As before, we assume that our stock grows according to:

P(n) = C (1 + R)ⁿ

where we start with a price $C and, after n months (days? years?), the price is $P(n), where R is some "Best Fit" monthly return.
In order to find the "Best" values for C and R, we set k = 1+R and consider the sum of the squares of the deviations of the actual prices, y_n, from our "Best Fit", and take this as a measure of our "Error", which we'll call E(C,k):

    E(C,k) = Σ{ y_n - C kⁿ }²

As before, we set the two derivatives to zero, like so:

(1)     d/dC E(C,k) = 2Σ{ y_n - C kⁿ }kⁿ = 0
and
(2)     d/dk E(C,k) = 2Σ{ y_n - C kⁿ }C n k^n-1 = 0

These may be rewritten, like so:

(3)     Σ y_n kⁿ - C Σk²ⁿ = 0
and
(4)     Σ n y_n k^n-1 - C Σn k^2n-1 = 0

Hence we can solve for C, like so:

(5)     C = Σ y_n kⁿ / Σk²ⁿ = Σ n y_n k^n-1 / Σn k^2n-1

from which we get an equation to solve for k = 1 + R

If we define:
f(k) = Σ y_n kⁿ   and   g(k) = Σk²ⁿ   then we can then rewrite equation (5) like so:

(6)     C = f(k) / g(k) = f '(k) / [(1/2) g '(k)]

Now we cross-multiply, in equation (6), and define:

F(k) = f(k) g '(k) - 2 f '(k) g(k)     and we wish to solve   F(k) = 0   for k.

We can solve this using Newton's Method, which requires iterating: k - F(k) / F '(k).

Note that:

F '(k) = f(k) g ''(k) - f '(k) g(k) - 2 f ''(k) g(k)

Let's recall our definitions:

1. f(k) = Σ y_n kⁿ
2. g(k) = Σk²ⁿ     ignoring the fact that there's a formula for the sum of this geometric series
3. f '(k) = Σ n y_n k^n-1 = (1/k) Σ n y_n kⁿ
4. g '(k) = Σ2n k^2n-1 = (2/k) Σn k²ⁿ
5. f ''(k) = Σ n (n-1) y_n k^n-2 = (1/k²) Σ n² y_n kⁿ - (1/k²) Σ n y_n kⁿ = (1/k²) Σ n² y_n kⁿ - (1/k) f '(k)
6. g ''(k) = Σ2n (2n-1) k^2n-2 = (4/k²) Σn² k²ⁿ - (2/k²) Σn k²ⁿ = (4/k²) Σn² k²ⁿ - (1/k) g '(k)

Now, using Newton's Method, starting with an initial guess given by the Excel functions:

k = INDEX(LOGEST(y_n, n), 1)     C = INDEX(LOGEST(y_n, n), 2)     we get both "Best Fit" charts  

Note: The Excel function LOGEST(y:array, x:array) gives the parameters for the "Best Fit to the logarithm".