Thursday, April 1, 2010

Explanation of How a Zero Boundary Causes a Steep Yield Curve

This post will hopefully explain Paul Krugman's great interpretation of the yield curve... it's caused by the zero bound on the front-end of the curve. First to Paul...

As I tried to explain last time, to a first approximation you can think of the long term rate as reflecting an average of expected future short-term rates. Short-term rates, in turn, tend to reflect the state of the economy: if the economy improves, the Fed will raise short-term rates, if the economy worsens, the Fed will cut. So long-term rates can be either above or below short rates.

Except that now they can’t. If the economy improves, short rates will rise; but if it worsens, well, they’re already zero, so there’s nowhere to go but up. This implies that there has to be a positive term spread.

At first this confused me and apparently I was not alone. Crossing Wall Street details their personal misunderstanding:
Where I really lose Krugman is when he seems to imply that the steep yield curve is the result of nominal short rates being near zero. I have no problem accepting that the curve should be positive but I don’t get how that ought to impact its unusual steepness. After all, the two-ten spread recently hit an all-time record.
Lets see if I can explain with an example and (of course) a few charts.

Yield Curve Due to Zero Bound Rates

Assuming longer term rates only reflect futures expectations of rates, lets also assume there is no view of interest rates... an investor thinks that for each period from here on out, there will be a 50% chance of a rate hike and a 50% chance of a rate cut (lets assume 25 bps).

Non-Zero Bound

If the Fed Funds rate was 4%, then next period will either be 3.75% or 4.25%. The period after that 3.5% or 4% (if there is a cut period 1) or 4% or 4.5% (if there is a hike in period 1). In this case there is no expectation of a yield curve steepening because each possibility is as likely (a 50 bps drop in 2 periods or a 50 bps hike in 2 periods). The four outcomes are 4% (cut, hike), 4%, (hike, cut), 3.5% (cut, cut), or 4.5% (hike, hike). An average of.... 4% (i.e. rates went nowhere on average).

Zero Bound

This is where things get interesting. The Fed Funds rate is zero. They cannot go negative. Thus, flip your coin for a 25 bps hike or cut. If you get hike, rates rise to 25 bps. If you get a cut... well, you can't cut any more. Go to period two. Now under the initial hike, rates can go back to 0% or jump to 50 bps. Under the initial cut, they can only go to 25 bps or stay at 0%. Thus, the four outcomes in period 2 are 0, 0, 25 bps, and 50 bps. An average of 18.75 bps (they went up).


Starting at the zero bound and running this same scenario for 200 periods (and averaging the outcome of 15 samples), we get the following outcomes.

Example 1

Example 2

Running this hundreds of times (built a nice little spreadsheet I must say), the charts all looked roughly the same (just different scales).

I hope this was helpful...


  1. Jake,
    good stuff and I have no problem with the mathematical excercise. Real world applications make it a probelm, as with most math models of things economic. From TAE the other day a pefect example:
    "Ilargi: White then continues with a video interview BI's Aaron Task had with James Altucher at Formula Capital (I swear I was going to write Fantasy Capital), who claims there are no problems with muni and state bonds, specifically because in the instance of for example California, the constitution states that bondholders have to be paid before anyone else, including employees, once the 40% of the budget allocated to education is paid. So bondholders must be paid ahead of policemen and firemen.

    Altucher continues to say that both the housing markets and the employment situation are stabilizing, and we are very far removed from any kind of "breakdown of social order". Really, when you have to fire your police force in order to pay your debtholders? "Very far removed"?

    At least that makes us realize where he stands. And/or dreams. Mr. Altucher may have an inkling of knowledge about capital and/or bond markets, but he's entirely out of whack when it comes to the real world.

    Look, if Greece didn't have to pay its employees, there would be no problem there either, from a purely financial point of view. The problem with this "analysis" is that both Sacramento and Athens DO have to pay their employees. The alternative is to face ever more protesters, who grow ever more angry, with ever less police, who grow ever less motivated."

    Its a good point: on paper muni bonds look great because after all is states get in real trouble they can just stop paying the state employees and pensions and pay off bondholders at 100%. Of course this is excludign the federal bailout which will happen, but you see what I mean.

  2. The main flaw here is that you assume rates reflect future expectations. And, of course, they do to some degree, but not in totality.

    Institutional investors purchase treasuries with specific yields to match maturities on expected redemptions, payouts etc.

    Liquidity Preference Theory also plays a big role in the yield curve and I imagine there's a host of people out there who want a more liquid, short-term investment right now than want to hold the long bond. Why? Well, with signs (generally) pointing to some sort of recovery, investors don't want to get locked into long-term, low-yield treasuries if they feel a better return is just around the corner.

    I must admit that Expectations Theory is the most widely-held view of why the yield curve is shaped the way it is and why it's steep or flat. But, in reality, there has to be accounting for the other two theories as well.

  3. I agree with Krugman about this:

    "As many people have noticed, the term spread — the difference between short-term and long-term interest rates — is very high. The last time I wrote about this, people were taking this as proof that the economy would recover soon. Now they’re taking it as bad news — as somehow suggesting fears of default. But there’s a reason for a high term spread that has nothing to do with either explanation."

    But I thought the issue was this:

    "A sign of the strains across US fixed income markets was this week’s historic rupture between the 10-year Treasury yield and its close derivative, the interest rate swap.

    "For the first time since swaps emerged in the mid-1980s, the 10-year swap rate traded below that of the “risk free” 10-year Treasury yield. Analysts say this reflects how government debt issuance has altered the dynamics between “risk-free” yields and swaps, which reflect borrowing costs for non-sovereign borrowers."

    What I had argued was that this was not good because it reflected concern that we would have problems going forward politically solving our budget problems. This occurred at the time Health Care was being voted on because the bill was not bipartisan, in my view, signaling danger in dealing ahead.

    Since I'm a Democrat, I saw the GOP as the main cause of the problem, and not a concern about how the bill will play out, since it's a wash in my view.

    Did I miss something obvious? That's certainly possible.

    Don the libertarian Democrat

  4. Befuddles me that anyone still buys the "bond market can see X" or "the stock market can see X". They can see tomorrow and maybe noon on saturday, but thats about it.

  5. This Monte Carlo type analysis is an interesting theoretical exercise.
    But, the reality is of course somewhat more complex, confirmed by the fact that neither finance nor macro models adequately explain the yield curve.
    Specifically, if your measure of steepness is 2-10, who's to say that the 0-2 spread can't capture the majority of the steepness. (If, for example, the market expects a couple of years of exceptionally low rates, but then market normalisation - possible if inflation expectations are successfully contained.)
    It seems to me that this hypothetical analysis misses some key points.

  6. nowhere do i claim that this explains the steepness in its entirety. all this attempts to do is explain what krugman lays out... that on the margin, a zero bounds causes the curve to be steeper all else equal