More recently, Case and Shiller (1989) estimated the single-index market model from rates of change in a quarterly-average price index for single-family homes.
Applying this test to rates of change in both annual geometric and arithmetic means of monthly S&P 500 values for the years 1950 through 1985 (Ibbotson Associates 1987) produced an intercept estimate of -0.
The CAPM for rates of change in weighted geometric or arithmetic means of values at regular intervals within periods, for example, are extensions of expressions  and , respectively.
The CAPM relationship for rates of change in period-average stumpage price depends on the averaging process.
The same comparison of alpha estimates (remember that only the relative magnitude of the alpha values is of interest as they are calculated from rates of change in stumpage price alone) reveals a similar story.
Rates of change in period-average stumpage prices, and rates of return for forest assets calculated from them, are average rates over two periods of time.
This note derives the CAPM relationship for rates of change in period-average asset values, and outlines procedures to estimate CAPM parameters from averaged data.
Thus, the procedures for rates of change in the arithmetic average of asset values are directly applicable to estimation of the CAPM for forest assets.