Mat-2.108 Independent Research Project in Applied Mathematics

November 15, 2006

Principal Component Analysis on Term Structure of Interest

Rates

Helsinki University of Technology Systems Analysis Laboratory Antti Malava 64705M Department of Engineering Physics and Mathematics

1. Introduction..............................................................................................................................1

2. Principal Component Analysis.................................................................................................1

2.1 Mathematical formulation................................................................................................2

2.2 Choosing the number of Principal Components ..............................................................6

2.3 Interpretation of Analysis.................................................................................................6

2.4 Geometrical Interpretation ...............................................................................................7

2.5 Discussion ........................................................................................................................8

3. Modelling Term Structure of Interest Rates with PCA............................................................8

3.1 Term Structure .................................................................................................................8

3.2 Literature Review of Principal Component Analysis on Term Structure........................9

3.3 Applications in Finance .................................................................................................10

4. Implementation and Results...................................................................................................13

4.1 Data Preparation.............................................................................................................13

4.2 Preliminary Analysis......................................................................................................14

4.3 Problem Formulation .....................................................................................................16

4.4 Individual Currency Zone Results .................................................................................16

4.5 Combined Currency Zone Results .................................................................................21

5. Conclusions and Discussion ..................................................................................................24

References......................................................................................................................................25

1

1. Introduction

Term structure of interest rates has for long posed interesting challenges with regards to modelling, explaining and predicting its behaviour. However, analysing term structure often involves dealing with huge data sets that may cause the calculation processes to become slow and cumbersome and the results difficult to be interpreted and used in further applications. On the other hand, interest rates of different maturities exhibit distinguishable common behaviour. Therefore it may be very useful to simplify the data or the data structure by identifying factors of common behaviour such that not much of the contained information is lost.

This paper introduces principal component analysis (hereafter referred to as PCA) as a powerful tool of identifying patterns in data of high dimension. PCA is a statistical technique in which the original variables are replaced by a smaller number of artificial variables that preserve as much as possible of the variability of the original variables. There are two objectives of data simplification: to reduce the number of variables and to detect a structure in the relationships between variables.

The paper discusses PCA with focus on constructing a market model for different currency zones’ term structures of interest rates. PCA is applied to four interest rate curves, namely EUR, USD, JPY and GBP curves including both short and long term interest rates. Two different approaches are compared: performing PCA separately for each curve and performing one PCA for all curves combined. With regards to interest rates, the markets typically show three distinct patterns that are represented by first three principal components (hereafter referred to as PCs): “level” or “shift”, “slope” or “twist” and “curvature” or “bow”.

The rest of this paper is structured as follows. Section 2 presents the methodology of standard PCA in detail. Section 3 introduces term structure of interest rates, presents some of the PCA modelling performed on it in literature and discusses cases where it may be useful to apply PCA. In section 4, PCA is performed on term structure and the results are analysed. Finally, section 5 draws together conclusions and discussion arisen from the study.

2. Principal Component Analysis

PCA is a statistical technique for simplifying a set of data. The key idea of PCA is to perform a linear transformation of the original data to a new orthogonal coordinate system such that that axes are ordered in terms of the amount of variance in a dataset. In other words, the greatest variance by any projection of the original data set comes to lie on the first axis called the first PC, the second greatest variance on the second axis, and so on. PCA therefore finds the ‘true dimensionality’ of the data set by compressing correlations in the data to single variables. Figure 1 illustrates the transformation graphically for two-dimensional data. The obvious distinction to other linear transformations is that PCA’s basis vectors are not fixed but depend on the data set.

2

X2

X1

2. PC

1. PC

Figure 1. Geometrical representation of PCA

The linear transformation allows describing the original data set exactly by the uncorrelated artificial variables called principal components that are ordered with decreasing explanatory power. However, the real purpose of PCA is to select those PCs that explain the variability of the data to a required degree and accuracy. This allows considerably reduction in the dimension of variables, which in turn simplifies calculation processes.

PCA is often discussed with relation to factor analysis. However, although these two methods are similar statistical tools, they differ in the methodologies employed and the focus of the analysis. While PCA attempts to find a series of independent linear combinations of the original variables that provide the best possible explanation of diagonal terms of the matrix analysed, factor analyses focuses on the off-diagonal elements of the correlation matrix (Jorion, 2002). What this means is that PCA is not based on any particular statistical model whereas factor analysis is.

2.1 Mathematical formulation

The following mathematical formulation of PCA is mainly based on Mellin (2004).

Consider a set of n variables x1, …, xn as a random vector x with zero empirical mean and nonsingular covariance matrix S:

0)(

0)(≥=

=Σx

xCovE

3

)

1st Principal Component

The objective is to find the linear combination of random variables x1, …, xn that contains as much of the variability of the random variables x1,…,xn as possible. In other words, for a linear combination

∑=

=n

iii x

1βxβT

maximize its variance

(2 xβTD

subject to

12 == βββ T

Vector b is a weight vector that tells us by what weight does each of the variables xj affect the variance of the linear combination . The condition xβT

12 == βββ T

is a norming condition.

It can be shown that that

12

1

)(max λ===

1T1

βββ

ΣββxβTDT

where

l1 = largest eigenvalue of the covariance matrix S

= eigenvector corresponding to the largest eigenvalue of the covariance matrix S 1β

Therefore the 1st principal component is

xβT1=1y

4

)

2nd Principal Component

The next step is to find the linear combination of random variables x1, …, xn that is uncorrelated with the 1st principal component and contains as much of the variability of the random variables x1,…, xn as possible. In other words, for the linear combination

∑=

=n

iii x

1

βxβT

maximize its variance

(2 xβTD

subject to

12 == βββ T

and

0),( 1 =xβTyCov

Again it can be shown that

2222

0),(1

)(max

1

λ==

==

Σββxβ T

xβββ

β

T

yCov

D

T

T

where

l2 = second largest eigenvalue of the covariance matrix S

2β = eigenvector corresponding to the second largest eigenvalue of the covariance matrix S

Therefore the 2nd principal component is

xβT22 =y

Remaining Principal Components

Continuing the same way produces all n linear combinations of x with the following properties:

• The variance of the linear combinations is the largest possible

5

)

• The linear combination is uncorrelated with the previously identified linear combinations

• These linear combinations form the principal components of random vector x.

From above it is trivial that the principal components are obtained from the eigenvalue decomposition of the covariance matrix S:

TBDBΣ =

in which

,...,,( 21 ndiag λλλ=D

is the diagonal matrix composed of the eigenvalues

nλ...λλ ≥≥≥ 21

of the covariance matrix S, and the matrix

[ ]n21 β...ββB MMM=

is an orthogonal matrix

IBBBB == TT

consisting of the corresponding eigenvectors as its columns.

Therefore the principal components can be expressed as

NNii xxy ββ ++== K11i xβTi

representing the i.th principal component, i = 1, 2, …, n

The eigenvectors with the largest eigenvalues correspond to the dimensions that have the strongest correlation in the dataset. This means that the more correlated the data, the bigger share of the total variation is explained by the first principal component.

After finding the PCs, we can express the original variables as a linear combination of principal components:

NiNii yyBx β++== K11yβ i

6

2.2 Choosing the number of Principal Components

There is no single criterion against which the number of useful principal components can be evaluated, because the explanatory power of the principal components can differ arbitrarily from case to case. However, a couple of basic rules are often employed. Threshold criterion guides to select as many PCs as is necessary (i.e. the first K PCs) for the cumulative explanatory degree to exceed certain percentage, such as 90% (e.g., Kreinin et al., 1998.):

ThresholdN

ii

K

ii

>

∑

∑

=

=

1

1

λ

λ

Another criterion first sorts the eigenvalues in order of magnitude, then finds from the spectrum a break point that divides the eigenvalues in “large” and “small” and finally chooses the PCs corresponding to the “large” eigenvalues (Mellin, 2004).

If the original variables are standardized for PCA (see section 2.5), each of them will have a unit variance. Therefore any PC with an eigenvalue of at least 1 explains more of the total variance than any of the original variables. Thus a simple heuristic would be select those PCs that have an eigenvalue of at least 1 (Clustan, 2006).

Deciding to keep only the first r PCs allows us to replace the previous exact relationship by a close approximation:

ririi yyBx β++=≈ K11yβ i

2.3 Interpretation of Analysis

The computed eigenvalues represent the variance of each PC, while the eigenvectors are their “loadings”.

We can compute an n x r -principal component matrix, also referred to as factor structure :

[ ]

1/2rr

1r

DB

βββF

=

=

221 rrλλλ MLMM

where

[ ]r21r βββB MLMM=

( )rdiag λλλ ,,, 21 K=1/2rD

7

If PCs are calculated from standardized data (that is, from the correlation matrix), the PC matrix elements represent the correlation between the r chosen PCs and the n original variables.

We can also compute principal component scores that represent the “value” of each PC with respect to each observation:

nj ,...,2,1, == jrj xBy

where the i.th component

riy ji ,...,2,1, =

of r-vector

),...,,( 21 jrjj yyy=jy

is thus the value of i.th principal component yi at observation j=1,2,…,n, i=1,2,…,r

2.4 Geometrical Interpretation

The geometric interpretation of principal component analysis is the following. The j-dimensional linear subspace spanned by the first j principal components gives the best possible fit to the data points as measured by the sum of squared perpendicular distances from each data point to the subspace. In other words, let

x = (x1, x2, … , xn)

be a random vector for which the following holds:

0)(

)(≥=

=Σx

0xCovE

Quadratic form expression

cf T == − xΣxx 1)(

where c is a constant determines an ellipsoid of n-dimensional space as a function of variable x. It can be shown that the principal components are the main axis of the ellipsoid

c=− BxΣx 1T

The main axis of the ellipsoid coincide with the directions of the eigenvectors of the covariance matrix S and the lengths of the main axis relate to each other as numbers

8

iλ

where li is the ith eigenvalue of the covariance matrix S. The geometric interpretation is illustrated in figure 1, where all principal component vectors are drawn for 2-dimensional data.

2.5 Discussion

Although PCA appears to have many benefits, few things should be kept in mind when considering performing PCA to a particular data set. Firstly, PCA is not a statistical method from the viewpoint that there is no probability distribution specified for the observations. Therefore it is important to keep in mind that PCA best serves to represent data in simpler, reduced form.

Another challenge to be kept in mind is that it is often difficult, if not impossible, to discover the true economic interpretation of PCs since the new variables are linear combinations of the original variables.

In addition, for PCA to work exactly, one should use standardized data so that the mean is zero and the unbiased estimate of variance is unity:

x

xii

xz

σµ−

=

where

zi = i.th standardized variable

This is because it is often the case that the scales of the original variables are not comparable and that (those) variable (variables) with high absolute variance will dominate the first principal component.

There is one major drawback to standardization, however. Standardizing means that PCA results will come out with respect to standardized variables. This makes the interpretation and further applications of PCA results even more difficult.

3. Modelling Term Structure of Interest Rates with PCA

3.1 Term Structure

Term structure of interest rates, one of the most followed curves in money and capital markets, describes the time structure of cost of borrowing (or required rate of return). More specifically, term structure tells the market cost of borrowing (or required rate of return) for zero-coupon bonds of different maturities, i.e. the rate of return required by market for periods of different length.

9

Term structures are calculated for different classes of bonds, most typically for risk free government bonds. The most important and widely used interest rates, however, consist of Interbank Offered Rates and swap rates that derive from high credit rating banks borrowing money from each other. These curves typically lie a little higher than government curves.

There are numerous, ongoing challenges offered by the term structure: how to estimate it, how to use it to evaluate implicit interest rates in future (forward rates) and how to explain the shape and the movements of the curve, the most demanding challenge.

3.2 Literature Review of Principal Component Analysis on Term Structure

There are numerous examples in literature of PCA performed on interest rates with different setups and focuses. PCA has been applied to short term (money market) and long term (capital markets) interest rates separately, and on all maturities combined (for example, Heidari et al., 2000). Some research has not only considered interest rates but has combined data from equities, foreign exchange (FX) and interest rates markets (for example, Loretan, 1997). In addition, PCA has been performed to not only levels and differences but also to derivative features of interest rates such as option volatility surfaces (for example, Cont et al, 2002).

Even though the setups of the PCAs on interest rates in literature vary quite extensively, the results and conclusions, perhaps little surprisingly, are similar. It seems, in general, that interest rates levels and differences can be explained with first three PCs to a sufficient extent. Similarly, there is scope for significant dimension reduction when combining data from equity, exchange rates and interest rates (Loretan, 1997).

Considering interest rate derivatives features, such as option volatility surfaces, some studies have identified the need for additional PCs (“independent volatility factors”) in order to explain the variability of the surface (Heidari & Wu, 2001) while other similar studies have coped with standard three PCs (Cont et al). The reason why implied volatilities or credit spreads are more challenging to estimate in sufficient magnitude and accuracy with few PCs derives from their nonlinear nature (Frye 1998).

In addition, it is noteworthy that the quality and the focus of research vary extensively. While some papers concentrate on presenting the results, they overlook the real interpretation of PCA and do not take into account the limitations of PCA. In addition, the usability of PCA results is highly dependant on the choice of setup of analysis. For example, choosing to perform PCA on levels rather than differences forgoes the opportunity to construct meaningful and informative market scenarios that for many end users might be the real value of PCA. Continuing the same argument, standard PC selection criteria does not work effectively for all cases (Kreinin et al, 1998), discussion of which is often bypassed.

10

3.3 Applications in Finance

Simpler and Faster VaR Processes

PCA is often used in the field of portfolio risk management to reduce the dimensionality of the risk factor space (Loretan, 1997). Since a large share of positions held by market participants is interest rate related derivatives, modelling term structure of interest rates, or different yield curves, with PCA may result in significant simplification of risk models. Instruments not related to interest rates, such as common stocks, have also successfully been modelled with PCA.

The risk of asset portfolios is typically measured as Value at Risk (VaR)1:

“VaR measures the worst expected loss over a given horizon under normal market conditions at a given confidence level” (Jorion, 2004).

VaR can be defined in many forms. For example, relative VaR is defined as the loss in currency units relative to the mean, i.e. to the expected value:

)()( *0

* µ−−=−= RWWWEVaRMEAN

where

E(W) = expected value of the portfolio at the end of the target horizon

W * = lowest portfolio value at the given confidence level

W0 = initial investment

R * = lowest portfolio return at the given confidence level

From the above definition, it is easy to compute for example parametric VaR as

tWVaRMEAN ∆= ασ0

where

α = standard normal deviate corresponding to desired left-tail confidence level

σ = standard deviation of portfolio return

1 A thorough discussion of VaR -methodology itself is beyond the scope of this study. A reader interested to explore the field of VaR is referred to a comprehensive guide to VaR by Philippe Jorion (Jorion, 2004).

11

2/)1(*

∆t = time adjustment factor

It is obvious that “correlations are essential driving forces behind portfolio risk” (Jorion, 2004). However, the number of correlations increases geometrically with the number of assets (figure 2): with n assets, the number of correlations is +nn .

50 100 150 200assets

5000

10000

15000

20000corrs Number of correlationsas a function of number of assets

Figure 2. Number of correlations as a function of number of assets in a portfolio

This poses two problems with large portfolios. Firstly, when the number of assets increases, it is more likely that some correlations will be measured inaccurately or incorrectly. Secondly, the computation time of covariance matrix and the subsequent VaR calculations can increase dramatically, which is not feasible for making quick decisions on trading portfolio positions in fast-changing markets. Thirdly, the VaR of the portfolio may not be positive2.

The benefit of performing PCA is that to examine the behaviour of original variables we can simulate the movements of principal components. Not only is the number of PCs much smaller but also the covariance matrix is positive definite, because the eigenvalue decomposition produces uncorrelated variables.

Consider a portfolio z = w’R mapped into its exposures on the first K principal components:

2 VaR is proportional to portfolio variance, which is positive only if the covariance matrix is positive definite. This requires the number of observations to be larger than the number of variables, and the series cannot be linearly correlated. The problem of positive-definiteness occurs more likely when portfolio consists of a large number of highly correlated assets such as zero-coupon bonds.

12

KK

KNKNKNN

KNKNNKKii

yyywwyww

yywyywRw

δδββββ

ββββ

++=++++++=

++++++≈= ∑

... )...(...)...(

)...(...)...(

11

1111111

1111111z

where

represents the weighted exposure to i:th principal component iTβw=iδ

Furthermore, the variance of the portfolio can be computed easily:

)(...)(

...

...)(

221

221

21

21

2

KK

K

K

yy σδσδ

λλ

λλσ

++=

++=

++==

)β(w)β(w

wββwwββwωΣωz

KT

1T

TK´K

TT11

T*Τ

In other words, the variance of the portfolio z is given by sum of the squared exposures times the variance of each PC. This is a remarkable simplification compared to the variance calculated with original variables because instead of requiring all of the variances and covariances of the original variables, it is enough to use K independent variables. In other words, for a portfolio of m variables the covariance matrix of dimension m*m can be replaced with just a few variables.

However, “it is not difficult to construct a portfolio which has a large position sensitive to risk factors that appears unimportant in the PCA” (Kreinin et al., 1998). This means that to perform efficient PCA on a portfolio, one should select the PCs not based on how much they explain of the total variability of the data but on how much of the variability of the particular portfolio. For example Hull (2005) presents a portfolio that has little exposure to the first component but significant exposure to the second component (calculated for U.S. Treasury data). Using only one component to hedge the position, which is similar to duration-based hedging that considers a parallel shift in term structure, would dangerously understate VaR. This important remark is surprisingly often not mentioned in literature regarding principal component analysis.

More Efficient Hedging Strategies

In financial analysis duration, the first derivative of interest rate related asset’s price function with respect to the interest rate, is a basic but extensively employed tool for hedging, that is, for trying to insure that (small) changes in interest rates will not cause the value of an investment portfolio (of fixed-income securities) to fluctuate significantly. However, relying on simple duration is over simplistic because the changes in interest rate levels are not always parallel shifts, in which case zero-duration portfolio will not be immunised.

Identifying first few principal components allows achieving a better-hedged position because the components explain almost all of the return variability across the whole spectrum of maturities

13

(Litterman et al, 1991). Given the simple computation of portfolio returns and variance, it is easy to create portfolios that are immune to a factor by selecting asset holdings that make the sensitivity of the portfolio equal to zero.

Macroeconomic Analysis

Empirical research in literature suggests that although PCA produces artificial variables that explain the variability of interest rates, one can associate rational macroeconomic interpretations to these variables (Wu, 2003). For example, there is tendency for strong correlation between ‘surprises’ of monetary policy and the subsequent movement of the ‘slope’ component. This area of study is vital among central bankers and agents involved in and directly affected by central bank actions.

In addition, PCA overcomes some computational problems often confronted in macroeconomic analysis. Typical macroeconomic models try to explain or predict variations in response variables by variations in prediction variables trough, for example, Multiple Linear Regression (MLR). However, macroeconomic variables often tend to be somewhat linearly dependent which leads to problems of multicollinearity. PCA resolves this problem by creating uncorrelated variables on which the original response variables can be regressed. The methodology of combining PCA with MLR is called Principal Component Regression.

4. Implementation and Results

In this section, PCA is applied to the term structure of interest rates of four different currency zones: Eurozone (EUR), United States (USD), Japan (JPY) and Great Britain (GBP). The motivation makes it possible to examine and describe different currency zone interest rate structures with a few-factor model, and see whether global term structure movements can be explained by a set of relatively few common factors. In addition, combining short term and long term interest rates allows seeing if, how and why money markets (and thus money market instruments) behave differently from markets of longer maturities.

4.1 Data Preparation

The data used represents first differences of daily last quotes of Interbank Offered Rates (Euribor for EUR and Libor for USD, JPY and GBP) for money market data and interest rate swap rates of each currency zone. Time series of 3 months is used, from 7/4/2006 to 7/7/2006, gathered from Bloomberg.

The choice of Interbank Offered rates and swap rates is based on the previously stated fact that they are the widest used rates in financial and capital markets. In addition, these rates do not have the problem of demand and supply that might distort the informational content of the data (Heidari et al, 2000). The chosen currency zones represent large part of most influential economic zones in the world.

14

In the cases studied in this paper, all of the variables represent interest rates with variances of comparable magnitude. Evidence suggests that with such variables standardized and non-standardized data produce very similar results (Rodrigues, 1997). Therefore we have not standardized the data (except for the mean) to facilitate the interpretation of the results. Analysing first differences, rather than levels, has the advantage that it allows constructing different interest rate curve scenarios that are often very useful considering applications of PCA. The disadvantage, however, is that taking first differences has a tendency to artificially increase noise and therefore decrease the efficiency of estimation (Heidari et al., 2002).

I performed PCA on two different setups. Firstly, I analysed each currency zone term structures independently and compared the correlations between PCs specific to different currency zones. In the second setup I performed one aggregate analysis.

4.2 Preliminary Analysis

During the observation period the general trend of EUR interest rates was slightly upwards (figure 3). It can be seen that long-term interest rates behave very much in accordance with each other, but money market rates (and thus money market instruments) exhibit slightly different characteristics. This suggests that it is likely to find a rather well explaining EUR market model based on a few PCs, although money market rates need special considerations.

Similar behaviour can be distinguished in and USD and JPY interest rates during the observation period. Two differences, however, exists. Firstly, USD interest rates curve exhibit a small twist at the boundary of money market so that 2Y, 3Y and 4Y swap rates are actually at lower level than 12M Libor in the middle of the observation period. This hints about expectations of market participants about near future development of interest rates and thus the monetary policy moves. Secondly, JPY interest rates are at very low level with short rates almost at zero. Long rates are much higher implying that contractionary monetary policy actions are expected in the near future3.

While the general trend of GBP interest rates development over the observation period is similar to EUR and USD rates, the GBP interest rate curve looks quite different to them. Short GBP interest rates are significantly lower than medium term interest rates, which in turn are higher than long interest rates, thereby forming a hump-shaped curve. This indicates that monetary policy, future expectations and even economic situation were quite different in UK when compared to other currency zones analysed during the observation period.

3 In fact, on 14th July 2006, during the writing of this paper and slightly after the end of the observation period used, Bank of Japan ended its 6-year period of effectively zero interest rates by raising the key interest rate to 0.25%.

15

EUR Interest Rates Development

2,50

2,90

3,30

3,70

4,10

4,50

4,90

7.4.20

06

14.4.

2006

21.4.

2006

28.4.

2006

5.5.20

06

12.5.

2006

19.5.

2006

26.5.

2006

2.6.20

06

9.6.20

06

16.6.

2006

23.6.

2006

30.6.

2006

7.7.20

06

Date

Inte

rest

Rat

e (%

)

1M

3M

6M

12M

2Y

3Y

4Y

5Y

6Y

7Y

8Y

9Y

10Y

12Y

15Y

20Y

25Y

30Y

EUR Interest Rates, 23.5.2006

2

2,5

3

3,5

4

4,5

5

1M 3M 6M 12M 2Y 3Y 4Y 5Y 6Y 7Y 8Y 9Y 10

Y12

Y15

Y20

Y25

Y30

Y

Maturity

Perc

ent

USD Interest Rates Development

4,80

5,00

5,20

5,40

5,60

5,80

6,00

7.4.20

06

14.4.

2006

21.4.

2006

28.4.

2006

5.5.20

06

12.5.

2006

19.5.

2006

26.5.

2006

2.6.20

06

9.6.20

06

16.6.

2006

23.6.

2006

30.6.

2006

7.7.20

06

Date

Inte

rest

Rat

e (%

)

1M

3M

6M

12M

2Y

3Y

4Y

5Y

6Y

7Y

8Y

9Y

10Y

12Y

15Y

20Y

25Y

30Y

USD Interest Rates, 23.5.2006

4,804,905,005,105,205,305,405,505,605,705,80

1M 3M 6M 12M 2Y 3Y 4Y 5Y 6Y 7Y 8Y 9Y 10

Y12

Y15Y 20

Y25

Y30

Y

Maturity

Perc

ent

GBP Interest Rates Development

4,304,404,504,604,704,804,905,005,105,205,30

7.4.06

14.4.

06

21.4.

06

28.4.

065.5

.06

12.5.

06

19.5.

06

26.5.

062.6

.069.6

.06

16.6.

06

23.6.

06

30.6.

06

Date

Inte

res

Rat

e (%

)

1M

3M

6M

12M

2Y

3Y

4Y

5Y

6Y

7Y

8Y

9Y

10Y

12Y

15Y

20Y

25Y

30Y

GBP Interest Rates, 23.5.2006

4,204,304,404,504,604,704,804,905,005,105,20

1M 3M 6M 12M 2Y 3Y 4Y 5Y 6Y 7Y 8Y 9Y 10

Y12

Y15

Y20

Y25

Y30

Y

Maturity

Perc

ent

JPY Interest Rates Development

-0,50

0,00

0,50

1,00

1,50

2,00

2,50

3,00

3,50

7.4.06

14.4.

06

21.4.

06

28.4.

06

5.5.06

12.5.

06

19.5.

06

26.5.

06

2.6.06

9.6.06

16.6.

06

23.6.

06

30.6.

06

7.7.06

Date

Inte

rest

Rat

e (%

)

1M

3M

6M

12M

2Y

3Y

4Y

5Y

6Y

7Y

8Y

9Y

10Y

12Y

15Y

20Y

25Y

30Y

JPY Interest Rates, 23.5.2006

0,00

0,50

1,00

1,50

2,00

2,50

3,00

1M 3M 6M12

M 2Y 3Y 4Y 5Y 6Y 7Y 8Y 9Y 10Y

12Y

15Y

20Y

25Y

30Y

Maturity

Perc

ent

Figure 3. Graphical representation of the raw data used in analysis.

16

4.3 Problem Formulation

Considering each currency zone separately, let x = (x1, x2,…, x18) be a vector of the 18 zero-mean random variables representing absolute daily changes of interest rates of each maturity and let S be the corresponding covariance matrix. Performing eigenvalue decomposition

TBDBΣ =

yields 18 eigenvectors

[ ]181 βββB MLMM 2=

and 18 eigenvalues

T),...,,( 1821 λλλ=D

We get 18 principal components

xβxβ T18

T1 == 181 ,..., yy

that explain completely the variability of the original variables. The aim is to be able to explain at least 90% of the total variability of interest rates across the term structure with as few PCs as possible for each currency zone. In order to determine how many PCs are sufficient, we need to analyse the explanatory power of them.

Similar analysis is conducted for aggregate data with 4 (currency zones) * 12 (maturities) variables x = (x1, x2,…, x72) and corresponding 72 principal components.

4.4 Individual Currency Zone Results

Explanatory Power of PCs

Figure 4 plots cumulative percentage of total variance data explained by principal components for each currency zone. It can be seen that for EUR and USD the first 3. PCs are enough to explain at least 99% of the total variability of term structure, which is in accordance with evidence in literature. However, two more PCs are required in order to explain similar percentage of GBP and JPY interest rate curves. This suggests that the daily changes in GBP and JPY term structures exhibit a more complicated behaviour during the observation period.

17

Explanatory Power - EUR

88.00 %

90.00 %

92.00 %

94.00 %

96.00 %

98.00 %

100.00 %

102.00 %

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

Principal Component

Cum

ulat

ive

% o

f Var

ianc

e

Explanatory Power -USD

88.00 %

90.00 %

92.00 %

94.00 %

96.00 %

98.00 %

100.00 %

102.00 %

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

Principal Component

Cum

ulat

ive

% o

f Var

ianc

e

Explanatory Power - GBP

86.00 %

88.00 %90.00 %92.00 %

94.00 %96.00 %98.00 %

100.00 %102.00 %

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

Principal Component

Cum

ulat

ive

% o

f Var

ianc

e

Explanatory Power - JPY

82.00 %84.00 %86.00 %88.00 %90.00 %92.00 %94.00 %96.00 %98.00 %

100.00 %102.00 %

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

Principal Component

Cum

ulat

ive

% o

f Var

ianc

e

Figure 4. Cumulative percentage of total variance explained by the principal components

Breaking down the explanatory power of principal components on the variability of original variables reveals further insight (figure 5). Considering the wish to explain at least 90% of the variability of the interest rates across all maturities, we require 5 first PCs for each currency zone. Therefore the model for each currency zone is:

5544332211 yyByByByBx iiiiii β++++=≈ yβ i

The 1st PC explains very little of money market interest rates, almost all of medium-term swap rates and quite well very long swap rates. The 2nd PC shows opposite behavior: money market and long-term rates are more explained than medium-term rates by it. 3rd PC, similarly, is responsible for explaining some of the variation in short and long-term ends of the interest rate curve. However, the variation patterns differ from that of 2nd PC (otherwise the 3rd PC variation would be included in 2nd PC). Interestingly, 4th and 5th PCs are significant in describing movements of short-term interest rates - a result not often confronted in literature.

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Breakdown of Explanatory Power - EUR

0,00 %10,00 %20,00 %30,00 %40,00 %50,00 %60,00 %70,00 %80,00 %90,00 %

100,00 %

1M 6M 2Y 4Y 6Y 8Y 10Y

15Y

25Y

Maturity

Perc

enta

ge

1. PC2. PC3. PC4. PC5. PCTotal

Breakdown of Explanatory Power - USD

0,00 %10,00 %20,00 %30,00 %40,00 %50,00 %60,00 %70,00 %80,00 %90,00 %

100,00 %

1M 6M 2Y 4Y 6Y 8Y 10Y

15Y

25Y

Maturity

Perc

enta

ge

1. PC2. PC3. PC4. PC5. PCTotal

Breakdown of Explanatory Power -GBP

0,00 %10,00 %20,00 %30,00 %40,00 %50,00 %60,00 %70,00 %80,00 %90,00 %

100,00 %

1M 6M 2Y 4Y 6Y 8Y 10Y

15Y

25Y

Maturity

Perc

enta

ge

1. PC2. PC3. PC4. PC5. PCTotal

Breakdown of Explanatory Power - JPY

0,00 %10,00 %20,00 %30,00 %40,00 %50,00 %60,00 %70,00 %80,00 %90,00 %

100,00 %

1M 6M 2Y 4Y 6Y 8Y 10Y

15Y

25Y

Maturity

Perc

enta

ge

1. PC2. PC3. PC4. PC5. PCTotal

Figure 5. Breakdown of explanatory power of principal components

PC coefficients

Figure 6 shows to coefficients of the first five principal components for each currency zone. The coefficients describe “loadings” of each principal component on a particular variable, i.e. the effect of each PCs on different maturity. The distinction of loadings to curves in figure 4 is that loadings only describe what effect each PC has on each maturity without considering the absolute magnitude of the effect, whereas in figure 4 the curves represent the amount of total variation explained by each component. In general the 1st PC shows quite flat behavior and therefore explains “shift” of the interest rate curves. Similarly, the 2nd PCs for each currency zones show clear downward trends, therefore explaining “twist” of the interest rate curves. The 3rd PCs shows a bell-shaped trend and therefore explains the “bow” of the interest rate curves4. The 4th and 5th PCs are important for short-term interest rates, but their rational interpretation is more difficult.

4 In literature the first three components are most often referred to as ”level”, ”slope” and ”curvature”. This notation is meaningful when performing PCA on levels, rather than differences. Unfortunately, however, the notation is employed for differences quite often, too. This is an example of findings that PCA is “a black box that is widely used but poorly understood”.

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Loadings - EUR

-0,9

-0,7-0,5

-0,3

-0,10,1

0,3

0,50,7

0,9

1M 3M 6M 12M 2Y 3Y 4Y 5Y 6Y 7Y 8Y 9Y 10Y 12

Y15

Y20Y 25Y 30

Y

Maturity (months)

coef

ficie

nt v

alue 1. PC

2. PC

3. PC

4. PC

5. PC

Loadings - USD

-0,9-0,7-0,5-0,3-0,10,10,30,50,70,9

1M 3M 6M 12M 2Y 3Y 4Y 5Y 6Y 7Y 8Y 9Y 10

Y12

Y15

Y20

Y25

Y30

Y

Maturity (months)

Coef

ficie

nt v

alue 1. PC

2. PC3. PC4. PC5. PC

Loadings - GBP

-0,9-0,7-0,5-0,3-0,10,10,30,50,70,9

1M 3M 6M 12M 2Y 3Y 4Y 5Y 6Y 7Y 8Y 9Y 10

Y12

Y15

Y20

Y25

Y30

Y

Maturity (months)

coef

ficie

nt v

alue 1. PC

2. PC3. PC4. PC5. PC

Loadings - JPY

-0,9-0,7

-0,5-0,3

-0,1

0,10,3

0,50,7

0,9

1M 3M 6M 12M 2Y 3Y 4Y 5Y 6Y 7Y 8Y 9Y 10

Y12

Y15

Y20

Y25

Y30

Y

Maturity (months)

coef

ficie

nt v

alue 1. PC

2. PC3. PC

4. PC5. PC

Figure 6. Loadings of first 3 principal components as features of interest rate curve: shift, twist and bow

Scenarios

Changes or shocks in PCs cause the interest rate structure to shift relative to the loadings of particular PC in question. For example, if there is a “twist shock”, the effect will be larger on the mid-term (2-5y) interest rates than short (-12m) or very long interest rates (+5y). Therefore one can use loadings to construct several “market shock scenarios” and investigate how well the market model composing of the chosen components captures these scenarios.

There are different methods for creating market scenarios. For example, one can assume that PCs can move up or down some amount (for example 2.33 standard deviations corresponding to 1st and 99th percentiles of standard normal distribution). This method allows constructing 2n different scenarios as possible combinations of chosen n PCs. I decided to create scenarios based on Monte Carlo simulation. I create random vectors whose elements are drawn from multinormal distribution with zero mean and covariance matrix equal to diagonal matrix of chosen eigenvalues (that correspond to variances of chosen PCs) and multiply this vector with PCs to get different linear combinations representing daily interest rate movement scenarios. The advantage of Monte Carlo method is that one can choose the number of scenarios arbitrarily, the formation of scenarios (i.e. the combinations of changes in chosen PCs) randomly and examine the correlations between scenarios and original data, for example. Compared to simulating the original data, we now only need to simulate 5 variables each time instead of 18, which clearly shows the advantage of PCA in saving computational time.

20

Figure 7 plots the different scenarios for each currency zones. Thicker long-dashed curves representing 5th and 95th quantiles are drawn on the same diagram together with a particular day’s observed changes shown in short-dashed curves.

The scenarios seem to vary as expected within the quantiles. Only in few occasions are the quantiles exceeded by a particular scenario (such as for JPY). This suggests that the market model with the chosen PCs performs quite well in estimating the daily changes.

Scenarios - EUR

-8-6-4

-2024

68

1M 3M 6M 12M 2Y 3Y 4Y 5Y 6Y 7Y 8Y 9Y 10

Y12

Y15

Y20

Y25

Y30

Y

Maturity

Bas

is P

oint

s

Scenarios - USD

-10-8-6-4-202468

10

1M 3M 6M 12M 2Y 3Y 4Y 5Y 6Y 7Y 8Y 9Y 10

Y12

Y15

Y20

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Y30

Y

Maturity

Bas

is P

oint

s

Scenarios - GBP

-8-6-4

-2024

68

1M 3M 6M 12M 2Y 3Y 4Y 5Y 6Y 7Y 8Y 9Y 10

Y12

Y15

Y20

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Y

Maturity

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is P

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s

Scenarios - JPY

-12-10

-8-6-4-202468

1M 3M 6M 12M 2Y 3Y 4Y 5Y 6Y 7Y 8Y 9Y 10

Y12

Y15

Y20

Y25

Y30

Y

Maturity

Bas

is P

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s

Figure 7. Scenarios

Levels vs Differences

As noted before, taking first differences may artificially increase noise and therefore decrease efficiency of PCA. To study this effect, I performed PCA on levels of interest rates, too. The results suggest that 3 PCs is enough to explain most of the variation of all maturities (figure 8), compared to 5 PCs required for differenced data. This supports the disadvantage of taking first difference. Therefore one is faced with a trade-off between efficient estimation (low dimensionality of model) and applicability of results (scenarios and other applications where standard deviation of PCs is required). However, more extensive research, analysis and discussion on levels vs. differences are beyond the scope of this study.

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Breakdown of Explanatory Power - EUR

0,0 %10,0 %20,0 %30,0 %40,0 %50,0 %60,0 %70,0 %80,0 %90,0 %

100,0 %

1M 3M 6M12M 2Y 3Y 4Y 5Y 6Y 7Y 8Y 9Y10

Y12

Y15

Y20

Y25

Y30

YMaturity

Perc

enta

ge 1. PC2. PC3. PCTotal

Breakdown of Explanatory Power - USD

0,0 %10,0 %20,0 %30,0 %40,0 %50,0 %60,0 %70,0 %80,0 %90,0 %

100,0 %

1M 3M 6M12M 2Y 3Y 4Y 5Y 6Y 7Y 8Y 9Y10

Y12

Y15

Y20

Y25

Y30

YMaturity

Perc

enta

ge

1. PC2. PC3. PCTotal

Breakdown of Explanatory Power - GBP

0,0 %10,0 %20,0 %30,0 %40,0 %50,0 %60,0 %70,0 %80,0 %90,0 %

100,0 %

1M 3M 6M12M 2Y 3Y 4Y 5Y 6Y 7Y 8Y 9Y10

Y12

Y15

Y20

Y25

Y30

YMaturity

Perc

enta

ge

1. PC2. PC3. PCTotal

Breakdown of Explanatory Power - JPY

0,0 %10,0 %20,0 %30,0 %40,0 %50,0 %60,0 %70,0 %80,0 %90,0 %

100,0 %

1M 3M 6M12M 2Y 3Y 4Y 5Y 6Y 7Y 8Y 9Y10

Y12

Y15

Y20

Y25

Y30

YMaturity

Perc

enta

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1. PC2. PC3. PCTotal

Figure 8. Breakdown of explanatory power of principal components for levels of interest rates

4.5 Combined Currency Zone Results

Evidence presented in literature reveals that there may exist “consistent, common influences across the markets” (Rodriques, 1997) offering opportunities to further reduce the dimension of market models. I investigated correlations between PCs for different countries and perform PCA for aggregate data.

Correlations

Research and analysis performed in literature proposes that “some correlation between the first components extracted from different country bond data” exists although “correlation between second and third components” is less evident (Rodrigues, 1997).

I investigated the correlations between first five principal components estimated for different countries. Correlations are calculated from principal component scores for the observation period.

It can be seen that for 1. PC, all of the currency zones correlate quite strongly with each other, highest being 0.79 for EUR/GBP and lowest being 0.53 for USD/JPY and GBP/JPY (table 1). This means that the relative levels of interest rate changes have been quite stable between the currency zones.

Only EUR/USD shows moderate correlation (0.33) for 2. PC, while other combinations correlate much less obviously. This suggests that the “twist” effect (and therefore possibly monetary policy

22

actions) has been quite inconsistent between the currency zones during the observation period. Similar small correlations are found for 3. PC. These findings support the evidence presented in literature.

Interestingly, there is evidence of some EUR/USD, USD/GBP, USD/JPY and GBP/JPY correlation for 4. PC. Similarly, 5. PC shows correlations of comparable magnitude. Taking into account correlations for all PCs, it can be concluded that the interest rates fluctuate to some degree within same dimensions. In other words, several variables can be found that explain movements of interest rates across different currency zones, which suggests there exists some common global interest rate movements.

1st Principal Component 2nd Principal Component Correlation EUR USD GBP JPY Correlation EUR USD GBP JPY

EUR 1,00 0,77 0,79 0,64 EUR 1,00 0,20 0,00 0,06 USD 0,77 1,00 0,65 0,53 USD 0,20 1,00 0,12 -0,17 GBP 0,79 0,65 1,00 0,53 GBP 0,00 0,12 1,00 -0,15

JPY 0,64 0,53 0,53 1,00 JPY 0,06 -0,17 -0,15 1,00

3rd Principal Component 4th Principal Component Correlation EUR USD GBP JPY Correlation EUR USD GBP JPY

EUR 1,00 0,33 -0,12 0,02 EUR 1,00 -0,10 0,04 0,05 USD 0,33 1,00 -0,17 -0,18 USD -0,10 1,00 -0,11 -0,08 GBP -0,12 -0,17 1,00 0,12 GBP 0,04 -0,11 1,00 0,00

JPY 0,02 -0,18 0,12 1,00 JPY 0,05 -0,08 0,00 1,00

5th Principal Component

Correlation EUR USD GBP JPY EUR 1,00 0,17 0,14 0,00 USD 0,17 1,00 -0,05 -0,16 GBP 0,14 -0,05 1,00 -0,19 JPY 0,00 -0,16 -0,19 1,00

Table 1. Correlations of principal components between currency zones

However, it should be noted that the results are dependent on the observation period. Empirical evidence has shown that correlations across different currency zones may change significantly over a few-year observation period (Rodrigues, 1997). This suggests that multiple currency zone models are less stable than single country models.

PCA on Combined Data

Performing PCA on data that includes all individual currency zones’ data combined reveals slightly different pictures of global interest rate structures.

1. PC explains about 67% of the total variability of data that includes daily changes of interest rates for all four currency zones (figure 9). This figure rises to 86% with 3 PCs but 14 PCs is required to

23

capture 99% of the total variability. In other words, quadrupling the dimensions about quadruples the number of PCs needed to explain the variability of the data. This suggests that there is not much scope for additional dimension reduction by combining different interest rate markets.

Explanatory Power - Currency Zones Combined

0,00 %

20,00 %

40,00 %

60,00 %

80,00 %

100,00 %

120,00 %

1 5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69

Principal Components

Cum

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% o

f Var

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e

Figure 9. Cumulative percentage of total variance explained by the principal components

Breaking down the explanatory power of PCs reveals that PCA on aggregate data seems most efficiently work to long-term interest rates, while short-term interest rates show less common behavior across currency zones. For example, while first common 5 PCs explain 99% of JPY 4 year swap variability, the figure is less than 30% for 6 month Libor for the same currency zone (table 2). However, the differences across currency zones are so large that further general conclusions are hard to be extracted.

Currency Zone EUR USD

Maturity First 3 PCs

First 5 PCs

First 10 PCs

First 20 PCs

First 3 PCs

First 5 PCs

First 10 PCs

First 20 PCs

1M 3,4 % 8,0 % 23,0 % 94,5 % 5,9 % 27,5 % 37,6 % 95,8 % 3M 0,3 % 1,4 % 62,5 % 88,6 % 2,8 % 73,8 % 87,1 % 97,5 % 6M 6,7 % 12,4 % 83,1 % 97,7 % 6,5 % 83,8 % 97,8 % 99,2 %

12M 16,4 % 22,1 % 90,3 % 99,1 % 9,5 % 84,4 % 98,2 % 99,7 % 2Y 74,4 % 84,0 % 98,3 % 99,5 % 85,7 % 90,6 % 98,9 % 99,8 % 3Y 78,7 % 89,6 % 98,6 % 99,6 % 90,4 % 94,0 % 99,3 % 99,8 % 4Y 83,8 % 94,5 % 99,4 % 99,9 % 94,1 % 96,8 % 99,5 % 99,7 % 5Y 85,4 % 96,6 % 99,4 % 99,8 % 96,6 % 98,3 % 99,8 % 99,9 % 6Y 87,4 % 98,4 % 99,5 % 99,9 % 97,8 % 99,0 % 99,7 % 99,8 % 7Y 88,1 % 99,4 % 99,6 % 99,8 % 99,0 % 99,8 % 99,9 % 99,9 % 8Y 87,9 % 99,7 % 99,7 % 99,9 % 99,2 % 99,8 % 99,8 % 99,9 % 9Y 87,8 % 99,4 % 99,6 % 99,8 % 99,3 % 99,8 % 99,8 % 99,9 %

10Y 87,3 % 99,1 % 99,6 % 99,9 % 99,0 % 99,5 % 99,7 % 99,8 % 12Y 86,5 % 98,4 % 99,7 % 99,9 % 98,5 % 99,2 % 99,8 % 99,9 % 15Y 83,9 % 97,0 % 99,7 % 99,9 % 97,5 % 98,4 % 99,8 % 99,9 % 20Y 80,8 % 94,7 % 99,2 % 99,8 % 95,8 % 97,2 % 99,8 % 99,9 % 25Y 80,2 % 93,9 % 98,9 % 99,8 % 95,0 % 96,6 % 99,7 % 99,9 %

30Y 78,4 % 92,5 % 98,8 % 99,8 % 93,7 % 95,6 % 99,5 % 99,8 %

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Currency Zone GBP JPY

Maturity First 3 PCs

First 5 PCs

First 10 PCs

First 20 PCs

First 3 PCs

First 5 PCs

First 10 PCs

First 20 PCs

1M 7,3 % 14,3 % 26,1 % 38,7 % 20,2 % 22,5 % 49,1 % 67,8 % 3M 18,1 % 26,2 % 50,4 % 68,3 % 13,0 % 19,6 % 59,7 % 95,4 % 6M 20,5 % 42,1 % 76,3 % 96,2 % 19,2 % 28,0 % 68,1 % 97,2 %

12M 27,5 % 52,5 % 85,7 % 99,3 % 19,4 % 27,4 % 62,7 % 98,3 % 2Y 76,7 % 78,9 % 96,0 % 99,4 % 70,3 % 74,4 % 95,3 % 99,5 % 3Y 81,2 % 82,3 % 96,5 % 99,6 % 80,7 % 84,3 % 97,7 % 99,6 % 4Y 90,0 % 91,2 % 98,9 % 99,6 % 89,4 % 93,5 % 99,0 % 99,4 % 5Y 93,2 % 94,5 % 98,8 % 99,5 % 93,7 % 95,9 % 99,1 % 99,6 % 6Y 94,9 % 96,9 % 98,7 % 99,6 % 94,7 % 96,8 % 99,3 % 99,7 % 7Y 96,2 % 98,6 % 99,2 % 99,7 % 95,8 % 97,6 % 99,3 % 99,9 % 8Y 95,7 % 98,9 % 99,1 % 99,6 % 96,5 % 97,9 % 99,3 % 99,9 % 9Y 94,9 % 98,8 % 99,0 % 99,7 % 96,8 % 97,9 % 99,1 % 99,7 %

10Y 94,9 % 98,8 % 99,0 % 99,5 % 96,3 % 97,1 % 98,8 % 99,5 % 12Y 95,1 % 98,1 % 98,8 % 99,1 % 88,8 % 91,2 % 97,5 % 98,4 % 15Y 90,1 % 95,5 % 97,4 % 99,6 % 86,4 % 88,3 % 96,4 % 99,6 % 20Y 89,1 % 94,6 % 98,1 % 98,8 % 81,7 % 85,5 % 98,0 % 99,5 % 25Y 79,7 % 83,7 % 90,7 % 99,9 % 78,2 % 82,7 % 98,8 % 99,7 %

30Y 80,0 % 89,0 % 95,3 % 99,6 % 74,9 % 80,0 % 98,2 % 99,7 %

Table 2. Cumulative explanatory power of principal components calculated from data consisting of all currency zones’ interest rate curves.

5. Conclusions and Discussion

This study has shown that PCA offers the opportunity to significantly reduce data dimensions of term structure of interest rates while retaining most of the information held by the data. In addition, the results of PCA form a basis for further applications and analysis, such as effective Monte Carlo Simulation regarding the measurement of market risk deriving from interest rate movements.

The study has revealed that although there are clear correlations between currency zones’ results, combining the data does not allow further considerable reduction in dimensions. However, the existence of correlations allows one to think of other ways of depicting global movements with fewer variables.

An interesting result of PCA performed in this study is that five PCs are required to explain at least 90% of the movements of interest rate curves whereas, for the same original data set, three PCs are enough to explain the levels of interest rates to similar degree of accuracy. This finding may form a basis for further investigation given that both methods have advantages and disadvantages.

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