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Diversified minimum-variance portfolios

Abstract

We build on a one parameter family of weighting schemes arising from \(L^2\)-constrained portfolio optimization problems. The parameter allows to fine tune the trade-off between the volatility and the diversification of the portfolio. We propose two criteria in order to determine two unique portfolios: the first criterion requires that no weights be negative while the second one imposes a target diversification which is median between full concentration and full diversification. Both portfolios are empirically compared to classical benchmarks. The first one behaves very much like other popular Long-Only weighting schemes while the second displays a more aggressive profile, while generating moderate turnover. We also discuss implementation issues, as well as estimation related problems.

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Fig. 1

Notes

  1. See for instance Klein and Bawa (1976), Jobson and Korkie (1980), Merton (1980), Jorion (1985), Jorion (1991), Best and Grauer (1991), Chopra and Ziemba (1993), Britten-Jones (1999) and Kondor et al. (2007).

  2. The literature on norm constraints in portfolio optimization is rapidly growing. To cite but a few of the latest references: Gotoh and Takeda (2011), Fan et al. (2012), Behr et al. (2013) and Lin (2013).

  3. This is also true for many other Long-Only weighting schemes which allocate non-zero weights to most, if not all, constituents: for instance, the ERC of Maillard et al. (2010), the inverse volatility portfolio, the inverse variance portfolio, among others.

  4. Green and Hollifield (1992), in their article on the diversification of optimal mean-variance portfolios, use the \(L^1\) norm. In this article, diversification is only assessed through the concentration of portfolio weights. Whether it is the best way to assess diversification is clearly out of our scope.

  5. A more technical explanation of this fact is provided in “Appendix A”.

  6. Apart from the S&P500, we have chosen the same universes as in DeMiguel et al. (2009).

  7. These three estimators are not linked by the rotation-equivariant property described in Sect. 4 but are simply well known benchmarks for covariance matrix estimation.

  8. This value can only be reached asymptotically when the leverage goes to infinity, that is, when the magnitude of some positive and negative weights increases to infinity. If the weights are positive, then the minimum value is \(1/N\).

  9. Note that this is not always true, since the lowest turnovers are usually achieved by cap-weighted portfolios which are rather concentrated. However, in the case of norm-constrained portfolios, this is often verified. Indeed, the principal source of turnover is the change in covariance matrix (rather than changes in asset prices). In the remark after the proof of Theorem 1, we argue that tight constraints - and hence strong diversification - lead to a lower sensitivity of weights to variations in the covariance structure.

  10. Given the first half of the proof, it suffices to replace all the values \(\varLambda _{k,k}\) or \(\varLambda _{i,i}\) which are numerators by 1, so that \(\varvec{w'\varSigma w}\) is replaced by \(\varvec{w'I_Nw}=\varvec{w'w}\).

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Correspondence to Guillaume Coqueret.

Appendices

Appendix A: The formal proof of why \(\gamma ^{LO}\) exists and is unique

The main difficulty in (3) is the matrix inversion but it can be handled in the following way. We factorize \(\varvec{\varSigma }\) using its eigendecomposition: \(\varvec{\varSigma =P\varLambda P'}\). Basic algebra then implies

$$\begin{aligned} \varvec{Q}=(\varvec{\varSigma }+\gamma \varvec{I_N})^{-1}=\varvec{P}(\varvec{\varLambda }+\gamma \varvec{I_N})^{-1}\varvec{P'}, \end{aligned}$$
(6)

where the inverse matrix is diagonal with values \((\varLambda _{i,i}+\gamma )^{-1}\) for \(i=1,\ldots ,N\). We use the standard notation \(M_{i,j}\) for the elements of the matrix \(\varvec{M}\). Further computations lead to:

$$\begin{aligned} Q_{i,j} = \sum _{k=1}^N P_{i,k} (\varLambda _{k,k}+\gamma )^{-1} P_{j,k}. \end{aligned}$$

Therefore, summing the lines of \(\varvec{Q}\), we see that the weights in (3) are proportional to

$$\begin{aligned} \sum _{j=1}^N Q_{i,j}=\sum _{j=1}^N \sum _{k=1}^N P_{i,k} (\varLambda _{k,k}+\gamma )^{-1} P_{j,k}, \quad i=1,\ldots ,N. \end{aligned}$$

Because we are interested in Long-Only portfolios, we require that

$$\begin{aligned} \sum _{j=1}^N \sum _{k=1}^N P_{i,k} (\varLambda _{k,k}+\gamma )^{-1} P_{j,k} \ge 0, \quad i=1,\ldots ,N. \end{aligned}$$

Getting rid of the denominators (the inverse parts), this is equivalent to

$$\begin{aligned} \sum _{j=1}^N \sum _{k=1}^N P_{i,k}P_{j,k} \prod _{l \ne k} (\varLambda _{l,l}+\gamma ) \ge 0, \quad i=1,\ldots ,N. \end{aligned}$$
(7)

The function on the l.h.s. of (7) is polynomial in \(\gamma \) with degree \(N-1\). The leading term (in \(\gamma ^{N-1}\)) has coefficient

$$\begin{aligned} \sum _{j=1}^N \sum _{k=1}^N P_{i,k}P_{j,k}=\sum _{j=1}^N(\varvec{PP'})_{i,j}=1, \quad i=1,\ldots ,N, \end{aligned}$$

because \(\varvec{P}\) is orthogonal. This latter expression ensures that as \(\gamma \) increases, all of the weights will progressively become positive. Some of them are already positive, even for \(\gamma =0\), because they correspond to the long positions of \(\varvec{w}_{MV}\).

Appendix B: Proof of Proposition 1

For notational ease, we will write \(\varvec{w}\) instead \(\varvec{w}_\gamma \) during the proof.

Using the eigendecomposition \(\varvec{\varSigma =P\varLambda P'}\) and the fact that \(\varvec{P'P=PP'=I_N}\),

$$\begin{aligned} \varvec{w' \varSigma w}&=\frac{\varvec{1'(\varSigma +\gamma I_N)^{-1}\varSigma (\varSigma +\gamma I_N)^{-1} 1}}{(\varvec{1'(\varSigma +\gamma I_N)^{-1}1})^2} = \frac{\varvec{1'P(\varLambda +\gamma I_N)^{-1} \varLambda (\varLambda +\gamma I_N)^{-1} P'1}}{(\varvec{1'P(\varLambda +\gamma I_N)^{-1}P'1})^2}\\&= \frac{\displaystyle \sum \nolimits _{i=1}^N\sum _{j=1}^N\sum _{k=1}^N P_{i,k}P_{j,k} \varLambda _{k,k}( \varLambda _{k,k}+\gamma )^{-2}}{\displaystyle \left( \sum \nolimits _{i=1}^N\sum \nolimits _{j=1}^N\sum \nolimits _{k=1}^N P_{i,k}P_{j,k}(\varLambda _{k,k}+\gamma )^{-1}\right) ^2} = \frac{\displaystyle \sum \nolimits _{k=1}^N \varLambda _{k,k}(\varLambda _{k,k}+\gamma )^{-2}\left( \sum \nolimits _{l=1}^N P_{l,k}\right) ^2}{\displaystyle \left( \sum \nolimits _{k=1}^N(\varLambda _{k,k}+\gamma )^{-1} \left( \sum \nolimits _{l=1}^N P_{l,k}\right) ^2\right) ^2} \end{aligned}$$

We now introduce the following quantity, which will be ubiquitous in the remainder of the proofs:

$$\begin{aligned} L_k=\left( \sum _{l=1}^N P_{l,k}\right) ^2. \end{aligned}$$

Straightforward differentiation gives

$$\begin{aligned}&\frac{\partial }{\partial \gamma }\varvec{w' \varSigma w}\nonumber \\&\quad = \frac{\displaystyle -2\sum \nolimits _{k=1}^N\frac{ \varLambda _{k,k}}{(\varLambda _{k,k}\!+\!\!\gamma )^{3}}L_k \!\times \! \sum \nolimits _{k=1}^N(\varLambda _{k,k}\!\!+\!\gamma )^{-1}L_k\!\!+\!2 \sum \nolimits _{k=1}^N\frac{ \varLambda _{k,k}}{(\varLambda _{k,k}\!\!+\!\gamma )^{2}}L_k \!\times \! \sum \nolimits _{k=1}^N(\varLambda _{k,k}\!+\!\gamma )^{-2}L_k }{\displaystyle \left( \sum \nolimits _{k=1}^N(\varLambda _{k,k}\!\!+\!\gamma )^{-1} L_k\right) ^3}. \end{aligned}$$

The denominator is always positive, therefore, the sign of \(\frac{\partial }{\partial \gamma }\varvec{w' \varSigma w}\) is equal to that of

$$\begin{aligned}&\sum _{k=1}^N \sum _{i=k+1}^NL_kL_i \left[ \frac{\varLambda _{k,k}}{( \varLambda _{k,k}+\gamma )^2} (\varLambda _{i,i}+\gamma )^{-2}-\frac{\varLambda _{k,k}}{(\varLambda _{k,k}+\gamma )^3}(\varLambda _{i,i}+\gamma )^{-1} \right. \\&\qquad \left. + \frac{\varLambda _{i,i}}{(\varLambda _{i,i}+\gamma )^2}(\varLambda _{k,k}+\gamma )^{-2} - \frac{\varLambda _{i,i}}{(\varLambda _{i,i}+\gamma )^3}(\varLambda _{k,k}+\gamma )^{-1}\right] \\&= \sum _{k=1}^N \sum _{i=k+1}^NL_kL_i (\varLambda _{i,i}+\gamma )^{-1}(\varLambda _{k,k}+\gamma )^{-1} f_\gamma (\varLambda _{i,i},\varLambda _{k,k}) \end{aligned}$$

where

$$\begin{aligned} f_{\gamma }(x,y)=(x+\gamma )^{-1}(y+\gamma )^{-1}(x+y)-x(x+\gamma )^{-2}-y(y+\gamma )^{-2}. \end{aligned}$$

Further, we have

$$\begin{aligned} \frac{\partial }{\partial x}f_{\gamma }(x,y)=\frac{2\gamma (x-y)}{(x+\gamma )^3(y+\gamma )}, \quad \frac{\partial ^2}{\partial ^2 x}f_{\gamma }(x,y)=\frac{2\gamma (\gamma -2x+3y)}{(x+\gamma )^2(y+\gamma )}, \end{aligned}$$

and consequently the mapping \(x \mapsto f_{\gamma }(x,y)\) reaches its minimum value (which is zero) at \(x=y\). Therefore, \(\frac{\partial }{\partial \gamma }\varvec{w' \varSigma w} \ge 0\). The derivative can only reach zero if all eigenvalues are equal, which is excluded because \(\varvec{\varSigma }\) is not a multiple if \(\varvec{I_N}\). The derivative of \(\varvec{w' w}\) with respect to \(\gamma \) is dealt with in the same fashionFootnote 10 except that \(f_{\gamma }(x,y)\) must be replaced by

$$\begin{aligned} g_{\gamma }(x,y)=2(x+\gamma )^{-1}(y+\gamma )^{-1}-(x+\gamma )^{-2}-(y+\gamma )^{-2}=\frac{-(x-y)^2}{(x+\gamma )^2(y+\gamma )^2}, \end{aligned}$$

which is always strictly negative for \(x,y,\gamma > 0\) and \(x \ne y\) (which occurs because \(\varvec{\varSigma }\ne \alpha \varvec{I_N}\)). Moreover, it is obvious that \(\frac{\partial }{\partial \gamma }\varvec{w' w} \) bounded for \(x,y>0\) and \(\gamma \ge 0\).

Appendix C: Proof of the lemmas

We start with Lemma 1.

Proof

For the first assertion, if we consider \(\gamma >U\), then we can write \(\varvec{\varSigma }+\gamma \varvec{I_N}=\gamma (\varvec{\varSigma }/\gamma + \varvec{I_N})\). By the definition of \(U\), the largest eigenvalue of \(\varvec{\varSigma }/\gamma \) is strictly smaller than one and hence, the Neumann series of the inverse is given by

$$\begin{aligned} (\varvec{\varSigma }+\gamma \varvec{I_N})^{-1}&=\gamma ^{-1} \sum _{k=0}^\infty (-\varvec{\varSigma }/\gamma )^k \\&=\gamma ^{-1} \sum _{k=0}^\infty \left[ (\varvec{\varSigma }/\gamma )^{2k}-(\varvec{\varSigma }/\gamma )^{2k+1} \right] \end{aligned}$$

We now write \(s_{i,j}^{(2k)}\) for the elements of \((\varvec{\varSigma }/\gamma )^{2k}\). The \(i\)th row sum of \((\varvec{\varSigma }/\gamma )^{2k+1}\) is then given by

$$\begin{aligned} \sum _{j=1}^N s_{i,j}^{(2k+1)}&=\sum _{j=1}^N \sum _{l=1}^N s_{i,l}^{(2k)} \times \varSigma _{l,j}/\gamma \\&= \sum _{l=1}^N s_{i,l}^{(2k)} \sum _{j=1}^N \varSigma _{l,j}/\gamma \\&\le \sum _{l=1}^N s_{i,l}^{(2k)}. \end{aligned}$$

In the last inequality, we have used the definition of \(U\) (as well as the symmetry of \(\varvec{\varSigma }\)) and the fact that by the assumption of the lemma, the row sums of even powers of \(\varvec{\varSigma }\) are positive. The inequality shows that all of the row sums of \((\varvec{\varSigma }+\gamma \varvec{I_N})^{-1}\) will be positive, thereby fulfilling the condition in the definition of \(\gamma ^{LO}\) in (4).

We now turn to Lemma 2.

Proof

We use the notations of “Appendix B”, and by replacing \(\varvec{\varSigma }\) by \(\varvec{I_N}\) in the computation of the ex-ante variance, we get

$$\begin{aligned} D(\varvec{w}_\gamma )=\frac{\displaystyle \sum \nolimits _{k=1}^N (\varLambda _{k,k}+\gamma )^{-2}L_k}{\displaystyle \left( \sum \nolimits _{k=1}^N(\varLambda _{k,k}+\gamma )^{-1} L_k\right) ^2}. \end{aligned}$$

First, we have

$$\begin{aligned} \displaystyle \sum _{k=1}^N (\varLambda _{k,k}+\delta \varLambda _{1,1})^{-2}L_k \le (\delta \varLambda _{1,1})^{-2}\sum _{k=1}^N L_k, \end{aligned}$$

while at the same time

$$\begin{aligned} \displaystyle \left( \sum _{k=1}^N(\varLambda _{k,k}+\delta \varLambda _{1,1} )^{-1} L_k\right) ^2 \ge \displaystyle \left( \sum _{k=1}^N((1+\delta )\varLambda _{1,1})^{-1} L_k\right) ^2. \end{aligned}$$

Therefore,

$$\begin{aligned} D(\varvec{w}_\gamma )^{-1} \ge \frac{\delta ^2}{(\delta +1)^2}\sum _{k=1}^N L_k=\frac{\delta ^2}{(\delta +1)^2}N, \end{aligned}$$

where the equality stems from the definition of \(L_k\) and the fact that \(\varvec{P}\) is orthogonal (the sum of all elements of \(\varvec{PP'}\) is equal to \(N\)).

Appendix D: Proof of Proposition 2

We start by looking at the \(L^2\)-norm difference of the two portfolios (3), with common \(\gamma \), but different \(\varvec{\varSigma }^{(i)}\). With the notations of the former proof:

$$\begin{aligned} \epsilon&=|(\varvec{w}_\gamma ^{(1)})'\varvec{w}_\gamma ^{(1)}-(\varvec{w}_\gamma ^{(2)})'\varvec{w}_\gamma ^{(2)}| \nonumber \\&=\left| \frac{a(\varLambda ^{(1)},\gamma )}{b(\varLambda ^{(1)},\gamma )^2}-\frac{a(\varLambda ^{(2)},\gamma )}{b(\varLambda ^{(2)},\gamma )^2} \right| , \nonumber \\&=\frac{a(\varLambda ^{(1)},\gamma )[b(\varLambda ^{(2)},\gamma )^2\!-\!b(\varLambda ^{(1)},\gamma )^2]+b(\varLambda ^{(1)},\gamma )^2[a(\varLambda ^{(1)},\gamma )-a(\varLambda ^{(2)},\gamma )]}{b(\varLambda ^{(1)},\gamma )^2b(\varLambda ^{(2)},\gamma )^2}, \end{aligned}$$
(8)

with

$$\begin{aligned} a(\varLambda ^{(i)},\gamma )=\sum _{k=1}^N (\varLambda ^{(i)}_{k,k}+\gamma )^{-2}L_k, \quad \quad b(\varLambda ^{(i)},\gamma )=\sum _{k=1}^N (\varLambda ^{(i)}_{k,k}+\gamma )^{-1}L_k. \end{aligned}$$
(9)

Basic computations further yield

$$\begin{aligned} |a(\varLambda ^{1},\gamma )-a(\varLambda ^{2},\gamma )|&= \sum _{k=1}^N \frac{(\varLambda ^{(2)}_{k,k}+\gamma )^2-(\varLambda ^{(1)}_{k,k}+\gamma )^2}{(\varLambda ^{(1)}_{k,k}+\gamma )^2(\varLambda ^{(2)}_{k,k}+\gamma )^2} L_k \nonumber \\&= \sum _{k=1}^N \frac{(\varLambda ^{(2)}_{k,k}-\varLambda ^{(1)}_{k,k})((\varLambda ^{(2)}_{k,k}+\varLambda ^{(1)}_{k,k}+2\gamma )}{(\varLambda ^{(1)}_{k,k}+\gamma )^2(\varLambda ^{(2)}_{k,k}+\gamma )^2} L_k \nonumber \\&\le \sum _{k=1}^N \frac{2|\varLambda ^{(2)}_{k,k}-\varLambda ^{(1)}_{k,k}|}{\left( \varLambda ^{(1)}_{k,k}+\gamma \right) \left( \varLambda ^{(2)}_{k,k}+\gamma \right) \left( \min \left( \varLambda ^{(1)}_{k,k},\varLambda ^{(2)}_{k,k}\right) +\gamma \right) } L_k \nonumber \\&\le \frac{2\eta N}{\left( \varLambda ^{(1)}_{N,N}+\gamma \right) \left( \varLambda ^{(2)}_{N,N}+\gamma \right) \left( \min \left( \varLambda ^{(1)}_{N,N},\varLambda ^{(2)}_{N,N}\right) +\gamma \right) } \nonumber \\&:= \kappa \end{aligned}$$
(10)

where the first inequality stems from the fact that \(x+y+2z \le 2(\max (x,y)+z)\) for \(x,y,z \ge 0\). In the second inequality, we have set \(\eta = {\text {max}}_{i=1,\ldots ,N} \left| \varLambda ^{(2)}_{i,i}-\varLambda ^{(1)}_{i,i}\right| \) and used the identity \(\sum _{k=1}^N L_k=N\).

Moreover,

$$\begin{aligned}&|b(\varLambda ^{(1)},\gamma )^2-b(\varLambda ^{(2)},\gamma )^2|\nonumber \\&\quad =(b(\varLambda ^{(1)},\gamma )+b(\varLambda ^{(2)},\gamma ))|b(\varLambda ^{(1)},\gamma )-b(\varLambda ^{(2)},\gamma )| \nonumber \\&\quad = \left( \sum _{k=1}^N \frac{L_k}{\varLambda ^{(1)}_{k,k}+\gamma }+\sum _{k=1}^N \frac{L_k}{\varLambda ^{(2)}_{k,k}+\gamma }\right) \left| \sum _{k=1}^N \frac{L_k}{\varLambda ^{(1)}_{k,k}+\gamma }-\sum _{k=1}^N \frac{L_k}{\varLambda ^{(2)}_{k,k}+\gamma }\right| \nonumber \\&\quad =\left( \sum _{k=1}^N\frac{\varLambda ^{(1)}_{k,k}+\varLambda ^{(2)}_{k,k}+2\gamma }{\left( \varLambda ^{(1)}_{N,N}+\gamma \right) \left( \varLambda ^{(2)}_{N,N}+\gamma \right) }L_k \right) \left| \sum _{k=1}^N\frac{\varLambda ^{(2)}_{k,k}-\varLambda ^{(1)}_{k,k}}{\left( \varLambda ^{(1)}_{N,N}+\gamma \right) \left( \varLambda ^{(2)}_{N,N}+\gamma \right) }L_k \right| \nonumber \\&\quad \le \frac{2N}{\min \left( \varLambda ^{(1)}_{N,N},\varLambda ^{(2)}_{N,N}\right) +\gamma } \!\times \! \frac{\eta N}{\left( \varLambda ^{(1)}_{N,N}\!+\!\gamma \right) \left( \varLambda ^{(2)}_{N,N}\!+\!\gamma \right) } \nonumber \\&\quad = \kappa N \end{aligned}$$
(11)

Combining (8), (10) and (11), we get

$$\begin{aligned} \epsilon \le \frac{a(\varLambda ^{(1)},\gamma )\kappa N+b(\varLambda ^{(1)},\gamma )^2 \kappa }{b(\varLambda ^{(1)},\gamma )^2b(\varLambda ^{(2)},\gamma )^2}. \end{aligned}$$

Lastly, the simple bounds

$$\begin{aligned} \frac{N}{(\varLambda ^{(i)}_{1,1}+\gamma )^2} \le a(\varLambda ^{(i)},\gamma )\le \frac{N}{(\varLambda ^{(i)}_{N,N}+\gamma )^2} \end{aligned}$$

and

$$\begin{aligned} \frac{N}{\varLambda ^{(i)}_{1,1}+\gamma } \le b(\varLambda ^{(i)},\gamma )\le \frac{N}{\varLambda ^{(i)}_{N,N}+\gamma }, \end{aligned}$$
(12)

lead to

$$\begin{aligned} \epsilon \le \frac{2 \kappa }{N^2} \frac{(\varLambda ^{(1)}_{1,1}+\gamma )^2(\varLambda ^{(2)}_{1,1}+\gamma ) ^2}{(\varLambda ^{(1)}_{N,N}+\gamma )^2} :=c \eta /N, \end{aligned}$$
(13)

for some constant \(c>0\) (which is all the more large that the eigenvalues are dispersed).

We now consider the functions \(h_{i}(\gamma )=(\varvec{w}^{(i)}_\gamma )'\varvec{w}^{(i)}_\gamma \) for \(i=1,2\). Recall from the proof of Proposition 1, that the derivatives \(h_i'\) are strictly negative. Therefore, the inverse functions \(h^{-1}_i\) are also strictly decreasing and Lipschitz continuous on the interval \((1/N+\varepsilon ,h_i(0))\) for any strictly positive \(\varepsilon \), as is depicted in the right side of Fig. 2.

Fig. 2
figure 2

Graphical illustration of the functions \(h_i\) and their inverses \(h_i^{-1}\)

Since \(\varvec{w}^{(i)}_{\gamma _i}\) both satisfy (2), then

$$\begin{aligned} 0&=(\varvec{w}^{(1)}_{\gamma ^{(1)}})'\varvec{w}^{(1)}_{\gamma ^{(1)}}-(\varvec{w}^{(2)}_{\gamma ^{(2)}})'\varvec{w}^{(2)}_{\gamma ^{(2)}}\\&=\left[ (\varvec{w}^{(1)}_{\gamma ^{(1)}})'\varvec{w}^{(1)}_{\gamma ^{(1)}}-(\varvec{w}^{(1)}_{\gamma ^{(2)}})'\varvec{w}^{(1)}_{\gamma ^{(2)}}\right] +\left[ (\varvec{w}^{(1)}_{\gamma ^{(2)}})'\varvec{w}^{(1)}_{\gamma ^{(2)}}-(\varvec{w}^{(2)}_{\gamma ^{(2)}})'\varvec{w}^{(2)}_{\gamma ^{(2)}}\right] \end{aligned}$$

Now, because the \(h^{-1}_i\) are Lipschitz on \((\delta ,h_i(0))\), the first bracket satisfies

$$\begin{aligned} |\gamma ^{(1)}-\gamma ^{(2)}| \le c \left| \varvec{w}^{(1)}_{\gamma ^{(1)}})'\varvec{w}^{(1)}_{\gamma ^{(1)}}-(\varvec{w}^{(1)}_{\gamma ^{(2)}})'\varvec{w}^{(1)}_{\gamma ^{(2)}} \right| , \end{aligned}$$

while the second, as we have shown in (13) is \(O(\eta /N)\), therefore it must also hold that

$$\begin{aligned} |\gamma ^{(1)}-\gamma ^{(2)}| \le C \eta /N, \end{aligned}$$

for some other constant \(C\) which depends on the maximum absolute values of \((h_i^{-1})'\) on the interval \([\delta ,h_i(0)]\) - note that it is not a priori obvious (or true) that the \(h_i\) are convex. Overall, we therefore expect \(|\gamma ^{(1)}-\gamma ^{(2)}|\) to be smaller for larger values of \(\delta \). When the norm constraint is loose, the \(\gamma ^{(i)}\) will usually be very small and so will their difference.

Appendix E: Proof of Theorem 1

Simplifying the notations of the preceding proof, we will henceforth write \(b_i= b(\varLambda ^{(i)},\gamma ^{(i)})\) (defined in (9)).

$$\begin{aligned} \epsilon&= (\varvec{w}^{(1)}-\varvec{w}^{(2)})'(\varvec{w}^{(1)}-\varvec{w}^{(2)}) \nonumber \\&=(\varvec{w}^{(1)})'\varvec{w}^{(1)}+(\varvec{w}^{(2)})'\varvec{w}^{(2)}-2(\varvec{w}^{(1)})'\varvec{w}^{(2)} \nonumber \\&= \sum _{k=1}^N X_k \times L_k \end{aligned}$$
(14)

where

$$\begin{aligned} X_k =&\frac{1}{\left( \varLambda _{k,k}^{(1)}\!+\!\gamma ^{(1)}\right) ^2b_1^2} \!+\! \frac{1}{\left( \varLambda _{k,k}^{(2)}+\gamma ^{(2)}\right) ^2b_2^2}-2\frac{1}{\left( \varLambda _{k,k}^{(1)}\!+\!\gamma ^{(1)}\right) \left( \varLambda _{k,k}^{(2)}\!+\!\gamma ^{(2)}\right) b_1b_2} \nonumber \\ =&\frac{\left( \varLambda _{k,k}^{(1)}+\gamma ^{(1)}\right) ^2b_1^2\!+\!\left( \varLambda _{k,k}^{(2)}+\gamma ^{(2)}\right) ^2b_2^2\!-\!2\left( \varLambda _{k,k}^{(1)}+\gamma ^{(1)}\right) \left( \varLambda _{k,k}^{(2)}+\gamma ^{(2)}\right) b_1b_2}{\left( \varLambda _{k,k}^{(1)}+\gamma ^{(1)}\right) ^2\left( \varLambda _{k,k}^{(2)}+\gamma ^{(2)}\right) ^2b_1^2b_2^2} \nonumber \\ =&\frac{\left( \left( \varLambda _{k,k}^{(1)}+\gamma ^{(1)}\right) b_1-\left( \varLambda _{k,k}^{(2)}+\gamma ^{(2)}\right) b_2\right) ^2}{\left( \varLambda _{k,k}^{(1)}+\gamma ^{(1)}\right) ^2\left( \varLambda _{k,k}^{(2)}+\gamma ^{(2)}\right) ^2b_1^2b_2^2} \nonumber \\ =&\frac{\left( b_1(\varLambda _{k,k}^{(1)}-\varLambda _{k,k}^{(2)})+b_1(\gamma ^{(1)}-\gamma ^{(2)})+ \varLambda _{k,k}^{(2)}(b_1-b_2)+\gamma ^{(2)}(b_1-b_2)\right) ^2 }{\left( \varLambda _{k,k}^{(1)}+\gamma ^{(1)}\right) ^2\left( \varLambda _{k,k}^{(2)}+\gamma ^{(2)}\right) ^2b_1^2b_2^2} \end{aligned}$$
(15)

The denominator of \(X_k\) is bounded from below by

$$\begin{aligned} d=N^4\frac{\left( \varLambda _{N,N}^{(1)}+\gamma ^{(1)}\right) ^2\left( \varLambda _{N,N}^{(2)}+\gamma ^{(2)}\right) ^2}{\left( \varLambda _{1,1}^{(1)}+\gamma ^{(1)}\right) ^2\left( \varLambda _{1,1}^{(2)}+\gamma ^{(2)}\right) ^2}, \end{aligned}$$

and this constant, given Lemma 2, can be increased so as to be unrelated to the \(\gamma ^{(i)}\).

Moreover, if we recall that \(\eta = {\text {max}}_{i=1,\ldots ,N} \left| \varLambda ^{(2)}_{i,i}-\varLambda ^{(1)}_{i,i}\right| \), the following bound also holds

$$\begin{aligned} |b_1-b_2|\!\le \!\sum _{k=1}^N\frac{|\varLambda ^{(2)}_{k,k}\!-\!\varLambda ^{(1)}_{k,k}|}{\left( \varLambda ^{(1)}_{k,k}\!+\!\gamma ^{(1)}\right) \left( \varLambda ^{(2)}_{k,k}\!+\!\gamma ^{(2)} \right) }L_k\!\le \! \frac{\eta N}{\left( \varLambda ^{(1)}_{N,N}+\gamma ^{(1)}\right) \left( \varLambda ^{(2)}_{N,N}+\gamma ^{(2)} \right) }. \nonumber \\ \end{aligned}$$
(16)

We now reconnect the pieces. There are 4 terms in the squared numerator in (15):

  • the first one is bounded by a multiple of \(\eta N\)

  • the second one, by Proposition 2 and (12) is \(O(\eta )\)

  • the third one, by (16) is \(O(\eta N)\) like the first one

  • lastly, by (16) and Lemma 2, the fourth one is also \(O(\eta N)\)

Consequently, going back to (14) and plugging \(\sum _{k=1}^N L_k=N\), we get

$$\begin{aligned} \epsilon = (\varvec{w}^{(1)}-\varvec{w}^{(2)})'(\varvec{w}^{(1)}-\varvec{w}^{(2)})=O(\eta ^2/N). \end{aligned}$$

Remark Instead of looking at the parameters \(\eta \) and \(N\), one may wonder what is the impact of \(\delta \) on \(\epsilon \). It can essentially be measured through the magnitude of the \(\gamma ^{(i)}\). First, by the last remark of the proof of Proposition 2, we expect \(|\gamma ^{(1)}-\gamma ^{(2)}|\) to be a decreasing function of \(\delta \). But this term in (15) is associated with the only term which does not increase with \(N\). Moreover, the terms in \(b_i\) and \(|b_1-b_2|\) are all bounded above and below by functions which decrease \(\gamma ^{(i)}\) and hence increase with \(\delta \) (by Proposition 1). In the end, it is very likely that \((\varvec{w}^{(1)}-\varvec{w}^{(2)})'(\varvec{w}^{(1)}-\varvec{w}^{(2)})\) will be most of the time an increasing function of \(\delta \). This seems logical because as \(\delta \) decreases to \(1/N\), both \(\varvec{w}^{(1)}\) and \(\varvec{w}^{(2)}\) will converge to the equally weighted portfolio. When the constraint is tight, there is not much room for a large differences between \(\varvec{w}^{(1)}\) and \(\varvec{w}^{(2)}\), even if \(\varvec{\varSigma }^{(1)}\) and \(\varvec{\varSigma }^{(2)}\) are very different.

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Coqueret, G. Diversified minimum-variance portfolios. Ann Finance 11, 221–241 (2015). https://doi.org/10.1007/s10436-014-0253-x

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  • DOI: https://doi.org/10.1007/s10436-014-0253-x

Keywords

  • Portfolio optimization
  • Minimum variance
  • Diversification

JEL Classification

  • G11
  • C61