Modeling and Stressing the Interest Rates Swap Curve
This article presents a twostep modeling and stress testing framework for the term structure of interest rates swaps that generates sensible forecasts and stressed scenarios out of sample. The results are shown for the euro, the US dollar, and British pound swap curves.
In recent years, modeling and forecasting interest rates and yields has acquired a central role for central banks, policymakers, regulators, and practitioners. It is of crucial importance for central banks and policymakers to understand the effects of their actions on the different segments of the interest rates curve, especially the short and long ends, that will ultimately anchor expectations and transmit monetary and fiscal policy. Needless to say, that interest rate risk and the movements of the fullterm structure are among the more important areas of risk management and stress testing for banks and regulators.
The academic literature has developed a nonnegligible number of models of the term structure that have been later adopted by practitioners. These models could be divided into two groups whose foundation is the reduction of the dimension of the cross section of maturities to a lower number of unobserved factors that summarizes the dynamic properties of the whole cross section. However, these two approaches differ on the assumptions about the underlying determinants of the term structure as well as on their technical treatment. The first group of models streamed from the work of Vasicek (1977) and Cox, Ingersoll, and Ross (1985) are built on risk neutrality and the noarbitrage condition.
To the second group belongs the socalled macrofinance stream of models that do not necessarily impose risk neutrality or the noarbitrage condition but explicitly model the relationship of the macroeconomic variables with the term structure of yields and interest rates. These models stem from the dynamic version of the Nelson and Siegel (1987) work and are wellrepresented by Diebold and Li (2006) or Diebold, Rudebusch, and Aruoba (2004). Even though both streams started early on and seemed to not intersect, they were eventually connected by Christensen, Diebold, and Rudebusch (2009), who show how the Dynamic NelsonSiegel models of the term structure can be extended to be made arbitragefree and therefore equivalent to the term structure models used in the riskneutral finance area. Therefore, this paper reviews only the methodology followed by the macrofinance approach.
This article seeks to contribute in the realm of methodology for forecasting and stress testing the interest rates curve. Although great progress has been made in understanding interest rates, and refined models have been developed, their forecasting and stress testing performance remains less encouraging. During the last decade, efforts have been made in several directions to incorporate macroeconomic factors to models of the term structure – Ang and Piazzesi (2003), Diebold and Li (2006), Diebold et al (2006), Ang et al (2007), and Rudebusch and Wu (2008). Such efforts were initially undertaken in order to relate movements in the curve to factors that were more easily interpretable and to increase the insample fit. However, no attempt at forecasting or stress testing for a significant time horizon and in a dynamic environment was made at that stage.^{1}
In fact, whether for business planning or for regulatory compliance, practitioners would normally need to forecast and stress test the term structure for longer horizons: two, three, or even five years. Presented here is a twostep approach to modeling and stressing the interest rates curve over long horizons. The goal is to develop a methodology that is capable of generating sensible forecasts by targeting two features of the data. On the one hand, current models appear to have difficulty in reproducing the dynamics of the spread across maturities as economic conditions evolve. In particular, it is observed in the data that under certain conditions the spread across maturities widens considerably, whereas in other environments the spread is significantly reduced. On the other hand, to the best of these authors’ knowledge, no methodology for interest rates swap curves looks at the fact that certain swap rates tenor points bear a close relationship to their corresponding government yield tenor. It is the belief of these authors that it would be a desirable property that the outcome from the model reflected this relationship.
Methodology
The nature of a stress test exercise is unidirectional, as defined by regulation, modeling a risk metric as a function of the economic variables. This approach implies allowing for the economic drivers to impact the swap rates in this case, but not otherwise. More important, there is evidence from different setups that there is a significant effect from macroeconomic variables on the term structure but not so much in the reverse direction (Diebold et al [2006], Ang et al [2007], Dewachter and Lyrio [2002], and Rudebusch and Wu [2003]).
Furthermore, Joslin, Priebsch, and Singleton (2012) argue that current macrofinance models may impose strong and counterfactual constraints on how the macroeconomy interacts with the term structure. They maintain that one should model macroeconomic risks that are distinct from yield curve risks, and they propose an asymmetric treatment of yields and macro variables in which the economic factors are not spanned by any portfolio of bond yields.
In line with these observations, our proposed framework to conduct stress testing of swap rates is a twostage process. The first stage involves forecasting the dynamic paths of key macroeconomic indicators such as GDP, money rates, and government yields under different scenarios. These projections are generated by means of a macroeconometric model that will be discussed below. The dynamics of these macro models are driven by a set of simultaneous equations built upon economic theory and econometric methods. By including some key financial variables such as government yields, this study accounts for the presence of feedback loops between the macroeconomy and the financial sector. In the second stage, a factor model is developed for the full curve of interest rates that explicitly integrates the macroeconomic drivers generated in the first stage. Because these drivers are forecast under alternative assumptions, we will be able to project the term structure of interest rates over those different scenarios.
As part of this exercise, a comparison is made between the forecasting properties of this modeling approach with other dynamic models of the term structure such as Diebold and Li (2006). That model imposes functional forms on the way the different maturities load on the factors while leaving the factors free.^{2} This model does not impose any structure on either loadings or the factors.
Macroeconomic scenarios
Part of the literature on interest rates generates forecasts for the macroeconomic factors along with those for the interest rates by estimating them jointly in a vector autoregressive system. This branch of the literature often focuses purely on shortterm forecasting accuracy. However, the main interest in this paper lies in stress testing, and for that purpose conditional forecasts are considered. In short, the interest rates curve will be linked to a set of economic factors whose forecasts under alternative scenarios are derived separately.
In order to forecast macroeconomic variables, a macroeconometric model represented by the system of simultaneous equations inspired by the Cowles Commission's approach, is employed.^{3} Such models are still widely used among practitioners despite some criticisms (Simon, Pouliquen, Monso, Lalanne, Klein, ErkelRousse and Cabannes [2012]) thanks to their practical usefulness and a balance between consistency with economic theory and actual data fit. These are nonstructural models in that they are built from many equations that describe relationships derived from empirical data, yet they are structural models in that they also use economic theory to postulate the relationships.
In the broadest sense, the macro model used describes aggregate economic activity determined by the intersection of aggregate demand and supply. In the short run, fluctuations in economic activity are primarily determined by shifts in aggregate demand, while the level of resources and technology available for production is taken as a given. Prices and wages adjust slowly to equate aggregate demand and supply. In the long run, changes in aggregate supply determine the economy’s growth potential. The rate of expansion of the resource and technology base of the economy is the principal determinant of the pace of economic growth.
This model is composed of a set of equations for “core” and “auxiliary” endogenous variables. The core variables are the most important and decisive variables such as GDP and its components, trade, labor market, prices, and monetary policy. The system also includes exogenous variables such as population growth, global GDP, and global energy prices, which are forecast outside the macro model.^{4}
These exogenous variables relate to foreign demand, international competitiveness and foreign prices affecting a small, open, domestic economy and are the starting point of our forecast process. Also important, they are key sources of where exogenous shocks could originate from. In turn, the auxiliary variables may be driven by the core and exogenous variables but are not allowed to determine the core variables. Examples of such secondtier endogenous variables are price deflators and industrial production.
Formally, the reduced form for the system of simultaneous equations can be written as:
where Y_{t} is the vector of endogenous variables; X_{t} is the vector of exogenous variables, and β_{k}, β_{p} are coefficient matrices. The specification of each individual equation is selected based on statistical properties, backcast performance, evaluation of shortterm and longterm forecasts, the system’s stability, and parsimony. The whole macro model is also shocked with stress scenarios of exogenous or endogenous variables to ensure that the responses of the system to impulses are within a reasonable range.^{5}
Forecasts are obtained from simulations on these models where regressions are used to estimate coefficients based on historical relationships and theoretical a priori. Our scenario generation begins with our baseline forecast, from which we develop the basic outlines of alternative scenarios by running multiple simulations to develop a probability distribution of economic outcomes. We then produce alternative scenarios that align with this probability distribution.
Modeling swap rates
When modeling the term structure, the correlated dynamics of the cross section of maturities plays an important role, as it allows data to be compressed into a lowerdimensional vector of unobserved factors. A very popular specification frames the interest rates in a statespace form:
The first equation models the different interest rates as a function of N factors, F, and the
second equation models the dynamics of the swap rates curve through a number, K, of lags of the factors. denotes a (M × 1) vector of swap rates observed at time t for M different maturities; F_{t} denotes a (N × 1) vector of factors obtained from the interest rates data with N<M. A is a constant matrix that may generally be zero, and L is the matrix that defines how the interest rates depend on the factors. are approximation errors that will be described below, and Vt are standard regression errors. and V_{t} are mutually orthogonal.
The state space representation in (1) and (2) nests most of the existing models for modeling and forecasting the term structure commonly used in the literature as well as by practitioners. In our model, however, we include a set of economic drivers, , obtained from our macro models, that enter the second equation as exogenous determinants of the factors dynamics:
The system (3) is then estimated as a VAR of typically order 1 (K=1). While in most of the literature the macroeconomic drivers and their relationships with the factors are estimated as endogenous variables in the for a number H of economic drivers, we focus here on the stronger directional causality from macroeconomic variables to the interest rates curve, as discussed before and reported in the literature.
Even though most modern models of the term structure consider three factors, that are interpreted as the level, slope and curvature of the interest rates curve, we will follow here more recent studies that consider only the first two of those factors, as the curvature factor tends to show little variability and almost no relation to economic variables. Although such is the most widely used approach, modern models differ in the way they extract the factors and the loadings of the different maturities on those factors. The macrofinance approach streaming from Diebold and Li (2006) and Diebold et al (2006) places structure on those loadings, leaving the factors to be determined in the following system of equations:
Where r_{t} is the interest rate at time t for maturity m; are the level, slope and curvature factors; and is a parameter controlling the decay of the dependence on the factors.
In contrast, the principal component analysis does not place any structure on the loadings or on the factors, other than the latter being orthogonal. This technique extracts the factors through the diagonalization of the correlation matrix of the data—that is, they are the eigenvectors of the data covariance matrix and therefore are purely datadriven. Thus, interest rates are a linear combination of these eigenvectors (factors):
where is the matrix of eigenvectors and is the matrix of the loadings of the eigenvectors on the interest rates. PCA produces orthogonal factors by construction, therefore The set of M eigenvectors explains all the variance in the set of M interest rates. However, since our aim is to reduce the dimension of the model, we want to consider only the set of first N eigenvectors (factors, F_{t}) that would still explain most of the variance of the dataset. The choice of PCA is based on the fact that the orthogonality of the factors allows the reduction of the dimension without generating a bias from omitting some of the factors, or from modeling rates as a function of factors that are not independent. Also, using independent factors extracted from the correlation matrix will better capture the underlying structural relationships in the data, and each factor will explain a different part of the data.
In line with most of the recent literature, we find that the first two factors (level and slope) account for about 98% of the variance in the data, and therefore we will focus on the modeling of these two. This implies that with N=2. Thus, estimating the curve of interest rates, R_{t}, as a function of these two factors — equation (1) — will always carry an approximation error, as there will always remain a small fraction (about 2%) of the data unaccounted for. Finally, depending on the default transformations to the matrix of loadings, applied by the different software, it might be convenient to reestimate the linear function in equation (1), L, that relates the interest rates to the two factors.^{6}
It is important to note that the factors extracted in the macrofinance literature are not guaranteed to be independent. Also, factors estimated through the Kalman filter may impose normality. PCA instead is a neutral technique in that sense, respecting the properties of the data whatever they may be. As a final note, this approach based on PCA is silent about the noarbitrage condition. This article follows advice in Duffee (2012) and Diebold and Rudebusch (2013)^{7} that if the noarbitrage condition is embedded in the data, imposing it does not improve the forecasts, whereas if it is not present in the data, imposing it will create a bias. It is precisely in stressed times that noarbitrage may be less likely to hold, and the goal is to develop a methodology for stress testing.
Estimating the dynamics of the curve
Next, the monthly data for interest rates swaps for the euro is considered. The sample period is 2000:1 to 2013:2. The cross section of maturities includes the spot swap contract rates for tenor points one, two, three, six and nine months, and forward swap contract for one, two, three, four, five, six, seven, eight, nine, 10, 15, 20 and 30year tenor points. Data have been retrieved from Bloomberg. The rates are modeled in logs in order to ensure strictly positive forecasted interest rates. Chart 8 illustrates the evolution of euro and GBP interest rates swaps over the sample period (see Charts 89).
Sharp upswings in the euro shortterm rates between 2006 and 2008 reflected the European Central Bank’s controlling of thriving euro zone’s economies with tightmoney policies. In this expansionary period, the spread between shortand longterm rates is very narrow. Following the peak in 2008, shortterm rates fell sharply with economies in recession and policy rate cuts, while the longerterm rates formed a relatively smoother downtrend. This created a wider spread between short and longterm rates, increasing sharply the slope of the swap rates curves, that is the difference between the longand shortterm rates. It is this behavior of the spread across maturities that other models fail to capture and what we will use as a criterion of the forecasting ability of our approach.
In this section, we want to compare the estimation and forecasting results of the macrofinance family of models, based on the Dynamic NelsonSiegel approach, with the results from the model. Also, the results are analyzed in terms of our ability to capture both the dynamics of the spread across maturities and the alignment of the key swap rates to the corresponding yields.
Chart 10 shows that there may be significant differences between the two main factors, level and slope, extracted from the DNS model and those extracted via PCA. The time series of the DNS factors are extracted as described in equation (4) using the cross section of yields for each month, while fixing lambda (see Charts 1213).^{8}
The following figures display connections between the latent factors and macroeconomic variables, providing some intuitive support for our models for the level and slope. Charts 14 through 21 show that the level factor appears to be closely linked to money market rate and 10year sovereign yields. They also show the relation of economic growth and the term premium (defined here as the difference between the 10year yield and the threemonth money market rate) with the PCA slope factor.
Now the dynamics of the factors are modeled in (6) following different approaches: (a) separate autoregressive integrated moving average (ARIMA) models for each factor, (b) separate ARIMA models with autoregressive conditional heteroskedasticity innovations, and (c) VAR models for the factors with the economic variables as exogenous drivers and the first lag of the factors. The following system is representative of the models tested:
The parameter estimates signs and magnitude are mostly as expected by economic theory. Both the level and slope factors are highly persistent. The longterm and shortterm interest rates are significant determinants of the level factor, which is typically interpreted as reflecting the evolution over time of the perceived mediumterm inflation target. By doing this, the calibration of the short end of the swap curve to the shortterm bond yields is achieved, as the money market rate moves very closely with the threemonth yield rate. Moreover, 10year sovereign yields are also incorporated as part of this equation, as they reflect the longerterm inflationary expectation, which also allows aligning the long end of the curve.
The slope factor responds with some lag to the output deviation from its trend as well as to the term premium. The latter is included in the slope equation to complete the calibration of the whole curve: the difference between 10year and the threemonth yield rates. In other words, the level is a medium to longterm variable, whereas the slope reflects adjustments to shortterm fluctuations.
Baseline forecasting and stress testing
Models of type (b) do not seem to bring much extra value that could not be captured through seasonaltype effects, so the focus on the results for models (a) and (c). As discussed in an earlier section, the loadings in equation (1) are reestimated with a simple ordinary least squares regression of the swap rates at each maturity on the level and slope factors.^{9} In contrast, the DNS swap rates are calculated using the fixed functional form associated with the factors defined in equation (4).
Given a set of parameter estimates from models (a) and (c) we compute conditional dynamic forecasts of endogenous variables (level and slope) for the period 2013:3 through 2018:3. Forecasts for the swap rates conditional on the macro variables projections under the baseline and the euro zone crisis scenarios are shown in Chart 22. The PCA approach seems to be able to replicate the historical behavior of the spread across maturities based on macroeconomic fundamentals. The PCAbased model forecasts a narrowing of the spread in the baseline scenario, which features a recovery of the economy, as the swap rates increase; that is, the swap curve becomes less steep. Under the more severe scenario, however, the spread is kept wide for the whole scenario horizon as indicated by the term premium; in other words, the curve remains quite steep for a long time (see Charts 2225).
This approach also seems to produce a fair alignment of the 10year and threemonth tenor points to the corresponding government yields (see Charts 2633).
Finally, results presented in Charts 34 through 37 suggest that modeling the PCA factors with a VAR or two separate ARIMA processes produces very similar results, which makes sense given that cross lags of the factors in the equations are not included.
This article introduced a twostep modeling and stress testing framework for the term structure of interest rates swaps that is able to generate forecasts that reflect two important features of the data: the dynamics of the spread across maturities and the alignment of the key swap rates tenor points to their corresponding government yields. Modern models of the term structure of interest rates are designed to produce accurate projections only to some extent for a short time horizon, thus normally failing to replicate such behavior in the data. These authors favor the extraction of factors via Principal Component Analysis, as it helps reduce estimation biases and it is free from any structure or model imposition. PCA is also appropriate for reverse stress testing, as it ensures that the mapping of a stress testing process can be inverted. Future research will be directed to the modeling of dynamic loadings as a function of the economy.
Chart 1. EUR Swap Curve: Baseline
Chart 2. Euro zone GDP Growth, Alternative Scenar
Chart 3. US GDP Growth, Alternative Scenarios
Chart 4. ECB Rate & Money Market Rate, Alternative Scenarios
Chart 5. Fed Funds Rate & USD Libor Rate & 3month yields, Alternative Scenarios
Chart 6. German 10yr Government Yields, Alternative Scenarios
Chart 7. US 10yr Government Yields, Alternative Scenarios
Chart 8. Euro Swap Rates, Maturities: 1M360M
Chart 9. USD Swap Rates, Maturities: 1M360M
Chart 10. Level Factor, EUR Swap Curve
Chart 11. Slope Factor, EUR Swap Curve
Chart 12. Level Factor, USD Swap Curve
Chart 13. Slope Factor, USD Swap Curve
Chart 14. Level Factor vs Money Market Rate, EUR Swap Curve
Chart 15. Level Factor vs German 10yr Yield, EUR Swap Curve
Chart 16. Level Factor vs Monetary Policy Rate, USD Swap Curve
Chart 17. Level Factor vs US 10yr Yield, USD Swap Curve
Chart 18. Slope Factor vs Euro Zone GDP Growth, EUR Swap Curve
Chart 19. Slope Factor vs Term Premium, EUR Swap Curve
Chart 20. Slope Factor vs US GDP Growth, USD Swap Curve
Chart 21. Slope Factor vs Term Premium, USD Swap Curve
Chart 22. EUR Swap Curve vs Term Premium, Baseline
Chart 23. EUR Swap Curve vs Term Premium, Euro Zone Crisis
Chart 24. USD Swap Curve vs Term Premium, Baseline
Chart 25. USD Swap Curve vs Term Premium, Euro Zone Crisis
Chart 26. Baseline
Chart 27. Chart 27 Euro Zone Crisis
Chart 28. Baseline
Chart 29. Euro Zone Crisis
Chart 30. Baseline
Chart 31. Euro Zone Crisis
Chart 32. Baseline
Chart 33. Euro Zone Crisis
Chart 34. EUR Swap Curve vs Term Premium (VAR), Baseline
Chart 35. EUR Swap Curve vs Term Premium (VAR), Euro Zone Crisis
Chart 36. USD Swap Curve vs Term Premium (VAR), Baseline
Chart 37. USD Swap Curve vs Term Premium (VAR), Euro Zone Crisis
Sources
1 Some models, such as Ang and Piazzesi, feature static factors, while models with dynamic factors and macroeconomic variables perform outofsample exercises for only very short forecast horizons (Pooter et al [2007]).
2 The models commonly used in finance place further structure by restricting both loadings and factors.
3 Cowles Commission approach can be thought of as specifying and estimating approximations of the decision equations (Simon, Pouliquen, Monso, Lalanne, Klein, ErkelRousse and Cabannes [2012]).
4 The forecasts of exogenous variables, such as population projections, are sourced from international agencies, including the International Monetary Fund and the World Bank.
5 Cointegration and error correction methods are used when appropriate in separate equations.
6 Dauwe and Moura (2011) mention that “any set of vectors that can span the subspace generated by the loadings is then equivalent to the loadings without loss of accuracy.”
7 Christensen, Diebold, and Rudebusch (2009) adjust the NelsonSiegel model to make it consistent with arbitragefree models. Although they show that it forecasts well outofsample, Carriero, Kapetanios, and Marcellino (2009), using a longer forecasting sample, report that the performance of the arbitragefree DNS model is not that different from the twostep NelsonSiegel model.
8 The main role played by lambda is to determine the maturity at which the loading on the curvature factor is at its maximum. In Diebold and L (2006), the value of lambda that maximizes the curvature loading at 30 months is 0.0609.
9 As we mentioned before, as the principle components are independent, omitting additional components while leaving only two factors does not cause bias in the coefficient estimates.
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