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Finance Research Letters
www.elsevier.com/locate/frl
On the nature of mean-variance spanning
C. Sherman Cheung, Clarence C.Y. Kwan
∗, Dean C. Mountain
DeGroote School of Business, McMaster University, Hamilton, Ontario, Canada L8S 4M4
a r t i c l e i n f o a b s t r a c t
Article history: Received 24 April 2008 Accepted 8 December 2008 Available online 13 December 2008 JEL classification:
D81 G10 G11 G12 Keywords: Asset spanning Portfolio choice
Asset spanning tests are very useful tools for the determination of which asset classes belong to an investor’s portfolio. There are numerous applications of such tools in the finance literature. What is not so obvious is the proper decision an investor should make if the extra asset classes are spanned by some existing assets. Should the investor make a conscious decision not to invest in them as they add no value? Should the investor invest in them anyway as they do no harm? This study provides an analytical solution to the puzzle and also offers an economic rationale.
2008 Elsevier Inc. All rights reserved.
A major issue confronting a mean-variance optimizing investor is whether new asset classes should be added to an existing portfolio. The alternative way to ask the same question is whether the additional asset classes improve the efficient frontier for the investor.Huberman and Kandel (1987) in-troduced the concepts of asset spanning for the very purpose of addressing the above issue. If diversi-fying into additional asset classes has no impact on the efficient set for aspecificquadratic utility func-tion, the efficient frontier with the additional assets intersects the efficient frontier without the addi-tional assets. If the addiaddi-tional assets offer no improvement in the efficient frontier foranyquadratic utility function, the efficient frontier of the initial assets spans the efficient frontier of the enlarged set of assets. Huberman and Kandel proposed regression-based tests of spanning and intersection. Their contribution has already found its way to the textbook byCampbell et al. (1997). As pointed out by
Jobson and Korkie (1989), asset set intersection and spanning are related to performance measures. The concepts of asset spanning has been applied to a variety of settings. DeSantis (1994) and
Bekaert and Urias (1996) considered whether the expanded frontier with emerging market stocks is spanned by the initial frontier with only developed markets. De Roon et al. (2001)derived the tests
withoutshort sales and empirically found the spanning of emerging markets by developed markets.
*
Corresponding author.E-mail address:kwanc@mcmaster.ca(C.C.Y. Kwan).
1544-6123/$ – see front matter 2008 Elsevier Inc. All rights reserved.
What are the implications for an investor who is confronted with the evidence that certain asset classes are spanned? This may appear to be a trivial question not worthy of further analysis. One quick answer is not to include the new asset classes in the existing benchmark portfolio. Intuitively, this is by no means the only answer. A combination of the existing benchmark portfolio and the new asset classes may also be a rational decision.1 While spanning implies equal performance of the benchmark portfolio and the expanded portfolio, equal performance, in turn, implies that the investor should be indifferent to these portfolios. Given the indifference, it is impossible to tell whether one should invest in the extra asset classes, thus resulting in an ambiguous situation. This is very different from a conscious decision not to invest in the extra asset classes. Thus, when an asset is judged to be spanned by a benchmark portfolio, there are two potential outcomes to consider. One is that the optimal investment in the spanned asset is uniquely zero. The other outcome is that the optimal investment in the spanned asset is indeterminate. The literature is completely silent on the correct decision an investor should make when spanning prevails. This puzzle calls for further investigation of the nature of spanning in the performance framework.
The study examines this issue, heretofore not discussed in the spanning literature. We use a setup in which no assumptions are made regarding any possible positions in the new asset classes. Instead, we assume as a starting point of our analysis the benchmark frontier to coincide with the expanded frontier when spanning occurs. This agnostic assumption concerning possible positions in the new asset classes is extremely flexible since it allows for infinite combinations of the benchmark portfolio and the new asset classes as possible outcomes. We show that spanning implies a decision not to in-vest in the extra asset classes. We offer an analytical proof and economic intuition why the conscious decision not to invest is indeed the only rational outcome.
1. Preliminaries
For ann-asset case, let
µ
be ann-element column vector of expected returns andVbe ann×
ncovariance matrix, with individual elements labeled as
μi
andσi j
, respectively, fori,
j=
1,
2, . . . ,
n. In addition, letxbe ann-element column vector of portfolio weights, with individual elements beingxi, and letι
be a column vector of ones with the same dimension.The optimization problem with frictionless short sales can be stated as minimization ofx′Vx, sub-ject tox′
µ
=
q andx′ι
=
1, where each prime indicates matrix transposition and q is the required expected return. The Lagrangian of the optimization problem isL
=
x′Vx−
λ(
x′µ
−
q)
−
θ (
x′ι
−
1),
(1)where
λ
andθ
are Lagrange multipliers. FollowingRoll (1977), minimization ofLleads tox
=
V−1MM′V−1M−1q,
(2)whereM
= [
µ ι
]
is ann×
2 matrix andq= [
q 1]
′ is a two-element column vector. Thus, given the expected return vectorµ
, the covariance matrixV, and the required expected returnq, the efficient portfolio weight vectorxcan be determined directly.The variance of returns of the efficient portfolio corresponding to each given value ofqis
σ
2 p=
x′
Vx
=
q′M′V−1M−1q.
(3)As shown below, Eqs.(2) and (3)enable us to establish the conditions for spanning.
2. Spanning ofN
+
K assets byK assetsAs mentioned earlier, the coincidence of the two efficient frontiers is the starting point of our analysis. IfK benchmark assets are able to span N
+
K assets, the frontier based on theK assets mustbe identical to the frontier based on theN
+
K assets. For all values of the required expected returnq, the corresponding values ofσ
2p for the two cases must be the same. Thus, in case of spanning, the 2
×
2 matrixM′V−1Mmust remain the same, regardless of whether it is based on theK assets or theN
+
K assets. As shown in the following, this analytical property ensures that no investment funds be allocated to theN assets and that the allocations among theK assets in an(
N+
K)
-asset portfolio be the same as those based on the K assets alone.In order to establish the conditions of spanning for the case where the covariance matrix based on the expanded set of assets is invertible, we partition the expanded covariance matrix as follows:
V
=
VN N VN K VK N VK K
,
(4)whereVN N isN
×
N,VN K isN×
K,VK NisK×
N, andVK K isK×
K. We also partition the expanded expected return vectorµ
into column vectorsµ
N andµ
K with N and K elements, respectively. Likewise, the corresponding(
N+
K)
-element vectorι
is partitioned into column vectorsι
N andι
K.Following the algebra of block matrix inversion, we have
V−1
=
E
−
EVN KV−K K1−
V−K K1VK NE(
V−K K1+
V−K K1VK NEVN KV−K K1)
,
(5)where
E
=
VN N−
VN KV−K K1VK N−1.
(6)PartitioningV−1in this manner allowsM′V−1Mto be directly compared with the corresponding 2
×
2 matrix for the K benchmark assets alone. As spanning requires equality of the two cases, it implies thatµ
′N
−
µ
′KV−K K1VK NE
µ
N−
VN KV−K K1µ
K=
0,
(7)µ
′N
−
µ
′KV−1 K KVK N
E
ι
N−
VN KV−K K1ι
K=
0,
and (8)ι
′N
−
ι
′KV−1 K KVK N
E
ι
N−
VN KV−K K1ι
K=
0.
(9)Notice that the left-hand side of each of these three equations is of the form where the square matrix Eis pre-multiplied by anN-element row vector and post-multiplied by anN-element column vector. In the first case as well as the third case, the two vectors are transposes of each other.
Recall that the covariance matrix Vfor the
(
N+
K)
-asset case is invertible.2 This means that V is positive definite. So is its inverse, V−1. All principal submatrices of a positive definite matrix are positive definite. As Eis a principal submatrix ofV−1,Eis positive definite as well. For any nonzero column vectoruwithN elements, we must haveu′Eu>
0. Ifu′Eu=
0, then the vectorumust be a vector of zeros. Thus, asEis positive definite, Eq.(7)implies thatµ
N=
VN KV−K K1µ
K,
(10)and Eq.(9)implies that
ι
N=
VN KV−K K1ι
K.
(11)Eqs.(10) and (11)ensure that Eq.(8)holds as well.
With the expanded vector of portfolio weights, x
,
partitioned intoxN andxK, column vectors ofN andK elements, respectively, substituting Eqs.(10) and (11)into the partitioned version of Eq.(2)
leads to xN
=
0N, for all values ofq. Here,0N is N-element column vector of zeros. Accordingly, the weights on the K benchmark assets in an(
N+
K)
-asset portfolio are the same as those based on the K assets alone. That is, spanning unambiguously implies a conscious decision not to invest in the extra asset classes and rules out any combination of the benchmark portfolio and the extra asset classes. What is not obvious from the above analysis is the economic rationale for not investing in the extra asset classes in a perfect world free of transaction costs. To shed light on this issue, we also perform the analysis from another perspective.2 The invertibility of
3. Rationale based on a tangency portfolio perspective
An alternative to the above approach in the construction of the efficient frontier is by maximizing the slope of a tangent line for each given intercept on the expected return axis, on the plane of expected return and standard deviation of returns. Each intercept can be viewed as the expected return of a portfolio (a zero-beta portfolio) that is uncorrelated in returns with the tangency portfolio. Given the equivalence of the two approaches, the spanning conditions that Eqs.(10) and (11)provide also apply to the tangency portfolio results.
Let
μ0
be the intercept of a tangent line. The tangency portfolio weights can be obtained fromy
=
V−1(
µ
−
μ0
ι
)
(12)by scaling the column vectoryin such a way that the portfolio weights sum to one, i.e.,x
=
y(
ι
′y)
−1. For the(
N+
K)
-asset case, we partitionyintoyN andyK, two column vectors withNandK elements, respectively. If the N+
K assets are spanned by theK assets and the expanded covariance matrix is invertible [implying Eqs.(10) and (11)], Eq.(12)becomesyN
=
Eµ
N−
VN KV−K K1µ
K−
μ0E
ι
N−
VN KV−K K1ι
K=
0N (13)and
yK
=
V−K K1(
µ
K−
μ0
ι
K).
(14)Therefore, spanning implies that each tangency portfolio has unique portfolio weights, with invest-ment funds allocated only among theK assets, and that theN assets always have zero weights.
The tangency portfolio setting allows us to explore further what the spanning conditions in Eqs.(10) and (11) mean. The analytical detail, which draws on the work of Stevens (1998) regard-ing the inverse of the covariance matrix, is provided inAppendix A. We start with a
(
K+
1)
-asset portfolio consisting of the K assets and an arbitrary asset, say asseta, from the set of N assets. By regressing the random return of asseta on the random returns of the remaining assets in the port-folio, we are able to establish that, if spanning occurs, its expected return is a weighted average of the expected returns of the K benchmark assets. More specifically, the weights are the correspond-ing regression coefficients. We then extend the idea recursively by augmentcorrespond-ing the portfolio with the remaining assets from the set of N assets, one at a time, until all N assets are accounted for. Re-gressing the random return of each additional asset on the random returns of the assets already in the portfolio yields an analogous result. That is, its expected return is always a weighted average of the expected returns of theK benchmark assets, with the weights being the corresponding regression coefficients.4. Further characterization of mean-variance spanning
In view of the regression approach above, the spanning conditions as provided by Eqs. (10) and (11) can be stated equivalently as follows3: Define the column vectors of random returns as r
N
=
(
r1,
r2, . . . ,
rN)
′andRK=
(
R1,
R2, . . . ,
RK)
′, with the elements there being the individual asset returns.Lemma.For a return generating process,4
ri
=
αi
+
β
′iRK+
ǫi
,
(15)where
αi
is the intercept term,β
iis a K -element column vector of parameters, andǫi
is random noise withE
(
ǫi
)
=
Cov(
ǫi
,
Rj)
=
0, for i=
1,
2, . . . ,
N and j=
1,
2, . . . ,
K , the optimal investment weight in each asseti is zero if and only if
αi
=
0andβ
′iι
K=
1.Proof. SeeAppendix A.
✷
An implicit condition here, as in any regression model, is that each error term
ǫi
has a nonzero variance. Under this condition, which ensures that the covariance matrix based on N+
K assets is invertible, the spanning conditions in Eqs. (10) and (11) can be stated in terms of the regression results. The redundancy of rN is indicated by zero investments being the optimal choice of the Nassets. In contrast, if each
ǫi
has a zero variance, as the corresponding ri is a linear combinationof R1
,
R2, . . . ,
RK, the determinant of the expanded covariance matrix is zero, thus rendering thematrix non-invertible. In such a case, for any expected return requirement, the corresponding portfolio weight vector
β
isatisfyingβ
′i
ι
K=
1 is not unique.The spanning condition in the above lemma can also be deduced from the two-fund separation condition in Huang and Litzenberger (1988, p. 85). Consider the case of N
+
K assets for portfolio construction. According to two-fund separation, given an efficient portfolio pwith random returnrp and its orthogonal portfolio z with random returnrz, satisfying the condition of Cov(
rp,
rz)
=
0, the random returnrsof any portfolioscan be stated asrs
=
β
sprp+
(
1−
β
sp)
rz+
ǫs
,
(16)where
β
sp=
Cov(
rs,
rp)/
Var(
rp)
andE(
ǫs
)
=
0, withǫs
being random noise. Using E(
rp)
andE(
rz)
for two different values ofq in Eq.(2), the corresponding allocations of investment funds in portfolios pand z can be determined. If spanning occurs, then both p andz are portfolios of the K benchmark assets. That is,
rp
=
wpK ′RK and (17)
rz
=
wzK ′RK
,
(18)where wpK and wzK are K-element column vectors of portfolio weights, satisfying the condition of
(
wpK)
′ι
K
=
(
wzK)
′ι
K=
1. Accordingly, Eq.(16)becomesrs
=
β
spwpK+
(
1−
β
sp)
wzK ′RK
+
ǫs
.
(19)As the condition of
[
β
spwpK+
(
1−
β
sp)
wzK]
′ι
K=
1 always holds andscan be any portfolio based on the N+
K assets, including any of theN individual assets, Eq.(15), withαi
being zero andβ
i being a portfolio-weight vector, follows directly.3 We thank the reviewer for this suggestion.
5. Conclusion
While spanning tests are very useful tools for the determination of which asset classes belong to a portfolio, the detection of redundant asset classes also leads to an indeterminate outcome for an investor. Since investing in the redundant asset classes causes no harm, an investor can invest or not invest in them.
We use a setup in which coincidence of the benchmark frontier with the expanded frontier is the starting point of our analysis when spanning occurs. No initial assumptions are made regarding any possible positions in the new asset classes. The decision to invest or not to invest in the extra asset classes is the outcome of our analysis. This study demonstrates that spanning implies zero weighting for the redundant asset classes. In other words, spanning leads to a unique decision not to invest. The rationale is due to the presence of residual risk associate with the extra asset class. Residual risk makes the extra asset class less attractive (for either holding long or selling short) when compared with investments in the existing benchmark assets alone. That is, as long as residual risk is present, the extra asset, which offers an expected excess return just like a portfolio of the benchmark assets, is not worth holding, either long or short.
Acknowledgments
Research support was provided by the Social Sciences and Humanities Research Council of Canada. The authors wish to thank an anonymous reviewer for helpful comments and suggestions.
Appendix A
IfK assets span theN
+
K assets, they must spanK+
1 assets, where the extra asset can be any of theN assets. Let us consider an arbitrary asseta among theN assets, with an expected returnμa
and a variance of returnsσaa
. The column vector of covariances of returns between theK benchmark assets and this asset is VK a. The transpose ofVK a is the row vector VaK. For a set of K+
1 assets spanned by theK benchmark assets, letA
=
Aaa AaK AK a AK K
=
V−1,
(A.1)the inverse of the covariance matrix of the K
+
1 assets. Here, Aaa is a scalar, AK a is a K-element column vector,AaK is the transpose ofAK a, andAK K is aK×
K matrix.The scalarAaa, according to Eq.(6), can be written as
Aaa
=
σaa
−
VaKV−K K1VK a −1.
(A.2)Likewise, the row vectorAaK, according to Eq.(5), can be written as
−
AaaVaKV−K K1. If we regress the random returns of assetaon the random returns of theK benchmark assets, with a residual errorǫa
, the row vector of the multiple regression coefficients is asymptotically5β
′a=
VaKV−K K1.
(A.3)The part of the variance of returns of asset a that is unexplained by the regression, the residual variance, is
σ
ǫ2a=
σaa
1−
R2a,
(A.4)whereR2a is the asymptotic R-square of the regression. Drawing onStevens (1998), we have directly the following results:
5 If
AaaVaKV−K K1
=
σaa
R2a,
(A.5)Aaa
=
1σaa
(
1−
Ra2)
=
1σ
ǫ2a,
and (A.6)AaK
= −
Aaaβa′=
−
β
′
a
σaa
(
1−
R2a
)
=
−
β
′
a
σ
2ǫa
.
(A.7)Eq.(13)can be written for assetain the
(
K+
1)
-asset case asya
=
(
μa
−
μ0
)
−
β
a′(
µ
K−
μ0
ι
K)
σ
2 ǫa=
0.
(A.8)As
σ
ǫ2a>
0, spanning of the K+
1 assets by the K benchmark assets impliesμa
−
μ0
=
β
′a(
µ
K−
μ0
ι
K),
(A.9)a single condition requiring that the expected excess return of asset a be a linear combination of the expected excess returns of the K benchmark assets. The choice of
μ0
to establish the tangency portfolio being arbitrary, Eq.(A.9)can be written asμa
=
β
′aµ
K and (A.10)1
=
β
′aι
K,
(A.11)which imply that, with
β
a being a K-element column vector of portfolio weights,μa
is a weighted average of the expected returns of the K benchmark assets.Eqs.(A.10) and (A.11)jointly represent the conditions for uniqueness of the portfolio weights when spanning occurs. The invertibility of the covariance matrix of an expanded set of assets, and subse-quently ya
=
0, require that the random return of the extra assetanot be a perfect linear combination of the random returns of the K benchmark assets. A perfect linear combination would result inRa2=
1 andσ
2ǫa
=
0. With bothμa
−
μ0
−
β
a′(
µ
2−
μ0
ι
2)
=
0 andσ
ǫ2a=
0, Eq. (A.8)would render ya, and accordingly the weight xa on asseta, indeterminate. Having bothμa
−
μ0
−
β
′a(
µ
2−
μ0
ι
2)
=
0 andσ
ǫ2a>
0 ensures a zero weight on asseta.Noting that the extra assetain the
(
K+
1)
-asset case can be any of theN assets, we can extend the same idea recursively. Suppose that the portfolio now consists ofK+
2 assets instead; they include the K benchmark assets, asset a, and another arbitrarily selected asset b, also among the N assets. With assetbviewed as the only extra asset in a portfolio consisting ofK+
2 assets, it will have a zero weight ifμb
is a weighted average of the expected returns of assetaand theK benchmark assets. Asμa
itself is a weighted average of the expected returns of the K benchmark assets, so isμb
. We can successively add more assets to the portfolio, one at the time, until all N assets are accounted for.Regardless of when an asseti, among i
=
1,
2, . . . ,
N, is selected recursively for the above regres-sion runs,μi
will still be the same weighted average of the expected returns of the K benchmark assets as that in the(
K+
1)
-asset case, where assetiis the extra asseta. Thus, for spanning to hold, we must haveµ
N=
β
′µ
K and (A.12)ι
N=
β
′ι
K,
(A.13)where
β
= [
β
1β
2. . .
β
N]
is a K×
N matrix. As Eqs. (A.12) and (A.13) confirm, each row i ofβ
′ represents the portfolio weights on the K benchmark assets that make the corresponding asset ifrom the set ofN assets redundant.
To prove the necessity condition of Lemma in the main text, consider the following: If
αi
=
0 andβ
′iι
K=
1, the underlying modelri=
αi
+
β
′iRK+
ǫi
with E(
ǫi
)
=
0 implies thatμi
=
β
′iµ
K, fori
=
1,
2, . . . ,
N. Withβ
′that
µ
N=
VN KV−K K1µ
K andι
N=
VN KV−K K1ι
K. Given Eq. (13), we have yN=
0N, and thus the redun-dancy of each of theN assets is assured. To prove the sufficiency condition, suppose that theN assets are redundant. This is equivalent to Eqs.(A.12) and (A.13), which jointly indicate that each rowiofβ
′ is a set of portfolio weights on theK benchmark assets corresponding to asseti among theN assets. It follows directly from the underlying modelri=
αi
+
β
′iRK+
ǫi
withE(
ǫi
)
=
0 thatμi
=
αi
+
β
′
i
µ
K,fori
=
1,
2, . . . ,
N. Thus, given Eq.(A.12), we haveαi
=
0, fori=
1,
2, . . . ,
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