variance-skewness-kurtosis frameworks. Using data on fourteen stocks of the Euronext Paris, from January 1999 to December 2005, we conclude that the high frequency portfolios outperform the low frequency portfolios for every out-of-sample measure, irrespectively to the relative risk aversion coefficient considered. The empirical results also suggest that for moderate relative risk aversion the best performance is always achieved through the jointly use of the realized variance, skewness and kurtosis. This claim is reinforced when trading costs are taken into account.

The Utility Value of Longevity Risk Pooling: Analytic Insights

The consensus among researchers is that (some) longevity risk pooling is the optimal strategy for drawing down wealth in retirement, and a robust literature has developed around its measurement via annuity equivalent wealth. However, most of the published work is conducted numerically, and authors usually report only a handful of limited values. In this article we derive closed-form expressions for the value of longevity risk pooling with fixed life annuities under constant relative risk aversion preferences. We show, for example, that this value converges to when the interest rate happens to be the inverse of life expectancy, remaining lifetimes are exponentially distributed, and utility is logarithmic. In general the various formula we derive match previously published numerical results when properly calibrated to discrete time and tables. More importantly, we focus attention on the incremental utility from annuitization when the retiree is already endowed with preexisting pension income such as Social Security benefits. Indeed, because of the difficulty in working with the so-called wealth depletion time in lifecycle models, we believe this is an area that hasn’t received proper attention in the literature. Overall, our article offers an assortment of tools to help explain the value of longevity risk pooling.

HETEROGENEITY IN RISK PREFERENCES LEADS TO STOCHASTIC VOLATILITY

This paper studies the price processes of a claim on terminal endowment and of a claim on firm book value when the underlying variables follow a bivariate geometric Brownian motion. If the state-price process is multiplicatively separable into time and endowment functions, our main result shows that firm (endowment) price volatility is stochastic (state-dependent) if, and only if, the endowment function is not a power function. In a pure exchange economy populated by two agents with constant relative risk aversion (CRRA) preferences we confirm the separability, and we show furthermore that firm (endowment) price volatility is stochastic (state-dependent) if, and only if, both agents are heterogeneous in risk-preferences.

Stocks for the Log-Run and CRRA Preferences

Almost Stochastic Dominance has been employed to support the case of stocks for the long-run. It implies that stocks are preferred over bonds in the long-run for all preferences with bounded marginal utility. Unfortunately, this analysis excludes the very central family of Constant Relative Risk Aversion (CRRA) preferences, which have unbounded marginal utility. In this paper, we derive a simple closed-form solution for the portfolio-choice problem of CRRA investors. This solution and the empirical return parameters imply that stocks are preferred over bonds for all CRRA investors with relative risk aversion smaller than 3.9. As most empirical and experimental estimates of relative risk aversion are much smaller than this value, the stocks-for-the-long-run argument turns out to also hold for most CRRA investors. As a by-product, the analysis offers an alternative derivation of the continuous-time CAPM, that does not rely on dynamic programming, and allows for ambiguous investment horizons.

The Utility Value of Longevity Risk Pooling: Analytic Insights

The consensus among scholars is that (some) longevity risk pooling is the optimal strategy for drawing down wealth in retirement and a robust literature has developed around its measurement via annuity equivalent wealth (AEW). However, most of the published work is conducted numerically and authors usually report only a handful of limited values. In this paper we derive some closed-form expressions for the value of longevity risk pooling with fixed life annuities under constant relative risk aversion preferences. We show, for example, that this value converges to the square root of Euler's e -1 (=65%), when the interest rate is the inverse of life expectancy, lifetimes are exponentially distributed and utility is logarithmic. In general the various formulae we derive match previously published numerical results, when properly calibrated to discrete time and tables. More importantly, we focus attention on the incremental utility from annuitization when the retiree is already endowed with pre-existing pension income such as Social Security benefits. Indeed, due to the difficulty in working with the so-called wealth depletion time (WDT) in lifecycle models, we believe this is an area that hasn't received proper attention from actuarial researchers. Our paper offers tools to explain the value of longevity risk pooling.

Prospect Theory and the Investment Horizon

Prospect Theory (PT) and Constant Relative Risk Aversion (CRRA) have clear-cut implications for the optimal asset allocation between stocks and the risk-free asset as a function of the investment horizon. While CRRA preferences imply that the allocation should be independent of the horizon, we show that PT implies a dramatic and discontinuous “jump” to the maximal possible allocation to stocks as the horizon increases. We experimentally test these predictions at the individual level. We find very little support for PT with at most 6% of the choices supporting it. We find much stronger support for CRRA with 32%-44% of the choices conforming to it. Moreover, the aggregate allocation, which is relevant for asset pricing, conforms to CRRA. These findings provide some justification for economic models employing CRRA preferences, and suggest that the equity premium puzzle is yet to be solved.

Stochastic continuous time growth models that allow for closed form solutions

We find a closed form solution that maximises the expected utility of an agent’s inter-temporal consumption subject to a stochastic technology, which is a linear combination of AK and Cobb–Douglas technologies. Additionally, we consider two cases of agent preferences: (i) Constant Relative Risk Aversion (CRRA) preferences, which treat optimal consumption as a linear function of capital, and (ii) Hyperbolic Absolute Risk Aversion (HARA) preferences, which treat optimal consumption as an affine function of capital. By establishing a minimum (subsistence) level of consumption in the HARA model, we are able to create a framework that more accurately represents real-world circumstances than previous studies have done. Furthermore, for both the CRRA and HARA cases we show the suitable, consistent stochastic differential equation which describes the capital dynamics. Finally, we perform a numerical simulation based on the CRRA case and calibrate US data for the HARA case.

Optimal investment, consumption and life insurance under stochastic framework

In this paper, we consider an optimal control problem of investment, consumption and life insurance for a wage earner who has constant relative risk aversion (CRRA) preferences. The wage earner can invest in zero-coupon bond, stock and life insurance. The interest rate is Vasicek model, the stock follows an extension of Heston stochastic volatility model. We use a generalized Black-Scholes model to characterize labor income. Risks of the incomes increasing rate and interest rate are cointegrated. The optimal strategies of the problem are derived by dynamic programming method and technique of solving associated HJB equations. We also present a sensitivity analysis to explore the impact of economical parameters on the optimal strategies.

An evolutionary explanation for risk aversion

Recent models of the evolution of preferences have provided profound new insights into the origins of risk attitudes. In most of these models the evolutionary “objective function” is the maximization of the expected number of offspring, or alternatively, the maximization of the geometric-mean growth rate. We suggest that careful consideration should also be given to the objective of maximizing the probability of Having Descendants Forever, p(HDF). We show that the p(HDF) criterion implies risk aversion. Moreover, it leads to preferences that are very closely approximated by the constant relative risk aversion preferences. Thus, constant relative risk aversion can be viewed as an evolutionary-developed heuristic aimed to maximize the probability of having descendants forever.

Positive and Negative Externalities in Innovation, Trade, and Regional Economic Growth

We analyze the interactions between positive and negative externalities in innovation and trade for economic growth in a region when this region is part of an aggregate economy consisting of two regions. In both regions consumers have constant relative risk aversion preferences, there is human capital use, and there are three kinds of manufacturing activities involving the production of blueprints for inputs or machines, the inputs or machines themselves, and a single final good for consumption. We study two cases. In the first case, there is no growth in the human capital stock but innovative activities give rise to positive externalities or knowledge spillovers in two ways. In this setting, we study whether and under what circumstances opening a region to trade results in an increase in this region’s equilibrium growth rate. In the second case, there is growth in the human capital stock but there are negative externalities in innovation. In this scenario, we show that opening a region to trade leads to more innovation but to no change in its long run growth rate.

Can CRRA Preferences Explain CAPM-Anomalies in the Cross-Section of Stock Returns?

A large number of empirical studies find systematic deviations from the CAPM. The CAPM tends to understate the returns on low-beta stocks and overstate the returns on high-beta stocks, i.e. the security market line is too steep. Other well-documented anomalies are the size premium and the value premium. The CAPM is a special case of the consumption-based CAPM. This study adresses the question whether the consumption-based CAPM with constant relative risk aversion preferences and lognormally distributed dividends can explain CAPM-anomalies. An example of an economy with power utility and lognormal dividends is examined that can be solved in closed form. The model leads to a security market line that is flatter than in the CAPM and generates a size and a value premium. The comparative statics suggest that cross-sectional anomalies and the equity premium puzzle are tightly linked.

CRRA utility maximization under dynamic risk constraints

The problem of optimal investment with CRRA (constant, relative risk aversion) preferences, subject to dynamic risk constraints on trading strategies, is the main focus of this paper. Several works in the literature, which deal either with optimal trading under static risk constraints or with VaR{based dynamic risk constraints, are extended. The market model considered is continuous in time and incomplete, and the prices of nancial assets are modeled by It^o processes. The dynamic risk constraints, which are time and state dependent, are generated by a general class of risk measures. Optimal trading strategies are characterized by a quadratic BSDE. Within the class of time consistent distortion risk measures, a three{fund separation result is established. Numerical results emphasize the eects of imposing risk constraints on trading.