On the nature of mean variance spanning
Finance Research Letters 6 (2009) 106–113
Contents lists available at ScienceDirect
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
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.
Keywords:
Asset spanning
Portfolio choice
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) introduced the concepts of asset spanning for the very purpose of addressing the above issue. If diversifying into additional asset classes has no impact on the efficient set for a specific quadratic utility function, the efficient frontier with the additional assets intersects the efficient frontier without the additional assets. If the additional assets offer no improvement in the efficient frontier for any quadratic
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 by Campbell 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
without short 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
doi:10.1016/j.frl.2008.12.003
2008 Elsevier Inc. All rights reserved.
C.S. Cheung et al. / Finance Research Letters 6 (2009) 106–113
107
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 invest 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 an n-asset case, let µ be an n-element column vector of expected returns and V be an n × n
covariance matrix, with individual elements labeled as μi and σi j , respectively, for i , j = 1, 2, . . . , n. In
addition, let x be an n-element column vector of portfolio weights, with individual elements being xi ,
and let ι be a column vector of ones with the same dimension.
The optimization problem with frictionless short sales can be stated as minimization of x′ Vx, subject to x′ µ = q and x′ ι = 1, where each prime indicates matrix transposition and q is the required
expected return. The Lagrangian of the optimization problem is
L = x′ Vx − λ(x′ µ − q) − θ(x′ ι − 1),
(1)
where λ and θ are Lagrange multipliers. Following Roll (1977), minimization of L leads to
−1
x = V−1 M M′ V−1 M
(2)
q,
[q 1]′ is a two-element column vector. Thus, given the
where M = [µ ι] is an n × 2 matrix and q =
expected return vector µ, the covariance matrix V, and the required expected return q, the efficient
portfolio weight vector x can be determined directly.
The variance of returns of the efficient portfolio corresponding to each given value of q is
−1
σ p2 = x′ Vx = q′ M′ V−1 M
q.
(3)
As shown below, Eqs. (2) and (3) enable us to establish the conditions for spanning.
2. Spanning of N + K assets by K assets
As mentioned earlier, the coincidence of the two efficient frontiers is the starting point of our
analysis. If K benchmark assets are able to span N + K assets, the frontier based on the K assets must
1
In fact, there may be infinitely many rational combinations of the benchmark portfolio and the new asset classes. We
assume no transaction costs throughout this paper. The existence of market imperfections can favor one asset class over the
others. Our concern here is portfolio decisions based purely on the risk-return characteristics of various asset classes.
108
C.S. Cheung et al. / Finance Research Letters 6 (2009) 106–113
be identical to the frontier based on the N + K assets. For all values of the required expected return q,
the corresponding values of σ p2 for the two cases must be the same. Thus, in case of spanning, the
2 × 2 matrix M′ V−1 M must remain the same, regardless of whether it is based on the K assets or the
N + K assets. As shown in the following, this analytical property ensures that no investment funds
be allocated to the N assets and that the allocations among the K 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
VK N
VN K
VK K
(4)
,
where V N N is N × N, V N K is N × K , V K N is K × N, and V K K is K × K . We also partition the expanded
expected return vector µ into column vectors µ N and µ K with N and K elements, respectively.
Likewise, the corresponding ( N + K )-element vector ι is partitioned into column vectors ι N and ι K .
Following the algebra of block matrix inversion, we have
V−1 =
where
1
−EVN K V−
KK
E
−1
−1
−1
−1
−V K K V K N E (V K K + V K K V K N EV N K V K K )
1
E = V N N − V N K V−
K K VK N
−1
,
(5)
(6)
.
Partitioning V−1 in this manner allows M′ V−1 M to 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
that
µ′N − µ′K V−K 1K V K N E µN − VN K V−K 1K µ K = 0,
′
′
−1
−1
µN − µ K V K K V K N E ιN − VN K V K K ι K = 0, and
′
′
−1
−1
ιN − ι K V K K V K N E ιN − VN K V K K ι K = 0.
(7)
(8)
(9)
Notice that the left-hand side of each of these three equations is of the form where the square matrix
E is pre-multiplied by an N-element row vector and post-multiplied by an N-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 V for 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 E is a principal submatrix of V−1 , E is positive definite as well. For any nonzero
column vector u with N elements, we must have u′ Eu > 0. If u′ Eu = 0, then the vector u must be a
vector of zeros. Thus, as E is positive definite, Eq. (7) implies that
µN = VN K V−K 1K µ K ,
(10)
and Eq. (9) implies that
ιN = VN K V−K 1K ι K .
(11)
Eqs. (10) and (11) ensure that Eq. (8) holds as well.
With the expanded vector of portfolio weights, x, partitioned into x N and x K , column vectors of
N and K elements, respectively, substituting Eqs. (10) and (11) into the partitioned version of Eq. (2)
leads to x N = 0 N , for all values of q. Here, 0 N 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 V is essential to the portfolio optimization problem and empirical spanning tests.
C.S. Cheung et al. / Finance Research Letters 6 (2009) 106–113
109
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 from
y = V−1 (µ − μ0 ι)
(12)
by scaling the column vector y in such a way that the portfolio weights sum to one, i.e., x = y(ι′ y)−1 .
For the ( N + K )-asset case, we partition y into y N and y K , two column vectors with N and K elements,
respectively. If the N + K assets are spanned by the K assets and the expanded covariance matrix is
invertible [implying Eqs. (10) and (11)], Eq. (12) becomes
yN = E
µN − VN K V−K 1K µ K − μ0 E ιN − VN K V−K 1K ι K = 0N
(13)
and
1
y K = V−
K K (µ K − μ0 ι K ).
(14)
Therefore, spanning implies that each tangency portfolio has unique portfolio weights, with investment funds allocated only among the K assets, and that the N 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) regarding the inverse of the covariance matrix, is provided in Appendix A. We start with a ( K + 1)-asset
portfolio consisting of the K assets and an arbitrary asset, say asset a, from the set of N assets. By
regressing the random return of asset a on the random returns of the remaining assets in the portfolio, 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 corresponding regression coefficients. We then extend the idea recursively by augmenting the portfolio with the
remaining assets from the set of N assets, one at a time, until all N assets are accounted for. Regressing 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 the K benchmark assets, with the weights being the corresponding regression
coefficients.
What is implicit in each of these regression runs is the presence of residual risk. Its presence
makes each of the N assets less attractive (for either holding long or selling short) when compared
with investments in the K benchmark assets alone. The extra asset, which offers an expected return just like a portfolio of the K benchmark assets, is not worth holding, either long or short. With
the expected return of the extra asset being neither higher nor lower than what can be achieved by
investing in the K benchmark assets alone, taking a nonzero position in the extra asset inevitably increases the portfolio risk and weakens the portfolio’s risk-return trade-off. Thus, having a zero holding
of the asset is the only rational decision. In the absence of residual risk, however, the augmented covariance matrix is no longer invertible. Then, the equivalence (in terms of risk-return trade-off) can be
established between investing in portfolios of the K benchmark assets alone and investing also in the
extra asset. The absence of residual risk would require that the random return of each of the N assets
be a perfect linear combination of the random returns of the K benchmark assets, thus rendering the
portfolio decision non-unique.
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C.S. Cheung et al. / Finance Research Letters 6 (2009) 106–113
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 , . . . , r N )′ and R K = ( R 1 , R 2 , . . . , R K )′ , with the elements there being the individual asset returns.
Lemma. For a return generating process,4
r i = αi + β ′i R K + ǫi ,
(15)
where αi is the intercept term, β i is a K -element column vector of parameters, and ǫi is random noise with
E (ǫi ) = Cov(ǫi , R j ) = 0, for i = 1, 2, . . . , N and j = 1, 2, . . . , K , the optimal investment weight in each asset
i is zero if and only if αi = 0 and β ′i ι K = 1.
Proof. See Appendix 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 r N is indicated by zero investments being the optimal choice of the N
assets. In contrast, if each ǫi has a zero variance, as the corresponding r i is a linear combination
of R 1 , R 2 , . . . , R K , the determinant of the expanded covariance matrix is zero, thus rendering the
matrix non-invertible. In such a case, for any expected return requirement, the corresponding portfolio
weight vector β i satisfying β ′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 p with random return r p
and its orthogonal portfolio z with random return r z , satisfying the condition of Cov(r p , r z ) = 0, the
random return r s of any portfolio s can be stated as
r s = βsp r p + (1 − βsp )r z + ǫs ,
(16)
where βsp = Cov(r s , r p )/ Var(r p ) and E (ǫs ) = 0, with ǫs being random noise. Using E (r p ) and E (r z ) for
two different values of q in Eq. (2), the corresponding allocations of investment funds in portfolios p
and z can be determined. If spanning occurs, then both p and z are portfolios of the K benchmark
assets. That is,
p ′
r p = wK RK
and
(17)
z ′
(18)
rz = wK RK ,
p
wK
and wzK are K -element column vectors of portfolio weights, satisfying the condition of
where
p ′
(w K ) ι K = (wzK )′ ι K = 1. Accordingly, Eq. (16) becomes
p
′
r s = βsp w K + (1 − βsp )wzK R K + ǫs .
p
(19)
As the condition of [βsp w K + (1 − βsp )wzK ]′ ι K = 1 always holds and s can be any portfolio based on
the N + K assets, including any of the N individual assets, Eq. (15), with αi being zero and β i being
a portfolio-weight vector, follows directly.
3
We thank the reviewer for this suggestion.
To characterize mean-variance spanning via this return generating process is consistent with Huberman and Kandel (1987).
The spanning conditions (which are αi = 0 and β ′i ι K = 1, for i = 1, 2, . . . , N) also match those in their study.
4
C.S. Cheung et al. / Finance Research Letters 6 (2009) 106–113
111
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
If K assets span the N + K assets, they must span K + 1 assets, where the extra asset can be any
of the N assets. Let us consider an arbitrary asset a among the N assets, with an expected return μa
and a variance of returns σaa . The column vector of covariances of returns between the K benchmark
assets and this asset is V K a . The transpose of V K a is the row vector VaK . For a set of K + 1 assets
spanned by the K benchmark assets, let
A=
A aa
AK a
AaK
AK K
= V−1 ,
(A.1)
the inverse of the covariance matrix of the K + 1 assets. Here, A aa is a scalar, A K a is a K -element
column vector, AaK is the transpose of A K a , and A K K is a K × K matrix.
The scalar A aa , according to Eq. (6), can be written as
A aa =
σaa − VaK V−K 1K V K a
−1
.
(A.2)
1
Likewise, the row vector AaK , according to Eq. (5), can be written as − A aa VaK V−
K K . If we regress the
random returns of asset a on the random returns of the K benchmark assets, with a residual error ǫa ,
the row vector of the multiple regression coefficients is asymptotically5
1
β a′ = VaK V−
KK.
(A.3)
The part of the variance of returns of asset a that is unexplained by the regression, the residual
variance, is
σǫ2a = σaa 1 − R a2 ,
(A.4)
where R a2 is the asymptotic R-square of the regression. Drawing on Stevens (1998), we have directly
the following results:
5
′
If
β a is the sample based estimate of the regression coefficient, it can be shown that with Cov(ǫa , R j ) = 0, for j =
′
1
1, 2, . . . , K , p lim
β a = β a′ = VaK V−
KK .
112
C.S. Cheung et al. / Finance Research Letters 6 (2009) 106–113
1
2
A aa VaK V−
K K = σaa R a ,
A aa =
1
σaa (1 − R a2 )
AaK = − A aa β a′ =
=
(A.5)
1
σǫ2a
,
and
(A.6)
−β a′
−β a′
=
.
σaa (1 − R a2 )
σǫ2a
(A.7)
Eq. (13) can be written for asset a in the ( K + 1)-asset case as
ya =
As
(μa − μ0 ) − β a′ (µ K − μ0 ι K )
σǫ2a
= 0.
(A.8)
σǫ2a > 0, spanning of the K + 1 assets by the K benchmark assets implies
μa − 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 as
μa = β a′ µ K and
′
1 = βa ιK ,
(A.10)
(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 subsequently ya = 0, require that the random return of the extra asset a not be a perfect linear combination
of the random returns of the K benchmark assets. A perfect linear combination would result in R a2 = 1
and σǫ2a = 0. With both μa − μ0 − β a′ (µ2 − μ0 ι2 ) = 0 and σǫ2a = 0, Eq. (A.8) would render ya , and
accordingly the weight xa on asset a, indeterminate. Having both μa − μ0 − β a′ (µ2 − μ0 ι2 ) = 0 and
σǫ2a > 0 ensures a zero weight on asset a.
Noting that the extra asset a in the ( K + 1)-asset case can be any of the N assets, we can extend
the same idea recursively. Suppose that the portfolio now consists of K + 2 assets instead; they include
the K benchmark assets, asset a, and another arbitrarily selected asset b, also among the N assets.
With asset b viewed as the only extra asset in a portfolio consisting of K + 2 assets, it will have a zero
weight if μb is a weighted average of the expected returns of asset a and the K benchmark assets. As
μa itself is a weighted average of the expected returns of the K benchmark assets, so is μb . We can
successively add more assets to the portfolio, one at the time, until all N assets are accounted for.
Regardless of when an asset i, among i = 1, 2, . . . , N, is selected recursively for the above regression 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 asset i is the extra asset a. Thus, for spanning to hold,
we must have
µN = β ′ µ K and
′
ιN = β ι K ,
(A.12)
(A.13)
where β = [β 1 β 2 . . . β N ] is a K × N matrix. As Eqs. (A.12) and (A.13) confirm, each row i of β ′
represents the portfolio weights on the K benchmark assets that make the corresponding asset i
from the set of N assets redundant.
To prove the necessity condition of Lemma in the main text, consider the following: If αi = 0
and β ′i ι K = 1, the underlying model r i = αi + β ′i R K + ǫi with E (ǫi ) = 0 implies that μi = β ′i µ K , for
i = 1, 2, . . . , N. With β ′i ι K = 1, β i becomes a vector of portfolio weights and μi = β ′i µ K is, accordingly,
the expected return of the corresponding portfolio. In the context of multiple regressions under the
condition of E (ǫi ) = Cov(ǫi , R j ) = 0, for i = 1, 2, . . . , N and j = 1, 2, . . . , K , the row vector of regres1
−1
′
sion coefficients is asymptotically β ′i = Vi K V−
K K , as Eq. (A.3) indicates. With β = V N K V K K , it follows
C.S. Cheung et al. / Finance Research Letters 6 (2009) 106–113
113
−1
1
that µ N = V N K V−
K K µ K and ι N = V N K V K K ι K . Given Eq. (13), we have y N = 0 N , and thus the redundancy of each of the N assets is assured. To prove the sufficiency condition, suppose that the N assets
are redundant. This is equivalent to Eqs. (A.12) and (A.13), which jointly indicate that each row i of β ′
is a set of portfolio weights on the K benchmark assets corresponding to asset i among the N assets.
It follows directly from the underlying model r i = αi + β ′i R K + ǫi with E (ǫi ) = 0 that μi = αi + β ′i µ K ,
for i = 1, 2, . . . , N. Thus, given Eq. (A.12), we have αi = 0, for i = 1, 2, . . . , N.
References
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Contents lists available at ScienceDirect
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
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.
Keywords:
Asset spanning
Portfolio choice
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) introduced the concepts of asset spanning for the very purpose of addressing the above issue. If diversifying into additional asset classes has no impact on the efficient set for a specific quadratic utility function, the efficient frontier with the additional assets intersects the efficient frontier without the additional assets. If the additional assets offer no improvement in the efficient frontier for any quadratic
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 by Campbell 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
without short 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
doi:10.1016/j.frl.2008.12.003
2008 Elsevier Inc. All rights reserved.
C.S. Cheung et al. / Finance Research Letters 6 (2009) 106–113
107
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 invest 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 an n-asset case, let µ be an n-element column vector of expected returns and V be an n × n
covariance matrix, with individual elements labeled as μi and σi j , respectively, for i , j = 1, 2, . . . , n. In
addition, let x be an n-element column vector of portfolio weights, with individual elements being xi ,
and let ι be a column vector of ones with the same dimension.
The optimization problem with frictionless short sales can be stated as minimization of x′ Vx, subject to x′ µ = q and x′ ι = 1, where each prime indicates matrix transposition and q is the required
expected return. The Lagrangian of the optimization problem is
L = x′ Vx − λ(x′ µ − q) − θ(x′ ι − 1),
(1)
where λ and θ are Lagrange multipliers. Following Roll (1977), minimization of L leads to
−1
x = V−1 M M′ V−1 M
(2)
q,
[q 1]′ is a two-element column vector. Thus, given the
where M = [µ ι] is an n × 2 matrix and q =
expected return vector µ, the covariance matrix V, and the required expected return q, the efficient
portfolio weight vector x can be determined directly.
The variance of returns of the efficient portfolio corresponding to each given value of q is
−1
σ p2 = x′ Vx = q′ M′ V−1 M
q.
(3)
As shown below, Eqs. (2) and (3) enable us to establish the conditions for spanning.
2. Spanning of N + K assets by K assets
As mentioned earlier, the coincidence of the two efficient frontiers is the starting point of our
analysis. If K benchmark assets are able to span N + K assets, the frontier based on the K assets must
1
In fact, there may be infinitely many rational combinations of the benchmark portfolio and the new asset classes. We
assume no transaction costs throughout this paper. The existence of market imperfections can favor one asset class over the
others. Our concern here is portfolio decisions based purely on the risk-return characteristics of various asset classes.
108
C.S. Cheung et al. / Finance Research Letters 6 (2009) 106–113
be identical to the frontier based on the N + K assets. For all values of the required expected return q,
the corresponding values of σ p2 for the two cases must be the same. Thus, in case of spanning, the
2 × 2 matrix M′ V−1 M must remain the same, regardless of whether it is based on the K assets or the
N + K assets. As shown in the following, this analytical property ensures that no investment funds
be allocated to the N assets and that the allocations among the K 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
VK N
VN K
VK K
(4)
,
where V N N is N × N, V N K is N × K , V K N is K × N, and V K K is K × K . We also partition the expanded
expected return vector µ into column vectors µ N and µ K with N and K elements, respectively.
Likewise, the corresponding ( N + K )-element vector ι is partitioned into column vectors ι N and ι K .
Following the algebra of block matrix inversion, we have
V−1 =
where
1
−EVN K V−
KK
E
−1
−1
−1
−1
−V K K V K N E (V K K + V K K V K N EV N K V K K )
1
E = V N N − V N K V−
K K VK N
−1
,
(5)
(6)
.
Partitioning V−1 in this manner allows M′ V−1 M to 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
that
µ′N − µ′K V−K 1K V K N E µN − VN K V−K 1K µ K = 0,
′
′
−1
−1
µN − µ K V K K V K N E ιN − VN K V K K ι K = 0, and
′
′
−1
−1
ιN − ι K V K K V K N E ιN − VN K V K K ι K = 0.
(7)
(8)
(9)
Notice that the left-hand side of each of these three equations is of the form where the square matrix
E is pre-multiplied by an N-element row vector and post-multiplied by an N-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 V for 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 E is a principal submatrix of V−1 , E is positive definite as well. For any nonzero
column vector u with N elements, we must have u′ Eu > 0. If u′ Eu = 0, then the vector u must be a
vector of zeros. Thus, as E is positive definite, Eq. (7) implies that
µN = VN K V−K 1K µ K ,
(10)
and Eq. (9) implies that
ιN = VN K V−K 1K ι K .
(11)
Eqs. (10) and (11) ensure that Eq. (8) holds as well.
With the expanded vector of portfolio weights, x, partitioned into x N and x K , column vectors of
N and K elements, respectively, substituting Eqs. (10) and (11) into the partitioned version of Eq. (2)
leads to x N = 0 N , for all values of q. Here, 0 N 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 V is essential to the portfolio optimization problem and empirical spanning tests.
C.S. Cheung et al. / Finance Research Letters 6 (2009) 106–113
109
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 from
y = V−1 (µ − μ0 ι)
(12)
by scaling the column vector y in such a way that the portfolio weights sum to one, i.e., x = y(ι′ y)−1 .
For the ( N + K )-asset case, we partition y into y N and y K , two column vectors with N and K elements,
respectively. If the N + K assets are spanned by the K assets and the expanded covariance matrix is
invertible [implying Eqs. (10) and (11)], Eq. (12) becomes
yN = E
µN − VN K V−K 1K µ K − μ0 E ιN − VN K V−K 1K ι K = 0N
(13)
and
1
y K = V−
K K (µ K − μ0 ι K ).
(14)
Therefore, spanning implies that each tangency portfolio has unique portfolio weights, with investment funds allocated only among the K assets, and that the N 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) regarding the inverse of the covariance matrix, is provided in Appendix A. We start with a ( K + 1)-asset
portfolio consisting of the K assets and an arbitrary asset, say asset a, from the set of N assets. By
regressing the random return of asset a on the random returns of the remaining assets in the portfolio, 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 corresponding regression coefficients. We then extend the idea recursively by augmenting the portfolio with the
remaining assets from the set of N assets, one at a time, until all N assets are accounted for. Regressing 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 the K benchmark assets, with the weights being the corresponding regression
coefficients.
What is implicit in each of these regression runs is the presence of residual risk. Its presence
makes each of the N assets less attractive (for either holding long or selling short) when compared
with investments in the K benchmark assets alone. The extra asset, which offers an expected return just like a portfolio of the K benchmark assets, is not worth holding, either long or short. With
the expected return of the extra asset being neither higher nor lower than what can be achieved by
investing in the K benchmark assets alone, taking a nonzero position in the extra asset inevitably increases the portfolio risk and weakens the portfolio’s risk-return trade-off. Thus, having a zero holding
of the asset is the only rational decision. In the absence of residual risk, however, the augmented covariance matrix is no longer invertible. Then, the equivalence (in terms of risk-return trade-off) can be
established between investing in portfolios of the K benchmark assets alone and investing also in the
extra asset. The absence of residual risk would require that the random return of each of the N assets
be a perfect linear combination of the random returns of the K benchmark assets, thus rendering the
portfolio decision non-unique.
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C.S. Cheung et al. / Finance Research Letters 6 (2009) 106–113
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 , . . . , r N )′ and R K = ( R 1 , R 2 , . . . , R K )′ , with the elements there being the individual asset returns.
Lemma. For a return generating process,4
r i = αi + β ′i R K + ǫi ,
(15)
where αi is the intercept term, β i is a K -element column vector of parameters, and ǫi is random noise with
E (ǫi ) = Cov(ǫi , R j ) = 0, for i = 1, 2, . . . , N and j = 1, 2, . . . , K , the optimal investment weight in each asset
i is zero if and only if αi = 0 and β ′i ι K = 1.
Proof. See Appendix 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 r N is indicated by zero investments being the optimal choice of the N
assets. In contrast, if each ǫi has a zero variance, as the corresponding r i is a linear combination
of R 1 , R 2 , . . . , R K , the determinant of the expanded covariance matrix is zero, thus rendering the
matrix non-invertible. In such a case, for any expected return requirement, the corresponding portfolio
weight vector β i satisfying β ′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 p with random return r p
and its orthogonal portfolio z with random return r z , satisfying the condition of Cov(r p , r z ) = 0, the
random return r s of any portfolio s can be stated as
r s = βsp r p + (1 − βsp )r z + ǫs ,
(16)
where βsp = Cov(r s , r p )/ Var(r p ) and E (ǫs ) = 0, with ǫs being random noise. Using E (r p ) and E (r z ) for
two different values of q in Eq. (2), the corresponding allocations of investment funds in portfolios p
and z can be determined. If spanning occurs, then both p and z are portfolios of the K benchmark
assets. That is,
p ′
r p = wK RK
and
(17)
z ′
(18)
rz = wK RK ,
p
wK
and wzK are K -element column vectors of portfolio weights, satisfying the condition of
where
p ′
(w K ) ι K = (wzK )′ ι K = 1. Accordingly, Eq. (16) becomes
p
′
r s = βsp w K + (1 − βsp )wzK R K + ǫs .
p
(19)
As the condition of [βsp w K + (1 − βsp )wzK ]′ ι K = 1 always holds and s can be any portfolio based on
the N + K assets, including any of the N individual assets, Eq. (15), with αi being zero and β i being
a portfolio-weight vector, follows directly.
3
We thank the reviewer for this suggestion.
To characterize mean-variance spanning via this return generating process is consistent with Huberman and Kandel (1987).
The spanning conditions (which are αi = 0 and β ′i ι K = 1, for i = 1, 2, . . . , N) also match those in their study.
4
C.S. Cheung et al. / Finance Research Letters 6 (2009) 106–113
111
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
If K assets span the N + K assets, they must span K + 1 assets, where the extra asset can be any
of the N assets. Let us consider an arbitrary asset a among the N assets, with an expected return μa
and a variance of returns σaa . The column vector of covariances of returns between the K benchmark
assets and this asset is V K a . The transpose of V K a is the row vector VaK . For a set of K + 1 assets
spanned by the K benchmark assets, let
A=
A aa
AK a
AaK
AK K
= V−1 ,
(A.1)
the inverse of the covariance matrix of the K + 1 assets. Here, A aa is a scalar, A K a is a K -element
column vector, AaK is the transpose of A K a , and A K K is a K × K matrix.
The scalar A aa , according to Eq. (6), can be written as
A aa =
σaa − VaK V−K 1K V K a
−1
.
(A.2)
1
Likewise, the row vector AaK , according to Eq. (5), can be written as − A aa VaK V−
K K . If we regress the
random returns of asset a on the random returns of the K benchmark assets, with a residual error ǫa ,
the row vector of the multiple regression coefficients is asymptotically5
1
β a′ = VaK V−
KK.
(A.3)
The part of the variance of returns of asset a that is unexplained by the regression, the residual
variance, is
σǫ2a = σaa 1 − R a2 ,
(A.4)
where R a2 is the asymptotic R-square of the regression. Drawing on Stevens (1998), we have directly
the following results:
5
′
If
β a is the sample based estimate of the regression coefficient, it can be shown that with Cov(ǫa , R j ) = 0, for j =
′
1
1, 2, . . . , K , p lim
β a = β a′ = VaK V−
KK .
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C.S. Cheung et al. / Finance Research Letters 6 (2009) 106–113
1
2
A aa VaK V−
K K = σaa R a ,
A aa =
1
σaa (1 − R a2 )
AaK = − A aa β a′ =
=
(A.5)
1
σǫ2a
,
and
(A.6)
−β a′
−β a′
=
.
σaa (1 − R a2 )
σǫ2a
(A.7)
Eq. (13) can be written for asset a in the ( K + 1)-asset case as
ya =
As
(μa − μ0 ) − β a′ (µ K − μ0 ι K )
σǫ2a
= 0.
(A.8)
σǫ2a > 0, spanning of the K + 1 assets by the K benchmark assets implies
μa − 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 as
μa = β a′ µ K and
′
1 = βa ιK ,
(A.10)
(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 subsequently ya = 0, require that the random return of the extra asset a not be a perfect linear combination
of the random returns of the K benchmark assets. A perfect linear combination would result in R a2 = 1
and σǫ2a = 0. With both μa − μ0 − β a′ (µ2 − μ0 ι2 ) = 0 and σǫ2a = 0, Eq. (A.8) would render ya , and
accordingly the weight xa on asset a, indeterminate. Having both μa − μ0 − β a′ (µ2 − μ0 ι2 ) = 0 and
σǫ2a > 0 ensures a zero weight on asset a.
Noting that the extra asset a in the ( K + 1)-asset case can be any of the N assets, we can extend
the same idea recursively. Suppose that the portfolio now consists of K + 2 assets instead; they include
the K benchmark assets, asset a, and another arbitrarily selected asset b, also among the N assets.
With asset b viewed as the only extra asset in a portfolio consisting of K + 2 assets, it will have a zero
weight if μb is a weighted average of the expected returns of asset a and the K benchmark assets. As
μa itself is a weighted average of the expected returns of the K benchmark assets, so is μb . We can
successively add more assets to the portfolio, one at the time, until all N assets are accounted for.
Regardless of when an asset i, among i = 1, 2, . . . , N, is selected recursively for the above regression 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 asset i is the extra asset a. Thus, for spanning to hold,
we must have
µN = β ′ µ K and
′
ιN = β ι K ,
(A.12)
(A.13)
where β = [β 1 β 2 . . . β N ] is a K × N matrix. As Eqs. (A.12) and (A.13) confirm, each row i of β ′
represents the portfolio weights on the K benchmark assets that make the corresponding asset i
from the set of N assets redundant.
To prove the necessity condition of Lemma in the main text, consider the following: If αi = 0
and β ′i ι K = 1, the underlying model r i = αi + β ′i R K + ǫi with E (ǫi ) = 0 implies that μi = β ′i µ K , for
i = 1, 2, . . . , N. With β ′i ι K = 1, β i becomes a vector of portfolio weights and μi = β ′i µ K is, accordingly,
the expected return of the corresponding portfolio. In the context of multiple regressions under the
condition of E (ǫi ) = Cov(ǫi , R j ) = 0, for i = 1, 2, . . . , N and j = 1, 2, . . . , K , the row vector of regres1
−1
′
sion coefficients is asymptotically β ′i = Vi K V−
K K , as Eq. (A.3) indicates. With β = V N K V K K , it follows
C.S. Cheung et al. / Finance Research Letters 6 (2009) 106–113
113
−1
1
that µ N = V N K V−
K K µ K and ι N = V N K V K K ι K . Given Eq. (13), we have y N = 0 N , and thus the redundancy of each of the N assets is assured. To prove the sufficiency condition, suppose that the N assets
are redundant. This is equivalent to Eqs. (A.12) and (A.13), which jointly indicate that each row i of β ′
is a set of portfolio weights on the K benchmark assets corresponding to asset i among the N assets.
It follows directly from the underlying model r i = αi + β ′i R K + ǫi with E (ǫi ) = 0 that μi = αi + β ′i µ K ,
for i = 1, 2, . . . , N. Thus, given Eq. (A.12), we have αi = 0, for i = 1, 2, . . . , N.
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