Tax evasion, audits with memory, and portfolio choice

Should the proportion of risky assets in the risky part of an investor’s portfolio depend on the investor’s risk aversion? According to basic financial theory, in particular the mutual-fund separation theorem with a riskless asset, the answer is no. The theorem states that rational investors should divide their assets between a riskless asset and a risky mutual fund, the composition of which is the same for all investors. Risk aversion affects only the allocation between the riskless asset and the fund. However, Niko Canner et al. (1997), CMW hereafter, observed that popular investment advice does not conform to this theory. They reported the stocks, bonds, and cash allocations recommended by four advisors for conservative, moderate, and aggressive investors. As shown in Table 1, which is reproduced from CMW, the advisors recommend a bond/stock ratio that varies directly with risk aversion. For example, Fidelity recommends a bond/stock ratio of 1.50 for a “conservative” (more riskaverse) investor, a ratio of 1.00 for a “moderate” (less risk-averse) investor, and a ratio of 0.46 for an “aggressive” (still less risk-averse) investor. The inconsistency between such advice and the separation theorem is called an asset allocation puzzle by CMW. They attempted to solve the puzzle by relaxing key assumptions in the theory, but finally reached a negative conclusion: “Although we cannot rule out the possibility that popular advice is consistent with some model of rational behavior, we have so far been unable to find such a model” (p. 181). However, they suggested that consideration of intertemporal trading might help resolve the puzzle. In the present paper, we provide theoretical support for the popular advice. The two key insights are that the investor’s horizon may exceed the maturity of the cash asset and that the investor rebalances the portfolio as time passes. If the investor’s horizon exceeds the maturity of cash, which might be a money-market security with maturity of one to six months, then cash is not the riskless asset as is commonly assumed in the basic theory. In a theory allowing portfolio rebalancing, as opposed to a buy-and-hold framework, it is not unreasonable to assume that the investor can synthesize a riskless asset (a zero-coupon bond maturing at the horizon) using a bond fund and cash. Then bonds will be both in the (synthetic) riskless asset and in the risky mutual fund and we show that in this case the theoretical bond/stock ratio varies directly with risk aversion for any hyperbolic absolute risk aversion (HARA) investor. As an example of the type of results that a specific model can produce, we provide a continuous-time model with closed-form solutions, which produces theoretical bond/stock ratios similar to the popular advice. The present paper is organized as follows: in the next section, we analyze the popular advice in terms of the theory of mutual-fund separation of David Cass and Joseph E. Stiglitz (1970). We show that this theory is relevant both in static and dynamic frameworks and use it to analyze the popular advice in complete and incomplete markets. In Section II, we analyze the popular advice in the context of Robert C. Merton’s (1971) continuous-time statement of mutualfund separation and present an illustrative model in which a CRRA investor makes continuous-time portfolio decisions under interest rate and stock price uncertainty. In Section III, numerical results are compared with the popular advice. Section IV is a conclusion. * Bajeux-Besnainou: Department of Finance, School of Business and Public Management, George Washington University, 2023 G Street NW, Washington, DC 20052; Jordan: National Economic Research Associates, 1255 23rd Street NW, Washington, DC 20037; Portait: CNAM and ESSEC, Finance Chair CNAM, 2 Rue Conte, Paris, France. This research was supported by a grant from the Institute for Quantitative Investment Research. We thank two anonymous referees for their comments. 1 HARA functions include quadratic utility, which is one way of justifying mean-variance preferences, and constant relative risk aversion (CRRA) utility. Both quadratic and CRRA utility were considered in the CMW analysis.

Purpose This study develops a comprehensive discrete numerical model for option valuation that explicitly incorporates risk preferences, which may deviate from risk neutrality. Unlike the traditional binomial tree models – strictly under the risk-neutral paradigm – our framework embeds a constant relative risk aversion (CRRA) utility specification, capturing heterogeneous attitudes toward risk while preserving the arbitrage-free pricing rule. Design/methodology/approach The model extends the multiplicative binomial recombination tree (MBRT) by adjusting key parameters – transition probabilities, growth factors, discount rates and drift/diffusion terms – to reflect the investor's degree of risk aversion. The classical Cox-Ross-Rubinstein binomial tree (CRR) emerges as a special case when risk aversion is set to zero. The methodology remains consistent with geometric Brownian motion (GBM) dynamics and is benchmarked against a modified Monte Carlo simulation to ensure robustness. Findings Results show that option values can be consistently derived under both traditional risk-neutral settings and preference-driven settings. Sensitivity analysis highlights the impact of time to maturity, volatility, strike price and the risk-free rate under varying levels of risk aversion. Research limitations/implications While this research offers significant theoretical and practical contributions, certain limitations warrant further study. Computational complexity: the CRRA-based valuation method introduces additional numerical challenges, requiring precise calibration and advanced optimization techniques. Dependence on risk aversion estimates: the model assumes that investor risk preferences can be accurately measured and remain stable, which may not always reflect dynamic market conditions. Absence of a closed-form solution: our proposed approach lacks an analytical closed-form solution. Therefore, it is crucial to dedicate efforts to its development. Practical implications The integration of CRRA utility functions into derivative valuation represents a key innovation, as it explicitly accounts for investor risk preferences beyond the traditional risk-neutral paradigm. This framework advances the literature on utility-based and nonlinear risk-adjusted pricing by demonstrating how variations in the relative risk aversion (RRA) coefficient shape option values. From a practical perspective, the model offers a flexible tool for portfolio managers, traders and policymakers by aligning valuations with observed market behavior while preserving consistency with classical models under specific conditions. Accurate calibration of risk preferences thus becomes essential for reliable pricing and policy design. Originality/value The novelty of this research lies in bridging utility-based preferences with recombining lattice valuation: while prior studies focused exclusively on risk-neutral or arbitrage-based approaches, our model incorporates explicit risk aversion into the numerical structure. By deriving general algebraic expressions and validating the framework through numerical experiments, this study offers a tractable and versatile tool for analyzing option prices under heterogeneous risk attitudes, without losing the analytical clarity of traditional methods.

Although farmers in developing countries are generally thought to be risk averse, little is known about the actual form of their risk preferences. In this paper, we use a relatively large lab-in-the-field experiment to explore risk preferences related to sweet potato production among a sample of farmers in northern Mozambique. A unique feature of this experiment is that it includes a large subsample of husband and wife pairs. After exploring correlations between husband and wife preferences, we explicitly test whether preferences follow the constant relative risk aversion (CRRA) utility function, and whether farmers follow expected utility theory or rank dependent utility theory in generating their preferences. We reject the null hypothesis that farmers' preferences follow the CRRA utility function, in favor of the more flexible power risk aversion preferences. If we make the common CRRA assumption in our sample, we poorly predict risk preferences among those who are less risk averse.

Role of index bonds in an optimal dynamic asset allocation model with real subsistence consumption

Business activities in the agricultural sector, especially rice farming, will always be faced with a high risk of uncertainty. The risks experienced by farmers come from the natural environment, natural disasters, climate, and plant-disturbing organisms. To avoid this situation, the government is currently providing the best solution in the form of a Rice Farming Insurance program (AUTP), which is expected to provide protection against the risk of crop failure that farmers may experience. The purpose of this research is to analyze rice farmers' risk preferences in determining rice farming insurance premiums. The research method for rice farmers' risk preferences was analyzed using constant relative risk averse (CRRA) utility theory. Based on the research results, rice farmers in Majalaya District, Bandung Regency has a very risk averse risk preference. The risk preferences of farmers participating in AUTP and non-AUTP are very risk averse. The policy implications that can be explained based on the results of this study are increasing farmers' understanding regarding the description and benefits of agricultural insurance through counseling and assistance by the Agriculture Service and PT. Jasindo, so that rice farmers in Majalaya District, Bandung Regency have awareness of the benefits of insurance. Encouraging the participation of rice farmers in Majalaya District in the AUTP program can also be carried out by prioritizing rice farmers with very risk averse risk preferences. The policy implications that can be explained based on the results of this study are increasing farmers' understanding regarding the description and benefits of agricultural insurance through counseling and assistance by the Agriculture Service and PT. Jasindo, so that rice farmers in Majalaya District, Bandung Regency have awareness of the benefits of insurance. Encouraging the participation of rice farmers in Majalaya District in the AUTP program can also be carried out by prioritizing rice farmers with very risk averse risk preferences. The policy implications that can be explained based on the results of this study are increasing farmers' understanding regarding the description and benefits of agricultural insurance through counseling and assistance by the Agriculture Service and PT. Jasindo, so that rice farmers in Majalaya District, Bandung Regency have awareness of the benefits of insurance. Encouraging the participation of rice farmers in Majalaya District in the AUTP program can also be carried out by prioritizing rice farmers with very risk averse risk preferences.

This study investigates the distinctive modeling of regret utility when compared with common utility. I also introduce the interplay between common utility and regret utility. Using this model, I examine the differences in decision making, which encompasses issues such as risk sharing and principal–agent dilemmas. Regret utility is set so that its risk aversion shows common utility’s prudence (i.e., downside risk aversion). This paper reveals, both qualitatively and quantitively and with a concrete model, that regret utility leads to a more balanced and optimal ratio of agent payouts to outputs compared with common utility, meaning when major outputs are kept by principal, there are relatively larger agent payouts, and when major outputs are kept by the agent, there are relatively smaller agent payouts. This means that regret makes a more balanced distribution, and regret utility is more conservative (not biased). In addition, preliminary empirical research was performed in which people were asked risk preference or averseness questions, and their risk averseness was calculated by using the CRRA (Constant Relative Risk Aversion) utility function. The regret condition leads to a more conservative attitude. Furthermore, the regret model can be used in other areas, like in conservative investment portfolio optimization.

The decision to transfer or share an insurable risk is critical for the decision maker’s economy. This paper deals with this decision, starting with the definition of a function that represents the difference between the expected utility of insuring, with or without deductibles, and the expected utility of not insuring. Considering a constant relative risk aversion (CRRA) utility function, we provide a decision pattern for the potential policyholders as a function of their wealth level. The obtained rule applies to any premium principle, any per-claim deductible and any risk distribution. Furthermore, numerical results are presented based on the mean principle, a per-claim absolute deductible and a Poisson-exponential model, and a sensitivity analysis regarding the deductible parameter and the insurer security loading was performed. One of the main conclusions of the paper is that the initial level of wealth is the main variable that determines the decision to insure or not to insure; thus, for high levels of wealth, the decision is always not to insure regardless of the risk aversion of the decision maker. Moreover, the parameters defining the deductible and the premium only have an influence at low levels of wealth.

Comparison of optimal portfolios with and without subsistence consumption constraints

Risk Aversion Under Preference Uncertainty

Risk aversion under preference uncertainty

In this thesis, we investigate a pensioner’s gains from access to annuities. We observe the optimal asset allocation and annuitization strategies for a pensioner whose retiring age is 65, with an individual pension wealth at retirement and with a guaranteed income from social security during the retirement period. We also observe with particular personal risk preferences towards risk, with a certain amount to buy more annuities after retirement and with certain limitations on pensioner’s asset allocation and annuitization strategies. The pensioner’s objective is to maximize the utility drawn from consumption during retirement with a Constant Relative Risk Aversion utility function. We develop and solve two main models on stochastic volatility for the pensioner who receives an income after retirement from life annuities and investment performance which are under the Constant Elasticity of Variance model and Heston’s Model. We start with the model proposed by Milevsky and Young in 2007 under the Geometric Brownian Motion model and address using the change variable technique. We extend the model under stochastic volatility and solve it using the combination of Legendre transform, dual theory and change variable technique. By adopting the Legendre transform, dual theory and change variable approaches, the explicit solution for optimal investment, consumption and annuitization strategies is derived for the power utility. We use a numerical example to investigate the influence of life annuities and model parameters on the optimal strategy. The results show that the optimal strategy depends on model parameters and the presence of life annuities in the model affects the pensioner’s decisions regarding optimal investment and annuity income level strategies for a period after retirement under the stochastic market price of risk. Besides, an annuity income plays a role in altering the consumption rates for all levels of risk aversion.

We introduce a method to infer an investor's risk aversion based on the observed asset allocation of their pension savings. By assuming the actual allocation is optimal under a constant relative risk aversion (CRRA) utility, we invert Merton's optimal investment formulas to estimate the risk aversion parameter. The approach incorporates the present value of future premiums, resulting in strategies that align with life-cycle pension products. To ensure stability, we develop a customized risky fund matched with the investor's allocation, enabling reliable calibration across various asset classes. A numerical study on a Danish pension portfolio demonstrates the practical use. The findings show realistic and stable risk aversion levels consistent with the CRRA assumption and offer a tool to better understand and benchmark the implicit preferences embedded in pension product design.

<p>This thesis is based upon four very simple premises: 1. managers, not shareholders make the investment decisions for the firm; 2. managers do more than just say "yes" or "no" to investments, they can also exert effort that affects the payoff from investment; 3. executive compensation schemes can cause managers to hold more stock than is optimal for diversification purposes; and 4. many investments can be delayed and involve irreversible capital costs as well as uncertain payoffs. Combining these four premises gives the two central questions this thesis attempts to answer: 1. How does the level of managerial stock-ownership affect the investment decisions managers make for the firm? and 2. given the answer to (1), how does this affect the shareholder's decision to hire a manager? In this thesis I use a continuous time "Real Options" framework to answer these questions. The form of the utility function assumed for the manager has a huge impact on the tractability of the modelling. The assumption of Constant Relative Risk Aversion (CRRA) utility as opposed to Constant Absolute Risk Aversion (CARA) causes the manager's valuation of the cash flow (the very first step of the modelling) to become wealth dependent. This in itself is an interesting issue, but it also poses interesting numerical issues and makes the later steps of the analysis intractable. Because of this we split the substantive analysis of this thesis into two parts. In the first we assume CARA utility in order to remove wealth dependence from the valuation and obtain a "clean path" to the end goal of a dynamic model of hiring, effort and irreversible investment. In the second we focus on CRRA utility thus allowing the manager's valuation to depend on his financial wealth. We then explain the resultant numerical issues, and the appropriate approach to their solution.</p>

<p>This thesis is based upon four very simple premises: 1. managers, not shareholders make the investment decisions for the firm; 2. managers do more than just say "yes" or "no" to investments, they can also exert effort that affects the payoff from investment; 3. executive compensation schemes can cause managers to hold more stock than is optimal for diversification purposes; and 4. many investments can be delayed and involve irreversible capital costs as well as uncertain payoffs. Combining these four premises gives the two central questions this thesis attempts to answer: 1. How does the level of managerial stock-ownership affect the investment decisions managers make for the firm? and 2. given the answer to (1), how does this affect the shareholder's decision to hire a manager? In this thesis I use a continuous time "Real Options" framework to answer these questions. The form of the utility function assumed for the manager has a huge impact on the tractability of the modelling. The assumption of Constant Relative Risk Aversion (CRRA) utility as opposed to Constant Absolute Risk Aversion (CARA) causes the manager's valuation of the cash flow (the very first step of the modelling) to become wealth dependent. This in itself is an interesting issue, but it also poses interesting numerical issues and makes the later steps of the analysis intractable. Because of this we split the substantive analysis of this thesis into two parts. In the first we assume CARA utility in order to remove wealth dependence from the valuation and obtain a "clean path" to the end goal of a dynamic model of hiring, effort and irreversible investment. In the second we focus on CRRA utility thus allowing the manager's valuation to depend on his financial wealth. We then explain the resultant numerical issues, and the appropriate approach to their solution.</p>

This paper analyzes an optimal investment and management strategy for a bank under constant relative risk aversion (CRRA) and hyperbolic absolute risk aversion (HARA) utility functions. We assume that the bank can invest in treasuries, stock index fund and loans, in an environment subject to stochastic interest rate and inflation uncertainty. The interest rate and the expected rate of inflation follow a correlated Ornstein–Uhlenbeck processes and the risk premia are constants. Then we consider the portfolio choice under a power utility that the bank's shareholders can maximize expected utility of wealth at a given investment horizon. Closed form solutions are obtained in a dynamic portfolio optimization model. The results indicate that the optimal proportion invested in treasuries increases while the optimal proportion invested in the loans progressively decreases with respect to time.