Some Results on Correlation Matrices for Interest Rates

Acta Appl Math (2011) 115:291–318DOI 10.1007/s10440-011-9622-x

Some Results on Correlation Matrices for Interest Rates

Ernesto Salinelli · Carlo Sgarra

Received: 10 September 2010 / Accepted: 31 May 2011 / Published online: 17 June 2011© Springer Science+Business Media B.V. 2011

Abstract In this paper we systematize and develop some theoretical results about shift,slope and curvature for correlations matrices of interest rates. We provide a general inves-tigation on the relations among some standard features of correlation models for interestrates and the existence of shift, slope and curvature. Our results show how their presence,excluding some peculiar behavior strictly related to low dimensions, can not be directly con-nected to the usual assumptions on the interest rates correlations structure. We provide someestimates of the distance between a shift and the vector, named pure shift, having all entriesequal. We prove also that in a two-factor framework shift and slope are sufficient to justifythe usual properties of correlations between rates.

Keywords Correlation structures · Interest rates models · Principal component analysis ·Total positivity

Mathematics Subject Classification (2000) 91G30 · 15A18 · 62H25

1 Introduction

Correlation models play a fundamental role in the financial framework, for example inderivatives pricing and in risk management. Accurate valuation of basket options, creditderivatives and fixed income products requires a detailed knowledge of correlation structuresamong the underlying assets. In particular, for interest rates products, correlation structureshave been extensively studied during the last years, and several interesting models have beenprovided describing them [20, 22, 27].

E. SalinelliDipartimento di Scienze Economiche e Metodi Quantitativi, Università del Piemonte Orientale“A. Avogadro”, Via Perrone 18, 28100 Novara, Italye-mail: [email protected]

C. Sgarra (�)Dipartimento di Matematica, Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133 Milano, Italye-mail: [email protected]

mailto:[email protected]

mailto:[email protected]

292 E. Salinelli, S. Sgarra

Interest rates can be described in terms of the yield curve and their correlations shed lighton its evolution. It is of great interest for both academicians and practitioners to developstatistical methods suitable to describe the movements of the yield curve and to empiricallyjustify some important models for interest rate dynamics based on one or more factors.

When the yield curve is represented by a random vector, Principal Component Analysis(PCA) has turned out to be one of the most useful tools to find such factors, through the studyof the eigenvectors of the associated correlation matrix. Starting from the seminal work ofLitterman and Scheinkman [18], in all the empirical literature related to PCA for interestrates it is emphasized how the first three eigenvalues of the correlation matrix can explainalmost completely the variability of the yield curve: the corresponding three eigenvectorshave been respectively termed as shift, slope and curvature, because of the peculiar behav-ior of their entries. Nowadays, it is usual (see e.g. [27]) to consider their existence amongseveral different properties, as positivity or monotonicity with respect to the maturities, thata “good” correlation model for interest rates should exhibit.

From these forewords it is then evident the importance to understanding how the corre-lation structure of the yield curve is related to the existence of shift, slope and curvature,which are to be defined in a suitable way from a theoretical perspective.

Recently, Salinelli and Sgarra [28, 29] and Lord and Pelsser [21] have faced these prob-lems. In the first paper shift, slope and curvature were formally defined in two differentways: the first, based only on the sign changes of the eigenvectors entries, defines weakshift, weak slope and weak curvature (WSSC), the second one, based also on their orderingproperties, defines shift, slope and curvature (SSC). In both the above mentioned papers,under suitable hypotheses on the positivity of the minors of the correlation matrices, theauthors proved some results concerning the sign changes of their first three eigenvectors,exploring the relations between these positivity assumptions and some usual requirementson the correlation structures.

In Salinelli and Sgarra [29] some results on the existence of SSC are given for a rele-vant class of models, in which the correlation coefficients are exponentially decaying withdifference in maturities. Anyway a more general and systematic investigation of the inter-connections of both the properties mentioned above is still lacking.

The purpose of the present work is to fill this gap, clarifying the relations among somestandard properties of correlation models and the presence of SSC and WSSC, by adoptingthe assumptions on the correlation structure considered in Rebonato [27], Lord and Pelsser[21]. Our results will show how the presence of Shift, Slope and Curvature, excluding somepeculiar behavior strictly related to low dimensions, can not be directly connected to theusual assumptions on the interest rates correlations structure, and could be the starting pointfor the research of new and more fundamental properties of these correlations. In doing so,among other things, we show how to estimate the size of the hump of the shift in terms ofcorrelations between rates illustrating the application in a relevant model. Examples con-cerning some relevant models introduced in the literature of interests rates complete thispart of our analysis.

We also consider the inverse problem with respect to that investigated before. Since thefirst three principal components explain almost the full variation of the yield curve meansthat the correlation matrix has essentially rank 3, hence it can be written in a simple form interms of its eigenvectors: if they exhibit the typical features of Shift, Slope and Curvature,under which conditions the elements of the correlation matrix will enjoy the same propertiesof those empirically observed? We are able to answer to this question: it is sufficient toassume the rank of the matrix is two with the first eigenvector of the correlation matrix isof Shift type and the second is a Slope to obtain the monotonicity by rows and column and,under mild extra conditions, the positivity of the matrix elements is granted too.

Some Results on Correlation Matrices for Interest Rates 293

In order to make the presentation self-contained, in each section of the paper, our originalresults are often preceded, when necessary, by a synthetic and systematic presentation of thestate of art of the subject.

After introducing the main definitions and clarifying the notations used throughout thepaper, in Sect. 2 we start with an overview of the existing literature, extending the analy-sis to a further assumption concerning the non-homogeneous time behavior of interest ratescorrelations and to the existence of SSC. In Sect. 3 the same analysis is then performed in amore general context, clarifying the role of the problem dimension. Successively, in Sect. 4,we focus our analysis on the first eigenvector of a correlation matrix of interest rates, pro-viding some estimation results. In Sect. 5 we discuss the inverse problem mentioned beforein the case of rank 2. Finally, in Sect. 6, we outline some possible future developments andperspectives of the present work, whereas in the Appendix we collect some considerationson a particular model.

2 Notation, Definitions and General Results

In this section, after formalizing the problem we deal with, we present a brief survey of someknown results concerning the presence of SSC for correlation matrices of interest rates.

Several are the technical questions about the application of PCA to these models, asthe choice between spot or forward rates, level or differences between rates, covariance orcorrelations matrices and so on: for a survey see [16, 22, 27]; among the huge number ofempirical studies devoted to the subject, we recall here the paper by Longstaff et al. [20],in which several graphic representation of different shapes for the correlation surface forforward rates are presented. We shall develop our considerations on PCA with reference tocorrelation matrices for forward rates differences (for a motivation of this choice see [28]).

2.1 Shift, Slope and Curvature

The empirical literature has found that changes in the shape of the yield curve are substan-tially imputable to three (unobservable) factors, respectively termed shift (or level), slopeand curvature, each one operating in a different way. The first changing the interest rates ofall maturities by almost identical amounts, the second varying short-term interest rates inan opposite way with respect to long-term interest rates, the third presenting the main ef-fects on medium-term interest rates. These factors can be identified by some properties thatcharacterize the first three (in terms of eigenvalue magnitude) eigenvectors of a correlationmatrix of forward rates. As a first description, we could say that the first has approximatelyequal elements of the same sign and for this is called shift or level; the second has elementsof opposite sign with approximately the same magnitude at the opposite end of the maturityspectrum admitting an interpretation as slope of the yield curve; the third has elements ap-proximately with the same magnitude and signs at the opposite end of the maturity spectrumand twice as large and of opposite sign in the middle, interpreted as curvature of the yieldcurve.

The description just given admits at least two refinements. On one side the symmetricbehavior (elements with the same distance from the central one have more or less the sameabsolute value), although present in some popular models as the exponential one, can not bepresent. Furthermore, the elements of the first eigenvector are usually increasing and thendecreasing, pointing out a bigger contribution given by intermediate maturities forward ratesto the main movement of the yield curve. As pointed out by Golub and Tilman [8] “. . . in


particular, we do not enforce the first principal component as a parallel shock. . . ; on thecontrary, we believe that the “humped” shape of the first principal component is importantby itself and can be used as a tool for placing curve bets”.

The features just discussed are captured by the following definition introduced in Salinelliand Sgarra [28].

Definition 1 Given a n × n, n ≥ 3, correlation matrix R having its first three eigenvaluessimple, we define: its first eigenvector v1 weak shift if v1,i > 0 for i = 1, . . . , n; shift if isweak shift and there exists an integer i ∈ {2, . . . , n − 1} such that

v1,i ≥ v1,i−1 ≥ · · · ≥ v1,1; v1,i ≥ v1,i+1 ≥ · · · ≥ v1,n;pure shift if is shift with v1,i = v1,j for all i, j = 1, . . . , n; its second eigenvector v2 weakslope if there existsi ∈ {2, . . . , n − 1} such that

v2,i−s < 0 s = 1, . . . ,i − 1; v2,i+k > 0 k = 1, . . . , n −i;slope if

v2,1 ≤ v2,2 ≤ · · · ≤ v2,n−1 ≤ v2,n and v2,1 · v2,n < 0;its third eigenvector v3 weak curvature if there exist i1 and i2 such that

1 ≤ i1 < i2 ≤ n and

{

v3,i ≥ 0 if i ≤ i1 or i ≥ i2

v3,i ≤ 0 if i1 < i < i2;

curvature if there existsi ∈ {2, . . . , n − 1} such that

v3,i ≤ v3,i−1 ≤ · · · ≤ v3,1; v3,i ≤ v3,i+1 ≤ · · · ≤ v3,n

and

min{v3,1, v3,n} > 0, v3,i < 0.

Remark 2 Recalling that normalized eigenvectors are uniquely defined up to a sign change,the choice implicit in Definition 1 is coherent with the empirical evidence.

As suggested by the previous discussion, Definition 1 states the difference between SSCin the weak and strict form: the first refers only on the number of the sign variation ofthe eigenvector elements, the second one also requires an ordering property of them. Thus,a weak shift has all its elements positive, whereas a shift has its elements (positive and)increasing and then decreasing too. In particular, a pure shift vector is proportional to 1 =[1, 1, . . . , 1]T . The reasoning just exposed extends in a straightforward way to the slopeand curvature cases.

To conclude, observe that, given a vector v ∈ Rn, to compute the number of sign vari-

ations in their components, the zero elements, as usual (see [6]), can be considered bothpositive or negative. According to the sign attributed to each of them, a maximum S+

v andminimum S−

v number of sign variations can be defined: if S+v = S−

v we define this commonvalue the number of sign variations of v. Note that S+

v and S−v coincide only if the first and

the last components of a vector are not zero and if for every zero component the precedingand the following components are of different sign.


2.2 Correlation assumptions

It is common belief (see e.g. [27]) that a good correlation model for interest rates shouldenjoy the following few basic features that the empirical evidence suggests:

(a) interest rates at different maturities are positively correlated;(b) the correlation coefficients decrease when the distance between the indices increases;(c) the previous reduction in the correlation between variables corresponding to the same

difference in the indices tends to decrease as the maturities of both the variables aregreater.

We formalize (a), (b) and (c), referring to an n-dimensional correlation matrix R = [ρij ]with the following properties:

(P0): positivity, i.e. ρij > 0, ∀i, j = 1,2, . . . , n;(P1): strict decreasingness of superdiagonal row elements:

∀ fixed i ≥ 1, ρi,j decreases with respect to j ≥ i;

(P2): strict increasingness of superdiagonal column elements:

∀ fixed j ≥ 1, ρi,j increases with respect to i ≤ j ;

(P3): strict increasingness of secondary superdiagonal elements:

∀ fixed p > 0, ρi,i+p increases with respect to i.

Observe that, by the symmetry of R, properties (P1) and (P2) can be stated in terms of subdi-agonal elements, exchanging the rows and columns roles. Furthermore, the assumption thatR is a correlation matrix can not be weakened since properties (P0)–(P3) do not guaranteethat a symmetric matrix with diagonal elements equal to 1 is positive semidefinite, as shownby the following example:

A =⎡

⎣

1 0.9 0.60.9 1 0.980.6 0.98 1

⎤

⎦ , λ3 = −0.0669759.

We shall denote with �(i1, . . . , is) the set of n-dimensional correlation matrices satisfy-ing properties (Pi1)–(Pis ).

Some conclusions on correlation matrices R ∈ �(0,1,2,3) are collected in the following

Proposition 3 If R = [ρij ] is an n × n correlation matrix in �(0,1,2,3), then:

(i) ρij < 1 for all i �= j ;(ii) minij {ρij } = ρn1;

(iii) maxi �=j {ρij } = ρn−1,n;(iv) ρij > ρi−1,j+1 ∀i = 2, . . . , n; ∀j = 2, . . . , n − 1;(v) ∀i < j it holds

ρi+1,j+1 − ρi,j+1 > ρi+1,j+1 − ρi,j and ρi+1,j+1 − ρi,j+1 > ρi,j − ρi,j+1; (1)


(vi) if

ri =∑

j

ρij ; mr = mini

ri; Mr = maxi

ri; s =√

mr − 1

Mr − 1(2)

then the following estimate of the dominant eigenvalue λ1 of R holds:

mr +(

1

s− 1

)

ρn,1 ≤ λ1 ≤ Mr − (1 − s)ρn,1.

Proof Conclusion (i) trivially follows by (P1) and/or (P2).The (ii) is a consequence of properties (P1) and (P2), that jointly to (P3) imply (iii).Applying (P2) and (P1) for any i = 2, . . . , n and j = 2, . . . , n−1, we have ρij > ρi−1,j >

ρi−1,j+1 and (iv) follows.The inequalities (1) are a consequence of the identity

ρi+1,j+1 − ρi,j+1 = (ρi+1,j+1 − ρi,j ) + (ρi,j − ρi,j+1) ∀i < j

and properties (P0)–(P3).The last statement is a direct consequence of an estimate proved in [24] and (ii). �

Proposition 3 gives us several informations on R. First of all note that, although proper-ties (P1)–(P3) determine the minimum and maximum element of R, for n > 3 they do notallow to totally order the elements. For example, if n = 4 it is not possible to decide a prioriif ρ12 is greater or lower than ρ24. Property (iv) tell us that it is also present a monotonicbehavior along secondaries subdiagonals showing that correlations decay going far from themain diagonal in orthogonal direction. From property (v) we deduce that correlations on(superdiagonal) columns decrease (down to up) faster than correlations on the rows and onthe superdiagonal. The inequalities (1), for all p > 0, obviously generalize to

ρi+p,j+p − ρi,j+p > ρi+p,j+p − ρi,j and ρi+p,j+p − ρi,j+p > ρi,j − ρi,j+p.

Hence, given a fixed maturity, correlations between forward rates decrease faster proceedingbackward in time than proceeding forward; this is in turn an indirect consequence of thenon-invariant behavior of correlations with respect to translation in time.

Finally we observe that (vi) is sharper than the usual estimate of the dominant eigenvalueλ1 of R ∈ �(0) that, thanks to its symmetry, can be expressed only in terms of its row sums:

mini

ri ≤ λ1 ≤ maxi

ri . (3)

The latter, however, for R ∈ �(0,1,2,3) allows to obtain:

1 + (n − 1)ρn,1 ≤ λ1 ≤ 1 + (n − 1)ρn,n−1

which, since often in practice ρn,1 ≥ 0.5, justifies the fact that λ1 explains more than the50% of the total variability tr(R).

2.3 Weak SSC and Total Positivity

As we have outlined in Sect. 2.1, the definition of WSSC gives only a partial representationof what empirically are recognized as shift, slope and curvature. Nevertheless their role as


necessary condition for the existence of SSC, has risen some interest in the literature. Weexpose briefly some results on the existence of WSSC proved in [21, 28] referring to thetheory of total positive matrices. We complete this analysis by showing the connectionsbetween total positivity and SSC.

Observe preliminarily that property (P0) via the Perron-Frobenius Theorem, implies theexistence of a simple dominant positive eigenvalue with associated a positive eigenvector,that is the existence of weak shift eigenvector.

For what concerns the study of weak slope and weak curvature, we start with the follow-ing definition (see [6, 21]):

Definition 4 An n × n matrix A is called: (strictly) totally positive of order k, denotedby TPk (STPk), if all its minors of order ≤ k are nonnegative (positive); (strictly) totallypositive, denoted by TP (STP), if it is TPn (STPn); oscillatory of order k (Ok) if it is TPk ,non singular and for all i = 1, . . . , n we have ai,i+1 > 0 and ai+1,i > 0; oscillatory (O) if itis TP and there exists a q ∈ N\{0} such that Aq is STP.

It can be shown that if A is oscillatory of order k then An−1 is STPk . The main spectralproperties of TP and O matrices useful for our analysis are collected in the following resultof Lord and Pelsser [21] that extends some classical results (see [1, 6, 7]).

Theorem 5 Assume A is an n × n positive definite, symmetric matrix that is STPk . Thenwe have λ1 > λ2 · · · > λk > λk+1 ≥ · · · ≥ λN > 0 i.e. at least the first k eigenvalues aresimple. For s ∈ {1, . . . , k} the s-th eigenvector has exactly s − 1 changes of sign. The sameconclusion is true if A is an n × n positive definite, symmetric matrix having some finitepower that is O3.

The previous theorem allows to deduce that an STP3 correlation matrix has WSSC. Theconverse it is not true, as illustrated by the following

Example 6 The correlation matrix

R =

⎡

⎢

⎢

⎣

1 0.9 0.8 0.70.9 1 0.8 0.60.8 0.8 1 0.90.7 0.6 0.9 1

⎤

⎥

⎥

⎦

has WSSC:

v1 = [0.508 0.493 0.522 0.476 ]T

v2 = [0.384 0.560 −0.286 −0.676 ]T

v3 = [0.703 −0.435 −0.503 0.253 ]T

but is not STP2 (hence neither STP3) since it has at least a negative minor of order 2:R[1,2 | 3,4] = −0.08.

A second question concerns the relations between properties (P0)–(P3) and the total pos-itivity of the correlation matrices.


Salinelli and Sgarra [28] have shown that properties (P0)–(P2) do not imply R is oscilla-tory or STP3. The following example shows that adding property (P3) this conclusion doesnot change.

Example 7 The following correlation matrix in �(0,1,2,3)

R =

⎡

⎢

⎢

⎣

1 0.8 0.75 0.720.8 1 0.82 0.8

0.75 0.82 1 0.850.72 0.8 0.85 1

⎤

⎥

⎥

⎦

is not STP2 (hence it is not STP3), since R[1,2 | 2,3] = −0.094. Nevertheless, R has SSC:

v1 = [0.484 0.508 0.508 0.500 ]T

v2 = [0.762 0.144 −0.357 −0.520 ]T

v3 = [0.428 −0.841 0.134 0.302 ]T .

The relations between total positivity and properties (P0)–(P2) was studied in the abovementioned paper, whose main conclusions are resumed in the following:

Theorem 8 (i) An n × n STP2 correlation matrix R = [ρij ] satisfies properties (P0)–(P2).(ii) An n×n correlation matrix R with n ≥ 3, satisfying properties (P0)–(P2) has positive

all its minors R[i1, i2 | j1, j2] of the following types:

(a) i1 = j1 and i2 = j2;(b) i1 = j1 or j2 = i2;(c) i1 < j1 and i2 > j2, or j1 < i1 and i2 < j2.

(iii) An n × n correlation matrix R with n ≥ 3, satisfying property (P0) and such that fori = 2, . . . , n−1 and s = 1, . . . , i −1 its minors of order two R[s, s +1 | i, i +1] are positive,is STP2.

To complete previous analysis, we present two examples. The first shows that STP3

(hence also STP2) does not implicate the validity of (P3) too.

Example 9 The following correlation matrix

R =⎡

⎣

1 0.85 0.60.85 1 0.80.6 0.8 1

⎤

⎦

is STP3 but not satisfies (P3).

The second example illustrates how a correlation matrix having properties (P0)–(P2) butnot STP3, could not present weak slope or weak curvature.


Example 10 The following correlation matrix in �(0,1,2)

R =

⎡

⎢

⎢

⎣

1 0.9 0.8 0.60.9 1 0.85 0.70.8 0.85 1 0.90.6 0.7 0.9 1

⎤

⎥

⎥

⎦

is not STP2 (since R[1,2 | 2,3] = −0.035), has shift, slope but does not have weak curva-ture:

v1 = [0.489 0.511 0.526 0.472 ]T

v2 = [0.572 0.353 −0.25 −0.696 ]T

v3 = [0.578 −0.783 0.229 −0.005 ]T .

We furthermore observe that Example 7 tells us that STP3 is not necessary for a corre-lation matrix to admit SSC: we conjecture that it is not sufficient either, that is there existcorrelation matrices that are STP3 but having no SSC. We are not able to prove our conjec-ture, but we find that it is well motivated by the results stated in Theorem 11: for suitablevalues of ρ, it is possible to find matrices that are TP3 (and oscillatory) that does not admitSSC.

To conclude, as long as the financial interpretation of the total positivity of the correlationmatrix is concerned, this turns out not to be straightforward. Lord and Pelsser have pointedout that STP2 can be written in the following form:

ρjl − ρjk

ρjl

≥ ρil − ρik

ρil

, i ≤ j, k ≤ l

implying that the relative change from moving from k to l relative to the correlation ρjl

should be larger on the correlation curve denoted by j than on the curve denoted by i.This means that on the r.h.s. of the diagonal the relative change on correlation curves forlarge tenors should be flatter than for shorter tenors, while on the l.h.s. this behavior isreversed. In our opinion a detailed analysis of the financial meaning and an empirical well-established justification of total positivity for interest rates correlation matrices remain stillopen questions.

2.4 Total Positivity and Some Financial Models

Several correlation models of interest rates that capture the financial features recalled inSect. 2.2 are known in the literature. In spite of this, few attempts to systematically studythe presence of SSC (or WSSC) has been made, if we exclude a few examples. The mostrelevant perhaps concerns a classical model (see [27]), that we termed as exponential one,with the following correlation function of the maturities ts > 0:

ρi,j = exp{−β|tj − ti |}, β > 0. (4)

In this way it is assumed the homogeneity with respect to time, that is interest rates withthe same maturity differences exhibit the same correlation. This model although too simpleto be realistic, well captures the main features of interest rates correlation structure and


for this reason it is assumed as the basic starting point for many analyses together with itscontinuous-time counterpart (see [14]).

By setting ρ = e−β , one obtain the corresponding correlation matrix

R =

⎡

⎢

⎢

⎢

⎢

⎢

⎣

1 ρ ρ2 · · · ρn−1

ρ 1 ρ · · · ρn−2

ρ2 ρ 1 · · · ρn−3

......

.... . .

...

ρn−1 ρn−2 ρn−3 · · · 1

⎤

⎥

⎥

⎥

⎥

⎥

⎦

(5)

that satisfies (P0)–(P2) but not (P3), being R a Toeplitz matrix. In Salinelli and Sgarra [28]the oscillatory property of this class of correlation matrices has been proved, providing a suf-ficient condition for the existence of WSSC, while in Salinelli and Sgarra [29] the followingresult on the existence of shift and slope was proved:

Theorem 11 (i) For each ρ ∈ (0,1) and n ≥ 4, the correlation matrix (5) has dominanteigenvector v1 that is shift.

(ii) For each n ≥ 4, there exists a ρ∗ ∈ (0,1) such that the correlation matrix (5) hassecond eigenvector that is slope (in a strict sense) for all ρ ∈ (ρ∗,1).

Remark 12 The result (i) of Theorem 11 can be straightforward extended to the more generalmodel

ρij = exp{−β|i − j |q}, β ∈ R+, q ∈ N\{0}. (6)

The corresponding correlation matrix is TP, hence has WSSC. This last model describes acorrelation structure with very fast decay in maturity differences, although by keeping thefundamental feature of the exponential model, i.e. the time homogeneity.

Note that Theorem 11 gives us an example of an oscillatory (or TP) correlation matrixthat does not admit SSC.

A more general model has been considered in Brigo [3] and Rebonato [27]:

ρi,j = exp{−β|tγj − tγ

i |}, β ∈ R+, γ ∈ (0,1) (7)

which, with the usual identification between indices and maturities, becomes:

ρi,j = exp{−β|jγ − iγ |}, β ∈ R+, γ ∈ (0,1). (8)

In fact, model (7) achieves a relevant improvement with respect to the exponential modelconsidered before: it breaks the time-homogeneous behavior, describing correlations notonly enjoying properties (P0)–(P2), but also (P3).

Oscillatoriness property of correlation matrices of the form (8) can be proved in strictanalogy with [21], Theorem 4 and Corollary 4, p. 123, where oscillatoriness of theSchoenmacher-Coffey [30, 31] correlation matrices has been shown to follow by the ob-servation that these matrices belong to the class of Green matrices. We can conclude thatfor each choice of the parameters the correlation matrix with elements given by (8) exhibitsWSSC.

As far as the ordering of the factor loadings is concerned in the framework providedby model 8, proving the usual order of the eigenvector components turns out to be much


more difficult than for the exponential model, even for the first eigenvector. In the case ofthe latter model the high symmetry of the correlation matrix plays a fundamental role inproving that the “central” loading of the first eigenvector is the one with largest value, andthis was an important step to prove that the components are increasing towards the center,and then decreasing. For different classes of models it is hard to prove the same behavior,in particular for some of the models considered in the present section this could not evenhold true. The greatest component of the first eigenvector could be not the central one, butanother one (often between the central and the last one), although the canonical behavior(first increasing than decreasing) is respected; in order to prove the ordering property thetechnique providing the proof for the exponential model cannot be used any more.

A further refinement of the previous model, still exhibiting temporal nonhomogeneity,and including other desirable features like a proper asymptotic behavior for ρ1n, n → ∞, isconsidered by Brigo [3] and Rebonato [27]:

ρi,j = exp{[−β + α max(ti , tj )]|tγj − tγ

i |}.

By making the usual identification between indices and maturities, setting: ρ = e−β andξ = eα , the general element of this correlation matrix can be written in the following form:

ρi,j = ρ |i−j |ξ |i−j |max(i,j).

Brigo [3] pointed out that the matrix just introduced is positive definite (so it should be inorder to be a correlation matrix) only for some particular values assumed by the parametersinvolved. It is immediate to verify that this matrix has some 2-minors which are negative, sothe existence of WSSC is not guaranteed. We do not investigate further the spectral proper-ties of this class of correlations matrices.

3 Beyond Total Positivity

After presenting the results relating the existence of WSSC with the Total Positivity, weshall now make the attempt to look for tools of different kind with the aim of connectingexistence of SSC with other significant properties of the correlation structure of interestrates. We face these questions clarifying the relations between properties (P0)–(P3) and thespectral properties we are analyzing. To this purpose, the following four examples shed lighton this problem.

We start by observing that (see Example 6) properties (P0)–(P3) are not necessary forWSSC: we show now that they are not sufficient either (see also the matrix in (2.9) in Lordand Pelsser [21]).

Example 13 The correlation matrix in �(0,1,2,3)

R =

⎡

⎢

⎢

⎣

1 0.8 0.65 0.450.8 1 0.81 0.77

0.65 0.81 1 0.90.45 0.77 0.9 1

⎤

⎥

⎥

⎦

has not weak curvature:

v3 = [0.304 −0.74 0.589 −0.113 ]T .


Anyway there are examples (see (8) before) of correlation matrices that are in�(0,1,2,3) having WSSC.

Since SSC imply WSSC, previous results tell us that properties (P0)–(P3) are not suffi-cient for the existence of SSC. They are not necessary, too.


R =

⎡

⎢

⎢

⎢

⎢

⎣

1 0.9 0.7 0.71 0.60.9 1 0.89 0.8 0.720.7 0.89 1 0.94 0.790.71 0.8 0.94 1 0.780.6 0.72 0.79 0.78 1

⎤

⎥

⎥

⎥

⎥

⎦

without enjoying property (P0)–(P3) has SSC:

v1 = [0.422 0.466 0.468 0.458 0.419 ]Tv2 = [0.688 0.346 −0.224 −0.268 −0.534 ]Tv3 = [0.245 0.00831 −0.433 −0.462 0.734 ]T .

Two more remarks are in order. First there exist correlation matrices in �(0,1,2,3) withSSC.


R =

⎡

⎢

⎢

⎣

1 0.86 0.85 0.820.86 1 0.89 0.870.85 0.89 1 0.90.82 0.87 0.9 1

⎤

⎥

⎥

⎦

is in �(0,1,2,3) and has SSC:

v1 = [0.49 0.504 0.506 0.499 ]Tv2 = [0.795 0.0343 −0.269 −0.542 ]Tv3 = [0.348 −0.837 0.0896 0.412 ]T .

Second, a correlation matrix in �(0,1,2,3) having WSSC can not present SSC.

Example 16 The following correlation matrices

R(1) =

⎡

⎢

⎢

⎢

⎢

⎣

1 0.8 0.67 0.65 0.450.8 1 0.83 0.79 0.770.67 0.83 1 0.85 0.80.65 0.79 0.85 1 0.90.45 0.77 0.8 0.9 1

⎤

⎥

⎥

⎥

⎥

⎦

; R(2) =

⎡

⎢

⎢

⎣

1 0.6 0.59 0.580.6 1 0.92 0.805

0.59 0.92 1 0.930.58 0.805 0.93 1

⎤

⎥

⎥

⎦

are in �(0,1,2,3), have WSSC but have not shift and slope respectively:

v(1)

1 = [0.393 0.466 0.464 0.468 0.44 ]Tv(2)

2 = [0.908 −0.189 −0.278 −0.25 ]T .


Summing up, our analysis makes it evident how properties (P0)–(P3) and even total pos-itivity are not enough to justify the presence, empirically observed, of shift, slope and curva-ture for correlation matrices of forward rates. The existence of SSC must then be consideredan extra requirement until some more hidden properties of correlation structures implyingthem can be identified.

To complete our analysis some words have to be spent on the case of dimension n = 3.This is obviously the minimum required in order to have all the three eigenvectors on whichwe are concentrating our considerations: we now show that in this case properties (P0)–(P2)are sufficient for the existence of SSC.

Proposition 17 A 3-dimensional positive definite correlation matrix R ∈ �(0,1,2) hasthree simple eigenvalues, with corresponding eigenvectors that are SSC.

Furthermore if R ∈ �(0) has first eigenvector shift, then R ∈ �(0,1,2).

Proof Consider the correlation matrix

R =⎡

⎣

1 ρ12 ρ13

ρ12 1 ρ23

ρ13 ρ23 1

⎤

⎦ (9)

satisfying properties (P0)–(P2). Thus 0 < ρ13 < min{ρ12, ρ23} < 1 and 3−‖ρ‖2 > 0, havingset ρ = [ρ12 ρ13 ρ23]T . We prove the simplicity of the three positive eigenvalues of R. Thefirst (λ1) is simple by the Perron-Frobenius theorem. We show that λ2 �= λ3. To this purpose,it is sufficient to observe that, denoting by P(λ) the characteristic polynomial of R

P(λ) = λ3 − 3λ2 + (3 − ‖ρ‖2)λ − det R

the minimum zero of P ′(λ) is 1 −√

33 ‖ρ‖ and

P(

1 −√

3

3‖ρ‖

)

=√

3

9‖ρ‖3 − ρ12ρ13ρ23

is positive, as straightforwardly follows by properties (P0)–(P2) and the arithmetic-geometric inequality.

The first eigenvector v1 = [v11 v21 v31]T of R is weak shift (i.e. positive); subtracting thefirst row to the second one in the identity Rv1 = λ1v1, we obtain

v31(ρ23 − ρ13) = (λ1 + ρ12 − 1)(v21 − v11). (10)

By property (P2) the l.h.s. is positive, then v21 > v11 follows from λ1 > 1 and ρ12 > 0.Subtracting the third row to the second, by (P1) it is possible to prove that v21 > v31, hencev1 is shift.

Operating on the third and first rows as before, we conclude ρ12 � ρ23 if and only ifV11 � V31. We assume ρ12 < ρ23, the proof still holding with suitable modifications in thecase ρ12 ≥ ρ23.

The second eigenvector v2 = [v12 v22 v32]T having to be orthogonal to v1 must have atleast a negative element and can not have two null elements. We prove that v2 has no zeroelements. Suppose on the contrary that, for example, v32 = 0. By the first two relations inRv2 = λ2v2 it is easy to deduce v22 = −v12, that is inconsistent with the third and ρ13 < ρ23.The same reasoning applies to the other components of v2.


We set now

ρ = ρ13; δ = ρ23 − ρ13 > 0; ε = ρ12 − ρ13 > 0

observing that δ > ε, as follows by statement (v) in Proposition 3. Thus, we can rewrite Ras

R = A + B =⎡

⎣

1 ρ ρ

ρ 1 ρ

ρ ρ 1

⎤

⎦ +⎡

⎣

0 ε 0ε 0 δ

0 δ 0

⎤

⎦

where (see Example 29)

λ1(A) = 1 + 2ρ; λ2(A) = λ3(A) = 1 − ρ

and

λ1(B) =√

δ2 + ε2; λ2(B) = 0; λ3(B) = −√

δ2 + ε2.

Since⎡

⎣

v12 + ρv32 + v22(ε + ρ)

v22 + v12(ε + ρ) + v32(δ + ρ)

v32 + ρv12 + v22(δ + ρ)

⎤

⎦ ≡ λ2

⎡

⎣

v12

v22

v32

⎤

⎦

subtracting the second relation to the first one we obtain

(λ2 − 1 + ρ + ε)(v22 − v12) = δv32.

By a monotonicity property of eigenvalues (see e.g. [26], p. 322) it holds

λ2(R) ≥ λ3(A) + λ2(B) = 1 − ρ

hence λ2 − 1 + ρ + ε > 0. Thus

v32 > 0 ⇔ v22 > v12. (11)

To complete our analysis we subtract the second relation to the third, obtaining

(λ2 − 1 + ρ + δ)(v32 − v22) = −εv12

and

v12 > 0 ⇔ v32 < v22. (12)

Furthermore, subtracting the first relation to the third, we have

(λ2 − 1 + ρ)(v32 − v12) = (δ − ε)v22

and since δ > ε it follows

v22 > 0 ⇔ v32 > v12. (13)

Assume now that v12 > 0. By (12) it follows v32 < v22: if v22 < 0 then v32 < 0 and v2 isslope; if v22 > 0 then by (13) v32 > v12 > 0 contrary to the presence of at least a negativeelement. Operating in the same way, it is possible to verify that if v12 < 0 the only admissible


case is for v22 > 0 and v32 > 0, that is v2 is slope. Since it is not restrictive, in the sequel wewill assume v12 < 0 < v22 < v32.

The third eigenvector v3 = [v13 v23 v33]T , by the orthogonality of v1 and v2 respectively,has at least a sign change and its elements can not have the sign configurations +, −, − and−, +, +. To prove that v3 is curvature we have to show that configurations +, +, − and −,−, + are not possible too. To this aim, observe that operating as for v2 we are able to obtainonly

v23 > 0 ⇔ v33 < v13. (14)

Assume that elements of v3 have signs +, +, −. By (14), the inequality 0 < v22 < v32 and theorthogonality between v2 and v3, it descends |v23| > |v33| but this implies |v23v21| > |v33v31|contrary to the orthogonality between v1 and v3.

To prove the last statement, it is enough to revert the argument used to prove that v1 isshift. �

By the previous proof we can conclude that for n = 3 the existence of shift is a directconsequence of properties (P0)–(P2), whereas the presence of slope and curvature is a con-sequence of the ordering property of the first eigenvector and the orthogonality among them.Thanks to Theorem 8(i), we can also say that a 3-dimensional STP3 correlation matrix hasalways SSC. Finally, note that property (P3) does not play any role, giving only informationson the shape of the shift and on the number of positive elements of the slope.

For n = 4, operating as in the first part of the previous proof, it is simple to prove thatif R ∈ �(0,1,2) then λ1 is simple, v11 < v21 and v41 < v31, that is v1 is shift. Neverthe-less, properties (P0)–(P3) alone do not allow to conclude if v21 � v31, as illustrated by thefollowing example:

Example 18 The correlation matrices

R(1) =

⎡

⎢

⎢

⎣

1 0.7 0.65 0.60.7 1 0.8 0.75

0.65 0.8 1 0.90.6 0.75 0.9 1

⎤

⎥

⎥

⎦

; R(2) =

⎡

⎢

⎢

⎣

1 0.72 0.65 0.60.72 1 0.8 0.780.65 0.8 1 0.820.6 0.78 0.82 1

⎤

⎥

⎥

⎦

are in �(0,1,2,3) and have as shift the vectors

V(1)

1 = [0.454 0.508 0.525 0.51 ]T

V(2)

1 = [0.461 0.519 0.515 0.504 ]T .

No more general results are true: Examples 14 and 16 show as, for n = 4, slope andcurvature are not deducible by properties (P0)–(P3), and, for n = 5, the same conclusionholds for the shift.

Figure 1 resumes all the results presented here and in the previous section on correlationmatrices with dimension n ≥ 4.

4 Further Results on Shift

The empirical financial literature has given the name of shift to the first eigenvector of in-terest rates correlation matrix since it represents the main movement of the yield curve as a


Fig. 1 Relations among different properties of yields correlation matrices with dimension n ≥ 4. Strict totalpositivity of order 3 (STP3) implies (Definition 4) strict total positivity of order 2 (STP2) and this last implies(Theorem 8) properties (P0)–(P2) but not (P0)–(P3) (Example 9); these last properties do not imply STP3or STP2 (Example 7); STP3 implies (Theorem 5) the existence of WSSC whereas this last does not imply(P0)–(P3) (Example 6); for what concerns SSC no inclusion relations (apart from the trivial one with WSSC)with the other sets are true (Examples 14, 15 and 16): the question mark indicates the lack of an example ofan STP3 matrix having no SSC

rigid translation. This idea, in our framework, has been formalized by the concept of pureshift that indicates that all the factor loadings give the same contribution to the first PC. Onlyin the very special case of all forward rates perfectly correlated, i.e. ρ = 1, this rigid trans-lation is the only movement of the yield curve compatible with the model. This situationhappens for all one-factor models, like the well known Vasicek, CIR, Ho-Lee ones (see e.g.[9, 11, 32]).

The presence of a pure shift is strictly connected with the row sums behavior. Moreprecisely, a (positive) correlation matrix has first eigenvector v1 pure shift if and only if allits row sums ri are mutually equal: ri = rj ∀i �= j . In this case λ1 = ri for all i.

Since by (P1) and (P2) it follows r1 < r2 and rn < rn−1, we deduce that the first eigen-vector of a correlation matrix R ∈ �(0,1,2) can not be pure shift, that is v1 �= c1, c ∈ R

+.A more precise result is the following:

Proposition 19 If v1 = [v11, v12, . . . , v1n]T is the dominant eigenvector of R ∈ �(0,1,2),with n ≥ 3, then:

v11 < v12 and v1,n−1 > v1n.

Proof Subtracting the second row to the first one in the identity RV1 = λ1V1, with simplecomputations one obtain

(λ1 − 1 + ρ12)(v12 − v11) =n

∑

j=3

v1j (ρ2j − ρ1j ).

Since λ1 > 1 and ρ12 > 0 we have λ1 − 1 +ρ12 > 0; furthermore, since v1j > 0 for all j andproperty (P2) guarantees that ρ2j > ρ1j for all j ≥ 3, so the inequality v11 < v12 is proved.The same proof works (with respect to property (P1)) for the second inequality. �


The result of Proposition 19 can be obtained also studying the row sums of the power ofR applied to the pure shift vector 1. Indeed, since 1 is not an eigenvector of R ∈ �(0,1,2),using the power method (see [12]) we conclude that ‖Rp1‖Rp1 −→ v1 as p → +∞. De-noting with r

(p)

i the i-th row sum of ‖Rp1‖Rp1, it holds

Proposition 20 Given a correlation matrix R ∈ �(0,1,2), for any positive integer p thefollowing conclusions hold:

(i) r(p)n < r

(p)

n−1,

(ii) r(p)

1 < r(p)

2 .

Proof For p = 1 the conclusion is obvious. Assuming that r(p)n < r

(p)

n−1 for a given p, bysetting γ (p+1) = ‖Rp1‖−1‖Rp+11‖, we have:

γ (p+1)(r(p+1)n − r

(p+1)

n−1 ) = (1 − ρn−1,n)(r(p)n − r

(p)

n−1) +n

∑

j=1

r(p)

j (ρn,j − ρn−1,j )

and the conclusion follows by the induction assumption and property (P1). The proof of (ii)is similar. �

Remark 21 If also property (P3) holds the minimum row sum is r1. Indeed, if R ∈�(0,1,2,3), by the symmetry of R, we have

i∑

j=1

(ρi,j − ρ1,j ) =i

∑

j=1

(ρi−j+1,i − ρ1,j ) =i−1∑

j=2

(ρi−j+1,i − ρ1,j )

and then, for i > 1,

ri − r1 =i−1∑

j=2

(ρi−j+1,i − ρ1,j ) +n

∑

j=i+1

(ρi,j − ρ1,j ) .

The positivity of the two sums follows by properties (P3) (note that i − (i − j + 1) = j − 1)and (P2) respectively.

For what concerns max ri , apart from the trivial case n = 3 (for which, by Proposition 20,max ri = r2) for n > 3, if R ∈ �(0,1,2,3), it is not possible to forecast the maximum rowsum of R, how showed in Example 18 where for R(1) we have maxi ri = r3 = 3.35, whereasfor R(2) it holds maxi ri = r2 = 3.3.

The approach followed in the proof of Proposition 20 combined with the power method,turned out to be useful in studying the existence of SSC for the exponential model (5): therelated results are exposed in Theorem 11. This approach however, does not seem applicablein more general situations, and in any case it should be applied with some care, as illustratedby the following example.



R =

⎡

⎢

⎢

⎢

⎢

⎣

1 0.8 0.7 0.65 0.60.8 1 0.83 0.79 0.770.7 0.83 1 0.85 0.80.65 0.79 0.85 1 0.90.6 0.77 0.8 0.9 1

⎤

⎥

⎥

⎥

⎥

⎦

is in �(0,1,2,3) and its row sums are not increasing and then decreasing:

r1 = 3.75; r2 = 4.19; r3 = 4.18; r4 = 4.19; r5 = 4.07,

but has shift dominant eigenvector:

v1 = [0.407 0.459 0.459 0.461 0.447 ]T .

This last ordering anyway is exhibited by the elements of the vector R21.

As we have pointed out, the first eigenvector of an interest rates correlation matrix usuallyexhibits some kind of nonuniformity in their own components: the “canonical” behavior, thatwe have termed as shift, is usually characterized by the central components with a slightlybigger size and a slightly smaller size on the extreme with a humped shape. The financialexplanation of this property is that interest rates with intermediate maturities play a slightlymore relevant role with respect to those with short or long maturities in determining themain movement of the yield curve. To obtain a quantitative indication about how much the“humped” shape is pronounced, is equivalent to measure how much this eigenvector differsfrom a pure shift. A useful quantity to express this distance is given by the following indexγ originally introduced in [17]:

γ = maxi,j

v1,i

v1,j

.

Note that γ ≥ 1 and γ = 1 if and only if v1 is proportional to 1.With respect to the following notation:

M1 = maxi

aii M2 = maxi �=j

aij Mr = maxi

ri

m1 = mini

aii m2 = mini �=j

aij mr = mini

ri

for a positive matrix A = [aij ], some different upper estimations of γ based only on theelements of A are collected in Table 1 (see e.g. [2, 23–25]).

Depending on the particular matrix considered, each estimate in Table 1 can be thesharpest. For a correlation matrix R ∈ �(0,1,2,3), we are able to order some of theseestimates. In fact, by ρij ∈ (0,1), for n ≥ 3, it follows

γ3 = ρn,n−1

ρn,1<

1

ρn,1= γ4. (15)

Furthermore, since Mr > mr = r1 (see Remark 21), we obtain, for n ≥ 3,

γ2 = Mr − r1

2ρn,1+

√

(

Mr − r1

2ρn,1

)2

+ 1 <Mr − r1

2ρn,1+

√

(

Mr − r1

2ρn,1+ 1

)2

= γ1. (16)


Table 1 Estimations of γ

Ostrowski 1952 [24] γ ≤ Mr − mr + m2

m2= γ1

Brauer 1957 [2] γ ≤Mr − mr +

√

(Mr − mr)2 + 4m22

2m2= γ2

Ostrowski 1960/1961 [25] γ ≤ max(M1 + m2 − m1,M2)

m2= γ3

Minc 1970 [23] γ ≤ maxj,s,t

asj

atj= γ4

Finally, for n = 3 the equality γ1 = γ3 = ρ23/ρ13, holds, whereas, for n ≥ 4, any orderingbetween γ2 and γ3 (and also between γ1 and γ4) is possible, as showed by the following twoexamples:

R(1) =

⎡

⎢

⎢

⎣

1 0.8 0.7 0.650.8 1 0.83 0.790.7 0.83 1 0.840.65 0.79 0.84 1

⎤

⎥

⎥

⎦

; γ2 = 1.229; γ3 = 1.292

R(2) =

⎡

⎢

⎢

⎣

1 0.8 0.64 0.630.8 1 0.83 0.810.64 0.83 1 0.840.63 0.81 0.84 1

⎤

⎥

⎥

⎦

; γ2 = 1.336; γ3 = 1.333.

In the following proposition we obtain a complete information about the sign of the differ-ence γ2 − γ3 for the exponential model.

Proposition 23 For the correlation matrix (5) with ρ ∈ (0,1), the following inequalityholds: if n = 3,4 then γ2 < γ3 for all ρ ∈ (0,1); for n = 5,6,7 the sign of γ2 − γ3 is notconstant; if n ≥ 8 then γ2 > γ3, for all ρ ∈ (0,1).

Proof Observe preliminary that:

γ3 = ρ

ρn−1= 1

ρn−2.

We proceed by distinguishing the odd case from the even one.In the odd case, since Mr = r(n+1)/2 and mr = r1, we find

Mr − r1 = 1 + 2(n−1)/2∑

s=1

ρs −n−1∑

s=0

ρs =(n−1)/2∑

s=1

ρs −n−1∑

s=(n−1)/2+1

ρs

= (

1 − ρ(n−1)/2)

(n−1)/2∑

s=1

ρs = (

1 − ρ(n−1)/2)2 ρ

1 − ρ


by (16) we obtain

γ2 = 1

ρn−2

(

(1 − ρ(n−1)/2)2

2(1 − ρ)+

√

(1 − ρ(n−1)/2)4

4(1 − ρ)2+ ρ2(n−2)

)

.

Hence:

γ2 − γ3 = (1 − ρ(n−1)/2)2 − 2(1 − ρ) + √

(1 − ρ(n−1)/2)4 + 4(1 − ρ)2ρ2(n−2)

2(1 − ρ)ρn−2.

If n = 3 the conclusion follows immediately, by observing that for all ρ ∈ (0,1)

γ2 − γ3 = 1

2ρ

(−(1 + ρ) +√

(1 − ρ)2 + 4ρ2)

= 1

2ρ

(−(1 + ρ) +√

(1 + ρ)2 − 4ρ(1 − ρ))

< 0.

For n = 5, if ρ = 0.5 we have γ2 − γ3 = 1.109772 whereas for ρ = 0.8 we find γ2 − γ3 =−0.1369057; for n = 7, if ρ = 0.5 we have γ2 −γ3 = 17.02040 whereas if ρ = 0.99 we findγ2 − γ3 = −4.080132 × 10−3.

If n = 9, we have:

γ2 − γ3 = (1 − ρ4)2 − 2(1 − ρ) + √

(1 − ρ4)4 + 4ρ14(1 − ρ)2

2(1 − ρ)ρ7.

If (1 − ρ4)2 − 2(1 − ρ) ≥ 0 for ρ ∈ (0,1) the thesis follows. If (1 − ρ4)2 − 2(1 − ρ) < 0 (asnumerical simulations show is the case) then

γ2 − γ3 = (1 − ρ4)4 + 4ρ14(1 − ρ)2 − (2(1 − ρ) − (1 − ρ4)2)2

2(1 − ρ)ρ7(√

(1 − ρ4)4 + 4ρ14(1 − ρ)2 + 2(1 − ρ) − (1 − ρ4)2).

The numerator of the fraction with some computation can be rewritten as

4(1 − ρ)3(−ρ13 − ρ12 − ρ11 − ρ10 − ρ9 − ρ8 − ρ7 + ρ5 + 2ρ4 + 3ρ3 + 2ρ2 + ρ) > 0

and the positivity of the difference γ2 − γ3 follows.The positivity of γ2 − γ3 for any (odd) n > 9 follows by observing that for each fixed

ρ ∈ (0,1) the quantity 2(1 − ρ)ρn−2(γ2 − γ3) is increasing with respect to n.If n is even, then for ρ ∈ (0,1):

Mr − r1 =n/2∑

s=0

ρs +n/2−1∑

s=1

ρs −n−1∑

s=0

ρs = (1 − ρn/2)

n/2−1∑

s=1

ρs

= ρ(1 − ρn/2)(1 − ρn/2−1)

1 − ρ

and:

γ2 = 1

ρn−2

(

(1 − ρn/2)(1 − ρn/2−1)

2(1 − ρ)+

√

(

(1 − ρn/2)(1 − ρn/2−1)

2(1 − ρ)

)2

+ ρ2(n−2)

)

.


Thus, we have:

γ2 −γ3 = 1

2ρn−2

(

(1 − ρn/2)(1 − ρn/2−1)

1 − ρ−2+

√

(

(1 − ρn/2)(1 − ρn/2−1)

(1 − ρ)

)2

+ 4ρ2(n−2)

)

.

For n = 4, we obtain for all ρ ∈ (0,1):

γ2 − γ3 = 1

2ρ2

(−(1 + ρ2) +√

(1 − ρ2)2 + 4ρ4)

= 1

2ρ2

(−(1 + ρ2) +√

(1 + ρ2)2 − 4ρ2(1 − ρ2))

< 0.

For n = 6 if ρ = 0.5 we have γ2 − γ3 = 5.048 whereas for ρ = 0.9 we find γ2 − γ3 =−0.05753.

For n = 8 we have:

γ2 − γ3 = 1

2ρ6

(−2 + (1 + ρ + ρ2)(1 − ρ4) +√

(1 + ρ + ρ2)2(1 − ρ4)2 + 4ρ12)

.

If −2 + (1 + ρ + ρ2)(1 − ρ4) ≥ 0 the thesis follows; if −2 + (1 + ρ + ρ2)(1 − ρ4) < 0, weobtain with some computation (similar to the ones of case n = 9):

γ2 − γ3 = 4ρ(1 + ρ − ρ3 − ρ4 − ρ5 + ρ11)

2ρ6(√

(1 + ρ + ρ2)2(1 − ρ4)2 + 4ρ12 + 2 − (1 + ρ + ρ2)(1 − ρ4)).

Since for all ρ ∈ (0,1)

1 + ρ − ρ3 − ρ4 − ρ5 + ρ11 = (1 − ρ3) + ρ(1 − ρ3) + ρ8 − ρ5 − ρ8 + ρ11

= (1 − ρ3)(1 + ρ − ρ5 − ρ8) > 0

the thesis follows. The positivity of γ2 − γ3 for any (even) n > 8 follows by observingthat for each fixed ρ ∈ (0,1) the quantity 2(1 − ρ)ρn−2(γ2 − γ3) is increasing with respectto n. �

We have just observed that the best estimate of the shift “hump” is provided by γ3 = ρ2−n;although this is just an estimate from above, it is quite natural to remark that, once n > 2 hasbeen fixed, the bound for γ3 is monotonically decreasing in ρ. This suggests that for ρ → 1the hump will disappear (γ3 → 1) and the eigenvector will becomes a pure shift, turning thecorrelation model into a one factor model.

5 Two Factor Models

In this section we want to deal with the problem which is in someway the inverse of thatinvestigated in sections before: given a correlation matrix with rank k ≤ 3, assuming thatgenerating vectors are shift, slope and curvature, are the usual properties of positivity andmonotonicity of the matrix elements granted? This question arises in a natural way sincein the correlations models under consideration the first few factors explain almost all the


variability, then a reduced rank matrix seems to be a good approximation of the real correla-tion matrices. We do not devote any more attention to one factor models on which we havefocused the previous section.

In principle our investigation should involve three factors, but, as we shall see below,two factors are sufficient to justify all the relevant properties characterizing the correlationmatrices of interest rates.

We are now going to examine in full detail the correlation matrices of rank two gen-erated by two eigenvectors of shift and slope kind, verifying under which conditions theeigenvectors properties assure the positivity and monotonicity, i.e. properties (P0), (P2) and(P3) of Sect. 2.2, of the matrix elements. This problem arises not only as a natural exten-sion of the previous results, but also as a relevant issue related to financial applications:two-factor models received some attention in the literature on interest rate models and in-terest rate derivatives valuation, due to their better performances in modelling with respectto one-factor models and still keeping a good level of analytical tractability with respect tothree-factor models. Among the most popular two-factor models we just mention the Hulland White [10] and the Extended Longstaff and Schwarz models [19]. An extensive treat-ment of two-factor models is presented in Brigo and Mercurio [4], where a detailed study ofthe two-additive-factor Gaussian model G2++ is provided. Moreover it is shown there howtwo-factor models like G2++ or CIR++, with a suitable choice of the parameters, can giverise to a humped volatility curve of instantaneous forward rates, which cannot be obtained inone-factor models setting. The authors justify their choice to “. . . focus on two-factor modelsfor their better tractability and implementability, especially as far as recombining lattices areconcerned” ([4], p. 140).

Consider the n × n correlation matrices of the form

R = λ1V V T + λ2UUT

where V and U are shift and slope, respectively, in a strict sense, that is

0 < V1 < V2 < · · · < Vs

0 < Vn < Vn−1 < · · · < Vs

with s ∈ {1, . . . , n}, and

U1 < U2 < · · · < Uk < 0 < Uk+1 < Uk+2 < · · · < Un.

Thus the elements of R are

ρij = λ1ViVj + λ2UiUj , i, j = 1, . . . , n

with:

(1) λ1V2i + λ2U

2i = 1, i = 1, . . . , n

(2) λ1 + λ2 = n

(3) λ1 > λ2 > 0

(4) 〈U,V 〉 = 0

(5) ‖U‖2 = 1.

(17)

Note that obviously by (1), (2) and (5) it follows ‖V ‖2 = 1, and (1), (2) and (3) imply thatR is a correlation matrix.


Since Vi > 0 for all i, from (1) we obtain

Vi =√

1 − λ2U2i

λ1, i = 1, . . . , n (18)

and

ρij =√

1 − λ2U2i

√

1 − λ2U2j + λ2UiUj . (19)

Remark 24 Observe that by (18) we can conclude that, assuming j > i,

Vi < Vj ⇔√

1 − λ2U2i <

√

1 − λ2U2j ⇔ U 2

i > U 2j ⇔ |Ui | > |Uj |.

We prove that, under our assumptions, properties (P1) and (P2) are assured in a two factormodel.

Proposition 25 If V and U are shift and slope, respectively, in a strict sense and properties(17) hold, then the elements of R = λ1V V T + λ2UUT satisfy the monotonicity by rows andcolumns.

Proof Consider the monotonicity by rows. By (19) ρij , fixed i with i < j , and the assump-tion on U is sufficient to evaluate the sign of

∂ρij

∂Uj

= λ2√

1 − λ2U2j

(

Ui

√

1 − λ2U2j − Uj

√

1 − λ2Ui

)

.

If Ui < 0 and Uj > 0 we obtain∂ρij

∂Uj< 0. The same conclusion is true if Ui < Uj < 0 since

∂ρij

∂Uj

< 0 ⇔ −Uj

√

1 − λ2Ui < −Ui

√

1 − λ2U2j ⇔ U 2

j < U 2i

and for 0 < Ui < Uj since

∂ρij

∂Uj

< 0 ⇔ Ui

√

1 − λ2U2j < Uj

√

1 − λ2Ui ⇔ U 2i < U 2

j .

With a similar reasoning referred to ∂ρij /∂Ui , the monotonicity by columns follows. �

If the monotonicity is assured, this is not the case for the positivity, as illustrated by thefollowing

Example 26 If λ1 = 2.5, λ2 = 1.5 and

⎛

⎜

⎜

⎝

U1

U2

U3

U4

⎞

⎟

⎟

⎠

=

⎛

⎜

⎜

⎝

−0.7329−0.30.2

0.577

⎞

⎟

⎟

⎠

;

⎛

⎜

⎜

⎝

V1

V2

V3

V4

⎞

⎟

⎟

⎠

=

⎛

⎜

⎜

⎝

0.27880.58820.61320.4475

⎞

⎟

⎟

⎠


we obtain:

R =

⎛

⎜

⎜

⎝

1.0 0.7398 0.2075 −0.32250.7398 1.0 0.8117 0.39840.2075 0.8117 1.0 0.8591

−0.3225 0.3984 0.8591 1.0

⎞

⎟

⎟

⎠

.

However, since the monotonicity by rows and columns imply minij ρij = ρ1n a necessaryand sufficient condition for the strict positivity of R is

1 − λ2(U21 + U 2

n ) > 0. (20)

Indeed ρ1n > 0 if and only if

√

1 − λ2U21

√

1 − λ2U 2n > −λ2U1Un

equivalent to

(1 − λ2U21 )(1 − λ2U

2n ) > λ2

2U21 U 2

n

that is (20).

Remark 27 Since U is a normalized eigenvector, from (20) λ2 ≤ 1 is a sufficient conditionto obtain the strict positivity of R. Note furthermore that in practice interest rates are highlycorrelated, i.e. ρij > α, where usually α � 0.5. This happens if and only if:

1 − λ2U21 − λ2U

2n > α(α − 2λ2U1Un).

This last relation shows that, fixed n, high correlations are possible if λ2 is “small”.

A few words have to spent on property P3): it is not assured by the assumptions consid-ered, as the following example shows.

Example 28 If λ1 = 3.5 and

⎛

⎜

⎜

⎜

⎜

⎝

V1

V2

V3

V4

V5

⎞

⎟

⎟

⎟

⎟

⎠

=

⎛

⎜

⎜

⎜

⎜

⎝

0.27520.40660.51290.51020.4854

⎞

⎟

⎟

⎟

⎟

⎠

;

⎛

⎜

⎜

⎜

⎜

⎝

U1

U2

U3

U4

U5

⎞

⎟

⎟

⎟

⎟

⎠

=

⎛

⎜

⎜

⎜

⎜

⎝

−0.7−0.530.23

0.24330.3421

⎞

⎟

⎟

⎟

⎟

⎠

we have:

R =

⎛

⎜

⎜

⎜

⎜

⎝

1 0.9481 0.2524 0.2359 0.10830.9481 1 0.547 0.5327 0.41880.2524 0.547 1 0.9999 0.98930.2359 0.5327 0.9999 1 0.99160.1083 0.4188 0.9893 0.9916 1

⎞

⎟

⎟

⎟

⎟

⎠

.


6 Concluding Remarks

In this paper we have provided a systematic investigation on the relationships between theso-called shift, slope and curvature and some properties of the correlation matrices for in-terest rates. After recalling the formal definition of SSC that distinguish between the signchange and the ordering properties of the factor loadings, we have presented some knownresults of the literature giving some new insight on the properties of these correlation matri-ces. Next, we have analyzed the role of the time homogeneity assumption of the correlations,showing that the removal of this hypothesis does not add any information about the existenceof SSC. In doing this we also reported on some popular correlation models for interest rates.We have also completely clarified the role played by the matrices dimension together tosome standard financial properties assumed in ensuring the existence of SSC. Furthermore,in a reverse perspective, we have shown that two vectors of shift and slope type, under asupplementary mild condition, are sufficient to generate correlation matrices enjoying theusual properties of positivity and monotonicity.

Our work makes evident how many efforts should still be made before a complete charac-terization of Shift, Slope and Curvature based on the structural properties of the correlationmatrices could be achieved. Only sufficient or necessary conditions are available, never both(at least for significant dimension, n > 3), even for the weak case. Total positivity is a veryimportant property allowing to explain the sign changes in the eigenvector’s components,but it is not a necessary condition, moreover it cannot explain the ordering properties; it isthen only a little part of the full story. The ordering properties of the “factor loadings” canbe justified on the basis of the algebraic properties of the correlation matrices only in veryfew cases: the shift, and the slope (but only for a range of values for the correlation param-eter) in the exponential model. We could finally point out that the conjecture formulated byLord and Pelsser [21] is still open as far as the second eigenvector is concerned: while ouranalysis has proved a definite result on the relationship between properties (P0)–(P3) andthe existence of SSC, the sufficiency of the same properties in order to guarantee existenceof weak slope has not yet been proved. The aim of our future work will be that of look-ing for more general conditions granting the existence of shift, slope and curvature both inthe weak and in the strict sense in order to provide a complete and rigorous picture of thisdeep interconnection between empirical and spectral properties of such correlation matricesappearing in interest rate modelling.

Acknowledgements We thanks two anonymous referees for their useful suggestions, in particular one ofthem for a suggestion that stimulated the establishment and development of Sect. 5. Usual disclaimers apply.

Appendix

We now turn our attention to an interesting correlation model which exhibits a pure shift asfirst eigenvector (see e.g. [13, 15]).

Example 29 (Equicorrelation) If n ≥ 2 and ρij = ρ ∈ (0,1] for i �= j , then R admits thesimple dominant eigenvalue λ1 = 1 + (n − 1)ρ with associated eigenvector which is pureshift. The second eigenvalue λ2 = 1 − ρ has algebraic and geometric multiplicity n − 1,hence the slope and curvature definitions make no sense for n ≥ 3.

If n = 3, the equicorrelation is also a necessary condition for R having pure shift domi-nant eigenvector. In this case the second and third eigenvectors are:

v2 = [−1 0 1 ]T ; v3 = [−1 1 0 ]T .


Vector v2 is slope, while v3 is not of proper curvature kind since S+v = 2 but S−

v = 1. Henceit does not exist a three-dimensional positive correlation matrix with eigenvectors that arepure shift, slope and curvature.

Since in this model interest rates with every maturity exhibit the same correlation with allthe others, the only meaningful principal component turns out to be the first one, describinga simple rigid movement of the yield curve in a common direction.

The equicorrelation model represents the limit case in which all properties (P1)–(P3)hold with the equal sign, so it lies on the boundary of the set �(0,1,2,3) of correlationmatrices here considered.

We prove now an optimality property of equicorrelation matrices. The starting point is toobserve that given a correlation matrix R ∈ �(0,1,2,3) with dominant eigenvalue λ1(R),there are several other correlation matrices Rs ∈ �(0) having dominant pure shift eigenvec-tor and such that λ1(Rs) = λ1(R). In fact, if R has dimension n, n ≥ 3, the linear systemR · 1 = λ · 1 has n(n− 3)/2 freedom degrees. Hence, for n = 3 we have a unique correlationmatrix with pure shift dominant eigenvector (see Example 29), whereas for n ≥ 4 we haveinfinitely many matrices: we will denote the set of these matrices by S(n,λ1).

Example 30 The following correlation matrices

R1 =

⎡

⎢

⎢

⎣

1 0.5 0.3 0.10.5 1 0.1 0.30.3 0.1 1 0.50.1 0.3 0.5 1

⎤

⎥

⎥

⎦

R2 =

⎡

⎢

⎢

⎣

1 0.4 0.2 0.30.4 1 0.3 0.20.2 0.3 1 0.40.3 0.2 0.4 1

⎤

⎥

⎥

⎦

R3 =

⎡

⎢

⎢

⎣

1 0.3 0.3 0.30.3 1 0.3 0.30.3 0.3 1 0.30.3 0.3 0.3 1

⎤

⎥

⎥

⎦

have dominant eigenvalue λ1 = 1.9 and pure shift dominant eigenvector.

Note now that given a matrix R in S(n,λ1) its 2-norm, 1-norm and ∞-norm

‖R‖2 = maxx�=0

‖Rx‖‖x‖ ‖R‖∞ = ‖R‖1 = max

iri

coincide with λ1, hence elements in S(n,λ1) are not distinguable with respect these norms.The choice of the Frobenius norm

‖R‖F =√

√

√

√

n∑

i,j=1

ρ2ij

seems to be the most natural in this framework.

Example 31 For the correlation matrices of Example 30 we have

‖R1‖F = 2.3238; ‖R2‖F = 2.2716; ‖R3‖F = 2.2539.

In the previous example the equicorrelation matrix R3 presents the minimum Frobeniusnorm: this is a general fact. To prove this, we introduce a useful definition.


Definition 32 Given a correlation matrix R, the average correlation, denoted by ρ, is de-fined as

ρ = 1

n(n − 1)

∑

ij

i �=j

ρij .

Recalling that if R ∈ S(n,λ1) then λ1 = ri for all i, we can express the dominant eigen-value of R in terms of the average correlation

λ1 = 1 + (n − 1)ρ (21)

whereas, as showed in [5], if v1 �= 1, the r.h.s. in (21) is only a lower bound:

λ1 ≥ 1 + (n − 1)ρ.

Proposition 33 Fixed n ≥ 4 and λ1, for any n-dimensional correlation matrix R = [ρij ] inS(n,λ1) the (unique) matrix in this space with minimum Frobenius norm is the equicorrela-tion matrix with elements ρij = ρ.

Proof If R = [ρij ] is in S(n,λ1) with n ≥ 4, we have

‖R‖F = n(n − 1)Var[ρij ] + n[1 + (n − 1)ρ2]where

Var[ρij ] = 1

n(n − 1)

∑

i �=j

(ρij − ρ)2.

Since n and ρ are fixed, minimizing ‖R‖F is equivalent to minimizing Var[ρij ] and thishappens for ρij = ρ = (λ1 − 1)/(n − 1) for i �= j , where the last equality follows from(21). �

References

1. Ando, T.: Totally positive matrices. Linear Algebra Appl. 90, 165–219 (1987)2. Brauer, A.T.: The theorems of Ledermann and Ostrowski on positive matrices. Duke Math. J. 24, 265–

274 (1957)3. Brigo, D.: A note on correlation and rank reduction. Working paper for Banca IMI (2001)4. Brigo, D., Mercurio, F.: Interest Rate Models. Springer, Berlin (2001)5. Friedman, S., Weisberg, H.F.: Interpreting the first eigenvalue of a correlation matrix. Educ. Psychol.

Meas. 41, 11–21 (1981)6. Gantmacher, F.R.: The Theory of Matrices. Chelsea Publishing Company, New York (1964)7. Gantmacher, F.R., Krein, M.G.: Oscillation matrices and kernels and small vibrations of mechanical

systems. Dept. of Commerce, Washington (1961)8. Golub, B.W., Tilman, L.M.: Measuring yield curve risk using principal components analysis, value at

risk, and key rate durations. J. Portf. Manag. 23(4), 72–94 (1997)9. Ho, T., Lee, S.: Term structure movements and pricing interest rates contingent claims. J. Finance 41,

1011–1029 (1986)10. Hull, J.: Options, Futures and Other Derivatives, 3rd edn. Prentice-Hall, Upper Saddle River (1997)11. Hull, J., White, A.: Pricing interest-rate derivative securities. Rev. Financ. Stud. 3, 573–592 (1990)12. Isaacson, E., Keller, H.B.: Analysis of Numerical Methods. Dover, New York (1994)13. Jolliffe, I.T.: Principal Component Analysis. Springer, Berlin (2004)14. Kletskin, I., Lee, S.Y., Li, H., Li, M., Tolmasky, C., Wu, Y.: Correlation structures corresponding to

forward rates. Can. Appl. Math. Q. 12(3), 125–135 (2004)


15. Kotz, S., Pearn, W.L., Wichern, D.W.: Eigenvalue-eigenvectors analysis for a class of patterned correla-tion matrices with an application. Stat. Probab. Lett. 2, 119–125 (1984)

16. Lardic, S., Priaulet, P., Priaulet, S.: PCA of the yield curve dynamics: questions of methodologies. J.Bond Trading Manag. 1, 327–349 (2003)

17. Ledermann, W.: Bounds for the greatest latent roots of a positive matrix. J. Lond. Math. Soc. 35, 265–268(1960)

18. Litterman, R., Scheinkman, J.: Common factors affecting bond returns. J. Fixed Income 1, 54–61 (1991)19. Longstaff, F., Schwartz, S.: A two-factor interest rate model and contingent claims valuation. J. Fixed

Income 3, 16–23 (1992)20. Longstaff, F., Santa-Clara, P., Schwartz, S.: The relative valuation of caps and swaptions: theory and

empirical evidence. UCLA Working Paper (1999)21. Lord, R.: Pelsser, A.: Level, slope, curvature: fact or artifact? Appl. Math. Finance 14(2), 105–130

(2007)22. Martellini, L., Priaulet, P., Priaulet, S.: Fixed-Income Securities: Valuation, Risk Management and Port-

folio Strategies. Wiley, Chichester (2003)23. Minc, H.: On the maximal eigenvector of a positive matrix. SIAM J. Numer. Anal. 7, 424–427 (1970)24. Ostrowski, A.M.: Bounds for the greatest latent root of a positive matrix. J. Lond. Math. Soc. 27, 253–

256 (1952)25. Ostrowski, A.M.: On the eigenvector belonging to the maximal root of a non-negative matrices. Proc.

Edinb. Math. Soc. 12, 107–112 (1960/1961)26. Rao, C.R., Rao, M.B.: Rao Matrix Algebra and Its Applications to Statistics and Econometrics. World

Scientific, Singapore (1998)27. Rebonato, R.: Modern Pricing of Interest-Rate Derivatives. Princeton University Press, Princeton (2002)28. Salinelli, E., Sgarra, C.: Correlation matrices of yields and total positivity. Linear Algebra Appl. 418(1–

2), 682–692 (2006)29. Salinelli, E., Sgarra, C.: Shift, slope and curvature for a class of yields correlation matrices. Linear

Algebra Appl. 426(1–2), 650–666 (2007)30. Schoenmakers, J.G.M., Coffey, B.: Systematic generation of parametric correlation structures for the

LIBOR market model. Int. J. Theor. Appl. Finance 6(5), 507–519 (2003)31. Schoenmakers, J.G.M., Coffey, B.: Systematic generation of parametric correlation structures for the

LIBOR market model. WIAS Preprint No. 611 (2003)32. Vasicek, O.: An equilibrium characterization of the term structure. J. Financ. Econ. 5, 177–188 (1977)

Some Results on Correlation Matrices for Interest Rates

Documents