Where the last equality follows from the normalization of the eigenvectors achieved with the orthogonal transformation. What this means is that a unit change of the jth factor causes a change βjt for each spot rate t. Since the factors are independent of each other we may therefore express the total change of the random variable spot rates, rt, by
∆rt=
k
X
j=1
βjt∆fj, (2.3)
wherek is the number of factors, identified by the eigenvector analysis, used to approximate the variance of the portfolio.
To summarize the results derived in this section we now give a definition of the principal components of the term structure of interest rates and a definition of factor loading which the coefficient βjt will be called from now, taken from Practical Financial Optimization (2005).
Definition 2.5 Principal components of the term structure.
Letr˜= (˜rt)⊤t=1be the random variable spot rates andQbe theT×T covariance matrix. An eigenvector ofQis a vectorβj= (βjt)⊤t=1 such thatQβj=λjβj for some constantλj called eigenvalue ofQ. The random variablef˜j =P⊤
t=1βjtr˜t
is a principle component of the term structure. The first principal component is the one that corresponds to the largest eigenvalue, the second to the second
largest, and so on. 3
Definition 2.6 Factor loadings.
The coefficients βjt are called factor loadings, and they measure the sensitivity of thet-maturity ratet to changes of thejth factor. 3
2.3 Application of Factor Analysis
A principal component analysis, as formulated in sections 2.2.1and2.2.1, was implemented on the data set described in section 1.2in order to recognize the key factors of the Danish term structure. More precisely it was performed for yearly maturity steps dated from the 4. of January 1995 to the 4. of October 2006, all in all thirty maturities in 614 issue dates i.e. n= 614sets of p= 30 observed variables.
In appendixA.2the results of the factor analysis performed on data from 1995–
2006, beginning from 1995 and adding one year at time are displayed. From those figures it can be seen that the shape of the factors becomes stable when
data from 4-5 years are included. Therefore it is concluded that the factors found from data groups containing more than five years of data give a stable es-timation. The results displayed below are found from factor analysis performed on the years 1995-2006.
Table2.3shows the standard deviation, the proportion of the variance and the cumulative proportion of the seven most significant principal components found for the period. The first three components, or factors, explain 99.9% of the total variation and where as the first factor accounts by far for the most of the variation or 94.9%.
PC1 PC2 PC3 PC4 PC5 PC6 PC7
Std. 5.335 1.1902 0.30696 0.15000 0.05260 0.02704 0.01863
Pr. of Var. 0.949 0.0472 0.00314 0.00075 0.00009 0.00002 0.00001
Cum. Prop. 0.949 0.9960 0.99912 0.99987 0.99996 0.99998 0.99999
Table 2.3: The seven most significance components found applying PCA on Danish ZCB from 1995–2006. Std. is the standard deviation, Pr.of Var. is the proportion of the total variance and Cum. Pr. is the cumulative proportion of the variance.
Figure 2.4 shows the three factor loadings corresponding to the three largest principal components in table2.3(the loadings are listed in appendixA.1). The loadings we recognize as the shift, steepness and convexity factors identified by Litterman & Scheinkman (1991).
From looking at figure 2.4 it can be observed that the the first factor, forms almost a horizontal line over the whole time period, excluding approximately the first five to six years. This corresponds to a change of slope for the first five years and a parallel shift for the rest of the maturity horizon. Although the slope in the first five to six years of the first factor is a deviation from what was observed in the other experiments mentioned in the introduction of section2.2, the horizontal line is dominant for the rest of the term structure and hence the factor is recognized as the level factor.
The second factor, the slope, which corresponds to a change of the slope for the whole term structure accounts for 4.72% of the total variation. It can be seen from the plot that the slope is decreasing as a function of maturity which fits the description of a normal yield curve. This is in accordance to the fact that the yield curve the period investigated was for most parts a normal yield cure with marginally diminishing yields. It is also worth mentioning that the slope for the first ten years is much steeper.
The third factor, can be interpreted as the curvature factor since positive changes in it cause a decrease in yield for bonds with short and long maturities but cause an increase in yield for medium length maturities.
2.3 Application of Factor Analysis 21
0 5 10 15 20 25 30
−0.6−0.4−0.20.00.20.40.6
1995−2006
Maturity (years)
Factor loadings
Factor 1 Factor 2 Factor 3
Figure 2.4: The first three factor loadings of the Danish yield curves, the values of the factor loadings can be seen in appendixA.1.
In reference to equation2.2the three factors level, slope and curvature should be sufficient to form an estimation variance-covariance matrixQˆsince they can explain the variance of the term structure up to 99.9%.
Although the first two factors are sufficient, from a statistical point of view, to describe the term structure accurately the third factor, which describes the curvature, is beneficial to include in a model since changes in the curvature of the term structure do occur. Therefore a model which does not take this change of term into account has a potential weakness of not capturing possible movements of the yield curve. Because of this we will use three factors throughout the report.
Example of the effects of factors on rates
Equation2.3describes the relationship a change of the factors has on the level of rates, redisplay here for convenience
∆rt=
k
X
j=1
βjt∆fj.
As an example lets see what effect a unit change (∆f1 = 1) of the level factor (j= 1) has on the ten year rate (t= 10).
j 1 2 3
βj,10 0.1870124 -0.0003624621 0.213623944
Table 2.4: The values ofβj,10for the first three factors, taken from appendixA.1.
From table 2.4 we have βjt = β1,10 = 0.1869201so a unit change in factor 1 causes 0.1869201 change in the ten year rate, which means that if the ten year rate is 5% a unit change in the level factor causes it to become 5.1869%.
In the same manner a unit change of three most significance factors (∆fj = 1) forj= (1,2,3), again for ten years means:
∆r10=
3
X
j=1
βj.10∆fj = (0.1870−0.0004 + 0.2136)·1 = 0.4002
meaning that a 5% ten year rates would become 5.4002% if a unit change oc-curred for all the factors.