Level Of Relative Risk Aversion Research Articles

A new approach to the uniqueness of equilibrium with CRRA preferences

This paper investigates the uniqueness of equilibrium in the economy where agents have preferences with possibly distinct levels of relative risk aversion. Equilibrium prices exist in a price range determined by micro and macroeconomic information. Macroeconomic information is inferred from two representative-agent economies, which provide lower and upper bounds on aggregate demand, respectively. Analysis of the first and second-order price effects shows that individual demand has a unique inflection point at which the income effect maximally dominates the substitution effect. In contrast, aggregate demand may have multiple inflection points, which creates multiple equilibria. Sufficient conditions for the uniqueness of equilibrium are characterized by the local behavior of individual demand at the inflection point. They require that aggregate demand must have no more than two inflection points in the equilibrium price range. The result of this paper sheds light on the Sonnenschein-Mantel-Debreu theory that has negative implications for the uniqueness of equilibrium.

Transactions costs and the equity premium puzzle

Campbell and Cochrane (1999b) habit formation model is able to resolve the equity premium and riskless interest rate puzzles, but only for high values of relative risk aversion. In this paper, I incorporate transactions costs in the Campbell and Cochrane (1999b) model and find that the required level of relative risk aversion at the steady state reduces from 35 to 15. Thus, transactions costs seem able to reduce, but not completely solve, the remaining puzzle.

Estimates of households consumer behavior in Russia

In this paper, the consumption profiles of households in the Russian Federation are constructed and, on their basis, the parameters of future discounting and relative risk aversion are evaluated. The paper demonstrates that the age profiles are hump-shaped, the peak consumption level is reached at 38—40 years. The important determinant of the hump shape is the composition of the household: when consumption is recalculated for the average individual in a household, the hump shape becomes less pronounced. On average, households have a high level of relative risk aversion and a low level of future discounting. This configuration of parameters indicates the important role of the precautionary motive in the consumer behavior of households. Estimates are presented for households with different levels of education and number of people.

Ordering Arbitrage Portfolios and Finding Arbitrage Opportunities

Concepts are introduced for analyzing arbitrage portfolios in the face of ambiguity about investor risk preferences and initial portfolio holdings. A Stochastic Arbitrage Opportunity is a self-financing overlay portfolio which enhances every feasible host portfolio for all relevant utility functions. The logical alternative to the existence of such opportunities is the solvability of a system of asset pricing restrictions for the base assets. An asymptotic statistical theory is developed to analyze the behavior of the empirical optimal arbitrage portfolio if the latent parameters of the joint payoff distribution are estimated using time-series data. An empirical application to one-month S&P500 index options uncovers significant arbitrage opportunities in the form of short strip strangle spreads, for investors with reasonable levels of relative risk aversion. The optimal option combinations enhance not only passive index portfolios but also portfolios with existing volatility risk exposure.

Performance of Characteristics-Based Portfolio Choice

We compare the performance of the characteristics-based parametric portfolio approach introduced by Brandt, Santa Clara and Valkanov (RFS 2009) with standard optimal portfolio investments on the basis of S&P-500 stocks. We establish that the characteristics-based parametric portfolio approach can only be exactly justified as optimal investment under exceedingly strong assumptions. Moreover, we find that the empirical implementation of the parametric portfolio policy approach to US-data (for 1995-2013) runs into difficulties at moderate levels of relative risk aversion. Without refinements an interior maximum of the expected utility functional need not exist. Finally, and despite the strong implicit assumptions required by theory, the characteristics-based approach compares reasonably well with optimal portfolio investment for absolute risk aversion even in the absence of transactions costs, when we run horse-races in simulated and empirical data.

Measuring excess-predictability of asset returns and market efficiency over time

We build on the predictability bounds of Huang and Zhou (2017) and Potì (2018) to develop an index of informational market inefficiency. This index takes values given by the levels of relative risk aversion (RRA) of the marginal investor such that, net of sampling error at a given confidence level, the observed predictability does not exceed the predictability bound. We demonstrate the usefulness of our index in a study of the predictability of forward exchange rates of currencies of emerging and developed economies from 1994 to 2016, to shed light on how the efficiency of currency markets has evolved over this time. We find widespread evidence of excess-predictability, hence currency market inefficiency, in the early part of the sample period and then at specific times, such as the recent global financial crisis. In the more recent part of the sample period, the evidence of excess-predictability is largely limited to emerging market currencies.

Measuring Excess-Predictability of Asset Returns and Market Efficiency over Time

We build on the predictability bounds of Huang et al. (2017) and Poti (2018) to develop an index of informational market inefficiency. This index takes values given by the levels of relative risk aversion (RRA) of the marginal investor such that, net of sampling error at a given confidence level, the observed predictability does not exceed the predictability bound. We demonstrate the usefulness of our index in a study of the predictability of forward exchange rates of currencies of emerging and developed economies from 1994 to 2016, to shed light on how the efficiency of currency markets has evolved over this time. We find widespread evidence of excess-predictability, hence currency market inefficiency, in the early part of the sample period and then at specific times, such as the recent global financial crisis. In the more recent part of the sample period, the evidence of excess-predictability is largely limited to emerging market currencies.

Online-Appendix to: Making Parametric Portfolio Policies Work: Comments on Brandt, Santa-Clara and Valkanov (2009)

We provide examples of pitfalls for parametric portfolio policies as introduced by Brandt, Santa Clara and Valkanov. For the leading case of constant relative risk aversion (CRRA) strong assumptions on the properties of the returns, the variables used to implement the parametric portfolio policy and the parameter space are necessary to obtain a well defined optimization problem. As possible remedies for practical work various extensions of CRRA Bernoulli utility to the real line are discussed. Also prospect theory is suggested as an alternative approach. We observe that for low levels of relative risk aversion expected utility turns non-monotonic and an interior maximum need not exist. We provide economic conditions that overcome such empirical problems and that guarantee the effectiveness of the approach more broadly. We illustrate our concerns by applying parametric portfolio policies to a large universe of stocks. Full paper is available at: https://ssrn.com/abstract=3081100

Making Parametric Portfolio Policies Work: Comments on Brandt, Santa-Clara and Valkanov (2009)

We provide examples of pitfalls for parametric portfolio policies as introduced by Brandt, Santa Clara and Valkanov (RFS 2009). For the leading case of constant relative risk aversion (CRRA) strong assumptions on the properties of the returns, the variables used to implement the parametric portfolio policy and the parameter space are necessary to obtain a well defined optimization problem. As possible remedies for practical work various extensions of CRRA Bernoulli utility to the real line are discussed. Also prospect theory is suggested as an alternative approach. We observe that for low levels of relative risk aversion expected utility turns non-monotonic and an interior maximum need not exist. We provide economic conditions that overcome such empirical problems and that guarantee the effectiveness of the approach more broadly. We illustrate our concerns by applying parametric portfolio policies to a large universe of stocks. Appendix is available at: https://ssrn.com/abstract=3131081

Asset pricing puzzles in an OLG economy with generalized preference

AbstractWe seek to explain a number of asset pricing anomalies – the equity premium puzzle, the risk‐free rate puzzle, and portfolio allocation puzzle – in a parsimonious overlapping generations (OLG) model with two key features: borrowing constraint and Epstein–Zin–Weil (1989) preference. The model goes a long way towards the resolution of these puzzles, and is able to simultaneously match asset pricing moments and individual portfolio decisions using reasonable values of parameters governing behavior. We find that the main driver of savings behavior, equity returns, and asset allocation is the relative difference between the two parameters: the level of relative risk aversion and the inverse of the elasticity of substitution.

Weak Dependence of CRRA on Standard Deviation in the Case of Truncated Normal Distribution of Returns

This paper analyzes the dependence of the Certainty Equivalent Return of a Constant Relative Risk Aversion, CER[CRRA], on the Standard Deviation of the Return with the hypothesis of a Truncated Normal distribution of returns and for some level of Relative Risk Aversion (RRA) parameter. The paper compares this dependence with those detected by an Annualized Geometrical Return (AGR) and by CER of the Quadratic Utility Function, CER[Q]. The behavior of CER[CRRA] is more similar to AGR than CER[Q] and only for a higher value of RRA is it possible to find substantial differences, even if in this case we find values of Standard Deviation that have discontinuity points for the concavity. Using a ranking criteria equal to the one introduced by Morningstar for a set of Funds, the paper shows that, in a wide range for monthly Standard Deviation and Mean of the Returns, the ranking done by CER[CRRA] is similar to the one induced by AGR, and that a CER[Q] has essentially different behavior. It will be shown that Morningstar ranking may be considered a particular case of the CER[CRRA] and thus all the considerations can be applied to the well-known Morningstar Rating methodology. An application is made to Italian Pension Funds.

Consumption-based equity valuation

Using a CCAPM-based risk-adjustment model, we perform yearly valuations of a large sample of stocks listed on NYSE, AMEX, and NASDAQ over a 30-year period. The model differs from standard valuation models in the sense that it adjusts forecasted residual income for risk in the numerator rather than through a risk-adjusted cost of equity in the denominator. The risk adjustments are derived based on assumptions about the time-series properties of residual income returns and aggregate consumption rather than on historical stock returns. We compare the performance of the model with several implementations of standard valuation models, both in terms of median absolute valuation errors (MAVE) and in terms of excess returns on simple investment strategies based on the differences between model and market prices. The CCAPM-based valuation model yields a significantly lower MAVE than the best performing standard valuation model. Both types of models can identify investment strategies with subsequent excess returns. The CCAPM-based valuation model yields time-series of realized hedge returns with more and higher positive returns and fewer and less negative returns compared with the time-series of realized hedge returns based on the best performing standard valuation model for holding periods from 1 to 5 years. In a statistical test of 1-year-ahead excess return predictability based on the models’ implied pricing errors, the CCAPM-based valuation model is selected as the better model. Using the standard series of aggregate consumption and the nominal price index, a reasonable level of relative risk aversion, and calibrated growth rates in the continuing value at each valuation date, the CCAPM-based valuation model produces small risk adjustments to forecasted residual income and low continuing values. Compared with standard valuation models, it relies less on estimated parameters and speculative elements when aggregating residual earnings forecasts into a valuation.

Optimal Diversification in the Presence of Parameter Uncertainty for a Risk Averse Investor

We consider an investor who faces parameter uncertainty in a continuoustime financial market. We model the investor's preference by a power utility function leading to constant relative risk aversion. We show that the loss in expected utility is large when using a simple plug-in strategy for unknown parameters. We also provide theoretical results that show the trade-off between holding a well-diversified portfolio and a portfolio that is robust against estimation errors. To reduce the effect of estimation, we constrain the weights of the risky assets with an L1-norm leading to a sparse portfolio. We provide analytical results that show how the sparsity of the constrained portfolio depends on the coefficient of relative risk aversion. Based on a simulation study, we demonstrate the existence and the uniqueness of an optimal bound on the L1-norm for each level of relative risk aversion.

Optimal Diversification in the Presence of Parameter Uncertainty for a Risk Averse Investor

We consider an investor who faces parameter uncertainty in a continuous-time financial market. We model the investor's preference by a power utility function leading to constant relative risk aversion. We show that the loss in expected utility is large when using a simple plug-in strategy for unknown parameters. We also provide theoretical results that show the trade-off between holding a well-diversified portfolio and a portfolio that is robust against estimation errors. To reduce the effect of estimation, we constrain the weights of the risky assets with an L1-norm leading to a sparse portfolio. We provide analytical results that show how the sparsity of the constrained portfolio depends on the coefficient of relative risk aversion. Based on a simulation study, we demonstrate the existence and the uniqueness of an optimal bound on the L1-norm for each level of relative risk aversion.

THE RELATIONSHIP BETWEEN RELATIVE RISK AVERSION AND THE LEVEL OF EDUCATION: A SURVEY AND IMPLICATIONS FOR THE DEMAND FOR LIFE INSURANCE

AbstractThe expected utility hypothesis has been widely used in the construction of economic models and numerous difficulties are encountered in attempting to take into account preferences toward risk in a real‐world setting. More recently, attention has focused on comparative studies in economics and business in an international framework and problems related to the hypothesis of relative risk aversion (RRA). One ambiguous hypothesis is the relationship between the level of RRA and the level of education, which has been found either positive or negative. From a causality point of view, it may be argued that investors with a high level of education are less risk averse, but it may also be argued that less risk‐averse individuals choose to pursue a higher level of education. The purpose of this paper is to survey the empirical literature on this subject. It provides evidence that risk aversion is negatively correlated with higher education and human development. The results have important implications for macroeconomic empirical studies and the demand for financial assets and more specifically on the demand for life insurance. Assuming the same degree of RRA for utility‐maximizing consumers should be limited to homogeneous samples.

Optimal portfolio choice in real terms: Measuring the benefits of TIPS

In this paper we solve an optimal portfolio choice problem to measure the benefits of Treasury Inflation Indexed Securities (TIPS) to investors concerned with maximizing real wealth. We show how the introduction of a real riskless asset completes the investor asset space, by contrasting optimal portfolio allocations with and without such assets. We use historical data to quantify gains from availability of TIPS in the presence of other asset classes such as equities, commodities, and real estate. We draw a distinction between buy-and-hold long-term investors for whom TIPS fully displace nominal risk-free assets and short-term investors for whom TIPS improve the investment opportunity set of real returns. Finally, we show how gains from TIPS are tempered by the availability of alternative assets that covary with inflation, such as gold and real estate. ► We measure the empirical benefits that TIPS provide to investors. ► We differentiate investors according to their investment horizon. ► Short-term investors devote a fraction of their wealth to TIPS. ► The benefits to short-term investors diminish when they have a wider investment opportunity set. ► Long-term investors replace nominal bonds with TIPS for all levels of relative risk aversion.

Approximated Portfolio Choice - Do We Dance on a Pinhead?

HARA-utility investors allocate their money to a risk-free fund and to a risky fund (two fund separation). The paper shows that under weak conditions, the risky fund can be approximated by the risky fund derived from exponential utility, without material effects on the certainty equivalent of the portfolio payoff. Also, effects of changes in asset return parameters on the risky fund can be approximated using the exponential utility function, thereby simplifying the analysis. The approximation is of high quality if extreme portfolio returns have a very small probability and if the investor's level of relative risk aversion exceeds 2.

Is More Always Better? Empirical Evidence on Optimal Portfolio Size

A restriction on portfolio size results in welfare losses for investors. To measure these welfare losses, we compare n-asset optimal portfolios with 26-asset optimal portfolios using the concept of proportionate opportunity cost. The original historical asset returns data are used with a VAR in generating joint returns distributions for the portfolio formation period. We find that suboptimal diversification imposes substantial costs on investors with low levels of relative risk aversion. Investors with high levels of risk aversion incur very small or no cost at all diversifying sub-optimally. We show that investors with high levels of risk aversion place most of their initial wealth in the safe asset and, therefore, few stocks are needed to achieve optimal diversification.

The Opportunity Cost of Being Constrained by the Type of Asset: Bonds Only or Stocks Only

I explore investors' welfare losses when they restrict themselves to invest in either stocks only or bonds only, but not in both. The restriction gives investors sub-optimal asset allocations that result in welfare losses. To measure these welfare losses I compare “only stock indices and Treasury bills” optimal portfolios and “only bond indices and Treasury bills” optimal portfolios with “stock and bond indices and Treasury bills” optimal portfolios using the concept of proportionate opportunity cost along with various CRRA utility functions. The original historical asset returns data set is used with a VAR in generating joint returns distributions for the portfolio formation period. I show that for investors with low levels of risk aversion welfare losses do not exceed 1.8% of initial wealth when they invest sub-optimally. For investors with medium and high levels of relative risk aversion, suboptimal portfolios of only one type of assets, stocks only or bonds only, along with Treasury bills, give expected utility about as high as optimal portfolios that include both types of assets, stocks and bonds.

The equilibrium allocation of diffusive and jump risks with heterogeneous agents

We study a two-agent pure exchange equilibrium subject to both nondiversifiable diffusive and jump risks. Agents can trade in a financial market consisting of a stock market, a money market, and an insurance market for jump risk. Heterogeneity is introduced through different levels of relative risk aversion. In the framework of standard expected utility we find the surprising result that the less risk averse agent purchases insurance contracts against jump risk from the more risk averse agent. This equilibrium allocation is linked to the non-linear wealth sharing rule in such an economy, and preserves the wealth effects studied by Dumas [Rev. Financ. Stud. 2 (1989) 157] in the case of pure diffusive risk. Since the benchmark economy with homogeneous agents generates no excess uncertainty in the stock market, we study the effect on excess volatility and excess jump size solely due to different levels of relative risk aversion. We observe 3% excess uncertainty in jump sizes for a reasonable specification of economic fundamentals.