So, I’ve been thinking about getting into the consulting business (primarily in the U.S.), and a few high-profile wealth-management firms have advised me to move to the States in order to take advantage of its favorable tax plan. Saving money on taxes always sounds like a pretty good strategy, but is moving to a different country the best option?

We’ll get to that…but, first, we require a brief digression (or two).

Some of you might have heard of something called “Purchasing Power Parity” (PPP). It’s a concept from economic theory that “compares different…currencies through a…’basket of goods’ approach. [T]wo currencies are in equilibrium…when a…basket of goods (taking into account the exchange rate) is priced the same in both countries” (Investopedia). PPP basically compares the exchange rate () with the quotient of prices (*E*) for an identical item (or a basket of items) sold in both countries. It’s a speculative measure designed to “predict” the movement of the value of one currency against another. So, if a particular Einstein bobblehead costs USD and CAD, USD, which is significantly higher than the then-current exchange rate of (about) 1.29 USD. This means Canadians will prefer to purchase the bobblehead in the U.S. (@ $12.89) and use the other three-plus dollars for something else. In short, the Canadian price is too high, suggesting the USD will be appreciate (over some time frame) until *E* equals the exchange rate. When , we speculate the possible existence of an *arbitrage* opportunity. The concept of arbitrage is hardly new, yet it continues to drive the boldest (and, at times, most reckless) investment strategies on Wall Street and beyond. It’s often said, colloquially, that “there ain’t no such thing as a free lunch” (TANSTAAFL); the proven concept of arbitrage, however, completely belies that claim. I offer a quick and tailored primer for skeptical readers.

#### “TISATAAFL” or “How to Guarantee a Gambling Profit Using Mathematics”

In very basic terms, a betting-arbitrage opportunity arises when the sum of the *bettors’* odds of a successful outcome derived from the gaming odds is less than one.* In a two-system bet, this is very simply calculated as

,

where the odds of Bettor 1 to *win *are . The second bet placed with Bettor 2 follows similarly. An example: Suppose James desires to place a wager on the 2018 college football national title game between Alabama and Clemson. His brother, Joshua, is giving 2/1 odds on Alabama, and Michael is giving 1.25/1 odds on Clemson. James wants to determine if this represents a genuine arbitrage opportunity. The 2/1 line means Joshua believes Alabama has a 2/3 probability of winning (i.e., 2/(2+1)), and the 1.25 (= 5/4)/1 line means Michael believes Clemson has a 5/9 probability of winning (i.e., (5/4)/[(5/4)+1]). Calculating (their belief of) the probability that *James* will win each bet is . It is, in fact, an arbitrage opportunity.

How does arbitrage work? Let’s say James’s gambling allowance happens to be 100 dollars in total, and he’ll bet *x* dollars on Alabama and 100 – *x* dollars on Clemson. Let’s imagine Alabama wins. An Alabama victory means James loses –*x* dollars to Joshua and wins 5(100 – *x*)/4 dollars from Michael. This yields the first linear profit curve (red line, below): . Of course, we want the profit to be greater than zero, so we set the LHS of the equation accordingly and solve, yielding . Now, imagine Clemson wins. This would mean James loses -(100 – *x*) dollars to Michael and wins 2*x* dollars from Joshua, the sum of which yields the second linear profit curve (blue line, below): such that . Thus, the amount of money James needs to wager with Josh (*x*) and Michael (100 – *x*) to *guarantee* a profit no matter which team wins the game lies within the global inequality derived from both equations: . This is a range from (roughly) 33.34 dollars to 55.56 dollars.

Curious readers will want to know two things: (1) the *maximum* possible profit based on the gambling allowance and (2) the optimum wager James should place on Alabama to generate that amount. We’ve jumped the gun a bit by providing the above graphic, but the answer involves solving the system of linear equations in the previous paragraph; the linear profit curves cross each other at an equilibrium point—recall the supply-and-demand curves of elementary economics—and it is this intersection that represents the Cartesian coordinate that (a) reveals the maximum possible profit () and (b) the optimum bet () that guarantees that maximum profit amount. Fortunately, as a general principle, we don’t need to graph these functions. Setting both equations equal to each other and solving for is sufficient: Solving gives us an optimum bet value of dollars on Alabama and 100 – 42.87 = 57.14 dollars on Clemson. This betting profile generates a maximum guaranteed profit (*p*) of dollars based on a 100-dollar total wager—*no matter which team wins the game*. The above graphic provides the relevant visual representation.

This should give you a general sense of the power and seductiveness arbitrage offers and why it’s essentially the Holy Grail of any investment strategy. (For some readers, it might be cool enough to know arbitrage exists, and you may want to make a few bets with your friends. But do the math first!) To put it simply, there’s no better option available to you than the one that generates a financial profit no matter the outcome. (Sorry, Milton Friedman!) One might ask what this has to do with PPP or moving to a foreign country. It involves the notion of currency arbitrage as a “free lunch.” Recall the Canadian who was interested in the Einstein bobblehead. She essentially earns three dollars by making the (online) purchase from the States. It’s as if she bought the bobblehead in Canada and the government deposited three dollars into her account. Unfortunately, PPP doesn’t tell us when to make the purchase (we need the exchange rate for that), but we use that information to make inferences, like whether we’ll save money if we buy a book from Spokane rather than Vancouver. PPP, however, only involves “tradeable” commodities. “Immobile goods” like real estate and services are inaccessible to PPP calculations.

One such “inaccessible” item is tax liability. The professional advice I received was simple: Move to the United States in order to avail yourself of the more attractive federal tax rates. But can I get a “free lunch” by staying in my country of residence? PPP tells me whether there exists a currency imbalance, not whether I should move. An approach that *does* help me make this determination involves what I will call the “Net-Purchasing-Power Index” (NPPI). NPPI simply calculates the exchange rate () that represents the equilibrium point between two baskets of post-tax income portfolios and compares it to the current exchange rate (). We begin with first principles—the technical definition of *net income*—and derive the NPPI from there:Here, is the total value of the income portfolio *in the U.S.*, is (again) the equilibrium rate, is the relevant federal tax rate in the target country, and is the relevant federal tax rate in the U.S. Our goal is to calculate the NPPI by calculating the quotient of and . When NPPI < 1, the exchange rate is greater than equilibrium rate, and we have the *potential* for an arbitrage opportunity—but not yet a *guarantee*. For that, we need to do a bit more work. As the NPPI tends to zero (i.e., as the exchange rate gets larger), the portions of our potential free lunch grow significantly.

Let’s walk through an example.

Suppose an analysis suggests my U.S. corporation will generate $500,000 USD in consulting fees in 2018. Conventional wisdom, as we’ve seen, suggests relocating to the U.S. That is, a $500,000 portfolio at a U.S. federal tax rate of 39% leaves me with $500,000(1 – 0.39) = $305,000 dollars if I move to the States. If I bring that money into a target country with a federal tax rate of, say, 47%, I’ll only have an after-tax amount of $265,000. It seems as if I’m losing money by choosing not to move. But what about ? Let’s imagine the exchange rate between the U.S. and the target country is USD. Is an arbitrage opportunity possible? Using the equation above, , which is the rate that “equalizes” the post-exchange purchasing power between both countries. Because , I would be (really) be losing money. Clearly, the after-tax, after-conversion portfolio of $291,500 is less than the $305,000 I’d be able to spend on goods and services in the U.S. If I think the extra $13,500 I’d save by moving to the U.S. is worth the time and effort, I should relocate. But what if . Then, and I’ve have a real chance to make some free money by staying put: In this case, I create $332,098 by bringing my U.S. income into the target country, and I enjoy a net-purchasing power of +$27,098. (Assume I’ve taken advantage of the legal means to minimize double-taxation issues.)

But isn’t this a guaranteed arbitrage opportunity? No, because we haven’t accounted for price differentials. What if prices are much more expensive in the target country? That is, what if PPP is severely unbalanced, as in the bobblehead example? In that case, the increased prices eat away at any NPPI surplus, though if we’re dealing with tradeable commodities, as we saw earlier, one would simply purchase those items from the States. Unfortunately, importing goods isn’t always a guarantor of profits. Let’s say, for argument’s sake, the amount of income we’re dealing with is $100. If , then NPPI < 1 and I’m left with $61 living in the U.S. and $66.41 in the target country. After buying the bobblehead at $9.99 USD, I have $51.01 in the U.S. and $50.42 if I buy it in the target country ($66.41 – $15.99). NPPI < 1, but I’ve still lost money. (For simplicity sake, assume the sales tax is equal and the federal tax rate applies to our $100 conversion.) This means I have to import the bobblehead from the States to have a chance at maintaining my NPPI advantage, and, in this case, I do still come out ahead: $66.41 – 9.99(1.2532) = $53.89, which means staying in the target country is still $2.88 better than if I’d moved to the U.S. and paid for the bobblehead in USD. It’s a very slight advantage, but that’s only because the amounts we’re dealing with are small. As the portfolio (*a*) grows, so does the advantage. (This ineluctably leads to the notion of leverage as an investment strategy, but we won’t address that here.) Unfortunately, as the price grows, the advantage decreases, and if the price is high enough, choosing not to move becomes a *disadvantage*.

The question, then, becomes this: Is there any way to evaluate an arbitrage opportunity given a specific constellation of values for the variables we’ve been discussing? Yes, there is. Such an evaluation involves solving a linear optimization problem that accounts for price levels. I will call this the Currency Arbitrage Price (CAP), and it utilizes both NPPI and PPP values. In what follows, however, we assume NPPI < 1. (Recall that if NPPI > 1, then no arbitrage opportunity is possible.) So, what do we need to know? We need to determine the maximum price level of a specific item in the target country that guarantees a post-purchase profit. We can calculate this by adding our price variables to the calculation of . Solving the necessary inequality for , we have:Notice the cancellation that occurs when . This last inequality tells us how much an (identical) item needs to cost in the target country in order to guarantee a profit given the other variables. We can visualize this inequality by graphing the linear CAP function

,

and we guarantee arbitrage when and . Armed with this information, let’s revisit the $100/bobblehead example. Solving the above inequality gives us . This means we are guaranteed an arbitrage opportunity when the bobblehead price is $9.99 in the U.S. and less than $15.41 in the target country. Let’s imagine it’s priced in the clearance bin (in the target country) at $11.99. In the U.S., paying in USD, we’d be left with the usual after-tax, after-purchase amount of $51.01, but in the target country, we’d now have an after-tax, after-purchase balance of $54.42. Despite the disparity in currency valuations and the higher tax rate, we enjoy an overall profit, which is an increase from the earlier amount of $53.89 we gained from importing.

Free lunch.

We can do a bit more. Imagine the target country decides all bobbleheads should be $11.99, and the U.S. decides it must reduce bobblehead prices to stay competitive. We love this Einstein bobblehead so much that we want to send it to all our friends. But the U.S. price keeps falling. How long can we purchase the bobblehead at $11.99 in the target country until we lose our arbitrage advantage? In other words, at what U.S. price does our profit reach zero? To solve this problem, we simply solve the above inequality for . This gives us:

.

The function for follows similarly. As long as the U.S. price is greater than , we retain our arbitrage advantage. So, if bobblehead prices remain fixed at $11.99 in the target country, the U.S. price can fall to $6.57 per unit and we’ll still earn a profit (as small as it might be at that price). You don’t need any extra information to calculate . PPI gives us the prices we need, and the values for all the other variables—exchange and tax rates—are easily accessible to the public. We’re simply doing some basic algebraic shuffling.

If we factor sales tax into the price differential, we add a layer of complexity to the problem of quantifying arbitrage. If and are the sales-tax rates in the U.S. and Canada, respectively, then our profit curves become

for the price in Canada and

for the price in the U.S., respectively. In this more complex case, imagine we import $5000 USD at an exchange rate of with federal income-tax rates of and and sales-tax rates of and in the U.S. and Canada, respectively. We note that NPPI < 1, and we want to purchase a new computer where USD and CAD. Do we have an arbitrage opportunity? Unfortunately, we don’t—not until the Canadian price is reduced to less than $3,165.

The function for the Canadian price (above) reveals this upper bound when (solid green). As you can see from the graph, we lose money at the current price ratio ($1407.22 – $1781.25 = -$374.03). This is shown by the gap between the red- and blue-dashed lines transversed by the constant function that represents the U.S. price ; this is the difference between values of the *individual* profit curves (and not the above functions that arise from setting those equations equal to each other and solving for and , which are represented by the solid lines on the graphs). Though we lose money if we purchase the computer in Canada at the current price of $3,499, we do gain a profit of $137.71 by importing it from the U.S. But let’s imagine we choose to wait for a local sale, and the Best Buy in Vancouver reduces the price to $2,999. Now, we *do* have an arbitrage opportunity:

We’ve now earned a better-than-importing profit of $1967.22 – $1781.25 = $185.97, despite the higher federal- and sales-tax rates, by bringing in the USD-based income and paying for the computer in CAD. The graph also reveals that we’ll continue to generate a profit until the U.S. price—in response to Canada’s competitiveness—drops to about $2,329 (purple), at which point the individual (dashed) profit curves intersect with each other at and the total profit drops to zero: Both the Canadian and U.S. consumers, at that point, would be left with a remaining balance of $1967.22 after purchasing the computer.

### Conclusion

So, that’s it. Currency arbitrage in a nutshell. Perhaps something like this exists somewhere in the literature—I imagine it might, even though I’ve never seen it explicitly during my study of economics—but we offer it here in the event it will pique general interest. It might be beneficial to review the basic process involved in calculating the CAP for a given portfolio combined with a certain collection of data points:

(1) Calculate the equilibrium rate

(2) Confirm NPPI < 1

(3) Determine and

(4) Purchase “identical items” in the target country if the price less than

(5) Purchase (4)’s items freely until its price in the host country reaches (assume remains constant)

Anticipated objections to the CAP model:

(I) *Availability of identical goods*

If a tradeable good in the target country is truly unique, you couldn’t have purchased it anywhere else; the notion of PPP is simply unimportant in those cases. Of course, you will have to decide whether you wish to (or, for some reason, must) pay for that uniqueness or if you’d prefer to choose an item that closely (but not precisely) matches the one you’re considering, assuming such an item is available. As far as price modeling is concerned, very closely matched items can be (and probably should be) considered “identical.” Variations among packages of Bic pens, for example, probably don’t mean very much with respect to the sticker price.

(II) *PPP applicability*

PPP only accounts for so-called tradeable goods, but it is possible to compare “immobile” goods using a number of objective metrics. For homes and real estate, for example, we could use price/sq.ft., location, year of construction, amenities, projected repairs, and many other measures of objective value. Much like the issue of identical goods, then, we can gain a pretty good comparison between immobile goods between countries that will allow us to use a generalized approach to PPP. Value is in the eye of the beholder, which means an eye toward equality of value between such goods is achievable.

So, what about the big question: Should I move to the U.S. based on my fanciful financial projections or remain in the target country and bring the money here? Well, if NPPI < 1, which means the exchange rate outstrips the taxation gap, then I’m guaranteed a free lunch (or two) as long as I purchase (near-)identical goods that fall below the upper bound. If I can do that through importing goods with a favorable exchange rate or by taking advantage of cheaper relative prices in the target country given a certain sales-tax profile, then it’s in my interest to eschew the idea of relocating, even though the tax rates are more favorable in the States.

* In wagers like these, everyone *must hold their money* until *after* the event is completed. In this way, an arbitrageur can cover her losses with her winnings and keep the remaining profit. Online betting sites require you to front the money as you make the wager, which is why this arbitrage strategy won’t work in those cases.