Here is an example that I highlight for you to understand the differences when it comes the amount at the end, when we calculate from the simple versus compound interest rate. So let's assume that I gave you $100, and for 10% per year as an interest rate for five year yields. And I'll ask you to try before I show you the solution here to calculate the interest or the future value of this $100 using first the simple interest calculations, second the compound interest calculations. Move forward with the example, and let's solve it then together. So from a simple interest calculation, we have, if you remember, the F5 or the future value of the amount, the $100, the principal. At the end of the fifth year, 100 times, between parenthesis, 1 plus that n number of years times the I. If you remember, that's what we did from the equations before and the table that we explained. So that will give us around $150, because there is no interest on interest, so you have the interest per year, you find it will say $10, and you have five years, so you have $50 interest on the 100 dollar for the total five years. So the total amount that you will have on hand at the end of the fifth year is $150. Let's say now for the compounded calculations where we apply interest on interest, we have a five future value equal the $100 times the one plus 10% to the power of five, because you are applying interest on interest, as I explained before. That will give you around $161 and around maybe five cents. So the conclusion we have here is the compound interest calculations will give you always greater interest and future value towards the end of the n period of times that you are doing the study for. In this example here, we have around 7% greater with the compound interest calculations than the simple interest calculations. So that's a clarification example of the simple versus compound interest rate calculations. So let's move forward. In this section, I would love to go through couple of terms that you might face during your career as a CM, construction manager, or a matter of fact, you can face these kind of terms in your day-to-day activity. I will highlight a couple of terms, and here, two specific forms of interest rates, which is the differences between a nominal interest rate versus effective interest rate. In finance, you can find the same terminology, or these same terms, but used in different ways or different wording for them. So here, I will highlight the word nominal and effective, and I will highlight another two terms in a minute. There are variety of interest rates offered on deposits banks and financial institutions. These could be in the form of an annual interest rate that is compounded annually or an annual interest rate that compounded semi-annually, or even a quarterly, or monthly, or even daily. Here, the importance on this topic you'll need to know and understand is that the greater the frequency, which is greater, I mean from annually to semi-annually. Semi-annually greater than the annually. Quarterly is greater than semi-annually. Monthly is greater than quarterly. Daily is greater than monthly. So the greater the frequency with which the interest is compounded, the higher the future value of the amount deposited. So moving forward, the nominal interest rate, sometimes it referred to as APR, annual percentage rate. As the effective interest rate is referred to as APY, annual percentage yield. Again, sometimes in finance, you can find different terminologies used for nominal and effective. In this module, I'm highlighting APR and APY or, nominal versus effective interest rate. Now, this is the important point that I want you to understand and learn here. The relationship between both the APY, effective, and the APR, nominal, is given by the following equations. Effective = 1 + nominal divided by c. All the two parenthesis here to the power of c minus 1. And c here is the number of interest periods per year. Monthly, you have then 12. C equal 12, because you have a 12 month per year. Quarterly, you have four. If you want semi-annually, you have two. There's two six months per year. Moving, and so on and so forth. So let's say if APR is 12% compounded monthly, that will give you then APY. Let's write it quickly here. So if you want to say APR = 12% compounded monthly. Or this is the a nominal. What would be the effective interest rate? Which will be APY? So APY would be equal to 1 + 12% divided by c. c here, because it's compounded monthly, you have 12 months per year. So you have 12 here to the power of 12 minus one. So APY here, if we do the calculations, will be 12.68 Percent. So this is the effective interest per year based on a nominal interest of 12% compounded monthly per year. Now, let's build on this. And I found always, the best way to explain the differences between both rates is through some examples and some numbers. So let's move forward with some examples here. So the first example here, I'm giving to better understand the differences between a nominal and effective, APR versus APY, is the following. Let's say you have $2,000 is invested in our account, which based 12%, and let's take, in this first example, and then we'll give you three examples of per year compounded monthly. Let's go step by step. The question here. What is the balance in the account, let's say you put this account in the bank or in any financial institutions, or in a project for investment, and what is the balance of that account after three years? First, you know that the nominal annual interest rate is equal to 12%. And that 12%, what we know from the question is per year compounded monthly. So the number of interest periods you have for all 3 years would be the 3 years times the 12 months. So in this case, it'll say that you have around 36 interest periods in this 3 years. And this is, we will use it from calculating using the nominal interest or the nominal interest rate. The interest rate I paired one month would be the 12% divided by the 12 months per year. So every month, we have an interest of 1%. So if we want to find the future value, and the balance in the account, after three years, which here, we can have it after 36 months, we can use the future value that I just explained in a previous module. The future value equal to the present value that you got the money, which is here $2,000 times the 1 plus the i to the power of the number of interest periods. So here, I took the 1%, because you have 1% per month interest, and you have 36 months. So the $2,000 after 36 months would be $2,861.54. So from an effective point of a view, that effective interest rate point of view, we can come up with the same conclusion and the same results If we apply the previous equation that I highlighted, which is the effective rate. The APY, it's equal to 1 plus the APR, which is the 12% here divided by 12, which is 12 months, which is c, what I just highlighted in the previous slide, to the power of C, which is a 12 minus 1, which will give you 12.6825%. So the effective rate, then we use the same equation, but in this case, the effective rate, we use it per year, not per month. So that will give you the future value after three years. The same equation, P, 2,000. You multiply it by the 1 plus the effective annual rate, which is we just found it here, to the power of 3, because it's for 3 years. And it will give you the same exact amount. So what we did here is two approaches. The first approach, which will be the 1 here, that approach just focused on that nominal annual interest rate 12%. Compounded monthly, so I found how much per month is the interest? 12% divided by 12 is 1% per month. And I will apply compounded interest rate calculations. The future value after three years, because I have 1% per month, so I want to see how many month I have per year in this three years? So number of interest periods would be then 3 years times 12, which would be 36 months, or 36 interest periods, based on the 1% interest rate per month. So applying that equation, you will find the number of the future value of the $2,000 after 36 months from now. You cannot apply the 12% directly in this equation because it's compounded monthly, as I explained. So if you want to use that interest rate annually for only 3 years, you have to convert the nominal interest from 12% compounded there monthly to an effective interest rate using this equation. Then you can apply that per year, the interest rate is actually 12.68, not 12% if you want to have it year by year. So these are the two approaches, and you have to have the same number.
