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Benchmark-based evaluation of portfolio performance: a characterization

Abstract

Benchmarking is a universal practice in portfolio management and is well-studied in the optimal portfolio selection literature. This paper derives axiomatic foundations of the relative return, which underlies a benchmark-based evaluation of portfolio performance. We show that the existence of a benchmark naturally arises from a few basic axioms and is tightly linked to the economic theory. Our method relies on the use of both axiomatic and economic approaches to index number theory. We also analyze the problem of optimal portfolio selection under complete uncertainty about a future price system, where the objective function is the relative return.

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Notes

  1. 1.

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

  2. 2.

    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).

  3. 3.

    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.

  4. 4.

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

  5. 5.

    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.

  6. 6.

    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).

  7. 7.

    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 \(\hbox {X}\).

  8. 8.

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

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Acknowledgements

The work is partially supported by the Russian Foundation for Basic Research (RFBR), Project 14-06-00347. The authors are grateful to Drs. J.W. Kolari and T. Swarthout for valuable suggestions and corrections on earlier drafts of this paper.

Author information

Correspondence to Mikhail V. Sokolov.

Appendix: Proofs

Appendix: Proofs

Proof of Proposition 1

(a)\(\Rightarrow \)(b). Let \(f(\varvec{p}):=\bar{{\varvec{p}}}\cdot \varvec{x}(\varvec{p})\) be the value function of a self-financing portfolio \(\varvec{x}\). Since \(f(\varvec{p})-{\varvec{p}}\cdot \nabla f(\varvec{p})=x_{n+1} (\varvec{p})\ge 0\) a.e., (5) holds. Let us show that f is non-decreasing. By (2) for almost all \(p_2 ,\ldots ,p_n \) the function \(f( \cdot ,p_2 ,\ldots ,p_n )\) has a.e. non-negative derivative. The restriction of \(f( \cdot ,p_2 ,\ldots ,p_n )\) to an arbitrary closed bounded interval is globally Lipschitz and hence absolutely continuous. An absolutely continuous function with a.e. non-negative derivative is non-decreasing. Therefore, \(f( \cdot ,p_2 ,\ldots ,p_n )\) is non-decreasing for almost all \(p_2 ,\ldots ,p_n \), and, thus, for all \(p_2 ,\ldots ,p_n \), since f is continuous.

(b)\(\Rightarrow \)(c). Let \(\hbox {G}\) be the set of non-differentiability points of f and let \(1_{\mathrm{G}} \) be the characteristic function of \(\hbox {G}\). By Rademacher’s theorem, \(\hbox {G}\) is a set of Lebesgue measure zero. Converting to polar coordinates in the Lebesgue integral \(\int \nolimits _{\mathrm{P}} {1_{\mathrm{G}} (\varvec{p})d{\varvec{p}}} =0\) yields that for almost all \(\varvec{p}\) the function f is differentiable a.e on the ray \(\{t{\varvec{p}}:t\in \hbox {R}_{++} \}\). Put \(g_{\varvec{p}} (t):=\ln f(t{\varvec{p}})\). By (5), for almost all \(\varvec{p}\)

$$\begin{aligned} {g}'_{\varvec{p}} (t)=\frac{p\cdot \nabla f(t{\varvec{p}})}{f(t{\varvec{p}})}\le 1/t\hbox { a.e}. \end{aligned}$$
(25)

The restriction of the function \(g_{\varvec{p}} \) to the interval \([1,\alpha ]\) is absolutely continuous. Integrating the inequality (25) over the interval \([1,\alpha ]\) yields (6).

  • (c)\(\Leftrightarrow \)(d). Proved in Rubinov (2000, theorem 3.1, p. 80).

  • (d)\(\Rightarrow \)(e). If \(\bar{{f}}\) is non-decreasing in its arguments, then

    $$\begin{aligned} \frac{f({\varvec{{p}}'})}{f(\varvec{p})}=\frac{\bar{{f}}({\bar{{\varvec{p}}}}')}{\bar{{f}}(\bar{{\varvec{p}}})}\le \frac{\bar{{f}}\left( {\max \left( {\frac{{\bar{{\varvec{p}}}}'}{\bar{{\varvec{p}}}}} \right) \bar{{\varvec{p}}}} \right) }{\bar{{f}}(\bar{{\varvec{p}}})}=\max \left( {\frac{{\bar{{\varvec{p}}}}'}{\bar{{\varvec{p}}}}} \right) . \end{aligned}$$
  • (e)\(\Rightarrow \)(a). A positive function f, satisfying (7), is subhomogeneous,

    $$\begin{aligned} f(\alpha {\varvec{p}})\le \max \left( {\frac{(\alpha {\varvec{p}},1)}{\bar{{\varvec{p}}}}} \right) f(\varvec{p})=\alpha f(\varvec{p}), \, \alpha >1, \end{aligned}$$

and is non-decreasing: if \({\varvec{p}}-{\varvec{{p}}'}\in \hbox {R}_+^n \) then

$$\begin{aligned} f({\varvec{{p}}'})\le \max \left( {\frac{{\bar{{\varvec{p}}}}'}{\bar{{\varvec{p}}}}} \right) f(\varvec{p})=f(\varvec{p}). \end{aligned}$$

Thus, (c) and (d) hold. A function f satisfying (c) is continuous Rubinov (2000, p. 78). Let \(\hbox {B}\subset \hbox {R}_{++}^n \) be a closed ball in \(\hbox {R}^{n}\). For any \({\varvec{p}},{\varvec{{p}}'}\in \hbox {B}\) we have

$$\begin{aligned} \left| {f({\varvec{{p}}'})-f(\varvec{p})} \right| \le \max \left( {\frac{\left| {{\varvec{{p}}'}-{\varvec{p}}} \right| }{{\varvec{p}}}} \right) f(\varvec{p})\le \max (\left| {{\varvec{{p}}'}-{\varvec{p}}} \right| )\frac{f(\varvec{p})}{\min (\varvec{p})}. \end{aligned}$$

Since the function \({\varvec{p}}\mapsto {f(\varvec{p})}/{\min (\varvec{p})}\) is continuous and thus bounded on \(\hbox {B},\,f\) is locally Lipschitz.

Put \({\varvec{x}}(\varvec{p}):=\nabla \bar{{f}}(\bar{{\varvec{p}}})\) if f is differentiable at \(\varvec{p}\), and \(x_i (\varvec{p}):=0,\,i=1,\ldots ,n,\,x_{n+1} (\varvec{p}):=f(\varvec{p})\) otherwise. Then \(\bar{{\varvec{p}}}\cdot \varvec{x}(\varvec{p})\equiv f(\varvec{p})\) and (2) holds. Since \(\bar{{f}}\) is non-decreasing, \({\varvec{x}}(\varvec{p})\in \hbox {X}\). \(\square \)

Proof of Theorem 1

(a)\(\Rightarrow \)(b). From internality (i) it follows that \(I({\varvec{x}},{\varvec{p}},t;{\varvec{x}},{\varvec{p}},{t}')=1\). Applying (ii) with \(({\varvec{x}},{\varvec{p}})=({\varvec{{x}}'},{\varvec{{p}}'})\) and with \(({\varvec{{x}}'},{\varvec{{p}}'})=({\varvec{x}''},{\varvec{{p}}''})\), we obtain that for each fixed \({\varvec{{x}}}\), \({\varvec{{p}}}\), \({\varvec{x}'}\), \({\varvec{{p}}'}\) the value \(I^{{t}'-t}({\varvec{x}},{\varvec{p}},t;{\varvec{{x}}'},{\varvec{{p}}'},{t}')\) is independent of t and \({t}'\). Define \(J({\varvec{x}},{\varvec{p}};{\varvec{{x}}'},{\varvec{{p}}'}):=I^{{t}'-t}({\varvec{x}},{\varvec{p}},t;{\varvec{{x}}'},{\varvec{{p}}'},{t}')\). By (ii), J satisfies the Sincov functional equation, \(J({\varvec{z}};{{\varvec{z}}}')J({{\varvec{z}}}';{{\varvec{z}}}'')=J({\varvec{z}};{{\varvec{z}}}''),\,{\varvec{z}},{{\varvec{z}}}',{{\varvec{z}}}''\in \hbox {X}\times \hbox {P}\). Its general solution is given by (Aczél 1966, section 8.1.3)

$$\begin{aligned} J({\varvec{x}},{\varvec{p}};{\varvec{{x}}'},{\varvec{{p}}'})=\frac{g({\varvec{{x}}'},{\varvec{{p}}'})}{g({\varvec{x}},{\varvec{p}})}, \end{aligned}$$

where g is an arbitrary positive function. Applying (i) with \({\varvec{{p}}'}={\varvec{p}}\), we get

$$\begin{aligned} \frac{g({\varvec{{x}}'},{\varvec{p}})}{g({\varvec{x}},{\varvec{p}})}=J({\varvec{x}},\varvec{p};{\varvec{{x}}'},{\varvec{p}})=I^{{t}'-t}({\varvec{x}},{\varvec{p}},t;{\varvec{{x}}'},{\varvec{p}},{t}')=\frac{\bar{{\varvec{p}}}\cdot {\varvec{{x}}'}}{\bar{{\varvec{p}}}\cdot \varvec{x}}. \end{aligned}$$
(26)

Since (26) holds for any \(\varvec{x}\) and \({\varvec{{x}}'}\), there exists a function \(f\in \hbox {F}\) such that \(g({\varvec{x}},{\varvec{p}})={(\bar{{\varvec{p}}}\cdot \varvec{x})}/{f(\varvec{p})}\). By (i), \(f\in \hbox {S}\).

(b)\(\Rightarrow \)(a). Straightforward. \(\square \)

Proof of Theorem 2

Let \(I=I_f ,\,f\in \hbox {S}\).

(a)\(\Rightarrow \)(d). Applying (iii) with \({\varvec{x}}=1/{\bar{{\varvec{p}}}},\,{\varvec{{x}}'}=1/{{\bar{{\varvec{p}}}}'},\,{t}'-t=1\), we obtain the multiplicative Cauchy functional equation with respect to the function f:

$$\begin{aligned} \frac{f(\varvec{p})}{f({\varvec{{p}}'})}=G\left( {\frac{{\varvec{p}}}{{\varvec{{p}}'}}} \right) . \end{aligned}$$

Its general positive non-decreasing solution is given by \(f(\varvec{p})=c\prod \nolimits _{i=1}^n {p_i^{\lambda _i } } \) (Aczél 1966, section 8.1.1), where \(c>0,\,{\varvec{\lambda }} =(\lambda _1 ,\ldots ,\lambda _n )\in \hbox {R}_+^n \). Since f is subhomogeneous, \({\varvec{\lambda }} \in \Delta ^{n}\).

(b)\(\Rightarrow \)(d). From (iv) with \(({\varvec{\alpha }} ,\alpha _{n+1} )=\bar{{\varvec{p}}}\), we again obtain the multiplicative Cauchy functional equation with respect to f.

(c)\(\Rightarrow \)(d). By (v), for any \(i\in \{1,\ldots ,n\}\) the ratio \({f(\varvec{p})}/{f(1,{\varvec{p}}_{-i} )}\) is a function of only \(p_i \). Thus, there exist positive functions \(f_i ,\,i=1,\ldots ,n\) such that \(f(\varvec{p})=\prod \nolimits _{i=1}^n {f_i (p_i )} \). From (v) with \(x_{n+1} ={{x}'_{n+1}} =0\) it follows that the ratio \({f(\alpha {\varvec{p}})}/{f(\varvec{p})}\) is a function of only \(\alpha \) and, therefore, there is a constant \(\lambda \ge 0\) such that \({f(\alpha {\varvec{p}})}/{f(\varvec{p})}=\alpha ^{\lambda }\). Then

$$\begin{aligned} \prod _{i=1}^n {f_i (\alpha p_i )} =f(\alpha {\varvec{p}})=\alpha ^{\lambda }f(\varvec{p})=\alpha ^{\lambda }\prod _{i=1}^n {f_i (p_i )} \end{aligned}$$

and \({f_i (\alpha p_i )}/{f_i (p_i )}\) is again a function of only \(\alpha \). Thus, there exist constants \(c_i >0\) and \(\lambda _i \ge 0\) such that \(f_i (p_i )=c_i p_i^{\lambda _i } \). \((\lambda _1 ,\ldots ,\lambda _n )\in \Delta ^{n}\) since \(f\in \hbox {S}\).

(d)\(\Rightarrow \)(a), (b), (c). Trivial. \(\square \)

Proof of Theorem 3

(a)\(\Rightarrow \)(b). We have

$$\begin{aligned} ({\varvec{x}},{\varvec{p}},t;{\varvec{{x}}'},{\varvec{{p}}'},{t}')&\sim (\mathbf{0}_n \hbox {,}\bar{{\varvec{p}}}\cdot \varvec{x},{\varvec{p}},t;\mathbf{0}_n \hbox {,}{\bar{{\varvec{p}}}}'\cdot {\varvec{{x}}'},{\varvec{{p}}'},{t}') \\&\sim \left( {\mathbf{0}_n \hbox {,1,}{\varvec{p}},0;\mathbf{0}_n \hbox {,}\frac{{\bar{{\varvec{p}}}}'\cdot {\varvec{{x}}'}}{\bar{{\varvec{p}}}\cdot \varvec{x}},{\varvec{{p}}'},{t}'-t} \right) , \end{aligned}$$

where the first equality follows from (VI) with \({\varvec{y}}=(\mathbf{0}_n ,\bar{{\varvec{p}}}\cdot \varvec{x})\) and \({\varvec{y}'}=(\mathbf{0}_n \hbox {,}{\bar{{\varvec{p}}}'}\cdot {\varvec{{x}}'})\), while the second one follows from (III) and (IV).

Let q be the function that takes each vector \((d,{\varvec{p}},{\varvec{{p}}'},\tau )\in \hbox {R}_{++} \times \hbox {P}^{2}\times \hbox {R}_{++} \) to a solution \(x\in \hbox {R}_{++} \) of the equation

$$\begin{aligned} (\mathbf{0}_n ,1,{\varvec{p}},0;\mathbf{0}_n ,x,{\varvec{{p}}'},\tau ) \sim (\mathbf{0}_n ,1,\mathbf{1}_n ,0;\mathbf{0}_n ,d,\mathbf{1}_n ,1). \end{aligned}$$

By (I) (parts (A) and (C)) and (V) the function q is well defined and continuous by the implicit function theorem for continuous maps (Kumagai 1980).

From (II) it follows that

$$\begin{aligned} q(d,{\varvec{p}},{\varvec{{p}}'},\tau )q(d,{\varvec{{p}}'},{\varvec{{p}}''},{\tau }')=q(d,{\varvec{p}},{\varvec{{p}}''},\tau +{\tau }'). \end{aligned}$$
(27)

Equation (27) is a combination of the Cauchy and Sincov functional equations (Aczél 1966, sections 2.1.2, 8.1.3). Its general solution that is continuous in the last argument is given by

$$\begin{aligned} q(d,{\varvec{p}},{\varvec{{p}}'},\tau )=\alpha (d)^{\tau }\frac{\beta (d,{\varvec{{p}}'})}{\beta (d,{\varvec{p}})} \end{aligned}$$

for some functions \(\alpha :\hbox {R}_{++} \rightarrow \hbox {R}_{++} \) and \(\beta :\hbox {R}_{++} \times \hbox {P}\rightarrow \hbox {R}_{++} \). By the definition of \(q, q(d,\mathbf{1}_n ,\mathbf{1}_n ,1)=d\) and, hence, \(\alpha \) is the identity function. Let us prove that for every fixed \({\varvec{p}},{\varvec{{p}}'}\in \hbox {P}\) the function \(d\mapsto {\beta (d,{\varvec{{p}}'})}/{\beta (d,{\varvec{p}})}\) is identically equal to a constant. Indeed, for any \(d<{d}'\) we have

$$\begin{aligned} d^{\tau }\frac{\beta (d,{\varvec{{p}}'})}{\beta (d,{\varvec{p}})}=q(d,{\varvec{p}},{\varvec{{p}}'},\tau )<q({d}',{\varvec{p}},{\varvec{{p}}'},\tau )={d}'^{\tau }\frac{\beta ({d}',{\varvec{{p}}'})}{\beta ({d}',{\varvec{p}})}. \end{aligned}$$
(28)

Tending \(\tau \rightarrow 0+\) in (28), we obtain

$$\begin{aligned} \frac{\beta (d,{\varvec{{p}}'})}{\beta (d,{\varvec{p}})}\le \frac{\beta ({d}',{\varvec{{p}}'})}{\beta ({d}',{\varvec{p}})} \end{aligned}$$
(29)

The inequality (29) holds for all \({\varvec{p}},{\varvec{{p}}'}\in \hbox {P}\) if and only if it holds with equality.

Put \(f(\varvec{p}):=\beta (1,{\varvec{p}})\). Then (I) (part (B)) implies \(f\in \hbox {S}\). Now (b) follows from the definition of the function q.

(b)\(\Rightarrow \)(a). Straightforward. \(\square \)

Proof of Theorem 4

(a)\(\Rightarrow \)(b). By Theorem 3, there exists \(f\in \hbox {S}\) such that \(I_f\) represents \(\succeq \). From (VII) with \({\varvec{x}}={{\varvec{x}}'}={\varvec{y}}={\varvec{y}}'=(\mathbf{0}_n ,1),\,\alpha _{n+1} =1\), and \({t}'-t={\tau }'-\tau =1\) it follows that

$$\begin{aligned} \frac{f({\varvec{{p}}'})}{f(\varvec{p})}\ge \frac{f({\varvec{q}}')}{f({\varvec{q}})} \Rightarrow \frac{f({\varvec{{\alpha p}}'})}{f({\varvec{\alpha p}})}\ge \frac{f({\varvec{\alpha }}{{\varvec{q}}}')}{f({\varvec{\alpha }} {\varvec{q}})} \end{aligned}$$
(30)

for any \({\varvec{p}},{\varvec{{p}}'},{\varvec{q}},{{\varvec{q}}}',{\varvec{\alpha }} \in \hbox {P}\). The general positive and continuous solution of the system of functional inequalities (30) is given by (Aczél 1990, corollary 10)

$$\begin{aligned} f(\varvec{p})= & {} \exp \left( {a\prod _{i=1}^n {p_i ^{\lambda _i }} +b} \right) \hbox { and}\nonumber \\ f(\varvec{p})= & {} c\prod _{i=1}^n {p_i ^{\lambda _i }} , \end{aligned}$$
(31)

where \(a,\,b,\,c>0,\,{\varvec{\lambda }} =(\lambda _1 ,\ldots ,\lambda _n )\) are some constants. The inclusion \(f\in \hbox {S}\) holds only for the solution (31) with \({\varvec{\lambda }} \in \Delta ^{n}\).

(b)\(\Rightarrow \)(a). Trivial. \(\square \)

Proof of Proposition 2

(a)\(\Rightarrow \)(b). The identities \(A_u^{(\varvec{p}_{0} )} =I_f \) and \(e\left( {\varvec{p}_{0} ,v(\varvec{p}_{0} ,M)} \right) =M\) (Shah 2007, theorem 4.3) imply \(e\left( {\varvec{p}_{0} ,v({\varvec{p}},M)} \right) ={Mf(\varvec{p}_{0} )}/{f(\varvec{p})}\). Since \(e(\varvec{p}_{0} , \cdot )\) is non-decreasing and continuous (Shah 2007, theorem 2.4) and \(v({\varvec{p}}, \cdot )\) is upper semicontinuous (Shah 2007, theorem 3.2), we have

$$\begin{aligned} v({\varvec{p}},M)=g\left( {\frac{M}{f(\varvec{p})}} \right) , \end{aligned}$$

where the function \(g(z):=\sup \{y:e(\varvec{p}_{0} ,y)\le f(\varvec{p}_{0} )z\}\) is strictly increasing, right-continuous, and \(g(0)=0\). Since an indirect utility function is quasi-convex (Shah 2007, theorem 3.2), the function \(\bar{{f}}\), defined by (3), is quasi-concave and hence (since \(\bar{{f}}\) is homogeneous) concave. Therefore, \(f\in \hbox {K}\). Given the indirect utility function v, there is a unique direct utility function \(u\in \hbox {U}\) that induces v (Shah 2007, theorem 3.2), namely \(u(\varvec{x})=\sup \bigcap \nolimits _{\varvec{p}\in \mathrm{P}} {\{y:v({\varvec{p}},\bar{{\varvec{p}}}\cdot \varvec{x})\ge y\}} \). We get

$$\begin{aligned} u(\varvec{x})= & {} \sup \bigcap _{\varvec{p}\in \mathrm{P}} {\{y:v({\varvec{p}},\bar{{\varvec{p}}}\cdot \varvec{x})\ge y\}}\nonumber \\= & {} \sup \left\{ {\hbox {R}\backslash \bigcup _{\varvec{p}\in \mathrm{P}} {\{y:v({\varvec{p}},\bar{{\varvec{p}}}\cdot \varvec{x})<y\}} } \right\} =\inf \left\{ {\bigcup _{\varvec{p}\in \mathrm{P}} {\{y:v({\varvec{p}},\bar{{\varvec{p}}}\cdot \varvec{x})<y\}} } \right\} \nonumber \\= & {} \mathop {\inf }\limits _{\varvec{p}\in \mathrm{P}} \inf \{y:v({\varvec{p}},\bar{{\varvec{p}}}\cdot \varvec{x})<y\}=\mathop {\inf }\limits _{\varvec{p}\in \mathrm{P}} v({\varvec{p}},\bar{{\varvec{p}}}\cdot \varvec{x})=\mathop {\inf }\limits _{\varvec{p}\in \mathrm{P}} g\left( {\frac{\bar{{\varvec{p}}}\cdot \varvec{x}}{f(\varvec{p})}} \right) \nonumber \\= & {} g\left( {\mathop {\inf }\limits _{\varvec{p}\in \mathrm{P}} \frac{\bar{{\varvec{p}}}\cdot \varvec{x}}{f(\varvec{p})}} \right) , \end{aligned}$$
(32)

where we use the facts that \(\bigcup \nolimits _{\varvec{p}\in \mathrm{P}} {\{y:v({\varvec{p}},\bar{{\varvec{p}}}\cdot \varvec{x})<y\}} \) is the set of the form \((a,+\infty ),\,g\) is strictly increasing and right-continuous.

(b)\(\Rightarrow \)(c). The function \(\tilde{u}\) defined in (17) is homogeneous, unbounded above, concave, and, therefore, continuous, \(\tilde{u}(\mathbf{0}_{n+1} )=0\). Since u is strictly increasing, then so is \(\tilde{u}\). Thus, \(\tilde{u}\in \hbox {U}\) and \(u\in \tilde{\hbox {U}}\). For any \(\varvec{p}\) the function

$$\begin{aligned} ({\varvec{x}},{\varvec{{p}}'})\mapsto \frac{1}{f({\varvec{{p}}'})}\frac{{\bar{{\varvec{p}}}}'\cdot \varvec{x}}{\bar{{\varvec{p}}}\cdot \varvec{x}} \end{aligned}$$

is quasi-concave with respect to \(\varvec{x}\) for each fixed \({\varvec{{p}}'}\) and quasi-convex with respect to \({\varvec{{p}}'}\) for each fixed \(\varvec{x}\) (since \(f\in \hbox {K})\). By the minimax theorem (Sion 1958, corollary 3.3),

$$\begin{aligned} \mathop {\sup }\limits _{\varvec{x}\in \mathrm{R}_{++}^{n+1} } \frac{\tilde{u}(\varvec{x})}{\bar{{\varvec{p}}}\cdot \varvec{x}}= & {} \mathop {\sup }\limits _{\varvec{x}\in \mathrm{X}} \frac{\tilde{u}(\varvec{x})}{\bar{{\varvec{p}}}\cdot \varvec{x}}=\mathop {\sup }\limits _{\varvec{x}\in \mathrm{X}:\left\| {\varvec{x}} \right\| =1} \frac{\tilde{u}(\varvec{x})}{\bar{{\varvec{p}}}\cdot \varvec{x}}=\mathop {\sup }\limits _{\varvec{x}\in \mathrm{X}:\left\| {\varvec{x}} \right\| =1} \mathop {\inf }\limits _{{\varvec{{p}}'}\in \mathrm{P}} \frac{1}{f({\varvec{{p}}'})}\frac{{\bar{{\varvec{p}}}}'\cdot \varvec{x}}{\bar{{\varvec{p}}}\cdot \varvec{x}}\nonumber \\= & {} \mathop {\inf }\limits _{{\varvec{{p}}'}\in \mathrm{P}} \mathop {\sup }\limits _{\varvec{x}\in \mathrm{X}:\left\| {\varvec{x}} \right\| =1} \frac{1}{f({\varvec{{p}}'})}\frac{{\bar{{\varvec{p}}}}'\cdot \varvec{x}}{\bar{{\varvec{p}}}\cdot \varvec{x}}=\mathop {\inf }\limits _{{\varvec{{p}}'}\in \mathrm{P}} \frac{1}{f({\varvec{{p}}'})}\max \left( {\frac{{\bar{{\varvec{p}}}}'}{\bar{{\varvec{p}}}}} \right) =\frac{1}{f(\varvec{p})},\nonumber \\ \end{aligned}$$
(33)

where \(\left\| { \cdot } \right\| \) is the Euclidean norm on \(\hbox {R}^{n+1}\). The first equality in (33) follows from continuity and homogeneity of \(\tilde{u}\) and the last equality follows from part (e) of Proposition 1. Identity (33) implies (18).

(c)\(\Rightarrow \)(a). Let \(v\, (\mathrm{resp.}~\tilde{v})\) and \(e\, (\mathrm{resp.}~\tilde{e})\) be the indirect utility function and the expenditure function generated by \(u\, (\mathrm{resp.}~\tilde{u})\). Since \(\tilde{u}\) is homogeneous, \(\tilde{v}({\varvec{p}},M)=M/{f(\varvec{p})}\) and \(\tilde{e}({\varvec{p}},y)=f(\varvec{p})y\), where f is defined in (18). Then \(e\left( {\varvec{p}_{0} ,v({\varvec{p}},\bar{{\varvec{p}}}\cdot \varvec{x})} \right) =\tilde{e}\left( {\varvec{p}_{0} ,\tilde{v}({\varvec{p}},\bar{{\varvec{p}}}\cdot \varvec{x})} \right) ={(\bar{{\varvec{p}}}\cdot \varvec{x})f(\varvec{p}_{0} )}/{f(\varvec{p})}\) and \(I_f =A_u^{(\varvec{p}_{0} )} \) for any \(\varvec{p}_{0} \). \(\square \)

Proof of Theorem 5

Let \(f\in \hbox {K}\). Then the function \(\tilde{u}\) defined in (17) is concave and hence continuous. \(\tilde{u}(\mathbf{0}_{n+1} )=0\) and \(\tilde{u}\) is positive on \(\hbox {R}_{++}^{n+1} \):

$$\begin{aligned} \tilde{u}(\varvec{x})=\mathop {\inf }\limits _{\varvec{p}\in \mathrm{P}} \frac{\bar{{\varvec{p}}}\cdot \varvec{x}}{f(\varvec{p})}\ge \mathop {\inf }\limits _{\varvec{p}\in \mathrm{P}} \frac{\bar{{\varvec{p}}}\cdot \varvec{x}}{f(\mathbf{1}_n )\max (\bar{{\varvec{p}}})}=\frac{\min (\varvec{x})}{f(\mathbf{1}_n )}>0,\, \varvec{x}\in \hbox {R}_{++}^{n+1} , \end{aligned}$$
(34)

where the first inequality is implied by \(f\in \hbox {S}\). \(\tilde{u}\) is homogeneous and hence unbounded above. Finally, \(\tilde{u}\) is strictly increasing: for any \({\varvec{z}}\in \hbox {R}_{++}^{n+1} \)

$$\begin{aligned} \tilde{u}({\varvec{x}}+{\varvec{z}})=\mathop {\inf }\limits _{\varvec{p}\in \mathrm{P}} \frac{\bar{{\varvec{p}}}\cdot ({\varvec{x}}+{\varvec{z}})}{f(\varvec{p})}\ge \tilde{u}(\varvec{x})+\tilde{u}({\varvec{z}})>\tilde{u}(\varvec{x}). \end{aligned}$$

Therefore, \(\tilde{u}\in \hbox {U}\) and, by Proposition 2, \(I_f =A_{\tilde{u}}^{(\varvec{p}_{0} )} \) for any \(\varvec{p}_{0} \in \hbox {P}\). Since the expenditure function generated by \(\tilde{u}\) is given by \(e({\varvec{p}},y)=f(\varvec{p})y,\,I_f =KP_{\tilde{u}}^{(u_0 )} \) for any \(u_0 \in \hbox {R}_{++} \).

Now let \(\varvec{p}_{0} \in \hbox {P},\,u\in \tilde{\hbox {U}}\), and let g and \(\tilde{u}\in \hbox {U}\) be a strictly increasing function and a homogeneous utility function such that \(u=g\circ \tilde{u}\). Then f defined by (18) is positive by arguments similar to those used in (34). Thus, by Proposition 2, \(f\in \hbox {K}\) and \(A_u^{(\varvec{p}_{0} )} =I_f \).

Finally, given \(u\in \hbox {U}\) and \(u_0 \in \hbox {R}_{++} \), put \(f(\varvec{p}):=e({\varvec{p}},u_0 )\), where e is the expenditure function generated by u. Then \(KP_u^{(u_0 )} =I_f \) and, by the regularity properties of an expenditure function (Shah 2007, theorem 2.4), \(f\in \hbox {K}\). \(\square \)

Proof of Proposition 3

Given \(I=I_f \in \hbox {I}(\hbox {S})\), assume that the generalized product test holds. Define the vector-valued functions \({\varvec{g}}=(g_1 ,\ldots ,g_{n+1} ):\hbox {X}^{2}\rightarrow \hbox {X}\) and \(\bar{{\varvec{h}}}=(\bar{{h}}_1 ,\ldots ,\bar{{h}}_{n+1} ):\hbox {P}^{2}\rightarrow \hbox {P}\times \{1\}\) by \({\varvec{g}}({\varvec{x}},{\varvec{{x}}'}):={\varvec{x}}\circ {\varvec{{x}}'},\,\bar{{\varvec{h}}}({\varvec{p}},{\varvec{{p}}'}) :=({\varvec{p}}{\bullet }{\varvec{{p}}'},1)\).

From the generalized product test with \({t}'-t=1\), we have

$$\begin{aligned} \left( {{\frac{{\bar{{\varvec{p}}}}'\cdot {\varvec{g}}({\varvec{x}},{\varvec{{x}}'})}{f({\varvec{{p}}'})}}\Big /{\frac{\bar{{\varvec{p}}}\cdot {\varvec{g}}({\varvec{x}},{\varvec{{x}}'})}{f(\varvec{p})}}} \right) {*}\left( {\frac{\bar{{\varvec{h}}}({\varvec{p}},{\varvec{{p}}'})\cdot {\varvec{{x}}'}}{\bar{{\varvec{h}}}({\varvec{p}},{\varvec{{p}}'})\cdot \varvec{x}}} \right) ={\frac{{\bar{{\varvec{p}}}}'\cdot {\varvec{{x}}'}}{f({\varvec{{p}}'})}}\Big /{\frac{\bar{{\varvec{p}}}\cdot \varvec{x}}{f(\varvec{p})}}. \end{aligned}$$
(35)

Let \(\mathbf{1}^{j}:=(0,\ldots ,0,1,0,\ldots ,0)\) be the \((n+1)\)-dimensional vector, where 1 stands at the j-th position. Put \(\hbox {Y}:=\ln f(\hbox {P})\). Since \(f\in \hbox {S}\) and, hence, continuous, \(\hbox {Y}\) is a (possibly unbounded) real interval. We consider two cases:

  1. 1.

    \(\hbox {Y}\) is unbounded. Setting \({\varvec{x}}=\mathbf{1}^{n+1},\,{\varvec{x}}'=(0,\ldots ,0,{{{x}}'})\) in (35), we obtain

    $$\begin{aligned} \left( {\frac{f(\varvec{p})}{f({\varvec{{p}}'})}} \right) {*}{x}'=\frac{f(\varvec{p})}{f({\varvec{{p}}'})}{x}'. \end{aligned}$$

    Since \(\hbox {Y}\) is unbounded, the function \(({\varvec{p}},{\varvec{{p}}'})\mapsto {f(\varvec{p})}/{f({\varvec{{p}}'})}\) is onto \(\hbox {R}_{++} \).

  2. 2.

    \(\hbox {Y}\) is bounded. Setting in (35) \({\varvec{{p}}'}=\alpha {\varvec{p}},\,{\varvec{x}}=(x_1 ,\ldots ,x_n ,0),\,{\varvec{{x}}'}=\beta {\varvec{x}}\), where \(\alpha \) and \(\beta \) are positive scalars, we have

    $$\begin{aligned} \left( {\alpha \frac{f(\varvec{p})}{f(\alpha {\varvec{p}})}} \right) {*}\beta =\alpha \beta \frac{f(\varvec{p})}{f(\alpha {\varvec{p}})}. \end{aligned}$$

    Since \(\hbox {Y}\) is bounded, the function \((\alpha ,{\varvec{p}})\mapsto {\alpha f(\varvec{p})}/{f(\alpha {\varvec{p}})}\) is onto \(\hbox {R}_{++} \).

Therefore, the binary operation \({*}\) defined on \(I_f (\hbox {V})=\hbox {R}_{++} \) is multiplication and the identity (35) takes the form

$$\begin{aligned} \frac{{\bar{{\varvec{p}}}}'\cdot {\varvec{g}}({\varvec{x}},{\varvec{{x}}'})}{\bar{{\varvec{p}}}\cdot {\varvec{g}}({\varvec{x}},{\varvec{{x}}'})}\frac{\bar{{{\varvec{h}}}}({\varvec{p}},{\varvec{{p}}'})\cdot {\varvec{{x}}'}}{\bar{{{\varvec{h}}}}({\varvec{p}},{\varvec{{p}}'})\cdot \varvec{x}}=\frac{{\bar{{\varvec{p}}}}'\cdot {\varvec{{x}}'}}{\bar{{\varvec{p}}}\cdot \varvec{x}}. \end{aligned}$$
(36)

Setting in (36) \({\varvec{x}}=\mathbf{1}^{i},\,{\varvec{{x}}'}=\mathbf{1}^{j}\) and \({\varvec{x}}=\mathbf{1}^{j},\,{\varvec{{x}}'}=\mathbf{1}^{i},\,i\ne j\), we get

$$\begin{aligned} \frac{\bar{p}'_i g_i (\mathbf{1}^{i},\mathbf{1}^{j})+\bar{p}'_j g_j (\mathbf{1}^{i},\mathbf{1}^{j})}{\bar{p}_i g_i (\mathbf{1}^{i},\mathbf{1}^{j})+\bar{p}_j g_j (\mathbf{1}^{i},\mathbf{1}^{j})}\frac{\bar{{h}}_j ({\varvec{p}},{\varvec{{p}}'})}{\bar{{h}}_i ({\varvec{p}},{\varvec{{p}}'})}= & {} \frac{\bar{p}'_j }{\bar{p}_i },\, \frac{\bar{p}'_i g_i (\mathbf{1}^j ,\mathbf{1}^i )+\bar{p}'_j g_j (\mathbf{1}^j ,\mathbf{1}^i )}{\bar{p}_i g_i (\mathbf{1}^j ,\mathbf{1}^i )+\bar{p}_j g_j (\mathbf{1}^j ,\mathbf{1}^i )}\frac{\bar{{h}}_i ({\varvec{p}},{\varvec{{p}}'})}{\bar{{h}}_j ({\varvec{p}},{\varvec{{p}}'})}\nonumber \\= & {} \frac{\bar{p}'_i }{\bar{p}_j }. \end{aligned}$$
(37)

Combining the equalities (37), we obtain

$$\begin{aligned}&\frac{\left( {\bar{p}_i g_i (\mathbf{1}^{i},\mathbf{1}^{j})+\bar{p}_j g_j (\mathbf{1}^{i},\mathbf{1}^{j})} \right) \left( {\bar{p}_i g_i (\mathbf{1}^{j},\mathbf{1}^{i})+\bar{p}_j g_j (\mathbf{1}^{j},\mathbf{1}^{i})} \right) }{\bar{p}_i \bar{p}_j} \nonumber \\&\quad =\frac{\left( {{\bar{p}}'_i g_i (\mathbf{1}^{i},\mathbf{1}^{j})+{\bar{p}}'_j g_j (\mathbf{1}^{i},\mathbf{1}^{j})} \right) \left( {{\bar{p}}'_i g_i (\mathbf{1}^{j},\mathbf{1}^{i})+{\bar{p}}'_j g_j (\mathbf{1}^{j},\mathbf{1}^{i})} \right) }{{\bar{p}}'_i {\bar{p}}'_j }. \end{aligned}$$
(38)

The equality (38) holds for any \(\varvec{p}\) and \({\varvec{{p}}'}\) if and only if either \(g_i (\mathbf{1}^{i},\mathbf{1}^{j})=g_j (\mathbf{1}^{j},\mathbf{1}^{i})=0\) and \(g_j (\mathbf{1}^{i},\mathbf{1}^{j})>0,\,g_i (\mathbf{1}^{j},\mathbf{1}^{i})>0\) or \(g_j (\mathbf{1}^{i},\mathbf{1}^{j})=g_i (\mathbf{1}^{j},\mathbf{1}^{i})=0\) and \(g_i (\mathbf{1}^{i},\mathbf{1}^{j})>0,\,g_j (\mathbf{1}^{j},\mathbf{1}^{i})>0\). If the former alternative holds then \({\bar{{h}}_i ({\varvec{p}},{\varvec{{p}}'})}/{\bar{{h}}_j ({\varvec{p}},{\varvec{{p}}'})}={\bar{p}_i }/{\bar{p}_j }\); if the latter one holds then \({\bar{{h}}_i ({\varvec{p}},{\varvec{{p}}'})}/{\bar{{h}}_j ({\varvec{p}},{\varvec{{p}}'})}={{\bar{p}}'_i }/{{\bar{p}}'_j }\). Thus, since \(\bar{{h}}_{n+1} ({\varvec{p}},{\varvec{{p}}'})=\bar{p}_{n+1} ={\bar{p}}'_{n+1} =1\), we get that either \(\bar{{{\varvec{h}}}}({\varvec{p}},{\varvec{{p}}'}) =({\varvec{p}},1)\) or \(\bar{\varvec{h}}({\varvec{p}},{\varvec{{p}}'})=({\varvec{{p}}'},1)\) hold.

If \(\bar{{{\varvec{h}}}}({\varvec{p}},{\varvec{{p}}'})=({\varvec{p}},1)\) (the arguments in the case of \(\bar{{{\varvec{h}}}}({\varvec{p}},{\varvec{{p}}'})=({\varvec{{p}}'},1)\) are similar), then (36) reduces to

$$\begin{aligned} \frac{{\bar{{\varvec{p}}}}'\cdot {\varvec{g}}({\varvec{x}},{\varvec{{x}}'})}{{\bar{{\varvec{p}}}}'\cdot {\varvec{{x}}'}}=\frac{\bar{{\varvec{p}}}\cdot {\varvec{g}}({\varvec{x}},{\varvec{{x}}'})}{\bar{{\varvec{p}}}\cdot {\varvec{{x}}'}}. \end{aligned}$$
(39)

The identity (39) holds for any \(\varvec{p}\) and \({\varvec{{p}}'}\) if and only if \({\varvec{g}}({\varvec{x}},{\varvec{{x}}'})=\alpha ({\varvec{x}},{\varvec{{x}}'}){\varvec{{x}}'}\) for some function \(\alpha :\hbox {X}^{2}\rightarrow \hbox {R}_{++} \).

The converse implication is trivial. \(\square \)

Proof of Proposition 5

If \(\hbox {G}\) is a cone, then the map \(g\mapsto \ln g\) defines a one-to-one correspondence between \(\hbox {G}\cap \hbox {C}(f,{\varvec{p}},M)\) and \(\partial _{\ln \hbox {G}} \ln f(\varvec{p})\) modulo “differ by a constant” relation (two supergradients are considered equivalent if they are differ by a constant). If \(\hbox {G}\) is a cone closed under vertical shifts, then the map \(g\mapsto M^{-1}f(\varvec{p})g\) is a bijection between \(\hbox {G}\cap \hbox {C}(f,{\varvec{p}},M)\) and \(\partial _{\mathrm{G}} f(\varvec{p})\) modulo “differ by a constant” relation.

To establish a one-to-one correspondence between \(\hbox {G}\cap \hbox {C}(f,{\varvec{p}},M)\) and \(\partial _{\bar{\mathrm{G}}} \bar{{f}}(\bar{{\varvec{p}}})\) we prove that \(\bar{{g}}\) is a \(\bar{\hbox {G}}\)-supergradient of a function \(\bar{{f}}\) at a point \(\bar{{\varvec{p}}}\) if and only if \(g(\varvec{p})=f(\varvec{p})\) and \(g({\varvec{{p}}'})\ge f({\varvec{{p}}'})\) for all \({\varvec{{p}}'}\in \hbox {P}\). Indeed, if \(\bar{{g}}\) is a \(\bar{\hbox {G}}\)-supergradient of \(\bar{{f}}\) at \(\bar{{\varvec{p}}}\), then \(p_{n+1}^\prime g({\varvec{{p}}'})-g(\varvec{p})\ge p_{n+1}^\prime f({\varvec{{p}}'})-f(\varvec{p})\) holds for all \({\varvec{{p}}'}\in \hbox {P}\) and \({p}_{n+1}^\prime \in \hbox {R}_{++} \). Passing to the limit as \({p}_{n+1}^\prime \rightarrow 0+\), we get \(g(\varvec{p})\le f(\varvec{p})\). On the other hand, \(g({\varvec{{p}}'})-f({\varvec{{p}}'})\ge {(g(\varvec{p})-f(\varvec{p}))}/{{p}_{n+1}^\prime }\). Passing to the limit as \({{p}'}_{n+1} \rightarrow +\infty \), we obtain \(g({\varvec{{p}}'})\ge f({\varvec{{p}}'})\) for all \({\varvec{{p}}'}\in \hbox {P}\). The converse implication is obvious. \(\square \)

Proof of Proposition 6

(a)\(\Rightarrow \)(b). Let \(\varvec{x}^{*}\in \hbox {X}\) be the structure of a constant portfolio conservative with respect to \(I_f ,\,{\varvec{p}},\,M\) and let \(\tilde{v}({\varvec{p}},M)=M/{f(\varvec{p})}\) be the indirect utility function generated by \(\tilde{u}\) (17). Then \(\varvec{x}^{*}\in \hbox {B}({\varvec{p}},M)\) and

$$\begin{aligned} \tilde{v}({\varvec{p}},\bar{{\varvec{p}}}\cdot \varvec{x}^{*})\le \tilde{v}({\varvec{{p}}'},{\bar{{\varvec{p}}}}'\cdot \varvec{x}^{*})\hbox { for all }{\varvec{{p}}'}\in \hbox {P}. \end{aligned}$$

From (32) it follows that

$$\begin{aligned} \tilde{v}({\varvec{p}},M)=\tilde{v}({\varvec{p}},\bar{{\varvec{p}}}\cdot \varvec{x}^{*})=\mathop {\inf }\limits _{{\varvec{{p}}'}\in \mathrm{P}} \tilde{v}({\varvec{{p}}'},{\bar{{\varvec{p}}}}'\cdot \varvec{x}^{*})=\tilde{u}(\varvec{x}^{*}); \end{aligned}$$

thus \(\varvec{x}^{*}\) is a solution of the utility maximization problem \(\left\langle {\tilde{u},{\varvec{p}},M} \right\rangle \).

(b)\(\Rightarrow \)(a). Let \(\varvec{x}^{*}\) be a solution of the problem \(\left\langle {\tilde{u},{\varvec{p}},M} \right\rangle \). Since \(\varvec{x}^{*}\in \hbox {B}({\varvec{{p}}'},{\bar{{\varvec{p}}}}'\cdot \varvec{x}^{*})\) for any \({\varvec{{p}}'}\in \hbox {P}\), then

$$\begin{aligned} \tilde{v}({\varvec{p}},\bar{{\varvec{p}}}\cdot \varvec{x}^{*})=\tilde{v}({\varvec{p}},M)=\tilde{u}(\varvec{x}^{*})\le \tilde{v}({\varvec{{p}}'},{\bar{{\varvec{p}}}}'\cdot \varvec{x}^{*})\hbox { for all }{\varvec{{p}}'}\in \hbox {P}. \end{aligned}$$

(c)\(\Leftrightarrow \)(a), (a)\(\Leftrightarrow \)(d). The assertions follow from Propositions 4 and 5.

Since the sets \(\partial f(\varvec{p})\) and \(\hbox {L}\cap \hbox {C}(f,{\varvec{p}},M)\) are equivalent (the map (23) is a bijection), \(\hbox {L}\cap \hbox {C}(f,{\varvec{p}},M)\) is a singleton if and only if f is differentiable at \(\varvec{p}\) (Niculescu and Persson 2006, theorem 3.8.2, p. 136). Equality (24) now follows from part (d). \(\square \)

Proof of Proposition 7

All the assertions are the direct consequences of the equality

$$\begin{aligned} \mathop {\sup }\limits _{g\in \mathrm{G}} \frac{g({\varvec{{p}}'})}{g(\varvec{p})}=\max \left( {\frac{{\bar{{\varvec{p}}}}'}{\bar{{\varvec{p}}}}} \right) , \end{aligned}$$

which follows from part (e) of Proposition 1. \(\square \)

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Alekseev, A.G., Sokolov, M.V. Benchmark-based evaluation of portfolio performance: a characterization. Ann Finance 12, 409–440 (2016). https://doi.org/10.1007/s10436-016-0286-4

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Keywords

  • Portfolio performance
  • Compound annual growth rate
  • Benchmarking
  • Index number theory
  • Portfolio choice under uncertainty

JEL Classification

  • C43
  • G11