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CURRENCY CARRY TRADING
WITH MGARCH-BASED
CARRY-TO-RISK PORTFOLIO OPTIMIZATION∗
Harald Schmidbauer†/ Angi R¨osch‡/ Tolga Sezer§/ Vehbi Sinan Tunalıo˘glu¶
c
2010 Harald Schmidbauer / Angi R¨osch / Tolga Sezer / Vehbi Sinan Tunalıo˘glu
(Last compiled: July 21, 2010)
Abstract
Currency carry trading is an investment strategy which borrows in a low-yield currency
and/or invests in a high-yield currency, with the hope of profiting from the interest rate
differential (the carry) between two money markets, as well as from currency movements.
Technically, profit (or loss) will be generated by a difference between the exchange rate
written in a forward contract, and the spot exchange rate on the day when the contract
expires. The risk in this strategy stems from unforeseeable currency movements. Which
currencies, and with which weights, should be included in a portfolio? One possible approach
is to use carry-to-risk maximization. We extend the usual static approach by substituting
an MGARCH-based conditional covariance matrix capable of forecasting portfolio risk for
the next period. By comparing examples of time series of realized profits, our paper shows
the benefit of forecasting the conditional covariance structure of currency movements when
constructing a carry-trading portfolio.
Key words: Currency carry trading; portfolio optimization; carry-to-risk optimization;
multivariate GARCH
1 Introduction
The basic idea of currency carry trade strategies is to borrow in lower yielding currencies and
to invest in high yielding ones. These strategies are an outcome of the observation that the
Uncovered Interest Rate Parity (UIP) may fail to hold. UIP is an arbitrage condition which
states that there should be no profit opportunity from the differences in interest rates between
two currencies. The present paper tests whether the economic benefit for carry trade investors
can be enhanced by using advanced volatility forecasting tools.
Deviations from the UIP, also leading to what is termed “Forward Premium Puzzle”, have
been widely discussed in literature during the past 20 years. Surveys such as in Hodrick [13],
Froot and Thaler [11] and Engel [9] are widely cited. In this branch of literature, the failure
of the UIP to hold is attributed to the time-varying nature of risk premia, expectational errors
∗This research project was presented on the 30th International Symposium on Forecasting ISF2010, held in
San Diego, June 20-23. The paper is published in: The International Institute of Forecasters (ed.), Proceedings
of the 30th International Symposium on Forecasting ISF2010 , San Diego, June 20-23. ISSN: 1997-4124.
†Bilgi University, Istanbul, Turkey, & Ideal Analytix, Singapore; e-mail: harald@hs-stat.com
‡FOM University of Applied Sciences, Munich, Germany / Tai’an, China, & Ideal Analytix, Singapore; e-mail:
angi@angi-stat.com
§Ideal Analytix, Singapore; e-mail: tolga.sezer@ideal.sg
¶Ideal Analytix, Singapore; e-mail: sinan.tunalioglu@ideal.sg
1
about the future spot exchange rate, and market microstructure issues such as liquidity and bid-
ask spreads. However, a relatively small amount of literature has actually attempted to measure
the economic benefit that can be achieved by an investor who tries to take advantage of this
puzzle. This is astonishing from a practical point of view because investment practioners have
been applying strategies, notably carry trading, to exploit this anomaly ever since its discovery
by Bilson [3]. Evidence for the high volume of carry trading activity can be found in Galati
et al. [12].
Furthermore, Backus et al. [1], Villanueva [16] and Wagner [17] have contributed to a new
line of literature in this field. Villanueva [16] points out that “few studies examine forward bias
trading profits even though significant returns define the economic significance of an anomaly”.
In this line, Burnside et al. [4, 5, 6] have undertaken investigations aimed at tackling the literature
shortcomings mentioned by Villanueva [16]. More specifically, they show how these carry trading
strategies yield high Sharpe ratios on average. Using monthly data starting from 1976, they show
that significant profits can be made using currency carry trading for individual currencies as
well as for a portfolio formed in a classic Markowitz framework. However, they point out that
these returns are not a compensation for risk. They argue that in the presence of microstructure
frictions and limitations to speculation in foreign exchange, the marginal Sharpe ratios are zero
even though average Sharpe ratios are positive.
Della Corte et al. [7, 8] point out that in the context of dynamic asset allocation strategies,
there is no study investigating the economic value of the predictive ability of empirical exchange
rate models, which condition on the forward premium while allowing for “volatility timing”.
Further, they implicitly assume that the underlying returns are normally distributed. They
allocate wealth according to different models such as a stochastic volatility model, a simple
random walk, a monetary exchange rate model, and a GARCH-based model. To measure the
economic benefit, the authors compare the level of average utility generated by each estimate
of conditional mean and variance. Their findings lead them to conclude that there is significant
economic benefit to an investor who exploits deviations from UIP by forecasting currency returns.
In Mayer [14], the returns to forward bias-trading in dynamic multi-currency strategies are
investigated to assess the limits to speculation in foreign exchange markets, thereby challenging
the findings by Burnside [4]. The results show that bias-trading strategies allow for economi-
cally significant excess returns and represent attractive diversification devices. Furthermore the
author finds that carry trading portfolios containing emerging market currencies result in large
diversification gains.
Galati et al. [12] point out that carry trading strategies are very widely used in the investment
community and that a popular measure to gauge the attractiveness of the carry trade ex ante
is to use the carry-to-risk ratio. The latter adjusts the interest rate differential w.r.t. the risk of
future exchange rate movements, where this risk is proxied by the expected volatility (implied
by foreign exchange options) of the relevant currency pair.
In the light of the existing literature, our goal is to extend extant approaches in this field while
building on established properties of carry trading, and try to overcome certain shortcomings of
currency carry trading, as practiced up to now:
1. Carry trade returns tend to fall in times of high volatility. Forecasting future exchange
rate volatility by means of conditional volatility should enable us to better time when to
invest and to unwind the carry trade (volatility timing effect).
2. The scope of the trade should be extended to form an optimal portfolio of currencies
which maximizes the investor’s utility function by estimating the joint conditional risk of
the portfolio.
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3. A framework for the investment strategy should be formulated which allows the measure-
ment of the ex ante attractiveness of the carry trading strategy in terms of the widely
applied carry-to-risk ratio.
4. If carry trade returns tend to fall in times of high volatility, one should be able to short
the high yielding currency, which is the currency prone to post negative returns.
This paper is organized as follows. Section 2 sets the stage for our investigation by introduc-
ing the relevant notions for the case of currency carry trading involving a single foreign currency.
Portfolio construction and optimization, when investing in two foreign currencies, is the topic of
Section 3. We obtain the dynamic risk forecast for the portfolio based on a bivariate GARCH
model; details are explained in Section 4. Empirical results when investing in the Brazilian real
and the Mexican peso, with the USD as base currency, are given and discussed in Section 5.
Finally, Section 6 summarizes and draws some conclusions.
2 Currency Carry Trading: The Case of a Single Currency
In the following, we shall show the connection between different notions of carry,profit, volatility
forecasts, and possible investment criteria. We also show the behaviour (i.e. realized profits and
losses) of a monthly sequence of investments, based on the USD (US dollar) and the BRL
(Brazilian real).
Our strategy
We take on the perspective of a USD investor, who contracts in a 1-month USD-BRL forward on
Tuesday every four weeks. Our numerical example assumes that investments start in January
2005. The maturity date of each forward contract we enter is assumed to be the Tuesday four
weeks later. On the date of maturity we exercise our contract, i.e. buy BRL at the contracted
forward price, and immediately sell them at the observed spot price on this day. With this
strategy, our realized (=observed) gross profit/loss at each date of maturity will be:
contracted forward price
observed spot price on the day of maturity (1)
The sequence of realized gross profits/losses of this strategy, when performed from January
2005 onwards, is the red line in Figure 1.
Expectations at the beginning of the contract
According to the interest rate parity theorem (that is, according to UIP), the contracted forward
price is supposed to be chosen so as to offset the interest rate differential between the USD and
the BRL money market, which means that, on the date of maturity, spot prices are expected to
equal the contracted forward prices. In this case, the profit/loss of our strategy will be zero, in
other words, the gross profit/loss at expected spot prices will be one. The black line in Figure 1
refers to our expected gross profits/losses according to UIP.
Empirical studies, however, contradict this theory. They show that expected profits of carry
trade investments are positive on the average. For example, in the case of constant spot prices
(that is, the observed spot price on the day of maturity equals the spot price at the beginning
of the contract), our gross profits/losses on the day of maturity will be
contracted forward price
spot price at the beginning of the contract ,(2)
which can be interpreted as the gross profit/loss at constant spot prices. This ratio is called the
carry of the contract, in the sense of a contract parameter. Since its value is available at the
3
realized gross profit
2005 2006 2007 2008 2009 2010
0.8 0.9 1.0 1.1
Figure 1: USD-BRL gross profit/loss sequences: realized (red line) and hypothetical (black and
blue line)
date of decision whether to enter the contract or not, it may serve as an investment criterion,
leading to a simple investment strategy: enter the contract if its carry is not smaller than one,
otherwise don’t. The sequence of the carries of the contract is the blue line in Figure 1. — This
strategy clearly neglects the volatility of spot prices.
Our observation on the day of maturity
Our realized gross profits/losses (the red line in Figure 1) show much more volatility than in
either of the two hypothetical scenarios at the beginning of the contract (black and blue line).
Although our strategy provides an average gross profit significantly greater than one (1.014928,
p-value: 2.343e−07), it does not prevent us from entering the contract during the second half
of 2008, where we encounter huge losses. Therefore, a simple decision strategy on the basis of
the amount of the carry of the contract alone is insufficient.
Volatility forecast at the beginning of the contract
On each date of decision, we apply a GARCH(1,1) model to estimate the conditional volatility
of the next week’s return on spot prices.1From that we calculate a four-week-forecast of the
return’s volatility, matching the date of contract maturity. The sequence of volatility forecasts
(assigned to the date of maturity, again in a weekly time setting) is shown in Figure 2.
Carry-to-risk forecast at the beginning of the contract
We combine carry of the contract and volatility forecasts to obtain a more sophisticated invest-
ment criterion, the so-called carry-to-risk forecast:
carry of the contract
1 + volatility forecast/100 (3)
The sequence of carry-to-risk indicators (assigned to the date of maturity, again in a weekly
time setting) is shown in Figure 3. A possible investment criterion related to it is: Enter the
contract if the carry-to-risk exceeds a given threshold, otherwise don’t.
1All computations for this paper were carried out in R [15].
4
volatility forecast (in percent)
2005 2006 2007 2008 2009 2010
123456
Figure 2: USD-BRL four-week volatility forecast
3 Portfolio Construction
This section deals with the problem of forming a currency carry trade portfolio when investing
in two currencies. The performance of the portfolio is measured as the profit/loss generated
after four weeks.
Let tdenote our decision point of time. Then, the carry of the portfolio is the gross profit/loss
of the portfolio we would expect at constant spot prices. It is composed of the single currency
carries, here denoted in percent by
ci,t =fi,t
pi,t
−1·100%,(4)
where i= 1,2 indicates the currency.
A portfolio is characterized by the weights φiinvested in each currency. The carry-to-risk
for such a portfolio can be computed as
carry of the portfolio
1 + portfolio volatility forecast/100 =1 + φ0·ct/100
1 + pφ0·ˆ
Ht+4 ·φ/100,(5)
where φ0= (φ1, φ2), ct= (c1,t, c2,t)0, and ˆ
Ht+4 designates the covariance matrix forecast of the
return vector at time t+ 4, i.e. four weeks after the initial investment has been made. The
optimal portfolio is then determined by maximizing the carry-to-risk w.r.t. weights φ1and φ2
under the constraints
|φ1|+|φ2|= 1,−1≤φ1, φ2≤1,(6)
where a negative value of φ1or φ2indicates short selling. There are several approaches to obtain
the covariance matrix forecast ˆ
Ht+4. Two obvious ways are:
•“static approach”: Use rolling estimation (e.g. the 50 weeks preceding t) to obtain an
“average” historical covariance pattern. The historical covariance matrix is taken as the
forecast.
•“dynamic approach”: Use a conditional covariance matrix, obtained from a multivariate
GARCH model.
We shall compare these two approaches empirically in Section 5 below. The model to obtain
the conditional covariance matrix forecast is described in the next section.
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carry−to−risk
2005 2006 2007 2008 2009 2010
0.95 0.96 0.97 0.98 0.99 1.00
Figure 3: USD-BRL four-week carry-to-risk forecast
4 The Conditional Covariance Matrix Forecast
Let rt= (r1,t, r2,t) be the vector of returns (exchange rate changes in percent) in week t. We
assume that
rt=µ+t,(7)
where µis a constant vector and (t) is modeled as a bivariate GARCH-BEKK process.2This
mean specification is appropriate if no significant autocorrelation is found in the return series.
The process (t) is specified as
t=H1/2
t·νt,(8)
where (νt) is a bivariate white noise process with cov(νt) = I(the 2 ×2 unity matrix) and
Ht=C0C+A0t−10
t−1A+B0Ht−1B(9)
with parameter matrices C= (cij) (c21 = 0), A= (aij ), B= (bij ). This is a symmetric model
insofar as any t−1will lead to the same Htas −t−1.
The one-period ahead forecast of the covariance matrix, ˆ
Ht+1, can be computed from equa-
tion (9) at time t. According to (5), a four-period ahead forecast for the covariance matrix is
needed. This can be obtained recursively by substituting the respective conditional covariance
matrix. The conditional expectations of the ·0terms in (9) are estimated from averages over
the four weeks preceding week t:
ˆ
Ht+i=C0C+A0¯t¯0
tA+B0ˆ
Ht+i−1B, i = 2,3,4.(10)
5 Empirical Results — The Case of USD-BRL and USD-MXP
We assume a USD-based investor investing in a portfolio of 1-month-forward contracts in BRL
(Brazilian real) and MXP (Mexican peso). Investments start on a Tuesday in January 2005.
2See Engle and Kroner [10], Bauwens and Rombouts [2].
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Four weeks later, the contracts are exercised, and gross profits/losses can be observed in terms
of the portfolio of realized gross profits/losses at observed spot prices on the day of maturity,
fi,t
pi,t+4
,(11)
where i= 1,2 indicates the currency. On the same day, the portfolio is restructured with respect
to the weights invested in each currency.
We compare two strategies of portfolio construction. Both strategies are based on maximizing
carry-to-risk (5), but they use different input with respect to forecasting risk (via the covariance
matrix) four periods ahead: In the static approach, the covariance matrix is estimated from
historical data, starting 50 weeks previous to the date of the contract decision. Alternatively,
we apply an MGARCH-BEKK model (9) to data from 2000 onwards to obtain a conditional
covariance matrix.
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weight of USDBRL
2005 2006 2007 2008 2009 2010
−1.0 −0.5 0.0 0.5 1.0
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static approach
symmetric MGARCH
Figure 4: USD-BRL — sequences of optimal weights: static approach vs. MGARCH-BEKK
Figure 4 shows the way in which the portfolio is restructured from month to month, beginning
in week 3 of January 2005. The plot shows only the weights of BRL, not those of MXP. Weights
can be negative (short selling). A gap in the sequence of points indicates a month with no
investment, since the portfolio carry was forecast negative for these months. The red dots refer to
the MGARCH-BEKK approach. An investment criterion based on MGARCH obviously involves
much more restructuring than a static approach based on an average covariance matrix (the black
dots). This is no surprise, since conditional volatility and correlation change dynamically. In
other words, the MGARCH-based portfolio will be more responsive to news.
The portfolio risk an investor encounters from month to month with either approach is shown
in Figure 5 in a weekly time setting. The lines constitute the denominator of the carry-to-risk
in equation (5). A comparison of this plot with Figure 2 reveals the decrease in risk when this
portfolio is considered, rather than an investment in BRL alone. Again, the MGARCH reacts
more promptly to new information, thus providing a more realistic evaluation of the risk involved
than the static model.
The sequence of gross portfolio profits/losses, which could have been realized from January
2005 onwards (performed in a weekly time setting), as compared to one-month US treasury
bonds, is shown in Figure 6. Each return is computed w.r.t. the investment made one month
earlier, for each Tuesday. The greatest difference turns out to appear in 2009, in times of high
volatility.
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portfolio risk
2005 2006 2007 2008 2009 2010
1.01 1.02 1.03
static approach
symmetric MGARCH
Figure 5: USD-BRL and USD-MXP — sequences of portfolio risks: static approach
vs. MGARCH-BEKK
Return characteristics, as compared to a benchmark, can be described by Sharpe ratios. The
Sharpe ratio can be defined as:
E(R−Rf)
pvar(R−Rf),
where
R= (realized gross profit/loss −1) ·100%,
Rf= benchmark return
We use the Sharpe ratio empirically (ex post), not as a decision criterion. Values are shown in
Table 1.
no restriction no trading in case of
w.r.t. trading negative portfolio carry
symmetric MGARCH 0.14531 0.14652
static approach 0.00505 0.00074
Table 1: Sharpe ratios
The total profit accrued from month to month with either approach is shown in Figure 7.
Here, our investment is beginning in week 3 of January 2005. A month with no investment
results in a horizontal line for total accrued profit.
Figures 4, 5, 6 and 7 suggest that the effect of using a dynamic volatility model in portfolio
optimization may be volatility-specific. To obtain more insight, weekly portfolio volatility, as
predicted by the MGARCH, can be classified by magnitude: very low,lower ,higher , and very
high, each containing 25% of the observations. (The border between higher and lower defines
the median.) Figure 8 shows the average surplus profit for the corresponding weeks when using
the dynamic model vs. the static model, together with 95% confidence intervals for the expected
surplus. It turns out that expected surplus is significantly positive when volatility is either very
low or very high, and expected surplus is still positive in case of moderate volatility, even though
the increase is not significant.
8
realized gross profit
2005 2006 2007 2008 2009 2010
0.85 0.90 0.95 1.00 1.05 1.10
static approach
symmetric MGARCH
riskfree bond
Figure 6: USD-BRL and USD-MXP — sequences of total gross profits/losses: static approach
vs. MGARCH-BEKK
The better performance of the dynamic model in times of very high volatility is in accordance
with item 1 (end of Section 1). A glance at Figure 4 provides empirical evidence for the conjecture
in item 4 concerning a short position in times of high volatility.
6 Summary and Conclusions
The Uncovered Interest Rate Parity (UIP) condition states that there should be no arbitrage
opportunity from the interest rate differential (the carry) between two currencies. From the
observation that the UIP may fail to hold, a small branch of literature on currency carry trad-
ing strategies has emerged, aiming to measure the economic benefit to an investor who takes
advantage of the failure of UIP.
The goal of our study is to try to overcome certain shortcomings of currency carry trading
strategies as practised up to now. These shortcomings are related to the limited responsiveness
to volatility movements of the usual static approach of volatility forecasting.
An MGARCH-BEKK model is used to obtain a conditional covariance matrix capable of
forecasting portfolio risk for the next period. We compare this dynamic to the usual static
approach by measuring realized profits/losses from the perspective of a USD-based investor
who contracts in a portfolio of two currencies: the Brazilian real and the Mexican peso. The
portfolio is restructured each month by means of carry-to-risk maximization. We found that this
dynamic approach leads to a significant surplus profit in times of either very low or very high
volatility. Conditionality, on which the dynamic model rests, is superior in capturing disruptions
of continuity and triggering trading signals.
Further research needs to be undertaken in at least two directions. Firstly, from a practical
point of view, it is desirable not to limit the portfolio to only two currencies. Secondly, from
a more methodological point of view, an asymmetric MGARCH model may be superior in
exploiting the information structure of past events and thus lead to more differentiated trading
signals.
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total accrued profit
2005 2006 2007 2008 2009 2010
1.0 1.1 1.2 1.3 1.4 1.5
static approach
symmetric MGARCH
Figure 7: USD-BRL and USD-MXP — sequences of total accrued profit: static approach
vs. MGARCH-BEKK
●
●
●
●
0.0 0.5 1.0 1.5
portfolio volatility
surplus profit in percentage points
●
●
●
●
very low lower higher very high
Figure 8: USD-BRL and USD-MXP — 95% confidence intervals for the surplus profit with
MGARCH-BEKK volatility forecasts
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