Here are two charts.   

The first shows the annual USD CPI rate in January of each year going back to 1980.  

The second shows the Fed funds rate in January of each year since 1980.   

Both figures have been trending lower from 1980 till now.  Why?  Lower rates from the Federal Reserve bank are said to promote inflation.  If the Fed is cutting rates, and that activity on its own represents an economic stimulus of sufficient effect, then we should see the rate of inflation stay the same or increase.  This has not been the case.   

The CPI can be said to measure the relative impact of two forces in the economy:  Those that are inflationary, and those that are deflationary.  In other words, the CPI could be said to measure the net impact of all the inflationary and deflationary forces in the economy.  If the inflationary forces overwhelm the deflationary ones, you get growth in CPI; if not, CPI shrinks. 

Since 1980, the Fed has been engaged in two activities that are inflationary:  They’ve been slowly lowering the Fed funds rate, and since 2008 they’ve expanded the money supply via Quantitative Easing.  Each attempt to unwind QE and raise rates has been abandoned, and the inflationary activities continued.  If these activities were sufficient to indefinitely prevent deflation, then I would expect the inflation rate to remain positive and stable or to increase over time.  However, the rate of inflation is still trending towards zero. Why has inflation been slowing down while the Fed is reducing rates and expanding the money supply? 

 
My answer to this question is this:  Rather than interest rates influencing inflation, inflation is influencing interest rates, so to speak.  The Fed has been lowering rates and printing money to prevent deflation that’s already occurring.  While this has kept inflation positive, inflation will at some point become deflation if the current trends continue. 

I think there are two possible paths from here: Either the Fed will inaugurate a negative interest rate regime officially while maintaining QE, or they will keep rates at zero and exponentially increase QE to keep deflation at bay. 

This puts the higher-than-normal rates of inflation in summer 2021 into perspective.  A few months, or even a couple years, of higher-than-normal rates of inflation are not significant in the context of a multi-decade downtrend.  However, the increase in inflation could persist if the Fed ramps up its inflationary activity to such an extent that it overcomes the deflationary forces at work.  If that happens, we should see increasing rates of inflation (what I would consider “hyper-inflation”), or at least inflation reach a higher but stable plateau. 

How will inflation, or the lack of it, impact the stock market?  My view is that if either of the two scenarios above comes to pass, this will be bullish for the market.  A negative interest rate regime is bullish for the market because many stock valuation models utilize the risk-free rate of interest (the Federal Funds rate, in practice) as the discount rate.  If that rate is negative, that means future cash flows from stocks will be worth more in real terms the further into the future they occur.  Negative rates, therefore, will raise the present value of all stocks. 

 A regime of exponentially increased QE will also be bullish for the market, simply because there will be more money in the economy which will find its way into people’s pockets and thus into the top lines of corporate income statements.  Increased corporate revenues are bullish for stocks.  Additionally, the added money in the economy may end up resulting in increased demand for stocks, as more people have more money available for investment. 

So don’t freak out about inflation right now.  I think the concerns about it are overblown.  We shall see.