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This section details the evolution of both research and results of the literature on :

- Currency portfolio
- Carry trade and its origin: failure of the uncovered interest rate parity
- Portfolio risk measurements with an emphasis on the Value-at-Risk, Expected Shortfall and Reward-to-VaR

I. Currency portfolio

Efficient portfolios of financial assets find their origin with Markowitz’s Modern Portfolio Theory (1952). Markowitz wants to determine the investment behaviour of investors as he seeks the behaviour that is the closest to reality. Several rules are proposed but rejected as, according to these rules, only returns’ maximization is sought or because there is a lack of diversification. The one rule that is accepted and prevails is the expected returns (mean) – variance of returns rule, also known as E-V rule. From the expected returns and variance, an investor can build a set of attainable E-V combinations of portfolio. Reaching an attainable combination does not mean that it should be picked. Indeed, investors should pick a combination that is both attainable and efficient. An efficient combination being a combination where, for a given level of risk (illustrated by the variance) that is unique and dependent to each investor according to his risk profile and preference, expected returns are maximized. Or, for a given level of expected returns, variance is minimized (Markowitz, 1952).

This theory of Modern Portfolio has been applied not only to equities and bonds but also to currencies. In 1989, Dumas and Jacquillat determined whether, using several strategies and returns measurements, an investor holding a currency portfolio would be able to generate a return in excess of the market return, in other words if he could beat the market. Even though the currency portfolio generates satisfactory risk-return results, Dumas and Jacquillat come to the conclusion that the portfolio would not be able to generate excess returns. They argued that their experiment was implemented on currencies that have a high correlation and therefore, their portfolio might lack diversification. Bauwens et al. (2006) designed currency portfolios and sought the model that would, according to a Value-at-Risk constraint, allocate in an optimal way the currencies. De Macedo (1983) used eight major currencies to build optimal portfolios with different ways to optimally distribute the weight of each currency component, according to two different risk profiles that an investor could have. In the article of De Macedo, a finding that could be also seen in this paper is that the American dollar, in both risk profile, receives a large allocation of the portfolio.

In the following literature review, an exploration of the literature related to the carry trade is performed.

II. Carry trade and its origin: failure of the uncovered interest rate parity

The uncovered interest rate parity (UIP) is one of the major assumptions of several foreign exchange rate theories, its aim being to try to explain the movements of currency pairs exchange rate. According to this parity, the difference in the nominal interest rates (called interest rates spread) between two currencies should actually be equal to the expected change in the exchange rate (Darvas, 2008). Therefore, following this theory, any investor wanting to apply a carry trade strategy should not be able to generate a continuous profit.

However, from the beginning of the 1980s, empirical studies from a growing consensus of researchers have been debating this parity. Bilson (1981) tests the “speculative efficiency” which is an hypothesis arguing that forward prices are the best available forecasts of future spot prices. In the examined sample, from 1974 to 1980, the future spot rates are closer to current spot rates than to the current forward rates. Bilson even claims that the exchange rate evolves as a random walk without drift.

From the 1980s, a large literature grew in order to explain the reasons behind the failure of UIP, known as the “forward premium puzzle”. Several research articles aim at trying to explain the reasons behind this failure.

First, an opinion grew within the community of financial researchers that forward exchange rates have limited, if any, power to forecast changes in spot rates (Fama, 1984).

Then, Fama (1984) emphasizes the importance of risk premia by highlighting that forward exchange rates variation is mostly due to a variation in the premiums. Also, McCallum (1994) points out the effect that monetary authorities have on interest rates management. Interest rate policy makers shape their intervention in order to smooth any sharp movement in the exchange rate (McCallum, 1994).

From these anomalies and deviation from the UIP, Brunnermeier et al. (2008), among others, highlight one of the sources of profits for investors implementing a carry trade strategy. Considering that there is no appreciation or depreciation in any of the currency used in the strategy, the interest rate differential between the target currency and the funding currency is the first source of profit for the investor (Brunnermeier et al., 2008).

Then, as more investors enter into the same carry trade strategy, there will be a rise in the demand for the target high interest rate currency resulting in an appreciation of this currency. The investor will be able to make an additional profit on this currency which was originally bought at a forward discount (Cavallo, 2006). The carry trade strategy is derived from this puzzle.

Working at a portfolio level, Burnside et al. (2006) provide information about the profits generated from carry trades. They opt both for an equally weighted portfolio and for an optimally weighted one, implying that they are seeking to build a portfolio where the risk-adjusted measure Sharpe ratio will be maximized. Results from their studies show that :

- Both strategies’ cumulative returns dominate the S&P 500
- The volatility of the cumulative returns of the optimally-weighted carry trade portfolio is much smaller that the one of the S&P 500
- Using a three-year rolling period, both strategies deliver a high Sharpe ratio in the early 1980s.

From these points rises the attractiveness of the carry trade strategies : the volatility of returns of these strategies is lower than the volatility of the returns of the S&P500 index.

Nevertheless, Baillie and Cho (2014) perform an analysis indicating that the desirability of carry strategies had declined for many currencies and may not be an attractive strategy following the financial crisis of 2008 (Baillie and Cho, 2014).

III. Carry trade portfolio risk exposure

Burnside et al. (2006) use a risk-adjusted return measurements, the Sharpe ratio, to examine the performance of the different strategies implemented. It is noticeable that the current literature has not dig deeper in the risk return measurement methods, therefore one of the contributions of this thesis is to conduct other measurements on the portfolio that are constructed.

Portfolio risk exposure through its measurements has been widely discussed in the past literature. There are basically two schools of thoughts that have been animating the debate and controversy on the way to measure risk.

On one side, the earliest views on risk with Markowitz (1959), state that risk can be defined as the uncertainty that the expected return, the forecasted return would differ from the realized return. Following this view, one should use the variance of returns to measure both the positive or negative dispersion of returns around the mean. The square root of this variance is known as the volatility and is the standard deviation of returns (Markowitz, 1959). One assumption behind this measure is that it assumes a normal distribution of returns, in other words there would be a symmetry between the probability to generate a loss or a gain.

On the other side, in the current area, risk managers put an emphasis on the left side of the returns distribution: the probability of generating a loss. Even though Markowitz suggested the mean-variance model, Sharpe (1964) reminds that Markowitz also advocated that according to the circumstances, the approach could result in irrelevant outputs. Indeed, Markowitz emphasises that a model based on the semi-variance where only potential losses would be examined could be preferable (Sharpe, 1964).

The non-normal distribution of financial assets’ returns was first discussed by Mandelbrot in 1963 when commodity and equity prices were examined. An alternative distribution with fat-tails was proposed and then followed by Barnea and Downes (1973) and Blattberg and Gonedes (1975). The same examination of normality of returns has also been performed on currencies. Westerfield (1977) examines the variability risk of currencies trading under both fixed and floating regimes. Before this publication, it was assumed that exchange rates were normally distributed but Westerfield shows that this assumption cannot hold and that models using this normality assumption are inaccurate.

Since the beginning of the 1990, the main risk management tool that was used in the financial industry is the Value-at-Risk (Christoffersen, 2009). The Value-at-Risk gives a single number that summarizes the worst possible loss that won’t be exceeded according to confidence level over a specific and given future period of time. Even though this measure is widely used within the industry, several critics have arisen. One of the main deficiencies is highlighted by Basak and Shapiro (2001). They show that this risk measure may provide spurious information about the distribution tail. Therefore, should an investor use this measure as part of their risk management, he might end up with a position that would generate extreme loss. In other words, VaR only gives a loss information if a tail even does not occur. But what if a tail event does occur ? Then, the investor could expect to lose more than the amount given by the VaR but he does not give the information about this expected loss.

Another deficiency that is generated by the use of Value-at-Risk is its lack of sub-additivity (Artzner et al., 1999) : the risk of an entire portfolio should be lower than the sum of the risk of each portfolio component.

These different deficiencies, coupled with a more volatile financial environment, have motivated the search of other risk measurement methods. The Expected Shortfall (ES) is one of them and is defined as the expected value of loss beyond a specific confidence level. ES answers to the specific deficiencies that were mentioned in the last paragraphs as ES is a coherent risk measure.

Linking risk measurement methods and currency portfolios, Yamai and Yoshiba (2002) compared the VaR and expected shortfall of both emerging and developed countries currencies under market stress in order to find whether these extreme market conditions would alter the properties of the two risk measures. The data used is the exchange rate as a unit of the American rate from three industrialized countries currencies and eighteen emerging economies currencies from November 1, 1993 to October 29, 2001. Under the univariate analysis, it is shown that emerging economies have fatter tails compared to developed countries.

However, Yamai and Yoshiba (2002) also highlight that VaR presents tail risk when comparing the risk of some emerging countries with developed countries : VaR at 95% confidence level for Japan is greater than the VaR of eleven emerging countries. This analysis is also true for the expected shortfall: the ES at the 99% level of confidence is smaller for six emerging currencies than the one of Japan. This finding implies that both VaR and expected shortfall may set too low the risk of securities with fat-tailed properties and therefore it could generate a higher potential for large losses.

At a portfolio level, Yamai and Yoshiba (2002) introduce the possibility that ES and VaR may neglect the tail dependence of asset returns, the tail dependence being the level of dependence in the tail of a bivariate analysis. For both of the previous results, it is nevertheless non negligible to highlight the fact that ES has less of a problem in neglecting both the fat tails and tail dependence compared to Value-at-Risk.

Brunnermeier et al. (2008) provide evidence that carry traders are exposed to high “crash risk”, that is, when there is a large and positive interest rate differential between the target currency and the funding currency, carry trade returns are negatively skewed.

In practice, it is often assumed that the rate of returns of portfolio are normally distributed (Hull and White, 1998). However, as just discussed, carry trade returns are negatively skewed. Therefore, if in the case of this thesis, returns are also not normally distributed, using a risk measurement method where the assumption of normality is dropped makes sense. Gordon et al. (2003) introduced a measurement method where normality does not longer belong to the assumptions and that is close to the Sharpe ratio, reward-to-VaR. When the assumption of normality of returns is taken into consideration, both the Sharpe ratio and reward-to-VaR deliver the same ranking while, however, when the normality assumption is dropped, this may not be true (Gordon et al., 2003).

From this broad literature review, the objective of this paper is to connect several subjects altogether : by constructing different strategies of currency carry trade portfolios, an assessment of their returns and risks through various risk measures is conducted in order to determine if such strategies still compete against the returns of the equity market. This examination is conducted from the financial crisis of 2008 to 2017, a time period that raises doubts about the relevance of entering into a carry trade strategy (Baillie and Cho, 2014). Furthermore, as most of the literature focuses on G10 currencies, an additional expansion brought by this thesis is the diversification brought by adding other currencies belonging to developed economies but that are outside the current G10 currencies.

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