Over a 30-year period an asset had an arithmetic return of 13 percent and a geometric return of 10.5 percent. Using Blume's formula, what is your best estimate of the future annual returns over the next 10 years?

12.22% will be the forecast rate over the next 10-years framework.

Explanation:

Blume's formula is used to do an average of both, the arithmetic and geometric mean. The formula is as follow:

(T-1) / (N-1) * Geometric average + (N-T) / (N-1) * Arithmetic average

Where:

T = forecast period = 10 years for this case

N = the data we have samples = 30 years for this particular project

Then we have thegeometric and arthmetic mean

We place them on the formula and solve for Return on Equity

Return = (10-1) / (30-1) * 0.105 + (30-10) / (30-1) * 0.13

Return = (9/29) *.105 + (20/29) *.13 = 0.122241379 = 12.22%

12.22% will be the forecast rate over the next 10-years framework.