A general test for SSD portfolio efficiency

Abstract

We develop and implement a Linear Programming test to analyze whether a given investment portfolio is efficient in terms of second-order stochastic dominance relative to all possible portfolios formed from a set of base assets. In case of efficiency, the primal model identifies a sub-gradient vector of a utility function that rationalizes the evaluated portfolio. In case of inefficiency, the dual model identifies a second, efficient portfolio that dominates the evaluated portfolio. The test gives a general necessary and sufficient condition, and can deal with general linear portfolio restrictions, inefficiency degree measures, and scenarios with unequal probabilities. We also develop a compact version of the test that substantially reduces computational burden at the cost of losing information about the dual dominating portfolio in case of inefficiency. An application to US investment benchmark data qualifies a broad stock market index as significantly inefficient, and suggests that no risk-averse investor would hold the market index in the face of attractive premiums offered by some more concentrated investment portfolios.

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References

  1. Abdellaoui M, Barrios C, Wakker PP (2007) Reconciling introspective utility with revealed preference: experimental arguments based on prospect theory. J Econ 138(1):356–378

    Article  Google Scholar 

  2. Banz RW (1981) The relationship between return and market value of common stock. J Financ Econ 9:3–18

    Article  Google Scholar 

  3. Barrett G, Donald SG (2003) Consistent tests for stochastic dominance. Econometrica 71:71–104

    Article  Google Scholar 

  4. Bawa V, Lindenberg E (1977) Capital market equilibrium in a mean-lower partial moment framework. J Financ Econ 5:189–200

    Article  Google Scholar 

  5. Bawa VS, Bodurtha JN Jr, Rao MR, Suri HL (1985) On determination of stochastic dominance optimal sets. J Financ 40:417–431

    Article  Google Scholar 

  6. Dentcheva D, Ruszczyński A (2003) Optimization with stochastic dominance constraints. SIAM J Optim 14:548–566

    Article  Google Scholar 

  7. Dentcheva D, Ruszczyński A (2010) Inverse cutting plane methods for optimization problems with second-order stochastic dominance constraints. Optimization 59:323–338

    Article  Google Scholar 

  8. Dentcheva D, Martinez G (2012) Two-stage stochastic optimization problems with stochastic ordering constraints on the recourse. European J Oper Res 219:1–8

    Article  Google Scholar 

  9. Fábián CI, Mitra G, Roman D (2011) Processing second order stochastic dominance models using cutting plane representations. Math Program 130:33–37

    Article  Google Scholar 

  10. Gibbons MR, Ross SA, Shanken J (1989) A test of the efficiency of a given portfolio. Econometrica 57:1121–1152

    Article  Google Scholar 

  11. Hardy GH, Littlewood JE, Polya G (1934) Inequalities. Cambridge University Press, Cambridge

    Google Scholar 

  12. Harlow V, Rao R (1989) Asset pricing in a generalized mean-lower partial moment framework: theory and evidence. J Financ Quant Anal 24(3):285–311

    Article  Google Scholar 

  13. Hogan W, Warren J (1974) Toward the development of an equilibrium capital-market model based on semi-variance. J Financ Quant Anal 9(1):1–11

    Article  Google Scholar 

  14. Kopa M, Chovanec P (2008) A second-order stochastic dominance portfolio efficiency measure. Kybernetika 44(2):243–258

    Google Scholar 

  15. Kopa M, Post T (2009) A portfolio optimality test based on the first-order stochastic dominance criterion. J Financ Quant Anal 44(5):1103–1124

    Article  Google Scholar 

  16. Kuosmanen T (2004) Efficient diversification according to stochastic dominance criteria. Manag Sci 50(10):1390–1406

    Article  Google Scholar 

  17. Kuosmanen T, Post T (2002) Nonparametric efficiency analysis under price uncertainty: a first-order stochastic dominance approach. J Prod Anal 17(3):183–200

    Article  Google Scholar 

  18. Levy H (1985) Upper and lower bounds of put and call option value: stochastic dominance approach. J Financ 40(4):1197–1217

    Article  Google Scholar 

  19. Levy H (1992) Stochastic dominance and expected utility: survey and analysis. Manag Sci 38(4):555–593

    Article  Google Scholar 

  20. Levy H (2006) Stochastic dominance. Investment decision making under uncertainty. Springer, New York

    Google Scholar 

  21. Levy H, Hanoch G (1970) Relative effectiveness of efficiency criteria for portfolio selection. J Financ Quant Anal 5(1):63–76

    Article  Google Scholar 

  22. Linton O, Maasoumi E, Whang Y-J (2005) Consistent testing for stochastic dominance under general sampling schemes. Rev Econ Stud 72:735–765

    Article  Google Scholar 

  23. Linton O, Post T, Whang Y-J (2014) Testing for the stochastic dominance efficiency of a given portfolio. Econ J 17:S59–S74

    Google Scholar 

  24. Luedtke J (2008) New formulations for optimization under stochastic dominance constraints. SIAM J Optim 19:1433–1450

    Article  Google Scholar 

  25. Nelson RD, Pope R (1991) Bootstrapped insights into empirical applications of stochastic dominance. Manag Sci 37:1182–1194

    Article  Google Scholar 

  26. Post T (2003) Empirical tests for stochastic dominance efficiency. J Financ 58:1905–1932

    Article  Google Scholar 

  27. Post T, Kopa M (2013) General linear formulations of stochastic dominance criteria. Eur J Oper Res 230(2):321–332

    Article  Google Scholar 

  28. Post T, Fang Y, Kopa M (2014) Linear tests for DARA stochastic dominance. Forthcoming in Management Science.

  29. Post T, Versijp P (2007) Multivariate tests for stochastic dominance efficiency of a given portfolio. J Financ Quant Anal 42:489–515

    Article  Google Scholar 

  30. Post T, Van Vliet P, Levy H (2005) Risk aversion and skewness preference. J Bank Financ 185(3):1564–1573

    Google Scholar 

  31. Rockafellar RT, Uryasev S (2000) Optimization of conditional value-at-risk. J Risk 2(3):21–41

    Google Scholar 

  32. Roman D, Darby-Dowman K, Mitra G (2006) Portfolio construction based on stochastic dominance and target return distributions. Math Program 108:541–569

    Article  Google Scholar 

  33. Rubinstein M (1974) An aggregation theorem for security markets. J Financ Econ 1(3):225–244

    Article  Google Scholar 

  34. Yaari ME (1987) The dual theory of choice under risk. Econometrica 55:95–115

    Article  Google Scholar 

Download references

Acknowledgments

This work was partially supported by Czech Science Foundation (Grant P402/12/G097). Financial support by Koç University Graduate School of Business is gratefully acknowledged.

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Correspondence to Miloš Kopa.

Appendix

Appendix

Proof of Lemma 1:

\(\Leftarrow \)” We will first prove that a violation of Definition 1 implies a violation of Definition 1’. One possible violation occurs in the rare case when

$$\begin{aligned} \sum _{s=1}^{S}p_{s} \left( u(\mathbf {r}^{s}\varvec{\lambda }) - u(\mathbf {r}^{s}\varvec{\tau }) \right) = 0 \quad \forall u\in U_{2}. \end{aligned}$$

Since \(V_{2}\subset U_{2}\), this implies a violation of Definition 1’. We can therefore restrict attention to the more common cases with

$$\begin{aligned} \sum _{s=1}^{S}p_{s} \left( u(\mathbf {r}^{s}\varvec{\lambda }) - u(\mathbf {r}^{s}\varvec{\tau }) \right) = \nu < 0 \end{aligned}$$
(18)

for some \(u \in U_{2}\). Since \(clo(V_{2}) = U_{2}\), for every \(\varepsilon >\)0 and \(u\in U_{2}\) we find

$$\begin{aligned} | \sum _{s=1}^{S}p_{s} \left( u(\mathbf {r}^{s}\varvec{\lambda }) - u(\mathbf {r}^{s}\varvec{\tau }) \right) - \sum _{t=1}^{T} p_{s}\left( v(\mathbf {r}^{s}\varvec{\lambda }) - v(\mathbf {r}^{s}\varvec{\tau }) \right) | \le \varepsilon \end{aligned}$$
(19)

for some \(v \in V_{2}\). Choosing \(0<\varepsilon <-\nu \), (18) and (19) imply that

$$\begin{aligned} \sum _{t=1}^{T} p_{s}\left( v(\mathbf {r}^{s}\varvec{\lambda }) - v(\mathbf {r}^{s}\varvec{\tau }) \right) \le \varepsilon + \nu < 0 \end{aligned}$$

for some \(v \in V_{2}\), a violation of Definition 1’.

\(\Rightarrow \)” We will now prove that a violation of Definition 1’ implies a violation of Definition 1. The most common case is

$$\begin{aligned} \sum _{t=1}^{T} p_{s}\left( v(\mathbf {r}^{s}\varvec{\lambda }) - v(\mathbf {r}^{s}\varvec{\tau }) \right) < 0. \end{aligned}$$

for some \(v\in V_{2}\). Since \(V_{2}\subset U_{2}\), it is clear that Definition 1 is violated. What remains is the less common case with

$$\begin{aligned} \sum _{s=1}^{S}p_{s} \left( v(\mathbf {r}^{s}\varvec{\lambda }) - v(\mathbf {r}^{s}\varvec{\tau }) \right) = 0 \end{aligned}$$
(20)

for some \(v \in V_{2}\). This case arises naturally if \(\mathbf {r}^{s}\varvec{\lambda }= \mathbf {r}^{s}\varvec{\tau }\), \(s=1,2,...,S\), or, more generally, if the values for \(\mathbf {r}^{s}\varvec{\lambda }\) are a permutation of the values for \(\mathbf {r}^{s}\varvec{\tau }\), a situation when Definition 1 is clearly violated. In all other situations, choose a scenario \(t \in \{1,2,...,S\}\) such that \(\mathbf {r}^{t}\varvec{\lambda }\ne \mathbf {r}^{s}\varvec{\tau }\), \(s=1,2,...,S\). Consider a small neighborhood of \(\mathbf {r}^{t}\varvec{\lambda }\): \(\mathcal {N} = (\mathbf {r}^{t}\varvec{\lambda }- a, \mathbf {r}^{t}\varvec{\lambda }+ a)\), a \(>\) 0, such that \(\mathbf {r}^{s}\varvec{\tau }\notin \mathcal {N}\), \(s=1,2,...,S\) and \(\mathbf {r}^{s}\varvec{\lambda }\notin \mathcal {N}\) for all \(s\) satisfying \(\mathbf {r}^{s}\varvec{\lambda }\ne \mathbf {r}^{s}\varvec{\tau }\). For \(v\in V_{2}\) satisfying (20) let w(x) be equal to v(x) outside \(\mathcal {N}\) and a linear approximation of v(x) inside \(\mathcal {N}\):

$$\begin{aligned} w(x)&= v(x), \, x \notin \mathcal {N} \\&= v(\mathbf {r}^{t}\varvec{\lambda }- a) + \textstyle \frac{(x - \mathbf {r}^{t}\varvec{\lambda }+ a)}{2a}\left( v(\mathbf {r}^{t}\varvec{\lambda }+ a) - v(\mathbf {r}^{t}\varvec{\lambda }- a)\right) \!, \, x \in \mathcal {N}. \end{aligned}$$

By construction, \(w \in U_{2}\). Since v(x) is strictly concave, \(w(\mathbf {r}^{s}\varvec{\lambda }) < v(\mathbf {r}^{s}\varvec{\lambda })\) for all \(s\) satisfying \(\mathbf {r}^{s}\varvec{\lambda }= \mathbf {r}^{s}\varvec{\tau }\). Moreover, the definition of \(\mathcal {N}\) gives \(w(\mathbf {r}^{s}\varvec{\tau }) = v(\mathbf {r}^{s}\varvec{\tau })\), \(s=1,2,...,S\) and \(w(\mathbf {r}^{s}\varvec{\lambda }) = v(\mathbf {r}^{s}\varvec{\lambda })\) for all \(s\) that satisfy \(\mathbf {r}^{s}\varvec{\lambda }\ne \mathbf {r}^{s}\varvec{\tau }\). Hence,

$$\begin{aligned} \sum _{s=1}^{S}p_{s} \left( w(\mathbf {r}^{s}\varvec{\lambda }) - w(\mathbf {r}^{s}\varvec{\tau }) \right) < \sum _{s=1}^{S}p_{s} \left( v(\mathbf {r}^{s}\varvec{\lambda }) - v(\mathbf {r}^{s}\varvec{\tau }) \right) , \end{aligned}$$

and (20) implies

$$\begin{aligned} \sum _{s=1}^{S} p_{s}\left( w(\mathbf {r}^{s}\varvec{\lambda }) - w(\mathbf {r}^{s}\varvec{\tau }) \right) < 0, \end{aligned}$$

a violation of Definition 1.\(\square \)

Proof of Lemma 2:

\(\Rightarrow \)” We will first prove that SSD dominance requires \(\omega (\varvec{\lambda },\varvec{\tau }|\mathbf {w})>0\). Using Definition 1’ and (3), if portfolio \(\varvec{\lambda }\) does not dominate portfolio \(\varvec{\tau }\), then

$$\begin{aligned} T^{-1}\sum _{t=1}^{T} \left( v((X\varvec{\lambda })^{[t]}) - v(\mathbf {x}^{t}\varvec{\tau })\right) \le 0 \end{aligned}$$
(21)

for some \(v\in V_2\). Using (6), we can assume a strictly positive lower bound \(\varepsilon >0\) for the sub-gradient decrements: \( v'(z_{t}) - v'(z_{t+1}) > \varepsilon \), \(t=1,2,...,T-1\) and \(v'(z_{T}) >0\). Let \(\vartheta = \max _{t}w_{t}/\varepsilon \) and \(g(x) = \vartheta v(x)\). For this function, we have \( g'(z_{t}) - g'(z_{t+1}) > \max _{t}w_{t} \) and \(g'(z_{T}) >\max _{t}w_{t}\) which obeys (7.1) and (7.2) and therefore \( g'(z_{t}) \), \(t=1,2,...,T\) is a feasible solution of (7). Using (5) and (21), we find a non-positive value for the objective function (7) when \(\beta _t = g'(z_{t})\), \(t=1,2,...,T\):

$$\begin{aligned} T^{-1}\sum _{t=1}^{T} \left( g'(z_{t})((X\varvec{\lambda })^{[t]} - \mathbf {x}^{t}\varvec{\tau })\right) = T^{-1}\sum _{t=1}^{T} \left( g((X\varvec{\lambda })^{[t]}) - g(\mathbf {x}^{t}\varvec{\tau })\right) \le 0. \end{aligned}$$

\(\Leftarrow \)” We will now prove that SSD relation implies \(\omega (\varvec{\lambda },\varvec{\tau }|\mathbf {w})>0\).

If \(\omega (\varvec{\lambda },\varvec{\tau }|\mathbf {w})\le 0\) then

$$\begin{aligned} T^{-1}\sum _{t=1}^{T} \beta _{t}\left( (X\varvec{\lambda })^{[t]} - \mathbf {x}^{t}\varvec{\tau }\right) \le 0 \end{aligned}$$

for some \(\beta _{t}\), \(t=1,2,...,T\) satisfying (7.1) and (7.2). Consider \(\alpha _t = \beta _{t}\) if \(t \notin \Theta (\varvec{\lambda },\varvec{\tau })\) and

$$\begin{aligned} \alpha _t = \textstyle \frac{\sum ^{t_{2}+1}_{t=t_{1}}\beta _{t}}{t_{2}-t_{1}} \end{aligned}$$

for \(t\in \Theta (\varvec{\lambda },\varvec{\tau })\) and \(t_{1} = \min _{s} \{s\in \Theta (\varvec{\lambda },\varvec{\tau }) : (X\varvec{\lambda })^{[s]} = (X\varvec{\lambda })^{[t]} \wedge \mathbf {x}^{s}\varvec{\tau }= \mathbf {x}^{t}\varvec{\tau }\}\), \(t_{2} = \max _{s}\{s\in \Theta (\varvec{\lambda },\varvec{\tau }):(X\varvec{\lambda })^{[s]} = (X\varvec{\lambda })^{[t]} \wedge \mathbf {x}^{s}\varvec{\tau }= \mathbf {x}^{t}\varvec{\tau }\}\). By construction, \(\alpha _{t}\) obeys the required ranking of (6) and

$$\begin{aligned} T^{-1}\sum _{t=1}^{T} \alpha _{t}\left( (X\varvec{\lambda })^{[t]} - \mathbf {x}^{t}\varvec{\tau }\right) = T^{-1}\sum _{t=1}^{T} \beta _{t}\left( (X\varvec{\lambda })^{[t]} - \mathbf {x}^{t}\varvec{\tau }\right) . \end{aligned}$$
(22)

Therefore, there exists a utility function \(v\in V_{2}\) and points \(z_t =\varphi _{t}(X\varvec{\lambda })^{[t]} + (1-\varphi _{t})\mathbf {x}^{t}\varvec{\tau }\), \(0\le \varphi _{t} \le 1\), \(t=1,2,...,T\), such that \(v'(z_{t}) = \alpha _t\) satisfy (5) and (6), and, in addition

$$\begin{aligned} T^{-1}\sum _{t=1}^{T} \left( v((X\varvec{\lambda })^{[t]}) - v(\mathbf {x}^{t}\varvec{\tau })\right) = T^{-1}\sum _{t=1}^{T} \alpha _{t}\left( (X\varvec{\lambda })^{[t]} - \mathbf {x}^{t}\varvec{\tau }\right) \le 0 \end{aligned}$$

that is, portfolio \(\varvec{\lambda }\) does not dominate portfolio \(\varvec{\tau }\). The last inequality follows from (22). \(\square \)

Proof of Theorem 1:

Let \(\zeta (\varvec{\tau }|\mathbf {w}) = \max _{\varvec{\lambda }\in \Lambda }\omega (\varvec{\lambda },\varvec{\tau }|\mathbf {w})\). Lemma 2 implies that portfolio \(\varvec{\tau }\) is efficient if and only if \(\zeta (\varvec{\tau }|\mathbf {w}) =0\). Our strategy is to prove that \(\zeta (\varvec{\tau }|\mathbf {w}) = \xi (\varvec{\tau }|\mathbf {w})\). \(\zeta (\varvec{\tau }|\mathbf {w})\) has a max–min structure that minimizes expected utility over utility for every portfolio. An alternative min–max formulation maximizes expected utility over portfolios for every utility function.

$$\begin{aligned} \zeta (\varvec{\tau }|\mathbf {w})&= \min _{\gamma _{t,s}} \max _{\varvec{\lambda }\in \Lambda } T^{-1} \sum \limits ^{T}_{t=1} \sum \limits ^{T}_{s=1}\gamma _{t,s}\mathbf {x}^{t}\varvec{\lambda }- T^{-1}\sum \limits ^{T}_{t=1} \sum \limits ^{T}_{s=t}\textstyle \frac{1}{s}\sum ^{T}_{k=1} \gamma _{k,s}\mathbf {x}^{t}\varvec{\tau }\nonumber \\&\mathrm {s.t.} \ \ (8.1),\, (8.2),\,(3.3). \end{aligned}$$
(23)

Both formulations are equivalent because the optimization is over two convex sets; see Sion (1958). Since the portfolio set (1) is a polytope, the embedded LP maximization over the portfolio space

$$\begin{aligned} \max _{\varvec{\lambda }}&T^{-1}\sum ^{T}_{t=1} \sum ^{T}_{s=1}\gamma _{t,s}\mathbf {x}^{t}\varvec{\lambda }\\ \mathrm {s.t.}&\sum ^{N}_{j=1}a_{ij}\lambda _{j}\le b_{i}, \ \ i=1,2,...,M \nonumber \end{aligned}$$

involves the following LP dual minimization problem:

$$\begin{aligned} \min _{\theta _{s}}&\sum ^{M}_{i=1}b_{i}\theta _{i} \\ \mathrm {s.t.}&\sum ^{M}_{i=1}a_{ij}\theta _{i} - T^{-1}\sum ^{T}_{t=1} \sum ^{T}_{s=1}\gamma _{t,s}x_{j}^{t} = 0, \ \ j=1,2,...,N \nonumber \\&\theta _{i} \ge 0, \ \ i=1,2,...,M. \nonumber \end{aligned}$$

Making the substitution in (23) gives (9).\(\Box \)

Proof of Theorem 2:

The cumulative returns \(\Omega (\varvec{\lambda },s)\) can be expressed as follows:

$$\begin{aligned} \Omega (\varvec{\lambda },s) = \min _{\alpha _{k}}T^{-1}\sum _{k=1}^{T}\alpha _{k}\mathbf {x}^{k}\varvec{\lambda }\\ \mathrm {s.t.} \ \ \sum _{t=1}^{T}\alpha _{t} = s \\ 0 \le \alpha _{t}\le 1,&t = 1,2,...,T \end{aligned}$$

The optimal values \(\alpha _{k}^{*}\) are equal to one if \(\mathbf {x}^{k}\varvec{\lambda }\le (X\varvec{\lambda })^{[s]}\) and are equal to zero otherwise. Taking the dual of this program, we find

$$\begin{aligned} \Omega (\varvec{\lambda },s) = \max _{z_{s},\nu _{t,s}}\left( sz_{s} - \sum _{t=1}^{T}\nu _{t,s} \right) \\ \mathrm {s.t.} \ \ z_{s} - \nu _{t,s} \le T^{-1}\mathbf {x}^{t}\varvec{\lambda }&t = 1,2,...,T \nonumber \\ \nu _{t,s} \ge 0&t = 1,2,...,T. \nonumber \end{aligned}$$
(24)

Substituting (24) for each s into (13), we find:

$$\begin{aligned} \xi (\varvec{\tau }|\mathbf {w}) = \max _{d_{s},z_{s},v_{t,s},\varvec{\lambda }} \sum ^{T}_{s=1}w_{s}d_{s}&\\ \mathrm {s.t.} \ \ sz_{s} - \sum ^{T}_{t=1}v_{t,s} -T^{-1}\sum _{k=1}^{s}\mathbf {x}^{k}\varvec{\tau }&\ge d_{s}, \ \ s=1,2,...,T \nonumber \\ z_{s}- v_{t,s} -T^{-1}\mathbf {x}^{t}\varvec{\lambda }&\le 0, \ \ t,s = 1,2,...,T \ \ \nonumber \\ v_{t,s}&\ge 0, \ \ t,s = 1,2,...,T \ \ \nonumber \\ d_{s}&\ge 0, \ \ s = 1,2,...,T \ \ \nonumber \\ \varvec{\lambda }&\in \Lambda .\nonumber \end{aligned}$$
(25)

Since we maximize the weighted average of \(d_{s}\), the first set of inequality constraints must be fulfilled as equations. Hence,

$$\begin{aligned} z_{s} = \textstyle \frac{1}{s}\sum ^{T}_{t=1}v_{t,s} + T^{-1}\textstyle \frac{1}{s}\sum _{k=1}^{s}\mathbf {x}^{k}\varvec{\tau }+ \textstyle \frac{1}{s}d_{s}, \ \ s=1,2,...,T. \end{aligned}$$

Substituting these equalities into the second set of inequality constraints, we find (10). \(\square \)

Proof of Theorem 3:

Theorem 1 implies that a given portfolio \(\varvec{\tau }\) is SSD efficient if and only if \(\xi (\varvec{\tau }|\mathbf {w})=0\). If \(\xi (\varvec{\tau }|\mathbf {w})>0\), then Theorem 2 shows that solution portfolio \(\varvec{\lambda }^{*} \) SSD dominates \(\varvec{\tau }\). Since \(\varvec{\lambda }^{*} \) maximizes the weighted sum of cumulative returns among all dominating portfolios, using Definition 1”, \(\varvec{\lambda }^{*} \) is SSD efficient.\(\Box \)

Proof of Theorem 4:

To simplify the proof, we say that portfolio \(\varvec{\lambda }\) is co-monotone with portfolio \(\varvec{\tau }\) if

$$\begin{aligned} (\mathbf {x}^{t_{1}}\varvec{\lambda }- \mathbf {x}^{t_{2}}\varvec{\lambda })(\mathbf {x}^{t_{1}}\varvec{\tau }- \mathbf {x}^{t_{2}}\varvec{\tau }) \ge 0 \quad \forall t_{1},t_{2}\in \{1,2,...,T\}. \end{aligned}$$

This definition means that the ranking of \(X\varvec{\lambda }\) may differ from the ranking of \(X\varvec{\tau }\) only inside \(\Phi _k\).

If portfolio \(\varvec{\tau }\) is SSD inefficient then there exists a portfolio \(\varvec{\lambda }\in \Lambda \) that SSD dominates \(\varvec{\tau }\). Moreover, every portfolio \(\bar{\varvec{\lambda }}(\alpha ) = (1-\alpha )\varvec{\tau }+ \alpha \varvec{\lambda }\), \(\alpha \in [0,1]\) must also SSD dominate \(\varvec{\tau }\). Furthermore, for sufficiently small \(\alpha \in [0,1]\) a portfolio \(\bar{\varvec{\lambda }}(\alpha ) = (1-\alpha )\varvec{\tau }+ \alpha \varvec{\lambda }\) is co-monotone with portfolio \(\varvec{\tau }\). Hence, there exists a portfolio \(\bar{\varvec{\lambda }}(\alpha )\) such that \(\bar{\omega }(\bar{\varvec{\lambda }}(\alpha ),\varvec{\tau }|\mathbf {w}) > 0\), where

$$\begin{aligned}&\bar{\omega }(\bar{\varvec{\lambda }}(\alpha ),\varvec{\tau }|\mathbf {w}) = \min _{\beta _{t}}T^{-1}\sum _{t=1}^{T} \beta _{t}\left( \mathbf {x}^{t}\bar{\varvec{\lambda }}(\alpha ) - \mathbf {x}^{t}\varvec{\tau }\right) \\ \mathrm {s.t.}&\beta _{t} - \beta _{s} \ge w_{k}, \ \ \ \ t\in \Phi _k, \ \ s\in \Phi _{k+1}, \ \ k = 1,2,...,K-1 \nonumber \\&\beta _{t} \ge w_{K}, \ \ \ \ t\in \Phi _K \nonumber \end{aligned}$$
(26)

Since both \(\mathbf {x}^{t}\bar{\varvec{\lambda }}(\alpha )\) and \(\mathbf {x}^{t}\varvec{\tau }\) take the same values for all \(t \in \Phi _k^{l_k}\), we may replace the variables \(\beta _{t}, t \in \Phi _k^{l_k}\) by a single variable \(\beta _k^{l_k}\) and rewrite (26) as follows:

$$\begin{aligned}&\bar{\omega }(\bar{\varvec{\lambda }}(\alpha ),\varvec{\tau }|\mathbf {w}) = \min _{ \beta _{k}^{l_k}} T^{-1}\sum ^{K}_{k=1}\sum ^{L_k}_{l_k=1}\beta _k^{l_k} \sum _{t \in \Phi _k^{l_k}} \left( \mathbf {x}^{t}\bar{\varvec{\lambda }}(\alpha ) - \mathbf {x}^{t}\varvec{\tau }\right) \\ \mathrm {s.t.}&\beta _{k}^{l_k} - \beta _{k+1}^{l_{k+1}} \ge w_{k}, \ \ k = 1,...,K-1, \ \ l_k = 1,...,L_k,\ \ l_{k+1} = 1,...,L_{k+1} \nonumber \\&\beta _{K}^{l_K} \ge w_{K}, \ \ l_K = 1,...,L_K \nonumber \end{aligned}$$

This means that \(\kappa (\varvec{\tau }|\mathbf {w})>0\), where

$$\begin{aligned}&\kappa (\varvec{\tau }|\mathbf {w}) = \min _{ \beta _{k}^{l_k}}\max _{\varvec{\lambda }\in \Lambda }T^{-1}\sum ^{K}_{k=1}\sum ^{L_k}_{l_k=1}\beta _k^{l_k} \sum _{t \in \Phi _k^{l_k}} \left( \mathbf {x}^{t}\varvec{\lambda }- \mathbf {x}^{t}\varvec{\tau }\right) \\ \mathrm {s.t.}&\beta _{k}^{l_k} - \beta _{k+1}^{l_{k+1}} \ge w_{k}, \ \ k = 1,...,K-1, \ \ l_k = 1,...,L_k, \ \ l_{k+1} = 1,...,L_{k+1} \nonumber \\&\beta _{K}^{l_K} \ge w_{K}, \ \ l_K = 1,...,L_K \nonumber \end{aligned}$$

We used the Minimax theorem to reverse the order of the two optimization operations. Similar to the proof of Theorem 1, taking the dual to the embedded maximization problem

$$\begin{aligned} \max _{\varvec{\lambda }\in \Lambda }T^{-1}\sum ^{K}_{k=1}\sum ^{L_k}_{l_k=1}\beta _k^{l_k} \sum _{t \in \Phi _k^{l_k}}\mathbf {x}^{t}\varvec{\lambda }\end{aligned}$$

we find that \(\upsilon (\varvec{\tau }|\mathbf {w}) = \kappa (\varvec{\tau }|\mathbf {w}) >0\).

If portfolio \(\varvec{\tau }\) is SSD efficient then no co-monotone dominating portfolio exists and hence, \(\kappa (\varvec{\tau }|\mathbf {w}) \le 0\). Since \(\kappa (\varvec{\tau }|\mathbf {w})\) is always non-negative, \(\upsilon (\varvec{\tau }|\mathbf {w})=\kappa (\varvec{\tau }|\mathbf {w}) = 0\).\(\Box \)

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Kopa, M., Post, T. A general test for SSD portfolio efficiency. OR Spectrum 37, 703–734 (2015). https://doi.org/10.1007/s00291-014-0373-8

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Keywords

  • Stochastic dominance
  • Portfolio analysis
  • Market portfolio efficiency
  • Linear programming