Patent Grant
7324978
1. Technical Field
The present invention relates to risk model aggregation. More particularly, the present invention relates to a method and apparatus for a mathematical technique that forces an aggregate risk model to be consistent with embedded standalone models.
2. Description of the Prior Art
Shakespeare reminds us that the perils of investment have been with us always. Investors have long appreciated the importance of diversification. The idea took quantitative form with the work of H. Markowitz, The Birth of Mean-Variance Optimization (7.1, 77-91), Portfolio Selection, Journal of Finance (1952). Since then increasingly sophisticated mathematical and statistical tools have been brought to bear on the problem of estimating the aggregate risk of a portfolio.
The aggregate risk of a portfolio depends crucially on the covariances of the portfolio's constituent assets. Unfortunately, in practical situations so many covariances come into play that it is impossible to estimate all of them directly from historical data. Factor models overcome this difficulty by expressing the large number of asset covariances in terms of a small number of factor covariances.
A factor model is defined through a linear regression, as follows:
y=A·x+ε. (1)
Here y is the vector of asset returns whose variances and covariances require estimation, x is a vector of factor returns whose variances and covariances can be reliably estimated, and A is a matrix, specified a priori, that describes the sensitivities of the assets to the factors. The vector ε of errors is usually assumed to be normally distributed with a diagonal covariance matrix D. This model estimates the asset covariance matrix, Σ(y), as
Σ(y)≈A·Σ(x)·AT+D (2)
where Σ(x) is the matrix of factor covariances. The dimension of the matrix Σ(x) is small enough that historical data allow reasonable estimates of the covariances.
As quantitative risk management has become more sophisticated, factor models have become more finely detailed. At the same time, models have broadened; large models encompassing many asset classes and markets have become necessary for large firms to forecast their “total risk.” Both of these trends have forced the number of factors upward to the point where, once again, more covariances are required (this time, between factors instead of assets) than can be accurately estimated.
This problem can be addressed by iterating the idea that worked before. Factor models themselves can be built up from smaller factor models. However, this approach introduces a new set of difficulties related to consistency. To illustrate the problem, imagine building a factor model to estimate risk for a portfolio composed of US equity and fixed income securities. Suppose further that there are already excellent standalone factor models that separately treat equities and fixed income securities. An aggregate factor model will almost certainly be inconsistent with the standalone factor models. The discrepancies might result from the iterated factor structure, from differences in the amount and frequency of data, from clashes in statistical methods specific to each standalone model, or from other sources. Such discrepancies are undesirable in part because they can cause different levels of a firm to have different views of the same source of investment risk.
It would be advantageous to provide a mathematical technique that enforces consistency between an aggregate model and the standalone models, i.e., to achieve breadth without sacrificing meaningful detail. More specifically, it would be advantageous to revise the aggregate risk model to be consistent with the standalone models. Unfortunately, enforcing consistency almost always involves the destruction of covariance Information in the aggregate model. The problem is to minimize the damage.
The present invention relates to a method and apparatus for a mathematical technique that forces an aggregate risk model to be consistent with embedded standalone models. More specifically, the present invention provides an optimized method and apparatus for revising submatrices, also referred to herein as subblocks, of a covariance matrix. A particular linear change of variables, called a revising transformation, is introduced whereby each diagonal submatrix corresponding to a standalone model is overwritten with a revised version. The space of revising transformations is shown to be parameterizable by a product of orthogonal groups via a particular parameterization. A revising transformation is provided that least changes the cross blocks (i.e., the off-diagonal blocks) of the covariance matrix, thereby preserving as much information in the original cross block correlations as possible, consistent with a prescribed overwriting. An objective function that quantifies the loss of cross block information is specified and minimized over the orthogonal group.
The present invention relates to a method and apparatus for a mathematical technique that forces an aggregate risk model to be consistent with embedded standalone models. More specifically, the present invention provides an optimized method and apparatus for revising the diagonal submatrices that correspond to standalone models, so that agreement is achieved. A particular linear change of variables is introduced whereby each diagonal submatrix is overwritten with a revised version; the revision of a particular diagonal submatrix leaves the others unaffected. The space of simple revising transformations is shown to be parameterizable a product of orthogonal groups via a particular parameterization. A revising transformation is sought that least changes the cross blocks (Le., the off-diagonal blocks) of the covariance matrix, thereby preserving as much of the original information in the cross block correlations as possible. An objective function that characterizes information loss is provided and minimized over the product of orthogonal groups.
An important part of understanding the invention is understanding the concept of an aggregate risk model and knowing some relevant definitions.
An Aggregate Risk Model and Examples
In many cases, the vector of factors underlying an aggregate risk model can be decomposed into meaningful subvectors. For example, a factor model may cover more than one market or asset type. Alternatively, statistical properties may distinguish a particular group of factors.
Definitions. An aggregate factor model is a factor model whose vector of factors f decomposes into subvectors:
Let Σ be the covariance matrix of the factors f. The decomposition in equation (3) imposes a block structure on the covariance matrix. The ith diagonal block Σi of Σ contains the subset of covariances between factors in the subset fi. The cross-block covariances are the elements of Σ not contained in one of the Σi's.
The problem of revising an aggregate risk model to be consistent with the standalone models is illustrated below in two examples: a currency block example and a global equities model example.
A Currency Block Example
The first example arises in connection with currency risk. A substantial amount of the variation in the return of an unhedged international portfolio of investment-grade bonds is due to changes in currency exchange rates. Hence international risk models need factors to account for currency risk, as well as risk due to changes in interest rates and credit spreads. However, the statistical properties of the two types of factors are quite different. Currencies are very volatile and subject to regime changes. Risk estimates for these factors make use of recent, high frequency data. For local market factors, however, it is desirable to use lower frequency data and a model with a longer memory.
One approach to combining the models is to build a “first draft” of the aggregate model based on lower frequency data and then to embed a time-scaled currency risk block that has been estimated separately. Unfortunately, the easiest way to do this, i.e. simply overwriting the first draft of the currency block with the high frequency block, usually leads to an inconsistent covariance matrix. The inconsistency can arise in many ways. It might appear because the variance ascribed to a factor is too small to account for its large covariances with factors of another standalone model, or because it is now very strongly correlated with another factor from the same standalone model, but the off-diagonal blocks indicate the two factors behave quite differently in relation to a third factor from another standalone model. The “covariance matrix” resulting from the simple overwrite will characteristically make negative variance forecasts (or imaginary forecasts of standard deviation) for some unfortunate portfolios, and forecasts for other portfolios will be hopelessly compromised. It follows that any realistic approach to overwriting the currency block must disturb cross-block correlations.
The preferred embodiment of the invention provides a simple approach that produces consistency. It assumes that there is a linear relationship between the returns forecast by the “first draft” of the aggregate model and the returns forecast by the standalone model, and transforms from the original to the revised diagonal block following this relationship. In our example, a linear change of variables overwrites the first draft of the currency block with the high quality standalone model version. As is necessarily the case, this change of variables disturbs the cross-block correlations.
This raises the question of how small the disturbance can be made. Using a naive linear change of variables, one can faithfully embed the currency block and in so doing typically change the cross-block correlations by ±0.1. Using the optimized approach described below, the average disturbance is cut to approximately ±0.05.
A Global Equities Example
The global equities example arises in connection with a risk model for a portfolio of global equities. An example of such risk model is one currently in use at Barra Inc., Berkeley, Ca. Barra's global equity model covers 56 markets and has 1274 factors. The structural approach of “iterated factor models” mentioned above is employed. The result is a “first draft” of the risk model that avoids spurious correlations. However, it leaves a family of embedded submodels that are inconsistent with single country equity models. Here the problem is more complicated than in the previous example, since many diagonal blocks require revision simultaneously. This results in an optimization problem with over 12,000 variables, as opposed to the roughly 700 variables in the global fixed income and currency embedding problem above. Thus, the overwhelming size of the problem has led to explorations of modifications of the optimization routine, achieving nearly the same result, but with a large reduction in the number of variables. Several simplified approaches to multiple block embedding are discussed below in the Section, Multiple Block Optimization.
Optimized Single Block Embedding
One preferred embodiment of the invention optimizes embedding blocks one at a time, as if each block were the only one requiring attention. Just as in the currency/global fixed income example, the invention offers a significant improvement over unoptimized methods.
To understand this embodiment of the invention, consider a simple example for vectors {tilde over (t)} and u, where ({tilde over (t)}) represents, currency and (u) represents fixed income factors. Then let {tilde over (Σ)} be the covariance matrix of such random vector, {tilde over (s)}, where
for vectors {tilde over (t)} and u.
This embodiment of the invention comprises overwriting a diagonal subblock of the covariance matrix, {tilde over (Σ)} without affecting other diagonals. However, cross-blocks are affected, as shown in the linear change of variables section below.
A Linear Change of Variables
Suppose that {tilde over (Σ)} is a covariance matrix of a random vector {tilde over (s)}. This means that {tilde over (Σ)} is symmetric and positive semi-definite. Let L be a nonsingular linear transformation satisfying
s=L·{tilde over (s)}. (4)
Then the matrix
Σ=L·{tilde over (Σ)}·LT (5)
is the covariance matrix of the random vector s. Consequently, Σ is symmetric and positive semi-definite.
We decompose the vector {tilde over (s)} of random variables into two parts:
This breakdown induces a block structure on the covariance matrix:
Here {tilde over (Θ)} and Y are the covariance matrices of the vectors {tilde over (t)} and u respectively. The off-diagonal block C contains covariances between members of {tilde over (t)} and members of u. Recall that an example to keep in mind is the breakdown into currency ({tilde over (t)}) and fixed income factors (u).
According to the preferred embodiment of the invention, the diagonal submatrix {tilde over (Θ)} can be overwritten with a revised version Θ. This can be accomplished with a transformation L of the following form:
The symbol {tilde over (Θ)}1/2 denotes the symmetric square root of Θ. It is the unique positive semi-definite symmetric matrix M satisfying
M·M=Θ.
The symbol {tilde over (Θ)}−1/2 denotes the multiplicative inverse of {tilde over (Θ)}1/2. The matrices {tilde over (Θ)}1/2 and {tilde over (Θ)}−1/2 are guaranteed to exist, because Θ and {tilde over (Θ)} are symmetric positive semi-definite.
Due to the special form of the matrix L, transformation (5) preserves the block structure of {tilde over (Σ)}. The matrix {tilde over (Θ)} is replaced by Θ and Y is unchanged. The revised covariance matrix is
The considerations above show that by taking a linear transformation that preserves the block structure of {tilde over (Σ)}, a diagonal subblock can be overwritten without affecting the other diagonal subblocks. However, the cross blocks are affected.
The discussion above raises two interesting issues:
These topics are addressed in the following two sections, Revising Transformations and The Optimal Revising Transformation, respectively.
Revising Transformations
As in the previous section of the aggregate factor model, let {tilde over (Σ)} be a covariance matrix of
Let {tilde over (Θ)} be the subblock corresponding to {tilde over (t)}, and let Θ be a revised version of {tilde over (Θ)}. Hence, the set of linear transformations that revise only one diagonal block can be characterized by a simple revising transformation, as defined below. Furthermore, the space of simple revising transformations of such set of linear transformations can be parameterized by a particular orthogonal group, as shown in the discussion below.
The preferred embodiment of the invention introduces two more definitions.
Definition. A simple revising transformation of ({tilde over (Σ)}; {tilde over (Θ)}, Θ) is a linear map L: {tilde over (s)}s that overwrites {tilde over (Θ)} by Θ in Σ=L·{tilde over (Σ)}·LT and restricts to the identity on u, so that it leaves the complementary diagonal block unchanged.
Definition. A square root of an n×n matrix Θ is an n×n matrix √{square root over (Θ)} such that
√{square root over (Θ)}√{square root over (Θ)}T=Θ.
It is a fact that a matrix Θ has a square root if and only if Θ is positive semi-definite symmetric. It is also not difficult to show that √{square root over (Θ)} is a square root of such Θ if and only if √{square root over (Θ)}=Θ1/2·O for some orthogonal matrix O.
The preferred embodiment of the invention provides a characterization of simple revising transformations, presented below as Theorem 1 in two parts, and the corresponding proof thereof.
Theorem 1
for some choice of square roots √{square root over (Θ)} and √{square root over ({tilde over (Θ)}.
for OεO(n).
Proof of Theorem 1:
Direct calculation demonstrates that each such L or L(O) is a simple revising transformation of ({tilde over (Σ)}; {tilde over (Θ)},Θ).
Conversely, suppose that L is a simple revising transformation of ({tilde over (Σ)}; {tilde over (Θ)},Θ). By assumption L takes the form
Transformation (5) implies that
Θ=M·{tilde over (Θ)}·MT. (12)
Since Θ and {tilde over (Θ)} are symmetric and positive semi-definite, equation (12) can be rewritten as
id=Θ−1/2·M·{tilde over (Θ)}1/2·{tilde over (Θ)}1/2·MT··{tilde over (Θ)}−1/2 (13)
Set O=Θ−1/2·M·{tilde over (Θ)}1/2. Then equation (13) becomes
id=O·OT, (14)
establishing that OεO(n), and so L is of the form L(O). Since L is clearly one-to-one, this finishes the proof of (b).
For (a), if L is a simple revising transformation, then, by (b), L=L(O) for some OεO(n), and, thus, √{square root over (Θ)}=Θ1/2·O and √{square root over ({tilde over (Θ)}={tilde over (Θ)}1/2.
The Optimal Revising Transformation
To address the second question posed in the linear change of variables section, it is desirable to find a revising transformation that least changes the cross-block correlations of the covariance matrix {tilde over (Σ)}. Such a revising transformation preserves as much information in the original cross-block correlations as possible consistent with the prescribed subblock Θ.
A preferred embodiment of the invention treats such modeling problem numerically as a nonlinear minimization. An objective function is chosen to measure the degree of change to the cross blocks due to L. Such choice of objective function generally depends on the relative importance of the different variables. It has been found that it is reasonable to make a generic choice for this analysis, e.g., the root-mean-square change in components of the cross block due to L(O):
f(O)=(1/m)∥Θ1/2·O·{tilde over (Θ)}−1/2C−C∥,
where m is the number of elements of C, and ∥·∥ refers to the matrix norm given by the square root of the sum of the squares of the components.
It is equivalent and more convenient to minimize the function
g(O)=m2f2=∥Θ1/2·O·{tilde over (Θ)}1/2C−C∥2.
Thus, the preferred embodiment of the invention provides the solution to the numerical problem of minimizing g over the orthogonal group O(n). It should be appreciated that although this group is disconnected, attention can be restricted to the connected component containing the Identity SO(n), the group of orthogonal matrices with determinant +1.
The form of the objective function g as a sum of squares makes it suitable to the Levenberg-Marquardt numerical method, a hybrid of Newton's method and steepest descent (see W. Press, et al, Numerical Recipes in C++, Cambridge, 2002). Because this method requires computation of derivatives of g with respect to the parameters in O, it is desirable to have a parameterization of SO(n) for which it is convenient to compute the partial derivatives. Such a parameterization need not be one-to-one, but should cover the whole group.
The preferred embodiment of the invention provides such a function, defined as follows:
Let N=n(n−1)/2 denote the dimension of SO(n). For 1≦i<j≦n, denote by Ri,j(θ) the rotation by angle θ in the (i,j) plane (right hand rule) in Rn. There are N distinct such coordinate planes. The preferred embodiment of the invention makes use of the following assertion.
Every element of SO(n) may be expressed as a product of N rotations,
h(θ1, . . . ,θN)≡R1,2(θ1)R1,3(θ2) . . . R1,n(θn) . . . Rn−1,n(θN),
for some angles θ1, . . . , θN.
Because it is relatively easy to compute the partial derivatives of h, the Levenberg-Marquardt minimization technique is applied to
G=g(h(θ1, . . . , θN))
in a straightforward way. If the method converges, the result is a value O*εSO(n) at which g has at least a local minimum, and therefore a simple revising transformation L(O*) likely to be better than L(id) or another random choice.
Multiple Block Optimization
In the full problem, it is desirable to adjust all of the submodel submatrices of the first-draft covariance matrix. One approach is to simultaneously optimize a sum of all the single block objective functions. In this case, the revising transformation is a product of simple revising transformations.
This leads to the following result.
Theorem 2
In practice the dimension of the product of orthogonals may be too large to be viably searched by our algorithm. Consequently, we explore approaches that reduce the dimensionality of the problem.
A preferred approach is to solve a sequence of single block problems, one problem for each submodel.
Accordingly, although the invention has been described in detail with reference to particular preferred embodiments, persons possessing ordinary skill in the art to which this invention pertains will appreciate that various modifications and enhancements may be made without departing from the spirit and scope of the claims that follow.
| Filing Document | Filing Date | Country | Kind | 371c Date |
|---|---|---|---|---|
| PCT/US02/17924 | 6/4/2002 | WO | 00 | 6/7/2004 |
| Publishing Document | Publishing Date | Country | Kind |
|---|---|---|---|
| WO02/099684 | 12/12/2002 | WO | A |
| Number | Name | Date | Kind |
|---|---|---|---|
| 5093869 | Alves et al. | Mar 1992 | A |
| 5761442 | Barr et al. | Jun 1998 | A |
| 5812987 | Luskin et al. | Sep 1998 | A |
| 5930762 | Masch | Jul 1999 | A |
| 6219805 | Jones et al. | Apr 2001 | B1 |
| 6249775 | Freeman et al. | Jun 2001 | B1 |
| 6278981 | Dembo et al. | Aug 2001 | B1 |
| 6539351 | Chen et al. | Mar 2003 | B1 |
| Number | Date | Country | |
|---|---|---|---|
| 20040236546 A1 | Nov 2004 | US |
