So, let's take another example or another two examples to better digest this information and understand it better. So example number two. Also $2000, exactly the same question, but instead of compounded monthly, I change it to compounded quarterly. And I ask you the same question. What is the balance In the account after 3 years from now. The nominal interest rate we know is 12%. The number of interest periods would be 3 years and every year you have 4 quarters per year. So the number of interest periods you have for all 3 years are 12 quarters. So how much the interest per quarter? We take the nominal which is 12% divided by 4, because you have 4 quarters per year. So the interest per quarter is a 3%. So the future value after 3 years or after, the 3 is equivalent to 12 quarters which will be F12, will be P times 1 plus the 3%, because 3% per quarter, to the power of 12, which is the 12 quarters or the 3 years. And it will give you an answer for what's the future value of the 2000 after three years of 12% interest rate per year compounded quarterly. Instead of using then quarter by quarter to have the interest in each quarter calculated, like here, 3%, I want to see like no, I want to have it per year, then I have to convert the 12% compounded quarterly to the effective interest rate yearly, annually. Using the APY which the one plus the 12%, the nominal interest, APR, divided by 4, how many quarters you have per 1 year? Is 4 to the power of 4 minus 1, so 12.55 is the effective interest rate per year. So since we found the effective interest rate per year, we can then use the same equation the future value after the 3 years knowing the effective interest rate per year, to the power of 3 and you have to have to get the same answers, either using this approach or this approach here. So this is a second question. So the first question we got was 12% per year compounded monthly, which right now 12% per year compounded quarterly. Let's take a last example of the same exact number but compounded semi-annually. The semi-annually would be the following. What exactly what we want to understand is what is the balance in the account after 3 years from now? Nominal interest rate, 12%. So I will ask you a question. Try before I move forward with the solution. I would ask you to try to solve this based on the two examples I gave before and let's solve it then together. So as we mentioned, we have a nominal annual interest rate at 12%. The number of interest periods will be 3 years times 2 because you have how many 6 months per year, you have 2. So that will give you 6 months periods in 3 years. So the interest rate you have in each period, which is each 6 months, will be 12% divided by 2, then you have 6% per one interest period. And how many interest periods we have? We have 6. So the future value after six interest periods based on an interest rate of 6% per period, it will give you the $2,000 times 1 plus the i to the power of 6, will give you this number, $2837. And if we want to calculate this on an effective interest rate annually then we have to convert the 12% nominal to effective yearly, which would be here the C equal two. We start with that C equal to 12 for monthly, and then, C equal 4 from quarterly, and then C equal 2 from semi-annually. And it'll give you an effective interest rate of 12.36 person. If you convert that nominal to effective, then you can use the effective interest rate to calculate based on a year by year basis as F3 equal the P to the times 1 plus the effective interest rate to the power of 3 years. And you have to get the same number you get from the previous one here. So, if you recall what I said in the introduction to the nominal versus effective, that the importance of this topic, you need to understand and to know very clearly is that, the greater the frequency with which the interest is compounded, the higher the future value of the amount deposited. So, if you recall what I said in the introduction to the nominal versus effective, that the importance on this topic, you need to understand and to know very clearly is that the greater the frequency with which the interest is compounded, the higher the future value of the amount deposited. So, let's emphasis more into it and you can recall from the slides before. When with the question was the amount was compounded monthly, the $2000 we had a future value at the end of the 3 years around $2,861, I think 54. Sorry, and this is 54. And that was compounded monthly, right? So when it was compounded quarterly which give you less frequency than the monthly, so you have 12 quarters or periods per 3 years versus 36 months or 36 periods in 3 years, right. In this case, the end result was less which is 2851. I think,06, and that was quarterly. So, the first one, monthly. And you have 36 interest periods. And here, quarterly, where you have 12 periods or interest periods. As for the much less frequency, as we can see in the last example I gave for 6 periods, we got $2,837, right? And that was, semi-annually. So, $2,861, $2, 851, it reduced, $2,837, it reduced even more, because here you have 36 periods of time, here you had around 12 periods of interest, and here you have only 6 periods of interest. So, the greater the frequency, the higher the future value of the amount deposited in that bank or in that project. As we can see from here, monthly, quarterly, semi-annually. In this section, I would love also to highlight another term of interest rate that I want you to pay attention to and understand which is the Minimum Attractive Rate of Return or the MARR. It's a very important term and I want to pay very attention to it. The definition, let's start from the definition point of view. The definition of the MARR is the minimum interest rate that is accepted by a person or an organization to accept the financial burden of building a project or buying the product. Again, it is the minimum interest rate that is accepted by an individual or an institution or a company to accept the financial burden of building a project, taking to build the project, or buying a specific product or equipment. Some of the questions we ask ourselves to better understand the MARR would be the following. Comparing receiving an amount of money two years from now versus one year from now. This is one specific question that we build to understand the MARR or compare and offer to receive $100 now or x dollar a year from now. These questions will trigger for us to think more about what MARR actually means. So, let's take for example and I'll ask you in a specific here. For example let's say, I have $100 and I want to compare it with several amount of x dollar of amounts a year from now. So I came to you, I told you, okay, what do you think? I will give you $100 now or you should maybe wait and I will give you $400 a year from now. Which offer are you going to take? You might tell me, no 400 is worth the wait, worth the burden, I will wait. Or I will tell you 100 versus 300, so you will start shaking your head like, yeah, maybe I will wait, 300 is still worth it. So maybe I will reach a point to keep reducing from 400 to 300 to 200 and you start shaking your head even more and do more calculations, in the beginning it was piece of cake. The answer, until you reached a point, I will ask you, what about $100 now versus $150 a year from now. Then you pause a little bit. And you can say, when you start saying, okay, you know what, it will not make any difference. Both are okay for me. So once you reach that there is no difference in the two amounts, then that would be your MARR, or that MARR, the Minimum Attractive Rate of Return. So let's say, if the $100 now would be the same as receiving $150 at the end of the year, that means the total interests, that fee would be how much? Would be the 50 Dollar, right? Let's say, I'm comparing down $100 versus the $150. And we said the same, in this case, that interest would be equal to the $50. And the interest rate would be the $50 divided by the $100. In this case, you have 50%. That will then tell us that the MARR, the Minimum Attractive Rate of Return in your case would be 50%. So as a rule of thumb, you need to remember the following, very, very, very carefully. If the interests on your investment is less than your MARR, then the investment to you is not profitable to your company, it's not profitable. To your business, it's not profitable. If i tell you, I will give you instead of $150, I will give you $140. Then that is an interest of $40, which is an interest rate of 40%, which is less than the attractive rate of the 50% you have, or that you are looking for to get. So in this case, it will be non-profitable for you. So, I kept dropping from 400, to 300, 200, until I reached the amount you told me there is no differences, so you found the MARR. So any interest I tell you on an investment opportunity to invest in you would let me know. If it is not 50% and more, I'm not going to invest that. So, I hope this now then clears what is an MARR and any interest that below your MARR, that investment to you is not profitable.
