In this series, I am attempting to demonstrate and describe the profound and unprecedented way in which the relationship between yields and inflation has changed over the past 140 years, from the time of Gibson's Paradox to the post-Bretton Woods order.

In the first article, I asserted that the relationship expressed in the simple equation

EY - DY - 10y + 1y = 0

did a remarkable job of expressing the movements of the S&P earnings yield (EY), the S&P dividend yield (DY), the ten-year Treasury yield (10y), and the one-year Treasury yield (1y).

It also did a fine job of modeling the relationship used as the basis of the so-called "Fed model" (EY-10y).

Unfortunately, as I previously recounted, the yield curve (10y-1y), highlights two fundamental and related problems with the equation.

First, the equation is useful for modeling *trends* in the particular yields, but not *levels*, as you can see in the charts in the last article.

The second problem is that under the gold standard, a) the model does a relatively indifferent job of modeling movements in the yield curve, and b) since the establishment of the Federal Reserve, it cannot even get the movements right.

Paradoxically, the problem is to some degree the solution, because the equation at the top (which I have deigned to call 'boe1') is incomplete.

EY - DY - 10y + 1y does not equal zero.

Since the 1960s or thereabouts, we can say that

EY - DY - 10y + 1y = CPI%.

This equation constitutes "boe2".

*(Note on sources: all values in this article are based on data from Robert* *Shiller and my calculations**)*.

Where boe1 is unable to account for levels, by simply including the rate of inflation, boe2 appears to resolve that problem for the period of the last half-century.

And, if that is the case, then we should see better results for the "Fed model" spread and, most importantly, the yield curve. And, since the rate of inflation is included, we will have to convincingly model the rate of inflation itself, real interest rates, and the absolute level of CPI.

I will include a table of correlations and a table of the respective equations below, but I think that the degree to which boe2 is able to approximate levels for both the yields and their spreads strongly suggests that whatever problems boe1 has, it has do primarily with the problem of how to incorporate inflation.

Before we address that problem, there still remains the yield curve.

As you can see, whatever the correlations might say, the clear improvement in the approximation of levels suggests that boe1 was not a fluke, and the fact that the EY-DY spread has tended to become inversely correlated with the 10y-1y spread since the dollar began to displace gold speaks to the profound change in the monetary order that has occurred over the last century.

Correlations by period, yield, spread or inflation, & modelCorrelations |
1872-1912 | 1913-1960 | 1961-2011 | 1872-2011 |

EY vs CPI |
0.5 | 0.07 | -0.39 | -0.3 |

EY vs boe1 | 0.39 | 0.51 | 0.34 | 0.5 |

EY vs boe2 |
0.19 |
0.51 |
0.77 |
0.46 |

DY vs CPI |
0.8 | -0.21 | -0.6 | -0.69 |

DY vs boe1 | 0.35 | 0.68 | 0.8 | 0.69 |

DY vs boe2 |
0.36 |
0.17 |
0.37 |
0.39 |

10y vs CPI |
0.85 | -0.19 | -0.24 | 0.4 |

10Y vs boe1 | 0.4 | 0.47 | 0.83 | 0.76 |

10y vs boe2 |
0.26 |
0.14 |
0.68 |
0.22 |

1y vs CPI |
0.59 | -0.04 | -0.39 | 0.1 |

1y vs boe1 | 0.36 | 0.35 | 0.57 | 0.58 |

1y vs boe2 |
0.36 |
0.3 |
0.85 |
0.49 |

CPI% vs CPI |
n/a | n/a | n/a | n/a |

CPI% vs boe1 | n/a | n/a | n/a | n/a |

CPI% vs boe2 |
0.51 |
0.45 |
0.71 |
0.43 |

Fed model vs CPI |
n/a | n/a | n/a | n/a |

Fed model vs boe1 | 0.22 | 0.58 | 0.37 | 0.72 |

Fed model vs boe2 |
0.24 |
0.53 |
0.58 |
0.46 |

Real 1yr vs CPI |
n/a | n/a | n/a | n/a |

Real 1yr vs boe1 | n/a | n/a | n/a | n/a |

Real 1yr vs boe2 |
0.38 |
0.46 |
0.56 |
0.4 |

Yield curve vs CPI |
*-0.19 | *-0.08 | *0.41 | *0.39 |

Yield curve vs boe1 | 0.18 | 0.04 | -0.23 | 0.1 |

Yield curve vs boe2 |
0.46 |
0.18 |
0.42 |
0.13 |

CPI vs CPI |
n/a | n/a | n/a | n/a |

CPI vs boe1 | n/a | n/a | n/a | n/a |

CPI vs boe2 | -0.34 | 0.9 | 0.99 | 0.99 |

* *"n/a" denotes relationships that were not implied by the respective models; t**he asterisk in the chart denotes a relationship that is not implied by either Gibson's Paradox or the two models I am presenting but which was interesting nonetheless.*

In a word, insofar as a) boe2 significantly "outperforms" boe1 over the course of the 1961-2011 period and b) that boe2's ability to model the behavior of yields and prices increases roughly to the degree to which Gibson's Paradox fails, I would suggest that boe2 represents a close approximation of the yield-inflation equilibrium under the dollar standard.

Regarding point (A) from the previous paragraph, most of the correlations do improve, some significantly. Two correlations are weaker, notably the dividend yield, but I think that the improvement in boe2's ability to approximate (although roughly) the dividend yield's actual level makes up for, if it doesn't trump, this slackening in correlations.

Regarding point (B), it is clear that Gibson's Paradox no longer holds, at least in its archaic formulation whereby Y=P (yields equal prices), so the only real question is whether or not the boe2 correlations have improved to significant levels over the last century and whether or not they will hold in the future. It is possible, after all, that this is just another transitional period to some other arrangement.

Without a convincing qualitative explanation as to why Gibson's Paradox existed and endured for centuries despite so much socioeconomic, political, and technological upheaval, and why this new equilibrium should then have taken its place, it is impossible to be certain that this boe2 equation constitutes a definitive statement of how yields and prices interact in a fiat currency system, although it is suggestive.

Ultimately, I believe that to the degree to which we have a combination of equations for the gold standard period, wherein

Y = CPI (a la Gibson's Paradox)

and

EY - DY - 10y + 1y = 0 (boe1),

and an equation for the dollar standard wherein

EY - DY - 10y + 1y = CPI% (boe2),

and insofar as boe1 appears to be a relatively stable and robust formulation, it seems that there is a general (and more sophisticated) equation that may account for the relationship between prices and yields under all historical conditions waiting to be discovered.

In the next article in this series, I intend to review some of the problems remaining in these two formulations as well as some of the curiosities and questions that they present. And beyond that article, I hope to individually examine each of the yields and spreads mentioned with greater detail, as well as corporate bonds, and to introduce the strange way in which the producer price index (PPI) relates both to yields and CPI.

For reference:

boe1 & boe2 equations by yield, spread, or inflationEquations |
boe1 | boe2 |

Earnings yield (EY) | DY-1y+10y | boe1 +CPI% |

Dividend yield | EY+1y-10y | boe1 -CPI% |

Ten-year Treasury yield (10y) | EY-DY+1y | boe1 -CPI% |

One-year Treasury yield (1y) | DY-EY+10y | boe1 +CPI% |

CPI% | n/a | EY-DY-10y+1y |

"Fed model" spread (EY-10y) | DY-1y | boe1 +CPI% |

Real One-year yield (1y-CPI%) | n/a | DY-EY+10y |

Yield curve (10y-1y) | EY-DY | boe1 -CPI% |

**Disclosure: **I 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 am short December S&P 500 futures. I might initiate a short on December gold over the next 72 hours.