Average Return for Commodity

Suppose I have a function of a varying values represented by randomness. A stock or price of a commodity would be a good example.

What is the average return for that stock on any given interval of prices?

Let average return be defined as the most likely return achieved given two bounds.

I've come up with an initial hypothesis for my average value function.

Let $$A$$ be the average value.

Let $$x_i \in [0,M]$$ Where $$M$$ is the most recent index for a price.

$$A = \sum_{x=0}^M \frac{1}{M} \sum_{y=x}^M \frac{1}{M -y+x} f(x,y)$$

Let $$f(x,y)$$ be the return using an arbitrary strategy between the intervals $$[x,y]$$

Is this a good indicator for any return? Does this say more about the stock or commodity or the strategy itself?

• I am not downvoting (or vtc) your question, but it is very unclear.Why and how are you setting an "interval of prices"? what does that mean? "Defining" average as the mode (3rd sentence) seemingly disrupts essential notions of probability. It is also unclear how M relates to the asset price or to [x,y]. Lastly, there is no indication about any "strategy" whatsoever (e.g., f is completely unspecified), whence it is impossible for your definition of average A to say anything about "the strategy itself". – Iñaki Viggers Mar 5 '19 at 12:21
• it sounds to me like you might be interested in calculating some expectation of the return over some period. In that case, people usually resort to commodity market efficiency arguments ( law of one price or possibly interest rate parity ) or actually calculating the expectation of the assumed underlying DGP. That's the only thing I can think of because, like the other commenter, I can't make sense of your expression. – mark leeds Mar 5 '19 at 17:28
• The idea is that $f(x,y)$ is any arbitrary strategy (possibly random). Suppose a strategy $f(x,y)$ exists from another function with other parameters. $$f(x,y) = g(x,y,a_1,a_2)$$ A good strategy would be one that chose $a_1$ and $a_2$ such that A is maximal. M is the number of prices over the lifetime of the asset or stock. – Jeremy Mar 6 '19 at 2:42
• Assume you have a segment of price data and can find return between the interval $[x,y]$ The expression calculates a value that represents the most likely return to be attained by choosing any two arbitrary $x$ and $y$. As an example suppose you have a dataset of $[1,3,5,7,4,3]$ where $M=6$. An interval of size 6 will be given the weight of $\frac{1}{6}$. An interval of size 5 will be given the weight of $\frac{1}{6 (2)}$ because there are two of them. An interval of size 3 will consist of three seperate choices for $x$ and $y$ and thus are given a weighting of $\frac{1}{6(3)}$. – Jeremy Mar 6 '19 at 2:56