We now turn to illustrating the use of principal components in describing changes in the term structure of interest rates. Table 8.1 shows a hypothetical correlation matrix of the changes in the yields of zero-coupon bonds of 10 different maturities ranging from three months to 30 years, along with their annual volatilities (in basis points). These yields are naturally thought of as either the yields on zero-coupon government bonds, or as zero-coupon yields constructed from benchmark interest-rate swap quotes. They are the under- lying basic risk factors, which will be further summarized through the use of principal components. Often, when the underlying risk factors are the yields of zero-coupon bonds, they are called key rates (Ho 1992), and we use this terminology below.
Through this choice of risk factors, the value-at-risk will fail to include the risks of changes in credit or liquidity spreads. If spread and liquidity risks are important for the portfolio, one would need to expand the corre- lation matrix to include the correlations and volatilities of these spreads and also to measure the exposures of the various instruments to these spreads.
Table 8.2 shows the covariance matrix corresponding to the correla- tions and volatilities in Table 8.1. One feature of the choice of maturi- ties in Tables 8.1 and 8.2 is that the “short end” of the term structure is not captured in great detail, the three-month maturity being the only maturity of less than one year. This choice is reasonable for many port- folio management applications involving measuring the risk of portfo- lios that include medium- and long-term bonds, but would be less useful for bank portfolios in which short-term instruments play a more impor- tant role.
⌺–iI
( )i = 0 i = 1 2 . . . , K,, ,
p = V′x, p = (p1 p2 . . . pK)′
V = (1 2 . . . K) K×K
⌺
TABLE 8.1 Hypothetical volatilities and correlations of changes in yields of zero-coupon bonds
Maturities 3 mos.
1 year
2 years
3 years
5 years
7 years
10 years
15 years
20 years
30 years Volatilities
(basis
points) 72 98 116 115 115 112 107 102 100 85
Correlations:
3 mos.
1 year
2 years
3 years
5 years
7 years
10 years
15 years
20 years
30 years 3 mos. 1.00 0.75 0.65 0.64 0.55 0.52 0.50 0.46 0.43 0.41 1 year 0.75 1.00 0.85 0.84 0.77 0.74 0.71 0.68 0.65 0.61 2 years 0.65 0.85 1.00 0.99 0.97 0.95 0.92 0.88 0.86 0.83 3 years 0.64 0.84 0.99 1.00 0.98 0.97 0.94 0.91 0.89 0.86 5 years 0.55 0.77 0.97 0.98 1.00 0.99 0.97 0.95 0.93 0.91 7 years 0.52 0.74 0.95 0.97 0.99 1.00 0.98 0.97 0.96 0.95 10 years 0.50 0.71 0.92 0.94 0.97 0.98 1.00 0.99 0.98 0.97 15 years 0.46 0.68 0.88 0.91 0.95 0.97 0.99 1.00 0.99 0.98 20 years 0.43 0.65 0.86 0.89 0.93 0.96 0.98 0.99 1.00 0.99 30 years 0.41 0.61 0.83 0.86 0.91 0.95 0.97 0.98 0.99 1.00 TABLE 8.2 Hypothetical covariances of changes in yields of
zero-coupon bonds Maturity 3
mos.
1 year
2 years
3 years
5 years
7 years
10 years
15 years
20 years
30 years 3 mos. 5184 5292 5429 5299 4554 4194 3852 3378 3096 2509 1 year 5292 9604 9663 9466 8678 8123 7445 6797 6370 5081 2 years 5429 9663 13,456 13,207 12,940 12,342 11,419 10,412 9976 8184 3 years 5299 9466 13,207 13,228 12,962 12,489 11,564 10,675 10,234 8410 5 years 4554 8678 12,940 12,962 13,226 12,748 11,934 11,144 10,694 8898 7 years 4194 8123 12,342 12,489 12,748 12,552 11,749 11,079 10,754 9037 10 years 3852 7445 11,419 11,564 11,934 11,749 11,451 10,804 10,487 8818 15 years 3378 6797 10,412 10,675 11,144 11,079 10,804 10,405 10,098 8498 20 years 3096 6370 9976 10,234 10,694 10,754 10,487 10,098 10,000 8413 30 years 2509 5081 8184 8410 8898 9037 8818 8498 8413 7231
Table 8.3 shows the factor loadings of the principal components, that is, the eigenvectors. These explain the changes in yields through equation (8.12); for example, the entries in the column headed “1” indicate the effect of the first principal component on the 10 different yields. As pointed out above, the sign of each vector i is arbitrary, implying that switching the signs of the entries in any column of Table 8.3 (or any set of columns) would yield an equivalent decomposition.
Table 8.4 shows variances of the principal components (the eigenval- ues), the fraction of the total variance explained by each of the principal components, the percentage of the total variance explained by each, and the cumulative percentages. The first principal component explains 89.24% of the variance, the first two together explain 96.42%, and the first three together explain a total of 98.15%. This finding that the first three principal components together explain more than 95% of the total variance of yields is a common one, and as a result users of principal com- ponents often restrict attention to the first three components. A second con- sideration reinforcing this is that the first three principal components have intuitive interpretations as corresponding to changes in level, slope, and curvature, discussed next.
Figure 8.1 shows the term structure shifts due to 100 basis point changes in each of the first three principal components. The plain solid line shows the assumed initial yield curve, rising from a yield of 6% for the shortest maturity to a 7% yield at the longest. The dashed line above it
FIGURE 8.1 Yield curve shifts due to first three principal components 5.0
5.5 6.0 6.5 7.0 7.5
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 Maturity (years)
Yield of zero-coupon bond(percent per year)
100 b.p. shock to 1st prin. comp.
Initial yield curve 100 b.p. shock to 3rd prin. comp.
100 b.p. shock to 2nd prin. comp.
127 TABLE 8.3Factor loadings of principal components (eigenvectors) of covariance matrix Factors Loadingsi of Principal Components (Eigenvectors) Maturity123456789 3 mos.0.1390–0.5369–0.68640.46020.0148–0.06790.0264–0.04790.0077– 1 year0.2563–0.58320.0122–0.75860.0393–0.08080.07690.02980.0444– 2 years0.3650–0.22960.37060.2343–0.14980.58340.2813–0.2099–0.36860 3 years 0.3669–0.16490.26370.19010.12320.0415–0.68540.29330.18300 5 years0.36940.02820.26560.1957–0.0893–0.57070.3722–0.32610.34590 7 years0.36030.12010.11410.11570.5593–0.22550.02130.1740–0.2062– 10 years0.34120.1925–0.1193–0.0102–0.58930.12580.11130.48160.3358– 15 years0.32000.2542–0.2289–0.1503–0.3460–0.3236–0.2429–0.1017–0.66390 20 years0.30930.2973–0.2755–0.18060.12380.3246–0.2699–0.61640.3356– 30 years0.25770.2940–0.3094–0.12160.39690.21060.40170.3285–0.01720
shows the effect of a 100 basis point realization of the first principal com- ponent, with the other principal components fixed at zero. The changes in the 10 key rates are computed using (8.12) with p1 100 basis points, the elements of set equal to the values in the column headed “1” in Table 8.3, and p2 p3 . . . pK 0. Doing this results in a relatively small change of x1 0.1390×100 basis points 13.90 basis points for the three-month maturity and larger changes in the range of 25.63 to 36.94 basis points for the other key rates. The changes for the maturities in between those of the key rates are then computed by linear interpolation;
for example, the change for the six-month maturity was interpolated from the changes for the three-month and one-year maturities. These shifts were then added to the initial yield curve to create the dashed curve labeled
“100 b.p. shock to 1st prin. comp.” This curve illustrates that the first prin- cipal component can be interpreted as corresponding with a shift in the overall level of interest rates. Because the signs of the factor loadings are arbitrary, this shift can be either an increase or decrease in rates.
The two curves labeled “100 b.p. shock to 2nd prin. comp.” and “100 b.p.
shock to 3rd prin. comp.” were computed similarly, except that p2 and p3 were set equal to 100 basis points and 2 and 3 were used with elements set equal to the values in the columns headed “2” and “3” in Table 8.3. Due to the pattern of negative and positive factor loadings on the second principal component, the curve corresponding to the second principal component is below the initial TABLE 8.4 Variance explained by principal components
Principal Component
1 2 3 4 5 6 7 8 9 10
Variance
(eigenvalues i) 94900 7636 1834 1338 203.1 186.9 122.1 70.8 36.7 11.0 Annual volatility
(basis points) 308.06 87.38 42.82 36.57 14.25 13.67 11.05 8.41 6.05 3.32 Monthly
volatility
(basis points) 88.93 25.22 12.36 10.56 4.11 3.95 3.19 2.43 1.75 0.96 Percentage of
variance
explained 89.24 7.18 1.72 1.26 0.19 0.18 0.11 0.07 0.03 0.01 Cumulative
percentage 89.24 96.42 98.15 99.41 99.60 99.77 99.89 99.96 99.99 100
=
1
= = = =
= =
yield curve for the short maturities and above it for the long maturities, illus- trating that the second principal component can be interpreted as correspond- ing to a change in the slope of the yield curve. Due to the pattern of negative and positive factor loadings on the third principal component, the curve corre- sponding to the third principal component is below the initial yield curve for the very shortest maturities, above it for intermediate maturities, and then again below it for the long maturities. This pattern is often called a change in the curvature of the yield curve.
One caveat worth mentioning is that Figure 8.1 overstates the relative importance of the second and third components, because the shocks to the principal components p1, p2, and p3 were each set equal to the same value of 100 basis points. Table 8.4 shows that, on a monthly basis, the volatility (stan- dard deviation) of the first principal component p1 is 88.93 basis points, so a change of 100 basis points is only standard deviations and is not unusual. However, the monthly volatilities of the second and third princi- pal components are 25.22 and 12.36 basis points, respectively, so changes of 100 basis points correspond to 100 25.22 3.97 and 8.09 standard deviations, respectively. Thus, the relative importances of the second and third factors are slightly more than one-fourth and one-eighth as large as the differences between the curves in Figure 8.1 suggest.