So, as stated in the previous blog post, we have the following known information.

T - Total Investment amount

Market value is Last Price times Current Shares, .

The New Market value after our Investment Buy will be, .

As we explained before, these two values should be equal for our new market value, .

Our Total Investment amount is the Asking Price times the Shares Bought plus the Commission Cost.

The individual investments are, , without commission cost.

To get the commission cost added, we multiply by 1.005 to get the 0.5% and then add 7.00 to each individual investment.

The individual investments plus commission are, .

The Total Investment amount then becomes, .

So, at this point we have two equations,  and , to work with.

We can solve for either X or Y by eliminating X or Y if we setup an equation where either X or Y is by itself in each equation and then set the two equations equal to each other.

For this first step we’ll solve for X by eliminate Y in isolating Y for each equation and setting them equal to each other.

becomes  and   becomes .

Since Y = Y, then the right sides of the equations   and   can be made equal to each other, .

The equation   can now be solved for X.

We can follow the same process for finding Y and that equation is the following.

At this point we haven’t given any values for Stock A or Stock B, so which stock is which?

What stock has a greater market value than the other?

This is where the duality comes in.

Since at this point we do not know the values, Stock A’s value can be inserted in to either A or B and likewise in the opposite, Stock B’s values can be inserted in to B or A.

However, keep in mind that once a stock value is assigned, the other must go in to the unassigned stock values, meaning if Stock A’s values are inserted in to B’s values, then B’s values must be inserted in to A’s values.

It’s kind of like the Pauli Exclusion Principle, you can only occupy one value at a time and the other must go someplace else, however the two are interchangeable in the quantum aspect of it.

If this is true, then when we give some values, we should see that when we swap the values, the X and Y values should be the same but swapped as well.

Let’s say our Total Investment is, T = 2000.00, and Stock A’s values are, A_L = 0.25, A_P = 0.265, A_C = 6780; Stock B’s values are B_L = 0.07, B_P = 0.08, B_C = 2950.

Using the equations we get,

and

Since shares are purchased in mostly whole amounts, we drop the fractional part and have X = 499 and Y = 23047.

Now, if the duality of the functions holds true, we should be able to swap the values for each stock and see if the buy values are the same but swapped for X and Y.

and

Dropping the fractional part gives, X = 23047 and Y = 499.

Compared to our original calculation, X = 499 and Y = 23047.

The table below shows the comparison.

This shows that when we solved for X, we were also solving for Y simultaneously; another indication of the duality in the function.

We have an Excel file you can examine yourself and change values to see the equation in action.

Keep in mind, this is for Scottrade and for stocks under $1.00.

Your investment firm may have a different commission schedule.

http://www.jadexcode.com/EqualizeInvestmentSolution.xls