There has been a phenomenal slide in the same is then extended to delve on the

interest rates which has been of interest to methodology to derive the sensitivity of a

the corporates and academicians alike. To portfolio of Fixed Income Securities to

put the facts in perspective, the following change in interest rates.

graph depicts the level of the yield curve on What are the traits associated with a brand

six sample dates:

like ‘Wheel’ of HLL? While the movement in the interest rates can

be loosely explained in terms of a fall of What are the features one desires when

around 400 bps. in the 10-year segment and buying a C-Class car? What are the

the like, one is tempted to enquire about a psychological traits that cause a person to

statistical framework to validate such contribute to a charity institution?

developments.

M a r k e t r e s e a r c h f i r m s d e v e l o p The following note attempts to explore the

questionnaires seeking responses on yield curve movement by introducing one of

personal background, social status, tastes, the statistical methods used extensively. The

financial strength, et. al. to identify various

YIELD CURVE ANALYSIS USING PRINCIPAL COMPONENTS

Yield Curve Movement

4

5

6

7

8

9

10

11

12

1 Year 5 Year 7 Year 10 Year 15 Year 20 Year 25 Year 30 Year

SpotR

ate

(%)

03-Apr-01

01-Oct-01

01-Apr-02

01-Oct-02

01-Apr-03

01-Aug-03

Figure. 1

*Swamynathan V.

*Swamynathan V. is Assistant Manager, Risk Management,The Clearing Corporation of India Limited.

factors that result in a person perceiving a security 7.37% GS 2014. This implies that:

brand, preferring a particular feature of a car while there may be 30 Factors (for the sake of

or contributing selflessly to a social cause. convenience) causing the daily movement in

the curve, there seems to be actually fewer Factor Analysis – a stream in Statistical

variables (Components) that drive the yield research – is used widely to collate and

curve. Hence, an effort has been made by us analyse responses and arrive at certain

to identify the underlying factors through common Factors which cause a particular

Principal Components Analysis (PCA). result. It’s a multi-variate statistical

technique for combining of seemingly The Principal Components Decomposition

diverse material responses into few unique, is a particular kind of Factor Model. Its role

unrelated (and perhaps abstract) underlying is to reduce the dimensionality of the subject

Factors. (in our case, the yield curve factors) i.e. to

reduce the number of underlying sources of The work on application of a technique of

uncertainty or market factors. These factors Factor Analysis to yield curve movements is

are not specified in advance; rather they are credited to Robert, Litterman and

derived out of the data analysed. Sheinkman. The trio published an article

‘Common Factors Affecting Bond returns’ The number of Principal Components

in the Journal of Finance (1991) employing derived from the yield curve analysis will be

the technique of Principal Component equal to the number of factors which drive

Analysis to the yield curve movements. the yield curve (i.e. 30 nos.). At first read, it

suggests that there may not be much The Indian Sovereign yield curve starts from

advantage using the PCA decomposition. the overnight rate and extends upto 30 years.

However, in the area of Finance, the data are Every tenure may be considered as a Factor

so highly correlated with each other that the which contributes towards the movement in

first few principal components explain most the yield curve; for the sake of convenience,

of the variability of the yield curve. we may state that there are only 30 factors

Thenceforth, one may work with the (corresponding to the 30 tenures) which

Principal Components which explain most effect the yield curve.

of the changes; than working with all the

But we know that the movement in the causal factors.

7.95% GS 2032 is related to the change in the Our analysis covers the change in yield curve

YTM of 6.01% GS 2028. The change in the since Jan-2000 till Feb-2004, a period

prices of the relatively illiquid securities in spanning just over 3 years. The data yield

the maturity bucket 7 to 12 years are driven curve has been developed by the Nelson-

by the change in the YTM of the benchmark Siegel model.

We initiated a process to identify the

Principal Components in the Matlab

environment by feeding data of the

yields of past 3 years. Instead of 30

yield points as factors as stated above,

for the sake of our analysis, data

relating to the yields of the tenures 0.5,

1, 5, 10, 15, 20, 25 and 30 years which is

9 in number were entered in the

software. The software has derived 9

components (as expected, it’s equal to

the number of tenure points of

analysis which is 9 in no.) which have

caused the movement in the yield curve. PCA also gives the contribution of The effect of each of the components on each variability by the principal components to tenure point of the yield curve is tabulated the yield movements.below:

It’s comforting to note that PC1 has PC1 contributes around 97.5% to the change

consistent negative values for all tenures. in the overall yield curve; followed by PC2 of

PC2 has negative values for the shorter around 2.23%. The third component

tenures; it increases for the medium to changes the curve to the extent of 0.1%;

longer terms. PC3 has a negative impact on remaining components have a negligible

the yield curve for the shorter and longer effect on the curve dynamics.

tenures while its effect is positive for the

medium tenures.

PC1 PC2 PC3 PC4 PC5 PC6 PC7 PC8 PC9

0.5 Year -0.3102 -0.5499 -0.3851 0.4135 -0.4438 -0.2705 0.1037 0.0341 0.0094

1 Year -0.3080 -0.5151 -0.2529 -0.2214 0.5785 0.4026 -0.1632 -0.0570 -0.0164

5 Year -0.3260 -0.2159 0.3800 -0.5558 0.0082 -0.4078 0.3825 0.2626 0.1107

7 Year -0.3356 -0.0894 0.4473 -0.1055 -0.3108 0.0352 -0.4286 -0.5479 -0.2941

10 Year -0.3437 0.0548 0.3926 0.3133 -0.1470 0.4673 -0.1271 0.4614 0.3969

15 Year -0.3469 0.2034 0.1587 0.4152 0.3081 0.0616 0.5272 -0.0821 -0.5073

20 Year -0.3451 0.2866 -0.0798 0.1934 0.3357 -0.4006 -0.1192 -0.3696 0.5754

25 Year -0.3423 0.3373 -0.2748 -0.0989 0.0471 -0.2668 -0.4807 0.4851 -0.3795

30 Year -0.3396 0.3707 -0.4268 -0.3772 -0.3745 0.3788 0.3054 -0.1866 0.1048

Table 1

Effect of Principal Components

-0.6000

-0.4000

-0.2000

0.0000

0.2000

0.4000

0.6000

0.5 1 5 7 10 15 20 25 30

Tenures (Yrs.)

Valu

es

PC1 PC2 PC3Figure 2

PC3 is the Curvature effect. It has an effect

similar to changing the curvature of the

curve of a day. This may be interpreted as the

curve taking a longer (or shorter) time to

reach the longest tenure rate.

The values of the first three Components as rd

on 23 Feb’04 are as follows:

We may, hence, conclude that the

dimensionality of the yield curve has even

got reduced from 9 to 2 (or 3). While we

haven’t assigned any name to the

components, it is interesting to realize that

PC1 effects a change in the level of the curve.

It has had a negative impact in the yield

curve in the past 3 years i.e. the fall in yield of The Principal Components derived are 1 year (of 400 bps.) through 30 years (of 500 linear combinations of the variables (30 bps.) is explained by PC1 (parallel yield curve tenures) under study. Each movement) to the extent of 97.59%. From component is, thus, arrived at by the Figure 2, the effect of PC1 is uniform across summation of the products of the Zero Rates the tenures indicating a parallel movement and the contribution of the component to in the curves. This is also diagrammatically the respective tenures.depicted in Figure 1.

To get an intuitive feel of the Principal PC2 can be considered as one that effects the Components, we have plotted the effects of a Slope of the curve. PC2 seems to cause a fall 100 bps. change in the Components on the

rdin the short term (due to its negative values yield curve of 23 Feb’04 in Figure 3. in the shorter tenures) and a rise in the

As is evident, PC1 results in a change in the longer term (due to the positive values in LEVEL of the curve; PC2 effects the SLOPE Table 1). This is evident from the Figure 2. of the curve; PC3 results in a change in the where a dip in rates is observed in the shorter CURVATURE of the curve.term and rise in the longer term.

Variance % Contribution Cum. Contribution

PC1 0.0037189000 0.9759 0.9759

PC2 0.0000849730 0.0223 0.9982

PC3 0.0000064787 0.0017 0.9999

PC4 0.0000002734 0.0001 1.0000

PC5 0.0000000364 0.0000 1.0000

PC6 0.0000000028 0.0000 1.0000

PC7 0.0000000001 0.0000 1.0000

PC8 0.0000000000 0.0000 1.0000

PC9 0.0000000000 0.0000 1.0000

Tenures Zero Rates

PC1 PC2 PC3

0.5 4.29% -0.3102 -0.5499 -0.3851

1 4.34% -0.3080 -0.5151 -0.2529

5 4.91% -0.3260 -0.2159 0.3800

7 5.20% -0.3356 -0.0894 0.4473

10 5.56% -0.3437 0.0548 0.3926

15 6.01% -0.3469 0.2034 0.1587

20 6.29% -0.3451 0.2866 -0.0798

25 6.48% -0.3423 0.3373 -0.2748

30 6.61% -0.3396 0.3707 -0.4268

-0.1664 0.0184 -0.0053

Table 2

Table 3

A Risk Manager would be interested in using for a 1 bp. change in each of the components

the concept of PCA to arrive at the volatility is computed. This is i.e. the change in the thof a portfolio of fixed income securities. price of the k bond for a basis point

thOne finds it convenient to deal with a change in the i Principal Component. This volatility matrix of 3 factors as compared to is similar to the Duration of a bond (which is a volatility matrix of 30 x 30. the change in a price of a security for a small

change in its yield). The values of are We have tried to implement the PCA model tabulated below for a 1 bp change in each of to a portfolio comprising of Zero Coupon the principal components.bonds - 25% of face value in a 1 year, 35% in 5

year and 40% in 20 year Zero coupon bonds It is notable that the effect of the change in rd

based on the yields of 23 Feb, 2004. The PC1 is all negative (LEVEL), change in slope

present value of the portfolio is as follows: due to PC2 (SLOPE) and change in curvature

caused by PC3 (Curvature).

The Variance of the portfolio is derived

from the table above using the formula:

Portfolio Variance = The sensitivity of the yield curve to changes

in the components is given by the respective Where is the Variance of the respective contributions to volatility of the Principal Principal Component.Components as listed in Table 2.

The above equation can be regarded as a Firstly, the change in the price of the bonds product of the Duration and Volatility of a

Component Effects

0.03

0.04

0.04

0.05

0.05

0.06

0.06

0.07

0.07

0.08

1 2 3 4 5 6 7 8 9Tenures

Rate

s(%

)

YC 23/02/04 YC shift due to PC1YC shift due to PC2 YC shift due to PC3

Zero Rates 4.34% 4.91% 6.29%

Present Value (Rs.) 95.84 95.32 94.08

Portfolio weight

0.25

0.35

0.4

kib

3

232

221

21 lblblb ppp ++

l

Figure 3

Table 4

bond to arrive at the risk exposure. be a classical case for implementing PCA.

Our results have identified three factors This results in the Variance of the portfolio

which can explain the change in the which can then be extended to arrive at the

complete yield curve. These factors can be Value-at-Risk figures. Assuming a Normal

used to compute the Variance of a portfolio.Distribution, the VaR can, then, be

computed by adopting the parametric One may extend the implementation of PCA

method. to Stress a portfolio by changing the

Principal Components with respect to the Conclusion

anticipation of yield changes in future.

The Indian yield curve movement seems to