Abstract
Global fixed income returns exhibit highly structured correlations across maturities and economies (data modes), and their modeling therefore requires analysis tools that are capable of directly capturing the inherent multi-way couplings present in such multimodal data. Yet, current analyses typically employ “flat-view” multivariate matrix models and their associated linear algebras; these are agnostic to the global data structure and can only describe local linear pairwise relationships between data entries. To address this issue, we first show that global fixed income returns naturally reside on multi-modal lattice data structures, referred to as tensors. This serves as a basis to introduce a multilinear algebraic approach, inherent to tensors, to the modeling of the global term structure underlying multiple interest rate curves. Owing to the enhanced flexibility of multilinear algebra, statistical descriptors, such as correlations, exist between tensor columns and rows (fibers), as opposed to between individual entries in standard matrix analysis. This allows us to express the covariance of global returns as a joint multilinear decomposition of the maturity-domain and country-domain covariances. This not only drastically reduces the number of parameters required to fully capture the global returns covariance structure, but also makes it possible to devise rigorous and tractable global portfolio management strategies; we tailor these specifically to each of the data domains and thereby fully exploit the lattice structure of global fixed income returns. The ability of the proposed multilinear tensor approach to compactly describe the macroeconomic environment through economically meaningful factors is validated via empirical analysis that demonstrates the existence of maturity-domain and country-domain covariances underlying the interest rate curves of eight developed economies.
TOPICS: Fixed income and structured finance, quantitative methods, statistical methods, portfolio construction, global markets
Key Findings
▪ We show that global fixed income returns naturally reside on multi-modal lattice data structures, referred to as tensors, which exhibit highly structured correlations across maturities and economies.
▪ We introduce a multilinear algebraic approach to the modeling of the global term structure underlying multiple interest rate curves, which allows us to express the covariance of global returns as a joint multilinear decomposition of the maturity-domain and country-domain covariances.
▪ This drastically reduces the number of parameters required to fully capture the global returns covariance structure, and makes it possible to devise rigorous and tractable global portfolio management strategies which are tailored to each of the data domains.
Participants in economic markets have long recognized the importance of identifying common factors (often of a hidden nature) in financial data. These co-movements then can be used to explain the returns of securities within an asset class. In such a task, it is critical to distinguish the common risks, which have a general impact on the returns of most securities, from the idiosyncratic risks, which influence securities individually. To this end, following from the seminal work in Litterman and Scheinkmann (1991), a significant portion of the fixed income literature has been devoted to the technique of principal component analysis (PCA) (Jolliffe 1986), which provides a parsimonious interpretation of the term structure dynamics of interest rate curves through latent factors, called the principal components (PCs). Empirical results suggest that the first three latent factors, referred to as:
1. Level – the parallel shift of the yield curve,
2. Slope – the tilting of the yield curve, and
3. Curvature – the flexing of the yield curve
are sufficient to almost fully reflect the behavior of the entire term structure. Moreover, the principal components in data covariance matrixes also frequently admit economically meaningful interpretation. This has made PCA a fundamental tool for characterizing single-economy interest rate curves. Overall, notable benefits of PCA include its:
▪ Analyticity and mathematical tractability, since it is formulated through classical linear algebra techniques
▪ Ability to parsimoniously describe economic factors in terms of the level, slope, and curvature components
▪ Application to stress-testing and scenario analysis using eigenvalue and eigenvector perturbation methods
▪ Direct applicability to portfolio hedging, owing to the orthogonality between principal components.
Despite the advantages of using PCA for domestic (single-country) fixed-income investing, financial institutions routinely invest globally where the increasingly growing interconnectedness of the international debt and money markets presents a major challenge for PCA-based risk management. This is largely due to the legacy PCA-based analytics that employ “flat-view” linear algebra methods, whereby trades are typically hedged by offsetting their domestic-curve principal components. The high correlation between interest rates across both maturities and countries leaves such strategies with limited avenues for diversification, and unprepared for cross-country co-movements arising from global macroeconomic events. The most recent credit crisis in 2007–2008, for example, shows how macroeconomic shocks can be transmitted across multiple interest rate curves. This all suggests that a parsimonious model for describing the co-movement of interest rates at the relevant maturities and in the relevant countries is necessary for global fixed income investors to adequately identify and manage risk.
These requirements have spurred modifications and variations of PCA for joint term structure analysis, with a variety of existing solutions largely stemming from the ambiguity in the problem formulation. Indeed, problem set-ups tend to differ widely with regard to the data processing, estimation procedure, number of latent factors required to explain the joint dynamics of multiple yield curves, and economic interpretation of the global and domestic factors obtained. For example, approaches that apply PCA to vectorized data obtained from multiple interest rate curves (Rodrigues 1997; Driessen et al. 2003; Novosyolov and Satchkov 2008) completely ignore the multi-curve lattice structure inherent to global interest rate curves, shown in Exhibit 1, and therefore yield factors that may be difficult to interpret and still tend to reflect local idiosyncratic and domestic behaviors. Furthermore, the numerical instability issues associated with covariance estimation lead to a counter-intuitive result, whereby the more collinear the asset returns, the greater the need for diversification. The corresponding increase in the number of assets, N, to compensate for such ill-posedness complicates the problem even further, as more data samples are required to obtain a positive-definite covariance estimate, with at least 
independent and identically distributed (i.i.d.) observations needed.
The Lattice Data Structure Inherent to the Interest Swap Ratesa for Eight Developed Countries: Switzerland, Euro Area, United Kingdom, Japan, Australia, New Zealand, Canada, and the United States, with Maturities of {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 15, 20, 25, 30} Years, Observed on July 22, 2017
SOURCE: Bloomberg.
For enhanced intuition, an approach referred to as common PCA (Flury 1988) aims to extract the eigenvectors that span a space that is identical across the interest rate curves of multiple countries (common latent factors). However, this method relies on the simultaneous diagonalization of multiple covariance matrixes, which is in general both: 1) not tractable for more than two matrixes, and 2) unable to model the co-variation between assets across countries (Juneja 2012). To this end, inter-battery factor analysis (Tucker 1958) was employed to capture all common factors across domestic term structures (Perignon et al. 2007). The method is computationally prohibitive, and implicitly assumes that idiosyncratic co-variations can occur only domestically, a restrictive and unrealistic assumption.
All in all, the main limitation underpinning PCA-based techniques lies in their reliance on standard matrix methods developed for multivariate analysis, which are agnostic to the intrinsic multi-curve lattice data structure inherent to real-world global fixed income returns, as shown in Exhibit 1. Such a flattened view of the data, and the limitations of the use of intrinsically pair-wise (scalar-to-scalar) based linear matrix algebra in multivariate analysis, make current strategies by and large inadequate and ineffective.
In order to address this void and account for the inter-related maturity vs. country nature of fixed income returns, we propose a multilinear algebraic approach, with the aim of matching the underlying data structure. We show that this allows for a rigorous model of global fixed income returns, spanning multiple maturities and countries, as in this way the rigid lattice structures where such data reside are accounted for in a natural and compact way. Indeed, multilinear algebra is a backbone of multiway tensor analysis that has found its place in a number of real-world applications across science and engineering (Kolda and Bader 2009; Cichocki et al. 2015, 2017a,b; Siridopoulos et al. 2017), as these techniques allow us to develop appropriate models for capturing the structured linear couplings and interactions between data fibers (multidimensional generalization of matrix rows and columns), which are intrinsic to lattice-structured data. In addition, by virtue of the proposed multilinear model, the covariance of returns between two assets becomes linearly separable, that is, it factorizes into a product of a maturity-to-maturity covariance and a country-to-country covariance, thus giving the analysis enhanced physical meaning. This is in stark contrast to the standard linear algebraic approach, which only can model each pair-wise asset-to-asset covariance. In other words, the proposed multilinear approach allows us to decompose the covariance matrix of global fixed income returns into a multilinear (Kronecker) product of the two matrixes associated with the independent covariance domains:
1. Maturity-domain covariance exhibited within all countries and
2. Country-domain covariance displayed at all maturities.
Operating within each domain in parallel makes it possible to:
▪ Devise novel rigorous and tractable global portfolio management strategies, which are specifically tailored to each covariance domain
▪ Parametrize the global structure of the covariance of fixed-income returns with fewer decision parameters, compared to standard linear models
▪ Provide a rigorous and parsimonious alternative to the classical unrestricted estimate, which is unstable or even unavailable if the number of assets considered is large compared to the sample size.
Our empirical analysis confirms the utility of the maturity-domain and country-domain covariances so derived, underlying the interest rate curves of eight developed economies, in providing a compact and physically meaningful insight into the global macroeconomic environment.
PRELIMINARIES OF MULTILINEAR ALGEBRA
A class of multi-dimensional arrays referred to as tensors, which are endowed with multilinear properties, is next introduced following the notation in Kolda and Bader (2009), whereby scalars are denoted by a lightface font, e.g., x; vectors by a lowercase boldface font, e.g., x; matrixes by an uppercase boldface font, e.g., X; and tensors by a boldface calligraphic font, e.g., 
.
The order of a tensor is defined as the number of its dimensions, also referred to as modes, i.e., the tensor 
has N modes and 
entries in total. Tensors can be reshaped into lower-dimensional representations (unfoldings) that can be manipulated using standard linear algebra. The vector unfolding, also known as vectorization, is denoted by
1with its elements given by
2Note that we use overlined subscripts to denote the linear indexing (or Little-Endian) convention (Kolda and Bader 2009), whereby
3The mode-n unfolding (matricization), denoted by 
, is obtained by reshaping a tensor into a matrix in the form
4where the column vector, 
, is the i-th mode-n fiber (a multi-dimensional generalization of matrix rows and columns), with elements defined as
5with i as the linear index 
, that is
6The operation of mode-n unfolding also can be viewed as a rearrangement of the mode-n fibers into column vectors of the matrix, X(n), as illustrated in Exhibit 2. Notice that the considered order-3 tensor, 
, has alternative representations in terms of mode-1 (left panel), mode-2 (middle panel), and mode-3 (right panel) fibers, that is, its columns, rows, and tubes.
Mode-n Matrix Unfolding, X(n), of an Order-3 Tensor, 
, as a Rearrangement of the Mode-n Fibers, 
We next consider the second-order moments of a tensor-valued random variable, which, without loss of generality, is assumed to have a zero-mean. The variance of a tensor-valued random variable, 
, is defined as its expected squared Frobenius norm, that is
7Using the fiber-based mode-n unfolding representation in Equation 4, it is possible to define the mode-n covariance matrix as follows
8Intuitively, the mode-n covariance in Equation 8 can be viewed as the expected covariance between mode-n fibers, owing to the total expectation theorem (Weiss et al. 2005), since
9where Ei{·} denotes the expectation over the indexes i. Intuitively, the i-th mode-n fiber, 
, can be viewed as the i-th realization of a stable generic i.i.d. random variable, f(n). Note that we assume ergodicity over the space associated with the indexes, i, which allows us to introduce an expectation operator through summation.
It is important to highlight that, although we consider order-2 tensors (matrixes), the multilinear matrix model proposed in this work largely differs from existing standard linear matrix models, as shown next.
A MULTILINEAR MODEL FOR GLOBAL FIXED INCOME RETURNS
In standard multivariate data analysis, multiple measurements are collected at a given trial, experiment, or time instant, to form a vector-valued data sample, x ∈ ℝK. A prior assumption typically adopted in statistical modeling is that the observables are distributed according to a normal distribution, 
, which implies that the mean vector, μ ∈ ℝK, and covariance matrix, Σ ∈ ℝK×K, are in general of an “overall” nature and thus unstructured. However, if the entries in x lie on a regular lattice in an N-dimensional coordinate system, then it is desirable, and often even necessary, to assume that the mean and covariance exhibit a more structured form thart arises from relevant physical considerations (Hoff 2011; Gupta and Nagar 2000; Scalzo et al. 2020). This is precisely the case for global fixed income returns that, as shown in Exhibit 1, reside in a two-dimensional coordinate system (maturity × country), as demonstrated next.
Multilinear Model of Global Fixed Income Returns
Consider a zero-mean random variable, xt(m, c) ∈ ℝ, which represents the return of a fixed income asset with a maturity, m, in a country, c, at a time instant t. In other words, the fixed income return, xt: ℝ × ℝ ↦ ℝ, is a bilinear map that maps a maturity, m, and a country, c, to a realization of a random variable xt(m, c) (a return). It can be interpreted as a random scalar field in a two-dimensional coordinate system spanned by the maturity and country axes.
For generality, we assume that the returns are distributed according to the coordinate-dependent distribution
10so that the covariance operator, σ: ℝ2 × ℝ2 ↦ ℝ, of the return field is defined as
11with σ(m, c, m, c) ≡ σ2(m, c).
Such a random field is said to exhibit a linearly separable covariance operator if it can be factorized into a product of covariance operators of constituent random variables, that is, each associated with an axis (country or maturity) of the underlying coordinate system. Then, a linearly separable covariance operator of the global returns in Equation 10 takes the form
12where 
is the covariance between asset returns with maturities mi and mk (which is independent of the countries), while 
is the covariance between asset returns in countries cj and cl (which is independent of the maturities).
Reshaping of the Global Return Data
When jointly considering the returns of assets at Im maturities within Ic countries at a time instant t, the returns can be shaped up to form an order-2 tensor (matrix-valued) random variable, 
, with its (i, j)-th entry defined as
13Therefore, each realization of the matrix, Xt, contains ImIc entries in total; an illustration of this matrix in a tensor-valued formalism at a time-instant, t, is given in Exhibit 3.
Illustration of an Order-2 Tensor (Matrix) Sample, Represented in Terms of: (i) the Individual Returns, xt(mi, cj), as in Matrixes (Left Panel); (ii) Mode-1 Fibers of a Tensor (Matrix Columns), 
for j = 1, …, Ic, (Middle Panel); and (iii) Mode-2 Fibers of a Tensor (Matrix Rows), 
for i = 1, …, Im, (Right Panel)
Tensor Unfoldings of Global Returns
We next demonstrate that the vector and matrix unfoldings of the considered tensor in Exhibit 1 admit a physically meaningful interpretation. To this end, observe that the mode-1 and mode-2 unfoldings of Xt, that is, 
and 
, take the form (see Exhibits 2 and 3)
14
15with the j-th mode-1 fiber, 
, and i-th mode-2 fiber, 
, being respectively defined as
16In other words, a j-th mode-1 fiber, 
, contains the returns of all maturities associated with the j-th country (the returns of an entire domestic curve). Similarly, the i-th mode-2 fiber, 
, contains the returns of all countries associated with the i-th maturity; see Exhibit 3 for an illustration. Finally, observe from Exhibit 3 that the vector unfolding of global returns, 
, can be expressed in terms of mode-1 fibers (domestic yield curves) as shown in Exhibit 4.
Illustration of the Vectorisation of an Order-2 Tensor (Matrix) Sample, Represented in Terms of Mode-1 Fibers (Domestic Yield Curves), 
for j = 1, …, Ic
Second-order Moments of Global Returns through Multilinear Algebra
At the core of our approach is the assumption that if data exhibit the lattice structure in Exhibit 1, then multilinear algebra can be used to naturally decompose the covariance matrix of global fixed income returns, 
, into its maturity-domain and country-domain components. To justify this assumption, observe that, from Exhibit 1 and the condition in Equation 12, if the entries in Xt are sampled from a linearly separable field, then the vectorized form of tensor-valued random variable, that is xt = vec(Xt), exhibits a Kronecker separable covariance structure of the form (Hoff 2011; Scalzo et al. 2020)
17with 
as the maturity-domain covariance matrix and 
the country-domain covariance matrix. From Equation 12 we also have
18
19and therefore the Kronecker separable structure in Equation 17 can be expanded as
20The linear separability condition in Equation 12 also implies the following relationships concerning the second-order moment of every mode-n unfolding
21
22This can be proved by inspecting the entries of the second-order moments of the matrix unfoldings in Exhibit 3 and Equations 14–15. For example, for the mode-1 unfolding, we have
23Similarly, we arrive at the result in Equation 22 for the mode-2 unfolding, from
24Therefore, [E{XXT}]jk is proportional to the covariance between asset returns with maturities mj and mk for a given country, averaged over all countries, while [E{XTX}]jk is proportional to the covariance between asset returns in countries cj and ck at a given maturity, averaged over all maturities.
Remark 1. The concept underpinning the proposed multilinear model is that the covariance matrix, E{xxT}, of the global returns admits a characterization in terms of its fiber-to-fiber (multilinear) covariance parameters. This contrasts the entry-to-entry (linear) covariance parameters implied by the multivariate normal distribution and assumed in standard linear algebra based matrix models.
In addition, the Kronecker separability condition in Equation 12, which is inherent to multilinear algebra, provides a stable and parsimonious alternative to an unconstrained standard estimate of E{xxT}, which is unstable or even unavailable if the number of variables (assets) are large compared to the number of samples. For example, consider the unconstrained covariance matrix, 
, calculated based on a standard multivariate treatment of the global returns data, which (owing to its symmetry) has 
distinct entries. In contrast, its Kronecker separable counterpart, Σ(c) ⊗ Σ(m), reduces to only 
distinct parameters. To fully appreciate the scale of dimensionality reduction achieved by the proposed multilinear model, consider fixed income returns for Im = 15 maturities and Ic = 8 countries, that is, with Im Ic = 120 returns in total at any time instant. Then, the standard multivariate covariance matrix will have 
distinct parameters, whereas the Kronecker separable multilinear counterpart reduces the model to 
parameters, a reduction by a factor of 97.9%. For more insight into the practical utility of the so achieved parameter reduction, see Exhibit 5.
Parameter Dimensionalities of the Standard Matrix Model and of the Proposed Multilinear Model, for a Varying Number of Countries, Ic, and Maturities, Im
Implied Domestic and Cross-country Covariance
Now that we have illustrated the benefits of Kronecker separability conditions, we proceed to investigate the domestic and cross-country fixed income return interactions implied by the proposed multilinear model. To this end, observe that the covariance matrix of global returns, 
, takes the form of a block matrix containing both domestic and cross-country covariance matrixes, that is
25where 
is the domestic interest rate curve covariance matrix associated with the i-th country, while 
is the cross-country interest rate curve covariance matrix between the i-th and j-th countries. From the Kronecker separability condition in Equation 20, the multilinear model asserts that each domestic and cross-country covariance matrix of returns takes the form
26which implies that all countries share the same maturity-domain covariance component, Σ(m). In other words, all countries exhibit the same level, slope, and curvature factors, as demonstrated empirically in the Global Analysis Section below.
A Statistically Identifiable Formulation
It is important to highlight that, although the introduced parametrization of the covariance decomposition in Equation 17 is physically meaningful, it is statistically non-identifiable, that is, different values of Σ(m) and Σ(c) may generate the same Kronecker product, Σ(c) ⊗ Σ(m) (Dutilleul 1999). On the other hand, the identifiability condition is absolutely necessary for a parameter estimator to be statistically consistent (Newey and McFadden 1994), as it asserts that the associated log-likelihood function has a unique global maximum (Huzurbazar 1947).
To overcome this issue, Scalzo et al. (2020) introduced a statistically identifiable formulation of the Kronecker separable covariance parameters, to yield statistically consistent estimators. For our case, this yields the following statistically identifiable re-parametrization of the covariance parameters
27where σ2 ∈ ℝ denotes the variance of the tensor-valued random variable of global returns, defined as
28while 
and 
denote respectively the maturity-domain and country-domain covariance density matrixes, which, being density measures, are constrained to have unit-trace, that is
29Now, under the identifiable parametrization in Equation 27, the second-order moments of the tensor-valued random variable become
30
31with
32
33Intuitively, σ2 is the expected Frobenius norm of the tensor of global fixed income returns, while the covariance density matrixes, Θ(m) and Θ(c), designate respectively the proportion of the total variance, σ2, associated with each mode-1 fiber (domestic curve) and mode-2 fiber (cross-country constant-maturity assets).
Estimation Procedure
Using the proposed identifiable framework, the estimate of the “tensor variance” parameter, σ2, over T time instants is given by
34while the maturity-domain and country-domain covariance density matrixes are obtained as
35
36Observe that these are sample estimators of a method of moments type, which are also statistically consistent (Scalzo et al. 2020).
GLOBAL FIXED INCOME FACTOR ANALYSIS
The maturity-domain and country-domain covariance density matrixes established in the previous section now can be employed to yield orthogonal basis functions that are obtained through their corresponding eigenvalue decompositions
37
38Notice that, owing to the properties of the Kronecker product (Magnus and Neudecker 1985), while the principal component analysis of the global returns covariance matrix still yields the standard eigendecomposition
39the matrixes of eigenvectors, U, and eigenvalues, Λ, now become Kronecker separable, that is
40
41Remark 2. We refer to the decomposition in Equations 39–41 as multilinear PCA. This result is intrinsically linked to the well-known multilinear singular value decomposition (Tucker 1966 and De Lathauwer et al. 2000).
The maturity-domain eigenvector matrix, 
, contains column vectors, 
, that describe the orthogonal directions of the maturity-domain covariance. These vectors include the well-known level, slope, and curvature factors. Similarly, 
contains column vectors, 
, which describe orthogonal directions of the country-domain covariance. These also have an economically meaningful interpretation, as shown in the sequel.
Note that for both factors, U(m) and U(c), the eigenvectors are orthogonal, that is, U(m)T U(m) = I and U(c)TU(c) = I. Similarly, the diagonal matrix, 
, contains the eigenvalues, 
, which describe the fraction of the total variance, σ2, explained by factor 
, while 
, contains the eigenvalues, 
, on its diagonal, which describe the fraction of the total variance, σ2, explained by factor 
. As such, the eigenvalues of both Λ(m) and Λ(c) sum up to unity, i.e., tr (Λ(m)) = tr (Λ(c)) = 1.
Owing to the Kronecker structure in Equation 39–41, upon inspecting the i-th eigenvector, 
, of the matrix 
in Equation 39, observe that it also exhibits the following Kronecker structured form
42with i = 1 + (j − 1)Im, and 
as the l-th element of the vector 
. Consequently, the eigenvalue, λi, is also separable and is given by
43From the structure of the eigenvector in Equation 42, we can assert that the eigenvector ui represents a concatenation of the maturity-domain components, 
, which are the same for each country, each weighted by their associated country-domain scaling factor, whereby the l-th country is scaled by 
. This follows from the result in Equation 26, that is, the (i, j)-th cross-country covariance matrix admits the spectral expansion in the form
44GLOBAL PORTFOLIO MANAGEMENT AND HEDGING
The application of principal components for a domestic interest rate curve analysis has been widely studied and implemented by the investment community for both the evaluation of market risk and construction of hedged portfolios (SalomonSmithBarney 2000; Credit Suisse 2012; Standard Chartered 2013; TD Securities 2015). However, existing approaches do not account for the sources of risk common to multiple interest rate curves—a critical concept required for an appropriate assessment of the potential for international diversification within a global fixed income portfolio.
Consider a global fixed income portfolio, consisting of assets with Im distinct maturities within Ic distinct countries, that is, ImIc assets in total. As the measure of portfolio risk, we consider the portfolio variance, 
, which is a function of the global covariance matrix of returns, 
. Given a vector of portfolio weights, 
, the portfolio variance can be expressed as
45We have demonstrated in the previous section that, when considering an international basket of fixed income assets, the covariance, Σ = σ2(Θ(c) ⊗ Θ(m)), exhibits a Kronecker separable structure. In the same spirit, since Exhibit 1 suggests that our portfolio vector also exhibits a Kronecker separable structure, we can formalize this as
46where 
and 
are respectively the maturity-domain and country-domain portfolio weights. As shown in Exhibit 5, this yields parsimonious and compact solutions for global portfolio management schemes. The corresponding reduction in parameters required to optimize the portfolio—from the standard ImIc to (Im + Ic) in the proposed scheme—is only made possible through the multilinear decomposition of the global returns covariance into the maturity-domain and country-domain covariances, as shown in Equation 31.
Minimum Variance Portfolio through Multilinear Analysis
Consider the standard capital-constrained portfolio setup, which attains the minimum variance through the widely studied optimization problem, given by (Markowitz 1952)
47with 1 as a vector of ones. The solution is given by the well-known minimum variance portfolio
48From Equation 31, for the considered Kronecker separable case, the optimal multilinear portfolio allocation reduces to
49Remark 3. The result in Equation 49 asserts that portfolio optimization can be separated into parallel sub-problems that operate within their respective maturity and country domains. This is equivalent to solving for w(m) and w(c) independently through the following individual minimizations
and
The asset allocation associated with a fixed income asset within the j-th country and with the k-th maturity now can be expressed in an intuitive and physically meaningful way as
50with i = k + (j − 1)Im. Also notice that the proposed model significantly alleviates the computational burden associated with the minimum-variance portfolio solution, since the original inversion of the matrix Σ with dimensions (IcIm × IcIm) is replaced with the inversion of two smaller matrixes, Θ(c) and Θ(m), with the respective dimensions (Ic × Ic) and (Im × Im).
Hedging
The task of hedging fixed income securities remains one of the most challenging problems faced by financial institutions. In the context of hedging, the i-th principal component of the global returns covariance matrix, 
, can be viewed as a portfolio allocation vector associated to a latent market risk factor. Therefore, the aim of hedging is to minimize the risk exposure of a portfolio allocation, w, with respect to an i-th principal component, ui, that is given by their inner product, 
. As a consequence, a hedged portfolio is attained through the condition of orthogonality, that is, 
.
Based on Equation 42, within the considered multi-country setup every principal component of the global covariance matrix decomposes into 
. In this way, upon constraining the portfolio vector, w, to be Kronecker separable, the risk exposure simplifies to
51Remark 4. The multilinear decomposition of the portfolio risk exposure in Equation 51 asserts that the desired condition of orthogonality for hedging, 
, can be attained by either achieving orthogonality within the maturity domain or within the country domain, and thus it is not necessary to achieve orthogonality within both domains simultaneously. This is because, if either 
or 
, the global exposure in Equation 51 vanishes.
We next illustrate the advantages of this multilinear risk exposure result in real-world applications.
Hedging long-term bonds. Consider hedging a long-only portfolio of international long-term fixed income assets, all of which have maturity mi (e.g., 30 years), using an international portfolio of short-term assets. The hedged portfolio therefore must satisfy the following well-known conditions within the maturity domain:
52
53
54where 
is a vector of zeros except for the i-th element which equal to 1. Intuitively, the first condition reflects the long-only position in assets with the i-th maturity; the second condition constrains the strategy to be self-financing; and the last condition enforces orthogonality with respect to the leading n maturity-domain principal components.
Hedging domestic bonds. Conversely, consider hedging a domestic long-only portfolio within country ci, using an international portfolio. The hedged portfolio must satisfy the following constraints within the country domain:
55
56
57where 
is a vector of zeros except for the i-th element which equal to 1. The first condition above reflects the long-only position in the i-th country; the second condition constrains the strategy to be self-financing; and the last condition enforces orthogonality with respect to the leading n country-domain principal components.
The above portfolio hedging problems reduce to solving the following linear systems
58
59Based on Equations 58 and 59, the optimal maturity-domain and country-domain weights are therefore given by w(m) = A(m)+ b(m) and w(c) = A(c)+ b(c), where (·)+ denotes the Moore-Penrose inverse operator.
EMPIRICAL ANALYSIS
We next provide a validation of the proposed multilinear factor model through an empirical analysis of the global term structure of the international interest rate swaps (IRS) market. The data comprise weekly IRS rate curves,1 ranging from January 1, 2015, to July 1, 2019, for eight developed economies: Switzerland, Euro Area, United Kingdom, Japan, Australia, New Zealand, Canada, and the United States. Each domestic IRS curve consisted of swaps with maturities {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 15, 20, 25, 30} years. Therefore, every week we observed Im = 15 IRS returns for each of the Ic = 8 economies, resulting in ImIc = 120 weekly observations in total. Exhibit 7 displays the historical IRS rates employed in the analysis. Observe the similar collective behavior of the historical data that confirms the importance of global factors in driving the co-movement of fixed income securities across advanced economies.
Domestic Analysis
As a complementary and preliminary assessment of the commonality of returns within different country IRS curves, we first performed principal component analysis for each of the eight economies considered, to obtain their dominant domestic principal components, that is, their domestic level, slope, and curvature factors. The three leading factors within each domestic IRS curve are shown in Exhibit 7, and the percentage of variance explained by each component is given in Exhibit 6.
Explanatory Power [%] of PC1 (Level), PC2 (Slope) and PC3 (Curvature) for the Eight Economies Considered
Weekly Swap Ratesa for the Eight Economies Considered, with the Maturities of {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 15, 20, 25, 30} Years (Respectively Color-Coded from Blue to Red) for the Period 2015-01-01 to 2019-07-01 (Left Panel) and Their Corresponding PC1 (Level), PC2 (Slope) and PC3 (Curvature) Obtained from the Standard PCA of the Weekly Swap Returns (Right Panel). The similarity of the leading three principal components across all economies indicates the existence of latent global factors
Weekly Swap Ratesa for the Eight Economies Considered, with the Maturities of {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 15, 20, 25, 30} Years (Respectively Color-Coded from Blue to Red) for the Period 2015-01-01 to 2019-07-01 (Left Panel) and Their Corresponding PC1 (Level), PC2 (Slope) and PC3 (Curvature) Obtained from the Standard PCA of the Weekly Swap Returns (Right Panel). The similarity of the leading three principal components across all economies indicates the existence of latent global factors
SOURCE: Bloomberg.
In agreement with the existing literature, here too all economies exhibited similar leading principal components. Moreover, the explanatory power of each component was consistent across all economies, whereby the first principal component (level) explains ≈ 90%, the second principal component (slope) explains ≈ 5%, and the third principal component (curvature) explains ≈ 1% of the total variation in interest rate changes. This supports our proposed multilinear approach to global fixed income returns analysis, as it serves as an indication of the existence of three leading dominant factors, that is, the global level, global slope, and global curvature factors. With these preliminary and intuitive results, we now proceed to evaluate the common global risk factors using the proposed multilinear model.
Global Analysis
We now proceed to evaluate the performance of the proposed multilinear factor analysis when applied to the international IRS dataset. The implementation procedure is summarized as follows:2
1. The weekly IRS returns at the t-th week were arranged to form the matrix-valued sample, 
, as described in Equation 13.
2. We estimated the parameters of the model (σ2, Θ(m), Θ(c)) using the analytic estimators in Equations 34–36.
3. We obtained the global maturity-domain and country-domain factors, U(m) and U(c), and their associated eigenvalues, Λ(m) and Λ(c), from the eigendecompositions of Θ(m) and Θ(c), as shown in Equations 37–38.
The three leading maturity-domain factors, 
, are plotted in Exhibit 8a, with their corresponding explanatory powers, 
, presented in Exhibit 9. Observe that the maturity-domain factors resemble the components obtained from traditional domestic principal components (see Exhibit 7), and therefore can be interpreted analogously, thus confirming the existence of a common set of bases shared by all economies. Furthermore, the explanatory powers of these factors are on par with those observed from the domestic analyses, which further validates our findings.
The Three Leading Maturity-Domain Global Factors (Top Panel) and the Country-Domain Global Factors (Middle and Bottom Panels). Observe that the maturity-domain factors resemble the components obtained from traditional domestic principal components in Exhibit 7, while the country-domain factors represent interpretable macroeconomic factors concerning subsets of the considered economies. See also Exhibits 9 and 10
Economic Interpretation and Explanatory Powers of the Three Leading Global Factors in the Maturity-domain
Economic Interpretation and Explanatory Powers of the Eight Leading Global Factors in the Country-domain
Remark 5. The maturity-domain factors obtained from the multilinear analysis, shown in Exhibit 8a, clearly serve as a stencil for describing the term structure within each domestic IRS curve, and as such we refer to these as the global level, global slope, and global curvature.
The country-domain factors, 
, are visualized in Exhibits 8a–8c, and their corresponding explanatory powers, 
, are presented in Exhibit 10. The most dominant factor, 
, had positive entries across all economies, and can be thought of as the global risk premium, analogous to the level factor in the maturity-domain. Notice that this factor also explains a significant portion of the international IRS variance. The remaining country-domain factors represent interpretable macroeconomic factors concerning subsets of the considered economies. These results demonstrate the ability of the proposed approach to yield enhanced physical insight into the global macroeconomic environment. This is achieved in a straightforward and compact manner, owing to the small number of parameters required to fully describe the global fixed income universe.
Observe that the maturity-domain (listed in Exhibit 9) and country-domain (listed in Exhibit 10) factors also can be directly employed for global macroeconomic hedging and risk management, as elaborated in the Global Portfolio Management and Hedging Section above.
Practical advantages of the proposed multilinear framework therefore include:
▪ A reduction in the number of parameters required to optimize the global portfolio, which for Im = 15 maturities and Ic = 8 countries reduces from ImIc = 120 to (Im + Ic) = 23 portfolio weight parameters, and
▪ A parsimonious description of the global risk in terms of a multilinear decomposition into the smaller-scale maturity-domain and country-domain risk factors that can facilitate the investor’s decision-making process.
CONCLUSIONS
We introduce a novel multilinear algebraic framework for modeling the covariance of global fixed income returns in order to cater for the highly structured correlations across maturities and economies. By virtue of the multilinear approach inherent to tensors, as opposed to the standard “flat-view” multivariate matrix ones based on linear algebra, we show the covariance structure of international fixed income returns to be factorizable analytically into the maturity-domain and country-domain covariances. In this way, the proposed analysis: 1) achieves a significant reduction in the number of parameters required to fully describe the international investment universe; and 2) offers a mathematically tractable platform for estimating and identifying physically interpretable global risk factors.
As a natural extension of the proposed framework, we have derived analytic solutions to the global hedging and portfolio management paradigms, which allow the investor to gain enhanced control over the portfolio risk within the two independent domains spanning the global returns landscape: maturity and country. We have performed empirical analysis on the interest rate swaps curves for eight developed economies, and confirmed the existence of common global risk factors and their practical utility. The results are supported by our own Python toolbox for multilinear tensor analysis (Kisil et al.).
ENDNOTES
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