Another thought on the distribution of forward interest rates

Recently, I was thinking about the distribution of expectations about forward interest rates, and how that would bias the market rates in the interest rate futures market.  I had been modeling the Eurodollars market as if expectations are normally distributed.  But, recently I considered the possibility that there is a negative skew.  The skew would cause the mode expectation to be above the median and the mean expectation (which we would expect to estimate the market rate) to be below the median.  But, the zero lower bound (ZLB) truncates the bottom portion of the expectations distribution.  This truncation would have the effect of pushing the mean expectation back above the median expectation.

I think it is plausible that these effects are at work in forward rate markets.  But, within my model, where I estimate the date of the first hike and the slope of the yield curve, I had only thought of the effect this would have on the yield curve slope and on rates at the very long end of the curve.

Previously, conceiving of the expectations of the rate hike in terms of time as having a normal distribution, we could estimate the mean, median, and mode date, which would all be the same date.  But, a skewed distribution means that the mode date is earlier than the median date and the mean date is later.  And, on the date of the mode expected first rate hike, the distortion of the ZLB would eliminate almost all of the information coming from expectations below the mode.  On the contract date that is the mode expectation, we would just see the distribution of rate expectations above the ZLB, so that it wouldn't look like a skewed distribution.  It would look a lot like a normal distribution with a very low variance.  So, the yield curve would turn up at the mode date.  The mode date would look just like the mean date of normally distributed expectations.

So, if this skewed distribution is accurate, then the mean expected date of the rate rise could be later than the date estimated by my model, and for that matter, than the date that would look obvious just by eyeballing the yield curve.  June could be the mode expected date, and that would cause the yield curve to start moving up in June.  But, as we move past June in forward rate contracts, the mode rate would move up off of the ZLB, and the market rate would move down, away from the mode, because the ZLB would have less of an effect on the mean.  The yield curve would begin rising earlier (June, the mode date, instead of the mean date, which is probably after September) and would rise at a less steep rate than it would if the mean expectations were the same but if there were no skew in expectations.

In fact, there is quite a negative skew in the Fed's dealer survey.  In the December dealer survey, the mode was June, the median was September, and the mean is likely even later.

Rates are now lower than they were when QE3 began.  Recently, both the expected date and the slope of the curve have fallen.  I have been attributing this to uncertainty.  But, if part of the lower measured slope of the yield curve could be from increasing skew, then the mean expected rate hike is actually later than it appears.

Possibly a Eurodollar position that was long (gains from falling rates) at the short end and short (gains from climbing rates) on forward dates could serve two purposes:

1) in the mode scenario, the near term position would not have significant gains or losses, and the forward position would have gains.

2) in the mean scenario, contingent on rates rising, the position might gain at both contracts.

3) in the mean scenario contingent on rates remaining at the ZLB persistently, then the two positions would act as a hedge for one another.  The forward position would lose more than the near term position would gain.  But, if most of the change in market expectations was from a delay in the first rate hike, or if this led to an expectation of a more dovish Fed, then the yield curve would shift to the right instead of shifting down, and the losses for the forward contracts would not necessarily be higher than the gains for the near term contracts.

In effect, this is a kind of arbitrage on the de-skewing of the expectations curve as information reduces uncertainty over time.  I haven't totally thought this through, but maybe my imagined gains here come from the inevitable shift to either a normal distribution if rates rise or a positive skew if rates remain at the ZLB.  A third moment arbitrage.  Is that something traders do?  Can you arbitrage the third moment?


So, instead of a June rate hike followed by a 20 basis point per quarter climb to 1.3% by September 2016, the true mean expectation might be a rate hike in December 2015, with the market divided between thinking rates will climb closer to 50bp per quarter, as is typical, to 2.25% (and rising) by December 2016, and expecting that we will still be sitting at the ZLB by then.

Possibly there are two types of positions here:

1) The Fed will be committed to raising rates:

Rates begin to rise in June (with very little variance in expectations), and rise to 3-4%, typical of past experience.  This creates the kink in the curve in June, and, before the influence of the other position, would have rates at 3% or more by the end of 2016.  (Rates in the graph are equal to 100 minus the y-axis number.)

2) The Fed will be willing to move rate hikes back:

There is a huge amount of variance in expectations, with many participants expecting rates to remain at ZLB for a long time.  This has the effect of pulling the market rate down from the level implied by the first position, across the yield curve.  But, since this is a proportional effect pulling the yield down, it doesn't pull the apparent date of the first hike to the right (forward in time).

I think this, more or less, describes my skewed distribution model, with maybe a sharper hump in the expectations curve than I have drawn.  Note that the change in slope after 2016 is subtle.  The ZLB effect on the mean rate expectation would slowly decrease over time as the ZLB truncates a smaller portion of expected outcomes.  In addition, there might start to be some positive skew in expectations from investors with long term inflation fears as the Fed unwinds their balance sheet.

I was hoping for confirmation of this week's MBA report of increased mortgage activity.  Friday's Federal Reserve H.8 report didn't have any sign of increasing closed end residential mortgage credit.  We have seen 2 weeks of increased home equity credit, though.  If next week shows an increase, that would be the first time since the crisis that it increased 3 weeks in a row.