Rationale based on a tangency portfolio perspective

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 μ be the intercept of a tangent line. The tangency portfolio weights can be obtained from y = V − 1 µ − μ 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 y N = E µ N − V N K V − 1 K K µ K − μ E ι N − V N K V − 1 K K ι K = N 13 and y K = V − 1 K K µ K − μ ι K . 14 Therefore, spanning implies that each tangency portfolio has unique portfolio weights, with invest- ment 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 regard- ing 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 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 augmenting 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 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 re- turn 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 in- creases 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 co- variance 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. 110 C.S. Cheung et al. Finance Research Letters 6 2009 106–113

4. Further characterization of mean-variance spanning