The graph shows Shiller’s Cape from Shiller’s on-line data file.

**Shiller’s CAPE**

This article is not a consideration of the merits of Shiller’s CAPE ratio as a measure of the fair pricing of stocks. Nor am I concerned with whether CAPE is a useful tool for timing the market (it isn’t), and I’m not arguing whether stocks are over-priced or fairly priced. My focus is on whether PE10 is persistently higher than its historical average since 1985, and, if yes, why is that so? The ‘why’ will be taken up in later articles.

My conclusions are that PE10 has been elevated for a long time, and the reasons are not because of higher anticipated rates of growth in earnings, which, because of lower expected rates of inflation can be discounted at lower rates of interest. The reasons are contextual, and PE10 will probably average much higher levels than it did prior to 1985 for many years to come. Incidentally, there is nothing special about the year 1985. The choice of that year is quite arbitrary. It gives me a significant number of years since then, and prior to the crash of 1987, dividend yields on the Dow went below 3% for the first time since 1929, and so, at that time, many analysts viewed stocks as over-valued, prior to the crash.

The staple measure of stock valuations is the price-earnings ratio. Because annual earnings can fluctuate greatly, the PE value based on last year’s earnings or next year’s estimated earnings is not a good metric for valuing stocks in the long-term. After recessions when earnings are depressed, the PE ratio for stocks is generally high, even though stocks are cheap from a long-term perspective. Shiller’s CAPE (cyclically-adjusted price-earnings) ratio is therefore preferred for valuing stocks in the long-term. All metrics for measuring the valuation of stocks (like the total market capitalization to GDP ratio) are correlated quite closely to PE10, and they arrive at similar conclusions on whether stocks are valued at levels above or below historical averages. The other metrics of valuation also agree with PE10 on degrees of over-valuation or under-valuation, but saying stocks are over-priced is another logical step and assumes that the way stocks should be valued is never different from the past. I think that things are always different.

Shiller’s CAPE divides the real price of the S&P 500, by the average annual real earnings of the index over the previous 10 years. Therefore, CAPE is often referred to as PE10. Calculating the CAPE back to 1880, when there was no S&P 500 index, meant that Shiller had to construct the index, and estimate prices, corporate earnings, and inflation back to 1870. He won the Nobel Prize for this. The data set is available on-line and is up-dated monthly.

Since 2014, many stock market analysts have argued that high values of CAPE are justified because of low interest rates, such that future earnings are discounted by only a small amount. However, Hussman argues that low interest rates exist because inflation expectations are low, and therefore the bond market is expecting low rates of future economic growth and low rates of future growth in corporate earnings. Therefore, low rates of interest do not justify high values of PE10. In Hussman’s view the stock market is over-valued when PE10 is well above its historical average no matter what the levels of interest rates are.

**Relationships and Context**

PE10 is a relationship. It’s the relationship between the price of the S&P 500 index and the earnings over the last 10 years. The relationship could be written as:

P = a + b(E10),

where ‘P’ is the price of an index, ‘E10’ is the average 10-year real earnings of the index, and ‘b’ is the parameter that describes how much ‘P’ changes (on average) if ‘E10’ is increased by one. The parameter ‘a’ indicates the value of ‘P’ when ‘E10’ equals zero, which may never have actually occurred. The parameters are estimated using a technique called linear least-squares regression, applied to historical data.

The value for b is presumed to describe an **independent** relationship between P and E10. ‘b’ tells us how much an increase in E10 has changed the value of P all by itself. **Other variables that influence P and E10 do not vary**. Therefore, ‘b’ does not change over time because of some other variable. If ‘b’ does change over time, then that must be the result of changes in **context**.

When the relationships between variables change, then there has been a change in the context in which those relationships occur. This is what I mean by **context**. A change in relationship between two variables may either maintain the same direction, but weaken or strengthen, or the direction may actually reverse, or the correlation may go to zero. **The change in context cannot be described by a variable, by definition**, because there are changes in the system, which have changed the inter-relationships between the variables that describe that system.

So, when we account for the variables that cause PE10 to vary, has PE10 averaged higher or lower values since the arbitrary year of 1985 that would suggest that there has been a change in context?

**Determinants of PE10**

The average for PE10 from January 1881 to December 2018 was 16.92, but from 1881 to December 1985 that average was 14.56. Since 1985 the average has been 24.43. Is this the result of a change in context, which can’t be measured by a variable, or is it the result of values of the variables that determine both P and E10 being at levels that merit a higher PE10 from 1985 to 2018?

The generally accepted historical determinants of PE10 are rates of economic growth, inflation rates and interest rates. Inflation rates and interest rates should be closely correlated. Higher values of both should have lowered PE10 because future earnings were discounted at higher interest rates. Also, higher interest rates should depress real corporate earnings because consumers consume less. So higher interest rates should lower PE10. But higher inflation should increase consumption because theory and history indicates that consumers spend their money before it loses value. Therefore, inflation and interest rates should have both joint and independent effects on PE10.

When economic growth rates slow, or the economy falls into recession, then earnings should be depressed and this should lower PE10. Economic growth rates were, of course, inter-correlated with inflation rates and interest rates, but the independent effects of these three variables on PE10 are not the issue. The issue is: has PE10 changed because of some contextual change that is independent of the effect that these three variables have on PE10?

The Shiller data set provides monthly values for real and nominal earnings, the CPI index, the real and nominal prices of the S&P 500, and the 10-year Treasury bond yield. It would have been nice to have the yield on 3-months T-bills so that I could have calculated yield spreads, but I did not have that data. I could not calculate economic growth rates because GDP was not measured until after WWII, but annual growth rates in real corporate earnings are a good surrogate for GDP growth rates.

**The Regressions**

My observations were monthly values. Because monthly values fluctuated quite a bit, I used the average of the previous 12 months to calculate my independent variables. I did not use the 12-month average for the S&P 500. I used the value reported by Shiller for each month. That value is not the end of month value, but it is the daily average for the month. The dependent variable, therefore, was PE10 as shown in Shiller’s data set, and the independent variables were:

- The 12-months moving averages (for the previous 12 months) of the inflation rate (IR);
- The 12-months MA of the growth rate (%) in real corporate earnings (RCEGR);
- The 12-months MA of the yield on the 10-year Treasury bond (TNX).

Then I added annual changes in these three variables to measure if corporate profit growth (RCEGRCH) or inflation (IRCH) were accelerating or decelerating, and to measure if TNX (TNXCH) had increased or decreased over the past year.

I estimated four multiple regression equations, one for 1881 to 2018 (equation 1), one for 1881 to 2018 (equation 2), which included a variable for TIME. TIME had the value 0 for dates prior to 1986, and a value of 1 for dates after 1985. Then I ran the regressions for 1881 to 1985 (equation 3) and 1986 to 2018 (equation 4). These are the results.

**Constant (a) and Regression Coefficients (b) for PE10 Regressions**

Variables | Equation 1 1881 - 2018 | Equation 2 1881 - 2018 | Equation 3 1881 - 1985 | Equation 4 1985 - 2018 |

Constant (a) | 18.17 | 17.81 | 16.77 | 34.58 |

TIME | NA | 10.864 | NA | NA |

TNX | -0.142 | -0.585 | -0.38 | +0.08 |

TNXCH | -0.773 | +1.30 | -0.45 | +0.76 |

IR | -0.325 | -0.305 | -0.30 | -3.58 |

IRCH | +0.15 | +0.136 | +0.105 | +2.56 |

RCEGR | +0.031 | -0.0041 | +0.054 | -0.032 |

RCEGRCH | +0.014 | +0.0021 | +0.0256 | +0.0074 |

r-squared | .09 | .485 | .207 | .293 |

**Changes in the Relationships**

The changes in many of these relationships are of minor importance. The discussion of these changes can be skipped. Many of the relationships with PE10 changed notably between the two time periods, although I wouldn’t read too much into that. The data is highly auto-correlated, and especially in the post-1985 regression there isn’t much fluctuation in the variables. Still, we see that the effect of 10-year Treasury yields (with inflation constant) has changed. Before 1986 a higher T-bond yield, and a greater annual increase in the yield, resulted in a lower PE10, as one would theoretically expect. This was also true for all years, that is, from 1881 to 2018. However, since 1985, the effect has been reversed. Higher yields and an increase in yields, with constant inflation, were associated with a higher price for stocks relative to earnings. This occurred primarily because TNX (the 10-year yield) was at much higher levels prior to 2000 when stock prices were elevated. With the 2002 and 2008 recessions, stock prices were depressed, as were Treasury yields.

Higher levels of inflation (with T-bond yields constant) were associated with lower values of PE10 during both time periods, although the effect of inflation was stronger after 1985. PE10 was low during the stagflation years of 1970 to 1982, but it was also low during the deflationary years of the Great Depression. Since 1986 inflation was highest in 1991 during the first Iraq War at a time of recession, which lowered stock prices. The inflation rate was also above average in 2001, and in 2008 during the market declines. An increase in the inflation rate during the previous year was associated with higher stock valuations in both time periods. This makes theoretical sense since an increase in inflation is to be expected when the economy is expanding rapidly, the stock market is doing well, and Treasury yields have not yet risen markedly in response to higher inflation rates.

If inflation and Treasury yields were constant, one would expect that, during economic booms before recessions, the earnings growth rate would be relatively high, and that stock prices would be high, resulting in a high PE10. Early in the recovery from a recession, the earnings growth rate should be low and stock prices should be low causing a low PE10. Therefore, the earnings growth rate should have a positive relationship with PE10. PE10 should be high when earnings growth rates are high, and low when earnings growth rates are low.

Near the end of an economic boom, while the earnings growth rate is high, the growth in that growth rate should be slowing or negative, at a time when stock prices are high and we should have a high PE10. During the recovery from recession, while the earnings growth rate is low (or negative) but the growth in the annual earnings growth rate is high, stock prices should be low resulting in a low PE10 value. Therefore we would expect the relationship between PE10 and the change in the earnings growth rate to be negative. When the earnings growth rate is slowing or declining PE10 should be high, and when the earnings growth rate is increasing PE10 should be low (early in a bull market).

These relationships are evident in the regression for 1881-1985 (equation 3), but the relationships are reversed in equation 4. From 1986 to 2018, above average real corporate earnings growth rates were associated with lower values of PE10, but when the earnings growth rates increased, the data indicates below average values of PE10 for that time frame. Remember that the average PE10 was very high for this time frame compared with the years before 1986.

I can’t explain why the reversal in the relationships occurred. Inspection of the data indicates that in 1989 to 1991 when the earnings growth rates and the changes in the earnings growth rate were both negative, PE10 remained higher than in the years before 1990. Both variables had negative values in 1996, and again in 1998, but the stock market kept booming. But, trying to explain these anomalies is not the point of the exercise. In general, **when corporate earnings growth rates are high and slowing, the stock market is near a peak**, unless there has been a change in context.

**Changes in PE10 as a Result of Context**

The regressions have shown that the relationships between PE10 and the variables which should influence PE10, were different in the two time periods. But the most notable feature of the regressions is that **the average PE10 since 1985 was much higher than its historical average for reasons other than inflation rates, interest rates, and corporate earnings growth rates**. When TIME was included in the regression (equation 2), the r^{2} value jumped from .09 to .48 and the average PE10 increased by 10.64 (the coefficient for the TIME variable), with the values of the variables that should account for PE10 held constant. Equation 3 shows that if the values of all the predictive variables were zero, then PE10 averaged 16.77. Equation 4 indicates that if all variables had zero values, then, PE10 averaged 34.58 in recent years, an increase of 18 points.

So since 1985 stocks have been valued higher for reasons unrelated to rates of growth in corporate profits, inflation rates, and interest rates. I ascribe the reason for the increased average valuation in stocks to changes in context; that is, there were changes in the way the economy and the financial sector work. These changes **cannot** be described by variables, and these changes changed the relationships between key variables.

Context may either be an on-going irreversible process, like the demographic transition and the aging US population, or a reversible process that may take decades to reverse like the new form of quantitative easing (QE) that combined lower interest rates with large increases in the Fed’s balance sheet. A change in context may also be the result of a singular event, which cannot be repeated, like a piece of legislation that changes how markets work. An example is ERISA, which created the first tax-deferred individual retirement programs.

There may be a change in context that we don't recognize, such that we don’t know what has happened to change the relationships between the variables, but the relationships have changed, although it may take decades to recognize it. In my next article I will explore some of the known changes in context that I think have changed the relationship between stock prices and earnings, and changed how the economic system works.

**Disclosure:** I/we have no positions in any stocks mentioned, and no plans to initiate any positions within the next 72 hours. I wrote this article myself, and it expresses my own opinions. I am not receiving compensation for it. I have no business relationship with any company whose stock is mentioned in this article.

**Additional disclosure: **I try to time the market.