According to this traditional model, we should then expect that the lower the interest rate, the higher the CAPE value* in a continuous inverse relationship. The actual empirical evidence, however, presents a somewhat different historical relationship between CAPE and interest rates, which this post explores.

*CAPE, the Shiller P/E, or P/E 10 ratio, is a valuation measure usually applied to the U.S. S&P 500 equity market. It is defined as the market price divided by the moving average of the trailing ten years of earnings, adjusted for inflation.

**REAL RATES & INFLATION:**Since interest rates are made up of a real yield and an inflation component, we can look at each of these separately first, to observe their past impact upon CAPE values. In the two charts below, U.S. monthly 10-year real Treasury yields and trailing 12-month inflation rates are sorted in rank order, from lowest to highest, with their associated CAPE values.

NOTE: Horizontal axis is not linear. Real yield = nominal yield - 12-month trailing inflation.

NOTE: Horizontal axis is not linear. Inflation = 12-month trailing rate.

Source: Shiller Data

**COMBINED IMPACTS:**By integrating the real yield and inflation data, Research Affiliates in a 2017 paper developed a continuous function that describes CAPE values for any given level of real yield and inflation. This two-dimensional model was applied to both U.S. stocks since 1880 (at left below) and developed-market international stocks since 1972 (at right). In their model, the combination of real yields and inflation produced R^2 values over 50%, explaining over half of the variation in CAPE values.

Again, this combined model shows the highest CAPE values occurring when real rates were in the 2%-4% range and inflation was 1%-3%. Contrary to popular belief, reductions in real yields or inflation below these ranges did not boost stock valuations.

Source: Arnott et al

**DISCUSSION:**The key takeaway for me is that we've been close to the "sweet spot" for high equity valuations for quite a few years, with moderate real interest rates and inflation. A sustained rise in either one of these would likely be a shock to P/E multiples.

One caution: In a recent thread, we saw that CAPE values are not a stationary time series and — beyond real rates and inflation — investors are willing to pay different amounts for the same dollar of earnings in different eras. Thus the shape of the "topographic CAPE maps" above may remain constant, but the CAPE values of the contour lines seem bound to shift in different time periods.

Other thoughts?