# A world without a risk free rate

#### by

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.

Is market incompleteness related to correlation both within and between asset markets? The housing markets in a country are correlated regionally and market behavior is more loosely correlated around the globe. Certainly the same for equity markets – Australian markets reflect events in the US and China as much as Australia and there is substantial correlation between different disparate sectors within a single market. It does seem difficult to diversify intelligently so you cannot evade risk. The correlation is related to international capital market integration which improves efficiency but means that shocks will have greater amplitude.

Of course, if you can’t find a portfolio which gives a positive fixed return in all states, then there is market incompleteness. So effectively you can’t get any form of law of large numbers, with or without correlated assets. Correlation in itself does not imply incompleteness.

The kind of incompleteness that I’m predicting here in this post is even worse. The idea is that there will be states where all asset returns are the same and trivial. It’s a much stronger condition than market incompleteness.

How is cash (notes and coins or equivalent electronic entries, issued by a monetary authority) treated in your model? I ask this question because in a 2004 paper, Model of Money Tokens:A Uniform Description of Monetary Objects and their Circulation (Furche, Gross)*, we argue that as long as cash is be redeemed by cash, there is no default and the monetary (numerical value) return on the security is constant but the rate of return on this security is always zero.

When you use the term ‘return’, do you mean pay-off or rate of return (as in much of the finance literature)?

*The full paper happened to be published on the registration web-site of the South American Econometric Society Conference although it was not presented.

cash is not explicitly treated. What I’m basically saying is that a world without a risk free rate is situation that precludes any reasonable way of calculating risk free rates.

So if we take a world without a risk free rate as something serious, then there must be events that we cannot foresee. In your model, someone with an AK47 takes all our cash.

Thank you for your quick response.

If we take a security, called cash, out of our model then we have a private ownership economy (no government which has the authority to issue legal tender).

For example, the CAPM is a characterisation of a solution of a special case of a model of a private ownership economy with complete securities markets.

But if we leave cash in the model then we do have a risk free rate of return. It just happens to be zero.

It is not clear to me whether you mean a strictly positive risk free rate or a non-negative risk free rate

We would say the habit of using, say the US treasury rate, as an empirical approximation of a ‘risk free rate’ lacks a theoretical foundation.

Zero risk free rate is fine, you pay one dollar now and you get a dollar back next period.

But

1. cash is not risk free. treasury bills are more risk free.

2. I’m talking about states in which you pay a dollar now and get zero back. That’s what not being able to compute a risk free rate must mean.

Suppose I keep 1 dollar bill (cash) for 1 period. Then my risk free rate of return is zero.

Suppose I exchange my 1 dollar bill (cash) for a treasury note and the issuer of the treasury note defaults. In this case the rate of return is strictly negative (but computable).

In what sense is a treasury note less risky than cash? (I recall you don’t have cash in your model.)