A Leverage Cap on an Inconsistent Investor

The mean–variance criterion does not satisfy Bellman’s principle, so “optimal” splits into three incompatible strategies: a pre-commitment plan that requires a commitment device, a naive agent who re-solves continually and follows none of his plans, and an equilibrium from which no future self wishes to deviate. We compute all three for a market whose Sharpe ratio is an Ornstein–Uhlenbeck factor, under a leverage constraint of the kind every mandate carries.

Unconstrained, the equilibrium value is affine in wealth and the policy splits into a myopic term and a hedging term that exists only because the problem is inconsistent. The correlation between the asset and the factor enters through that term and nowhere else: an uncorrelated factor leaves the policy unchanged to within solver noise, however volatile it is. The term’s share of the policy is governed by $\kappa T$, the number of times the opportunity set turns over inside the horizon, falling from thirty-six per cent to six across a factor of twenty-four — time-inconsistency costs something only when there is something to hedge over one’s horizon.

The leverage cap destroys that structure. The binding set has two components and occupies a third of the state space throughout the horizon, while the value it destroys falls by an order of magnitude; a mandate is therefore expensive in proportion to the runway it removes rather than to how often it binds. It also changes the policy in regions it never touches, by nearly four per cent at wealths well clear of the boundary. Simulated on common noise under the cap, the three strategies rank naive, pre-commitment, equilibrium — the reverse of what unconstrained optimality suggests, separated by tens of standard errors and by under one per cent of objective. A constraint charges a strategy in proportion to the leverage it wanted, and the most modest rule survives it best.