Preliminaries then the vector u must be a vector of zeros. Thus, as E is positive definite, Eq.

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 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 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 μ i and σ i 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 x i , 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, sub- ject 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 x = V − 1 M M ′ V − 1 M − 1 q , 2 where M = [ µ ι ] is an n × 2 matrix and q = [ q 1 ] ′ is a two-element column vector. Thus, given the 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 σ 2 p = x ′ Vx = q ′ M ′ V − 1 M − 1 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 2 p 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 = V N N V N K V K N V K 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 = E − EV N K V − 1 K K − V − 1 K K V K N E V − 1 K K + V − 1 K K V K N EV N K V − 1 K K , 5 where E = V N N − V N K V − 1 K K V K N − 1 . 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 − 1 K K V K N E µ N − V N K V − 1 K K µ K = , 7 µ ′ N − µ ′ K V − 1 K K V K N E ι N − V N K V − 1 K K ι K = , and 8 ι ′ N − ι ′ K V − 1 K K V K N E ι N − V N K V − 1 K K ι K = . 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 µ N = V N K V − 1 K K µ K , 10 and Eq. 9 implies that ι N = V N K V − 1 K K 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 = 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