Notes Toward Modeling a Risk-Free Rate with Default Possibilities

Brad DeLong asks why it hasn’t been done, if it hasn’t been done.  The biggest problem I can see is that you don’t know how insane the participants are—and that will have a major effect on how much damage is done when.

Don’t get me wrong; the damage is already being done; it has been since at least May, and if Barack H. Obama weren’t an idiot, he would have been mentioning that over the past two months.  Unfortunately, the sun is yellow on our world, and counterfactuals are masturbatory, not participatory, acts.

So let’s start with what we know:

i= r + π^e

Nick Rowe apparently would have us believe his (completely understandable) claim that i would not be directly affected by a short-term default. This strikes me as absurd.
Even when the economy is working on all cylinders–where G contributes something around 10-15% of growth at most—reducing G to zero for a week is about 2% of 15% or 0.2%-0.3%—noticeable, but arguably rounding error against the difference between π and π^e. So, if you assume a short-term issue, you get something like those legendary two weeks from 11-22 September 2001, when only the Saudi Royal Family was spending anything, writ somewhat smaller only because Gunderstates the effect on r.)

We can concede that inflation expectations themselves aren’t going to go up independently: any additional borrowing cost will be a drag on r, so it’s not unreasonable to assume that i will be fairly steady—again, working a very short-term issue.

But, as often happens, we leave out a variable in our assumptions, simply because we define i as the risk-free rate of return.  Let’s put it back in:

i= (r + π^e)*(1+ P_d)

Where (1+P_d) is 100% plus the probability that there will not be a default. (Note that in a model environment, P_d=0, otherwise, we would not call it risk-free: actors have the power to make and manage budgets, including the power to tax to pay for services desired by their plutocrats constituents.)

Finance people will recognize this reduced form equation:.  (1+ P_d)= β, the risk of the stock or portfolio in excess of the risk of the market.  For convenience, let’s just call this version Ω. So,

i= (r + π^e)*Ω

Next comes the hard part: term structure.  Or, as Robert said in a similar context, are you talking about the Federal Funds rate, or the rate on three-month Treasury Bills?

Well, that depends on how long we expect the issue to be an issue.  If Barack “I’m an idiot who stands for nothing and you’ll vote for me anyway because my opponent will be insane” Obama treated this “crisis” the way he treated the last (real) one, he would insist on getting a clean bill raising the debt ceiling passed through both Houses and on his desk for signing by the end of next week.  If he takes it as another chance to blow Cass Sunstein and the rest of his University of Chicago buddies, then it’s a complicated bill that will get a few Congressmen killed* and several others de-elected, and we might be talking weeks.

Right now, the markets are assuming the former.  Let’s be optimistic and assume they’re correct.

Four weeks ago, there was a Treasury Bill auction that produced a yield of  0.00%.  Extending that bill does no harm at all—not even to expected debt totals. (Investors mileage may vary, but they bought it with full knowledge of the timing.  And there were 10% again behind them bidding at the same rate.)

Some specific Notes and probably Bonds—it is August—will have coupon payments due on the 15th. But those are coupons, not principal repayment, so again we’re not talking much value of the Note or Bond itself, once you hit five years or so.

Bills will be a problem.  Short-term notes will be a problem.  Fed Funds is uncharted territory.  Tripartite Repo specifically, and Repo in general will be a major problem due to questions of collateral value.  And guess who uses those the most?  Hedge funds.  The people who have been financing John Boehner’s and Eric Cantor’s campaigns.

So the term structure looks like it would if you’re going into a recession: short-term rates rise significantly, while the longer term securities shift upward a bit. (Select Notes and Bonds with near-term coupons kink the curve, but there’s no certain arbitrage there, especially with transaction costs.  Cheapest-to-deliver calculation is also affected in the futures market. I could go on, but let’s just pretend—correctly—that these are minor issues.)

Because now our “baseline” rate is no longer risk-free—and we’re not certain what P_d is over time.  We know it will return to zero at some point, and we presume (at least at the beginning) that it will be soon. But we also know that there already are follow-on effects, and that they will only get worse. Even if we ignore the effect on G (and therefore i) of a short-term default, we lose our bearings for a while.

So the big question is collateral and spreads.  Been posting Treasuries to borrow against?  Yields up due to P_d > 0, so prices down, so less flow. And probably haircuts due to uncertainty of any return to “risk-free.” Posting Treasuries with a coupon due?  Haircut! Posting Treasuries with a near-term coupon?  Haircut! Posting Munis?  Think Michael Jordan (or Telly Savalas, if you’re Of a Certain Age).  So you can borrow less, and probably have to sell some of your assets.

Which ones?  If we’re lucky, it’s longer-term Treasuries, and some of the yield curve inversion mentioned earlier is reduced.  But the market is going to be less liquid than usual, so maybe some of those other bonds get sold—corporates, for instance.  The bid-offer on Munis is basically going to be zero-coupon bonds at a high discount. (Think fast about how many state and local municipal projects depend on some form of Federal funding.  Then realize that your estimate is probably low by an order of magnitude.)  Or corporations that are dependent on government funds (DoD providers, automobile fleets, interstate paving contractors, power supply and distribution companies, etc.)

In a ridiculously oversimplified model, the spreads simply expand by Ω, with a possible adjustment downward based on direct exposure to government financing. This, again, probably understates the effect.

So, in a closed economy, everyone gets to pretend things are close to the same—just more expensive, with a lot of damage to hedge funds and municipalities and borrowing costs and credit lines. So money supply drops significantly (multiplier effect reduction) even without Fed intervention.  You get an Economic Miracle: reduced supply and higher yields.

But we don’t live in a closed economy.  So there’s another factor.  And I’m running out of Greek letters, so let’s just use an abbreviation everyone knows:

i= (r + π^e)*Ω*d(FX_d)

where d(FX_d) is the change in the FX rate due to the default adding uncertainty to cash flows.

That’s right: we get not one but two economic miracles:  (1) domestically, a reduced supply of risk-free securities produces higher yields and (2) internationally, higher yields lead to a depreciation of the domestic currency.

Anyone still wonder why no one wants to build the full model?

*No, I don’t want this to be the scenario.  But if you offered me the bet, I wouldn’t take the under at 0.99.