AboutJosh Expertise When I work through problems, I emphasize principles and key ideas which I believe are worth noting. I will try to answer questions in the following areas, but not at the advanced level. Algebra. Sequences & Series. Trigonometry. Functions & Graphs. Coordinate Geometry. Quadratic Polynomials. Exponentials & Logarithms. Basic Calculus. Probability, Permutations and Combinations. Mathematical Induction. Complex numbers. Physics problems.
Experience
Experience: I have worked as a teaching assistant in college. My hope is that more people will share knowledge without boundary, give help without seeking recognition or monetary rewards.
Supplementary Website: See a selection of past questions in my maths repository under "Question Archive"
Education Credentials: Bachelor degree in Engineering Science."Everyone struggles with something."
Not too long ago I asked the following question, and you replied with the answer that follows my question:
An investor 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?
--------------
In your reply, you used the amount of $44.00 in your equation. Should this amount be $40.00, $20.00 + $20.00 commission to buy and to sell?
Here is an excerpt:
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.”
Your complete reply is as follows:
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.
I thank you for your reply.
ANSWER: The right hand side of Equation 4 represents F=1.1*C.
Replacing C with C=50N+40, we get F=1.1(50N+4)=55+44.
There is nothing wrong as far as I can see.
---------- FOLLOW-UP ----------
QUESTION: Hello:
I want to thank you for you reply.
If the selling price for 100 shares is $55.44/share, the total amount received will be $5,544 minus $20.00 (commission) equals $5,524. Subtract the "break even" amount of $5,020 from $5,524 and the amount is $524. Divide this amount by $5020, and the percentage is 10.438...%. However, if the selling price is $55.42 per share, the result is different:
$5,542 - $20 = $5,522 - $5,020 = $502/$5020 = 10%
So, in all due respect, I think you need to tweak your equation.
Simple Return
Simple return calculations are executed after you have sold the investment. The formula you use to calculate it is:
Simple Return = (Net Proceeds + Dividends) / Cost Basis - 1
Example: John purchased a stock for $2,000. He paid a commission fee to his broker amounting to $15. Therefore, the cost basis of John is $2,015 ($2,000 + $15).
John sold the stock for $3,000. Again he paid his broker a commission of $15. Therefore, his net proceeds are $2,985 ($3,000 - $15). He was also paid dividends amounting to $100. Therefore, the simple return of the investment is:
Simple Return = ($2,985 + $100)/$2,015 - 1 = 0.53
Thus, the simple return of John's investment is 53%.
Simple return doesn't give you an idea on the time period during which the investment was held. In order to get after-tax returns, net proceeds should be substituted by net proceeds after taxes and dividends should be substituted by after tax dividends.
It appears that the investor in the above received a 48.13895...% return. Can you create an equation from the above example in which the investor desires a 48.13895..% return on his investment. You can ignore the dividend payment if you want or you can use it.
I thank you for your reply.
Answer Dear Kenneth,
$55.44 vs $54.42. The differences are minor. I don't think we need to debate over this. The discrepancy is due to the way that the final cost of the commission is factored into the equation. Clearly, which assumption to use, what is right and proper etc. are matters outside the scope of algebra. They are governed by rules and regulations in financial practices which I am not qualified to comment on.
By all means, modify the equations as you see fit. The whole purpose of these exchanges is to help you set up the problem, understand what needs to be done so that you can formulate these yourself. No matter what perspective you take, the mathematical approach is essentially the same. I am more interested in illustrating the technique than getting the answer (or definition) right. You can tweak the equations, change the variables as much as you like.
It can be easily adjusted as follows:
Using the same definitions as before,
Upfront cost: C=50N+20
Future value: F=XN
Objective function: F-20=1.1C. [Eq.1]
Substituting the expression of C and F in [Eq.1]
XN-20=1.1(50N+20). After a few steps, X=(55N+42)/N.
You should use similar techniques when you approach the problem mentioned in the second half of your follow-up. Assign a symbol to represent each quantity. Same idea as before...
Let R=Simple Return,
P=Net Proceeds,
D=Dividends
C=Cost Basis
I=Initial value of stock
F=Future value of stock
N=Number of stock
X=Cost/share
A=Initial Commission
Z=Final Commission
