Empirical Evidence of Jump Behavior in the Colombian Bond Market
Evidencia empírica del comportamiento de saltos en el mercado de bonos Colombiano
Nicolás Romero Díaz*
Carlos Alberto Castro Iragorri**
Sebastián Vélez Hernández***
*PhD(C) Accounting & Finance. Doctoral Researcher at Vlerick Business School-Ghent University, Brussels (Belgium). [nicolas.romero@vlerick.com]; [ORCID ID: 0000-0002-5636-9347].
** PhD in Economics. Associate professor, Faculty of Economics, Universidad del Rosario, Bogotá (Colombia). [carlos.castro@urosario.edu.co]; [ORCID ID: 0000-0003-27962368].
*** MSc in Quantitative Finance. Research and Development Analyst, Precia S.A., Bogotá (Colombia). [svelez@precia.co]; [ORCID ID: 0000-0002-9405-4389].
The authors would like to thank Precia S.A. for providing a sample of historical bond prices. Thank you also for the comments and suggestions from Hugo Ramirez, Rafael Serrano, an anonymous referee and participants at the Finance Workshop of Universidad del Rosario, Universidad de los Andes, CESA Business School and Universidad Javeriana.
Artículo recibido: 22 de marzo de 2023 Aceptado: 30 de junio de 2023
Para citar este artículo: Romero, N., Castro, C., Vélez, S. (2023). Empirical Evidence of Jump Behavior in the Colombian Bond Market. Odeon, 119-147. DOI: https://doi.org/10.18601/17941113.n24.07
Abstract
Simulations and empirical studies suggest that incorporating a discontinuous jump process in asset pricing models improve volatility forecasting, pricing of instruments, and hedging positions in a portfolio. In this paper we analyze high frequency market data of Colombian sovereign bonds to study the presence or absence of discontinuities in the price generating process. We find that Colombian sovereign debt experiments jumps across all maturities but with different frequencies, in particular, we do not find that long term bonds jump less frequently than short term bonds. Furthermore, bonds with closer maturities cojump in greater magnitude than those with a greater distance between them. Finally, we find significant day-of-the-week effects, as well as an important increase in the jump frequency due to surprises in economic information related to US monetary policy, and no effect due to direct monetary policy announcements in Colombia.
Keywords: jumps; realized variance; high frequency; preferred habitat theory; monetary policy announcements.
JEL classification: G12, E43, C58.
Resumen
La incorporación de procesos con saltos en la modelación de precios se ha demostrado que mejora el pronóstico de volatilidad, la valoración de activos y las coberturas de un portafolio. El estudio encuentra que en el mercado local de bonos soberanos de Colombia se observan saltos en la formación de precios a lo largo de toda la curva, con diferentes intensidades. Contrario a lo esperado, no se identifica una frecuencia de saltos menor en los bonos de largo plazo en comparación con los bonos de corto plazo. Además, se encuentra que los bonos con periodos de maduración similares tienen una mayor frecuencia de saltos en comparación con aquellos que tienen periodos al vencimiento más distantes. Esto indica una relación entre la proximidad en los periodos de maduración y la ocurrencia de saltos en los precios de los bonos soberanos. En cuanto a las estacionalidades, se encuentran patrones semanales persistentes en la frecuencia de los saltos. Asimismo, se observan aumentos significativos en la frecuencia de los saltos asociados a sorpresas en la información económica que afecta la política monetaria de Estados Unidos. Sin embargo, no se encuentran efectos similares asociados a anuncios específicos de política monetaria interna.
Palabras clave: saltos; volatilidad realizada; alta frecuencia; teoría de Habitat preferido; anuncios de política monetaria.
Clasificación JEL: G12, E43, C58.
Introduction
The mathematical modeling of financial assets is a key aspect of quantitative portfolio management. Stock market participants use it for pricing instruments, hedging positions, and forecasting uncertainty. Pricing models assume that an asset's log-price follows a time-continuous diffusion process, usually a geometric Brownian motion. However, empirical studies and simulations suggest that incorporating pure jump processes is necessary for a correct specification of these models Johannes (2004). Additionally, Johannes (2004) and Andersen et al. (2007) find that explicitly expressing discontinuities in price models improves volatility forecasting, while Piazzesi (2005) finds improvements in the pricing of US treasuries when incorporating the Federal Open Market Committee (FOMC) news announcements as determinants of potential jump times.
Recent literature extends the notion of price jumps to include cojumps, i.e., simultaneous jumps present in different assets; these cojump phenomena were first studied in Barndorff-Nielsen and Shephard (2004a). Bollerslev et al. (2008) find strong evidence for modest-sized but highly significant co-jumps in a panel of high-frequency stock return data. Additionally, Novotny and Urga (2017) find common discontinuities in stock prices within a portfolio. They prove these cojumps can be diversified by means of a correct combination of assets, though a method to find the combination which eliminates these jumps is left as a future endeavor.
Most of the work previously cited is focused on the equity market. Unlike stocks, fixed-income instruments share many characteristics among themselves and are usually only differentiated by maturity and coupon rate. Dungey et al. (2009) find "significant evidence of jumps and cojumps in the US term structure" in response to macroeconomic news announcements. Furthermore, around one-fifth of cojump activity occurs independently of the news. The authors look at this cojump activity and interpret their findings in light of several theories about the formation and evolution of the term structure of the yield curve.
To test the presence of jumps, much of the previous literature uses the statistical test developed by Barndorff-Nielsen and Shephard (2004b) in which two measures of realized volatility are compared and contrasted: realized variance (RV) and bi-power variation (BV). By taking the difference between the former and the latter, we can obtain a notion of the size of a potential discontinuity (see Barndorff-Nielsen and Shephard (2004b), Andersen et al. (2003a), Huang and Tauchen (2005)). Intuitively, jumps are interpreted as the discrepancy between these two measures of realized volatility.
In this paper, we test for the presence of jumps using high-frequency Colombian sovereign bond data. Second, jump behavior is described and characterized by analyzing the frequency and magnitude of its activity. Third, following the procedure presented in Dungey et al. (2009), cojumps across various assets are compared in the context of the two main theories of term structure formation: liquidity preference theory of the term structure and the market segmentation/preferred habitat model. Finally, we look at whether there are day-of-the-week effects or the relationship between jump frequency and economic announcements and surprises.
Results indicate that bonds distributed throughout the Colombian yield curve commonly experience jumps independently of maturity, this is different from what is found in US data where long-term bonds show less jump activity than short-term bonds Dungey et al. (2009). Furthermore, our results indicate that an average of 46.989% of jumps occur simultaneously across two assets. Most commonly, the bonds in the shorter end of the term structure jump simultaneously, though illiquidity hinder a robust analysis of assets on the long end of the yield curve as is the case for the US Goyenko et al. (2011). Daily seasonalities are found in both univariate and multivariate jump activity, with both types of jumps being least likely to occur on Monday. Depending on the sampling frequency, cojumps are most likely to occur on Wednesdays or Thursdays. Furthermore, a panel logit model finds a persistent Thursday effect of an increase of 7% in the frequency of jumps for almost all sampling frequencies. In addition, we find that investors in the Colombian sovereign bond market are more sensitive to external surprises that may impact a change in US monetary policy than local changes in monetary policy or any other economic announcements. In particular, during 2017-2018, unexpected changes in CPI inflation in the US created a 37% increase in the probability of observing a jump, using 5-minute data.
This paper contributes to understanding the dynamics of bond markets in emerging economies. Also, it provides empirical evidence regarding conflicting theories on the terms structure of interest rates (liquidity preference vs preferred habitat hypothesis). In particular, measuring the importance of co-jumps across different segments of the yield curve provides evidence regarding the behavior of investors along the curve.
The rest of the document is organized as follows. Section 1 discusses how different theories regarding the term structure of interest rate can lead to different hypotheses regarding the timing frequency of jumps in different maturities along the yield curve. Section 2 presents the methodologies used to quantify and test for jumps using high-frequency transaction data. Section 3 contains an overview of the bond transaction database and considerations about sampling frequencies and methods. Section 4 presents an in-depth showcasing of results and the corresponding discussion. Finally, section 5 concludes.
1. Investor Preference and the Yield Curve
Measuring jumps on bonds has to consider the term structure of interest rates. Whereas jumps in specific stocks can be analyzed in isolation, jumps in bonds must consider the relationships among the different maturities. Term structure models are based on the idea that there exists a lower-dimensional set of variables (factors) that capture most of the movements across the different maturities. Therefore, it is important to consider how much of this co-movement is related to the discontinuity component of the data-generating process. Although this is beyond the scope of the paper, we empirically test for the relationship among jumps in different maturities.
The theories of liquidity preference and preferred habitat/market segmentation are two theories about how the term structure of the yield curve forms and evolves over time. Liquidity preference argues that yields of longer-dated bonds are higher due to a liquidity risk premium. This liquidity risk premium arises from the greater possibility of capital loss in long-term bonds compared to shorter-term debt. Consequently, a greater risk of loss would imply that long-dated bonds are more reactive to macroe-conomic news announcements and external shocks than short bonds. Thus, according to liquidity preference theory, in any country, we would expect to find greater jump activity in bonds of longer maturities.
On the other hand, the preferred habitat hypothesis argues that individual investors operate in different segments of the term structure according to their particular interests. Thus, movements in the yield curve respond to the supply and demand pressures of investors who populate different market sections. For example, speculators who want to maximize profits may be more interested in trading short-maturity bonds due to their liquidity. In contrast, pension funds or insurance companies may choose to trade long-term bonds to fund future liabilities. Originally, this model assumed a rigid segmentation of markets. Modigliani and Sutch (1966) argue against this premise, proposing that investors may operate outside their preferred segments if a risk premium compensates their aversion to reinvestment risk.
In this context, since prices respond to the local behavior of different investors, the short, medium, and long-term yields would be independent of one another. Thus, it is reasonable to expect that if speculators and arbitrageurs tend to operate in the short end of the term structure, news and announcements may cause a greater impact on short yields. At the same time, long bonds would be reactive to news relevant to the long-term state of the economy. This qualitative overview of two theories of term structure behavior will give us the guiding principles in our analysis of jump behavior. In addition, under the preferred habitat hypothesis, we expect bonds with similar maturities to "jump together" more frequently than bonds that are further apart. We explore this specific hypothesis in section 4.3.
2. Measuring and Testing for Jumps
Continuous time diffusion models are a vital tool in modeling the price evolution of financial instruments. Their analytic convenience makes them an extremely useful tool for drawing interpretations and simplifying hedging calculations based on modern financial derivatives. These models commonly assume that the change of an asset's log-price pt follows the stochastic differential equation:
where µt is the instantaneous drift given by a locally bounded variation process and σt is a strictly positive volatility process with well-defined limits. Wt is a Brownian motion. Under the premise of equation (1) the j-th intraday log-return is defined as rt,j = pt,j - pt,j-1. The associated quadratic variation of this model is given by:
In what follows, we assume that the data-generating process for a bond's log price is given by:
The new third term is a pure jump Levy process, where LJ(t) - LJ(s) = Σs≤τ≤t k(τ) is the jump size. We assume this is a case of the Levy process, known as a Poisson compound process. In other words, the number of jumps Nt follow a Poisson distribution. We also assume a constant jump intensity λ related to the frequency of the jumps and jump size k(τ) as an identically distributed (i.i.d.) random variable related to the magnitude of the jump. Now, the quadratic variation for this model is:
In the more general process, expression (3), the quadratic variation includes the jump size.
Asymptotically realized variance (RV) could give us a good approximation of the quadratic variation:
Definition 1 Realized variance:
where M denotes the number of intraday returns that are used to measure the realized volatility of day t. This means that, for our jump-diffusion model, the realized variance converges to expression (4) in the limit:
Equation (5) gives us an estimate of daily volatility which captures the effect of the volatility process σt as well as the magnitude of variance attributed to discontinuous jumps, given by . Barndorff-Nielsen and Shephard (2004a), as well as their following extensions in Barndorff-Nielsen and Shephard (2005a) and Barndorff-Nielsen and Shephard (2005b) suggest that, under reasonable assumptions, bi-power variation enables a consistent estimator of quadratic variation that is robust to jumps:
Definition 2 Bi-power variation:
This definition of bipower variation (BV) is multiplied by a coefficient of standardization µk, which allows for a direct comparison with RV. This coefficient is given by µk Ξ 2k/2 [(k + 1)/2] /
(1/2) for k > 0. Asymptotically, we have:
We can use the fact that BV is robust to jumps, while RV is not, to obtain a notion of the size of a jump. By taking the difference between (5) and (6), asymptotically, we get:
Equation (7) implies that we can obtain a consistent estimate for the size of daily jumps. Despite this, for finite samples, the difference between RV and BV is not guaranteed to be positive. We can truncate its value at zero and consider only positive values.
Instead of analyzing the magnitude of jumps, studying the relative contribution of jumps to price variance is more interesting. Thus, an initial expression of the jump statistic (JS) in Barndorff-Nielsen and Shephard (2004a) is given by:
Where the original difference in volatilities is now divided by a coefficient that standardizes the statistical distribution. This coefficient introduces the term , which determines the scale of equation (7) in units of conditional standard deviation (see Huang and Tauchen (2005)). A jump-robust estimate of this term is given by tri power quarticity (TQ):
Definition 3 Tripower quarticity
TQ is accompanied by the scale normalizing constant M since each absolute return is of the order . Since M is of order
, the whole expression approaches to one well-defined limit.
Even so, Huang and Tauchen (2005) find that simply using TQ tends to over-reject the null hypothesis of no jump. Instead, they propose the following modification:
Several authors (Barndorff-Nielsen and Shephard (2005a), Andersen et al. (2001), Andersen et al. (2003b)) argue that finite sample performance may be improved by basing the jump test on the log-difference of the realized measures, i.e.:
This implies that the numerators of equations (8) and (9) have the same asymptotic distribution. According to Huang and Tauchen (2005) this is due to the fact that the first-order Taylor expansion term of both numerators, centered around the asymptotic mean of BV (i.e. ds), have the same distribution. Then, the difference of both realized (and log-realized) measures generate the same asymptotic distribution. Thus, equation (9) is the expression used to test the presence of jumps in our empirical application.
3. Data
Our database consists of intraday transactions on the Mercado Electrónico Colombiano (MEC) operated by Bolsa de Valores de Colombia (BVC). The dataset includes price information from January 2, 2017 to December 28, 2018. This includes operations for a total of 485 trading days. Colombian sovereign debt is issued in Colombian peso (COP) and Unidad de Valor Real (UVR)1. Despite having data for both types of assets, only COP issuances are considered since they are more liquid.
Mnemonic conventions for Colombian debt titles enconde information about the bond's coupon, year of issuance, and maturity. For example, TFIT16240724 is a fixed coupon treasury (TFIT) issued in 2016 (TFIT16) with an expiration date on July 24, 2024 (TFIT16240724). For the sake of brevity we will denote bonds only by their expiration year in our discussions, i.e., we will refer to TFIT16240724 as T24.
3.1. Bond Selection Criteria
Bonds were selected for analysis according to the following criteria: i) Liquidity: Since the jump detection approach detailed in the theoretical framework is based on the asymptotic distributions of realized measures of variance, the most active assets will return the best results; ii) Maturity: The two theories discussed in section 1 provide some hypothesis on the jump behavior for bonds of different maturities. Thus, choosing bonds with maturities distributed along the term structure allows for an interesting comparison of jump behavior in light of those hypotheses.
For Colombia's sovereign bonds, these two criteria present a serious challenge. The local market has few agents trading day to day, which means liquidity is generally low. Additionally, most of these agents trade mainly short and medium-term bonds. This means long-term debt is much more illiquid since market participants buy or sell long-term bonds mostly to comply with regulations and to match long-term liabilities. Consequently, an analysis at the shorter end (less than 5 years) of the term structure will be much richer in comparison to the longer end (more than 10 years).
Table 1 displays daily descriptive statistics for all bond transactions. Maturity, total trading days, and average and median transactions are presented, as well as average and median Inter-Arrival Times (IAT). IAT is defined as the time interval between transactions, thus, IAT is lower for more liquid assets and greater for illiquid ones. Values reported in this table help us quantify the daily liquidity of each title. For example, the T24 bond has on average 189.971 transactions per day. Furthermore, IATs suggest that each transaction occurs every 89 seconds on average.
This means that this bond is much more liquid than the T20 bond, which trades around 24 times each day, with each transaction occurring every 7 minutes and 18 seconds on average. The bonds chosen for analysis are: TFIT06211118, TFIT06110919, TFIT15240720, TFIT-10040522, TFIT16240724 and TFIT16300632, hereafter T18, T19, T20, T22, T24, T32. In other words, if we take 2017 as a base year we are considering bonds with 1,2,3,5,7 and 15 years to maturity. Even though these bonds are the most traded, illiquidity remains a real challenge. Only T18, T20 and T24 have on average more than 10 transactions per day, while the only long-term bond (T32) has on average 4.25 transactions per day.
3.2. Data Sampling and Microstructure Noise
To apply the jump test in equation (9), our trade data must be sampled at equal discrete time intervals Dungey et al. (2009). Sampling high-frequency data entails the following trade-off: c hoosing a h igh sampling frequency captures more information about the evolution of the real-time price formation process at the cost of greater microstructure noise. On the other hand, a lower sampling frequency minimizes noise, at the expense of masking information about the asset's instantaneous market price.
Even though optimal sampling frequency tests exist, their results differ for different bond maturities (Zhang et al., 2005; Bandi and Russell, 2006). Different sampling frequencies for different bonds make comparisons across different assets impossible. For this reason, instead of using optimal frequency tests, the empirical literature cited so far applies several sampling frequencies for assets under consideration to compare and contrast the effects of sampling frequency on the jump test. We will replicate this procedure, sampling at 5, 10, 15, and 30 minutes intervals.
The optimal sampling method is also a source of debate among academics. Dungey et al. (2009) take the last price within a time interval as representative of the market price within that interval. Sheppard (2006) argues that this approach may lead to scrambling problems2 and could also skew the covariance of returns to zero for larger sampling frequencies.
On the other hand, Lee and Mykland (2012) propose a non-parametric approach which assumes that market noise has a zero-mean distribution. This way, taking local averages of prices within time intervals asymptotically removes noise from the underlying market price. Even though the authors assume that data is of ultra high frequency, we will adopt this method as our sampling procedure since scrambling problems are of greater magnitude for the more illiquid assets we are studying.
3.3. Additional Statistics
This section presents additional information about daily bond transactions. Tables 2 and 3 present the same statistics as table 1 for each year in our sample. As previously mentioned, IAT for more liquid assets are smaller than for illiquid assets since the time between transactions is shorter, thus, their values would cluster near zero in the distribution. We have decided to crop IAT values at 3600 seconds since intervals larger than an hour are uncommon.
Figures 1 through 6 showcase the number of transactions of the selected bonds during all trading days of 2017-2018. Additionally, IAT distributions for the selected bonds are included. This information on the trading activity in the bond market also shows the impact on expected changes on the incentives on market makers in the bond market. At the end of 2018, the treasury reduced the financial incentives for financial institutions participating in the primary bond market3. The incentive system in the Colombian bond market was established in the late nineties to foster the development of the bond market. However, recent studies indicated that the level of incentives was not necessary and created unnecessary trading activity from financial institutions in the secondary market in order to obtain the incentives in the primary market4. In particular, in Figure 5 we observe a large drop in the number of transactions at the end of 2018 for the most actively traded bond, T24.
4. Empirical Results
4.1. Univariate Jumps
Table 4 summarizes the results of applying equation (9) for 5, 10, 15, and 30 minutes sampling frequencies at a 5% significance level. Despite trading for 472 out of the 485 total days, the T20 bond exhibits the most active jump behavior at all frequencies except for 30 minutes, jumping 68.4% of the time at a 5-minute frequency and 55.2% on average. On the other hand, the T24 and T32 bills are among the least likely to jump. T24 jumps 55.7% of the time at the 5-minute sampling frequency, but this rejection rate quickly drops below 30% for all other frequencies. Meanwhile, the T32 rejection frequency increases for lower sampling frequencies.
This trend of lower sampling frequency resulting in higher rejection rates is unexpected since the presence of noise in higher sampling frequencies should generate more rejections of the test statistic. Out of the six bonds studied, this inverse relationship between frequency and rejection is present in the more illiquid assets: T19, T22, and T32. Table 4 reveals that these assets increase the number of detected jump days when the sampling frequency decreases. This may indicate that the lower sampling frequency captures more information about transaction dynamics in illiquid assets. Thus, when the average of the time interval is taken, the longer time intervals allow for a more representative average price.
In contrast, bonds characterized by higher liquidity demonstrate an upward trend in rejection rates as the sampling frequency increases. For instance, the rejection rate for T24 escalates from 0.14 to 0.245, 0.272, and 0.557 as the sampling frequency decreases from 30, 15, 10 to 5 minutes, respectively. This observation aligns with the notion that higher sampling frequencies tend to introduce greater noise, which is consistent with the findings reported by Dungey et al. (2009) for the United States. However, our research in the context of Colombia diverges from their conclusions as we discover that jumps are less prevalent in longer-term bonds when compared to short-term bonds. We do not identify any discernible relationship between bond maturity and the frequency of univariate jump rejections.
Graphical representations of jump test results for the 30-minutes sampling frequency are shown in figure 7. This plot shows the value of the jump statistic for each day in proportion to its critical value. It is clear by the observation that jumps are a common occurrence for fixed-income instruments, which suggests that simultaneous jumps across different assets are a real possibility. We study the cojumping behavior in detail in the next section. Univariate test results for different sampling frequencies are included in section 6.
4.2. Multivariate Jumps
As a complement to the univariate jump test, we can also consider the case of multiple bonds of different maturities jumping on a given day. This cojump behavior can be gauged by studying coexceedances, an approach developed by Bae et al. (2003) in the context of financial contagion and the occurrence of extreme events. A coexeedance occurs when, on a particular day, a bond of maturity i exceeds the jump statistic's critical value given that a bond of maturity j has also surpassed the critical value for the same day. This would imply that the assets have jumped synchronically at the daily level.
More formally, the procedure is as follows. We begin by looking at the individual time series of JSi,t values for each bond. A dummy variable di,tindicates if a bond of maturity i exceeds the statistic's critical value at day t:
With the series of dummy values for each bond, the number of coexceedances will be given by the sum of all di,t for i ≠ j given that j = 1
We have decided to limit the cojump analysis to the T18, T20, T24, and T32 emissions, since the first three are the most liquid and T32 is the longest-dated bond in our database. This means that the number of coexceedances will range in values from 0 to 3, where 0 indicates the number of unique jumps and 3 is the number of times when all bonds jump in a given day.
Table 5 presents the coexceedance results for all sampling frequencies and the total number of jumps. Interestingly, jumps of two assets (i.e. coexceedance iquals 1) are the most common event, followed by unique jumps (i.e. coexceedance iquals 0). The least common occurrence is the simultaneous jump of all four bonds. Furthermore, these results persist across all maturities and sampling frequencies, which may point to an underlying dynamic that causes this behavior in the Colombian bond market.
Averaging the 2 asset cojump proportions across bonds and maturities (except for T32 at 5 minutes) accounts for 46.989% of all jump activity. This implies that when the term structure experiences a jump, it generally does so in tandem with another part of the curve. Identifying exactly which pair of bonds cojump, will be analyzed in detail in section 4.3; for the moment, we observe that T18 and T20 experience many more 2-asset cojumps at all frequencies.
4.3. Cojump Pairs
By limiting our view to coexceedances of only two assets, we can see how their cojump behaviour evolves over time. We make this by defining a counter which keeps track of every time a coexceedance occurs for a pair of bonds. Everytime Ej,t|dj,t = 1= 1, the counter goes up by 1. When graphing this counter's values as a time series, this procedure has a convenient interpretation since the steepest curve indicates the most active pairing of cojumping bonds. Figure 8 shows the evolution of the cojump pairs for all sampling frequencies considered: T18-T20 as a green dashed and dotted line; T18-T24 as a solid orange line; and T20-T24 as a dashed blue line. Figure 9 graphs the same dynamic for the T24-T32 (dash and dot), T20-T32 (solid), and T18-T32 (dashed) pairs.
Our interest lies in comparing cojump behaviour of bonds distributed throughout the term structure. Thus, the following analysis is made clearer by referring to these pairs by the difference of their constituent's bond maturities. From smallest to largest difference, the pairs will be: T18-T20: 2Y pair; T20-T24: 4Y pair; T18-T24: 6Y pair. The second set would be T24-T32: 8Y pair; T20-T32: 12Y pair; T18-T32: 14Y pair.
The results at 5 minute sampling tend to align with the preferred habitat theory, since the two pairs of closest maturities, 4Y and 2Y, show the most (and second most) cojump activity. 4Y jumps 185 times, 2Y does so 161 times, while 6Y counts 129 coexceedances in our sample. Comparisons with the 10, 15 and 30 minute samplings show that 2Y is consistently the most active pair, with 4Y and 6Y being second and third. These results strengthen the case for cojump behavior following the market segmentation hypothesis, which foresees bonds of nearer maturities reacting similarly to external shocks. Yet, for sampling frequencies of 10, 15, and 30 minutes, the 4Y and 6Y pairs tend to move more in tandem with each other. This low cojump number is explained by the low univariate activity of the T24 bond at those frequencies, since T24 only jumps on 132, 119, and 68 days respectively (see table 4). Thus, pairs which contain T24 will have fewer days on which a possible coexceedance may occur.
Meanwhile, casual observation of figure 9 tells us that pairs of dissimilar maturities are much less active than those with similar maturities. Across all samplings, 12Y shows the most coexceedances, followed by 14Y (except at 5 minutes) and 8Y. Thus, we find no constructive evidence for either theory of the term structure of interest rates. Yet, we may replicate the argument that low univariate jump activity is responsible for the low cojump count for these pairs. In this case, the low activity of T32 constrains the number of days for a coexceedance to occur. Since T20 is the most active bond, it has the most chance of cojumping with the T32 bond. By the same logic, T24 is the least active bond, making the T24-T32 pair the least likely to cojump. Our results for sampling frequencies other than 5 minutes reflect that this is indeed the case.
4.4. Stylized Facts of the Colombian Bond Market
This section studies daily jump seasonalities: the daily distribution of the jump test results is studied in subsection 4.4.1. This allows us to more formally define a panel logistic regression model for a binary outcome of jump versus no jump. This approach lets us include central bank announcements. These results are presented in subsection 4.5.
4.4.1. Daily Distribution of Jumps
It is possible that both univariate and multivariate jumps exhibit daily seasonalities. For example, Das (2002) explicitly models day-of-the-week effects on jump behaviour and finds that jumps are more likely to jump on Wednesdays due to option expiry effects. Even though the procedure we have followed does not capture daily effects, we can observe the distribution of jumps and cojumps to check for daily patterns. Results of this analysis are presented in table 6.
For all sampling frequencies and almost all bonds, the least likely day for a jump to occur is Monday. Only T18, and T19 at 30 minute sampling, deviated from this behaviour. On the other hand, for all other weekdays, the assets studied did not reflect any particular seasonality that allow us to identify a most common jump day. For example, Wednesday, with 10 and 30 minute sampling frequencies, has on average 22.1% and 22.4% of jumps happening on this day of the week. At 5 and 15 minute sampling frequencies, Thursday has on average 22.5% of jumps occurring that day for both frequencies.
The results for cojumps exhibit some similarity to univariate jumps. Analyzing only jumps of more than one asset (coexceedance > 0) no particular day at any sampling frequency stands out as one where a cojump is most likely to happen. As was the case for univariate jumps, the least likely day for cojumps is once again Monday. The apparent Monday effect found in idiosyncratic jumps and cojumps contradicts the findings for US treasuries presented by Dungey et al. (2009), where the authors do not find any evidence of daily seasonalities for neither jumps nor cojumps. Day of the week effects are studied more formally in the next subsection, as well as the effect of economic announcements on jump activity.
4.5. Economic Announcements and Jump Activity
In this section, we are interested in estimating the impact of different economic announcements on jump activity. To do this, we define a panel logistic model which specifies the event of a jump occurring as a function of weekdays and economic announcements. The model is specified as follows:
where J*i,t is the result of the jump test applied to bond i at day t in proportion to the critical value. The identity function transforms the continuous values of the jump test into a binary outcome model which takes a value of 1 when the critical value is exceeded and zero otherwise. The Dk terms control for a day of the week, from Tuesday through Friday. We expect the βk coefficients to be positive since we found that Monday is the least likely day for a jump to occur. We estimate a random effects model,
εi,t = τi + ei,t
Where, ei,t ∼ iidN(0, σ2e) and τi ∼ iidN(0, σ2τ), captures the unobserved heterogeneity across the propensity of different maturities to jump.
We consider different types of announcements and sources. First, we consider announcement as an indicator variable (i.e, DAnnouncement takes value of 1 and 0 otherwise) on days that denote the date of news releases or the day after if the release is after the market closes. The announcements macroeconomic information from Colombia and the US Monetary policy (interest rate announcements and FOMC meetings), CPI, Unemployment rate, Underemployment rate, GDP, Consumer confidence, trade balances, durable goods, and rating changes on Colombian sovereign debt. Second, we also consider a different indicator variable that takes a value of 1 if the released indicator deviates from the expected value (from a survey of forecasters). This second definition provides a way to control the content of the announcement and whether the surprise contained in the information is related to the jump rather than just the type of information that is being released to the public.5
Table 7 presents logistic regression results and the average marginal effects for the most relevant variables in terms of statistical significance. Several day-of-the-week effects are found for 5, 10, and 15-minute samplings. We report positive Tuesday and Thursday effects, the former is especially important because it is consistently significant. At these frequencies, jumps are about 7.6% more likely to occur on Tuesdays and about 6% more likely to occur on Thursdays. The Thursday effect is robust to the introduction of different types of economic announcements. This result deviates from what is observed in Table 6 where we find a relatively similar distribution of jumps along weekdays, with a lower amount on Mondays.
With respect to economic announcements and surprises we have mixed results. Overall, we find that very few variables have an impact on the jump frequency, for example, at the 15 minute sampling frequency we do not find any significant effects. In particular, among the different sampling times we do not find common variables that increase the jump frequency, and we find that CPI inflation surprises regarding US data are more important for the 5 minute and 15 minute sampling frequencies. For the 10 minute sampling frequency surprises related to the Colombian trade Balance increase the probability of a jump by 8.7%. However, it is specific shocks rather than US (αNewsUS) or Colombian shocks (αNewsCOL) that are relevant because when we aggregate all types of announcement or surprises, by country, the effect is not statistically significant. When taking the sample we also observed two announcements regarding a stable and one negative outlook (by Fitch on February 22, 2018) on the Colombian sovereign rating. However, we find no statistically significant effect on the jump frequency and also there are mixed results regarding the sign across the different sampling frequencies.
Looking closely at the 5 minute sampling frequency and the 37% increase in the jump frequency due to the increase in the CPI inflation surprise in US, we find a possible explanation of the importance of external shocks to internal shocks. During the sampling period, 2017-2018, and further on in 2019, there was a succession of US CPI inflation reports that have been significantly above expectations; these reports raised questions regarding the tightening of monetary policy6. On the other hand, during the same period, 2017-2018, CPI inflation in Colombia was in line with the Central Bank's target. Therefore, it is not surprising that, during the period, investors in the Colombian sovereign bond market were more sensible to changes in the monetary policy in the US than any local shock.
We consider a broad range of announcements and surprises regarding economic conditions, both internal and external shocks (US), and find that the jump frequency is sensitive to specific shock that can have an incidence on monetary policy but not the policy announcement themselves. It is also important to note that external shocks seem to be more relevant than local shocks. Furthermore, we find systematic day-of-the-week effects that should be analyzed further to determine whether they provide arbitrage opportunities.
5. Conclusions
In this document, we have found that price discontinuities are a common occurrence for Colombian sovereign bonds. The results presented in sections 4.1 and 4.2 show the extent of this activity, though no relation was found between maturity and univariate jump presence. There is no bond that stands out above others in terms of jump activity, though T24 was the least active bond for all sampling frequencies except for 5 minutes. Further research should explore the relationship between jump activity and liquidity along the term structure.
By looking at the daily coexceedances, we can extend the notion of jumps to include simultaneous discontinuities across assets, which is interesting because of its effects on the yield curve. An analysis of the results shows that almost half of all jump activity consists of the cojumping of two bonds. In particular, the assets which cojumped the most were the ones with the shortest distance between maturities. This seems to suggest that the behavior of the local market falls more in line with the market segmentation theory, as opposed to the liquidity risk premium hypothesis.
The widespread presence of jumps in bond prices allows for an interesting study of their weekly distribution. For both univariate and multivariate jumps, the least common jump day is Monday. On all other weekdays, no strong evidence indicates more or less jump frequency between the days.
A panel logit model for 6 bonds allows for a formal study of daily jump seasonalities and the effects of economic announcements and surprises. As we expected, there are multiple positive and significant day-of-the-week effects that diminish in number and significance with s ampling frequency. In particular, a persistent Thursday effect was found for every sampling frequency except 30 minutes. We also find that jumps are determined by surprises and specific economic variables rather than just the monetary policy announcements. In addition, we find that investors in the Colombian sovereign bond market are more sensitive to external surprises that may impact a change in US monetary policy than local changes in monetary policy or any other economic announcement.
Notas
1 UVR represents the purchasing power of the Colombian peso and is defined as the price of a predetermined basket of goods and services.
2 Taking the last price in each time interval could result in intervals of uneven length. Since we need equal length intervals, this problem is known as scrambling.
3 In Colombia, Ministerio de Hacienda y Crédito Público is the institution in charge of issuing sovereign bonds.
4 Here is a recent post (in spanish) that describes the regulatory changes.
5 We obtain the dates of the announcement and the information regarding the observed and the expected macroeconomic indicator from Bloomberg.
6 A discussion by Gregory Mankiw in The New York Times.
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6. Appendix A: Complementary Results
This section includes the remaining results omitted in this chapter's previous discussions. Proportion of exceedance results are shown in figures 10, 11, and 12 for 5, 10, and 15 minute sampling frequencies. These results help highlight the interpretations given above, as well as illustrate the difficulty that liquidity imposes on our analysis.