# Bitcoin and the No-Arbitrage Condition

One of my favorite aspects of the Bitcoin phenomenon is that it has people talking about monetary economics and finance. Just as recessions tend to produce advances in academic macroeconomics, Bitcoin is forcing a wide cast of characters, from libertarian nerds to economic journalists, to think more deeply than they otherwise would about money and monetary institutions.

Nevertheless, monetary economics can be difficult, and there is a lot of confusion out there. It seems that most of the confusion is due to two errors. The first is that Bitcoin appreciation is deflationary, and therefore, recessionary. A variant is that Bitcoin volatility will create massive booms and busts in the Bitcoin economy. I think that this point has been decisively refuted by Jerry Brito. The macroeconomic effects of a currency have to do with its unit-of-account status, not with its medium-of-exchange status. Consequently, unless people begin (foolishly) denominating their long-term contracts in Bitcoin, the cryptocurrency won’t have any macroeconomic drawbacks.

As Bitcoin skeptics have come to terms with Jerry’s point, they have resorted to a second error, that Bitcoin’s long-run fixed supply would generate persistent, long-run deflation, which will cause hoarding of the currency. This would make it unsuitable as a medium of exchange, because no one would be willing to exchange it. Transactional demand for Bitcoin would be zero. Matt Yglesias and Matt O’Brien make versions of this argument.

I think that Tim Lee has done a good job of refuting this line of thinking both on Twitter and in two posts at Forbes. But his arguments haven’t satisfied everyone. O’Brien in particular seems to be doubling down on Twitter.

The problem is that in Yglesias’s and O’Brien’s posts on Bitcoin, I have not come across the word “arbitrage.” This is a pretty good sign that their claims about the long-run equilibrium are not rigorous. The long-run equilibrium must be defined by a “no arbitrage” condition—if arbitrage between currencies is possible, then we are not in equilibrium.

Let’s try to write down an equation that describes a first approximation of this condition:

$1 + i_\ = \dfrac{E_t(S_{t+k})}{S_t} - \pi_\sigma - \pi_l$

At a high level, this equation says that the expected return to holding dollars has to equal the risk- and liquidity-adjusted expected return to holding bitcoins. On the left side of the equation is the return to holding dollars, which is given by the nominal interest rate on dollars, $i_\$. On the right side of the equation, $\frac{E_t(S_{t+k})}{S_t}$ represents the expected appreciation in bitcoins from time t to time t + k, while $\pi_\sigma$ is the risk premium and $\pi_l$ is the liquidity premium. I am assuming for now that it is not possible to have a bitcoin-denominated loan, and therefore no interest rate on bitcoins, although I could imagine that there might be an overnight rate of some sort. But for now, assume that the equalization of returns has to happen via appreciation.

What this equation makes clear is that there is no free lunch from hoarding bitcoins. Bitcoin hoarders will be compensated for the risk they are bearing, for the illiquidity of the Bitcoin market, and for the opportunity cost of holding bitcoins, but for nothing else.

The fact that bitcoins are expected to appreciate in value does not increase the incentive to hoard bitcoins at the margin. Instead, all of the change in the value of bitcoins happens in the spot rate, $S_t$. The price of bitcoins adjusts now to accommodate any future expected increase in the value of bitcoins, and there are no further gains from hoarding bitcoins. There is therefore no disincentive to transactional use of bitcoins.

I expect further that in the future, forward contracts on the Bitcoin-dollar exchange rate will reduce or eliminate the risk premium, and more sophisticated entrants (read: hedge funds) into the Bitcoin market will supply additional liquidity, making even the nominal return to holding bitcoins about the same as that of holding other currencies. The model above is not meant to be a complete account of the market for bitcoins. But I think it serves as a good baseline for thinking about what claims about an incentive to hoard entail.

Finally, I’ll note that even if almost all of the eventual 21 million bitcoins are “hoarded,” a mere 1000 bitcoins would be more than adequate to supply the transactional needs of an economy as large as the United States. Each bitcoin is divisible into 10^8 “satoshis,” and there are only around 10^9 dollars, or 10^11 cents, in circulation. One thousand bitcoins would be 10^11 satoshis. If each satoshi equalled one cent, the market capitalization of bitcoin would have to rise to 2.1 quadrillion dollars. When the market capitalization exceeds that figure, I will concede that bitcoins have been over-hoarded.

# Globalize the Banks, Redux

Tyler Cowen’s essay on the inequality generated by the political economy of finance has received deserved attention. I think Tyler is exactly right that financiers are going short on volatility, that this is hard to detect and therefore regulate, that politicians have incentives to bail out the banks following increases in volatility, and that it leads to severe problems.

Please excuse the self-congratulation, but as far as I can tell, I am the only person to propose a reform that could plausibly reduce this problem. In April, I wrote,

[A]n important factor in the stability of the financial sector is its degree of globalization…

The second and deeper reason for more financial globalization is political. The doctrine of Too Big to Fail creates a moral hazard problem: banks reap the benefits of successful investments but don’t suffer the losses of failed investments, so their incentive—whether conscious or evolved—is to take a lot of risk. Politicians, as much as they claim to want to enact market discipline, face a time-consistency problem. They want banks to be sound, but they lack the political will to let unsound banks and their creditors suffer when the time comes to feel the pain.

Greater globalization reduces the time-consistency problem. American voters do not want to pay taxes to bail out banks that operate just as extensively in Europe, Asia, and Latin America as in the US. The fact that a bailout would convey these uncompensated externalities would strengthen the political will to let unsound banks fail. This is not to say that I hope lots of banks fail—rather, I want politicians to be able to make more credible claims that Too Big to Fail is over, so that banks take note and invest accordingly. In equilibrium, there would be fewer bank failures, not more of them.

My claim is not that globalization of banking is a panacea. But to extent that we can reduce the time-consistency problem that politicians face, we can reduce the incentive that financiers face to write naked puts on volatility.

# The Fragility of Eurozone Finance

Over at the FT, Gavyn Davies writes concerning the Eurozone,

Member states cannot print the euro, which automatically increases the risk that they will default on their debt. (Admittedly, it also reduces the risk that they will inflate their debt away. The markets are not too worried about this in these deflationary times, though one day they might be.)

I think about European sovereign debt in somewhat similar terms, but I want to elaborate on Davies’s framework a little.

First, refusing to pay up and paying up in inflated currency are both forms of default. The fact that EU member states cannot print the Euro therefore changes the form of the risk of default.

Second, the form of the risk of default matters. Think about the payout of a security under variable outcomes. When the outcome is very good, the security pays out a high amount, and when the outcome is very bad, the security pays out very little. We can imagine that in the middle of the outcome spectrum, there are a number of possible payout profiles. You can have payouts vary smoothly with outcome, or you can have a tipping point, above which the payout is high and below which the payout is low. When payouts vary smoothly with outcomes, the price of the security is not going to be very volatile. When payouts vary sharply with outcomes, the price of the security may be highly volatile when outcomes may fall on either side of the tipping point. I think this is what Nassim Taleb means when he says that debt is fragile and equity is robust: debt payouts vary sharply with outcomes and equity payouts vary gradually with outcomes.

Third, when a government can inflate away part of its debt, the real, inflation-adjusted payout can vary more smoothly with outcomes. This makes the debt a little less like debt and a little more like equity. When a government cannot inflate away its debt, the price of its securities are going to be very volatile around the outcome tipping point.

Fourth, since EU governments cannot inflate away their debt, the risk premia on their bonds are going to take a highly volatile form. This is going to lead to frequent debt crises unless the governments are highly responsible and avoid landing anywhere near the outcome tipping points.

Davies concludes his post with “Something, somewhere has to give.” I think that what has to give if people want to preserve currency union is that member countries are going to have to accept balanced budget requirements as the US states have. But at the same time, I am skeptical that the southern European countries can really be counted on to abide by such requirements if they are imposed. They failed to keep their deficits within the 3% of GDP limit the EU treaties currently require, and they have been and are going to be rewarded with bailouts.

Many commentators are talking about fiscal union, which is perhaps a subject for another post, but I am not too enamored of this option as it has many public-choicy downsides. Probably the thing to do, though difficult, is to give up on the common currency and instead focus on creating the largest free-trade and free-movement zone in the world. That is the robust road to prosperity.

# Thoughts on Sovereign Default

My new favorite website is this one, which monitors credit default swaps and displays some useful free data about them (you can pay to get more data). The most interesting statistic is the cumulative probability of default for various sovereign bonds. The site calculates these over a five-year period based on the term structure of interest rates and the price of various credit default swaps.

As of this writing, the five-year probability of default of selected EU countries is as follows:

• Greece: 43.21%
• Portugal: 24.25%
• Spain: not listed at the moment, but as I recall around 18%
• Italy: 17.18%

Looking at these numbers, you may not be alarmed. After all, what is the probability that all four of these countries will default? Multiplying the probabilities together yields 0.4321 × 0.2425 × 0.18 × 0.1718 ≈ 0.0032. That’s not so bad.

The problem is, it’s not correct. Multiplication assumes that each of these default probabilities is independent from the others. But during a financial crisis, all correlations go to one. The correct answer is about 17.18%, the probability that Italy will default. If Italy defaults, that means that the other four have probably already defaulted or are about to do so.

There is another implication of correlations going to one. The EU cannot bail out all of its members. They may be able to bail out Greece and Portugal, but I doubt they will be able to bail out Spain and Italy as well.

The other indicator I’ve been looking at is the risk premium on Greek bonds relative to German bonds. Earlier this month it went from 5.4 percent to 9.6 percent in four days. After the bailout package was announced, it dropped and is today around 5.1 percent (three years ago it was around 0.2 percent). I would not be surprised to see it spike again like it did earlier this month. If, when the crisis came, austerity was politically impossible, what makes anyone think that austerity will be politically feasible when there is no crisis? And why, when the next crisis comes, does anyone think the outcome will be any different?

In case you are wondering, the probability of the US defaulting in the next five years is around 3.51%. This means that the probability we will soon be living in caves is: 3.51%.

Finally, we have more evidence, as if we needed it, that politics isn’t about policy. Politicians say they want to regulate “systemic risk” so that we do not experience another financial crisis. What is the greatest source of systemic risk? You guessed it: government debt.

I’ve been trying to figure out the best investment strategy in light of these facts. Here’s how I’m leaning: invest in leisure—it can never be expropriated. Have a nice day.

# To Fix the Banks, Globalize Them

As politicians and pundits debate what regulations are needed to fix the financial industry, I want to draw attention to a neglected issue: an important factor in the stability of the financial sector is its degree of globalization. All the big American banks have some international dealings. But true globalization of banking would be an improvement for both financial and political reasons.

The first and more trivial reason that banks should be more globalized is simply the logic of diversification. When investments are diversified, investors get higher return for less risk—it is a free lunch. Failure to internationally diversify is one of the biggest mistakes that American investors make (I’ve heard, as a rule of thumb, 30 percent of your investments should be foreign). But the big banks make this mistake as well! For instance, 90 percent of Bank of America’s revenues come from the domestic market. And of course, smaller banks frequently have no international holdings.

International diversification reduces systemic risk just as national diversification reduces systemic risk. If all banks could borrow from and lend to people in only a single town, we would see bank failures every time a factory shut down or a large business failed. By spreading that risk over thousands of towns, banks become more stable. But the logic of geographic diversification does not stop at national boundaries. In fact, because there are greater differences between countries than between towns in the same country, international diversification may be even more beneficial than national diversification.

The second and deeper reason for more financial globalization is political. The doctrine of Too Big to Fail creates a moral hazard problem: banks reap the benefits of successful investments but don’t suffer the losses of failed investments, so their incentive—whether conscious or evolved—is to take a lot of risk. Politicians, as much as they claim to want to enact market discipline, face a time-consistency problem. They want banks to be sound, but they lack the political will to let unsound banks and their creditors suffer when the time comes to feel the pain.

Greater globalization reduces the time-consistency problem. American voters do not want to pay taxes to bail out banks that operate just as extensively in Europe, Asia, and Latin America as in the US. The fact that a bailout would convey these uncompensated externalities would strengthen the political will to let unsound banks fail. This is not to say that I hope lots of banks fail—rather, I want politicians to be able to make more credible claims that Too Big to Fail is over, so that banks take note and invest accordingly. In equilibrium, there would be fewer bank failures, not more of them.

A number of commentators have argued that to deal with this political dimension of our financial troubles, we must break up the big banks. The argument is that if the banks were smaller, they would have less sway over politicians and the regulatory agencies. I have a lot of respect for many of the people making this argument, but I believe they are mistaken. One need only look at the sway that car dealerships, which are decentralized, have over state legislatures to see that small does not mean apolitical.

How can we achieve a more globalized financial sector? In truth, I don’t know. I suspect that only the insiders could say for sure. Perhaps we should ask them. At a minimum, we should consider repealing regulations that make it difficult or unprofitable for banks to operate internationally. Instead, regulatory reform will probably consist of a lot of rhetoric about getting tough with the banks—further proof that politics is not about policy.