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7/29/2019 Studying the behaviour of Indian Sovereign Yield Curve using Principal Component Analysis
1/14
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%
-
-
r-
J
-
-
-
r-
J
-
-
-
r-
J
-
-
-
r-
J
-
-
-
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
Source: Bloomberg, Own Calculations
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
Source: Bloomberg, Own Calculations
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
Source: Bloomberg, Own Calculations
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
Source: Bloomberg, Own Calculations
0
20
40
60
80
100
0
1
2
3
4
5
6
7
8
9
10
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.