Factor investing is a broadly used approach in asset management, specially for the equity market, but, can we apply this idea in order to explain currency returns?

The idea at the core of factor investing is that there are different sources of risk in the market and that the exposure of the different assets to these sources of risk explain asset returns over the long term.

Let’s start the post understanding what are the core concepts behind this idea, recalling some important results in mathematical finance.

## CAPM

The capital asset pricing model (CAPM) was created by Sharpe in 1964 and it’s an equilibrium model based on the mean-variance optimization framework of Markowitz. Under this framework, asset excess returns are explained solely by its exposure to whole market excess returns.

Mathematically:

$$\mathbb{E}[R_i] – R_f = \beta_i^m ( \mathbb{E}[R_m] – R_f)$$

Despite its theoretical appeal, very soon several studies showed that the expected excess return on an asset is not just proportional to its market beta.

## APT

The arbitrage pricing theory is the theoretical cornerstone of factor investing. In 1967 Ross proposed this model in which several **risk factors **had to be taken into account in order to explain asset excess returns.

Mathematically:

$$ R_i = \alpha + \sum_{j=1}^{m} \beta_i^j F_j + \epsilon_i $$

Therefore, the return of asset i is a linear combination of m risk factors. The main difference between the APT and the CAPM is that APT doesn’t consider only market risk as a source of systematic risk but rather accepts more complex risk decompositions.

## How can we define factors?

The problem with APT is that it doesn’t make any kind of assumptions about what are the factors that drive asset returns, so they could be almost anything.

However, factors are usually constructed in one of three ways:

- By statistical factor decomposition (like for example using principal or independent component analysis) in order to build independent factors.
- Considering macroeconomic time series as factors that explain asset returns.
- Using market factors (sometimes also called style factors in the equity market). These factors are built as portfolios of assets from the universe, by ranking them in terms of some statistics or fundamental. The most famous examples are Fama-French factors.

Some market factors usually encountered in the factor literature for the equity market are size, value, momentum, low volatility, low beta.. . among a whole zoo of factors that have been discovered over time.

To know more about these factors (and some criticism to the factor approach) you can check this extraordinary paper by Thierry Roncalli: Facts and fantasies about factor investing.

## Currency factors

Unfortunately, factors are much less popular in the currency market and not that many of them have been uncovered so far. There’s only a very simple three-factor model that has been more or less broadly accepted so far.

In this factor model, for a basket of USD denominated currencies, we define the three factors as follows:

**USD index returns**: as all the currencies are USD denominated, this would be equivalent to the market returns in the equity market.**Carry factor**: this factor represents the excess total return for holding high-interest rates currencies while shorting low-interest rates currencies.**Momentum factor**: we can define a cross-asset momentum factor by getting long the recent winners while shorting the losers.

### Pure factor returns

We can now build the market factors with some specific implementation from their definitions, always capturing total returns for the currencies (i.e. taking into account the interest rates differential).

These are the returns obtained by these factors:

We can see some interesting results:

- The carry trade looks like a quite profitable factor.
- The cross-asset momentum not so much, without a clear mean-reverting or trending prevalence during our interval.
- The returns of holding USD are actually positive, if we take into account USD interest rates.

These results, specially the ones for the carry trade are very consistent with the academic literature on the topic, where carry trade returns are a puzzle for a lot of academics and economists since theory suggests that such opportunity should not exist in the currency market.

### The betas

Now that we have defined our factors, we can use rolling linear regressions, Kalman filter or any other dynamic regression tool in order to estimate the assets’ time-varying betas.

Once we have estimated these betas we have a decomposition of each currency returns in terms of the returns of the factors.

For example, the Brazilean Real is primarily exposed to the carry trade (unsurprisingly, because is one of the main components of our carry factor by construction).

The Swiss Franc, for example, has a quite different exposure to the carry factor:

It’s also clear that the exposure to the momentum factor changes with time, because the relative performance of the spot of the different currencies varies continuously, and clearly, this exposure seems to play a minor role compared to the carry, especially for high yielding currencies where interest yield is an important part of total currency performance.

It is important to note that, in order to check the validity of our factor model, our regression model should outperform a simple market model using only USD index returns, and, in addition, the intercept should be not significant.

Fortunately enough, the adjusted r2 improves on average when we consider the three-factor model rather than just the USD market model.

### Building a long short currency factor portfolio

To finish this post, let’s check some results. To obtain some simple long-short currency portfolios we can get long in those currencies with positive exposure to the factor and short the currencies with a negative exposure to it, sizing our positions proportionally to the size of the factor exposure.

Using this approach, we obtain three strategies: based on carry, momentum and combining both.

As we can see the carry strategy is the main source of alpha in these strategies. However, adding the momentum factor improves substantially the risk profile of the carry strategy.

This diversifying effect between carry and momentum makes a lot of sense conceptually since the return profile of both strategies is quite the opposite: carry strategy makes money steadily in risk-on environments and suffer large losses under market stress, whereas the momentum strategy is a natural hedge for this risk profile.

### Explanation

But, does all of this make sense? Is it actually possible that such a simple strategy yields such returns over time? There most broadly accepted explanation of the carry returns is that there exists a (time-varying) risk premium for holding carry risk.

That is, there’s an implicit risk in holding deposits of low-stability high-yield currencies that is rewarded in terms of return, even though historically the profits of the strategy have been much larger than the losses suffered during crisis, which means that there has been a time-varying but clearly positive risk premium in the market for the carry risk.

As a conclusion, I would like to highlight that factor investing is a bit controversial for different reasons: you can create whatever factor you want and market (or style) factors that are constructed profitable will always yield positive returns in strategies that get long the assets with positive exposure to such factor.

On the other hand, some factors are broadly accepted by practitioners because they make sense and have been obviously profitable over time, as it’s the case for the carry strategy.

Finally, even though we are all excited about the carry trade now, please think twice before running to your bank to take out a Japanese Yen mortgage 😉