I argued in October 2008, on Harry Clarke’s old blog, that the main theoretical story emerging from the global financial crisis is market incompleteness. At the time I summarised my views as follows:
what we are seeing is the effects of the incompleteness of the kind of instruments available to the market and not the result of too many complicated assets. Derivatives are not weapons of mass destruction on the contrary the price changes are due in large to the incompleteness of the derivatives market.
What that means is that it became apparent in 2008 that there can be events that cannot be insured against in the market. One dramatic example of this market incompleteness phenomenon has to do with possibility of that the US will default on its obligations next month. Indeed, US Treasury bills have been considered to be the canonical risk free asset. It has been used as the absolutely risk free asset for calculating the risk free rate. US Treasury bills are of course not risk free because without a doubt the risk of US default now is not trivial. For this reason one of the themes of the March IMF conference is how we understand a world without a risk free rate. I’ve been reflecting on this theme in terms of our economic models of market incompleteness. I’ll argue here that a world without a risk free rate is a world in which the market is unaware of certain states or events. Indeed, it must be a market characterized by complete unawareness of even the nature of the crises that may occur.
What does a world without a risk free rate mean? I understand it as follows:
- There are no risk free bonds. All government bonds have a non-trivial likelihood of default.
- There is no risk free portfolio of assets. That is, there is no way that you can replicate a risk free bond through diversification.
Both 1. and 2. seem to be a reasonable description of the state of the market. Every portfolio is risky. But what does this mean in terms of our simple models of incomplete markets?
I’ll confine my thinking to the very simple model describing market incompleteness of Ross, Options and efficiency (QJE, 1976). That paper shows that, in general, in an incomplete market you can find a portfolio such that if you add to the market all the possible call options using that portfolio, then you end up completing the market. In particular, after adding the options you can find a portfolio that gives you whatever payoff you want.
This insight was recently extended in a remarkable, very readable, yet relatively unknown paper by Alexandre Baptista On the Non-Existence of Redundant Options, (Economic Theory, 2007). Reworking Baptista’s result and proofs it is easy to show that, in general (conditions that are very realistic), if we are in a world without a risk free rate (1., 2., above), then it must be the case that for some states of nature every asset and every portfolio pays a trivial return. Conversely, if there are states of nature for which each asset and each portfolio pays zero return, then we are in a world without a risk free rate.
So essentially in this theoretical model a world without a risk free rate can be characterized as a world with uncertain events that are not priced by any of our assets. The only interpretation for this is that a “real” world without a risk free rate is a world with possibilities that the market is unaware of. These events can happen but they cannot be insured against because we don’t know what they are before they happen.
In short, the theme of the IMF conference is all about the prudential role of governments where there are non-trivial probabilities of things happening whose possibility we cannot foresee. A world with possibilities of a crisis whose effects cannot be mitigated before they realize. That to me seems to have always been the nature of the world that we live in.