Anyone who took arithmetic knows that you can't divide by zero. Those who asked their 3rd grade teacher that were told, "You just can't". Those who went into algebra, trig, and calculus know that when you divide by zero the function becomes undefined, meaning the product of a variable times zero is the same for every value of the variable and cannot be recovered by division.

y * 0 = 0, for every y

A function, y = m/x is undefined when x is equal to 0, no matter what m is. That equation is an example of a hyperbolic relationship. When a function is undefined it becomes asymptotic in nature - approaching either 0 or +/- infinity. Other functions are asymptotic as well. Polynomial and trigonometric functions have values where either they or their derivatives become undefined. When the derivative of a function equals 0 or is undefined it is said to be a "critical point". These points are very important and can either be stationary (dy/dx = 0) or inflection points (dy/dx undefined). In the picture below, the green dots are inflection points while the red marks are stationary.

At the inflection points, a function changes from concave to a convex, or vice versa. If a function is "concave up" it is increasing at a decreasing rate, and if "concave down" it is decreasing at an increasing rate. A convex function is the opposite. At stationary points the changes are just from up to down or down to up, but there are no changes from one to the other.

Lately, I have seen some very interesting curves develop in some very interesting markets. One is no surprise to many, China. If you look at the China ETF, the FXI you will begin to see an exponential relationship. I have posted the chart below.

In fact, I would go so far to say this is almost a textbook exponential function. It is very convex in nature and if a function, would look to be approaching an inflection point.

On the other side of the coin let's look at the S&P 500,

The S&P, on the other hand, has moved up but with a concave curvature. The entire move upwards has been somewhat concave, but more glaring is each push upwards has exhibited almost textbook concavity - compared to the equivalent moves in the FXI. All though the move in the S&P over the past few years has exhibited concavity, when you break down into individual sectors, the leaders of the past few years - the Industrials, Materials, and Energy - have not. Instead they have been convex in nature - much like China. The laggards however have behaved much more like the S&P and its concavity. These include the Financials and Consumer Discretionary sectors. Additionally, if you look at the S&P on its first trip to the 1500's in the 90's and on this second attempt, you can see the difference (below).

While this is nothing new that growth has been slowing here and rising in China. Commodities such as Oil and Gold have moved in a similar fashion. The last chart that needs to be looked at is then the currency which all of these things is denominated in - the US Dollar.As you can see, the dollar has been moving concave down. More interesting, although obvious to many, is that if you reflected over the X-axis (flipped the chart) it would look eerily similar to FXI, and very similar to the three sector leaders above.

So, what does this all have to do with dividing by zero? Well, in the world of the dollar standard, the dollar is the great denominator - the great divisor. If you take the the general form of an asset price as y = C*(M/X), where Y is the value of the asset today in local currency C is the intrinsic value of the asset, and M/X is the forward exchange rate for local currency in dollars. So, if you invest in a Chinese stock that you except to pay future dividends, you need to discount them by the future amount of dollars they can buy. If the yuan is expected to appreciate in the future, those future payments are worth more and therefore the asset is worth more. This is nothing new, but simply discounted cash flows using exchange rates instead of interest rates (since exchange rates already take interest rates into account). Commodities are easier to understand, as they don't have to consider where the flows are coming from. Since they are a futures contract for delivery of a commoditized asset, a weakening in the dollar will cause that futures contract to go up.Furthermore, lets look at US stocks. The Industrials, Materials, and Energy Firms are easy to understand. They own either own assets that are commodities, or in the case of industrials, the cash flows they generate are denominated in something other than dollars. In a global economy, there is hardly a difference for an US citizen to invest in a US company denominated in dollars, that does business exclusively in yuan, than for a Chinese Citizen to invest in a Chinese company that does business in China and is denominated in Yuan - therefore the behavior of both should exhibit some similarity. As the case with financials and consumer discretionary stocks,and other laggards, that do business mostly in dollars, the future exchange rate is rendered nearly meaningless because transactions are done mostly in dollars and dollars do not have to be exchanged for dollars for future cash flows. Hence why they have started to collapse.

More technically speaking, based on the fact that foreign exchange costs can be hedged and arbitrage opportunities are small. A European investing in American companies would not value a company the same in the context of a falling dollar, unless he could hedge his dollar exposure. This hedge is quantified by the term M/X, for a US dollar investor investing in US stocks M/X = X/X = 1. Therefore they only experience the dollar growth of C, Y=C. Please note that if someone in France is investing in Japan, M/X may be a series of multipliers in order, but each individual currency at some point will HAVE to be denominated in dollars.

This begs the question, what happens as the dollar approaches zero. To even try to answer this question seriously in a time of enormous amounts of derivatives, computer trading programs, and a interconnected global market would be silly. The only answer I can give is that it will be chaotic and unpredictable, if it even happens. What is scarier is at this point, the dollar is starting to speed up its downward descent towards zero, and at some point the thought that it could move in that direction very quickly may cause a reaction as if it were. This is not highly unlikely given the amount of physicists and quants managing large amounts of money, using many of the ideas behind these functions to model market movements or price derivatives. So, on an unusual day when China and Emerging Markets rallied, the US squandered, and Treasuries made enormous moves, it is important to keep this relationship in the back of your mind. Zero and infinity are some strange things to ponder, and for a market that doesn't like uncertainty, it sure as anything would hate "undefined".

ZB, another good post, but missing a couple points, I think:

1. You assume the CNY and USD will be pegged in perpetuity. The slow revaluation of CNY over the past two years suggests otherwise. Its slowness is specifically designed to avoid a "George Soros vs. Bank of England" repeat.

2. You cite as an identity that U.S. investors owning U.S. stocks doing business in yuan is equivalent to Chinese investors owning Chinese stocks doing business in yuan. Chinese investors have only very, very recently been allowed to invest *any* capital offshore. Chinese firms trading on the Shanghai and Shenzen exchanges (closed to foreign investors) are at whopping premia to supposedly identical shares trading in Hong Kong. Meanwhile, the Chinese have an extremely high savings rate, but they still (despite the bubbly stock markets) predominantly save in bank deposits, which must be invested in government bonds, which allows for more accumulation of currency reserves.

3. You neglect an important factor in considering your "dollar goes to zero" scenario. Foreigners own a tremendous amount of U.S. debt, predominantly in Treasuries, but also hundreds of billions in agencies and even corporate bonds. All of this debt is denominated in -- U.S. dollars! So a "dollar goes to zero" scenario also wipes out the foreign reserves. Think that'll happen?!? And what would the effects be? Note this is an important difference from when the U.S. was in debt up to an estimated 100%-plus of GDP in the 1890s. Back then, our debt was denominated in francs, pounds sterling and deutschemarks. Today, we can just inflate or devalue away our debt. Not a pretty "solution," but one that hurts the foreigners more than it hurts the U.S. consumer.

Posted by: Bond investor | October 18, 2007 at 12:51 AM

BI,

Thanks for the feedback. First off, in this post I tried to steer clear of interpreting anything beyond the actual dynamics and relationships. Any examples were just that examples, probably regarding China, I chose a poor example In response to your points...

1. I believe because of the peg, we have had to keep the dollar low - hence rising M3, etc. Therefore, although the Yuan/USD cross hasn't moved, the excess dollars have floated elsewhere. When the peg is lifted and the Yuan appreciates, it will not only do this against the dollar. There will be more of a redistribution of dollars as the system reaches some sort of equilibrium. Therefore, different measures of dollar indexes will change by different degrees. The trade weighted index will be different than G10 index. I think the yuan is overvalued because of dollars, but may be very undervalued against the Euro.

2. That was a poor example. I just tried to illustrate the idea of covered interest rate parity. You are right. I agree that in reality this is not the case, but may be offset by the yuan peg.

3. Note, I did not say go to zero but "approach" zero. It is the approach to zero or the approach to infinity that makes functions behave strangely as they never reach either. In reality the dollar CAN equal zero. Since a line can not gain conciousness while travelling on a plane, this will never happen in mathematics, but a market can - they can essentially shape their own critical points. If this path continues like it has without hitting some sort of critical point (which is what you mention above), then yes it could be zero, but some sort of intervention is more likely.

Posted by: Zero Beta | October 18, 2007 at 01:42 AM