Risk Limited Asset Allocation with a Budget Threshold Utility Function and Leptokurtotic Distributions of Returns
Abstract
An analytical solution to single-horizon asset allocation for an investor with a piecewise-linear utility function, called herein the "budget threshold utility," and exogenous position limits is presented. The resulting functional form has a surprisingly simple structure and can be readily interpreted as representing the addition of a simple "risk cost" to otherwise frictionless trading.
Summary
This paper presents an analytical solution for single-horizon asset allocation for an investor with a "budget threshold utility" function and exogenous position limits. The budget threshold utility function is piecewise linear, indifferent to wealth above a predefined threshold (the budget) and linear in the dis-utility of wealth below that budget. The distribution of asset returns is modeled using the Generalized Error Distribution (GED), allowing for leptokurtosis (fat tails) often observed in financial markets. The methodology involves deriving the expected utility function under the GED and maximizing it subject to the position limits. The key finding is a surprisingly simple closed-form solution for the optimal asset allocation. This solution shows that the optimal position is either at the maximum allowed limit (long or short) or zero, depending on whether the expected return exceeds a "risk cost" term. This "risk cost" is proportional to the standard deviation of returns and a scaling factor derived from the GED's kurtosis parameter. While the kurtosis parameter does affect the risk cost, the effect is small, suggesting that using a normal distribution assumption doesn't drastically alter the optimal policy. The paper bridges the gap between risk-neutral and risk-averse investment strategies by introducing a budget threshold. This matters to the field by providing a tractable and interpretable model for asset allocation that accounts for realistic constraints and non-normal return distributions, offering a practical alternative to more complex optimization frameworks.
Key Insights
- •The paper derives a closed-form solution for optimal asset allocation under a budget-threshold utility function and exogenous risk limits, a contribution to analytical solutions in portfolio theory.
- •The "risk cost" term, στ(κ), acts as a hurdle rate that the expected return must exceed for an investor to take a non-zero position.
- •The kurtosis parameter κ has a relatively small effect on the "risk cost" (Figure 3), suggesting that assuming normally distributed returns may not significantly degrade the optimal policy in this specific framework.
- •The optimal holding function exhibits a three-state system (long, short, or flat), contrasting with simpler binary (long or short) trading algorithms that prioritize maximizing gross profit.
- •The paper proposes a "semi-empirical" approach, replacing the parametric risk cost with a constant (K ≈ 0.4) to be determined from backtesting, acknowledging the non-stationarity of financial distributions.
- •The budget-threshold utility function, while acknowledged as potentially "irrational" by some, provides a mathematically tractable way to model a specific type of risk aversion.
- •The model assumes a single-period investment horizon, which limits its applicability to multi-period settings without further extensions.
Practical Implications
- •The trading algorithm (Equation 13) provides a simple and implementable rule for traders: take a maximum long/short position if the expected return exceeds a scaled standard deviation, otherwise stay flat.
- •Portfolio managers and quantitative analysts can use this model as a starting point for developing more sophisticated asset allocation strategies that incorporate risk limits and non-normal return characteristics.
- •The paper suggests that backtesting and empirical optimization are crucial for determining the constant K, highlighting the importance of adapting theoretical models to real-world data.
- •Future research could explore the extension of this model to multi-period settings, incorporate transaction costs, or consider alternative utility functions that address the limitations of the budget-threshold utility.
- •The model is particularly relevant for traders or investors who have strict loss limits or budget constraints, as it explicitly incorporates these factors into the optimization process.