The new valuation measure and dividend yield

Consider the new valuation measure which is supposed to be an improvement of Shiller CAPE and dividend yield. Previously, we considered this measure based on trailing 10-year earnings. Right now, we consider it based on 1-year dividend yield and use it to create a new simulator. But recently, it occurred to me that one can express this measure based on the history of annual dividend yields. How so? Let us recall that the valuation measure is defined as

$H(t) = \ln(W(t)/W(0)) - \ln(D(t)/D(0)) - ct$

where $W(t)$ is the wealth at end of year $t$ invested in stocks, and $D(t)$ is dividend paid in year $t$ for these stocks. And $c \approx 4-5\%$ is the linear trend. Assume now $S(t)$ is end-of-year level at year $t.$ Then $\triangle(t) = D(t)/S(t)$ is the dividend yield for this year. Total returns then are $Q(t) = \ln(W(t)/W(t-1)) = \ln(S(t)+D(t)) - \ln(S(t-1))$ and we can express $\ln(W(t)/W(0)) = Q(1) + \ldots + Q(t).$

The crucial insight:

$Q(t) = \ln(S(t)/S(t-1)) + \ln(1 + \triangle(t)).$

Plugging this into the main formula for $H(t)$ and denoting $A(t) := \ln(1 + \triangle(1)) + \ldots + \ln(1 + \triangle(t))$ we get:

$H(t) = \ln(S(1)/S(0)) + \ldots + \ln(S(t)/S(t-1)) + A(t) - \ln(D(t)/D(0)) - ct.$

Canceling these logarithms, we get

$H(t) = \ln(S(t)/S(0)) + A(t) - \ln(D(t)/D(0)) - ct$ which we can write as $H(t) = -\ln(\triangle(t)/\triangle(0)) + A(t) - ct$ which in turn, finally, we can write as $H(t) = -\ln \triangle(t) + \ln\triangle(0) + \sum\limits_{s=1}^t\ln(1 + \triangle(s)) - ct.$

If $\Delta$ is a strongly stationary process such that $d(t) = \ln(1 + \Delta)$ satisfies the Strong Law of Large Numbers (ergodic), then this measure converges to the stationary distribution if and only if $c$ is the mean of this $d(t).$