The final course of the specialization expands the knowledge of a construction project manager to include an understanding of economics and the mathematics of money, an essential component of every construction project. Topics covered include the time value of money, the definition and calculation of the types of interest rates, and the importance of Cash Flow Diagrams.

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The Mathematics of Money

Professor Ibrahim Odeh discusses the Mathematics of Money beginning with a definition of the Time Value of Money. Calculating simple and compound interest rates are covered along with distinguishing between nominal and effective interest rates. Illustrated in this module is drawing a cash flow diagram.

Instructor, Department of Civil Engineering and Engineering Mechanics, Columbia University Director of Research and Founder, Global Leaders in Construction Management

In this section, I'll be talking about a very basic concept when it comes to

the time value of money or the mathematics of money that we just explained

the introduction to it, which is that cash flow diagram.

Now by definition, a cash flow diagram, under the subject of the time

value mathematics of money, ends at its goal to provide

a graphical means, for describing situations in which

time intersect and interact with interests.

So the situation when there is an interest and

the time interact, that's what we as a cash flow diagram would

be able to help us understand better with these situations.

Moreover, developing an appropriate cash flow diagram is often

a very critical step in solving any type of engineering economy analysis.

For our course, we are more focusing on the construction management and

the construction engineering point of view.

Let's have a look at the basic concept or

what we can refer to as the main elements of the cash flow

diagram which we will use to build on for our coming modules later.

So we start by drawing the timeline of the diagram.

Which this would be divided into intervals or periods

of times or interest periods, similar to what I explained in previous modules.

So from the X axis, and there is only X axis, and then the bars in the Y axis.

Highlighting first the timeline as I explained, with, let's say, the number,

timeline from zero, one, two, three, four, all the way to an end period of time.

This periods could be as I said before days, or weeks,

or months, or quarters, semi annuals or yearly.

Most of the classes, or most of them we used in the examples,

I might be focusing either on monthly or annual.

But I highlighted if we have less or more intervals in that year,

what to do from that APY, from a nominal and effective interest rate.

So this is the first system.

Second is P n

as I said the number of compound periods the interest periods.

P is a present sum of money.

If you remember we talked about P a lot in previous modules and

this is what's the money we have today in hand.

This is the present worth of the money.

And after that I want to highlight another theorem, which will be

what we call or refer to uniform series or A value.

The A value here as you can see in the cash flow diagram, is an end of

period cash flow in a uniform

series counting for an n period of time.

And this A or installments are the same, the same value for it.

In order to say, okay, we have the same A for this number of periods of times,

and we will talk about it in much more details in the next slide.

Next, another term in the cash flow diagram we might be also dealing with

in addition to the P To the number of intervals and

A is the F value which is the future sum of money the future value.

And if you remember we also talk about the F value or the future value and

we connected the F value to the P value with an I interest rate

on this number of interest periods until M.

So, now, we have an idea about that P value,

which is the present value of the money today.

Let's us study next, how we can

calculate the equivalent A values,

the equivalent A values, of a given P value.

The A value here.

Sometimes we refer to it as the uniform series values.

Or if it is the timeline here if it is yearly, by year, year by year.

This sometimes we referred to it as annuities, or annuity.

So, after we will study [COUGH] excuse me,

after we're going to study how we can connect the A values to the P value, we

will then move forward to understand and to calculate

the equivalent future value F of an A series value.

So I will focus to connect the relation between the P and

how we can convert it to A's or vice versa.

And next we will work on study on how we can calculate or

find out equivalent values of an as with the link to an F.

So, let's move forward then to focus on the next topic which is

the uniform series values.

That definition of a uniform series values is a series

of equal or is a set of equal payments, or receipts,

paid out from a payment point of view, or received from receipts.

In a sequence over a period of time.

So, let's say I [INAUDIBLE] always to give example

with numbers to better digest this kind of information and better link what and

how we came up with formulas to calculate such A values, and P values, and

F values, similar to what I did in that previous slides and modules when

we talked about the simple interest rate and the compound interest rate, and so on.

So let's say on this to understand it better that you want to buy

a construction equipment or a tractor, or any piece of

equipment that you will use in your project or in your company [INAUDIBLE]

or any kind of tool that we'll be using.

Lets assume, that this piece of

equipment is worth around $30,000.

So, one option you have in mind here or in hand.

Is to pay the full amount of the $30,000 to buy that

specific piece of equipment today.

This is what we refer to in the previous slide

the value p from the present worth of the money.

From the previous slide, I'm going to highlight, I highlighted P and A and F.

And I will tell you that, okay, to buy an equipment worth $30,000,

one option is to pay for it today.

This is the 30,000.

Another option would be mathematically speaking to pay

in the future the entire amount, which is equivalent

to the future value or F from the cash flow

diagram before, of this amount 30,000, based

on a specific monthly interest rate or annual interest rate.

So if I tell you, at our example here, that the interest rate provided to you is.

Let's write it down so that we can follow up on that.

So let's say we have an interest

rate equal 10% and this is,

compounded monthly.

So, speaking of that, that's meaning each month,

because this is nominal interest rate, so this nominal interest rate

I can either convert it to an effective or find the interest rate per month.

To find the interest rate per month that means that each month that interest

i is equal to 10% divided by 12 months per year

which is equal 0.33%.

So i from monthly point of a view will be 10% divided by 12 months per year,

which will be equal point 833%.

Now, in this case,

in our example that we are building up here, we have a monthly interest rate

of .833%

So let's assume that the payment for the future value of

the 30,000 is due in 36 months from now.

So N will be equal to 36 months.

Then the future sum value, you need to pay in 36 months from today,

can be found using the equation that we explained earlier in the course.

Which is the following equation.

We have either F equal to the P

times one plus I to the power of M, or the other way around.

But for this value, we're trying to find, let's do this,

let's have that equation written here, P1 plus I, to the power of n.