Insper
Programa de Mestrado Profissional em Economia
Débora Darin
Carry and momentum in the
Brazilian yield curve
São Paulo
2020

Débora Darin
Carry and momentum in the
Brazilian yield curve
Dissertação apresentada ao Programa de
Mestrado Profissional em Economia do Insper
como parte dos requisitos para obtenção do
título de Mestre em Economia.
Área de concentração: Economia dos
Negócios
Linha de pesquisa: Finanças
Insper
Programa de Mestrado Profissional em Economia
Supervisor: Gustavo Barbosa Soares
São Paulo
2020
Darin, Débora.
Carry and momentum in the
Brazilian yield curve. / Débora Darin. – São Paulo, 2020.
52f.
Dissertação (Mestrado) – Insper
Programa de Mestrado Profissional em Economia, 2020.
Orientador: Gustavo Barbosa Soares
1. Risk Premia. 2. Yield Curve. 3. Carry. 4. Momentum. I. Gustavo Barbosa
Soares. II. Insper Instituto de Ensino e Pesquisa. III. Departamento de Economia.
IV. Carry and momentum in the Brazilian yield curve.
Débora Darin
Carry and momentum in the
Brazilian yield curve
Dissertação apresentada ao Programa de
Mestrado Profissional em Economia do Insper
como parte dos requisitos para obtenção do
título de Mestre em Economia.
Área de concentração: Economia dos
Negócios
Linha de pesquisa: Finanças
DATA DE APROVAÇÃO: São Paulo, 18 de dezembro de 2020.
BANCA EXAMINADORA
Prof. Dr. Gustavo Barbosa Soares
Insper Instituto de Ensino e Pesquisa
Prof. Dr. Michael Viriato Araújo
Insper Instituto de Ensino e Pesquisa
Prof. Dr. Rafael Porsani
BWGI Brasil Warrant Gestão de
Investimentos

AGRADECIMENTOS
Agradeço ao Prof. Dr. Gustavo Barbosa Soares, orientador desta dissertação, que me
auxiliou na escolha do tema e me deu suporte durante todo o desenvolvimento do projeto.
Aos meus pais, Eliana e Waldir Darin, que desde cedo me ensinaram a importância
da educação e me apoiaram durante toda minha formação acadêmica.
Aos meus colegas de turma, pelo companheirismo e pela rica troca de conhecimentos
ao longo de dois anos de curso.
Ao meu companheiro, Rafael Castro, por todo o incentivo para que eu alcançasse
meus objetivos nesta jornada.

ABSTRACT
The return premia of the yield curve can be comprehensively described by a perspective
of style factors, a set of characteristics that includes measures such as value, momentum
and carry, applied to bonds, according to Brooks and Moskowitz (2017). Based on the
referred article, this paper focuses on the information about expected returns that can
be extracted from the slope and the curvature of the yield curve (therefore, carry and
momentum strategies) across all levels of the yield curve, in a multi-maturity model. This
study combines carry and momentum strategies applied to the Brazilian yield curve, and
verifies whether it is possible to achieve superior returns by applying this combined style
factor trading strategy to the curve, regardless of the interest rates levels in the time-series.
Additionally, this study provides evidence on the application of carry and momentum
strategies to the Brazilian One-day Interbank Deposit Futures, and verifies its adherence
to the existing literature.
Keywords: Risk Premia. Yield Curve. Carry. Momentum.

RESUMO
O prêmio presente na curva de juros pode ser descrito de maneira abrangente através
de uma perspectiva de style factors, um conjunto de características que inclui medidas
como value, momentum e carry aplicadas a títulos de renda fixa, de acordo com Brooks e
Moskowitz (2017). Com base no artigo citado, o proposto estudo discorre sobre os retornos
que podem ser extraídos a partir da inclinação e da curvatura da curva de juros, em um
modelo que explora e compara contratos com diferentes vencimentos. O estudo combina
estratégias de momentum e carry ao longo da extensão da curva de juros brasileira, e
verifica se é possível obter retornos em excesso à taxa de juros livre de risco ao aplicar
uma estratégia combinada de fatores sistemáticos à curva, independentemente do nível
das taxas de juros e de outras variáveis econômicas na série temporal. Adicionalmente,
este estudo fornece evidências sobre a aplicação de estratégias de carry e momentum para
os contratos futuros de DI (Futuro de Taxa Média de Depósitos Interfinanceiros de Um
Dia), e verifica a aderência de seus resultados à literatura existente.
Palavras-chave: Prêmio de Risco. Curva de Juros. Carry. Momentum.

EXECUTIVE SUMMARY
In the financial market, traders benefit from finding and exploring price asymmetries.
As explored in the financial literature, Brooks and Moskowitz (2017) associate the yield
curve premia with characteristics that include measures such as value, momentum and
carry, a pattern that has been similarly found in the Brazilian yield curve by Shousha
(2005).
This study extends the present literature, applying concepts of the yield curve premia
to the Brazilian yield market in order to quantify its results through a set of performance
measures.
Through developing a quantitative trading model, this paper focuses on analyzing the
premium extraction from the Brazilian yield curve. The analysis will begin from extracting
information about expected returns from the slope and curvature of the curve across its
different levels (multi-maturity model), verifying if it is possible to achieve consistent
returns from these factors, and then proceeding to compare the achieved performances
between contracts with different maturities, not only within the same strategy, but also
between different strategies.
Much of the knowledge regarding trading strategies often keeps restrained to players
in the financial markets industry. Therefore, this field has not been completely explored in
the academic scope. The findings in this study should enrich the knowledge on both the
fields of trading strategies and of the Brazilian interest rates market, and consequently
open opportunities for further academic development in each of these areas.
The model focuses on the premium in the Brazilian yield curve that can be extracted
from alternative factors that are purely technical, and do not directly rely on monetary
policy decisions or economic forecasts. Therefore, the model elaborates on a Carry Strategy,
a Momentum Strategy, and a set of combinations of both models, in order to result in a
superior strategy. The combinations tested variations with and without risk control, and
three variations of trading frequency: daily, weekly and monthly.
An asset’s carry can be defined as its futures (or synthetic futures if none exist)
return, assuming that prices stay the same, as defined by Koijen et al. (2018). It is a
model-free characteristic that is directly observable ex-ante from futures prices, whereas
the expected price appreciation must be estimated using an asset pricing model.
The concept of time-series momentum is explored by Moskowitz, Ooi and Pedersen
(2012). Through this perspective, a security presents some persistence in returns for one to
12 months that partially reverses over longer horizons, consistent with sentiment theories
of initial under-reaction and delayed over-reaction.
The analyzed assets were the Brazilian interest rate futures denominated "DIs", and
traded on the Brazilian exchange "B3". The time series comprised data from ten years of
observations, from eight different length vertices from the Brazilian yield curve.
When compared to the Brazilian risk-free rate (CDI), and to a pure carry and a
pure momentum-based strategy, the combined strategies model proposed in this paper has
proven to overperform not only the considered benchmark, but also both of the individual
strategies. This exceeding performance has been further enhanced by the introduction of a
risk management tool in the model, named a volatility cap.
Additionally, the paper results indicated that the Low Volatility Anomaly, a widely
know phenomenon in the financial literature, can be observed among the Brazilian DI con-
tracts, in line with Baker, Bradley and Wurgler (2011) findings. The contracts with shorter
maturities, and shorter volatilities, have presented greater performances in comparison
with the contracts with longer maturities.
The present research contributed to the field of quantitative finance as it aimed to
verify if superior returns can be achieved through a combination of carry and momentum
strategies, which are strategies based on the expected returns from the slope and curvature
of the yield curve, in comparison to the application of each strategy separately or to the
benchmark risk-free rate.
Moreover, this paper provided information about the application of carry and
momentum strategies over the Brazilian yield curve, by means of a time series of exchange
traded future contracts, in line with the theory found in previous studies from the financial
literature that mainly focus on developed countries.
LIST OF FIGURES
Figure 1 – MACD - 1 Year Contracts (EWMA half lives = 12 and 26 periods) . . 26
Figure 2 – Interpolated Brazilian Yield Curve at 15/05/2020 . . . . . . . . . . . . 34
Figure 3 – Annualized Volatility (1 Month, 1 Year): 1 Year Contracts . . . . . . . 37
Figure 4 – Carry Model Backtest . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
Figure 5 – Carry Strategy Sharpe Ratios . . . . . . . . . . . . . . . . . . . . . . . 40
Figure 6 – Momentum Model Backtest (12, 26 MACDs) . . . . . . . . . . . . . . . 41
Figure 7 – Momentum Strategy Sharpe Ratios . . . . . . . . . . . . . . . . . . . . 42
Figure 8 – Brazil Selic Target Rate vs. Change of Signal (1 Year Contracts) . . . . 43
Figure 9 – Combined Strategies Backtest . . . . . . . . . . . . . . . . . . . . . . . 44
Figure 10 – Combined Strategies Sharpe Ratios . . . . . . . . . . . . . . . . . . . . 45
Figure 11 – Sharpe Ratios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

LIST OF TABLES
Table 1 – DIs contracts by maturity and price, with open interest greater than
zero, in 15th of May, 2020. . . . . . . . . . . . . . . . . . . . . . . . . . . 33
Table 2 – DIs contracts after open interest and liquidity filters, as of 2020 May 15th 35
Table 3 – Carry Strategy: Performance Measurement . . . . . . . . . . . . . . . . 39
Table 4 – Momentum Strategy: Performance Measurement . . . . . . . . . . . . . 41
Table 5 – Combined Strategy: Performance Measurement (Daily Frequency, With
Volatility Cap) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
Table 6 – Combined Strategy: Performance Measurement (Daily Frequency, With-
out Volatility Cap) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
Table 7 – Combined Strategy: Performance Measurement (Weekly Frequency) . . 46
Table 8 – Combined Strategy: Performance Measurement (Monthly Frequency) . . 46

CONTENTS
1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2 LITERATURE REVIEW . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.1 Term Structure of Interest Rates . . . . . . . . . . . . . . . . . . . . . 21
2.1.1 Brazilian Interest Rate Futures . . . . . . . . . . . . . . . . . . . . . . . 22
2.2 Alternative Risk Premia . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.2.1 Carry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.2.1.1 Carry Strategy Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.2.2 Time Series Momentum . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.2.2.1 Moving Average Convergence-Divergence . . . . . . . . . . . . . . . . . . 24
2.2.2.2 Momentum Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.3 Performance Measurement . . . . . . . . . . . . . . . . . . . . . . . . 28
3 DATA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.1 Financial Instrument Specifications . . . . . . . . . . . . . . . . . . . 31
3.2 Database . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
4 METHODOLOGY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
4.1 Model Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
4.1.1 Combined Strategies Algorithm . . . . . . . . . . . . . . . . . . . . . . 35
5 RESULTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
5.1 Backtest and Performance . . . . . . . . . . . . . . . . . . . . . . . . . 39
5.2 Carry Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
5.3 Momentum Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
5.4 Combined Strategies Model . . . . . . . . . . . . . . . . . . . . . . . 43
5.4.1 Alternative Frequencies . . . . . . . . . . . . . . . . . . . . . . . . . . 45
5.5 Additional Findings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
5.5.1 Low Volatility Anomaly . . . . . . . . . . . . . . . . . . . . . . . . . . 46
6 CONCLUSION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

19
1 INTRODUCTION
The return premia of the yield curve can be comprehensively described by a per-
spective of style factors, a set of characteristics that includes measures, such as value,
momentum and carry, applied to bonds, according to Brooks and Moskowitz (2017).
This combination of characteristics comprises pricing information of the yield curve’s
three principal components (PCs), which are level, slope and curvature, as well as other
important priced factors, such as macroeconomic variables (growth and inflation), and the
Cochrane and Piazzesi (2005) factor, which predicts excess returns on one- to five-year
maturity bonds based on a single tent-shaped linear combination of forward rates, as
explored in Brooks and Moskowitz (2017).
In Brooks and Moskowitz (2017), value corresponds to the yield on the bond minus
(maturity-matched) expected inflation (“real bond yield”) and provides information about
the level of yields in relation to a fundamental anchor – expected inflation; momentum
provides information about recent trends in yield changes; and carry provides information
about expected future yields assuming the yield curve stays the same. Therefore, value
subsumes a “level” factor, while momentum and carry subsume information about expected
returns from the slope and curvature of the yield curve, respectively. These characteristics
describe both the cross-section and time-series of yield curve premia and connect to return
predictability in other asset classes, suggesting a unifying asset pricing framework.
A similar pattern has been verified in the Brazilian yield curve by Shousha (2005),
in a study that decomposes the variation in bond yields into six principal components,
highlighting the components which appear to relate to the level and the slope of the yield
curve.
An asset’s carry can be defined as its futures (or synthetic futures if none exist)
return, assuming that prices stay the same, as defined by Koijen et al. (2018). It is a
model-free characteristic that is directly observable ex-ante from futures prices, whereas
the expected price appreciation must be estimated using an asset pricing model. The
concept of time-series momentum is explored by Moskowitz, Ooi and Pedersen (2012).
Through this perspective, a security presents some persistence in returns for one to 12
months that partially reverses over longer horizons, consistent with sentiment theories of
initial under-reaction and delayed over-reaction.
Based on the article of Brooks and Moskowitz (2017), and beyond the asset pricing
literature, this paper focuses on the field of applied quantitative finance, applying concepts
of the foreign literature to the Brazilian yield curve. Through calculating and analysing
momentum signals, and combining them with a carry trading strategy, the present study
aims to verify whether it is possible to achieve consistent returns in excess of the risk free
rate, at different levels of the interest rates. The concept of value was not considered in
this model, in order to produce a strategy that runs independently of economic variables
20 Chapter 1. Introduction
such as the expected inflation, a variable that would have to be input from proprietary
economic forecasts, or from sources such as the weekly "Focus Survey", published by the
Brazilian Central Bank. Therefore, the model in this article will extract information about
expected returns from the slope and curvature of the yield curve across all levels of the
yield curve (multi-maturity model).
Additionally, this study also analyzes the outcome of both the carry and momentum
strategies individually applied to the Brazilian market, and verifies its adherence to the
existing literature through outcomes which should correspond to the theory presented by
Moskowitz, Ooi and Pedersen (2012), Koijen et al. (2018) and Baz et al. (2015), which
cover other markets and financial assets.
The main objective sought is to design a more comprehensive approach for the yield
curve premium. The proposed model, combining both carry and momentum strategies,
makes possible the identification of yield premia for several different maturities across
the Brazilian yield curve, each of them represented by a one-day interbank deposit future
contract traded in Brazil.
The analysed premium should be obtained through the slope and curvature of the
yield curve or, in other words, this study focuses on the premia that can be obtained at
every level of yield in the market, across all different maturities along the yield curve.
This characteristic should be verified after backtesting the carry and momentum model.
By combining carry and momentum strategies across the curve, the ultimate goal is to
generate a trading strategy that is capable of extracting the most from interest rates
premium across different maturities, regardless of the level of the yields negotiated in the
country.
21
2 LITERATURE REVIEW
2.1 Term Structure of Interest Rates
This section describes the yield curve and its components, as this study’s models,
based on style factors, were performed within the Brazilian yield curve futures market.
An interest rate in a particular situation defines the amount of money a borrower
promises to pay the lender, as described by Hull et al. (2009). Moreover, Hull et al. (2009)
defines the n-year zero-coupon interest rate as the rate of interest earned on an investment
that starts today and lasts for n years, with all the interest and principal being realized at
the end of n years (there are no intermediate payments).
Assets of different maturities typically sell at different yields to maturity. When
these asset prices and yields are compiled, long-term assets sell at higher yields than short-
term assets. Bodie et al. (2013) summarize the relationship between yield and maturity
graphically in a yield curve, which is a plot of yield to maturity as a function of time to
maturity.
Forward interest rates are the future rates of interest implied by current zero rates
for periods of time in the future, according to Hull et al. (2009). Caldeira, Moura and
Portugal (2010) claim that the term structure of interest rates is represented by a set spot
rates for different maturities. A number of different theories have been proposed in order
to explain what determines the shape of the zero curve, as described by Hull et al. (2009).
The expectations theory conjectures that long-term interest rates should reflect
expected future short-term interest rates; more precisely, it argues that a forward interest
rate corresponding to a certain future period is equal to the expected future zero interest
rate for that period.
Another idea, market segmentation theory, conjectures that there need be no re-
lationship between short-, medium-, and long-term interest rates. Under the theory, a
major investor such as a large pension fund or an insurance company invests in bonds of a
certain maturity and does not readily switch from one maturity to another. The short-term
interest rate is determined by supply and demand in the short-term bond market; the
medium-term interest rate is determined by supply and demand in the medium-term bond
market; and so on.
The theory that is most appealing is liquidity preference theory. The basic assumption
underlying the theory is that investors prefer to preserve their liquidity and invest funds
for short periods of time. Borrowers, on the other hand, usually prefer to borrow at fixed
rates for long periods of time. This leads to a situation in which forward rates are greater
than expected future zero rates. The theory is also consistent with the empirical result
that yield curves tend to be upward sloping more often than they are downward sloping.
In fixed income, the term structure is, more often than not, upward sloping. Con-
22 Chapter 2. Literature Review
ventional explanations range from the liquidity theory of rates - investors should be
compensated for the higher risk of holding long bonds, hence the upward sloping yield
curve - to the theory of preferred habitats - investors prefer short bonds to long bonds.
Rates tend to roll down the curve: this translates into excess returns for fixed income
investors, as described by Baz et al. (2015).
At any point of time t, there will be a collection of zero-coupon bonds that differ
only in terms of maturity. However, in a given moment, there may not be a bond available
to all desired maturities as bonds are not negotiated for all possible maturities, as explain
Caldeira, Moura and Portugal (2010).
The interpolation method development can be found in McCulloch (1971) and
McCulloch (1975). The fundamental idea behind cubic spline interpolation is based on
the engineer’s tool used to draw smooth curves through a number of points. This spline
consists of weights attached to a flat surface at the points to be connected according to
McKinley and Levine (1998). A flexible strip is then bent across each of these weights,
resulting in a pleasingly smooth curve.
Real world numerical data is usually difficult to analyze. Any function which would
effectively correlate the data would be difficult to obtain and highly unwieldy. To this end,
the idea of the cubic spline was developed. Using this process, a series of unique cubic
polynomials are fitted between each of the data points, with the stipulation that the curve
obtained be continuous and appear smooth. As stated by McKinley and Levine (1998),
these cubic splines can then be used to determine rates of change and cumulative change
over an interval.
2.1.1 Brazilian Interest Rate Futures
The Brazilian interest rate futures market shows some particularities that distinguish
it from most markets worldwide, as noted by Santos and Silva (2015).
In Brazil, the contract traded represents the present value of a virtual bond whose
value at expiry is 100,000 BRL. Each DI contract has, as its underlying asset, the DI
interest rate compounded until the contract’s expiration date, for this purpose defined as
the capitalized daily DI rates verified in the period between the trade date and the last
trading day, as published by B3 (2020).
2.2 Alternative Risk Premia
According to Hamdan et al. (2016), alternative risk premia (ARP) are systematic
risk factors that can help to explain the past returns of diversified portfolios, and it is also
designated as non-traditional risk premia by Raiteri and Malagoli (2020). They may be
risk premia in a strict sense, but also market anomalies or common strategies.
2.2. Alternative Risk Premia 23
A style, as in Israel and Maloney (2014), is an alternative source of return which
consists on a disciplined, systematic method of investing that can produce long-term
positive returns across markets and asset groups, backed by robust data and economic
theory.
2.2.1 Carry
Carry is the tendency for higher-yielding assets to provide higher returns than
lower-yielding assets. (ISRAEL; MALONEY, 2014)
Koijen et al. (2018) define the carry of an asset as the return on a futures position,
assuming its price stays constant; hence, carry is the forward-looking dividend yield in
excess of rf . Baz et al. (2015) define carry as the difference between the spot the spot and
the forward price of an asset or, in other words, the profit in a long forward position if
prices do not change.
According to Koijen et al. (2018) the concept has been mostly studied in the currency
markets, but it can be applied to any asset. Any security or derivative expected return can
be decomposed into its “carry”, an ex-ante and model-free characteristic, and its expected
price appreciation. So, "carry" can be understood as the expected return of a security or
derivative if there is no change in underlying prices; that is, if prices do not move and only
time passes, that security or derivative will earn its "carry".
In line with Israel and Maloney (2014), the economic intuition behind carry is that
it balances out supply and demand for capital across markets. High interest rates can
signal an excess demand for capital not met by local savings; low rates suggest an excess
supply. As traditional economic theory would argue, in the case of currencies, these rate
differentials should be offset such that investor returns would be the same across markets,
but the evidence shows otherwise. This may be due to the presence of non-profit-seeking
market participants, such as central banks and corporate hedgers, introducing inefficiencies
to currency markets and interest rates (ISRAEL; MALONEY, 2014).
The concept of carry has been shown to be a good predictor of returns cross-sectionally
(going long securities with high carry while at the same time going short securities with
low carry) and in time series (going long a particular security when carry is positive or
historically high and short when carry is negative or historically low). So, the concept of
"carry" provides a unifying framework for return predictability and gives origin to carry
strategies across a host of different asset classes, including global equities, global bonds,
commodities, US Treasuries, credit, and options. Carry strategies are commonly exposed to
global recession, liquidity, and volatility risks, though none fully explain carry’s premium.
Koijen et al. (2018) is a great reference for discussing cross-sectional carry strategies
in several different markets. There are typically market-neutral carry strategies where we
are always long some currencies, rates, commodities and indices and short some others.
Baz et al. (2015) discusses in detail the differences between cross-sectional strategies
24 Chapter 2. Literature Review
and time series strategies for three different factors, including carry. Baltas (2017) also
looks at carry strategies, both cross-sectional and time series, across different futures
markets (commodities, equity indices and government bonds), and discusses the benefits
of constructing a multi-asset carry strategies
2.2.1.1 Carry Strategy Algorithm
The carry premium in this model corresponds to the performance accrued by hold-
ing each of the DI futures contracts, with different maturities. Therefore, the backtest
corresponds to a long-only strategy with DI futures contracts: long UP, short yield.
In the strategy algorithm, the DI future contract is held from the initial trade date
until 3 days before its expiry, being then rolled to the next available contract. The carry
trade performance accounts for the premium accrual for holding future contracts.
2.2.2 Time Series Momentum
Trend-following is one of the most prevalent quantitative strategies and it applies to
several markets such as equity indices, currencies, and futures markets such as commodities
and bond futures. Trend-following (TF), or Time Series Momentum (TSM), refers to the
predictability of the past returns on future returns and is the focus of several influential
studies. In the academic summary of trend-following, Moskowitz, Ooi and Pedersen (2012)
document significant momentum in monthly returns. There they find strongest results for
relatively short holdings periods (1 or 3 months) and mid-range lookback periods such as
(9 and 12 months).
Momentum consists on the tendency for an asset’s recent relative performance to
continue in the near future, as defined by Israel and Maloney (2014).
A univariate time-series momentum strategy is defined as the trading strategy that
takes a long or short position in a single asset based on the sign of the recent asset return
over a particular lookback period (BALTAS; KOSOWSKI, 2013). The classic definition of
momentum is simply the percentage change in the financial time series over h time periods.
Sometimes, this classic definition is calculated by looking at log-percentage changes by
calculating the the difference between the log of today’s price vs. the log of the price h
periods ago.
In this model, the momentum metric Momi (h) is associated with the concept of
moving averages convergence and divergence, in consonance with Appel (2003).
2.2.2.1 Moving Average Convergence-Divergence
The Cross-Over Moving Average (or sometimes called MACD, for Moving Average
Convergence-Divergence) is essentially a comparison between a short and a long moving
average. It consists of a technical analysis tool that can be used to determine future trends
2.2. Alternative Risk Premia 25
in the stock market by using actual stock market activity to establish patterns of strength
and weakness, as proposed by Appel (2003).
If the fast moving average is above the slow one, it means that the most recent
information is pulling the price upward, signaling a positive momentum.
Although the MACD signal can be constructed with traditional moving averages of
different periods, the most commonly used measure is the exponentially weighted moving
average (EWMA). So the MACD indicator is given by:
MACDi,t (m1,m2, α) = EWMAi,t (m1, α)− EWMAi,t (m2, α) (2.1)
where m1 < m2. The formula for the exponentially weighted moving average is:
EWMAi,t (m,α) =
∑m
l=1 α
lERi,t−l+1
∑m
l=1 αl
(2.2)
where m can take integer values from 1 to ∞, and 0 < α < 1 measures the speed of
decay of the weights. The most common way to set the value of α is based on the half-life
λ, which is interpreted as the number of periods it takes for the weight to reach 0.5.
α = 1− e−
ln(2)
λ . (2.3)
To illustrate the concept, in figure 1, the moving average convergence-divergence for
the DIs with maturities of 3 months have been plotted. The short average and the long
average have been calculated through the EWMA method, with half lives of 12 and 26
periods, respectively. These are the most common parameters, being recommended by
Appel (2003).
26 Chapter 2. Literature Review
Figure 1 – MACD - 1 Year Contracts (EWMA half lives = 12 and 26 periods)
2.2.2.2 Momentum Algorithm
In this study, the momentum signal is based on the cross over of two exponentially
weighted moving averages, in accordance with Appel (2003). The algorithm building that
signal is the following:
1. Select the set of time-scales, consisting of a short and a long exponentially weighted
moving average (EWMA);
2. In line with Appel (2003), the considered time-scales are Sk = (12) and Lk = (26).
As explained by Baz et al. (2015), those numbers are not look-back days or half-life
numbers. In fact, each number (n) translates to a lambda decay factor (λ) of nn−1 to
plug into the standard definition of an EWMA. The half-life (HL) is then given by:
HL =
log(0.5)
log(λ)
=
log(0.5)
log(1− 1n)
(2.4)
3. Interpolate the yield curve, in order to obtain the respective yield for each contract
and date in the time series;
4. For each contract and date, calculate both averages of the corresponding yields:
EWMA[P |Sk] (short period, or "fast" moving average) and EWMA[P |Lk] (long
period, or "slow" moving average);
2.2. Alternative Risk Premia 27
5. Fill in a dataframe with signals, for each trading date and contract maturity, at-
tributing value one where
EWMA[P |Sk] > EWMA[P |Lk] (2.5)
and value zero when the above condition is not met. In this dataframe, a signal 1
indicates a long position recommendation, while a signal 0 indicates a short position,
based on the market trend.
In this study, the momentum strategy will be calculated from the yield of the
contracts. Through this method, the model was conceptually built to be responsive to
changes in the expected rates at a future maturity, rather than changes in contract prices.
28 Chapter 2. Literature Review
2.3 Performance Measurement
For further performance calculation of the deployed trading strategy, a set of measures
have been selected based on each one’s specific characteristics.
1. Excess Returns
As in Cogneau and Hübner (2009), excess returns are contrasted with gain measures,
in the dimension of value creation. An excess return represents the result of a "zero-
investment strategy". So, it represents the payoff from a unit of investment financed
by borrowing. In this paper, the benchmark risk-free rate is the Brazilian CDI, the
interbank rate in Brazil.
2. Volatility
The volatility of historical returns account for their standard deviation (σ) for a
given time frame and frequency, as in Benninga and Czaczkes (2014). The time frame
commonly used varies wildly: Some practitioners use a short term of, say, 30 days,
while others use a much longer (up to 1 year) time frame. Similarly, the frequency of
returns can be daily, weekly, or sometimes monthly.
The higher the volatility in outcomes, the higher will be the average value of these
squared deviations. Therefore, variance and standard deviation provide one measure
of the uncertainty of outcomes, according to Bodie et al. (2013).
3. Sharpe Ratio
The Sharpe Ratio (SR) is the reward-to-volatility measure, first proposed by William
Sharpe in Sharpe (1966) and widely used to evaluate the performance of investment
managers, as stated by Bodie et al. (2013). Consistent with Sharpe (1994), the
Sharpe Ratio (SR) is calculated as:
Sharpe Ratio =
Rp −Rf
σ
(2.6)
In this formula, Rp represents the portfolio return, while Rf is the risk-free rate, and
σ is the standard deviation of the portfolio’s returns.
4. Sortino Ratio
The Sortino Ratio corresponds to a variant of the Sharpe Ratio, as described by
Bodie et al. (2013).
The use of standard deviation as a measure of risk when the return distribution is
non-normal presents two problems: (1) the asymmetry of the distribution suggests
we should look at negative outcomes separately; and (2) because an alternative to a
risky portfolio is a risk-free investment, we should look at deviations of returns from
2.3. Performance Measurement 29
the risk-free rate rather than from the sample average, that is, at negative excess
returns, as in Bodie et al. (2013).
According to Bodie et al. (2013), a risk measure that addresses these issues is the
lower partial standard deviation (LPSD) of excess returns, which uses only negative
deviations from the risk-free rate (rather than negative deviations from the sample
average), squares those deviations to obtain an analog to variance, and then takes
the square root to obtain a “left-tail standard deviation.” The LPSD is therefore the
square root of the average squared deviation, conditional on a negative excess return.
The Sortino Ratio consists of the ratio of average excess returns to LPSD (σd):
Sortino Ratio =
Rp −Rf
σd
(2.7)
5. Maximum Drawdown and Maximum Drawdown to Volatility
A drawdown is defined as the accumulated percentage loss due to a sequence of drops
in the price of an investment, according to Leal and Mendes (2005). The maximum
loss from a market peak to a market nadir, commonly called the maximum drawdown
(MDD), measures how sustained one’s losses can be, as follows:
MDD =
Trough V alue− Peak V alue
Peak V alue
(2.8)
One adjustment that can be made is to divide the MDD by the calculated volatility
of the returns. Scaling the MDD with respect to the standard deviation of returns
makes the ratio comparable across strategies, as in:
MDD − to− V ol =
MDD
σ
(2.9)

31
3 DATA
3.1 Financial Instrument Specifications
The DI rate, or DI-Cetip, is the average rate of interbank transactions carried out
through interbank certificates of deposits. Its value is close to the Selic rate, which guides
the monetary policy conduction in Brazil. The Brazilian interest rate futures market shows
some particularities that distinguish it from most markets worldwide, as reported by
Santos and Silva (2015).
In Brazil, the contract traded represents the present value of a virtual bond whose
value at expiry is 100,000 BRL. Each DI contract has, as its underlying asset, the DI
interest rate compounded until the contract’s expiration date, for this purpose defined as
the capitalized daily DI rates verified in the period between the trade date and the last
trading day, as published by B3 (2020).
The daily profits and losses of a position shall be settled in cash (payment of debits
and receipt of credits) on the following trading session. According to B3 (2020), open
positions at the end of each trading session, after being transformed into UP, shall be
settled according to the day’s settlement price. The variation shall be calculated up to the
expiration date, in accordance with the following formulas published by B3 (2020):
ADt = [(PAt − (PAt−1 × FCt)]×M ×N (3.1)
Where:
ADt = the daily settlement value, in Brazilian Reais, corresponding to day “t”;
PAt = the contract settlement price on day “t”, for the respective contract month;
PO = the trading price, in UP, calculated as follows, after the transaction has been
carried out:
PO =
100, 000
(
1 + i100
) n
252
(3.2)
Where:
i = the traded interest rate, expressed as a percentage;
n = number of business days between the trade date and the day preceding
the expiration date;
M = the Brazilian Real value of each UP point, as established by B3;
N = the number of contracts;
PAt-1 = the settlement price on day “t–1” for the corresponding contract month;
FCt = the correction factor on day “t”, defined by the following formula:
1. When there is one business day between the last trading session and the
32 Chapter 3. Data
day of the variation margin:
FCt = (1 +
DIt−1
100
)
1
252 (3.3)
Where:
DIt-1 = the DI rate, corresponding to the trading session preceding the day
to which the variation margin refers, to six decimal places.
2. When there is more than one business day between the last trading session
and the day of the variation margin and therefore there is more than one DI Rate published
for the period between two consecutive trading sessions, the correction factor will represent
the accumulation of all the published DI Rates, as below:
FCt =
n∏
j=1
(1 +
DIt−j
100
)
1
252 (3.4)
Where:
DIt-j = DI Rate, corresponding to each business day between the previous
trading session and the day before the settlement date, to six decimal places.
3.2 Database
The data examined in this study was extracted from the database provided by the
Brazilian exchange, "Brasil Bolsa Balcão", available at their website B3 (2020), through a
connection established with the AWS server from FinanceHub (accessed August 30, 2020).
The extracted database contains the historical series of daily settlement prices, for the
different maturities one-day interbank deposit future contracts, each of them denominated
a One-day Interbank Deposit Futures (DI) in the local financial market.
The period analyzed ranges from May-10 to May-20, amounting to 2.5k observations.
Table 1 shows part of the data obtained for the contracts available at B3, with open
interest greater than zero, in the most recent trading date considered for this analysis,
which is the 15th of May, 2020.
In sequence, the annualized unitary prices obtained in the data base were converted
into annualized rates, which would be used for further calculation in the model, where
UPt is the unitary settlement price of the contract in the date t, that is, the value (in
points) corresponding to 100,000 discounted by the interest rate. Bus is the number of
business days, in accordance with the Anbima (2020) calendar, between the considered
date and the contract’s date of expiry, and DIt is the corresponding rate of the contract
that should be calculated in order to construct the accurate yield curve.
The calculated rates for each DI contract available were, then, interpolated in order
to build the complete yield curve, comprising all possible maturities. The interpolation
method chosen was the spline method, based on its flexibility to adjust to the real data,
3.2. Database 33
time_stamp contract maturity_code open_interest_close settlement_price
150470 2020-05-15 DI1 M20 1100495 99875.16
150471 2020-05-15 DI1 N20 4056567 99653.76
150472 2020-05-15 DI1 Q20 336540 99435.65
150473 2020-05-15 DI1 U20 141165 99228.88
150474 2020-05-15 DI1 V20 816069 99026.15
150475 2020-05-15 DI1 X20 111820 98812.06
150476 2020-05-15 DI1 Z20 68885 98609.48
150477 2020-05-15 DI1 F21 3713326 98407.87
150478 2020-05-15 DI1 G21 67320 98168.26
150479 2020-05-15 DI1 H21 52957 97962.94
150480 2020-05-15 DI1 J21 605997 97698.97
150481 2020-05-15 DI1 K21 24395 97438.01
150482 2020-05-15 DI1 N21 980907 96840.57
150483 2020-05-15 DI1 V21 471595 95730.62
150484 2020-05-15 DI1 F22 2290014 94528.89
150485 2020-05-15 DI1 J22 371860 93171.64
150486 2020-05-15 DI1 N22 431745 91748.21
150487 2020-05-15 DI1 V22 53155 90091.32
150488 2020-05-15 DI1 F23 1646384 88589.54
150489 2020-05-15 DI1 J23 27455 86761.08
150490 2020-05-15 DI1 N23 821765 84971.49
150491 2020-05-15 DI1 V23 27485 83104.87
150492 2020-05-15 DI1 F24 593805 81255.15
150493 2020-05-15 DI1 J24 9860 79491.87
150494 2020-05-15 DI1 N24 39654 77582.91
150495 2020-05-15 DI1 V24 20840 75787.89
150496 2020-05-15 DI1 F25 848903 74064.02
150497 2020-05-15 DI1 J25 4170 72399.04
150498 2020-05-15 DI1 N25 11145 70817.43
150499 2020-05-15 DI1 F26 71438 67163.40
150500 2020-05-15 DI1 N26 9425 64040.56
150501 2020-05-15 DI1 F27 388313 60752.14
150502 2020-05-15 DI1 F28 25475 54699.27
150503 2020-05-15 DI1 F29 201417 49684.97
150505 2020-05-15 DI1 F31 31253 40824.11
Table 1 – DIs contracts by maturity and price, with open interest greater than zero, in
15th of May, 2020.
34 Chapter 3. Data
mentioned by McKinley and Levine (1998). A 2-order polynomial was chosen to implement
the spline, since many of the lines had few data to be interpolated, making it difficult to
use higher order polynomials.
Figure 2 – Interpolated Brazilian Yield Curve at 15/05/2020
The interpolation was smoother in the most recent periods, for which there is a large
volume of observations covering all vertices of the curve, both long and short. On the
other hand, the method can present disadvantages in the initial periods, where there are
fewer observations for longer vertices. In this way, the values are interpolated following
the trend of the short curve, which can generate an explosive trend for interest rates in
the long run - as can be seen in the high rates of rates for very long terms. In addition,
there is no parameter in the model for the level of long-term interest, to which the curve
should converge.
35
4 METHODOLOGY
4.1 Model Structure
Three codes were developed in Python, and the complete model is available online
(DARIN, 2020). Both models follow algorithms based on inputs from the finance literature,
as described below.
The work starts with data processing. Firstly, the database is obtained from B3,
with prices and contracts information on each of the available contracts at any date in the
ten year time-series. Then, the DIs unitary prices are converted into yields, and these are
interpolated in order to build the complete yield curve for each date in the series.
The database is then modelled according to three different algorithms. To all of the
models datasets, there have been applied a few filters. There were only included traded
vertices with a minimum of 150 daily open interest and maturity date in January, in order
to only select the most liquid contracts. After being subjected to these filters, the resulting
DI contracts on 2020 May 15th represented on Table 2. In the model, each DI contract is
represented by k, ordered from the shortest to the longest maturity-.
k order Maturity Contract Code
k=1 2022-01-03 DI1 F22
k=2 2023-01-02 DI1 F23
k=3 2024-01-02 DI1 F24
k=4 2025-01-02 DI1 F25
k=5 2026-01-02 DI1 F26
k=6 2027-01-04 DI1 F27
k=7 2028-01-03 DI1 F28
k=8 2029-01-02 DI1 F29
Table 2 – DIs contracts after open interest and liquidity filters, as of 2020 May 15th
The present model analyzes the latest 10 years of the Brazilian yield curve market.
This timeline has been selected after analysing the evolution of this specific market in
recent periods, seeking for a long horizon through which the market characteristics are
consistent. Although B3 provides slightly older data, the market presented comparably
lower liquidity and availability of tradable contracts.
For instance, each of the strategies generates signals in a daily basis, being traded
on closing prices. No costs have been taken into account, in order to obtain an analysis of
the premium in the curve solely.7
4.1.1 Combined Strategies Algorithm
Both models, Carry and Momentum, are combined in this paper into a single strategy.
36 Chapter 4. Methodology
In this model, the Momentum strategy is introduced as an enhancement. When,
in a given date, the momentum signal equals one (payer signal), the model assumes
levered long exposure to the contract, whose level should be calibrated according to the
following volatility algorithm. Otherwise, when the signal equals zero (receiver signal),
the momentum long exposure is neutralized and the model trades only based on carry
premium.
Hence, the carry model will account for 0.5 of the total portfolio risk exposure
through the entire time series (static strategy):
Carry Exposure = 0.5 (4.1)
When introducing the momentum exposure in the model, which would modify the
premium accrual strategy, there were two options of sizing. The first one, would be to
open the momentum position with the same size for both strategies: carry and momentum.
This way, when the momentum signal prescribed a long position, each of the strategies
(carry and momentum) would account for 0.5 of the total portfolio risk exposure. It is
important to highlight that attributing equal weights to both strategies would assume a
constant risk across all market scenarios.
For this reason, an alternative has been proposed in order to calibrate exposure
according to market volatility. The proposed alternative is idiosyncratic, and it attributes,
for each contract, the volatility on day t is calculated as the annualized standard deviation
(σ) for the returns in a given time frame. Two volatility time frames are considered for this
measure; the annual (long time frame) volatility, which comprises the latest 252 trading
days observations, and the monthly (short time frame) volatility, which considers the 21
most recent observations. All volatility calculations assume the same daily frequency of
observations.
Once the two annualized volatility levels have been calculated, they can be used for
sizing the total long exposure when the momentum signal equals 1, as follows:
Momentum Exposure =
V ol252d
V ol21d
∗ 0.5 (4.2)
The model assumes the one-year volatility as the pattern for the specific contract.
When the shorter volatility (21 days) level is greater than the longer level (252 days),
the net effect will be a reduction of the total risk exposure (Momentum Exposure < 0.5).
The opposite case also verifies; when the longer volatility (252 days) level is greater than
the shorter level (21 days), the net effect will be a leverage of the total risk exposure
(Momentum Exposure > 0.5).
This risk management procedure ensures that the momentum strategy will achieve
bigger exposure during periods when the market is rising consistently and the volatility
stays below its average one-year level, also working as a stop loss when market starts
to fall abruptly, the short term volatility rises above the long term volatility and the
4.1. Model Structure 37
portfolio suffers with losses. The model will backtest and compare both algorithms, one
with volatility cap rebalancing and one without it, to assess its advantages.
Figure 3 – Annualized Volatility (1 Month, 1 Year): 1 Year Contracts
Figure 3 plots two historical volatility measures, calculated for the 252 business days
to maturity yield; the blue line corresponds to the 21 trading days historical volatility,
while the orange line corresponds to the 252 business days historical volatility. As the chart
indicates, the short-term volatility not only varies considerably more, but it also spikes
more in comparison with the long-term measure, supporting the volatility cap control
intuition behind algorithm that guides the model.

39
5 RESULTS
5.1 Backtest and Performance
5.2 Carry Model
The Carry Model yields positive returns throughout the time series, as can be seen
in Figure 3. The highest premia concentrates on the longer vertices, in line with finance
theory of forward premium. Also, during market crashes, those longer maturities tended
to suffer more severe losses, relatively to the shorter maturities contracts.
Figure 4 – Carry Model Backtest
k=1 k=2 k=3 k=4 k=5 k=6 k=7 k=8
Frequency Daily Daily Daily Daily Daily Daily Daily Daily
Excess ret. 1.25% 1.85% 3.16% 4.09% 6.21% 5.24% 4.03% 6.07%
Vol. 0.75% 1.97% 4.08% 5.36% 5.99% 7.91% 9.87% 15.21%
Sharpe 1.67 0.94 0.77 0.76 1.04 0.66 0.41 0.40
Sortino 1.79 0.97 0.73 0.80 1.15 0.73 0.47 0.47
MaxDD -1.77% -7.22% -8.07% -16.33% -10.47% -18.38% -28.65% -46.81%
MaxDD to Vol -2.36 -3.67 -1.98 -3.04 -1.75 -2.32 -2.90 -3.08
No. Obs 2460 2460 2460 2460 2460 2460 2460 2460
Table 3 – Carry Strategy: Performance Measurement
40 Chapter 5. Results
The carry model presented Sharpe Ratios that ranged from approximately 0.40
(longest maturity contract), to 1.67 (shortest maturity contract), reaching 0.8310 on
average. The model also presented, for the longest maturity contract, relatively high
volatility (15.2%) and a Maximum DrawDown of 46.8 %. As shows Figure 5, the Sharpe
Ratios of the strategy tend to be smaller, the longest the contract maturity.
Figure 5 – Carry Strategy Sharpe Ratios
5.3 Momentum Model
The momentum model backtest presented a smoother return profile, where the
drawdowns are minimized, when compared to the carry strategy, indicating that the
MACD framework succeeded in identifying market trends. However, the Sharpe ratios are
considerably smaller on most maturities, indicating more volatility per return in excess of
the risk free rate.
5.3. Momentum Model 41
Figure 6 – Momentum Model Backtest (12, 26 MACDs)
k=1 k=2 k=3 k=4 k=5 k=6 k=7 k=8
Frequency Daily Daily Daily Daily Daily Daily Daily Daily
Excess ret. 0.76% 1.99% 1.64% 3.83% 5.43% 4.05% 5.04% 6.98%
Vol. 0.75% 1.95% 4.16% 5.25% 6.53% 8.34% 10.00% 12.89%
Sharpe 1.00 1.02 0.39 0.73 0.83 0.49 0.50 0.54
Sortino 1.13 1.15 0.38 0.80 0.91 0.52 0.63 0.69
MaxDD -1.66% -2.94% -12.91% -13.25% -9.23% -17.38% -14.77% -27.13%
MaxDD to Vol -2.20 -1.50 -3.10 -2.53 -1.41 -2.08 -1.48 -2.10
No. Obs 2460 2460 2460 2460 2460 2460 2460 2460
Table 4 – Momentum Strategy: Performance Measurement
As shown in Figure 7, the Sharpe ratios were still greater on the shorter maturity
contracts, ranging from 0.39 to 1.02 (0.6884 on average). In comparison with the Carry
model, the Momentum strategy performed better for the longer contracts, which tend to
be more susceptible to large drawdowns.
42 Chapter 5. Results
Figure 7 – Momentum Strategy Sharpe Ratios
Figure 8 plots the Selic Target Rate, the target interest rate set by the Brazilian
central bank as a monetary policy tool, which influences the level of the short-term interest
rates. The vertical lines correspond to dates where the model points to a change of signal,
for the 252 business days yield (1 year contracts). As the figure shows, the signal generation
is closely related to changes in the Selic rate path, alternating between "buy" and "sell"
recommendations accordingly.
5.4. Combined Strategies Model 43
Figure 8 – Brazil Selic Target Rate vs. Change of Signal (1 Year Contracts)
5.4 Combined Strategies Model
When combining the fundamentals of both the Carry and Momentum models, and
adding a volatility based rule for sizing positions, the yield trading model not only presents
higher excess returns adjusted to volatility, but also produces a trading strategy that is
less susceptible to drawdowns. This outcome is due to the combination of a carry accrual
strategy with a trend following one, adjusting the position sizing according to the volatility
measure. Through this strategy, the model reduces its long exposure during moments of
higher uncertainty, and leverages it when market trends are clearer.
44 Chapter 5. Results
Figure 9 – Combined Strategies Backtest
k=1 k=2 k=3 k=4 k=5 k=6 k=7 k=8
Frequency Daily Daily Daily Daily Daily Daily Daily Daily
Excess ret. 1.11% 1.89% 2.69% 3.96% 6.12% 4.98% 4.35% 6.99%
Vol. 0.64% 1.55% 3.63% 4.46% 5.44% 6.86% 7.16% 11.06%
Sharpe 1.73 1.22 0.74 0.89 1.12 0.73 0.61 0.63
Sortino 1.82 1.22 0.66 0.90 1.19 0.73 0.71 0.75
MaxDD -1.59% -4.46% -6.67% -9.92% -8.10% -10.70% -17.03% -29.37%
MaxDD to Vol -2.48 -2.88 -1.84 -2.22 -1.49 -1.56 -2.38 -2.66
No. Obs 2460 2460 2460 2460 2460 2460 2460 2460
Table 5 – Combined Strategy: Performance Measurement (Daily Frequency, With Volatility
Cap)
The Sharpe ratios in this model range from 0.61 to 1.73; both the lower and the
higher ratios are greater than the ones produced by the two base models. The model also
presented an average ratio of 0.9584, consistently superior than the ratios produced by the
other models.
5.4. Combined Strategies Model 45
Figure 10 – Combined Strategies Sharpe Ratios
In order to assess the effect of the volatility control, the combined strategies model
has been tested without the volatility cap control. As we can see from Table 6, even though
the combination produced superior Sharpe indexes relatively to the momentum strategy
k=1 k=2 k=3 k=4 k=5 k=6 k=7 k=8
Frequency daily daily daily daily daily daily daily daily
Excess ret. 0.75% 0.91% 1.99% 2.14% 3.57% 2.96% 1.80% 3.12%
Vol. 0.56% 1.56% 2.77% 4.03% 4.30% 6.04% 8.75% 13.01%
Sharpe 1.33 0.58 0.72 0.53 0.83 0.49 0.21 0.24
Sortino 1.43 0.60 0.74 0.57 0.95 0.57 0.22 0.28
MaxDD -1.65% -6.81% -7.21% -15.10% -10.24% -19.41% -28.10% -43.79%
MaxDD to Vol -2.96 -4.36 -2.61 -3.75 -2.38 -3.22 -3.21 -3.37
No. Obs 2460 2460 2460 2460 2460 2460 2460 2460
Table 6 – Combined Strategy: Performance Measurement (Daily Frequency, Without
Volatility Cap)
5.4.1 Alternative Frequencies
In order to assess the effect of frequency over the model, two other time frames have
been tested. For comparison purposes, the strategy was also backtested in a weekly and a
monthly trading frequency.
The purpose behind testing longer time frames in the model lies in the idea of testing
if a less responsive algorithm could produce superior results. Additionally, reducing the
46 Chapter 5. Results
trading frequency usually implies minimizing the trading costs associated with brokerage,
exchange fees and slippage, and consequently verifying the possibility of achieving superior
returns. In this model, though, trading costs have not been considered, since the brokerage
fees associated with future contracts are usually low, and the model usually suggested
holding each of the positions for long time frames.
Nevertheless, the results have been inferior, when compared to the daily frequency
model. The longer frequencies turned the models to be less responsive to market changes,
subtracting returns and adding volatility, therefore lowering the model’s overall Sharpe
ratios. Tables 6 and 7 present the results obtained with weekly and monthly frequencies,
respectively.
k=1 k=2 k=3 k=4 k=5 k=6 k=7 k=8
Frequency Weekly Weekly Weekly Weekly Weekly Weekly Weekly Weekly
Excess ret. 0.23% 0.39% 0.55% 0.80% 1.23% 1.01% 0.88% 1.40%
Vol. 0.29% 0.70% 1.65% 2.03% 2.47% 3.12% 3.25% 5.02%
Sharpe 0.78 0.55 0.33 0.40 0.50 0.32 0.27 0.28
Sortino 0.82 0.55 0.30 0.40 0.53 0.33 0.32 0.33
MaxDD -1.59% -4.46% -6.67% -9.92% -8.10% -10.70% -17.03% -29.37%
MaxDD to Vol -5.47 -6.34 -4.05 -4.89 -3.28 -3.43 -5.24 -5.85
No. Obs 2460 2460 2460 2460 2460 2460 2460 2460
Table 7 – Combined Strategy: Performance Measurement (Weekly Frequency)
k=1 k=2 k=3 k=4 k=5 k=6 k=7 k=8
Frequency Monthly Monthly Monthly Monthly Monthly Monthly Monthly Monthly
Excess ret. 0.05% 0.09% 0.13% 0.19% 0.28% 0.23% 0.20% 0.32%
Vol. 0.14% 0.34% 0.79% 0.97% 1.19% 1.50% 1.56% 2.41%
Sharpe 0.38 0.26 0.16 0.19 0.24 0.15 0.13 0.13
Sortino 0.40 0.26 0.14 0.19 0.25 0.16 0.15 0.16
MaxDD -1.59% -4.46% -6.67% -9.92% -8.10% -10.70% -17.03% -29.37%
MaxDD to Vol -11.38 -13.21 -8.43 -10.19 -6.82 -7.14 -10.90 -12.17
No. Obs 2460 2460 2460 2460 2460 2460 2460 2460
Table 8 – Combined Strategy: Performance Measurement (Monthly Frequency)
5.5 Additional Findings
5.5.1 Low Volatility Anomaly
Baker, Bradley and Wurgler (2011) have described the "Low Volatility Anomaly",
stating and proving cases in which high volatility assets substantially underperformed low
volatility assets. The idea behind this concept contradicts the common principle in finance,
that associates higher risks with superior returns. The authors highlight that, among the
5.5. Additional Findings 47
many candidates for the greatest volatility anomaly in finance, a particularly compelling
one is the long-term success of low-volatility and low-beta stock portfolios.
In an extension of the literature produced by the referred authors, in this study, a
similar pattern could be observed in the Brazilian yield future contracts, as depicted in
Figure 11.
Figure 11 – Sharpe Ratios
The contracts with shorter maturities and lower realized volatility tended to present
higher Sharpe ratios. This pattern was consistently found when analyzing both the
shortest and longest maturity DI contracts, but could not be precisely observed in the DIs
correspondent to the middle of the Brazilian yield curve.
The inverse relation between the superior performance, noted by higher Sharpe ratios,
and the volatility level has been observed in all variations of the model. Figure 11 plots
the Sharpe ratios obtained from the carry (or long only) strategy, from the momentum
strategy (pure trend following), and from the combined strategies model, which have
already been presented in previous sections of this study.

49
6 CONCLUSION
This paper elaborated on the development and backtesting of a trading strategy
combining different alternative risks among future yield curve contracts. The objective was
to compose a trading strategy capable of enhancing the premium extracted throughout
the curve regardless of the yield levels in the country, which have been anchored in this
study as the Selic target rate.
When compared to the Brazilian risk-free rate (CDI), and to a pure carry and a
pure momentum-based strategy, the proposed model has proven to overperform not only
the considered benchmark, but also both of the individual strategies.
The present research contributed to the field of quantitative finance as it aimed to
verify if superior returns can be achieved through a combination of carry and momentum
strategies, which are strategies based on the expected returns from the slope and curvature
of the yield curve, in comparison to the application of each strategy separately or to the
benchmark risk-free rate. Moreover, this paper provided information about the application
of carry and momentum strategies over the Brazilian yield curve, by means of a time series
of exchange traded future contracts, in line with the theory found in previous studies from
the financial literature that mainly focus on developed countries.
As the study presents, there is premium in the Brazilian yield curve that can be
extracted from alternative factors that do not directly include monetary policy decisions
or economic forecasts. Also, as the models show, the combination of a Carry model with a
Momentum model has provided superior returns, in comparison to the performance of each
of the strategies individually; this exceeding performance has been further enhanced by the
introduction of a risk management tool in the model, named a volatility cap. Additionally,
the paper results indicate that the Low Volatility Anomaly can be observed among the
Brazilian DI contracts, in line with Baker, Bradley and Wurgler (2011) findings.
Much of the knowledge regarding trading strategies often keeps restrained to players
in the financial markets industry. Therefore, this field has not been completely explored in
the academic scope. The findings in this study should enrich the knowledge on both the
fields of trading strategies and of the Brazilian interest rates market, and consequently
open opportunities for further academic development in each of these areas.

51
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