Risk-Constrained Kelly for Mutually Exclusive Outcomes: CRRA Support Invariance and Logarithmic One-Dimensional Calibration

TL;DR

This paper analyzes risk-constrained Kelly optimization with mutually exclusive outcomes, showing support invariance under certain conditions and providing a detailed calibration method for the logarithmic case.

Contribution

It establishes support invariance across CRRA parameters and develops a complete calibration framework for the logarithmic case in risk-constrained Kelly optimization.

Findings

Support support remains invariant across CRRA and drawdown parameters.

The logarithmic case reduces to a one-dimensional calibration with proven existence and uniqueness.

Numerical example demonstrates how risk constraints modify wealth profiles without changing support.

Abstract

We study the finite mutually exclusive outcome version of risk-constrained Kelly optimization with explicit state prices. The market has outcome probabilities $p_i>0$, state prices $q_i>0$, terminal wealths $W_i=c+x_i/q_i$, and a drawdown-surrogate constraint \[ \sum_{i=1}^n p_i W_i^{-\lambda}\le 1,\qquad \lambda>0. \] For constant relative risk aversion utility, we work primarily in the standard overround regime $\sum_i q_i>1$, where every optimizer is necessarily non-full-support. Under the usual unique likelihood-ratio prefix hypothesis for the unconstrained problem, we prove that the constrained optimizer has exactly the same active set. Thus, in the regime where the prefix theorem is meaningful, the risk constraint deforms the funded wealth profile but does not change the active set. The support is therefore invariant across both the CRRA parameter and the drawdown-surrogate parameter. We then isolate the logarithmic case $\gamma=1$. Once the common active prefix is known, the constrained problem reduces to a one-dimensional outer calibration together with independent one-dimensional inner equations on the active states. In this case we prove existence, uniqueness, and monotonicity for the inner solves, derive a complete calibration theorem, and record the resulting structured algorithm. We treat the fair and subfair regimes only as boundary cases: full-support phenomena can occur there, so the overround prefix theory no longer yields a parallel exact description of comparable sharpness. A numerical example illustrates how the risk constraint alters the funded wealth profile while leaving support unchanged.