A Critique of Factor Analysis of Interest Rates

FALL 2000 THE JOURNAL OF DERIVATIVES 1

Factor analysis concludes that three underlying eco-nomic factors affecting the level, steepness, and cur-vature of the term structure are sufficient to describethe dynamic evolution of interest rates. This study isa critique of these conclusions within the context of ano-arbitrage hypothesis that restricts spot interest ratesto be averages of the corresponding forward rates.

The author develops a model to demonstrate that theresults of factor analysis are a statistical artifact cre-ated by the restrictions that the no-arbitrage hypoth-esis imposes on the correlation matrix of spot interestrates. The effect of the no-arbitrage relationship is soprofound that even in an artificial economy with ran-dom and uncorrelated forward rates the correspond-ing spot rate correlations and spot rate factor structureare indistinguishable from those empirically observed.The use of forward rates, which avoid these problems,reveals a more relaxed factor structure.

Akey issue affecting the performanceof state-of-the-art interest ratemodels is the number and identityof the factors employed to describe

the dynamic evolution of the term structure.A number of researchers have applied princi-pal components analysis and factor analysis ofthe U.S. dollar and British pound spot inter-est rates in order to reduce the dimensional-ity of the vector space of the original variablesand to provide some insight into the nature ofthe common factors. Results tend to supportthe conclusion that three factors are sufficient

to explain most of the spot interest rate vari-ability. When they examine the pattern offactor loadings across maturities, researchershave interpreted these factors as a parallel shiftof the term structure, a change in its steepness,and a change in its curvature.

This article provides a critical reexami-nation of the results of factor analysis of inter-est rates. Factor analysis and principalcomponent analysis are data exploration tech-niques that attempt to capture the correla-tion structure of a particular term structurewith a small set of common factors. Theabsence of any economic theory in this anal-ysis can lead to misinterpretation of the results.The extent of this misinterpretation willbecome obvious when the results of factoranalysis are examined within the context of theno-arbitrage relationship that restricts spotrates to be averages of the corresponding for-ward rates.

When the effect of the no-arbitragerestrictions is isolated, we demonstrate thatno inference on the link between macroeco-nomic factors and changes of the shape of theterm structure can be drawn from the resultsof factor analysis. The factors identified fromforward rates that do not face the same restric-tions as spot rates are less structured. But,these results should be interpreted with cau-tion and the association between statisticalfactors and any underlying economic factorsremains undetermined.

Our factor analysis is not limited to a sin-

A Critique of Factor Analysisof Interest RatesILIAS LEKKOS

ILIAS LEKKOS

is a financial analyst inthe Monetary AnalysisDivision of theBank of England inLondon.

gle term structure. Instead, we perform factor analysis onthe term structures of the four most important currencies.Also, by performing factor analysis on both the spot andforward term structures, this study is able to demonstratethe distinct features of the factor structures of these yieldcurves.

I. PREVIOUS RESEARCH

Despite the importance of the issue, manyresearchers simply make ad hoc assumptions concerningthe number of factors that an interest rate model shouldemploy. As a result, the literature on identifying the num-ber and identity of the underlying factors using principalcomponents analysis or factor analysis is rather limited.Also, despite increased interest in modeling forward rates,mainly due to the work of Heath, Jarrow, and Morton[1992], there is very little research concerning the num-ber of factors in forward rates. As we shall demonstrate,the difference of factor structure between spot and forwardinterest rates can prove to be quite important.

Stambaugh [1988] uses the generalized method ofmoments to examine the number of state variables (or fac-tors) in the term structure of Treasury bills. He studies thenumber of latent factors required to describe expectedreturns on bills with maturities up to six months. He findsthat a two- or possibly three-latent factor model cannotbe rejected by the data.

Steeley [1990] uses principal components analysis tostudy the U.K. term structure between 1985 and 1987.He finds that the first component explains 87% of the vari-ability of U.K. spot interest rates. The addition of the sec-ond component raises the proportion of varianceexplained to 94%. Similarly, Carverhill and Strickland[1992] find that for the 1987-1990 period the first and sec-ond factors explain 96.0% to 98.7% of the U.K. termstructure and 94% to 99% of the U.S. term structure.

Dybvig [1997], using data for 1964-1987, reports thepresence of one dominant component in the U.S. termstructure that is responsible for 95% of the total variance.Using factor analysis, Litterman and Scheinkman [1991]find that for the 1984-1998 period, the first factor explains90% of the variability in the U.S. term structure, and thecombination of the first two factors accounts for 98% ofthe term structure’s variation.

The majority of these studies agree on the inter-pretation of the first three components (or factors). Theloadings of all spot interest rates on the first factor havethe same sign (either positive or negative) and roughly the

same magnitude, although they increase slightly withtime to maturity. This pattern of factor loadings hasprompted researchers to interpret the impact of the firstfactor as a roughly parallel shift in the term structure.Given the pattern of the factor loadings, the second andthird factors have been interpreted as a change in thesteepness and a change in the curvature of the term struc-ture.

II. METHODOLOGY

The starting point of factor analysis is the assump-tion that there are a small number of unobservable (latent)factors that can account for the observed covarianceamong the original variables. The general representationof the factor model with k factors applied to interestrates is:

ri = mi + bi1F1 + bi2F2 + ... + bikFk + ei (1)

where ri are the interest rates, for i = 1, ..., N, and Fj forj = 1, ..., k are the k common factors. The unobservablecommon factors are assumed to have unit variance and tobe orthogonal to each other. The coefficients bij are theloadings of the j-th factor on the i-th interest rate. Sincethe factors do not account for the total variance of eachvariable, there is an error term ei unique to each interestrate.

From the above, it should be clear that the varianceof ri can be reproduced by Var(ri) = b2

i1 + b2i2 + ... + b2

ik

+ var(ei). The sum b2i1 + b2

i2 + ... + b2ik is called the com-

munality of the i-th rate, and represents the portion of itsvariance attributable to the k common factors.

In a more compact matrix notation, we can rewrite(1) as follows:

R = mm + BF + ee (2)

where R¢ = (r1, r2, ..., rN) is the vector of interest rates; mmis an N ´ 1 vector of constants; B is an N ´ k matrix offactor loadings; F is a k ´ 1 vector of the unobservablecommon factors; and ee is an N ´ 1 vector of residuals withE(e) = 0 and cov(e) = YY, where YY is a diagonal matrix.

If the factors are restricted to be orthogonal to eachother, the covariance structure of this model is given by

WW = BB¢ + YY (3)

The next step is to estimate the matrix of factor

2 A CRITIQUE OF FACTOR ANALYSIS OF INTEREST RATES FALL 2000

loadings, B, by decomposing the variance-covariancematrix WW. There are several methods that can be used toperform this task. The most commonly used is the max-imum likelihood method (ML). One of the advantages ofML is that a likelihood ratio test can be performed to testwhether the number of factors is sufficient.

Starting from the most parsimonious model withonly one factor, we proceed by increasing the number offactors until we find evidence against the new augmentedmodel. When we perform this test on factor models forspot interest rates, we are not able to reject the aug-mented model even after the eighth factor.1 A possibleexplanation for the failure of ML to identify the numberof factors is a lack of normality in the data.2

To avoid the problems posed by a departure fromnormality in the data, we use the principal componentsmethod to estimate the factor loadings.3 Equation (3)holds exactly (i.e., YY = 0) when the number of factorsequals the number of variables. Since the objective of fac-tor analysis is to explain the covariance structure in termsof just few common factors, the covariance matrix isapproximated by

(4)

where l2, l3, ..., lk are the first k eigenvalues of WW, andgg1, gg

2, ..., ggk are the corresponding eigenvectors. Finally,

for 1, 2, ..., N

where the bij are the estimated factor loadings. The disadvantage of the principal components esti-

mation of the factor model is the lack of any statistical cri-terion to determine the number of factors employed bythe model. Intuitively we would like the right-hand side

y s bi i ijjk= - å =

21

WW YY

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gg

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BB

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of Equation (4) to approximate the left-hand side or,equivalently, the off-diagonal elements of YY to be closeto zero. A heuristic rule to ensure that the residual cor-relation is negligible is to ensure that the factors usedexplain a big part of the total covariation of the underly-ing variables.4 The proportion of total variance accountedfor by the first k factors is

The final step is to replace the covariance matrix inEquations (3) and (4) with the correlation matrix. Theadvantages of using the correlation matrix instead of thecovariance matrix are that 1) the loadings are not affectedby differences in the variances of the original variables, and2) correlations are more stable than covariances over thesample period.

We apply factor analysis to the correlation matrix ofspot and forward interest rate changes rather than inter-est rate levels. There are two reasons for not using inter-est rate levels. One is that interest rate levels are highlycorrelated. The effect is that factor analysis will considermost of the information embodied in the term structureas redundant. Hence a single trend variable is sufficient toaccount for the total variance.5 Taking first differencesreduces the correlation between interest rates and allowsus to study the factors influencing interest rate movements.

A second equally important reason for using differ-ences is that one of the assumptions inherent in factoranalysis is that the data under consideration representrandom, independent samples from a multivariate distri-bution. Interest rates are highly autocorrelated, with a cor-relogram very similar to that of a random walk series.Taking first differences reduces the autocorrelation of thedifferenced series drastically, although the problem is notcompletely removed.

The final step is to ensure that the results of factoranalysis are robust to the assumption about the underly-ing distribution. Given that the most common candidatesfor the distribution of interest rates are the normal and thelognormal distributions, we perform factor analysis onboth interest rate changes and log changes. In all cases theresults are essentially identical, so results on log changesare not reported.

III. DATA

One of the characteristics of interest rates on bonds

Variance explained by k factors = + + +l l l1 2 ... k

k


is that they cannot be directly observed in the marketplace;rather they have to be inferred from prices of interest rate-dependent instruments. The most common approach isto infer the term structure of interest rates using default-free bonds, as implemented by McCulloch [1971, 1975].An alternative approach is to combine money marketand swaps market data to infer the prices and thus theyields of zero-coupon bonds. This sequential extractionof interest rates from money and swap market instru-ments is also known as the bootstrap technique (see Galitz[1994]). For maturities up to twelve months the spotrates can be calculated from money market instruments,while for maturities between one year and ten years thespot rates can be inferred by application of the bootstrapmethod to interest rate swap rates.6

The database consists of daily money market andswaps market rates for the four main currencies: the U.S.dollar (USD), the German deutschemark (DM), theBritish pound (GBP), and the Japanese yen (JPY). Thematurities of the money market rates are three, six, andtwelve months. The maturities of the swap rates are two,three, four, five, seven, and ten years. The U.S. data coverthe period September 3, 1984-September 5, 1995. TheGBP and DM data are available March 24, 1987-Septem-ber 5, 1995, and the JPY data October 6, 1987-Septem-ber 5, 1995.

These rates and their corresponding maturities arethe input to the bootstrap method, described in theappendix. The output is daily spot and forward termstructures of interest rates. A preliminary examinationindicates that the daily data are exceedingly noisy. For thatreason we perform weekly sampling. The first day in thedataset is a Tuesday, so we sample money and swap mar-ket rates every Tuesday. If Tuesday’s rates are not availablefor a particular week, we use Wednesday’s rates instead.7

The spot term structures of all currencies consist of twelverates, with maturities ranging from three months to tenyears. The forward term structures consist of nine forwardrates with maturities ranging from one to nine years.The length of the forward period for the forward rates isone year.

IV. FACTOR ANALYSIS RESULTS

Spot Rates

The results of factor analysis of spot interest rates arereported in Exhibits 1-4. In the spot term structures, wefind that the first factor accounts for the largest portion


E X H I B I T 1Factor Analysis of Spot USD Interest Rates

1st 2nd 3rd 4th Factor Factor Factor Factor

Eigenvalue 10.25 1.19 0.25 0.10% Variance 85.4% 9.90% 2.05% 0.85%Cumulative 85.4% 95.28% 97.33% 98.18%

Factor Loadings

3 Months 0.735440 –0.63435 –0.12387 0.177566 Months 0.808738 –0.56254 –0.05832 –0.000041 Year 0.855267 –0.44410 0.03211 –0.255542 Years 0.950472 –0.02409 0.25545 0.032783 Years 0.966467 0.04824 0.19166 0.032034 Years 0.966545 0.09720 0.15860 0.049595 Years 0.966725 0.17537 0.07676 0.007476 Years 0.977437 0.18415 0.00309 0.005497 Years 0.966711 0.19095 –0.07615 0.003408 Years 0.968123 0.21017 –0.11995 –0.004779 Years 0.957839 0.22688 –0.16592 –0.0130810 Years 0.933311 0.24505 –0.21329 –0.02225

E X H I B I T 2Factor Analysis of Spot DM Interest Rates



Factor Loadings

3 Months 0.657941 –0.67404 –0.22625 0.236986 Months 0.756662 –0.60342 –0.14937 –0.090281 Year 0.816538 –0.51368 –0.00281 –0.215822 Years 0.927811 –0.21170 0.22822 –0.021423 Years 0.947868 –0.07732 0.25679 0.010964 Years 0.959059 0.05557 0.20498 0.067495 Years 0.964945 0.11262 0.15216 0.063696 Years 0.972141 0.20536 0.03172 0.042827 Years 0.934082 0.29844 –0.09981 0.018398 Years 0.933521 0.32505 –0.13556 –0.006569 Years 0.919812 0.34561 –0.17292 –0.0317810 Years 0.888428 0.36864 –0.20979 –0.05830

of the variability of interest rates, with explanatory powerthat varies from 80% in the case of German interest ratesto 85% in the case of U.S. interest rates. The combina-tion of the first three factors explains 97% of the total vari-ance in all four countries.

The factor loading patterns are similar to thosereported in previous research. The first factor can beinterpreted as a parallel shift in the term structure, the sec-ond as a change in the steepness, and the third as a changein the curvature of the term structure.

Forward Rates

The results of factor analysis of forward rates arereported in Exhibits 5-8. The factor structure is quite dif-ferent from that of spot rates. The first factor explainsbetween about 43% and 64% of the total variance; thecombination of the first and the second accounts forabout 70%; and addition of the third factor raises theexplanatory power to 81% to 86%. A total of five factorsis required to explain 95% of the total variability of for-ward term structures.

An examination of the factor loading patterns revealsthat, with the exception of JPY forward rates, the first fac-tor still seems to have roughly the same loadings across allrates. The second and third factors do not show any spe-cific pattern.

V. A CRITICAL EXAMINATION OF FACTOR ANALYSIS RESULTS

There are various reasons for different factor struc-ture results for spot and forward term structures. Factoranalysis and principal components analysis are data explo-ration techniques. We need a theoretical background toguard against misinterpretation of the results.

The starting point of our discussion is the relation-ship between spot and forward rates. It is well known thatif bn,t is the price at time t of a discount bond maturingat time t + n, the n-period yield to maturity is

rn,t = –n–1lnbn,t (5)

and the one-period forward rate from time t + n to t +n + 1 is

fn,t = ln(bn,t/bn+1,t) (6)

From (5) and (6), the spot rates are simple averages


E X H I B I T 3Factor Analysis of Spot GBP Interest Rates



Factor Loadings

3 Months 0.672035 –0.65968 –0.23591 0.224306 Months 0.773272 –0.59554 0.12366 –0.062411 Year 0.811508 –0.52357 –0.03383 –0.203282 Years 0.926964 –0.21498 0.22314 –0.041283 Years 0.954898 –0.07454 –0.24571 –0.009374 Years 0.966434 0.04264 0.20292 0.030895 Years 0.966704 0.11223 0.15113 0.067706 Years 0.970129 0.20687 0.05533 0.069417 Years 0.935906 0.30735 –0.05401 0.069878 Years 0.933767 0.33129 –0.11792 0.014879 Years 0.913814 0.35478 –0.18751 –0.0448510 Years 0.875731 0.37408 0.25549 –0.10416

E X H I B I T 4Factor Analysis of Spot JPY Interest Rates



Factor Loadings

3 Months 0.655080 –0.68088 –0.25150 0.171526 Months 0.737907 –0.62503 –0.05185 –0.067751 Year 0.841582 –0.45368 0.18395 –0.190392 Years 0.961713 –0.18554 0.18862 –0.024973 Years 0.967498 0.07729 0.17307 0.127344 Years 0.978148 0.12137 0.12183 0.111175 Years 0.971865 0.16425 0.06846 0.093396 Years 0.976604 0.19546 –0.00382 0.045407 Years 0.964062 0.22519 –0.08023 –0.005958 Years 0.961705 0.23970 –0.11685 –0.051339 Years 0.947170 0.25215 –0.15376 –0.0978410 Years 0.945866 0.25059 –0.15569 –0.10297

of the corresponding forwards

(7)

with f0,t = r1,t.

r n fn t j tj

n

, ,= å--

=

11

1

This averaging process can seriously bias the out-come of factor analysis, and it can be argued that it isresponsible for the high explanatory power of the first fac-tor for the spot rate, as well as the fact that the first fac-tor always represents a parallel shift (or, after rotation, thelong end of the term structure). Equation (7) indicates thatthe short-term forward rates are part of all the spot rates


E X H I B I T 5Factor Analysis of Forward USD Interest Rates

1st Factor 2nd Factor 3rd Factor 4th Factor 5th Factor

Eigenvalue 5.09 1.56 0.89 0.73 0.39% Variance 56.51% 17.38% 9.86% 8.12% 4.32% Cumulative 56.51% 73.88% 83.75% 91.86% 96.19%

Factor Loadings

1 Year 0.725139 –0.24888 –0.17307 –0.46163 0.010762 Years 0.763796 –0.07398 –0.37655 –0.03646 0.483293 Years 0.640599 –0.29528 0.33852 –0.51802 –0.145654 Years 0.602559 0.14039 –0.68586 0.06281 –0.358625 Years 0.773707 –0.56586 0.07412 0.27038 –0.039066 Years 0.665258 –0.61254 0.15793 0.39055 –0.048287 Years 0.919208 0.36676 0.12924 0.04518 –0.006548 Years 0.849435 0.47611 0.18223 0.08150 0.025759 Years 0.771447 0.57722 0.22494 0.09959 –0.02731

E X H I B I T 6Factor Analysis of Forward DM Interest Rates


Eigenvalue 4.55 1.56 1.21 0.79 0.50 % Variance 50.56% 17.31% 13.45% 8.77% 5.78% Cumulative 50.56% 67.88% 81.33% 90.10% 95.68%

Factor Loadings

1 Year 0.646067 –0.09332 –0.57804 0.15626 0.162272 Years 0.649865 –0.08962 –0.55038 –0.09195 0.342623 Years 0.602584 –0.10601 –0.13995 0.67803 –0.351754 Years 0.583588 0.12549 –0.38429 –0.52376 –0.470765 Years 0.674802 –0.70041 0.21080 –0.09163 –0.004376 Years 0.556272 –0.74507 0.34666 –0.11851 0.008157 Years 0.929827 0.30532 0.19533 7.22E-05 0.044768 Years 0.862549 0.38799 0.28011 –0.00974 0.096359 Years 0.796125 0.47448 0.35350 –0.01007 0.04131

— even though with declining weights. As a result, thereexist a number of common elements throughout the spotyield curve that contribute to the parallel shift effect.

An alternative way to look at the effect of this no-arbitrage relationship is that, as we move toward the longend of the term structure longer-maturity spot rates con-sist of almost the same number of forwards. This partly

explains why, after rotation, the first factor is associatedwith the long end of the curve.

The only input required for factor or principalcomponents analysis is the covariance or the correlationmatrix. To investigate the effect that the no-arbitragerelationship has on factor analysis, we analyze the influ-ence of Equation (7) on these matrices.


E X H I B I T 8Factor Analysis of Forward JPY Interest Rates


Eigenvalue 3.85 2.29 1.54 0.54 0.44 % Variance 42.83% 25.50% 17.09% 5.95% 4.93%Cumulative 42.83% 68.32% 85.41% 91.36% 96.30%

Factor Loadings

1 Year 0.657966 0.09311 –0.53038 0.26683 –0.276962 Years 0.632202 0.14816 –0.56584 0.30522 0.046093 Years 0.522804 –0.12787 –0.58444 –0.59950 –0.086824 Years 0.279202 0.76594 –0.19949 –0.03895 0.514575 Years 0.443181 –0.86375 –0.07385 0.05711 0.212896 Years 0.311731 –0.92475 0.03606 0.05792 0.198297 Years 0.952827 0.11064 0.27911 –0.02972 0.003978 Years 0.887710 0.14405 0.43305 –0.03387 –0.045689 Years 0.825252 0.16461 0.53204 –0.03539 –0.07826

E X H I B I T 7Factor Analysis of Forward GBP Interest Rates


Eigenvalue 4.59 1.69 1.25 0.58 0.47 % Variance 63.52% 6.28% 7.54% 8.11% 5.28% Cumulative 63.52% 69.80% 83.73% 90.14% 95.42%

Factor Loadings

1 Year 0.575994 –0.51891 –0.31201 0.12068 –0.374222 Years 0.646786 –0.41446 –0.32754 0.15324 –0.175473 Years 0.669883 –0.35611 –0.14985 0.34921 0.518774 Years 0.595983 –0.29885 –0.34025 –0.63991 0.173305 Years 0.754774 –0.28797 0.58555 –0.04635 –0.036356 Years 0.641345 –0.19484 0.73806 –0.06065 –0.032687 Years 0.902338 0.41435 –0.10261 0.02265 –0.032518 Years 0.822778 0.56012 –0.08724 0.02155 –0.029129 Years 0.751438 0.65367 –0.07481 0.01934 –0.02668

In matrix form, (7) can be written as

(8)

or, in a more compact notation

R = Wf (9)

where R is an N ´ 1 vector of spot rates, f is an N ´ 1vector of forward rates, and W is an N ́ N matrix of theweights of the forwards to the corresponding spot rates.Let WW

R and WWf denote the variance-covariance matrices

of the spot and forward rates, respectively. From Equation(9) it follows that the covariance matrix of the spot ratesis

WWR = WWW

fW¢ (10)

According to Equations (9) and (10), spot interestrates are in fact a “transformation” of the existing forwardrates. The result of this transformation is an augmentationof the spot rate covariances. Since this extra covariationis artificially created, any incremental informationextracted from spot rates that is not present in the forwardrates has no economic underpinnings and should beignored. The difference between factor analysis resultsfrom spot and forward rates is due to the effect of thisweighting matrix W. The effect of the no-arbitrage rela-tionship between spot and forward interest rates on theresults of factor analysis is equivalent to the effect of Won the covariance or correlation matrix of the spot inter-est rates.

To see the effect of W on factor analysis, assume thatthere is zero correlation among all forward rates. Also, forreasons of simplicity and without loss of generality, weassume that they all have the same variance s2

f .8 The

covariance matrix of the forward rates in this artificialeconomy is

r

r

rN N N

f

f

f

t

t

N t

t

t

N t

1

2

0

1

1

1 0 0

12

12

0

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and the correlation matrix will be equal to the identitymatrix, rr

f = I.For a covariance and a correlation matrix of this pat-

tern, nothing is gained by performing factor analysis. Alleigenvalues are equal, and the factor loading matrix is anidentity matrix. The second step is to employ Equation(10) to derive the covariance and correlation matrices ofthe corresponding spot rates.

If the term structure consists of ten rates, the spotcovariance matrix is given by

(11)

and the spot correlation matrix

WWR =

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s s s s

s s s

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ff f f

f f f

f f

f

22 2 2

2 2 2

2 2

2

2 3 10

2 3 10

9 10

10

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WWf =

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(12)

The correlation pattern in (12) reveals that evenwhen we start from an artificial economy where forwardrates are uncorrelated, the correlation matrix of the spotrates appears similar to that observed empirically, with highcorrelation between adjacent rates and declining correla-tion between distant rates. Equations (11) and (12) alsoserve as lower bounds for the covariance and correlationmatrices of the spot interest rates, conditional on thecorresponding forward rates having equal variances. Spotrate correlations or covariances lower than those reportedwould imply that the forward rates are negatively corre-lated.

The results of the factor analysis of the correlationmatrix in Equation (12) are given in Exhibit 9. As we havenoted, factor analysis is not relevant for forward ratesbecause there is no correlation or any common factors.9

The results of factor analysis of the spot correlation arevery different from that of forward rates. In Exhibit 9, thefirst factor explains 73% of the total covariation and thecombination of the first three factors explains 92%. Themost enlightening outcome is that, according to the pat-terns of factor loadings, the first factor can still be inter-preted as a parallel shift in the term structure, the secondas a change in the steepness, and the third as a change inthe curvature.

The implication here is that conventional factoranalysis of spot rates can be rather misleading. As wehave clearly shown, given zero correlation among forwardrates, the aggregation process alone is able to reproducespot rate factor loading patterns similar to these observedempirically.

VI. CONCLUSION

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Our exploration of the factor structure of spot andforward interest rates demonstrates the pitfalls in attemptsto assign an economic interpretation to the results of fac-tor analysis of spot rates. The traditional approach to iden-tifying common factors in spot interest rates is to employthe covariance or correlation matrix of interest rates toidentify clusters of highly intercorrelated interest ratesthat can be replaced by a single factor. When this approachis applied to USD, DM, GBP, and JPY spot rates, we findthat three factors can explain on average 97% of the totalcovariation.

Nevertheless, the factor analysis and principal com-ponent analysis used to derive these results are data explo-ration techniques that lack economic underpinning. Theirsole purpose is to reproduce the correlation structure ofthe original variables with fewer factors. The lack of eco-nomic intuition in the application of these techniques tothe study of spot interest rates has led to misinterpretationsof the spot rate factors. Researchers have associated theresults of factor analysis with three underlying economicfactors affecting the level, curvature, and steepness of thespot term structure.

We have examined these results within the contextof the no-arbitrage relationship that restricts spot rates tobe averages of the corresponding forward rates. Wedemonstrate that these results are not necessarily due to


E X H I B I T 9Factor Analysis of Artificial Spot Interest Rates


Eigenvalue 7.26 1.37 0.55 0.29% Variance 72.6% 13.7% 0.055% 0.029% Cumulative 72.6% 86.3% 91.8% 94.7%

Factor Loadings

1 Year –0.555 0.633 0.467 –0.2492 Years –0.731 0.569 0.063 0.2473 Years –0.832 0.394 –0.213 0.2044 Years –0.892 0.197 –0.302 –0.0215 Years –0.923 0.015 –0.254 –0.1816 Years –0.933 –0.135 –0.133 –0.2017 Years –0.929 –0.248 0.006 –0.1138 Years –0.912 –0.322 0.125 0.0139 Years –0.886 –0.362 0.203 0.11910 Years –0.852 –0.371 0.235 0.171

the impact of any economic factors affecting the spot termstructure. A factor structure like that reported by factoranalysis of spot interest rates could simply be a statisticalartifact created by the way the spot rates are linked to theforward rates. The effect of the no-arbitrage relationshipis so profound that, even in an artificial economy withcompletely random and uncorrelated forward rates, thecorresponding spot rate correlations and spot rate factorsare hard to distinguish from those observed empirically.

As an alternative, we examine the existence of com-mon factors in forward rates. Forward rates of equallength of forward period (one year in our case) are cross-sectionally non-overlapping, avoiding the problems inher-ent in spot interest rates. Factor analysis of forward ratesreveals a completely different factor structure. The first fac-tor explains as little as 43%, and five factors are needed toaccount for 95% of the total covariation.

Identification of the actual factor structure of inter-est rates is important not only in its own right; it has sev-eral implications for interest rate modeling. The fact thatthe factor structure is less constrained than previous resultssuggest means that more factors should be included inmodels describing the evolution of forward rates. Finally,our findings should prompt reexamination of interest ratemodels that are based on the level, steepness, and curva-ture properties of the spot term structure.10 The resultsreported here do not question the ability of these mod-els to provide a satisfactory fit to spot interest rate data.Nevertheless, using factor analysis on spot rates to modelforward term structure movements may prove to be prob-lematical.

A P P E N D I XImplementation of the Methodology

Term Structure and Money Market Data

We infer the yields of zero-coupon bonds from moneymarket rates with less than one year to maturity and swaps mar-ket data for longer maturities. We use rates quoted in themoney market for loans with maturities of three, six, andtwelve months.

Let the price of a pure discount bond that pays $1 at timet be b0,t. Also denote the money market rate for a loan of anidentical maturity as rt and the accrual factor a0,t.

11 Then we canestimate the discount bond prices for maturities up to one yearas:

(A-1)

Swaps Market Data

We infer prices of zero-coupon bonds from swap ratesby bootstrapping. In the case of USD and DM, where the swapcash flows occur annually, the prices of discount bonds impliedby the swap market are given by:

(A-2)

where st is the swap rate, i = 1, 2, ..., t, and the accrual factoris ai–1,i.

12 The only problem is that swaps are available one fortwo, three, four, five, seven, and ten years of maturity. Thus,a linear interpolation has to be used to get an estimate of themissing swap rates.

In the case of JPY and GBP swaps, where the swap cashflows occur semiannually the calculations are slightly morecomplicated.13 We have to use swap rates every half year to cal-culate the zero bond prices for the corresponding period, againwith linear interpolation to estimate the missing swap rates. Weare not able to calculate the s1.5 swap rate using linear interpo-lation since the one-year swap rate is not available, and the cor-responding one-year rate available from the money market isquoted on a different basis.

We thus make an adjustment:

(A-3)

and calculate s1.5 by interpolating between s1 and s2. Finally, the zero bond prices can be calculated using the

bootstrap method as in Equation (A-2), but now the index i inthe summation is calculated semiannually such that i = 1.0, 1.5,2.0, 2.5, ..., t.

According to these zero-coupon bond prices, we can esti-mate the implied annualized yields as14

(A-4)rbt

t

t

00

11

0

,,

,

=æ

èçö

ø÷-

-a

sb

b b10 1

0 0 5 0 0 5 0 5 1 0 1

1=

-+

,

. . , . . , ,a a

bs b

stt i i ii

t

t i i0

1 011

1

1

1,, ,

,=

- å( )+

-=-

-

a

a

brtt t

00

11,

,=

+ a


One-year forward rates can be estimated by

(A-5)

ENDNOTES

This article is based on the author’s Ph.D. dissertationat Lancaster University. He thanks his supervisor, R.C. Sta-pleton; the members of his Ph.D. committee, M. Martins andJ. Steeley, for their valuable motivation and suggestions; S.H.Poon for comments and corrections on earlier versions; andBZW and QRI for providing the data. The views expressed arethe author’s and do not necessarily reflect those of the Bank ofEngland.

1Similar problems are reported by Knez, Litterman,and Scheinkman [1994].

2The normality assumption has been tested with theKolmogorov-Smirnov and Jarque-Berra tests. The results, notreported here, clearly reject the presence of normality in thedata. For more details, see Lekkos [1998].

3This procedure should not be confused with othersimilar approaches to decompose the variance-covariance matrixof interest rates using principal components analysis. Principalcomponents attempts to reproduce the original covariancestructure with a small set of “factors” that are linear combina-tions of the original variables. The factor model in Equation (2)focuses on the part of total variance that is common to allvariables. Principal components is used within this context toestimate the loadings of the factor model without relying on theassumption of the normality of interest rates.

4The factor models estimated in Section IV alwaysaccount for at least 95% of the total covariation.

5Indeed by performing factor analysis on the levels ofinterest rates, we find that as a result of this common trend thefirst component explains 98% to 99% of the total variance.

6A detailed description of our implementation of thismethodology is provided in the appendix.

7Such substitutions occur in less than 0.5% of the num-ber of weekly observations.

8The advantage of using the same variance s2f for all for-

ward rates is that the correlation matrix of the spot rates (12)does not depend on sf, making the presentation more trans-parent.

9For a more detailed examination of the results of fac-tor analysis and principal components analysis of random data,see Stauffer, Garton, and Steinhorst [1985].

10See Litterman and Scheinkman [1991].11For DM, USD, and JPY, the interest rate accrual fac-

fb

bt t mt

t m

t t m

0 120

0 12

12

1, ,,

,

,

++

-

=æ

èçö

ø÷-

+a

tor is ai–1,i = (30/360). For GBP, the accrual factor is ai–1,i =(ti – ti–1)/365.

12For USD and DM interest rate swaps, the accrualfactor is defined as ai–1,i = (30/360).

13For GBP and JPY swap markets, the swap day-countconvention is 365 days per year. Thus, the accrual factor isdefined as ai–1,i = (ti – ti–1)/365.

14The accrual factors now refer to bonds, and as a resultthey are ai–1,i = (ti – ti–1)/365.

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Economy & Finance

A Critique of Factor Analysis of Interest Rates