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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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