What would Gödel do?
Posted on May 14, 2007
Kurt Gödel was a mathematical genius who has had immense impact on science, math and philosophy. Later in life he would become Albert Einstein’s best buddy. Gödel has long been a favorite of ours, perhaps because he was a logician. We never dreamed we would be writing about him in the context of financial markets and investments. But the deeper we delved into program trading, derivatives, globalization and volatility, the more we thought about Gödel’s famous incompleteness theorem. It takes a book to accurately and completely describe Gödel’s proof. Here we’re going to distill it to its basic ideas. Everyone knows there are high-powered mathematicians who create marvelous systems of rules (axioms) and statements (theorems) that are put forth as being complete and without defect. What Gödel showed is that any such system is actually full of holes (contradictions.) This is a profound result when you realize that it applies to any system sufficiently expressive to be of interest. Many gifted and famous intellects have spent their best efforts trying to deny the conclusions of Gödel’s work, all to no avail. Of course financial markets are not pure mathematical systems. Many prefer to apply behavioral and statistical methods for understanding the possibility of unexpected events (Fooled by Randomness by Nassim Nicholas Taleb, for example). The new players in this game seek to assure us that they too are familiar with all the tools and techniques available for managing risk. Furthermore they state that their systems are state-of-the-art and there is nothing to worry about. On another level we can be sure that these experts are unlikely to inform us of any concerns given their vested interests in ever more exotic instruments and larger bonus checks. It’s an over-played example, but the collapse of Long-Term Capital Management (LTCM) in 1998 is very illustrative of the problem of believing that formal systems can provide certain answers and sure profits. The events that unfolded for LTCM were deemed to be nearly impossible by the supposed experts running the firm, including the two Nobel Prize winners Myron Scholes and Robert Merton. In fairness the problems at LTCM may have had much to do with the partners overreaching for returns and moving out of their respective areas of true expertise. Still the situation illustrated that elements of risk believed to be static can change over time, invalidating previously accurate models. For example, one of the key inputs to determine Value at Risk (VaR) turned out to be such an element. This contributed to the evaporation of $4.6B in capital. There’s more to this story than LTCM. Errors in judgment will always happen and often lead to losses. When leverage is used, those losses can be spectacular. There is something deeper here. Our models of risk and volatility may be woefully inadequate. Moody’s is one of the leading credit rating agencies in the world. Recently it decided to change some of its rating criteria, which instantly raised many debt ratings. Industry experts were surprised at first and then appalled. Nobody thinks debt ratings are perfect but insofar as they measure the ability of the borrower to pay interest and principal based on the fundamentals of their business are certainly a valuable and desirable function. As we read about it today the Moody’s move seems bizarre. The idea was that some entities, notably banks, were not likely to be allowed to fail. Even if their businesses and practices led to a steady loss of capital and the inability to pay back debt, the governments would come and bail them out. While this may be true in a large sense it doesn’t seem consistent with a perfect (AAA) rating. If a bank faces insolvency it’s a good bet that their bonds are going to trade off par and show the kind of volatility more closely associated with lower graded bonds. Furthermore even if a government does step in to take on the liabilities, it doesn’t protect investors from losses from a weakened currency. Moody’s ended up capitulating to all the criticism and reverting back to its old system. The chain of events doesn’t project an image of rigorous and tested logic in describing risk in the debt markets. No comment on derivatives would be complete without touching on Collateralized Debt Obligations (CDOs) and their zany non-mark-to-market rules. When people such as Warren Buffet and Charlie Munger state that the true value of derivatives portfolios are totally unknowable one has to take note. Of course the companies that make fees in the market have no trouble telling everyone what the prices are. Most people have the quaint notion that the financial markets comprise investors. However, program trading is much larger than normal trading in the equity markets, and the size of the derivatives market on debt exceeds the actual debt underlying the market. So, which is the tail and which is the dog? Industry insiders counter that the global financial system is completely resilient these days and events such as the sub-prime meltdown and various hedge-fund implosions are minor costs to be expected in the system. They are no more worrisome than a few deadbeats are in the world of credit cards and consumer finance. The major players in the derivatives markets are very large indeed and provide assurances that they have huge investments in very sophisticated risk management systems that have been proven over time. Our global financial system has proven to be resilient but it is not invulnerable. We haven’t even mentioned the huge balance and foreign ownership of the U.S. debt, the near ubiquitous use of derivatives, rampant carry-trade behavior, incredibly lax underwriting standards, entitlement funding gaps around the world, higher crop failure risk due to decreasing biological diversity, new strains of flu or other pandemic-causing disease, political instability from a growing China or bolder Russia, and so on. Our nature is optimistic, and we have no special talents when it comes to being short. However the risks that are inherent in a complex system are undeniable when you think about them from a Gödel perspective. One doesn’t have to believe in any side to know that at some point everyone is likely to be very wrong. We think that this provides a strong basis for controlling your own investment risk profile. You shouldn’t rely on assurances from the experts. It’s also not just about expecting a major decline in markets; they could go up exponentially just as easily based on this argument. So being in cash isn’t a perfect solution either. You can still make investments, have a cash cushion and be prepared to deal with volatility. Gödel would probably believe in owning long-dated out-of-the-money call and put options, ideally with no expiration date. Positioning a portfolio properly can’t be done via a simple recipe but this argument suggests a large cash cushion, liberal use of long-dated puts and calls on stocks and indexes, an attention to positive cash flow and some ownership of liquid hard assets. It should go without saying that the use of leverage without these protections is very unwise indeed. – Kris Tuttle
Tags: Godel, Derivatives, Stock Market, Risk
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The word ‘word’ defines itself. ‘Word’ is a word that that defines itself. ‘Word’ is a self-defining word. ‘Word,’ also is a word in the group of words that define themselves.
What about ‘non-self-defining?’ Does ‘non-self-defining’ fit into that same group of words? ‘Non-self-defining’ is a word that does not define itself, since it’s non-self-defining. But then if it doesn’t define itself, it is non-self-defining! So is it part of the group – or set – or not?
The point is, you define a set, or a system of rules for say, words (or numbers) there’s always an exception.
This does not mean that ‘non-self-defining’ is not a word. It does not mean you can’t use the term ‘non-self-defining’ in a sentence. It does mean that when you create a system, a set of rules, or a formalized notion or ideas a flaw in that system exists, a priori.
This is a way to think of Godel’s Incompleteness Theorem. You can’t create a set of intellectual rules.
But…
This does not mean that finance as a mathematical construct is fundamentally flawed in its construction. Finance is “real” in that publicly traded pieces of paper – or bits on a computer screen - represents ownership of assets and cash flows.
I do not know how Godel would invest. I think Nassim Taleb would definitely want to “[own] long-dated out-of-the-money call and put options, ideally with no expiration date.” So would I, but only at a certain price. And you’d of course, have to find someone to sell them to you.
Then…
1) And if you did find someone to sell them to you, it would be interesting see how Moody’s evaluated that purveyor of credit derivatives selling permanent options and how they hedged their own position(s.)
2) A market would then – or could then be established, once the phone-based or floor-based traders were forced by the government to open up their club got out of the way – for these ‘perm options.’
Finance is not a formal mathematical system, but that does not mean there is not an optimal way to invest.
Owing the cheapest index product – Vanguard anyone? - is in fact a high capital way to own perma calls. Cash is a synthetic way to own the perma puts. Over time the price of the indices and cash goes up and down. So you should rebalance between them holding your asset allocation fixed. You should also, have your index held in the cheapest possible manner.
While there are tax issues and it would be great to generate interest income from short side players on your self-owned (not self-defined mind you) indices – you can’t do this because those asset managers take that from you unless you’re really rich – this is the optimal way to invest. Own the Vanguard US and international indices and rebalance among them and against your cash.