## Robust portfolios and weak incentives in long-run investments.(English)Zbl 1414.91339

The authors consider an agent who invests in assets $$S$$, thereby affecting total wealth $$X$$, so as to maximize expected utility from terminal wealth $$X_{T}$$ at time $$T$$: $$\max_{X}{\mathbf E}[U(X_{T})]$$. There are $$d+1$$ assets available. A safe asset, with price $$S^0$$, and $$d$$ risky assets, with price $$S=(S^1,\ldots,S^{d})$$. These assets are traded continuously, without frictions and no arbitrage opportunities are available. If $$x$$ denotes the initial capital, and $$(\phi^{i}_{t})_{0\leq t\leq T}^{1\leq i\leq d}$$ the number of shares of the $$i$$th asset at time $$t$$, the corresponding wealth equals $$X_{t}^{\phi}=S^0_{t}\left(x+\int_0^{t}\phi_{s}d(S_{s}/S^0_{s})\right)$$. The agent’s objective is described by a utility function $$U$$, which incorporates the combined effect of preferences and incentives. $$\tilde U$$ is the isoelastic utility function defined by $$\tilde U(x)=x^{p}/p$$, $$0\neq p<1$$, $$\tilde U(x)=\log x$$ for $$p=0$$. The main result is following. Let the safe asset $$S^0$$ is a deterministic, strictly positive function satisfying $$S^0_{t}\uparrow \infty$$ as $$t\uparrow\infty$$, the discount prices $$S/S_0$$ of the risky assets are local martingale under some probability. $$U(x)$$ is strictly increasing, concave, not necessarily differentiable or strictly concave for low wealth levels, but differentiable and strictly concave for large enough wealth, $$\lim_{x\uparrow\infty}U'(x)/\tilde U'(x)=1$$. For $$0<p<1$$, $$U$$ is bounded from below, for $$p=0$$: $$\lim\inf_{x\downarrow 0}U(x)/\tilde U'(x)>-\infty$$, for $$p<0$$ $$\lim_{x\uparrow\infty}U(x)=0$$ and $$\lim\inf_{x\downarrow 0}U(x)/\tilde U'(x)>-\infty$$. Let the isoelastic utility maximization problem is well posed. Then for any horizon $$T>0$$ there exist an optimal payoff $$X_{T}^{T}$$ for the generic utility $$U$$ and $${\tilde X}_{T}^{T}$$ for the isoelastic utility $$\tilde U$$. They satisfy $$\lim_{T\to\infty}U^{-1}({\mathbf E}[U({\tilde X}_{T}^{T})])/U^{-1}({\mathbf E}[U({X}_{T}^{T})])=1$$. As a consequence, optimal portfolios are robust to the perturbations in preferences induced by common option compensation schemes.

### MSC:

 91G10 Portfolio theory 91B16 Utility theory
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