With the frequent successes of backtests when discounting sentiment of the CurrencyShares Euro Trust ETF (NYSEARCA:FXE), this article succeeds this research by creating and testing a zero-beta portfolio of 5 major currency ETFs with regard to variance of returns from the Euro ETF.

Included is a walk-through on how to calculate and tabulate a matrix of betas and correlations based on 5 years of daily returns. Also provided is a tutorial of how to use solver in Excel to replicate a zero-beta portfolio of the major currency ETFs with regard to Euro.

In constructing the baseline portfolio, we will buy the Euro and British Pound (NYSE:FXB), while selling the Canadian Dollar (NYSEARCA:FXC), Swiss Franc (NYSE:FXF), and Japanese Yen (NYSE:FXY). All are exchange-traded funds tracking the 5 major currencies denominated in U.S. dollars. The reasoning for this is FXE and FXB typically perform the best with the baseline indicator, the CEFD indicator, as is depicted in the equity graphs below.

The Swiss Franc and Euro ETFs both have negative interest rates of -0.75% and -0.40%, respectively, relative to the U.S. Dollar. This interest is debited from the investment holdings every stated period. This gives the investments, or savings accounts, an interest rate risk that costs money to sit on. Naturally, it would be good to sell these investment assets when given the chance.

For purposes of this article, we will focus on performance returns, and stick to buying the Euro , as well as buying the British Pound , since these two performed well in backtests when testing our sentiment trading indicator. Here are the backtest results of the 5 majors going back five years with the CEFD sentiment indicator.

Figure 1: A 5 year comp graph of the trading strategy compiled on R.

**CREATING A BASELINE INDICATOR**

This is how to calculate the baseline indicator for CEFD(%), which is the daily Closed-End Fund Discount, expressed as a percentage discount, or premium, of the average price to the Net Asset Value NAV. This is a great way to discount sentiment in a portfolio by buying a closed-end fund or ETF at a discount from NAV, and selling at a premium. First, we must construct the indicator as follows:

This math formula defines the CEFD daily indicator as the natural log of the previous day's average price, divided by the previous day's NAV price. By gathering and compiling daily prices from Yahoo! Finance, and the Guggenheim fund company website, we are able to backtest this strategy for our asset allocation model.

Initially, we can construct a baseline portfolio of Beta equal to 1, where our benchmark stock is the Euro ETF. Assuming you have investments held in Euros, that's the capital amount you would want to focus on.

Here is a matrix of all the betas, given the Euro €. Previous 5-year data was compiled from Yahoo! Finance, to measure and hedge for market risk, in this case relative to the variance of the Euro 's daily returns.

Table 1 A 5-year Beta matrix of the given ETFs.

To create the beta matrix, first you would take the daily log returns of your portfolio assets for the previous 5 years, and calculate beta by taking the covariance of the daily returns for,and dividing it by the variance of returns for. The formula for beta is as follows.

In excel this can be calculated analytically as follows:

Figure 2 Spreadsheet calculation of beta.

Notice the array calculations are reduced to:

=COVAR(FXE,FXB)/VAR

Where FXE represents the entire array for the 5-year daily returns of the Euro ETF , and the same applies to FXB for its corresponding array. In order to do this, you would have to select the array you would like to define and name it in an adjacent box to the left.

Figure 3 Naming arrays in Excel.

Notice the highlighted cell on the upper-left corner of Figure 3. This is a named cell. Being able to codify your data makes life a whole lot easier when number crunching thousands of different data points.

**CORRELATIONS**

To create a correlation matrix like the one in Table 3, you could use the Data Analysis Toolpack in Microsoft Excel, found by scrolling to Data>Data Analysis. You should see 'correlation' as one of your menu options.

Table 3 Five-year correlation matrix of major currency ETFs.

This correlation matrix shows very similar results to the beta results, which isn't normally the case with differing investments. However, when checking your scatterplots, and sizing up the line fit regression, as well as the tightness of the overall variance in returns, you should know if your beta calculations are substantiated by the evidence from the data analysis. Here are the regression results from FXB, given FXE. Note, FXE is our x-predictor variable for every single other investment asset. In this case, it's our overall market proxy.

Chart 1 Scatterplot with line-fitting of FXB versus FXE

You'll notice that this line-fit plot of FXB vs. FXE corresponds to a beta of 0.57 in the above beta matrix. Testing the data, this looks to be an accurate reading with the slope of the fitted line to be about 0.50 exactly. You'll notice also that the data's variance is not too dramatic, minus a few obvious outliers. In this way, 5-year beta is represented linearly by the slope of the best-fit line.

**WORKING WITH ARRAYS**

Having your daily returns in an array format, like the following data sample, you should have no problem creating the correlation matrix with the Data Analysis Toolpack. It will ask if your data array has headers. If you have selected the data from the very top row where there are headers, you would simply select 'yes', and it should generate a correlation matrix that can be formatted like the one in Figure 3. Note, you want to have your data array in proper format, meaning all cells must contain numbers, and there can be no spaces between columns.

Figure 4 Data sample array of daily log returns.

If for whatever reason, you do not have the Data Analysis Toolpack activated, you can enable it by simply scrolling to File>Options>Add-Ins in Microsoft Excel. It will be there-either active, or disabled. If it is disabled, you can simply scroll down to it and press 'OK', you should be able to now see the tool pack when you restart the application.

Assuming an initial investment of $25,000 in Euros, we could solve for proper allocations, given the betas. Here is a currency-hedged portfolio of the five major currency ETFs, where FXE is our baseline holding. So in this case, we would have $12,500 of FXE in our portfolio, with nearly that amount in FXB and a short position on the 3 other majors. The share prices are based on closing prices from 5/21, and the betas are the same as the ones given in Table 1, they are five-year betas, with FXE being the market proxy.

Table 2 Asset allocation model of Currency ETFs, hedging for FXE.

We now have a zero-beta portfolio with regard to the Euro. The data used was based on 5-years of daily returns from the major CurrencyShares ETFs, with special attention on how to construct the betas in Excel.

Finally, here we have the asset allocation model backtested based on the adjusted beta proportions, represented by the purple line.

Figure 5 Backtest results including the zero-beta portfolio with regard to the €.

**Disclosure:** I/we have no positions in any stocks mentioned, and no plans to initiate any positions within the next 72 hours.

I wrote this article myself, and it expresses my own opinions. I am not receiving compensation for it (other than from Seeking Alpha). I have no business relationship with any company whose stock is mentioned in this article.