Let's move forward to the compound interest calculations. From compound interest calculations, it is actually almost universally compounded to include interest on interest. So we are rarely going to find a simple interest calculations in our industry. On matter of fact, in most of the industries, they use also compound interests, which is you have an interest on an interest I just mentioned. So let's do that same example before, but we do the calculations of having compound interest calculations and not simple interest calculations. And let's remember here, we borrowed around $1,000, and the interest was 10%. So let's say an i = 10%, just as a reminder, and we have. The principal of the amount borrowed was $1,000. So let's draw a table here similar to what we did before. And we said, the first column, we have the year, and the second column, we have what we call the base where we will apply the interest in every year to study it. And the third one. We said that we want to calculate the interest. And we have, at the end, the future value or the total amount owed towards the end of every year. And we said the first row would be now, the second row would be year 1. The third one will be year 2, and let's say three years similar to what we did of the simple interest calculations. And we said that principal is $1,000, and the $1,000 then for now, which will be the amount borrowed. For the first year when we apply the interest, the 10% on the $1000, so the base will be also $1,000, and the interest will be the 1000 times the 10%, which will give you the $100. So the total amount owed at the end of the first year would be the $1,000 plus the $100, which will be actually the same as, this is to say F1. Same as the simple interest calculations for the first year. Now, where is it becoming different? It will be in the second year and moving forward. The second here, the base to apply that interest would be then the total amount at the end of the previous year, which in this case, the $1,100 dollar F1. That is actually so it's going to be better understanding. Here, that will be F1, so F1 here would be the base for year 2, $1,100 dollar. And the $1100 You will apply the interest, which is the 10% on it, which this case will give you the $110. And here, what's happening, what we did is we took the base for year 2 is the total amount owed at the end of the previous year, which is 1,100 F1. So this will be F1, and the F1 times the 10% to calculate the interest for year 2, the interest for year 2 will be $110, because you must apply 1,100 times 10%. And the $110, you want to calculate what is the total amount owed towards the end of the second year will be 110 plus the F1, which is the 1,100. 1,100 plus 110 will give you $1,210 for the end of year 2, that's the amount owed. So from learning what did in year 2, the base for year 3 would be the total amount owed in the previous year, which is F2. So as you can see, you are actually paying interest on interest. In year 2 you included the amount of interest, which is the $100 from the previous year, and you multiply it by the 10%. So you put the 10% on the principal, the $1,000, and you put the 10% on the interest from year 1, which is the $100. For year 3, then you have a base of $1,210, and the $1,210 here, you multiply it by that interest. So that will give you $121. So what we did, the interest of year 3 will be the 10% times the total amount towards the end of the previous year, which is year 2, which is 1,210. 1,210 times the 10%, in the question we have, it will give you $121 interest in year 3. The interest in year 3, you added to the total amount on the previous year, which is the $1,210 plus the 121. It will give you $1,331, the total amount owed at the end of the third year, and then you can continue with the same as many years as you want. So what we did here that is different than the previous simple interest calculations was applying the interest on the interests from the previous year. So you always include the interest from the previous year, and you keep adding up to it. And we can take in a minute an example to see what's the differences in the results here. So let's move forward and look at the equations now. So based on what we just explained on the table of how we did the compound interest calculations, let's drive here the equations. The F1, which is the future value of the total amount owed towards the end of the first year would be that present or the principle that you borrowed plus that same amount times i, the interest that you have, which will be P (1+i). If you remember what we did, we applied interest on interest. So in this case for F2, we have F2 equal to F1 from the previous year plus F1 times i, because we have interest on the interest. And F1, you have the principle include in it the interest from the first year. So in this case, F2 would be equal to F1 (1+i), similar approach of what we are doing from the first bullet point. And if we want to substitute F1 with the equation from the first bullet, we have P ( 1 + i) to the power of 2, because you have 1 + i times 1 + i. Also, F3 the same. You have F2 (1+i), and F2 = P (1+i) to the power of 2. If we take this and have it here instead of F2, so we have 1 plus i, three times. So we have P (1+i) to the power of three. So if we have n number of years, then the future or the total value owed at the n number of years or n number of periods of times. We have the principal you borrowed, the money or the present value times 1+i to the power of that n number of periods. So from here, what I want you to remember and take from the compound interest calculations, which that will be our focus for the course. And matter of fact, I highly believe this is what you will experience in your career. I would like to summarize these two equations. The first one, which is we call it the future value. The future value of any value you borrowed, or you borrowed or loaned, at n number of years will be F will be equal to the P, that amount which P here we refer to as present value times the 1+i, the interest, to the power of n. So the interest that you have in the question to the power of the n number of years that you want to highlight, what is the future value of that in n number of period of times? If you want to reverse it, let's say I want to Put in the bank, or to have in the bank in three or four or five years from now, around $10,000. How much in the present I want to put in that bank if I have an interest rate equal to i percent, 2 or 3, 4, 5%? So from this equation, from the first equation, you just like to flip the two sides. So it will be then the present value equal to the future value that you want to find at the end divided by 1+i to the power of n. So it's either, or it's the same equation, but this is we refer to as the future value equation, and this is the present value. This is the basic first block in the time value in the mathematics of money topic that we are explaining in this course today. So let's move forward.