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...