Our Obsession with Second Derivatives

Please don’t let the calculus reference deter you from reading on – this is not a math-heavy piece.  I simply couldn’t think of a less verbose way of discussing that some of the most important market and economic concerns are more about how the rates of change of various key indicators might slow down or speed up than the directions of the changes themselves.

In August, we delved into the first derivative – the rate of change in an underlying index – when we explained how month-over-month inflation statistics could show as zero even if the year-over-year statistic rose sharply.  This was largely the result of a confusing set of terms that are used when we discuss prices.  The Consumer Price Index (CPI) is an index comprised of a wide range of price data points – not unlike an equity index – but the focus of the monthly CPI data is the change in that price index over a month or year.  Inflation is the amount that the CPI rose over a given period – the change in the price index.

We have now come to recognize that inflation – the first derivative of prices – may be above the Federal Reserve’s 2% target for some time.  Hoping for a quick reversion to stable prices is largely futile, so instead we are focusing more about whether inflation is slowing.  In other words, in our hunt for “peak inflation” we are seemingly more concerned about the second derivative of prices – whether the rate of change in prices (inflation) is slowing. 

This is not an idle concern, of course.  Assets prices are set at the margin and nuances matter.  For better or worse, most financial assets have discounted the notion that goods prices will be elevated for some time.  They have also discounted that the rate of change in those prices will be positive and even elevated for some period.  But it is also sensible for investors to look for signs that the rate of change will be reverting to a slower pace because it is logical to think that prices will need to accelerate more slowly.  In the Fed’s ideal world, prices will rise steadily at 2%.  The first derivative – inflation – would be 2%, while the second derivative – the change in inflation – would settle back to zero.  If inflation was unchanging at a 2% annual rate, that implies that the second derivative is zero.

The same notion now applies to Fed policy.  We spent a good portion of the summer obsessed about a “Fed pivot”, with hopes that the Fed would reverse its policy of raising interest rates.   A pivot is a first derivative change.  Rather than rates going up, they would go down.  That notion was disavowed quite spectacularly by Chair Powell at Jackson Hole.  Investors then began to switch to wondering about when the Fed might pause its rate hikes.  Again, a first derivative change, this time from a positive number to zero.  When that notion was stomped out by a series of Fed talking heads, the hopes began to switch to hopes for a slower pace of rate hikes.  This is a second derivative change.  Rates will still go up – first derivative – but more slowly – second derivative.  Having had their hopes for more aggressive changes to Fed policy dashed on various occasions, traders were more than happy to gravitate to Powell’s affirmation of a slower pace of rate hikes.  Enthusiasm about a second derivative change replaced the hopes for a first derivative change in the near term. 

Those of you who trade options, and many who don’t, should by now be quite familiar with the term “gamma squeeze”.  In options terminology, delta – the change in an option’s price for a given change in the underlying security– is the first derivative; while gamma – the change in delta that occurs for a given change in the underlying – is the second derivative.  We have seen several instances when gamma has seemingly been weaponized to lead to abrupt changes in the price of stocks, ETF’s, or even the market itself.  We won’t belabor the specifics – see the linked article – but bear in mind that second derivatives have now become part of the options’ market narrative on a frequent basis.