Smart financial decision using limits - how to do it

Mark Misin
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Aerospace & Robotics Engineer
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INTUITION MATTERS! - Applied Calculus for Engineers-Complete

Calculus + Engineering + PID: Functions, Limits, Derivatives, Vectors, differential equations, integrals: BEST CALCULUS

34:53:37 of on-demand video • Updated October 2020

  • You will develop very strong intuition & understanding in Calculus
  • You will learn how to apply Calculus in real life to the level not seen in other courses
English [Auto] Welcome back. I have a small financial exercise for you. Suppose that you have €1000 organized an orgy with PV it means present value €1000. Now why Piggy. Well it's a financial term. It means present value. It means that right now in the present if you have 1000 euros then it's worth 1000 euros. Now if you go into the future then because of interest rates and inflation rates the value of your €1000 can vary. Right. So if you have 1000 years and you have inflation or in other words if your prices go up then in one year you're 1000 euros is worth less because you're going to be able to buy less stuff for that one thousand euros. So your purchasing power will go down and on the other hand if you lend someone money for one year and that someone will pay you interest for that then you will have more money in in one year or in other words you can think of it like this. Your present money will be worth more in the future. Now the future value is denoted with F. V and then you go to a bank and you tell your bank Hey I've got €1000. And then the bank says look great if you deposit it in my bank for one year I'll pay you 10 percent interest and then and then you're going to be all right. So how much money would you get in one year. The formula is this future value equals present value one plus your interest rate and we're going to say that the interest rate is 10 percent right. Or it means zero point one. So if you're 1000 euros then your future value the money that you will get in one year or how much your present money's worth in one year. It's called The Well it can be written like this one thousand one plus 0.1 equals 1000 times 1.1 which is 1100 euros. So that would be your profit. RYAN Now suppose that your other bank calls you and tells you that Hey listen I've got a different offer for you. I'm still going to pay you an annual interest rate of 10 percent. However what I will do if this is your one year period and you get 10 percent in the end of the year. But what I'm going to do after six months I'm going to take five percent and then I'm going to take 5 percent and I'm going to calculate the interest of that money so it will be future value. Equals 1000 1 plus. Now the 0.1 divided by two. So two times a year. So that it will become 5 percent. Right. So I'm going to calculate how much you would get if you were to store your money in my bank for six months. Right. And I'm just going to take 10 percent divided by 2. And then in the next six months write I'm going to take another 5 percent I'm going to take another five percent and I'm going to calculate your second part of the year interest rate using using your original money and your previously earned interest as a base for calculating the new interest. So in other words what you're doing. You're first of all in the first case the bank will just calculate the interest rate one time from your 1000 euros but the second bank will do the operation twice in the first time. And then you will get some kind of interest in the first six months. The interest rate would be 5 percent but now the bank will calculate the second 5 percent. Now only from your original money but also from the previously earned interest. So you can also write this formula like this one thousand times one plus zero point one divided by two to the power of two. Right. So you will get your future value would equal one thousand one hundred oh two point five years. As you can see you receive a little bit of more money. That's because the second 5 percent is calculated not only from your present value but also from the previously earned interest and that is called Compound interest compound interest Okay. And of course it's a better deal for you. And if your third bank calls you and tells you Hey I'm going to make this calculation four times a year. So still 10 percent but I'm going to do it quarterly. Right. So I'm going to have 2.5 percent here 2.5 here 2.5 here. Well essentially the real thing would be that 2.5 2.5 the interval 2.5 and 2.5 percent. And of course you can guess that then your money would be bigger. So your future value would be 1000 one plus zero point 1 divided by four to the power of four. Now another Banquo's in tells you hey you know what I'm going to do. I'm going to calculate the interest rate continuously. So every time every millisecond every nanosecond every atomic second I'm going to calculate the new interest rate based not only on your base money but based on all the other interest rates that you had earned previously. So of course that would be the best deal for you but what would be the number. How would you calculate it. So poses and then I'll show it to you OK will in order to do that in order to do that you know that you have this equation right. And instead of this four we can't simply say future value equals 1000 one plus zero point one divided by an and then. And so the larger the number the more you chop your year right the more frequently the interest calculations happen. But does this equation remind you of something. Well it is. It's exactly the equation that I showed you in the previous video. So in fact what you can do you can simply calculate if the bank tells you that it calculates the interest rate continuously then that just means that just means that you're an approaches infinity. Right. So continuously Well you can't have infinite here but you can approach infinity. So continuous means that I'm going to chop my year into infinite infinitesimally small intervals very very thin intervals. It's continuous It's continued so the length of the interest calculation the length of the calculation will approach zero. In other words the amount of these intervals will approach infinity. So you will have future value limit and approaches infinity. Now 1000 one plus zero point one. And and so these ants will approach infinity and what will you get when it approaches infinity. Well from the previous video this function as function would become the Ilga number to the power of zero point 1. So this would be ill or number zero point one as and approaches infinity. And here you have 1000 which is this. Which means that your future value is 1000 times this number. So you will get approximately in one year you will get 100 one thousand one hundred five point one seven years. So that's an approximation because you use an irrational number so it would go to infinity but approximately you would get it that you will get that much. And as you can see not a big difference with small amounts of money but if you store millions and millions of dollars then it can be a game changer. All right. Thank you for attending this lecture. And see you next time.