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Creating Mathematical Models Many of the things we need to know about as architects, engineers and technologists are considered systems. Webster's dictionary defines a system as "a group of interacting, interrelated, or interdependent elements forming a complex whole." The difference between knowing a few facts about a system and really truly understanding it is in being able to describe the subtle relationship between its pieces. Luckily, we have a powerful tool to do that. The tool is mathematical modeling. Also to our advantage is the fact that most relationships in engineering are quantifiable, so they lend themselves well to mathematical manipulation. Lets take a look at a specific example. Suppose you want to design a commercial building and rent it out. You also want to have an apartment for yourself in the same building. Why would you do such a thing? The answer to this question is going to be one of the indicator variables for this system. The answer, of course is to make a profit. So, what is profit? Profit is defined as the difference between revenue and expenses. And, we can express it mathematically as follows. Profit = Revenue - Expenses Now, from a systems perspective, we have three variables. Profit is the indicating variable, which means that it is the result we are really concerned about. Revenue and Expenses are critical variables because they drive the indicator variable. The relationship between them is a mathematical model, all be it a very simple one. We still have a problem, we don't know the value of any of the variables, and to solve this equation we need to know at least two of them. Our next step must revolve around assigning a value to the variables. We know that profit is driven entirely by revenue and expense, so no amount of mathematical manipulation is going to tell us what the profit is if we don't figure out revenue and expenses. Which of these other two terms to tackle first is a matter of personal preference, as they both have to be solved eventually. Just for kicks, we'll look at revenue first. What does revenue equal? Well, we know that income per square foot is fairly predictable in a given geographic area. So, we could express revenue as the equation below. Revenue = Rental Square Footage X Income Per Sq. Ft. It might be tempting right now to put in a value for variables that we are able to estimate, but you have to wait. The reason is that we want to create the most flexible model possible. Now comes the part where you apply an algebra skill you never thought you'd use. We have two equations that share a variable and we're looking for a way to relate them. The way to accomplish this is through substitution. See the equation below. Profit = (Rental Square Footage X Income Per Sq. Ft.) - Expenses Looking at the equation above, it becomes clear that we've just substituted one variable for another. It would appear that we are no closer to our goal of getting the equation down to one variable, but we actually are because we are on the path to defining all of these ambiguous values. Now the question becomes, "what does the rental square footage equal?" The answer is that rental square footage equals total square footage minus apartment square footage. Substituting again, our equation would look like this: Profit = ([Total Square Footage - Apartment Sq. Ft.] X Income Per Sq. Ft.) - Expenses We've just made the problem worse by adding another variable. Apartment square footage is a value that there is no hard calculation for. It is a variable that can only be estimated. Eventually it is our goal to get the equation mathematically refined to the point that there is no more mathematical manipulation available, and the only thing left to do is estimate the remaining variables. Continuing down the path towards defining variables, we need to figure out what the critical variables for Total Square Foot are. Square footage can be calculated by taking the principle available for construction plus the down payment and dividing it by the cost per square foot. The equation now looks as follows. Profit = (Principle + Down Payment) / Cost per Sq. Ft.- Apartment Sq. Ft] X Income Per Sq. Ft.)) - Expenses Looking at the equation, it becomes apparent that it would be to our advantage to make the Principle a very large number. There is a naturally occurring phenomenon in all systems known as a limiting variable. In this case the limit to the principle is driven by the maximum monthly payment on the loan. Substituting in the mortgage formula, we get the following equation. Profit = ( Monthly Payment + Down Payment) / cost per Sq. Ft.- Apartment] X Income per Sq. Ft.) - Expenses ____J . 1 - ( 1 + J)-N Things start to look complex here, but many of the variables are easily definable. The variable J is a measure of the interest. It is equal to the Interest Rate / 1200. N is the number of months over which the loan is going to be amortized. For this variable there the domain is relatively limited. Mortgages are only offered in 5, 10, 15, 20, 25, and 30 year terms. Adding our knowledge of J and leaving N as the variable to solve for, we get: Profit = Monthly Payment + Down Payment) / cost per Sq. Ft.- Apartment] X Income per Sq. Ft.) - Expenses ____(I/1200) . 1 - ( 1 + [I/1200])-N To find the limit of this system, we must push the monthly payment to its highest possible point. This limit is created by the bank in the form of a rule that states that a loan payment may not exceed 30% of a person's monthly income. We could calculate using the following formula: Maximum Monthly Payment = (Salary / 12)*.3 Our mathematical model would look as follows. Profit = ([{ (Salary / 12)*.3 + Down Payment) / cost per Sq. Ft.- Apartment] X Income per Sq. Ft.) - Expenses ____(I/1200) . 1 - ( 1 + [I/1200])-N We are finally to the point where the variables in the revenue part of the equation can be reduced no more. The variables - salary, down payment, interest rate, term of the loan, cost per square foot, and apartment size must be estimated. Now onto expenses. Revenue and expenses could be calculated on any time interval, but monthly makes the most sense because that is the rate at which rent is collected and the mortgage is paid. There are several expenses that must be accounted for on a monthly basis. They include: mortgage payment, property taxes, insurance, maintenance, and utilities. Expressing it mathematically, we could say that: Expenses = mortgage payment + property taxes + maintenance + insurance + utilities The approach here is the same as in solving for profit. We just start eliminating variables until we get down to variables that must be guessed. Lets take a look at the mortgage payment. We already came up with a formula for that, so we will reuse it. Expenses = (Salary / 12)*.3 + property taxes + maintenance + insurance + utilities Property taxes may seem a little slippery at first, but remember, we have a formula for estimating them. Property taxes = assessed value x mill rate. Mill rate is something that must be estimated, but the assessed value can be assumed to be equal to the principle of the loan plus the down payment. To get this expense in monthly form, mill rate needs to be divided by 12. Principle is expressed as: (Salary / 12)*.3 Principle = ____(I/1200) . 1 - ( 1 + [I/1200])-N Which means: (Salary / 12)*.3 Property Tax =( ____(I/1200) . + Down Payment) X Mill Rate/12 1 - ( 1 + [I/1200])-N Substituting we get: Expenses = (Salary / 12)*.3 +( (Salary / 12)*.3 + Down Payment) X Mill Rate/12 + maintenance + insurance + utilities ____(I/1200) . 1 - ( 1 + [I/1200])-N You will recall that maintenance can be estimated as 1% of assessed value annually. This means that: (Salary / 12)*.3 Maintenance = ____(I/1200) . + Down Payment X .01 /12 1 - ( 1 + [I/1200]) -N If we put it all together, we get: Expenses = (Salary / 12)*.3+ (Salary / 12)*.3 + Down Payment X Mill Rate/12 + (Salary / 12)*.3 + Down Payment X .01 /12+ Insurance + Utilities ____(I/1200) . ____(I/1200) . 1 - ( 1 + [I/1200])-N 1 - ( 1 + [I/1200])-N We are again at a point where the only variables left must be estimated. Reference the insurance packet for a best guess on insurance. You can access the Excel file on the Share File for an estimate on utility costs. If we put all of the formulas together we get the mess that you see below. Profit = ([{ (Salary / 12)*.3 + Down Payment) / cost per Sq. Ft.}- Apartment] X Inc. per Sq. Ft.) - (Salary ____(I/1200) . 1 - ( 1 + [I/1200])-N Now, I know what your thinking, but it actually gets easier from here. The reason being that all we have to do from here on out is plug and chug. Lets take a look at the most confusing term: (Salary / 12)*.3 ____(I/1200) . 1 - ( 1 + [I/1200])-N In this case study, you were given a salary of $25,000.00. Your loan is available at 5% interest. Plugging these values in, we get: (25,000 / 12)*.3 ____(5/1200) . 1 - ( 1 + [5/1200])-N A little razzle dazzle and we have: $694.00 . ____(.0042) . 1 - ( 1 + .0042)-N Now, the only variable left is N, but N can only be 60, 120, 180, 240, 300, or 360 months. Running each of those numbers, we get the table below. Term Principle 60 months $36,739.86 120 months $65,310.79 180 months $87,529.13 240 months $104,807.33 300 months $118,243.81 360 months $128,692.75 Also, you know the following variables: Down Payment = $10,000.00 Cost Per Square Foot = $75.00 Income Per Square Foot = $0.80 Mill Rate = $25.84 Plugging these in, the above formula looks as follows: