 In this video, I want to talk about how we can use our formulas of compounded interest to make some comparisons to actually make decisions about what is a better investment. Because you have all these different things going into play, like the number of compounds and we want that to be big. The interest rate, we want that to be big. You know, we kind of see what's our good things, but what happens when you have these competing variables? How does one decide what the better investment is over time? So I want to show you some examples where we can actually analyze how good the investments are when we compare them apples to apples. So suppose we have a zero bond, a zero coupon bond that can be redeemed in 10 years for $1,000. So this is kind of like how government bonds work. Like my grandma would always send me like, here's a hundred dollar bond. And as a kid, you're like, wow, I'm five years old. Now I have a hundred dollars, which of course I, you know, I didn't realize as a kid that I don't get a hundred dollars until I'm like 35 or 45 or you know, whatever the type of bond was, right? It's not that it's worth a hundred dollars. Now is in the future, it'll be worth a hundred dollars or in this case, a thousand dollars, right? So this is zero coupon bond in 10 years. It's supposed to be worth a thousand dollars. Well, okay. Grandma had to buy this bond for me. How much should she have been willing to pay for a thousand dollars 10 years from now? How much would a thousand dollars be worth now? So to speak. Well, it depends on the interest rate. Let's suppose that at the time the bond was purchased, the, the, you could get an interest loan for 8% compounded monthly. Well, if that's your interest rate, R and our number of compounds is 12, right? And we want to invest this for 10 years and our principal was a thousand dollars. Then we can put this into our compounded interest formula. We get the amount is equal to the principal times one plus R over N to the NT. We can start plugging things in. We'd end up with a thousand. And I asked actually, that's where I miss, I miswrote on the screen earlier. That thousand isn't the principal, right? We're not investing a thousand dollars. That's how much it's supposed to be worth after 10 years. So the amount is a thousand dollars. And this is going to equal the principal. I don't actually know what that is. One plus 8% over 12 raised to the 12 times 10. So the thing is, we actually don't know what the principal is. Aha. Right? How much should you be willing to pay for it? Like how much should I pay for it now? That would be the principal of this loan. So we need to solve for P. But in order to do that, you probably want to divide this quantity on both sides. So you get one plus 0.08 divided by 12 raised to the 12.10 power right there. So you have to divide that on those sides like, ooh, that's a mess. But one thing to remember is when you divide by exponents, you actually could just stick a negative inside the exponent that actually makes it come up to the top right here. So if you divide both sides by this, instead of writing this big hunker over here in the denominator, we can actually just use a negative exponent and what we see is something like the following. The principal is equal to the amount times one plus r over n raised to the negative nt. I'd like to think of it as we're going back in time. So get your Mr. Fusion and jump into your DeLorean here. It's as if, okay, we're in the year 2020, right? Just as an example. And it's worth $1,000. If I went back in time to the year 2010, I probably should go back to 2015 if I continue with this back to future comparison here. But nonetheless, this video was recorded in 2020. If we go back in time 10 years, how much should it be? So we have this negative time because we're going back in time, all right? So let's now put this into our formula. The amount will be worth $1,000. The interest rate is 8%. The number of compounds will be 12 because we're doing monthly. And then the time frame will be 10 years. We're going back negative 10. So I really should think of this as 10, 12 times negative 10, if that helps you out. In terms of multiplications, it'll make much of a difference. It'll be negative 120 in the end anyways. Points are 8% divided by 12 will be .006 repeated. Add that to 1. You get 1.006 repeated. Just add a lot of 6s in that as you're trying to compute this. Take the negative 120th, the negative 120th power. You're going to end up with .4505. Again, try to keep this as much in your calculator of memory as possible. You don't want to round too early so that your calculation doesn't get error into it. Times this by 1,000. This means that grandma, if she were to buy this loan, if she bought this bond on my 10th birthday, she should be willing to pay $150 so that when I turn 20, again hypothetically, it's worth $1,000. That's how much you should pay for the bond. Of course, if the government's willing to sell it to you for less than $150, that's great. The problem is what they don't realize is like, oh, you buy it for $500 and then it'll be worth $1,000 later. It's like, well, but I have a bank account that I can invest in right now. I can invest my $500 at 8% compounded monthly. If I don't touch it for 10 years, it'll be worth more than $1,000. Your bond stinks. We need to compare this loan versus competing loans exist at the time. What if the bond was 7% compounded continuously? Look at that for a moment. The interest rate is lesser than say the previous one, but it compounds more frequently. It's compounding is infinity versus only 12. How does that affect things? Is it a better loan or not? Well, we can do the comparison here. In this situation, we have to use A equals PERT, principle times e to the RT, but the same thing as you divide by e to the RT on both sides, dividing by an exponential, you can just use a negative exponent and go back in time. The principle is going to equal the original amount times e to the negative RT, so plug in those values. You get 1,000 times e to the negative 0.07 times 10. The exponent's easy enough. You're just going to get negative 0.7, but then you're going to want to use a calculator whenever you do a calculation with e whatsoever. You're going to plug in e to the negative 0.7 exponent and then times that by 1,000. You'll get 496.59. What's the better bond? It's going to be this one, right? I can get the same $1,000 with less money. This is the bond you would want to buy if you were grandma buying it for your grandson on his 10th birthday. 8% monthly is better than 7% continuously. Let's continue with that game. Suppose that you want to open up a money market account. You invest three banks to determine their money market rates. This is what we're trying to describe right here is the idea of a certificate of deposit, sometimes called a CD. Compounded interest is a really good formula if you have a certificate of deposit or something equivalent to a CD here. The way that your typical CD works is it's a frozen asset. You deposit your original principal and then you don't touch it for a year, five years, 10 years until the loan, until the investment match your rates, right? In which case they don't collect interest and the interest will collect interest and the interest, interest will collect interest. So imagine that's what we're trying to do. Because the idea of like a checking account or savings account even have an interest rate, you're often making deposits and withdrawals. So the compound interest formula doesn't really do much for that. We need some more complicated formula. Certificate of deposits is what we're trying to talk about right here. So imagine we go to three different banks to see what type of CDs could we buy. You didn't know you're going to go to the store to a bank to buy CD, which of course, I know that joke doesn't make much sense in 2020, who buys CDs anymore. We download our music. Anyways, so we have a certificate of deposit. The bank A, it says we're going to offer you 6% annual interest compounded daily. Wow, that's a great, you know, three. So if you look at that, it's like my over here with bank A, you're going to get number of compounds is 365, that's great. And your interest rate is 6%. Okay, that's not, maybe that's good, maybe that's bad. Let's see. Bank B, right, it's going to offer you a 6.02% interest rate, but it's compounded quarterly. All right, well, how does that affect things? So your compounds will be four. So dramatically smaller than we saw for daily interest, but your interest rate's better, R equals 0.0602, but it's only a little bit better. How does that affect things? Well, we're going to see in a moment. And then bank C wants to offer you 5.98% interest compounded continuously. Wow, that's great, right? Your number of compounds is infinity, you can't do better than that. But your interest rate is smaller, you're going to get 0.598. So who's the best choice, right? When you look at these things, these interest rates are only different, they're only off by like 0.04%. So it's not a huge difference, but the number of compounds is way different. How do we compare these things apples to apples? And so what we want to do is we're seeking what's so-called the effective rate of interest, which is equivalent. Basically what we want to do is we want to turn this problem, all of these problems into a simple interest problem. So if we invested basically $1 for one year at a simple interest rate, what would be the rate of that simple interest? This effective rate of interest does not depend on the principle, it does not depend on the time. So again, if we turned all of these into simple interest problems so we could compare apples to apples, we invest $1 for one year, what rate would that have been? And then whoever has the best effective rate of interest will be the best investment. So how do you compute this effective rate of investment? Like I said, we're going to take RE right here, this is the, RE is going to be the effective rate of interest. So notice you get P times one plus rate times time. This is a simple interest problem, right? The principle, whatever, we're investing for one year at the effective rate of interest. Well, the problem is not actually, of course, a simple interest problem, it's a compounded interest problem. So we're going to set a simple interest equal to a compounded interest, where the principles, whatevs, the rate is given, the number of compounds is given, the time is going to equal to one. So as we simplify the right hand side while the X one becomes an N, we want to then solve for, we want to solve here. We can also divide both sides by P, because again, the principle doesn't matter. So we get one plus RE is equal to one plus R over N to the N. So track one, this is our formula for effective rate of interest. Of course, if you're doing continuously compounded interest, this would look like E to the R minus one, because again, T equals one in that situation for continuously compounded interest. So we compute this number right here. So for the first one, we're going to take one plus 6% divided by 365, raise that to the 365 minus one. You crunch the numbers, la, la, la, la, la, la, use your calculator, you end up with 6.183%. That's how good bank A is going to be. So then you look at bank B. So remember bank A was daily with a 6% interest rate. Bank B was 4%, it has the four quarters, excuse me, the number of compounds, but it had the best interest rate, 6.02%. When you put the numbers in here, we're going to take one plus 6.02% divided by four, raise to the fourth to track one, crunch the numbers, la, la, la, la, la, la, la, you're going to end up with 6.157%, which you can see when you compare that with bank A, bank A has a higher effective rate of interest. So that's a better investment between the two. What about continuously compounded interest? N equals infinity, right? But the interest rate was smaller, 5.98%. So you're going to take e to the r, so e to the 5.98%, that gives you about 1.061624, subtract one, and you end up with 0.061624, right, that is a percentage, right? You're going to hit 6.162, which that beats bank B, but actually bank A still is the winner here. So believe it or not, the bank with the highest interest rate actually was the worst investment because it had the fewest number of compounds. Then the highest number of compounds also was not the best investment here. Turns out the lukewarm investment was the best one, right? Not too many compounds, not too low interest, right? Bank A is the best deal. And this is what you have to do with finance. When you have these different options, like should I invest this way or this way? Do I take out this loan or what? When it comes to your money, it's worth spending a little bit of time and asking yourself, what is the best deal? And you need to make sure you are comparing apples to apples. The effective rate of interest is a tool you can use to actually compare like, over time, what's going to be the best investment? And we see here that bank A offers the best deal of these three certificates of deposit. Certificates of deposit.