Loans and money issues

To gain more insight into how exchange rates and local market project correlations matter, we will make up a simple example. As a representative of the NFL, living in a U.S. CAPM world, you consider investing in creating a German football league, the GFL. Your problem is determining the appropriate cost of capital for this project. You have to worry about currency movements. Moreover, you know that project returns are typically linked to their local stock markets more than to the U.S. stock market. We shall assume the following macroeconomic scenario:
• The U.S. market can go up 16% or down 8% (expected rate of return: +4%).
• The spot rate is 1.0886 $/€. The one-year forward currency rate today is 1.0783 $/€. In addition, we now assume that the actual exchange rate will be either 1.0000 $/€ or 1.1566 $/€ next year, averaging to 1.0783 $/€. It is important that we assume that currency movements are independent of stock market movements.
• The German stock market index, the DAX, returns whatever the U.S. market returns (adjusted for forward/spot rate movements), plus or minus 10%. For example, if the U.S. market appreciates 16%, then the German market is expected to appreciate by 7.1% or 27.1%. (I have not assumed that it will be exactly 27.0%, because of the expected currency rate change embedded in today’s forward rate. The extra 0.1% is not an important factor here.)
With two outcomes each, there are eight scenarios. We assume that they are equally likely.
Actually, this is not a bad macroeconomic model: it has reasonable realistic annual rates of return, exchange rates, standard deviations, and mutual correlations. When the U.S. stock market increases by 16%, the German stock market is expected to increase by (27.1% + 7.1%)/2 = 17.1% (in Euro returns!). When the U.S. stock market decreases by 8%, the DAX is expected to change by (2.9% − 17.1%)/2 = −7.1%. So, the DAX moves about one-to-one with the S&P500—though the DAX returns are in euros and the S&P500 returns are in dollars. More recent historical data suggest that this relationship is empirically higher than the 0.65 that I reported above, and perhaps now closer to 1.0. And finally, there is also good empirical evidence that currency movements are empirically not correlated with stock market movements.
Now consider the German project. Starting the German Football League costs €100 (million)  today. The rate of return on this project is assumed to be ˜ rp = 2.09% + ( ˜ r G M − 2.09%)·1.5 , (I.5) which really means that the GFL follows a German CAPM with a German market beta of 1.5 and a euro risk-free rate of 2.09%. For example, if the DAX were to return 7.1%, the GFL would return 9.6% in euros. This is not what you need to know, though—you are not representing a German corporation with German investors—you are representing a U.S. corporation with U.S. investors. What should be the project’s appropriate cost of capital and value for you?
The project costs you $108.86 today. Let us work through one of the branches—what happens if the U.S. stock market increases by +16%, if the exchange rate goes from 1.00886 today to 1.1566 next year, and if the DAX increases by 7.1%? Your project would then return 2.09% + (7.1% − 2.09%) · 1.5 = +9.6%. Having cost €100, your project would now be worth €109.62. At the 1.1566 $/€ exchange rate, this would be $126.79, equivalent to a dollar rate of return of 16.47% on your $108.86 investment. To determine the U.S. market beta, we need to find out what we can expect when the U.S. market goes up vs. when the U.S. market goes down. The table tells us that your average return is $134.38 (or +23.44%) if the U.S. market increases by 16%, and $95.19 (or −12.56%) if the U.S. market decreases by 8%. Draw a line between the points (X , Y ) = (+16%, +23.44%) and (X , Y ) = (−8%, −12.56%), and you will find that the slope is βP,S&P500 = 23.44% − (−12.56%) 16% − (−8%) = 1.5 . (I.6)
So, if the German stock market moves about 1-to-1 with the U.S. stock market (both in local currency, and even in the presence of extra volatility in the German market), and if exchange rate movements are uncorrelated with stock market movements, then the German project beta with respect to the DAX and quoted in euros is about the same as the project beta with respect to the S&P500 and quoted in dollars.
Now, we assumed that our project follows the German CAPM. Does the GFL also follow the U.S. CAPM? The U.S. CAPM would predict E ( ˜ rP ) = 1.12% + (4% − 1.12%) · 1.5 = 5.44% = r US F +