Benchmark-based evaluation of portfolio performance: a characterization

See Siegel (2003) for an extensive summary of the history and practice of benchmarking in the US.

This measure gives rise to the tracking problem: replicating the benchmark as closely as possible using a small number of assets (Barro and Canestrelli 2009; Beasley et al. 2003; Gilli and Këllezi 2002).

Earlier papers on benchmarks are concerned with the problem of maximizing expected utility subject to the constraint that wealth level does not fall below a deterministic threshold (Basak 1995; Grossman and Zhou 1996; Cox and Huang 1989). Later papers consider the case with a stochastic benchmark. Browne (1999) was probably one of the first to look at various performance goals associated with a stochastic benchmark. Teplá (2001) studies a more standard problem of maximizing expected utility subject to the constraint that a wealth level does not fall below a stochastic benchmark. Basak et al. (2006) relax the benchmark constraint by allowing for a given probability of a shortfall. Davis and Lleo (2008) consider a risk-sensitive control problem, in which the investor’s risk aversion enters the objective function directly. The benchmark portfolio approach to stochastic finance (Platen and Heath 2006) argues to use the “best” performing (growth optimal) portfolio as a benchmark.

As opposed to the absolute return that does not take benchmarks into account.

Recall that, according to Rademacher’s theorem (Niculescu and Persson 2006, theorem 3.11.1, p. 151), the differential of a locally Lipschitz function is defined a.e.

Note that the approximation is biased in the presence of probabilistic uncertainty as usually modeled in mathematical finance (e.g. see Brennan and Schwartz 1985 for a detailed analysis of the case of continuously rebalanced equally weighted portfolios).

Indeed, since \(u(\varvec{x})=\max \{y:({\varvec{x}},y)\in \hbox {cl}(\hbox {Hyp } u)\}\), then the hypograph of the extension is \(\hbox {cl}(\hbox {Hyp } u)\) and, therefore, the extension is upper semicontinuous. The extension is quasi-concave since the closure of the convex set \(\{\varvec{x}\in \hbox {X}:u(\varvec{x})\ge y\}\) is convex. Finally, if \(u(\varvec{x})\le u(\mathbf{0}_{n+1} )\) for some \(\varvec{x}\in \hbox {R}_{++}^{n+1} \), then, by quasi-concavity, \(u(\lambda {\varvec{x}})=u(\lambda {\varvec{x}}+(1-\lambda )\mathbf{0}_{n+1} )\ge \min \{u(\varvec{x}),u(\mathbf{0}_{n+1} )\}=u(\varvec{x}),\,\lambda \in (0, 1)\) which contradicts property (A) on the set \(\hbox {X}\).

Recall that \(\hbox {B}({\varvec{p}},M)\) is an investor’s budget set.