Studying the behaviour of Indian Sovereign Yield Curve using Principal Component Analysis

7/29/2019 Studying the behaviour of Indian Sovereign Yield Curve using Principal Component Analysis

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INDIAN INSTITUTE OF MANAGEMENT CALCUTTA

Studying the Behaviour ofIndian Sovereign YieldCurve Using PrincipalComponent Analysis

Fixed Income Markets

Group 3

9/10/2013

This paper analyses the factors responsible for affecting term structure changes inthe context of Indian sovereign yield curve. We apply Principal Component Analysison our data to determine these factors.


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Contents

Introduction ........................................................................................................ 2

Previous Literature .............................................................................................. 4

Data Source ........................................................................................................ 6

Methodology ........................................................................................................ 7

Results and Analysis ........................................................................................... 9

Conclusion ........................................................................................................ 13

Bibliography ...................................................................................................... 13


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Introduction

The yield curve is sometimes called the term structure of interest rates, and is often

presented as a simple chart showing the annualized yield investors receive for

loaning out their money for different time periods. These can range from overnight

to as long as 30 years in the case of some government bonds. Most often, analysts

look at the yield curve in terms of sovereign securities, since there should be (in

theory) no problems with credit quality that could distort the picture.

Golaka Nath (2012) identifies the following advantages of estimating the yield curve

reasonably:

The yield curve serves as a benchmark in the economy as private corporateentities raise funds by paying a credit spread for the risk inherent in them;

Investors use the sovereign yield curves to demand an appropriate price fortheir investment risk;

Banks and other financial institutions use the yield curves to not only pricethe illiquid securities in their books but also match the duration of their

assets and liabilities;

Central banks use the information from secondary market yield curves tomonitor the policy interest rate synchronization with the economic effective

rate in the inter-bank market;

At macroeconomic level, the yield curve has a predictive power for the stateof economy.

The Indian sovereign bond market is underdeveloped in comparison with the

developed markets of the world. A reasonably good measure of the level of

development of the bond market is the outstanding debt to GDP ratio. The followinggraph illustrates the outstanding debt as a percentage of GDP for the top 10

economies of the world (March 2013).


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Figure C1: Outstanding Debt as a percentage of GDP

Source: BIS Quarterly Review, IMF World Economic Outlook, CBonds.info

As shown above, India ranks 7th (out of 10) as far as the sovereign debt to GDP

ratio is concerned. The Bank for International Settlements (BIS) holds the view that

sovereign debt can be a deterrent to growth only if this ratio exceeds the 85% level.

India, with a ratio close to 38%, has a long way to go in this regard.

Another measure of the level of development of the bond market is liquidity,

measured in terms of number of bonds traded versus the total bonds outstanding.

Barring the benchmark securities and a few other bonds, sovereign bonds in India

are characterized by market illiquidity, especially the bonds with higher tenors

which are often held-till-maturity by insurance companies and pension funds.

These shortcomings pose significant challenges towards estimating the yield curve

for India. One popular estimation model is the Nelson-Siegel (NS) functional form

which has been used by NSE since 1999 to calculate the spot interest rates. The

main purpose of this paper is to study the term structure dynamics and to figure

out the common factors of the Indian term structure and its volatility as it helps to

understand the pricing mechanism of various OTC and other underlying and

derivative products. Previous research studies have indicated that the three biggest

determinants of term structure are level, slope and curvature of the yield curve. We

will use Principal Component Analysis (PCA) to analyze the effect of these factors

on the term structure.

0%

50%

100%

150%

84%

65%

92%

230%

22%

52%

35%

60%

38%

11%

Government

Debt/GDP


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Previous Literature

Influence of monetary policy on the term structure of interest rates:

1. A very important work done in this field was Ang, A., J. Boivin, S. Dong, and R.

Loo-Kung (2010): Monetary Policy Shifts and the Term Structure. The existence of

historical shifts in monetary policy, or different monetary policy experiments,

provided an opportunity to statistically estimate the effects of changes in the policy

rule on the term structure.

According to this model, short rates move due to movements in the

i) output gap component, andii)

the inflation component

The study found that the endogenous response of inflation to past changes in

inflation loadings is an important component of how bond prices reflect monetary

policy risk under the risk-neutral measure. In contrast, policy shifts in output gap

loadings exhibit little time series variation, so almost all changes in monetary policy

stances have been done with respect to inflation.

If investors assigned no value to monetary policy shifts, then the slope of the yield

curve would have been, on average, up to 50 basis points higher than the data. The

term spread would have also been significantly more volatile without activist

monetary policy that is priced by investors.

This valuable contribution of monetary policy discretion is due to the risk discount

assigned by investors for monetary policy shifts.

2. According to the research paper done by "Ying He and Carlos Medeiros" on "An

Assessment of Estimates of Term Structure Models for the United States", 2011 -

This paper assesses estimates of term structure models for the United States. In

this context, it first describes the mathematics underlying both the Nelson-Siegel

and Cox, Ingersoll and Ross family of models and estimation methodologies. It then

presents estimations of some of these models within these families of modelsa

three-factor, yield-only Nelson-Siegel model, a four-factor Svensson model, and a

preference-free, two-factor CIR modelfor the United States from 1972 to mid

2011.. These estimations encapsulate the changes in expectations of short-term

future interest rates, while confirming that the yield-factors of the term structure of


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interest rateslevel, slope and curvaturesprovide a good representation of the

term structure.

Effects ofTreasury Yields, Inflation, Inflation Forecasts, and Inflation Swap

Rates on the term structure of interest rates:

1.The flagship journal Estimating Real and Nominal Term Structures usingTreasury Yields, Inflation, Inflation Forecasts, and Inflation Swap Rates, was

published in 2011 by Haubrich, J., G. Pennacchi, and P. Ritchken.

Under this theory, the term structures are driven by state variables that include

the short term real interest rate, expected inflation, a factor that models the

changing level to which inflation is expected to revert, as well as four volatility

factors that follow GARCH processes. The study found that allowing for GARCH

effects is particularly important for real interest rate and expected inflation

processes, but that long-horizon real and inflation risk premia are relatively stable.

2.According to the research paper done by "Adrian, T., and H. Wu" on The TermStructure of Inflation Expectations, Working Paper, Federal Reserve Bank of New

York", 2009

"They presented estimates of the term structure of inflation expectations which was

derived from an affine model of real and nominal yield curves. They found out that

model-implied inflation expectations can differ substantially from breakeven

inflation rates when market volatility is high. These differences are highly

correlated with market volatility measures such as the VIX equity implied volatility

index. Intuitively, as implied volatility increases, risk premia increase, and

breakevens tend to overpredict inflation expectations."

Some other useful studies worth mentioning are:

1. According to Buraschi, A., A. Cieslak, and F. Trojani (2010): Correlation Risk

and the Term Structure of Interest Rates,

1) Within the economy, the predictability of excess bond returns is supportedby the single forecasting factor of Cochrane and Piazzesi (2005).

2) The dynamic correlations of yields and the co-movement in their volatilitieslead to realistic properties of conditional hedge ratios between bonds


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3) Some state variableswhile loading weakly on yieldshave an economicallysignificant impact on the prices of interest rate caps, thus evoking the notion

of un-spanned factors.

2. According to Fontaine, J.S., and R. Garcia (2011): Bond Liquidity Premia,

The pattern across interest rate markets and credit ratings is consistent with

accounts of flight-to-liquidity events. However, the effect is pervasive even in

normal times. The evidence points toward the importance of aggregate liquidity and

aggregate liquidity risk compensation in asset pricing.

They find that measures of changes in the stock of money and measures of the

availability of funds in the banking system are important determinants of our

measure of aggregate liquidity. To a lesser extent, the liquidity factor varies

positively with transaction costs and aggregate uncertainty.

Data Source

We performed our analysis on data collected from Bloomberg. PCA was applied to

monthly yield changes data from Jan08 to Sep13 (69 data points) for the set of

maturities given in the table below:

Table T1: Descriptive Statistics of Historical Term Structure of Interest Rates (in %)

3M 6M 1Y 2Y 3Y 4Y 5Y 7Y 10Y 15Y 20Y 30Y

Mean 6.91 7.12 7.01 7.29 7.52 7.71 7.83 7.96 7.92 8.49 8.58 8.91

StDev 1.85 1.81 1.59 1.18 0.94 0.79 0.74 0.61 0.71 0.48 0.49 0.56

Max 10.42 10.53 9.87 9.33 9.32 9.31 9.29 9.33 9.13 10.07 10.07 10.71

Min 3.53 3.78 4.06 4.81 5.30 5.75 5.73 6.19 5.79 7.07 7.29 7.62

Median 7.33 7.58 7.62 7.70 7.76 7.90 7.95 8.02 8.09 8.53 8.61 9.00

Source: Bloomberg

Also, the following figure illustrates the historical G-Sec rates in India for certain

key maturities using the data collected. Implications of T1 and C2 are discussed in

a later section.


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Figure C2: Historical G-Sec Rates in India

Source: Bloomberg

Methodology

Principal Component Analysis: An Overview

PCA is a useful statistical technique that can be used for finding patterns in data of

high dimension. These findings can then be used to highlight similarities and

differences between different data points. One chief advantage of PCA is that data

can be compressed after identification of patterns without much loss of

information. Since the PCA model explicitly selects the factors based upon their

contributions to the total variance of interest rate changes, it may help in hedging

efficiency when using only a small number of risk measures. In general, data

reduction and summarization is popularly done using a technique called factor

analysis. Factoranalysis is a statistical method used to describe variability among

observed, correlated variables in terms of a potentially lower number of unobserved

variables called factors. Factor analysis is done in the following circumstances:

4.00%

5.00%

6.00%

7.00%

8.00%

9.00%

10.00%

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r-

J

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r-

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-

-

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r-

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-

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r-

J

-

-

1-Y

3-Y

5-Y

10-Y

20-Y


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To identify underlying dimensions, or factors, that explain the correlationsamong a set of variables

To identify a new, smaller, set of uncorrelated variables to replace theoriginal set of correlated variables in subsequent multivariate analysis

(regression or discriminant analysis)

To identify a smaller set of salient variables from a larger set for use insubsequent multivariate analysis

Mathematically, a factor model with n factors can be represented as:

where,

Xi = ith standardized variableRi1 = standardized multiple regression coefficient of variable i on common factor j

Fi = common factor i

Ci = standardized regression coefficient of variable ion unique factorj

Ui = the unique factor for variable i

The common factors themselves can be expressed as linear combinations of the

observed variables:

where,

Fi = estimate of the ith factor

Wi = Weight or factor score coefficient

k = number of variables

In PCA, factor weights are computed in order to extract the maximum possible

variance, with successive factoring continuing until there is no meaningful variance

left.

PCA and the Indian Sovereign Term Structure

PCA can be successfully applied to determine the chief factors affecting movements

in the term structure. Thus, the yield curve shifts can be assumed to be a function

of different realizations of principal components. As mentioned before, height,

curvature and slope are considered to be the three major principal components


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affecting these shifts. According to PCA, not all components have equal significance

in explaining changes. The first component explains the maximum percentage of

the total variance of interest rate changes. The second component is linearly

independent (i.e., orthogonal) of the first component and explains the maximum

percentage of the remaining variance, the third component is linearly independent

(i.e., orthogonal) of the first two components and explains the maximum percentage

of the remaining variance, and so on. If yield curve shifts result from a few

systematic factors, then only a few principal components can capture yield curve

movements. Moreover, since these components are constructed to be independent,

they also help in simplifying the task of managing interest rate risk. The principal

components with low eigenvalues make little contribution in explaining the interest

rate changes, and hence these components can be removed without losingsignificant information. This not only helps in obtaining a low-dimensional

parsimonious model, but also reduces the noise in the data due to unsystematic

factors (Nawalkha, Soto and Beliaeva).

Results and Analysis

Table T1 on descriptive statistics shows that the difference between maximum and

minimum yields is far higher in the short-term than the long-term. This is because

short-term rates are guided by monetary policy rates and liquidity factors. For

instance, RBI recently increased short-term borrowing rates to curb the downfall of

rupee. In the aftermath of financial crisis in 2007-08, RBI supported the market by

infusing huge liquidity along with bringing down policy Repo rate and reserve ratios

for the Banks. This helped in lower interest rates at the shorter end but the longer

end remained more stable. Similarly, volatility (measured by standard deviation) isfar higher in the short-term than in the long-term. Again, this is because short-

term rates are the most affected by monetary policy changes. Hence, they

experience greater volatility than long-term rates. These observations can also be

observed through Figure C2. Both range and variability of interest rates fall sharply

with an increase in maturities.

Principal Component Analysis was applied on the different maturities specified

earlier from Jan08 to Sep13. The following table illustrates key results obtained.


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Table T2: Eigenvalues of the Covariance Matrix

Factors Eigenvalue Difference Proportion Cumulative

F1 9.3366 7.1760 0.7781 0.7781

F2 2.1607 1.8945 0.1801 0.9581

F3 0.2662 0.1482 0.0222 0.9803

F4 0.1180 0.0690 0.0098 0.9901

F5 0.0490 0.0063 0.0041 0.9942

F6 0.0427 0.0301 0.0036 0.9978

F7 0.0126 0.0075 0.0011 0.9988

F8 0.0051 0.0009 0.0004 0.9993

F9 0.0042 0.0015 0.0004 0.9996

F10 0.0028 0.0010 0.0002 0.9999

F11 0.0018 0.0015 0.0001 1.0000

F12 0.0002 0.0000 1.0000

Source: Bloomberg, Own Calculations

Table T-2 shows that the first three components account for 98.03% of the total

variance of interest rate changes. Component F1 accounts for 77.81% of the

variance, while F2 and F3 account for 18.01% and 2.22% respectively. Thus, F1,

F2, and F3 combined can be used to estimate the changes in term structure in

India with reasonable degree of confidence. Figure C3 shows the impact of these

three components on the yield curve.

Figure C3: Impact of F1, F2 and F3 on yield curve


F1 represents a parallel shift in the yield curve. Thus, it is also called the heightfactor. F2 represents a change in the steepness, and is called the slope factor. F3

-0.600

-0.400

-0.200

0.000

0.200

0.400

0.600

0.800

1.000

1.200

3m 6m 1y 2y 3y 4y 5y 7y 10y 15y 20y 30y

Factor 1

Factor 2

Factor 3


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affects the curvature of the yield curve by inducing a butterfly shift. It is called the

curvature factor. Thus, as discussed before, height, slope and curvature explain

about 98% of the shifts in yield curve. The height factor dominates the rest and its

coefficients are always positive. The slope factor dominates the remaining portion

and its coefficients turn from negative to positive with maturity. The curvature

factor accounts for the least of the 3 biggest factors. It carries a negative value

initially, becomes positive towards the middle, and again becomes negative at the

end part of the yield curve.

Table T3: Eigenvectors of 3 principal components

F1 F2 F3

3m 0.875 -0.451 0.118

6m 0.891 -0.431 0.098

1y 0.908 -0.401 0.082

2y 0.954 -0.278 0.041

3y 0.983 -0.157 0.008

4y 0.993 -0.056 -0.047

5y 0.988 0.043 -0.122

7y 0.981 0.117 -0.059

10y 0.944 0.101 -0.274

15y 0.655 0.670 0.322

20y 0.641 0.741 0.099

30y 0.650 0.695 -0.159


Table T3 shows that for a 10-year bond, a unit increase in factor 1 causes the 10-

year rate to increase by 0.944%. A unit increase in all 3 factors causes 10-year rate

to increase by 0.774%.

Figure C4 plots 2 factors against each other based on correlations with yield

changes in different maturities. On the left, F1 vs F2 plot shows high correlation of

F2 with yields of higher maturity securities. F1 has high correlation with yields of

all maturities. On the right, F5 shows hardly any correlation with any security.

Similar plots are observed for factors F6 to F12. Hence, this plot further proves the

dominance of F1, F2 and F3 as major principal components in the analysis of the

sovereign yield curve.


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Figure C4: Plots of Proportion explained by factors for Different Maturities


A scree plot is a plot of the Eigenvalues against the number of factors in order of

extraction. Experimental evidence indicates that the point at which the scree

begins denotes the true number of factors. In figure C5, one may choose to

consider 3 or more factors using the scree plot given. This is consistent with our

findings using the eigenvalue criterion.

Figure C5: Scree Plot of Eigenvalues and Cumulative Variability


0

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F1 F2 F3 F4 F5 F6 F7 F8 F9 F10 F11 F12

Cumulativevariability

(%)

Eigenvalue

axis

Scree plot


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Conclusion

A wide variety of research has been conducted to determine the factors contributing

to changes in yield curve. In this study, we used Principal Component Analysis to

identify these factors in the context of the sovereign yield curve of India. Our

results are in line with factors suggested by various research studies. Height, slope

and curvature are the three biggest factors responsible for changes in the yield

curve. Height accounts for 77.81% of the variability, while slope and curvature

account for 18.01% and 2.22% respectively. Together these three account for

98.04% of the total variability. Hence, one can estimate yield curve changes with

great certainty using information on these 3 changes.

Bibliography

Adrian, T., and H. Wu (2009): The Term Structure of Inflation Expectations, Working Paper, Federal Reserve Bank of New York.

Golaka Nath (2012): Estimating term structure changes using principalcomponent analysis in Indian sovereign bond market

A Tutorial on Principal Components Analysis (2002): Lindsay I Smith Fontaine, J.-S., and R. Garcia (2011): Bond Liquidity Premia, Review of

Financial Studies, forthcoming

Buraschi, A., A. Cieslak, and F. Trojani (2010): Correlation Risk and the TermStructure of Interest Rates, Working paper, University of Lugano.

Haubrich, J., G. Pennacchi, and P. Ritchken (2011): Estimating Real andNominal Term Structures using Treasury Yields, Inflation, Inflation Forecasts, and

Inflation Swap Rates, Working Paper, Federal Reserve Bank of Cleveland.

Ang, A., J. Boivin, S. Dong, and R. Loo-Kung (2010): Monetary Policy Shifts andthe Term Structure, Review ofEconomic Studies, forthcoming.

Studying the behaviour of Indian Sovereign Yield Curve using Principal Component Analysis

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