The Efficient Frontier for Weakly Correlated Assets

Comput Econ (2012) 40:355–375DOI 10.1007/s10614-011-9296-5

The Efficient Frontier for Weakly Correlated Assets

Michael J. Best · Xili Zhang

Accepted: 10 August 2011 / Published online: 20 August 2011© Springer Science+Business Media, LLC. 2011

Abstract For a general Markowitz portfolio selection problem with linear inequal-ity constraints, it is not possible to obtain a closed form solution. The number ofparametric intervals and corresponding segments of the efficient frontier is not knowna priori. In this paper, we analyze the structure of the efficient frontier under theassumptions of weakly correlated assets and no short sales constraints. By weaklycorrelated, we mean the off diagonal elements of the covariance matrix are small rel-ative to the diagonal ones. We obtain an explicit approximate solution for the entireefficient frontier. The error in the approximation is the order of the norm squared of theoff diagonal part of the covariance matrix. The assumption of weakly correlated assetsis restrictive. However, the explicit approximation of the efficient asset holdings in thepresence of bound constraints gives insight into the nature of the efficient frontier. Weprove that the efficient frontier is traced out in a monotonic fashion whereby assets arereduced to zero and subsequently remain at zero in order of their expected returns andthe number of parametric intervals is equal to the number of assets. This generalizesthe results of Best and Hlouskova (Math Methods Oper Res 52:195–212, 2000). Thederived structure and approximation are illustrated by a 3-asset example.

Keywords Mean-variance portfolio optimization · Efficient frontier · Quadraticprogramming · Approximate portfolios

M. J. Best (B) · X. ZhangDepartment of Combinatorics and Optimization, Faculty of Mathematics, University of Waterloo,Waterloo, ON, Canadae-mail: [email protected]

X. ZhangSchool of Business Administration, South China University of Technology, Guangzhou,People’s Republic of Chinae-mail: [email protected]

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356 M. J. Best, X. Zhang

1 Introduction

A portfolio consists of various amounts held in different assets. The investor is seekingto both maximize the expected return and minimize risk. In Markowitz (1959) andSharpe (1970), the Mean-Variance (M-V) portfolio selection problem is analyzed sub-ject to general linear constraints using quadratic programming and parameter quadraticprogramming methods. One way to formulate the M-V portfolio selection problem is1:

min

{−tμ′x + 1

2x ′�x | l ′x = 1, Ax ≤ b

}, (1.1)

where μ is an n-vector of expected returns where n is the number of assets, x is ann-dimensional vector of asset holdings to be determined, � is an (n, n) covariancematrix, l is an n-dimensional vector of one’s, t is a risk aversion parameter. Theconstraint l ′x = 1 is called the budget constraint. A is an (m, n) matrix and b isan m-dimensional vector, in which m is the number of constraints. The constraintsAx ≤ b represent general linear inequality constraints, such as nonnegativity con-straints (−x ≤ 0), upper or lower bound constraints (x ≤ bu , or, −x ≤ −bl , respec-tively), and any other linear constraints the investor may wish to impose.

The expected return of the portfolio is μ′x and its variance is x ′�x . A portfoliois called efficient if no other portfolio gives a higher expected return for fixed vari-ance or a lower variance for fixed expected return. Merton (1972) gave the functionalform of return-variance connection when only the budget constraint is imposed on thevariables x . Szegö (1980) studied Merton’s problem with a nonnegativity constrainton the allocation vector x and has given a procedure for revealing the explicit formof efficient portfolio frontier. Furthermore, Vörös (1986, 1987) provided a specializedactive set algorithm to solve a problem with only a budget constraint and no shortsales constraints. Lozza and Pellerey (2008) analytically discussed the link betweenstochastic choices and market bounds when the returns are elliptical distributed andno short sales are allowed.

Solving a portfolio optimization problem (1.1) in general is a difficult task. Itrequires a parametric quadratic programming algorithm, such as in Perold (1984) andBest (1996, 2010), which is iterative. Associated with (1.1) is a set of finitely manycritical intervals such that in each of these intervals, the components of the optimalsolution of (1.1) are linear functions of the parameter t . The optimal value of theobjective function in (1.1) as a function of t is piecewise quadratic. That is; there arev intervals [t0, t1], [t1, t2], . . . , [tv−1, tv] with 0 = t0 ≤ t1, t1 ≤ t2, . . . , tv−1 ≤ tv .In each interval, there exist n-dimensional vectors h0i and h1i , for i = 1, . . . , v, suchthat the components of the optimal solution is expressed as

x(t) = h0i + t h1i , t ∈ [ti−1, ti ], i = 1, . . . , v,

1 Throughout this paper, prime (′) is used to denote matrix transposition and any unprimed vector is acolumn vector.

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The Efficient Frontier for Weakly Correlated Assets 357

which is a piecewise linear function of t . Thus the expected return is μ′x(t) = μ′h0i +tμ′h1i (piecewise linear), and its variance is x(t)′�x(t) = h′

0i�h0i + 2th′1i�h0i +

t2h′1i�h1i (piecewise quadratic).However, the number of intervals v is not known. In this paper, we shall be con-

sidering that version of (1.1) with the general constraints ( Ax ≤ b ) being simplenonnegativity constraints ( −x ≤ 0 ). For any feasible point x for (1.1), the active set isthe set of indices i for which the i th component of x is zero. Each parametric intervalis determined by a particular active set. The difficulty in solving (1.1) is to determinethe correct active set at the optimal solution. Since there are two possibilities for eachnonnegativity constraint, there are a total of 2n possibilities. This number can be quitelarge. For example, for a 20 asset problem, there are v = 220 =1,048,576 possibleactive sets for any fixed value of t . Then it is difficult to determine the associatedvectors h0i , h1i (i = 1, . . . , v). It is often impossible to obtain a closed form solutionand analyze the expected return and the variance of the efficient portfolios. Green(1986) discussed that with a large number of assets, it can be difficult to translateone’s intuitive sense about how assets covary into statements about the inverse of thecovariance matrix. However, for the 20 asset problem mentioned above, we shall showthere are precisely 20 active sets required to formulate the entire optimal solution asa function of t .

Taking the covariance matrix as a diagonal matrix, Best and Hlouskova (2000) andHlouskova (2001) developed an explicit solution of the mean-variance portfolio prob-lem for uncorrelated and bounded assets. When the expected returns are all differentand are in increasing order, there are precisely n parametric intervals. The componentsof the efficient portfolios are reduced to zero and remain there in the order of smallestexpected return to largest expected return as the risk aversion factor, t , is increased.The end of the parametric interval is determined by the asset with the smallest expectedreturn among those still held positively, being reduced to zero. Xu (2006) gave a closedform solution for risky assets with a triple-branch covariance matrix, where the struc-ture of the efficient frontier is similar to the diagonal one in Best and Hlouskova (2000)under some technical assumptions. In general, for the portfolio selection problems inpractice, it appears rather reasonable that the assets are more or less correlated witheach other. Disatnik and Benninga (2006) and Disatnik (2010) constructed a globalminimum variance portfolio (GMVP) by using a block structure for the covariancematrix of asset returns.

In this paper, we discuss the efficient frontier for portfolio optimization with weaklycorrelated assets, that is; the off diagonal elements in the covariance matrix are smallrelative to the diagonal ones but all of them can be nonzero (i.e., dense). Althoughthe assumption of weakly correlated assets is restrictive, the explicit approximationof the efficient asset holdings in the presence of inequality constraints gives insightinto the nature of the efficient frontier.

The contribution of this paper is as follows. When the covariance matrix for therisky assets is positive definite with the off diagonal elements sufficiently smallerthan the diagonal ones, we obtain an explicit approximate solution for the entire effi-cient frontier. The error in the approximation is the order of the norm squared of theoff diagonal part of the covariance matrix. We prove that the number of parametric

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intervals is equal to the number of assets. Moreover, the efficient frontier is traced outin a monotonic fashion, in which, as the value of the risk aversion parameter increases,the components of the efficient portfolio are one by one reduced to zero and subse-quently remain at zero, in order of their associated expected returns, from the smallestto the largest. This generalizes the results of Best and Hlouskova (2000).

The paper proceeds as follows. In Sect. 2, we develop an approximation for theinverse of the covariance matrix. In Sect. 3, we obtain the exact optimal solution fora portfolio optimization problem with only a budget constraint. Next we develop anapproximate solution for it by using the approximation result in Sect. 2. We also showthe relationship between the exact and approximate solutions. Section 4 discusses thestructure of the efficient frontier for the portfolio optimization problem with no shortsales. Section 5 uses a 3-asset portfolio optimization problem to illustrate the obtainedresult and the approximate method by comparing with the diagonal approximation.Section 6 concludes.

2 Preliminaries

Before discussing the problems in this paper, we need the following results taken fromGolub and Loan (1989). Let � denote the set of real numbers. Let �m×n denote thevector space of all (m, n) real matrices.

Definition 2.1 Let A ∈ �m×n . Given any vector norm ||·||, the corresponding inducedmatrix norm of A, also denoted by || · ||, is defined by

||A|| = maxx∈Rn ,||x ||=1

||Ax ||.

This satisfies the following properties: (a) ||A|| ≥ 0, A ∈ �m×n, (b) ||A+B|| ≤ ||A||+||B||, A, B ∈ �m×n, (c) ||θ A|| = |θ | |||A||, θ ∈ �, A ∈ �m×n and (d)||AB|| ≤ ||A|| ||B||, A ∈ �m×l , B ∈ �l×n .

In this paper, we assume that all the induced matrix norms we use are associatedwith the ∞ norm, that is, || · || = || · ||∞.

Definition 2.2 Let A be an (n, n) matrix that is presumed to be sufficiently small. LetP(A) and Q(A) be matrices depending on A. For α > 0, we write

P(A) = Q(A) + O(||A||α),

if there is a constant γ such that

||P(A) − Q(A)|| ≤ γ ||A||α.

Lemma 2.1 Let A ∈ �n×n. If ||A|| < 1, then (a) I −A is nonsingular, (b) (I −A)−1 =∑∞i=0 Ai .

For the portfolio selection problems in practice, it appears rather reasonable that theassets are more or less correlated with each other. In this paper, our results will require

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the following assumption for the (n, n) covariance matrix � in the Mean-Varianceproblems to be satisfied.

Then the covariance matrix � = [σi, j ], i, j = 1, . . . , n, throughout this paper canbe written as

� = D + S,

where D = diag(σi ) is diagonal with each diagonal element being strictly positive.For the sake of brevity, we use the term σi for the diagonal elements of �, that is,

σi,i ≡ σi , i = 1, . . . , n.S ={

σi, j , i = j0, i = j

is an (n, n) symmetric matrix which is the

off-diagonal matrix of �. When S = 0, that is, � = D, the problem becomes thediagonal problem in Best and Hlouskova (2000).

Assumption 2.1 (a) � is positive definite. (b) ||S|| is sufficiently small.

Note that � positive definite implies � is nonsingular. Most of the quantities defin-ing the efficient frontier are in terms of �−1. Next, we want to show (D + S)−1 canbe expanded as an explicit functions of S and give an approximation for (D + S)−1.

Theorem 2.1 Let Assumption 2.1 be satisfied. We have the following results:

(a) Let A−1 = D−1 − D−1SD−1 and E(S) =∞∑

i=2(−D−1S)i D−1. Then

(D + S)−1 = A−1 + E(S).

(b) For ||S|| sufficiently small, (D + S)−1 can be approximated by A−1 and the errorof approximation satisfies E(S) = O(||S||2).

Proof (a) Lemma 2.1(a) and Assumption 2.1 imply that I + D−1S is nonsingular,so we have

(D + S)−1 = [D(I + D−1S)]−1 = [I − (−D−1S)]−1 D−1.

From Lemma 2.1(b), it follows that

(D + S)−1 =∞∑

i=0

(−D−1S)i D−1

= D−1 − D−1SD−1 +∞∑

i=2

(−D−1S)i D−1,

which completes the proof of part (a).

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(b) According to Definition 2.1, we have

||E(S)|| ≤∞∑

i=2

||(D−1S)i D−1||

≤ ||D−1||∞∑

i=2

||(D−1S)i ||

≤ ||D−1||∞∑

i=2

||D−1S||i .

Because∞∑

i=2||D−1S||i = ||D−1 S||2

1−||(D−1 S)|| , it follows that

||E(S)|| ≤ ||D−1||1 − ||D−1S|| ||D−1S||2

≤ ||D−1||31 − ||D−1S|| ||S||2.

For ||S|| sufficiently small, by Definition 2.2,

E(S) = O(||S||2),

which completes the proof of theorem. �Using Theorem 2.1, the principal tool of our result will be to approximate (D +

S)−1 � A−1. As ||S|| approach zero, the approximation error approaches zero qua-dratically.

Remark 2.1 As derived in Ravi and Wendell (1985) and Neralic and Wendell (2004),for small perturbations �B in B,

(B + �B)−1 � B−1 − B−1�B B−1

1 + tr(B−1�B), (2.1)

provided 1 + tr(B−1�B) = 0, where tr(X) denotes the trace of a matrix X .The approximation to the inverse of (D + S) in Theorem 2.1 is consistent with (2.1)

for tr(D−1S) = 0.

3 The Analysis of the Single-Constraint Case

First, we consider a problem with only a budget constraint:

min

{−tμ′x + 1

2x ′(D + S)x | l ′x = 1

}. (3.1)

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By solving (3.1), we can obtain the following lemma.

Lemma 3.1 Let Assumption 2.1 be satisfied. The optimal solution of (3.1) is

x = h0 + t h1,

where h0 = (D+S)−1ll ′(D+S)−1l

and h1 = (D + S)−1μ − l ′(D+S)−1μ

l ′(D+S)−1l(D + S)−1l.

The multiplier for the budget constraint is

u = −1

l ′(D + S)−1l+ t

l ′(D + S)−1μ

l ′(D + S)−1l.

Proof Because (D + S) is positive definite, the objective function for (3.1) is strictlyconvex. Furthermore, the sole constraint of (3.1) is linear. Therefore the Karush–Kuhn–Tucker (KKT) conditions are both necessary and sufficient for optimality.

The KKT conditions for (3.1) assert that

tμ − (D + S)x = ul and l ′x = 1, (3.2)

where u is the multiplier for the budget constraint.Solving for x gives

x = −u(D + S)−1l + t (D + S)−1μ. (3.3)

Applying the budget constraint gives a single equation for u, we obtain

u = −1

l ′(D + S)−1l+ t

l ′(D + S)−1μ

l ′(D + S)−1l.

Substituting for u in (3.3) gives

x = h0 + t h1,

where h0 and h1 are in the statement of lemma. The proof of the lemma is complete.�

In this paper, the symbol ( ˜) is used for approximation quantities compared with theexact ones and the symbol ( ˆ ) is used for diagonal quantities in Best and Hlouskova(2000). First, we introduce the approximation for 1

l ′(D+S)−1lin the following lemma.

Lemma 3.2 Let Assumption 2.1 be satisfied and A−1 be as in Theorem 2.1(a). Thenthe following holds:

1

l ′(D + S)−1l= 1

l ′ A−1l+ O(||S||2).

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Proof Applying Theorem 2.1 (a) to (D + S)−1 gives

1

l ′(D + S)−1l= 1

l ′(A−1 + E(S))l= 1

l ′ A−1l

1

1 + l ′ E(S)ll ′ A−1l

. (3.4)

Let e = l ′ E(S)ll ′ A−1l

. From Definition 2.1 it follows that |e| ≤ 1|l ′ A−1l| ||E(S)||. By

Assumption 2.1 and Definition 2.2, we have e = O(||S||2). Then (3.4) can be writtenas

1

l ′(D + S)−1l= 1

l ′ A−1l+

∞∑i=1

ei 1

l ′ A−1l.

Because

∣∣∣∣∞∑

i=1ei 1

l ′ A−1l

∣∣∣∣ ≤∣∣∣ 1

l ′ A−1l

∣∣∣ ∞∑i=1

|e|i =∣∣∣ 1

l ′ A−1l

∣∣∣ |e|1−|e| , Definition 2.2 implies

1

l ′(D + S)−1l= 1

l ′ A−1l+ O(||S||2),

from which the lemma follows. �The principal approximation for the efficient portfolios for (3.1) is formulated in

the following theorem.

Theorem 3.1 Let Assumption 2.1 be satisfied. Then

(a) Let

x = h0 + t h1,

where h0 = A−1ll ′ A−1l

and h1 = A−1μ − l ′ A−1μ

l ′ A−1lA−1l and let x be the exact solu-

tion for (3.1) given by Lemma 3.1. Then the exact efficient portfolio x can beapproximated by x and x − x = O(||S||2).

(b) Let

u = −1

l ′ A−1l+ t

l ′ A−1μ

l ′ A−1l

and let u be the exact multiplier for (3.1) given by Lemma 3.1. Then multiplier ucan be approximated by u and u − u = O(||S||2).

Proof From the discussion of x in Lemma 3.1, x in the statement of part (a), Defini-tion 2.1, Theorem 2.1 and Lemma 3.2, it follows that

||h0 − h0|| ≤∣∣∣∣ 1

l ′ A−1l

∣∣∣∣ O(||S||2) + ||A−1l|| O(||S||2).

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By Definition 2.2, we have

h0 − h0 = O(||S||2).

Similarly, we obtain

h1 − h1 = O(||S||2).

These results show that h0 and h1 can be approximated by h0 and h1 to withinO(||S||2), respectively. Hence, for any fixed t ,

x − x = (h0 − h0) + t (h1 − h1) = O(||S||2),

which verifies part (a).The proof of part (b) is similar to that of part (a). �

Remark 3.1 Theorem 3.1 gives an approximate solution for (3.1). It is straightforwardto see that the approximate solution x in Theorem 3.1 is also the optimal solution ofthe following n-dimension problem:

min

{−tμ′ x + 1

2x ′ Ax | l ′ x = 1

}, (3.5)

where A−1 = D−1 − D−1SD−1.The approximate multiplier u in Theorem 3.1 is the multiplier for the budget con-

straint l ′ x = 1.

We illustrate the key results with the following small numerical example.

Example 3.1 Consider a universe with n = 3 assets, μ = (1.1, 1.2, 1.3)′, D =

diag(0.01, 0.05, 0.07) and S =⎡⎣ 0 0.5 −0.9

0.5 0 −0.70.9 −0.7 0

⎤⎦×10−3. Note that ||S|| = 0.0016.

Then the exact optimal solution for (3.1) is obtained from Lemma 3.1 as

u = −0.0074 + 1.1375t,

x = (0.7417, 0.1419, 0.1164)′ + (−3.6073, 1.3187, 2.2886)′t.

From these we can obtain the expected return and variance of the portfolio as

μp = μ′x = 1.1375 + 0.5896t and σ 2p = x ′(D + S)x = 0.0074 + 0.5896t2.

From Theorem 3.1, the error of approximation in terms of A−1 of the inverse of(D + S) is obtained as ||(D + S)−1 − A−1|| = 0.1882. The approximate quantities

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6 7 8 9 10 11 12 13

x 10−3

1.13

1.14

1.15

1.16

1.17

1.18

Portfolio Variance σp2

Por

tfolio

Mea

n μ p

Efficient Frontier: Mean−Variance Space

ExactApproximateDiagonal

Fig. 1 The exact, approximate and diagonal efficient frontiers of a 3-asset problem with only a budgetconstraint, for every possible value of the risk aversion parameter t ≥ 0

u, x, μp and σ 2p can be obtained in terms of t :

u = −0.0074 + 1.1375t,

x = (0.7414, 0.1420, 0.1166)′ + (−3.6052, 1.3197, 2.2855)′t,μp = μ′ x = 1.1375 + 0.5891t and σ 2

p = x ′(D + S)x = 0.0074 + 0.5885t2,

respectively.We obtain an additional approximation by simply ignoring S and solving the result-

ing diagonal problem exactly. This gives the following quantities u, x, μp and σ 2p in

terms of t :

u = −0.0074 + 1.1362t,

x = (0.7447, 0.1489, 0.1064)′ + (−3.6170, 1.2766, 2.3404)′t,μp = μ′ x = 1.1362 + 0.5957t and σ 2

p = x ′ Dx = 0.0074 + 0.5957t2,

respectively. The error of diagonal approximation in terms of D−1 of the inverse of(D + S) is obtained as ||(D + S)−1 − D−1|| = 2.4259. Figure 1 shows that theapproximation of the optimal solution for (3.1) given by Theorems 3.1 is better thanthe diagonal approximation given by Best and Hlouskova (2000).

The portfolio selection problem with only the budget constraint, (3.1), is a specialcase of the portfolio selection problems with only equality constraints, which haveexplicit solutions. The exact solution for (3.1) is shown in Lemma 3.1. Theorem 3.1

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develops an approximate solution for (3.1), which is obtained without computingthe inverse of covariance matrix. Example 3.1 and Fig. 1 show the efficiency of theapproximation result. We will use the approximation result for the discussion of port-folio selection problem with inequality constraints in the following section.

4 Portfolio Optimization with no Short Sales

The problem to be analyzed is the following n-dimensional problem with no shortsales:

min

{−tμ′x + 1

2x ′(D + S)x | l ′x = 1, x ≥ 0

}. (4.1)

With Assumption 2.1, the KKT conditions for (4.1) are both necessary and sufficientfor optimality. These conditions are

tμ − (D + S)x = ul − v,

l ′x = 1, x ≥ 0,

v′x = 0, v ≥ 0,

⎫⎬⎭ (4.2)

where u is the multiplier for the budget constraint l ′x = 1 and v is n−dimensionalmultiplier vector for the no short sales constraints x ≥ 0.

Our analysis for (4.1) will require the following assumption to be satisfied in addi-tion to Assumption 2.1.

Assumption 4.1 (a) t ≥ 0, (b) μ1 < · · · < μn .

When Assumption 4.1 is satisfied, it is shown in Best and Hlouskova (2000), for(4.1) with � = D (i.e., S = 0), there are n intervals and the individual componentsof a portfolio are reduced to zero one by one and subsequently remain at zero as tincreases. This occurs in the order of the assets’ expected returns from the smallestto the largest (see the “Diagonal” part in Table 1). Motivated by the structure of theefficient frontier in Best and Hlouskova (2000), for k = 0, . . . , n − 1, we consideran (n − k)-dimensional problem with no inequality constraints and which is closelyrelated to (3.1) and (4.1):

min

{−tμ′−k x−k + 1

2x ′−k(D−k + S−k)x−k | l ′−k x−k = 1

}, (4.3)

where x−k is an (n − k)-vector, μ−k is an (n − k)-vector obtained from μ by deletingthe first k elements. (D−k + S−k) is an (n −k, n −k) matrix obtained from (D + S) bydeleting the first k rows and columns. l−k is an (n − k)-vector of one’s. When k = 0,problem (4.3) is problem (3.1). Then we have the following lemma.

Lemma 4.1 Let Assumption 2.1 be satisfied and let �0,−k = 1l ′−k (D−k+S−k )

−1l−kand

�1,−k = �0,−kl ′−k(D−k + S−k)−1μ−k . The optimal solution of (4.3) is

x−k = h0,−k + t h1,−k,

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Table 1 The exact, approximate and diagonal results of the whole intervals for the 3-asset problem withno short sales

Exact Approximate Diagonal

t j 0 0.2056 0.7070 t j 0 0.2057 0.7071 t j 0 0.2059 0.7

h0 j 0.7417 0 0 h0 j 0.7414 0 0 h0 j 0.7414 0 0

0.1419 0.5824 0 0.1420 0.5824 0 0.1489 0.5833 0

0.1164 0.4176 1 0.1166 0.4176 1 0.1064 0.4176 1

h1 j −3.6073 0 0 h1 j −3.6052 0 0 h1 j −3.6170 0 0

1.3187 −0.8237 0 1.3197 −0.8236 0 1.2766 −0.8333 0

2.2886 0.8237 0 2.2855 0.8236 0 2.3404 0.8333 0

μp(t j ) 1.1375 1.2587 1.30 μp(t j ) 1.1375 1.2587 1.30 μp(t j ) 1.1362 1.2588 1.30

σp(t j ) 0.0074 0.0323 0.07 σp(t j ) 0.0074 0.0323 0.07 σp(t j ) 0.0074 0.0327 0.07

where h0,−k = �0,−k(D−k + S−k)−1l−k and h1,−k = (D−k + S−k)

−1μ−k −�1,−k(D−k + S−k)

−1l−k .The multiplier for the budget constraint is

u−k = −�0,−k + t�1,−k .

Proof For k = 0, . . . , n − 1, Lemma 4.1 follows from Lemma 3.1 with the notation�0,−k and �1,−k . �

Remark 4.1 Consider the following n-dimensional problem

min

{−tμ′x + 1

2x ′(D + S)x | l ′x = 1, x1 = 0, x2 = 0, . . . , xk = 0

}, (4.4)

for k = 0, . . . , n − 1.The problem (4.4) is the portfolio selection problem with n risky assets, for the

first k assets are held of 0. Thus solving (4.4) is equivalent to solving (4.3). Theoptimal solution and multiplier for the budget constraint for (4.4) respectively arex = (0, 0, · · · , 0︸︷︷︸

k

, x ′−k)′ and u = u−k , where x−k and u−k are given in Lemma 4.1.

The multipliers for xi = 0, i = 1, . . . , k are

(v−k)i = u − tμi + (Di + Si )x, i = 1, . . . , k, (4.5)

where (Di + Si ) is the i th row of matrix (D + S).Let P0,−k = −�0,−k lk + Skh0,k and P1,−k = �1,−k lk + Skh1,−k, where lk is a

k-vector of one’s. Sk is a (k, n − k) matrix obtained from the first k rows of (D + S)

without the first k columns and Dk = 0. Then (4.5) can be expressed as

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v−k = P0,−k + t (P1,−k − μk), (4.6)

where μk is the vector of the first k components of μ.

For k = −1, . . . , n − 1, define

tk+1 =

⎧⎪⎨⎪⎩

0, k = −1,−(h0,−k)1(h1,−k )1

, k = 0, . . . , n − 2,

∞, k = n − 1,

(4.7)

For k = 0, . . . , n − 1, define

(xk(t))i ={

0, i = 1, . . . , k,

(h0,−k)i−k + t (h1,−k)i−k, i = k + 1, . . . , n,(4.8)

uk(t) = −�0,−k + t�1,−k, (4.9)

(vk(t))i ={

(P0,−k)i + t [(P1,−k)i − μi ], i = 1, . . . , k,

0, i = k + 1, . . . , n,(4.10)

where �0,−k,�1,−k, h0,−k and h1,−k are defined in Lemma 4.1. P0,−k and P1,−k aredefined in Remark 4.1.

Analogous to Theorem 2.1 and 3.1, for problems (4.3) and (4.4), we have thefollowing approximation results.

Theorem 4.1 Let Assumption 2.1 be satisfied and A−1−k = D−1

−k − D−1−k S−k D−1

−k . Let

�0,−k = 1l ′−k A−1

−kl−kand �1,−k = �0,−kl ′−k A−1

−kμ−k . Then for problems (4.3) and

(4.4), the following holds:

(a) Let x−k = h0,−k +t h1,−k, where h0,−k = �0,−k A−1−kl−k and h1,−k = A−1

−kμ−k −�1,−k A−1

−kl−k . The exact efficient portfolio x−k in Lemma 4.1 can be approxi-mated by x−k and x−k − x−k = O(||S||2).

(b) Let u−k = −�0,−k + t�1,−k . The multiplier u−k in Lemma 4.1 can be approxi-mated by u−k and u−k − u−k = O(||S||2).

(c) Let P0,−k = −�0,−k lk + Sk h0,−k, P1,−k = �1,−k lk + Sk h1,−k, and v−k =P0,k + t (P1,k − μk). The multiplier vector v−k in (4.6) can be approximated byv−k and v−k − v−k = O(||S||2).

Proof The proof of (a) and (b) is similar to the proof of Theorem 3.1.For part (c), because Q0,−k − Q0,−k = O(||S||2), Q1,−k − Q1,−k = O(||S||2),

h0,−k − h0,−k = O(||S||2) and h1,−k − h1,−k = O(||S||2), from the definition of v−k

and v−k , it follows that

v−k − v−k = O(||S||2),

which completes the proof of the theorem. �

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Remark 4.2 Theorem 4.1 gives an approximate solution for (4.3). We show the approx-imate solution x−k in Theorem 4.1 is also the optimal solution of the following (n−k)-dimensional problem:

min

{−tμ′−k x−k + 1

2x ′−k A−k x−k | l ′ x−k = 1

}, (4.11)

where A−1−k = D−1

−k − D−1−k S−k D−1

−k .The approximate multiplier u−k in Theorem 4.1 is the multiplier for the budget

constraint l ′ x−k = 1.

For k = −1, . . . , n − 1, define

tk+1 =

⎧⎪⎪⎨⎪⎪⎩

0, k = −1,

−(h0,−k )1

(h1,−k )1= �0,−k (1−∑n

i=k+2σk+1,i

σi)

�1,−k (1−∑ni=k+2

σk+1,iσi

)−(μk+1−∑ni=k+2

σk+1,i μiσi

), k = 0, . . . , n − 2,

∞, k = n − 1.

(4.12)

For k = 0, . . . , n − 1, define

(xk(t))i ={

0, i = 1, . . . , k,

(h0,−k)i−k + t (h1,−k)i−k, i = k + 1, . . . , n,(4.13)

uk(t) = −�0,−k + t�1,−k, (4.14)

(vk(t))i ={

(P0,k)i + t [(P1,k)i − μi ], i = 1, . . . , k,

0, i = k + 1, . . . , n,(4.15)

where �0,−k, �1,−k, h0,−k, h1,−k, P0,−k and P1,−k are defined in Theorem 4.1.From the KKT conditions (4.2) for (4.1), if we want to show the results (4.7–4.10)

are also the optimal solution for (4.1), we need to prove the following conditions: theend of the interval is increasing (tk < tk+1), the optimal solution xk(t) is nonnegative(xk(t) ≥ 0, for all t ∈ [tk, tk+1]) and the multiplier vector for the nonnegativity con-ditions are all nonnegative (vk(t) ≥ 0, for all t ∈ [tk, tk+1] ). However, because of theunknown form of (D−k + S−k)

−1, it is difficult to analyze the structure of the efficientfrontier of (4.1) directly. By applying Theorem 2.1, we can use the results (4.12–4.15)to analyze the exact results (4.7–4.10) approximately, in which the error is O(||S||2).

First, for t = 0 and k = 0, from (4.13), it follows that

x0(0) = h0,0 = 1

l ′ A−1lA−1l.

From Assumption 2.1 and Theorem 2.1, we have 1l ′ A−1l

> 0 and ||D−1S|| <

||I || = 1. Then x0(0) > 0, which imply all assets are held positively when t = 0.

123


Lemma 4.2 Let Assumption 2.1 and 4.1 be satisfied and let t0, . . . , tn be defined by(4.12). Then for k = 0, . . . , n − 1, we have

(a) tk < tk+1,

(b) xk(t) ≥ 0, for all t ∈ [tk, tk+1], where xk(t) is given by (4.13).(c) vk(t) ≥ 0, for all t ∈ [tk, tk+1], where vk(t) is given by (4.15).

Proof (a) From Assumption 2.1 and similar to the proof of Lemma 3.2, we obtain

�0,−k = 1

l ′−k D−1−k l−k

+ O(||S||) and �1,−k = l ′−k D−1−k μ−k


+ O(||S||).

From Theorem 4.1 and Definition 2.2, it follows that

(h0,−k)1 = σ−1k+1


+ O(||S||)

and

(h1,−k)1 = μk+1σ−1k+1 − l ′−k D−1

−k μ−k


σ−1k+1 + O(||S||).

Let

tk+1 = 1

l ′−k D−1−k μ−k − l ′−k D−1

−k l−kμk+1.

Then

tk+1 = tk+1 + O(||S||).

Note that tk+1 is the same as the end of the intervals for the diagonal problemobtained in Best and Hlouskova (2000). From the increasing order of the end ofintervals for the diagonal case, that is, tk < tk+1, it follows that

tk < tk+1

and this completes the proof of part (a).(b) From Theorem 4.1 and Definition 2.2,

h0,−k = D−1−k l−k


+ O(||S||)

and

h1,−k = D−1−k μ−k − l ′−k D−1

−k μ−k


D−1−k l−k + O(||S||).

123


Let

x−k = D−1−k l−k


+ t[

D−1−k μ−k − l ′−k D−1

−k μ−k


D−1−k l−k

].

Then

x−k = x−k + O(||S||).

Because x−k ≥ 0, for all 0 ≤ t ≤ tk+1, which is obtained in Best and Hlouskova(2000), from Assumption 2.1, we have

x−k ≥ 0,

for all 0 ≤ t ≤ tk+1 and this completes the proof of part (b).(c) Because P0,−k = −�0,−k lk + Sk h0,−k and P1,−k = �1,−k lk + Sk h1,−k, by

Assumption 2.1, we have

P0,−k = −�0,−k lk + O(||S||) = −1


+ O(||S||)

and

P1,−k = �1,−k lk + +O(||S||) = l ′−k D−1−k μ−k


+ O(||S||).

Let

(v−k)i = −1


+ t

[l ′−k D−1

−k μ−k


− μi

].

Then

v−k = v−k + O(||S||).

Following that v−k ≥ 0 for all t with t ≥ tk in Best and Hlouskova (2000), fromAssumption 2.1, we have

vk(t) ≥ 0.

for all t with t ≥ tk .�

The principal result for (4.3) with t ≥ 0 is the following theorem.

123


Theorem 4.2 Let Assumptions 2.1 and 4.1 be satisfied and let t0, . . . , tn be definedby (4.7). Then for k = 0, . . . , n − 1, we have

(a) tk < tk+1,

(b) x(t) = xk(t), for all t ∈ [tk, tk+1], is optimal for (4.1) with xk(t) being given by(4.8),

(c) the multiplier for the budget constraint is given by u(t) = uk(t), for all t ∈[tk, tk+1], where uk(t) is given by (4.9),

(d) the multipliers for x ≥ 0 are given by v(t) = vk(t), for all t ∈ [tk, tk+1], wherevk(t) is given by (4.10).

Proof From Assumption 2.1, Assumption 4.1, Q0,−k − Q0,−k = O(||S||2), Q1,−k −Q1,−k = O(||S||2), h0,−k − h0,−k = O(||S||2) and h1,−k − h1,−k = O(||S||2),it follows that

tk = tk + O(||S||2), k = 1, . . . , n − 1.

The result tk < tk+1 in Lemma 4.2 implies that

tk < tk+1,

and this completes the proof of part (a).Let 0 ≤ k ≤ n − 1 and xk, uk, vk be as in the statement of (4.8–4.10). It follows

directly from Lemma 4.1 and Remark 4.1 that l ′xk(t) = 1 and tμ − (D + S)xk(t) =uk(t)l − vk(t). The definition of xk(t) and vk(t) imply that x ′

k(t)vk(t) = 0.From Theorem 4.1 and Lemma 4.2, it follows that

(xk(t)) ≥ 0, t ∈ [tk, tk+1]

and

(vk(t)) ≥ 0, t ∈ [tk, tk+1],

which completes the proof the theorem. �

In Theorem 4.2, we show that the formulae (4.7–4.10) are the optimal solution for(4.1). Moreover, we also present an approximation solution for (4.1), in which, as ||S||approach zero, the approximation error approaches zero quadratically.

Our theory states that the holdings in the first asset will be reduced to zero at the endof the first interval. The other components will be nonzero. At the end of the secondinterval, both the first and second assets will be reduced to zero, and all the remainingwill be nonzero, and so on. At the end of the efficient frontier, all assets will be reducedto zero except the last (having the largest expected return) which has value unity.

123


Table 2 The exact, approximate and diagonal multipliers of the entire intervals for the 3-asset problemwith no short sales

Exact Approximate Diagonal

t j 0 0.2056 0.707 t j 0 0.2057 0.7071 t j 0 0.2059 0.7

�0 j −0.0074 −0.0288 −0.07 �0 j −0.0074 −0.0288 −0.07 �0 j −0.0074 −0.0292−0.07

�1 j 1.1375 1.2418 1.30 �1 j 1.1375 1.2418 1.30 �1 j 1.1362 1.2417 1.30

P0 j 0 −0.0289 −0.0709 P0 j 0 −0.0289 −0.0709 P0 j 0 −0.0292−0.07

0 0 −0.0707 0 0 −0.0707 0 0 −0.07

0 0 0 0 0 0 0 0 0

Q1 j 0 0.1406 0.2 Q1 j 0 0.1417 0.1991 Q1 j 0 0.1417 0.2

0 0 0.1 0 0 0.0993 0 0 0.1

0 0 0 0 0 0 0 0 0

5 Numerical Example

We consider the 3-asset problem discussed in Sect. 3 with no short sales restrictionsand discuss the entire efficient frontier and efficient portfolio for t ≥ 0. The exactresults for (4.1) are computed by using (4.7–4.10). The approximate results for (4.1)are computed by using (4.12–4.15). The diagonal results are obtained by the relatedformula in Best and Hlouskova (2000). All the results are summarized in Tables 1and 2.

In Table 1, for the ‘Exact’ part, t j in the 2nd row means the end of a interval;the following h0 j and h1 j are the two parts of the optimal solution, that is, x j (t) =h0 j + th1 j , for t ∈ [t j , t j+1]. Moreover, μp(t j ) is the expected return of the opti-mal portfolio when t = t j and σp(t j ) is the variance of the optimal portfolio whent = t j . t j , h0 j , h1 j , μp(t j ) and σp(t j ) in ‘Approximate’ part and t j , h0 j , h1 j , μp(t j )

and σp(t j ) in ‘Diagonal’ part are the approximations of the related values in the ‘Exact’part.

In Table 2, for the ‘Exact’ part, �0 j and �1 j are two parts of the multiplier for thebudget constraint, that is, u j (t) = �0 j + t�1 j , for t ∈ [t j , t j+1]. P0 j and Q1 j are twoparts of the multiplier vector for the no short sales constraint, that is, v j (t) = P0 j +t Q1 j , for t ∈ [t j , t j+1]. Similarly, in the ‘Approximate’ part, u j (t) = �0 j + t�1 j andv j (t) = P0 j + t Q1 j , for t ∈ [t j , t j+1]. In the ‘Approximate’ part, u j (t) = �0 j + t�1 j

and v j (t) = P0 j + t Q1 j , for t ∈ [t j , t j+1].Therefore, the exact efficient frontier is constructed by (μp(t), σp(t)) for t ∈

[0,∞), where μp(t) = μ′h0 j + tμ′h1 j and σp(t) = (h0 j + th1 j )′(D + S)(h0 j +

th1 j ), t ∈ [t j , t j+1]. In order to illustrate the structure obtained by (4.7–4.10), we alsoapply the PQP program presented in Best (1996) and (2010) to verify the exact opti-mal solution which has exactly the same result and structure. In the similar way, theapproximate efficient frontier is constructed by (μp(t), σp(t)) for t ∈ [0,∞), whereμp(t) = μ′h0 j +tμ′h1 j and σp(t) = (h0 j +t h1 j )

′(D+S)(h0 j +t h1 j ), t ∈ [t j , t j+1].The diagonal efficient frontier is constructed by (μp(t), σp(t)) for t ∈ [0,∞), whereμp(t) = μ′h0 j + tμ′h1 j and σp(t) = (h0 j + t h1 j )

′D(h0 j + t h1 j ), t ∈ [t j , t j+1].

123


0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08

1.12

1.14

1.16

1.18

1.2

1.22

1.24

1.26

1.28

1.3

t0

t1

t2

Portfolio Variance σp2

Por

tfolio

Mea

n μ p

Efficient Frontier: Mean−Variance Space

Exactthe end of the exact intervalsApproximatethe end of the approximate intervalsDiagonalthe end of the diagonal intervals

Fig. 2 Overview of the three kinds of efficient frontiers for 3-assets with no short sales

The three cases of efficient frontiers are summarized in Fig. 2. From Fig. 2, theexact, approximate and diagonal efficient frontiers have the similar structures. Weenlarge the three parts of efficient frontiers around the ends of intervals and displaythe results in Fig. 3. Figure 3 shows that the approximation derived in this paper iscloser to the exact efficient frontier and is a better approximation than the diagonal one.This verifies that our approximate error is smaller than the diagonal approximation inthis case.

6 Conclusion

For general portfolio optimization with inequality constraints, the components of theoptimal solution are piecewise linear functions of the risk aversion parameter and theoptimal value of the objective function is piecewise quadratic. However, the number ofparametric intervals is not known a priori and may be quite large. It is often impossibleto obtain a closed form solution and analyze the expected return and the variance ofthe efficient portfolios. In this paper, we discussed the structure of the efficient frontierfor a dense covariance matrix in which the off diagonal elements are small relative tothe diagonal ones. We developed an explicit approximate solution for the entire effi-cient frontier, in which the approximation error approaches zero quadratically as thenorm of its off-diagonal matrix approaches zero. By using the approximation results,we found that the structure of the efficient frontier of weakly correlated assets is asfollows. As the risk aversion parameter value increases, the asset holding componentsin a portfolio are reduced to zero one by one and remain there in the order of theassets’ associated expected returns from the smallest to the largest. The structure ofthe efficient frontier and the approximation derived in the paper were illustrated by a

123


7.2 7.4 7.6

x 10−3

1.136

1.137

1.138

1.139

1.14

1.141

1.142

1.143

t0

Around t0

0.03 0.032 0.034

1.258

1.2585

1.259

1.2595

1.26

1.2605

1.261

t1

Around t1

0.06 0.07

1.2975

1.298

1.2985

1.299

1.2995

1.3

1.3005

t2

Around t2

Fig. 3 The comparisons of the three kinds of efficient frontiers around the end of the intervals

3-asset portfolio optimization problem. It is shown that the derived approximation isbetter than the diagonal approximation obtained from ignoring the off diagonal terms.

Acknowledgements This research was supported by the National Science and Engineering ResearchCouncil of Canada and China Scholarship Council.

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The Efficient Frontier for Weakly Correlated Assets