Volatile markets and economic environments can significantly distort the shape and smoothness of yield curve movements. This study explores the influence of movements in United States interest rates on South African interest rates. This study aims to identify the main underlying movements present in the United States and South African yield curves and to further determine the dominant factors that are responsible for driving South African interest rate movements. The principal settings for the study were the United States and South African markets representing, respectively, a developed and developing market. Principal component analysis was used to discern the major drivers of developing and developed market interest rates. The findings show that the principal component analysis technique is able to effectively classify and quantify the movements of yield curves across both markets in terms of three main factors, namely level, slope and curvature shifts. During certain periods, South African yield curve changes were largely driven by variations in United States interest rates and the rand/dollar exchange rate. Results also demonstrated that a volatile market and economic environment can significantly distort the shape and smoothness of yield curve movements.
Explaining the volatility of financial assets has become a crucial part of portfolio management. Fund managers and investors retain a substantial amount of risk when they do not understand the volatility of their assets. In the fixedincome market, the main contributing factors to the volatility in bond prices are yield curves. Yield curves illustrate the relationship between bond yield rates at particular times and bond maturities with the same credit quality. This relationship is commonly known as the term structure of interest rates. Portfolio managers and investors seek to quantify yield curve changes to protect, or immunise, the fixedincome assets in their portfolios. The absolute level of the yield curve is generally considered less important to investors than changes of the curve (Baygun, Showers & Cherpelis
Yield curve shifts occur when the market revises their expectations on future interest rates. These rates fluctuate in response to variations in inflation expectations, risk premium and output growth. Empirical studies have revealed that yield curve movements can be characterised in terms of three factors, which are often termed as the ‘level’, ‘slope’ and ‘curvature’ factors. These terms describe how yield curves shift or change shape in response to a shock.
Yield curve shifts.
A parallel or level shift arises when interest rates across all maturities change by the same amount. Slope shifts develop when shortterm expectations change but longterm rates remain the same, or vice versa (Phoa
Principal component analysis (PCA) is a common approach considered to not only classify and quantify these yield curve movements but also to further manage the risk arising from groups of highly correlated market variables. In simple terms, the technique takes historical data on changes in market variables and attempts to define a set of uncorrelated components or factors that can effectively explain these movements in the most economical manner (Hull
Principal component analysis quantitatively describes yield curve variations by determining the percentage of variance each factor contributes in explaining these movements. This analysis turns out to be remarkably successful because the technique considers the variability present in the entire data set (Hull
This study employs the PCA technique to identify and quantitatively describe the main underlying movements present in yield curves. An established market and an emerging market will be used for this analysis, namely the United States and South Africa, respectively. This analysis also aims to relate fluctuations in the variance explained by each of these factors to macroeconomic influences. In addition, because PCA has the ability to explain a large proportion of yield curve variations in terms of only three uncorrelated factors, the technique allows regression analysis to be performed. In doing so, this study further attempts to ascertain the dominant factors responsible for driving South African interest rate movements.
The rest of this article proceeds as follows: Section ‘Literature review’ explores the relevant literature, section ‘Data’ discusses the data, and section ‘Methodology’ examines the PCA methodology. The results obtained are reviewed and discussed in section ‘Analysis’, and section ‘Conclusions and suggestions for future research’ concludes.
The quantitative decomposition of yield curve changes into submovements that can easily be described and analysed by investors and analysts is of fundamental importance in today’s financial markets (Colin
Traditional interest rate risk management focuses on duration, a technique which assumes that only parallel yield curve shifts are important (Phoa
Litterman and Scheinkman’s (
Zerocoupon yields can be estimated using functionbased methods including the form proposed by Nelson and Siegel (
Diebold, Ji and Li (
Litterman and Scheinkman (
Although the threefactor model performs admirably, the factors still have a degree of correlation between them (Su & Knowles
An alternate method to model yield curve dynamics is PCA. This method identifies the fewest number of new explanatory variables to account for almost all the variability of yield curve changes (Su
Johnson (
Principal component analysis works well in analysing yield curve changes; using three principal components explains up to 95% of bond yield variance, for example in the US market (Knez, Litterman and Scheinkman [
Phoa (
Phoa (
In addition, macroeconomic events could also illuminate the explained variance of the level, slope and curvature during historical periods. Rakotondratsimba and Jaffal (
Morita and Bueno (
Principal component analysis thus appears to enjoy an advantage over traditional durationbased approaches and factor analysis. PCA could identify and explain the fundamental variation of yield curves across a range of countries and historical periods. However, a research gap was identified with regard to explaining the relationship of yield curve changes between an established market and an emerging market in terms of principal components and macroeconomic factors across a range of historical time periods.
According to Su (
To determine if any relationship exists between US and South African yield curves, the following rates were selected:
the JPMorgan Emerging Market Bond Spread rate (JPEMBS);
the South African Rand to US Dollar exchange rate;
the US Federal Funds rate;
South African repo rate.
The JPEMBS provides an indication of the yield spread between the JPMorgan Emerging Markets Bond rate (denominated in US dollars) relative to the US Treasury bond rate. It could therefore identify relationships between the US market and an emerging market, such as South Africa. Monetary policies, such as the Federal Funds rate and the repo rate, will also affect yield curves and therefore it has been selected to aid in the analysis.
The respective swap curve rates were extracted from Bloomberg. Different time periods were used to perform certain investigations under specific economic conditions or because of restrictions with regard to the acquisition of data. The time period chosen for the analysis ranged between 1990 and 2014 so as to compare time periods of prefinancial crisis and postfinancial crisis. One of the main obstacles in a quantitative analysis was acquiring reliable and complete data. US sovereign curves were used, comprising six maturities between 1990 and 1999 as this time period resulted in a complete data set for various maturities. The data comprised 1year, 2year, 3year, 5year, 10year and 30year yield to maturities. Thereafter, US swap data spanning from 2000 to 2014 was extracted because of the availability of data. These comprised 1year, 2year, 3year, 4year, 5year, 7year, 10year, 12year, 15year and 20year maturity rates. According to Phoa (
Maturities comprising shortterm and longterm maturities were acquired to approximate the shape of yield curves. Maturities of less than a year were avoided to remove the idiosyncratic risk associated with those maturities. These data were extracted at a daily frequency. The daily frequency is also beneficial when performing an analysis on a short time period. The JPEMBSs were acquired from INet Expert, consisting of daily rates ranging between 2000 and 2014.
Additional rates such as the South African Rand to US Dollar exchange rate, repo rate and the Federal Funds rate were acquired from FRED Economic Data (FRED
Data preparation is a cumbersome task of structuring a large data set into a particular structure to conduct quantitative analysis. The data structure involved merging various maturity rates according to time (24h period) and any missing maturity rates with respect to time were linearly interpolated. Daily rates vary by a small amount and as this analysis uses daily data, linear interpolation should closely approximate the true value. Python’s
Oneyear and 30year swap rates for both countries.
United Sates treasury rates.
Principal component analysis summarises complex data sets by creating new, artificial variables called principal components. These are linear combinations of the original data and are constructed by exploiting the variability (i.e. spread) of each variable and the correlation between the variables. In addition, each of these new variables are assembled such that they are uncorrelated with one another. The components are derived by finding the eigenvectors and corresponding eigenvalues of the correlation or the covariance matrix between the variables in the original data set. These components effectively represent a rotation of the original data onto new axes. This new set of axes reveal more easily discernible patterns within the data.
The components are ordered according to the magnitude of their variance. The component with the largest corresponding eigenvalue is the first principal component – the linear combination that encapsulates most of the data variability. This component represents a rotation of the data along the axis representing the largest spread. The component with the second largest variance explains most of the remaining variability while being uncorrelated (perpendicular) to the first component. The same concepts apply to the remaining principal components. In the last step of this technique, PCA determines the number of ‘real’ dimensions present in the data. Eigenvectors are chosen such that the sum of the variances explained by these eigenvectors is sufficient to explain most of the variation present within the original data.
The theoretical insights gained from PCA can be used to make sense of yield curve dynamics. Eigenvectors and their corresponding eigenvalues are derived from a covariance matrix based on timeseries data of daily or monthly interest rate changes at each reference maturity.
The following sections explain the algebraic and graphical interpretation of PCA.
Suppose there are
Let
Principal component analysis is a decomposition of the covariance matrix.
Because
Because Λ is a diagonal matrix, the rotated variables are uncorrelated. The diagonal entries of Λ are the variances of the rotated variables. From ∑ = PΛP^{T}, Λ = P^{T} ∑ P. The trace of Λ is tr(Λ) = tr(P^{T} ∑ P) = tr(∑ P^{T}P)= tr(∑).
Let C be a (
Accordingly, (X –
A dimensionality reduction from
Suppose there exists a history of yield curves, each curve can be represented as a vector of
Projection of yield curve shifts across 10 maturities onto a threedimensional space.
Three principal components of yield curve shifts for entire sample.
The three most important factors driving movements in United States swap rates: 2000–2014.
The three most important factors driving movements in South African swap rates: 2000–2014.
The simulation results agree with the literature. Firstly, PCA can effectively derive three components that can explain most yield curve variations.
Threedimensional representation of annual US principal components for the period 1990–1999: (a) US annual PC1; (b) US annual PC2; and (c) US annual PC3.
Threedimensional representation of annual US principal components for the period 2000–2014: (a) US annual PC1; (b) US annual PC2; and (c) US annual PC3.
The form of the slope and curvature components during the 1990s are significantly smoother when compared with the components post2000. This could be because the 1990s were characterised by strong economic growth that resulted from a combination of rapid technological advancements and sound central monetary policy. In contrast, the market suffered with multiple crisis periods post2000 (e.g. the dotcom and credit crisis). The economic instability during this period is reflected in the variation of the Federal Funds rate, as shown in
United States federal funds rate post2000.
Variations in percentage of variance explained for each United States component for period: 2000–2014.
The changes in variance over the sample period, explained by the components, may be related to macroeconomic factors. Such an explanation may not be sensible during this unstable period, where movements in the term rates might have been attributable to many unexplained factors. However, extreme monetary policy changes might have been a possible factor contributing to increased slope shifts during certain years.
The slope factor is slightly more prominent post2010, a period devoid of monetary policy shocks. This suggests that during this quantitative easing phase, the slope factor is now rather explained by movements in longterm interest rate expectations.
Threedimensional representation of annual South African principal components for the period 1995–1999. SA, South Africa: (a) SA annual PC1; (b) SA annual PC2; (c) SA annual PC3.
Threedimensional representation of annual South African principal components for the period 2000–2014. SA, South Africa: (a) SA annual PC1; (b) SA annual PC2; (c) SA annual PC3.
Variations in percentage of variance explained for each South African principle component: 2000–2014.
Although the level shift is still dominant, South Africa’s level factor plays a minor role in comparison with the US level factor. As a result, the slope and curvature factors are observed to play a greater role in driving South African interest rate movements. This may be attributable to greater market volatility present in emerging markets as they are more vulnerable to external shocks. In addition, South Africa’s inflation expectations are more erratic than market inflation expectations in the United States.
Over the period 2000–2004, slope shifts explain a relatively high proportion of interest rate movements. This may be because of severe changes (substantial increases and decreases) in the repo rates (which in turn alters inflation expectations). These extreme changes are depicted in
South African repo rate post2000.
Based on the findings displayed in
The previous section illustrates the effect of US interest rates on the South African currency and interest rates. A relationship might thus exist between the changes of US yield curves and the changes of South African yield curves. The problem with performing a correlation and regression analysis on yield curve changes is that maturities in each data set are highly correlated. A few maturities can instead be selected and these used for correlation and regression analysis. However, those maturities will only explain a limited amount of variance. The advantage of PCA is that it provides a reduced number of uncorrelated principal components that explains most of the variance in the entire data set. The scores of the uncorrelated principal components can therefore be used in a correlation and regression analysis.
The JPEMBS could identify relationships between the US market and an emerging market, such as SA. If the spread increases, the risk premiums of emerging markets are increasing. This could be because of riskaverse investors taking investments out of emerging markets and placing them in established markets. It has also been suggested (Morita & Bueno
The following variables were used in the correlation and regression analysis:
scores of principal components 1, 2 and 3 of US yield curve changes;
scores of principal components 1, 2 and 3 of South African yield curve changes;
changes in the JPEMBS rate;
changes in the Federal Funds rate and repo rate;
percentage changes in the South African Rand to US Dollar exchange rate.
To build the multivariate regression models, a correlation analysis is first performed. Note that performing correlation over daily data is not sensible as correlation tends to underperform when applied to highfrequency signals. In addition, daily data are prone to more noise, which can affect the accuracy of correlation results considerably. To overcome these limitations, principal component scores were derived based on monthly mean swap rate changes. Monthly mean values are considered ahead of values occurring on the first or last day of each month as these values may themselves be outliers. Using mean values of each month allows average shifts to be encapsulated over that period. This in turn alleviates the problem of noise. Furthermore, using monthly values will overcome the issue of selecting accurate lag or lead values.
Sampling data using monthly mean values encapsulates the daily observations and hence would be a good approximation to an analysis done on daily observations. Accordingly, for the correlation and regression analysis to follow, all relevant data are resampled based on monthly mean values. In contrast, the Federal Funds rate and the repo rate data are based on actual monthly percentage changes.
The correlation results based on a monthly frequency for the period between 2000 and 2013 can be seen in
Correlation results for period between 2000 and 2013.
Variable  Component  SA 
US 
ZAR/USD  JPEMBS  Fed rate  Repo  

PC1  PC2  PC3  PC1  PC2  PC3  
SA  PC1  1  −0.02  −0.02  0.34  0.03  0.02  0.32  −0.01  0.07  0.26 
PC2  −0.02  1  0  0.04  0.05  0.01  −0.16  0.1  0.06  0.31  
PC3  −0.02  0  1  −0.04  0.04  0.24  0.02  −0.09  0.05  0.32  
US  PC1  0.34  0.04  −0.04  1  0.01  0.01  0.03  −0.2  0.19  0.04 
PC2  0.03  0.05  0.04  0.01  1  0.03  0.01  0.04  0.37  0.17  
PC3  0.02  0.01  0.24  0.01  0.03  1  −0.27  −0.11  0.42  0.07  
ZAR/USD    0.32  −0.16  0.02  0.03  0.01  −0.27  1  0.23  −0.16  0.05 
JPEMBS    −0.01  0.1  −0.09  −0.2  0.04  −0.11  0.23  1  −0.08  0.09 
Fed rate    0.07  0.06  0.05  0.19  0.37  0.42  −0.16  −0.08  1  0.04 
Repo    0.26  0.31  0.32  0.04  0.17  0.07  0.05  0.09  0.04  1 
SA, South Africa; US, United States; ZAR, South African rand; USD, United States dollar; PC1, Principal component 1; PC2, Principal component 2; PC3, Principal component 3; JPEMBS, JPMorgan Emerging Market Bond Spread rate; Fed rate, Federal Reserve Rate; Repo, South African Repo rate.
Also of interest is the negative correlation of 20% between the JPEMBS and the scores of the US level component. This could be because of riskaverse investors shifting investments to established markets. Tracey (
The significant correlations with respect to the scores of the South African principal components suggest a multivariate regression. In addition, correlation does not indicate causality. Therefore, a multivariate regression analysis needs to be performed to determine which rates drive South African yield curve changes. The analysis was performed using the ordinary least squares (OLS) method and a 95% confidence interval was selected. Two regression models were constructed based on the correlation results from
Regression models.
Model 1  Model 2  Models 1 & 2 

Dependant Variable  Dependant Variable  Independent Variables 
SA PC1 scores  SA PC2 scores  US PC1 Scores 
US PC2 Scores  
US PC3 Scores  
ZAR/USD exchange rate  
Emerging market bond spread  
Federal Funds rate  
Repo rate 
SA, South Africa; US, United States; ZAR, South African rand; USD, US dollar; PC1, Principal component 1; PC2, Principal component 2; PC3, Principal component 3.
The independent variables with resulting pvalues greater than 5% were removed and the analysis was thereafter repeated. This prevents the model from overfitting owing to the addition of insignificant dependent variables. In addition, if any of the regression models displayed insignificant results, they were neglected and only the significant results were presented.
The multivariate regression results for the period between 2000 and 2013 is shown in
Regression results for the period between 2000 and 2013.
Measure  Metric  Statistic 

Confidence interval  95%   
Dependant variable  SA PC1  
Method  OLS  Adj 
Independent variables    
US PC1    0.00 
ZAR/USD    0.00 
Repo rate    0.00 
SA, South Africa; US, United States; OLS, ordinary least squares; ZAR, South African rand; USD, US dollar; PC1, Principal component 1; Repo, South African repo rate.
The change in the gradient of this curve can therefore indicate different economic cycles of emerging markets. A regression analysis was therefore conducted on the different economic cycles of emerging markets. The following subperiods correspond to different economic cycles over the period 2000–2013.
Subperiod 1: 2000–2002 (characterised by the dotcom crash of 2000–2001).
Subperiod 2: 2003–2007 (the US recovery period).
Subperiod 3: 2008–2009 (includes the credit crisis of 2008).
Subperiod 4: 2010–2013 (characterised by US quantitative easing).
Multivariate regression results: Subperiod 1.
Model  Measure  Metric  Statistic 

Model 1  Confidence interval  95%   
Dependant variable  SA PC1  
Method  OLS  Adj 

Independent variables    
ZAR/USD    0.00  
Repo rate    0.00  
Model 2  Confidence interval  95%   
Dependant variable  SA PC2  
Method  OLS  Adj 

Independent variables    
JPEMBS    0.01 
SA, South Africa; OLS, ordinary least squares; ZAR, South African rand; USD, US dollar; PC1, Principal component 1; PC2, Principal component 2; JPEMBS, JPMorgan Emerging Market Bond Spread rate; Repo, South African repo rate.
The results in
Fortyone per cent of level shifts in South Africa were explained by the South African Rand to US Dollar exchange rate and the repo rate.
Eighteen per cent of South African slope shifts were explained by the JPEMBS rate.
Multivariate regression results: Subperiod 1.
Model  Measure  Metric  Statistic 

Model 1  Confidence interval  95%   
Dependant variable  SA PC1  
Method  OLS  Adj 

Independent variables    
ZAR/USD    0.01  
Repo rate    0.03  
Model 2  Confidence interval  95%  
Dependant variable  SA PC2  
Method  OLS  Adj 

Independent variables    
Repo rate    0.03 
SA, South Africa; OLS, ordinary least squares; ZAR, South African rand; USD, US dollar; PC1, Principal component 1; PC2, Principal component 2; PC3, Principal component 3; JPEMBS, JPMorgan Emerging Market Bond Spread rate; Fed rate, Federal Reserve rate; Repo, South African repo rate
The results in
Fourteen per cent of South African level shifts were influenced by exchange rate and repo rate fluctuations.
During this period, movements in the repo rate drove 20% of South African slope shifts.
Multivariate regression results: Subperiod 3.
Measure  Metric  Statistic 

Confidence interval  95%   
Dependant variable  SA PC1  
Method  OLS  Adj 
Independent variables    
US PC1    0.00 
SA, South Africa; US, United States; OLS, ordinary least squares; PC1, Principal component 1.
The results in
Fiftyfour per cent of level shifts in South Africa were explained by US level shifts.
Multivariate regression results: Subperiod 7.
Measure  Metric  Statistic 

Confidence interval  95%   
Dependant variable  SA PC1  
Method  OLS  Adj 
Independent variables    
US PC1    0.00 
JPEMBS    0.00 
SA, South Africa; US, United States; OLS, ordinary least squares; PC1, Principal component 1; JPEMBS, JPMorgan Emerging Market Bond Spread rate.
The results in
Subperiod 2 relates to a period of economic recovery in the United States. In addition, the period was characterised by protracted periods of strength in the rand (
Rand/US Dollar exchange rate post2000.
In contrast, subperiod 1 resembles a period where the rand was severely weakened (
A possible cause in the rand deterioration may have been attributable to the slowdown in global economic activity that began in 2000, where the demand for South African goods and services moderated. Accordingly, when the exchange rate volatility increases, investors require a larger yield compensation for holding emerging market bonds. This volatility coupled with a sharp decrease in the repo rate (
During the last two subperiods, the regression results suggest that movements in the US term rates played an important role in driving South African interest rates. This could largely be because of the impact of the credit crisis of 2008. In addition, the significant relationship seen during subperiod 4 (post2010) may stem from quantitative easing, a time when US investors are pushing funds to emerging markets that offer higher attractive yields.
This study used the PCA technique to identify and quantitatively describe the main underlying movements present in both US and South African yield curves. The analysis was further aimed at relating fluctuations in the percentage of variance explained by each of these factors to macroeconomic influences or events. The study further intended to determine dominant factors, if any, which were responsible for driving South African interest rate movements during certain economic periods.
The PCA technique classifies and quantifies yield curve movements across both markets with respect to three main factors, namely level, slope and curvature shifts. These factors encapsulate most of the yield curves variability, with the level shift representing the dominant component. Curvature movements only contribute a small change in yield curve variations.
In terms of relating these factors to macroeconomic variables, the findings revealed that monetary policy shocks across both markets played a prominent role in affecting the form and importance of slope and curvature shifts. In addition, the results illustrated that the shape and importance of each factor deviated during periods of economic instability. In particular, the form of the slope and curvature movements were erratic and distorted during these periods. In contrast, the shape of the slope and curvature shifts were significantly smoother during periods of economic stability. In the light of this, it can be argued that a volatile market and economic environment can significantly distort the shape, smoothness and percentage of variance explained of yield curve movements.
In terms of a brief comparison between the markets, the level shifts in South Africa were observed to play a far less dominant role, resulting in the slope and curvature factors driving South African interest rate movements to a greater degree. Furthermore, it was noticed that the form of South African slope and curvature shifts were far more erratic. These differences were suggested to be because of factors such as political instability, unstable monetary policy and volatile interest rate expectations, which are known to be prominent in emerging markets.
A regression analysis revealed that over the long term (2000–2013), South African rates were not largely driven by US interest rate movements. However, during certain subperiods, movements in the US term rates played a dominant role in driving South African rates. As an example, during the credit crisis period the regression models suggested that 54% of South African level shifts were driven by US level shifts. The results further illustrated that during certain periods, movements in the Rand/Dollar exchange rate played an important role in driving South African level shifts. In addition, the regression results were able to confirm the significant relationship between monetary policy shocks and corresponding slope and curvature movements across both markets.
Further research could involve the comparison of the impact of the credit crisis on the volatility of interest rates in both markets. An Ftest could be performed based on principal components scores (derived from yield curve changes) pre and postcrisis to compare the variances of yield curve movements during these periods.
Further research could also aim to verify the relationship suggested by the regression models between US and South African rates during certain periods. Alternate statistical methods can be researched to compare daily US and South African principal component scores, where methods such as auto correlation, cointegration or dynamic time warping could be used to accurately determine the lag between US and South African yield curve changes. Principal component scores for each subperiod using weekly mean changes in yields could also provide a possibility.
Lastly, an investigation could be performed on the factor loadings of the principal components across different subperiods. This will indicate the degree to which various interest rates affected the level, slope and curvature movements during certain economic periods.
The authors declare that they have no financial or personal relationship(s) that may have inappropriately influenced them in writing this article.
K.P. and A.M., the principal authors, instituted the study and undertook analysis and interpretation. G.W.v.V., the supervisor, recommended the project and oversaw the robustness of the results and interpretation.
General purpose and highlevel programming language.
Formally, for dimensionality reduction. In a data set where there are
A covariance matrix can be used since each variable (term to maturity) is measured in the same units and the variances between the variables (changes in yields across different maturities) do not vary greatly.
Because of the high correlation present in yields across different maturities, the first three components generally account for almost all the variability in yield curve changes.
The trace of a square matrix is the sum of the main diagonal elements.
The sample data considered are US treasury rates for the period 2000–2011.
This projection represents a dimensionality reduction from 10 dimensions to 3 dimensions.
See Footnote 2 for
Swap rates serve as an acceptable proxy for yield curves.
A slope shift can occur when shortterm rates change but longterm rates remain the same, or vice versa.
Changes in longterm inflation expectations will alter longterm interest rate expectations. This in turn affects the slope of interest rate curves.
The risk premium affects both the long and short ends of a yield curve. However, it is greater at longer durations because of more uncertainty and a greater chance of catastrophic events that can impact investments. Fluctuating risk premiums will therefore alter the slope of interest rate curves.
South African interest rates mostly follow movements in the US interest rates with a lag. This is confirmed by comparing
Financial markets worldwide do not have the same working hours. As a result, the study of correlation or causality between financial market indices becomes highly dependent on the issue of lagging or leading data. A lag of 1 day, for example may be chosen between US and South African time series data (i.e. when the US markets close, South African markets react the following day based on movements in the United States).