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Rui Dai, Zongxia Liang, Yang Liu · 2026-07-05
The paper tackles the practical problem that an investor's utility function (their attitude to risk and reward) is hard to specify. It proposes a 'preference-fitting' method where an investor supplies intuitive probability-wealth pairs, from which the authors reconstruct a fitted terminal wealth, portfolio, and utility function that provably converge to the investor's true optimal choices. They introduce a piecewise HARA (PHARA) utility approximation and prove convergence, applying it to derive near-explicit portfolios and to handle Value-at-Risk constraints.
Why it matters: For those building portfolios around explicit investor preferences, this offers a more intuitive way to elicit risk attitudes without guessing a utility form. The claimed analytical tractability and ability to handle VaR constraints without Lagrange multipliers could simplify certain optimization workflows, though this is a theoretical contribution rather than a tested trading edge.
⚠ This is a theoretical/mathematical framework with convergence proofs, not an empirically validated or live-tested strategy, and it assumes investors can reliably provide meaningful probability-wealth inputs.
The utility function plays a core role in portfolio selection, but its specific form is typically hard to elicit. We propose a definition of the elicited utility function and develop a preference-fitting method to obtain it. Basically, we use intuitive probability-wealth pairs to derive a fitted terminal wealth, a fitted portfolio and a fitted utility function, which converge to the optimal terminal wealth, the optimal portfolio and the elicited utility function of the investor, respectively. Specifically, we first establish a bijection between the utility functions and the terminal wealth functions, based on which we construct the fitted terminal wealth, and then obtain the fitted portfolio and the fitted utility function through the martingale-duality method. Next, we develop a piecewise hyperbolic absolute risk aversion (abbr. PHARA) utility approximation method, and verify the convergences in various senses: almost surely, $L^r$, uniform, etc. We demonstrate two applications of our method: obtaining asymptotically explicit portfolios and handling portfolio selection under Value-at-Risk (abbr. VaR) constraints, thereby illustrating its advantages including intuitiveness, analytical tractability, and ability to circumvent the Lagrange multiplier.
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