Expert: Josh - 5/21/2009

Question
QUESTION: Hello:

An investors buys 100 shares of stock where he is employed at $50.00 per share. Brokerage commission is $20.00 to buy and $20.00 to sell. At what prices must his investment become so that he can sell the 100 shares to realize a 10% profit?

I think that one of the solutions is correct, but I do not know which one:

(1)
$? - $20 divided by $5020 = 10%
10% X $5020 = $502 + $20 = $522 = $?
$522 - $20 = $502 divided by $5020 = 10%

(2)
$? - ($20 + $20) divided by $5000 = 10%
10% X $5000 = $500 + $20 = $520 = $?
$520 - $20 = $500 divided by $5000 = 10%

I'm not sure what to do with the first $20 commission fee.

I thank you for your reply.


ANSWER: Kenneth

Your setup does not produce an equation to be solved. It's confusing to set ?-$20/$5020 equal to 10%. Mathematically speaking, this is not on. It is never a good idea units on the left and right hand sides are INCOMPATIBLE. Please don't do it.

All equations should involve only numerals. Use 0.10*X, if you must, to represent 10% of something. This also make clear which quantity you are trying to find. I have made this point before, don't mix units in an equation. If you like to put your thoughts on paper, do it without mingling mathematical operators (= signs) and incompatible units in a single equation. It doesn't work here. Calculating in "maths mode", all quantities in the equation should be dimensionless.

I won't dwell upon this any further.

Look at it this way.

The total cost (C) consists of the following components:
- $20 (buy commission)
- $20 (sell commission)
- $50*N (cost of purchasing shares)

Therefore, C=50*N+40.   [Equation 1]

Future value (F) must be 10% higher than the initial cost.
So, F=1.1*C [Equation 2]

We don't know the price of the shares, so let this be X. Then, the future value is equivalent to the number of shares times the cost per share. Hence, F=N*X=55N+44. [Equation 3]

Both equation 2 and equation 3 represent the future value (F). Thus,
We set N*X=55*N+44. [Equation 4]

Finally, we were told the number of shares N=100.
Equation 4 becomes 100*X=5500+44.
Solving for X (the price of each share), we get X=5544/100=55.44.

Note: All quantities in the above equations are dimensionless, even though the unit is implicitly measured in dollar.

---------- FOLLOW-UP ----------

QUESTION: Hello Josh:

I want to thank you for your reply and answer.

Can you explain the following from your reply:

"Future value (F) must be 10% higher than the initial cost.
So, F=1.1*C [Equation 2]"

How did you get F=1.1*C? I know that 10% = 0.1, but how does future value become 1.1*C?

I thank you for your follow-up reply.  

Answer
Kenneth,

C is the amount of money you end up paying.
It includes:
1) the net value of the shares, i.e., N units at $50 per unit
2) commission (buy) $20
3) commission (sell) $20
Combining these three components, we get Equation 1.

The 10% gain is measured relative to the amount of money out of pocket, viz., the expression C. Hence, we have equation 2.