WORKING PAPER SERIES 141 - FORECASTING BONDS YIELDS IN THE BRAZILIAN FIXED INCOME MARKET ppt

ISSN 1518-3548 Forecasting Bonds Yields in the Brazilian Fixed Income Market Working Paper Series... Forecasting Bond Yields in the Brazilian FixedIncome Market ∗ The Working Papers shou

ISSN 1518-3548

Forecasting Bonds Yields in the Brazilian Fixed Income Market

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Forecasting Bond Yields in the Brazilian Fixed

Income Market ∗

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of the Banco Central do Brasil The views expressed in the papers arethose of the author(s) and not necessarily reflect those of the BancoCentral do Brasil

AbstractThis paper studies the predictive ability of a variety of models inforecasting the yield curve for the Brazilian fixed income market Wecompare affine term structure models with a variation of the Nelson-Siegel exponential framework developed by Diebold and Li (2006).Empirical results suggest that forecasts made with the latter method-ology are superior and appear accurate at long horizons when com-pared to different benchmark forecasts These results are importantfor policy makers, portfolio and risk managers Further research couldstudy the predictive ability of such models in other emerging markets.Keywords:term structure of interest rates; term premia; monetarypolicy; affine term structure models

JEL Code:E43; G12

∗ The views expressed are those of the authors and do not necessarily reflect the views of the Central Bank of Brazil Benjamin M Tabak gratefully acknowledges financial support from CNPQ Foundation.

† Banco Central do Brasil E-mail: jose.valentim@bcb.gov.br.

‡ Banco Central do Brasil E-mail: benjamin.tabak@bcb.gov.br.

1 Introduction

Accurate interest rates forecasts are essential for policy-makers, bankers,treasurers and fixed income portfolio managers These forecasts are mainingredients in the development of macroeconomic scenarios, which are em-ployed by large companies, financial institutions, regulators, institutionalinvestors, among others Nonetheless, to date there is very little research oninterest rates forecasting and specially on yield curve forecasting

Duffee (2002), Dai and Singleton (2002) and Ang and Piazzesi (2003)employ Gaussian affine term structure models and are successful in match-ing certain properties of the U.S term structure movement and generatingtime-varying term premia Recent literature has studied the joint dynamics

of the term structure and the macroeconomy in a general equilibrium work Wu (2006) for example develops an affine term structure model within

frame-a dynframe-amic stochframe-astic generframe-al equilibrium frframe-amework frame-and provides mframe-acroeco-nomic interpretations of the term structure factors The author argues thatchanges in the “slope” and “level” factors are driven by monetary policy andtechnology shocks, respectively However, these models focus on fitting termstructure models but provide poor forecasts of the yield curve

macroeco-Other researchers have studied the forecasting accuracy of interest ratessurveys and show that such forecasts correctly predicted the direction ofchanges in long-term interest rates for the US (see Greer (2003)) Bidakorta(1998) compares the forecasting performance of univariate and multivariatemodels for real interest rates for the US and finds that bivariate modelsperform quite well for short-term forecasting

In a recent paper Diebold and Li (2006) propose a model, which is based

on the Nelson-Siegel exponential framework for the yield curve, to forecastthe yield curve The authors present convincing evidence that their model issuperior to more traditional ones such as vector autoregression, random walkand forward rate and curve regressions They show that the model providesmore accurate forecasts at long horizons for the US term structure of interestrates than standard benchmark forecasts

Despite the advances in forecasting yields for the US economy there isvery little research applied to emerging markets However, some emergingcountries have large debt and equity markets and receive substantial inflows

of foreign capitals, playing an important role in the international capital kets Brazil deserves attention as it has large equity and debt markets, withliquid derivatives markets, and therefore represents interesting opportunities

mar-for both domestic and international investors Brazil has the largest stock

of bonds in absolute terms and as a percentage of GDP in Latin Americanbond markets In the Brazilian fixed-income market domestic federal publicdebt is the main asset, with approximately R$ 1 trillion (US$ 545 billions)

to the random walk benchmark1 This paper is the first that attempts tostudy interest rates forecast for the Brazilian economy, however it focuses onlong-term interest rates forecasts

Our paper contributes to the literature by estimating and calibrating a riety of models to the Brazilian term structure of interest rates and comparingtheir forecast accuracy The accuracy of out-of-sample forecasts is evaluatedusing usual mean squared errors and Diebold-Mariano statistics Empiricalresults suggest that the Diebold-Li (2006) model has good forecasting power

va-if compared with an affine term structure model and the random walk mark, especially for short-term interest rates Therefore, it provides a goodstarting point for research applied to emerging markets

bench-The remainder of the paper is organized as follows Section 2 presentsthe data and stylized facts, while section 3 discusses the the Diebold and Li(2006) methodology and an affine term structure model Section 4 presents

a comparison of forecasts made by each model while section 5 concludes

2 Data and stylized facts

The main data employed in this study are interest rates swaps maturing in 1,

2, 3, 6, and 12 months’ time In these swaps contracts, a party pays a fixedrate over an agreed principal and receives floating rate over the same prin-cipal, the reverse occurring with his counterpart There are no intermediatecash-flows and contracts are settled on maturity Therefore, we use as proxiesfor yields the fixed rates on swap contracts, negotiated in the Brazilian fixed

1

They, however, find that VAR/VEC models are able to capture future changes in the direction of changes in interest rates.

income market2.

The data is sampled daily and we build monthly observations by averagingdaily yields The sample begins in May 1996 and ends in November 2006,with 127 monthly observations

Table 1 presents descriptive statistics for yields The typical yield curve isupward sloping for time period under analysis The slope and curvature areless persistent than individual yields Both the slope and curvature presentlow standard deviations if compared to individual yields

Place Table 1 About HereFigure 1 presents the dynamics of yields for the period under study

Place Figure 1 About Here

It is important to note that the level and slope are not significantly related with each other (it is never larger than 30%) Curvature is also notsignificantly correlated with the level, however, it’s highly correlated with theslope (approximately -70%) This suggests that perhaps two factors (leveland slope) may explain well the term structure This is particularly true forthe Brazilian term structure of interest rates as for liquidity reasons we haveyields only up to 12-months maturity (which may be seen as the short-termpart of the term structure

cor-3 Yield Curve Models

Litterman and Scheinkman (1991) study the US yield curve, which has apronounced hump-shape, and conclude that three factors (level, slope andcurvature) are required to explain movements of the whole term structure ofinterest rates However, most studies have concluded that the level factor isthe most important in explaining interest rate variation over time

Most yield curve models include the three factors to account for interestrates dynamics Diebold and Li (2006) suggest the following three-factormodel:

2

Unfortunately we do not have information on Brazilian bond yields for long time periods Therefore, we are not able to employ Brazilian bond yields directly.

In order to estimate the parameters β1t, β2t, β3t, λ for each month t linear least squares could be used However, the λt value can be fixed andset equal to the value that maximizes the loading on the curvature factor.Inthis case, one can compute the values of the factor loadings and use ordinaryleast squares to estimate the factors (betas), for each month t We follow thisapproach and also let λ vary freely and compare the forecasting accuracy ofboth procedures.

non-The next step in the Diebold and Li (2006) approach is to assume that thelatent factors follow an autoregressive process, which is employed to forecastthe yield curve

The forecasting specification is given by:

Table 2 presents the results for the estimation of the three factors in theDiebold-Li representation of the Nelson-Siegel model All three factors arehighly persistent and exhibit unit roots, with the exception of β1t in which

we reject the null hypothesis at the 10% significance level These results aresimilar to the ones obtained in Diebold and Li (2006) and suggest that thefactor for the level is more persistent than the factors for slope and curvature

Place Table 2 About Here

3.2 Affine Term Structure Models

In recent years the class of affine term structure models (ATSMs) has becomethe main tool to explain stylized facts concerning term structure dynamicsand pricing fixed income derivatives Basically ATSMs are multifactor dy-namic term structure models such that the state process X is an affine diffu-sion3 and the short term rate is affine in X From Duffie and Kan (1996) weknow that ATSMs yield closed-form expressions for zero coupon bond prices(up to solve a couple of Riccati differential equations) and zero coupon bondyields are also affine functions of X4

In order to study problems related to admissibility5 and identification ofthese models, Dai and Singleton (2000) proposed a useful classification ofATSMs according to the number of state variables driving the conditionalvariance matrix of X For example, when there are three sources of un-certainty6, they group all three-factor ATSMs in four non-nested families:

Am(3), m = 0, 1, 2 and 3, where m is the number of factors that determinethe volatility of X When m = 0 the volatility of X is independent of Xand the state process follows a three-dimensional Gaussian diffusion On theorder hand (m = 3) all three state variables drive conditional volatilities

In this work we adopt a version of the A0(3) proposed by Almeida andVicente (2006)7 The short term rate is characterized as the sum of threestochastic factors:

rt= φ0+ Xt1+ Xt2+ Xt3,where the dynamics of process X under the martingale measure Q is givenby

dXt= −κXtdt + ρdWtQ,with WQ being a three-dimensional independent brownian motion under Q,

κ a diagonal matrix with κi in the ith diagonal position, and ρ is a matrixresponsible for correlation among the X factors

There is a consensus in the literature of fixed income that three factors are sufficient

to capture term structure dynamics See Litterman and Scheinkman (1991) for a seminal factor analysis on term structure data.

7

We remind the reader that our principal aim is to forecast bond yields Then A 0 (3) is

a natural choice since in this ATSM family all factors capture information about interest rate (there is no stochastic factor collecting information about the volatility process) Duffee (2002) tests the forecast power of ATSMs and shows that this intuition is true.

Following Duffee (2002) we specify the connection between martingaleprobability measure Q and physical probability measure P through an essen-tially affine market price of risk

dWP

t = dWtQ−¡λ0 + λ1Xt¢ dt,where λ0 is a three-dimensional vector, λ1 is a 3 × 3 matrix and WP is athree-dimensional independent brownian motion under P

On this special framework the Riccati equations, which defined bondsprices, have a simple solution Almeida and Vicente (2006) show that theprice at time t of a zero coupon bond maturing at time T is

P(t, T ) = eA(t,T )+B(t,T )′Xt

,where B(t, T ) is a three-dimensional vector with −1 − e

−κ i τ

κi in the ithelementand

proce-we assume observations with gaussian errors uncorrelated along time Tofind the vector of parameters which maximizes the likelihood function weuse the Nelder-Mead Simplex algorithm for non-linear functions optimiza-tion (implemented in the MatLabTMfminsearch function)9 Table 3 presentsthe values of the parameters as well as asymptotic standard deviations to

test their significance (parameters not shown on the table are fixed equal tozero).

Place Table 3 About Here

4 Comparison of Forecasting Models

In this section we examine the forecast performance of the Diebold-Li (2006)model assuming a constant λ and allowing for a variable λ with an affineterm structure model and the random walk benchmark

Our forecast accuracy comparisons are based on series of recursive casts, which are computed in the following way We use the first half ofthe sample to estimate the models and build forecasts from one-month totwelve-months ahead We then include a new observation in the sample andthe parameters are re-estimated and new forecasts are constructed This pro-cedure is repeated until the end of the sample These out-of-sample forecastsare used to compute the various measures of forecasts accuracy for yields ofdifferent maturities

fore-The forecasts of the affine term structure model follow Duffee (2002) andtherefore we employ half of the sample to estimate the model and then buildthe forecasts However, we do not re-estimate the parameters

Table 4 records the results for the mean squared error (MSE) of each

of the models used to forecast the term structure of interest rates It ispossible to see that for one-month ahead forecasts the MSE of the randomwalk benchmark is lower than most of the competitor models However,

as we increase the forecasting horizon the Diebold and Li (DL and DL withvariable λ) models seem to gain forecasting accuracy and present lower MSE

Place Table 4 About HereTable 5 presents Diebold-Mariano (1995) tests for equal forecast accu-racy Under the null hypothesis of equal forecast accuracy this statistic has

a standard normal asymptotic distribution We compare all the models withthe random walk benchmark Empirical results suggest that the DL modelprovides better forecasting accuracy than the random walk benchmark forinterest rates up to three-months for twelve-months ahead forecasts (long-term) However, evidence suggests that the random walk provides bettershort-term forecasts

Place Table 5 About Here

5 Final Considerations

This paper compares the yield curve forecasting accuracy of the Dieboldand Li (2006), affine term structure and random walk models The empir-ical results suggest that the Diebold and Li (2006) model provides superiorforecasts, specially at longer time horizons for short-term interest rates.This is the first paper that presents some evidence of forecasting accu-racy for the Brazilian yield curve, with promising results These results areimportant for fixed-income portfolio managers, institutional investors, finan-cial institutions, financial regulators, among others They are particularlyuseful for countries that have implemented explicit inflation targets and useshort-term interest rates as policy instruments such as Brazil

The models proposed in this paper may be used for policy purposes asthey may prove useful in the construction of long-term scenarios for the yieldcurve Nonetheless, more research is needed to develop models that mayprovide reasonable short-term forecasts

Further research could expand the set of models employed to compareforecasting accuracy and study other emerging markets Perhaps modelsthat incorporate other macroeconomic variables would perform well as well.Finally, it would be quite interesting to compare Asian and Latin Americanbond markets

[1] Ang, A and M Piazzesi (2003) A No-Arbitrage Vetor Autoregression

of Term Structure Dynamics with Macreconomics and Latent Variables.Journal of Monetary Economics, 50, 745-787

[2] Bidarkota, P.V (1998) The comparative forecast performance of variate and multivariate models: an application to real interest rateforecasting International Journal of Forecasting, 14, 457-468

uni-[3] Chen R.R and L Scott (1993) Maximum Likelihood Estimation for aMultifactor Equilibrium Model of the Term Structure of Interest Rates.Journal of Fixed Income, 3, 14-31

[4] Dai Q and K Singleton (2000) Specification Analysis of Affine TermStructure Models Journal of Finance, LV, 5, 1943-1977

[5] Dai Q and K Singleton (2002) Expectation Puzzles, Time-VaryingRisk Premia, and Affine Models of the Term Structure Journal of Fi-nancial Economics, 63, 415-441

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[8] Duffie D and R Kan (1996) A Yield Factor Model of Interest Rates.Mathematical Finance, Vol 6, 4, 379-406

[9] Filipovic, D (2001) A General Characterization of one Factor AffineTerm Structure Models Finance and Stochastics, 5, 389-412

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Maturity (Months) Mean Std Dev Maximum Minimum ρ(1) ρ(12) ρ(24)

12 and 24 months