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![Values of β0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta _{0}$$\end{document} and β1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta _{1}$$\end{document} across trading days derived from the Svensson model fitted by minimizing yield errors and using different approaches for initial values. Top panels show values when using approach #1 for initial values. Middle panels display values when using approach #2 for initial values. Bottom panels present values when using approach #3 for initial values. The approaches are defined in Sect. 3.2](publication/350894131/figure/fig4/AS:11431281116271428@1675221534463/Values-of-b0documentclass12ptminimal-usepackageamsmath-usepackagewasysym.png)
Values of β0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta _{0}$$\end{document} and β1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta _{1}$$\end{document} across trading days derived from the Svensson model fitted by minimizing yield errors and using different approaches for initial values. Top panels show values when using approach #1 for initial values. Middle panels display values when using approach #2 for initial values. Bottom panels present values when using approach #3 for initial values. The approaches are defined in Sect. 3.2
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We shed light on computational challenges when fitting the Nelson-Siegel, Bliss and Svensson parsimonious yield curve models to observed US Treasury securities with maturities up to 30 years. As model parameters have a specific financial meaning, the stability of their estimated values over time becomes relevant when their dynamic behavior is inter...
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... Consider for instance the MSE and RMSE metrics time series of Fig. 8: it is possible to notice that the 3F-DNS produced higher error peaks along the whole data sample, especially during turbulent periods. This is probably due to the well-known difficulties of the 3F-DNS model (Wahlstrøm et al., 2021) to fit more dynamic yield curve behavior, i.e. twisted and/or humped shapes, induced by the models lower flexibility. ...
The term structure of interest rates is a fundamental decision–making tool for various economic activities. Despite the huge number of contributions in the field, the development of a reliable framework for both fitting and forecasting under various market conditions (either stable or very volatile) still remains a topical issue. Motivated by this problem, this study introduces a methodology relying on optimal time–varying parameters for three and five factor models in the Nelson–Siegel class that can be employed for an effective in-sample fitting and out–of–sample forecasting of the term structure. In detail, for the in–sample fitting we discussed a two–step estimation procedure leading to optimal models parameters and evaluated the performances of this approach in terms of flexibility and fitting accuracy gains. For what it concerns the forecasting, we suggest an approach overcoming the well–known issue between the stability of factor models’ parameters and the optimal dynamic decay terms. To such aim, we use either autoregressive or machine learning techniques as local data generating processes based on the optimal parameters time series derived in the in–line fitting step. The so–obtained values are then employed to get day–ahead predictions of the yield curve. We assessed the proposed framework on daily spot rates of the BRICS (Brazil, Russia, India, China and South Africa) bond market. The experimental analysis illustrated that (i) time–varying parameters ensure a significant boost in the models fitting power and a more faithful representation of the yield curves dynamics; (ii) the proposed approach provides also stable and accurate predictions.
... Specifically, β 1 is inde-pendent of the time variable and, for this reason, is often interpreted as the long-term level of return; 2, 3, and 4, on the other hand, are dependent on the t variable's trend and decrease as the t variable increases, under the influence of the parameters λ 1 and λ 2 . According to the model, by setting λ, a fixed factor loading on a given maturity was imposed (Wahlstrøm et al. 2022). ...
Green bonds are an increasingly used instrument to catalyze cash flows towards a low-carbon economy. Nonetheless, the existence of an actual price advantage is still uncertain. This research paper aims to assess whether there is a green bond premium (“greenium”) for green bonds relative to conventional bonds with similar characteristics, and how liquidity may affect the determination of a price advantage. It analyzes the yield differentials between green and conventional bonds using three different methods. First, a Nelson-Siegel-Svensson method is executed, estimating the premium both as the yield spreads and as the differentials in Z-spreads. Using a matching method and creating a sample of green and synthetic conventional bonds, the second methodology consists in calculating the distances between each categories’ yield for the same duration. Finally, a fixed-effect regression is performed to better control the liquidity bias. In the first case, a positive premium emerges when analyzing the yield spreads (+37.89 basis points) and the Z-spreads (+10.62 basis points). The second method mitigates the liquidity risk by creating a sample of synthetic bonds and reveals a yield spread of –15.89 basis points. Lastly, the regression method shows a negative greenium equal to –17.1487 basis points. Thus, a greenium emerges from all the three different methods, but its nature, sign, and real determinants are still uncertain. It is, therefore, not possible to conclude a definite price advantage for issuers of green bonds.
... Оценка параметров моделей может осуществляться за счет минимизации средней квадратичной ошибки переоценки доходностей облигаций либо непосредственно на основе цен облигаций. Описание технических особенностей оценки модели можно найти, например, в работе [24]. Мы выбираем вариант оценки модели на основе цен облигаций, поскольку на рынке ОФЗ бескупонные доходности напрямую не наблюдаются. ...
Missing observations in market data is a frequent problem in financial studies. The problem of missing data is often overlooked in practice. Missing data is mostly treated using ad hoc methods or just ignored. Our goal is to develop practical recommendations for treatment of missing observations in financial data. We illustrate the issue with an example of yield curve estimation on Russian bond market. We compare three methods of missing data imputation — last observation carried forward, Kalman filtering and EM–algorithm — with a simple strategy of ignoring missing observations. We conclude that the impact of data imputation on the quality of yield curve estimation depends on model sensitivity to the market data. For non-sensitive models, such as Nelson-Siegel yield curve model, final effect is insignificant. For more sensitive models, such as bootstrapping, missing data imputation allows to increase the quality of yield curve estimation. However, the result does not depend on the chosen data imputation method. Both simple last observation carried forward method and more advanced EM–algorithm lead to similar final results. Therefore, when estimating yield curves on the illiquid markets with missing market data, we recommend to use either simple non-sensitive to the data parametric models of yield curve or to impute missing data before using more advanced and sensitive yield curve models.
Yield curve is a graphical representation of a relationship between interest rates and maturity. Shape of yield curve often attracts attention of analysts, because it represents market implied expectation of future interest rate path. However, the analysis of the yield curve shape often lacks theoretical foundation. It is based either on review of term spreads or on a simple visual investigation. In this article we formally define the shape of the yield curve in terms of function invariants. We use Nelson–Siegel model as a backbone for our classification and show that there exists only six possible shapes of yield curve. They are: a normal upward slopping yield curve, inverted yield curve, humped upward slopping, dipped upward slopping, humped inverted and dipped inverted. We analyze dynamics of zero coupon yield curve in Russian market based on real historical data. We show that transition from an upward slopping curve to an inverted one was always preceded by a hump at mid-tem maturities, while the transition back was always done through the dip. This highlights the importance of mid-term rates in reshapening the curve. We explain this behavior by short end of the curve being linked to the key rate and thus being more sticky. The contribution of the paper is twofold. First, it provides a formal framework to analyze the shape of the yield curve. Second, it describes the patterns in dynamic of the ruble yield curve that can be useful for bond investors in the Russian market.
In this article, we analyse the forecasting performance of several parametric extensions of the popular Dynamic Nelson–Siegel (DNS) model for the yield curve. Our focus is on the role of additional and time‐varying decay parameters, conditional heteroscedasticity, and macroeconomic variables. We also consider the role of several popular restrictions on the dynamics of the factors. Using a novel dataset of end‐of‐month continuously compounded Treasury yields on US zero‐coupon bonds and frequentist estimation based on the extended Kalman filter, we show that a second decay parameter does not contribute to better forecasts. In concordance with the preferred habitat theory, we also show that the best forecasting model depends on the maturity. For short maturities, the best performance is obtained in a heteroscedastic model with a time‐varying decay parameter. However, for long maturities, neither the time‐varying decay nor the heteroscedasticity plays any role, and the best forecasts are obtained in the basic DNS model with the shape of the yield curve depending on macroeconomic activity. Finally, we find that assuming non‐stationary factors is helpful in forecasting at long horizons.
This paper presents an alternative and straightforward two-step estimation method for the Nelson–Siegel yield curve model. The goal is to generate smoothed time series for the time-varying decay parameter and establish stable yield curve factors. To rectify excessive parameter estimates such as jumps or spikes, the decay parameter is adjusted towards its long-run mean using a closed-form expression. Empirical studies conducted with U.S. Treasury data reveal that this method generates stable and easily interpretable outcomes while the confounding effect, which is characterized by large magnitudes with opposite signs among parameters, is effectively mitigated. In out-of-sample forecasting exercises, the proposed model demonstrates comparable or modest performance compared to other competing models, including the random walk model. In particular, the shifting endpoints technique enhances the overall forecasting ability. Finally, the proposed model demonstrates an effective smoothing effect robustly even when applied to other countries.
In this paper, we present a robust predictive comparison of several continuous-time multi-factor models in the context of interbank rates. Recognizing the specific dynamics of the short-term segment of the yield curve, we examine the U.S. money market by extending two continuous-time frameworks with different factor structures, the Chan-Karolyi-Longstaff-Sanders (CKLS) model and the arbitrage-free dynamic Nelson-Siegel (AFDNS) model. A battery of formal forecasting accuracy tests is employed to select a subset of superior predictive models. Despite a better goodness-of-fit measure, additional factors improve the forecasting performance only for the CKLS family. With implications for monetary policy formulation, we found evidence of two separate maturity segments as the three-factor AFDNS and the five-factor CKLS models outperform parsimonious benchmarks in predicting the interbank rates for very short maturities. Our comparative forecasting results are re-confirmed with stronger out-of-sample performance for the five-factor CKLS model when the post global financial crisis sub-sample is analyzed.
