How to efficiently allocate capital lies at the heart of financial decision making. Portfolio theory, as developed by [35], provides a framework for this problem, based on the means, variances and covariances of the assets in the considered portfolio. The theory revolves around the trade-off between expected return and variance (risk), denoted by mean-variance optimization. In this setting, investors allocate wealth in order to maximize expected return given a certain level of risk or conversely allocate wealth to minimize the risk given a certain level of expected return. Although it has received a lot of criticism (see, e.g., [42] and [27]), the framework remains one of the most crucial components in portfolio management.

In this paper, we consider the tangency portfolio (TP) which is one of the most important portfolios in the financial literature. The TP weights determine what proportions of the capital to invest in each asset and are obtained by maximizing the expected quadratic utility function. For a portfolio of p risky assets, the TP weights are given by (1) w T P = α − 1 Σ − 1 ( μ − r f 1 p ) , where μ is a p-dimensional mean vector of the asset returns, Σ is a p × p symmetric positive definite covariance matrix of the asset returns, the coefficient α > 00$]]> describes the investors’ risk aversion,1¹

This value represents how willing an investor is to accept upward and downward risks on their investment. It can be determined through, e.g., qualitative assessment, such as interview questions posed to the investor.

Naturally, the TP weights w T P depend on knowledge of the mean vector μ and the covariance matrix Σ. In general, these quantities are not known and need to be estimated from data on N historical return vectors x 1 , … , x N . Plugging sample estimates of the mean vector and covariance matrix into (1) leads us to the sample estimate of the TP weights expressed as (2) w ˆ T P = α − 1 S − 1 ( x ¯ − r f 1 p ) , where S is the sample covariance matrix and x ¯ is the sample mean vector, respectively, of x 1 , … , x N .2²

It is worth to mention that a similar structure appears in the discriminant analysis. Namely, the coefficients of a discriminant function that maximizes the discrepancy between two datasets are expressed as a product of the inverse sample covariance matrix and the sample mean vector (see, for example, [6, 14]).

The common scenario considered is that the number of observations available for the estimation, denoted by N, is greater than the portfolio size, denoted by p. In this case the sample covariance matrix S is positive definite, and w ˆ T P can be obtained as presented in (2). However, when the considered portfolio is large, it is possible that the number of available observations is less than the portfolio dimension. This can be due to a lack of data for all the assets in the portfolio, but it may also occur due to the fact that covariance of asset returns tends to change over time. As such, the assumption of a constant covariance might only hold for limited periods of time, hence limiting the amount of data available for estimation. Many applications consider portfolios of large dimensions, containing up to 50, 100 or even 1000 assets (see, e.g., [41, 26, 34, 2, 20, 16, 22, 5, 12, 1]). If returns are measured on weekly or monthly intervals, data reaching back several decades might be required to ensure p ≤ N. Unless the considered assets can be assumed to have a constant covariance matrix over very long time periods, data spanning such long time intervals is not suitable to use in the estimation, or might simply not be available. Any such situations, where p > NN$]]>, would result in a singular sample covariance matrix S, which in turn is noninvertible, in the standard sense.

This issue can be remedied by estimating Σ − 1 in (1) with the Moore–Penrose inverse of S, which we will denote by S + . This generalized inverse has previously been successfully employed in portfolio theory for the p > NN$]]> case by [10, 11, 44, 15].4⁴

Instead of using the Moore–Penrose inverse, one can consider regularization techniques such as the ridge-type method [43], the Landweber–Fridman iteration approach [33], a form of Lasso [19], or an iterative algorithm based on a second order damped dynamical systems [24, 25].

The expectation and variance of an estimator are key quantities to describe its statistical properties. With the standard assumption of normally distributed asset returns, the stochastic components of w ˜ T P consists of S + and x ¯, which are independent under the assumption of normally distributed data (see, e.g., [10]). Unfortunately, there exist no derivation of the expected value or variance of S + , when p > NN$]]>. In [21] however, these quantities are presented in the special case of Σ = I p . The authors also provided approximate results, using moments of standard normal random variables, and exact results for moments of the generalized reflexive inverse, another quantity that can be applied as an inverse of S. Further, in a recent paper [28], several bounds on the mean and variance of S + are provided, based on the Poincaré separation theorem. Our paper builds on the results presented in [21] and [28] to provide bounds and approximations for the moments of the TP weights, E [ w ˜ T P ] and V [ w ˜ T P ], where E [ · ] and V [ · ] denote the expected value and variance, respectively. We also present a simulation study, where various measures compare the derived bounds with the equivalent sample quantities obtained from simulated data. Finally, we compare the moments obtained applying the reflexive generalized inverse and the sample moments based on the Moore–Penrose inverse.

The rest of this paper is organized as follows. Section 2.1 provides exact moment results for the case Σ = I p . Section 2.2 presents bounds for the moments of w ˜ T P in the general case, while approximate moments are derived in Section 2.3. Exact moments applying the reflexive generalized inverse are derived in Section 3. The simulation study is presented in Section 4 while Section 5 summarizes.

Let X be a p × N matrix with N asset return vectors of dimension p × 1 stacked as columns, where p > NN$]]>. Further, we assume that these return vectors are independent and normally distributed with mean vector μ and positive definite covariance matrix Σ. Thus X ∼ MN p , N ( μ 1 N , Σ , I N ), where MN p , n ( M , Σ , U ) denotes the matrix-variate normal distribution with p × N mean matrix M, p × p row-wise covariance matrix Σ and N × N column-wise covariance matrix U. Further, let the p × 1 vector x ¯ be the row mean of X. Now, define Y = X − x ¯ 1 N ′ , such that Y ∼ MN p , N ( 0 , Σ , I N ). Further, let S = Y Y ′ / n, such that rank ( S ) = n < p with n = N − 1, and n S ∼ W p ( n , Σ ), i.e. n S follows a p-dimensional singular Wishart distribution with n degrees of freedom and the parameter matrix Σ. Let S = Q R Q ′ denote the eigenvalue decomposition of S, where R is the n × n diagonal matrix of positive eigenvalues and Q is the p × n matrix with corresponding eigenvectors as columns. Further, define S + = Q R − 1 Q ′ . Then, S + constitutes the Moore–Penrose inverse of Y Y ′ / n, and S + is independent of x ¯ (see [10]).

In the following, let η = α − 1 ( x ¯ − r f 1 p ) and θ = E [ η ] = α − 1 ( μ − r f 1 p ). Consequently, from Corollary 3.2b.1 in [36], together with the fact that E [ x ¯ ] = μ and V [ x ¯ ] = Σ / ( n + 1 ), we obtain that (5) E [ η η ′ ] = θ θ ′ + Σ α 2 ( n + 1 ) , (6) E [ η ′ η ] = θ ′ θ + tr ( Σ ) α 2 ( n + 1 ) , (7) E [ η ′ Σ η ] = θ ′ Σ θ + tr ( Σ Σ ) α 2 ( n + 1 ) . Further, let s i j denote the element on row i and column j of S + , and let σ i j denote the element on row i and column j of Σ − 1 . Also let e i denotes a p × 1 vector where all values are equal to zero, except the i-th element, which is equal to one. Moreover, we assume that λ 1 ( M ) ≥ λ 2 ( M ) ≥ ⋯ ≥ λ p ( M ) are the ordered eigenvalues of a symmetric p × p matrix M, and that A ≤ L B denotes the Löwner ordering of two positive semi-definite matrices A and B.

An alternative to using the Moore–Penrose inverse S + to estimate Σ − 1 is an application of the reflexive generalized inverse, defined as S † = Σ − 1 / 2 Σ − 1 / 2 S Σ − 1 / 2 + Σ − 1 / 2 , where the elements of S † are denoted s i j † . Then, the TP weights vector can be estimated by w T P † = S † η , and we derive the following result.

An obvious drawback of w T P † is that Σ must be known in order to construct S † . Moreover, in the case of Σ = I p the results in Theorem 5 coincide with the results in Theorem 1, since in this case S † = S + .

The aim of this section is to compare the bounds on the moments of w ˜ T P derived in Section 2.2 with the sample mean and sample variance of this estimator. We will also investigate the difference between the moments of w T P † derived in Theorem 5 and the sample moments of w ˜ T P . Ideally, the bounds should not deviate from the obtained sample moments very much. To this end, define b l and b u as the p × 1 vectors with elements b i l = v i i μ i + ∑ j ≠ i p v i j μ j , b i u = z i i μ i + ∑ j ≠ i p z i j μ j , such that b l and b u represent the element-wise lower and upper bounds for the expected TP weights vector presented in Theorem 2, where we set α = 1 and r f = 0. Let B 1 = ( 2 c 1 + c 2 ) ( λ 1 ( Σ − 1 ) ) 4 k 1 Σ E [ η η ′ ] Σ + k 2 Σ E [ η ′ Σ η ] , B 2 = ( 2 c 1 + c 2 ) ( λ 1 ( Σ − 1 ) ) 4 E [ ( η ′ η ) ] I p represent the bounds in equations (17) and (18) in Theorem 3, respectively. Further, let m and V respectively denote the sample mean vector and sample covariance matrix of w ˜ T P based on an observed matrix X, as described in Section 2. Moreover, we define (23) t l = 1 p ′ | b l − m | p , (24) t u = 1 p ′ | b u − m | p , (25) t † = 1 p ′ | E [ w T P † ] − m | p , so that t l , t u and t † measure the element-wise difference between the sample mean vector and the lower and upper bounds on the mean, and mean of w T P † , respectively. Dividing by p allows comparing the measures between various portfolio sizes. Further, let (26) T 1 = 1 p ′ B 1 − V 1 p p 2 , (27) T 2 = 1 p ′ B 2 − V 1 p p 2 , (28) T † = 1 p ′ V [ w T P † ] − V 1 p p 2 , where 1 p is a p × 1 vector of ones. Then, T 1 and T 2 provide a measure of discrepancy between the sample covariance matrix and bounds presented in Theorem 3, while T † measures the discrepancy between the variance of w T P † presented in Theorem 5 and the sample covariance matrix of w ˜ T P . Since it is divided by p 2 , the number of elements in B 1 , B 2 , V [ w T P † ] and V, the measures T 1 , T † and T 2 again allow for comparison between different portfolio sizes. Moreover, define (29) f l = ‖ b l − m ‖ F 2 / ‖ m ‖ F 2 , (30) f u = ‖ b u − m ‖ F 2 / ‖ m ‖ F 2 , (31) f † = ‖ E [ w T P † ] − m ‖ F 2 / ‖ m ‖ F 2 , (32) F 1 = ‖ B 1 − V ‖ F 2 / ‖ V ‖ F 2 , (33) F 2 = ‖ B 2 − V ‖ F 2 / ‖ V ‖ F 2 , (34) F † = ‖ V [ w T P † ] − V ‖ F 2 / ‖ V ‖ F 2 , where ‖ M ‖ F 2 denotes the Frobenius norm of the matrix M. Hence f 1 , f 2 , F 1 and F 2 represent the normalized Frobenius norm of differences between the bounds and the sample variance, while f † and F † denote the differences between the moments of w T P † and the sample variance of w ˜ T P .