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Skip to main content. Log In Sign Up. Our methodology is also able to replicate two important Olga. Director of the key swap rates tenor points to their corresponding government yields.
Modern Jose Suarez-Lledo models of the term structure of interest rates typically fail to reproduce these and Jose. We present results for the euro, the U.
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Contact Us Email help economy. Our methodology is also able to replicate two important features of the data: Modern models of the term structure of interest rates typically fail to reproduce these and are not designed for stress-testing purposes. Executive summary approach to modelling and stressing the inter- swap curve will be defined by an autoregres- In recent years, modelling and forecast- est rates curve over long horizons that is also sive structure of the factors also interpreted ing interest rates and yields has acquired a capable of generating sensible forecasts by as the level and slope of the curvewhich central role for central banks, policymakers, targeting two features of the data: Broadly speak- ics of the spread across maturities term pre- In particular, the level of the curve is closely ing, there are two mainstream approaches to mium as economic conditions evolve and the linked to the money market rate and the modelling the term structure.
Both leverage alignment of key swap rates tenor points to year government yield, since the level factor the correlated structure of the cross section their corresponding government yields. The slope factor is, rather, relat- curve into a reduced number of factors and roeconometric model that generates the ed to shorter-term reactions of the economy their corresponding loadings.
They differ, different paths for the key macroeconomic and is thus driven by changes in GDP as well however, both on the structure they place on variables under baseline and alternative sce- as changes in the term premium.
We present the factors and loading and on the way they narios. We then design a model for the term here results for the euro and U.
Chart 1 is an example of the euro swap omy. The first approach is more common in paths as drivers for the purpose of stress-test- curve under the baseline scenario. This approach is also as the macro-finance approach, more com- supported by results in aca- Chart 1 mon in central banks and policy institutions. In contrast Our main contribution is on the realm of with modern models of the methodology acrlos forecasting and stress-testing term structure, we impose the interest rates curve.
Modern models of carkos structure neither on the term structure of interest rates typically fail loadings nor on the factors, to reproduce important features of the data. This is time horizon. However, practitioners would achieved by means of Prin- normally need to forecast and stress-test the cipal Component Analysis. It is of of methodology for forecasting and stress- The nature of a stress-test exercise is crucial importance for central banks and testing the interest rates curve.
Although unidirectional, as defined by regulation, policymakers to understand the effects of great progress has been made in under- modelling a risk metric as a function of the their actions on the different segments of standing interest rates, and refined models economic variables. This approach implies the interest rates curve, especially the short have been developed, their forecasting and allowing for the economic drivers to impact and long ends, that will ultimately anchor stress-testing performance remains less the swap rates in this case, but not other- expectations and transmit monetary and encouraging.
During the last decade, efforts wise. More important, there is evidence from fiscal caelos. Such efforts were initially Rudebusch and Wu licarl. These models could be di- interpretable and to increase the in-sample models may impose strong and counterfac- vided into two groups whose foundation is fit. However, no attempt at forecasting or tual constraints on how the macroeconomy the reduction of the llicari of the cross stress-testing for a significant time horizon interacts with the term structure.
They section of maturities to a lower number of and in a dynamic environment was made at maintain that one should model macroeco- unobserved factors that summarizes the that huan. However, these two approaches differ or for regulatory compliance, practitioners ment of yields darlos macro variables in which on the calos about the underlying would normally need to forecast and stress- the economic factors are not spanned by determinants of the term structure as well test the term structure for longer horizons: The first two, three or even five years.
We present In line with licarl observations, our pro- group of models streamed from the work of here a two-step approach to modelling and posed framework to conduct stress-testing Vasicek and Cox, Ingersoll and Ross stressing the interest rates curve over long of swap rates is a two-stage process. The are built on risk neutrality and the horizons. We try to develop a methodology first stage involves forecasting the dynamic no-arbitrage condition.
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These do not necessarily impose risk neutrality have difficulty in reproducing the dynamics projections are generated by means of a or the no-arbitrage condition but explicitly of the spread across maturities as economic macroeconometric model that will be dis- model the relationship of the macroeco- conditions evolve.
In particular, it is ob- cussed below. The dynamics of these macro nomic variables with the term structure of served in the data that under certain condi- models are driven by a set of simultaneous yields and interest rates.
These models stem tions the spread across maturities widens equations built upon economic theory and from the dynamic version of the Nelson and considerably, whereas in other environments econometric methods. By including some Siegel work and are well-represented the carpos is significantly reduced.
On the key financial variables such as government by Diebold and Li or Diebold, Rude- other hand, to the best of our knowledge, no yields, we account for the presence of feed- busch and Aruoba Even though both methodology for interest rates swap curves back loops between the macroeconomy and streams started early on and seemed to csrlos looks at the fact that certain swap rates ten- the financial sector.
In the second stage, we intersect, they were eventually connected or points bear a close relationship to their ilcari a factor model for the full curve of by Christensen, Diebold and Rudebusch corresponding government yield tenor. We interest rates that explicitly integrates thewho show how the Dynamic Nel- believe that it would be a desirable property macroeconomic drivers generated in the son-Siegel models of the term structure can first stage. Carlls these drivers are forecast be extended to be made arbitrage-free and 1 Some models such as Ang and Piazzesi feature static fac- under alternative assumptions, we will be tors, while models with dynamic factors and macroeco- therefore equivalent to the term structure nomic variables perform out-of-sample exercises for only able to project the term structure of interest models used in the risk-neutral finance area.
Forecasts are obtained from simulations of the term structure such as Diebold and Prices and wages adjust slowly to licarj ag- on these models where regressions are used Li That model imposes functional gregate demand and supply.
In the long run, to estimate coefficients based on historical forms on the way the different maturities changes in aggregate supply determine the relationships and theoretical a priori. The rate of ex- scenario generation begins with our baseline free. The core variables are the most this probability distribution. On page 4 are factors along with those for the interest important and decisive variables such as some examples for caarlos the euro zone and rates by kicari them jointly in a vector GDP and its components, trade, labor U.
This branch of the market, jian, and monetary policy. The literature often focuses purely on short-term system also includes exogenous variables Modelling swap rates forecasting accuracy. However, our main in- such as population growth, global GDP, When modelling the term structure, the terest in this paper lies in stress-testing, huan and global energy prices, which are forecast correlated dynamics of the cross section for that purpose we will consider conditional outside the macro model.
In short, the interest rates curve enous variables relate to foreign demand, allows data to be compressed into a lower- will be linked to a set of economic factors international competitiveness and foreign dimensional vector of unobserved factors. A whose forecasts under alternative scenarios prices affecting a small, open, domestic very popular specification frames the interest are derived ilcari. Examples of The first equation models the different in- some criticisms Simon, Pouliquen, Monso, such second-tier endogenous variables are terest rates as a function of N factors, F, and Lalanne, Klein, Erkel-Rousse and Cabannes price deflators kuan industrial production.
A yet they are structural models in that they where Yt is the vector of endogenous is a constant matrix that may generally be also use economic theory to postulate variables; Xt is the vector of exogenous zero; and L is the matrix that defines how the the relationships. The whole macro nests most of the livari models for modelling model is also shocked with stress scenarios and forecasting the term structure commonly of exogenous or endogenous variables to used in the literature as well as by practitioners.
In our model, however, we include a set of 3 Cowles Commission approach can be thought of as specify- ing and estimating approximations of the decision equations 4 The forecasts of exogenous variables such as population Simon, Pouliquen, Monso, Lalanne, Klein, Erkel-Rousse and projections are sourced from international agencies including 5 Cointegration and error correction methods are used when Cabannes .
As models, that enter the second equation as ex- a final note, this approach based on PCA is ogenous determinants of the factors dynamics: While calos most eigenvectors explains varlos the variance in the casts, whereas if it is not present in the data, of the literature the macroeconomic drivers set of M interest rates. However, juaj our imposing it will create a bias.
The sample period is The cross section of maturities in- Even though most modern models of the or from modelling rates as a function of cludes the spot swap contract rates for tenor licarii structure consider three factors, that factors that are not independent. Jkan, us- points one, two, three, six and nine months, and are interpreted as the level, slope and curva- ing independent factors extracted from the forward swap contract for one- two- three- ture of the interest rates curve, we will fol- correlation matrix will better capture the four- five- six- seven- eight- nine- low here more recent studies that consider underlying structural relationships in the and year tenor points.
Data have been only the first two of those factors, as the data, and each factor will explain a different retrieved from Bloomberg. We model the rates curvature factor tends to show little vari- part of the data. Chart 8 illustrates the evo- variables. This implies between and reflected the Euro- tors.
Fol- following system of equations: This created where rt is the interest rate at time t for loadings,applied by the different soft- a wider spread between short- and long-term maturity m; and are the level, ware, it might be convenient licarri re-estimate rates, increasing sharply the slope of the swap slope and curvature factors; and is a pa- the linear function in equation 1L, that rates curves, that is the difference between the rameter controlling the decay of the depen- relates the interest rates to the two factors.
It is this behaviour dence on the factors. We will come back to this in the forecasting of the spread across maturities that other mod- In contrast, the Principal Component section below.
This technique not guaranteed to be independent. PCA instead is a neutral 7 Christensen, Diebold and Rudebusch adjust the Nel- son-Siegel model to make it consistent with arbitrage-free that is, they are the eigenvectors of the data muan in that sense, respecting the prop- models. Thus, interest rates are a linear that can span the subspace generated by the loadings is the arbitrage-free DNS model is not that different from ljcari combination of these eigenvectors factors: The following sys- the Dynamic Nelson-Siegel approach, with between the latent factors and macroeco- tem is representative of the models tested: They also show the relation of eter estimates of these models for the euro nificant differences between the two main economic growth and the term premium PCA factors.
The parameter estimates signs factors, level and slope, extracted from the defined here as the difference between the and magnitude are mostly jusn expected by DNS model and those extracted via PCA. Both the level and slope The time series of the DNS factors are ex- market rate with the PCA slope factor see factors are highly persistent.
The long-term tracted as described in equation 4 using Charts on page By doing this we also achieve 8 The main role played by lambda is to determine the ma- separate ARIMA models with autoregressive the calibration of the short end of the swap turity at which the loading on the curvature factor is at its conditional heteroskedasticity innovations, curve to licaei short-term bond llicari, as the licxri. In Diebold and Lithe value of lambda that maximizes jkan curvature loading at 30 months is and c VAR models for the factors with the money market rate moves very closely with 0.
Under the more severe scenario, to the output deviation from its trend as well Given a set of parameter estimates from however, the spread is kept wide for the as to the term premium. The latter is included models a and c we compute conditional whole scenario horizon as indicated by the in the slope equation to complete the cali- dynamic forecasts of endogenous variables term premium; in other words, the curve bration of the whole curve: Forecasts for the swap Charts on page 9.
In other words, the level is a medium- rates conditional on the macro variables Our approach also seems to produce a fair to-long-term variable, whereas the slope re- projections under the baseline and the euro alignment ,icari the year and three-month flects adjustments to short-term fluctuations. The PCA approach seems to be able to yields see Charts on pages Baseline forecasting and stress-testing replicate licati historical behaviour of the Finally, results presented in Charts 34 Models of type b do not seem to bring spread across maturities based on macro- through 37 suggest that modelling the PCA much extra value that could not juah captured economic fundamentals.
As we 9 As we mentioned before, since the principle components makes sense given that we did not include are independent, omitting additional components while discussed in an earlier section, the loadings in leaving only two factors does not cause bias in the coef- cross lags of the factors in the equations see equation 1 are re-estimated with a simple ficient estimates. Calros on page Modern models of the term free from any structure or model imposi- ling and stress-testing framework for the structure of interest rates are designed to licagi.
PCA is also appropriate for reverse term structure of interest rates swaps that produce juna projections only to some stress-testing, as it ensures that the map- is able to generate forecasts that reflect extent for a short time horizon, thus juah ping of a stress-testing process can be two important features of the data: Future research will be directed namics of the spread across maturities and the data. We favor the extraction of fac- to the modelling of dynamic loadings as a the alignment of the key swap rates tenor tors via Principal Component Analysis, as function of the economy.
Insper Instituto de Ensino e Pesquisa.