WEBVTT 00:00:00.000 --> 00:00:01.940 Have you ever been staring at, say, two different 00:00:01.940 --> 00:00:05.500 ads for what seem like identical financial products, 00:00:05.660 --> 00:00:08.000 maybe a CD or a high -yield savings account? 00:00:08.220 --> 00:00:10.320 Yeah. And the numbers just don't quite line up. 00:00:10.400 --> 00:00:12.500 All the time. You'll see one bank advertising 00:00:12.500 --> 00:00:16.120 a deposit rate of, I don't know, 4 .65%. And 00:00:16.120 --> 00:00:18.359 then right next door, a competitor is offering 00:00:18.359 --> 00:00:22.480 4 .75%. And you're left wondering, OK, if they're 00:00:22.480 --> 00:00:25.079 both just savings accounts, why are these rates 00:00:25.079 --> 00:00:28.039 different? And more importantly, which one is 00:00:28.039 --> 00:00:30.230 the real number I should be using? Well, that 00:00:30.230 --> 00:00:33.890 confusion is, it's often deliberate. And it's 00:00:33.890 --> 00:00:35.490 exactly what we're here to cut through today. 00:00:35.649 --> 00:00:38.460 It feels like it. It is. Because when you, as 00:00:38.460 --> 00:00:40.600 a consumer, are comparing those two products, 00:00:40.840 --> 00:00:43.840 one institution might be quoting you the simple 00:00:43.840 --> 00:00:47.240 nominal rate, sort of the sticker price, while 00:00:47.240 --> 00:00:49.240 the other is quoting a rate that reflects the 00:00:49.240 --> 00:00:51.619 reality of compounding. And compounding is. Yeah. 00:00:51.700 --> 00:00:54.079 That's how frequently the interest is calculated 00:00:54.079 --> 00:00:55.920 and then added back to your principal, which 00:00:55.920 --> 00:00:58.119 then starts earning its own interest. Exactly. 00:00:58.119 --> 00:01:00.520 It's the engine of growth. And it's what makes 00:01:00.520 --> 00:01:02.979 those two numbers different. OK, let's unpack 00:01:02.979 --> 00:01:06.239 this. We are diving deep into that single vital 00:01:06.239 --> 00:01:09.040 number that's designed to unify all these different 00:01:09.040 --> 00:01:10.879 offerings. We're talking about the annual percentage 00:01:10.879 --> 00:01:13.939 yield or APY. It's the metric that's supposed 00:01:13.939 --> 00:01:15.959 to cut through all that noise and give you a 00:01:15.959 --> 00:01:19.400 true apples to apples comparison. And today we're 00:01:19.400 --> 00:01:21.439 armed with some foundational financial sources. 00:01:21.599 --> 00:01:25.640 We've even got the actual regulatory texts that 00:01:25.640 --> 00:01:27.680 govern how banks have to calculate and present 00:01:27.680 --> 00:01:30.359 this number to you. Our deep dive today is really 00:01:30.359 --> 00:01:33.480 a mission to demystify this whole concept. It 00:01:33.480 --> 00:01:36.260 often feels intentionally opaque. You know, all 00:01:36.260 --> 00:01:37.819 you want to know is how much money you're actually 00:01:37.819 --> 00:01:40.340 going to earn. So we need to understand not just 00:01:40.340 --> 00:01:43.480 what APY is, but the math behind it, specifically 00:01:43.480 --> 00:01:45.959 the mechanics of that compounding and the really 00:01:45.959 --> 00:01:48.420 strict regulations that dictate how it's used. 00:01:48.700 --> 00:01:51.260 My goal here is that you can walk away with a 00:01:51.260 --> 00:01:53.920 definitive understanding of why APY is the essential 00:01:53.920 --> 00:01:56.439 number when you're evaluating any deposit account. 00:01:57.169 --> 00:02:00.269 And maybe you'll be able to use the math to even 00:02:00.269 --> 00:02:02.189 check your bank's homework. And if you look at 00:02:02.189 --> 00:02:04.950 it from, say, a 30 ,000 foot view, what's so 00:02:04.950 --> 00:02:08.110 fascinating here is that the APY is this fundamental 00:02:08.110 --> 00:02:10.849 bridge. It's the numerical difference between 00:02:10.849 --> 00:02:14.129 simple interest, that theoretical rate, and the 00:02:14.129 --> 00:02:16.400 compounding reality. It provides the context. 00:02:16.599 --> 00:02:19.159 It provides the crucial context for the true 00:02:19.159 --> 00:02:21.960 cost or, in this case, the true benefit analysis 00:02:21.960 --> 00:02:24.580 of any deposit product you might see out there. 00:02:24.840 --> 00:02:26.479 All right. So let's start with the groundwork. 00:02:26.560 --> 00:02:29.419 When we talk about APY, what are we really talking 00:02:29.419 --> 00:02:32.479 about? Is this a global standard? It is, even 00:02:32.479 --> 00:02:34.439 if the regulatory details change from country 00:02:34.439 --> 00:02:37.280 to country. At its absolute core, the annual 00:02:37.280 --> 00:02:40.680 percentage yield is a normalized representation 00:02:40.680 --> 00:02:43.039 of an interest rate. Normalized representation. 00:02:43.180 --> 00:02:45.479 That's a key phrase. What does normalized mean 00:02:45.479 --> 00:02:47.719 here? It means it's standardized. It's based 00:02:47.719 --> 00:02:50.300 on a compounding period of one full year, 365 00:02:50.300 --> 00:02:53.560 days. Okay. You can almost think of it as a financial 00:02:53.560 --> 00:02:56.360 lingua franca for interest rates. It's a common 00:02:56.360 --> 00:02:58.379 denominator that all banks have to translate 00:02:58.379 --> 00:03:00.719 their own specific rates into, no matter how 00:03:00.719 --> 00:03:02.439 they structure their calculations on the back 00:03:02.439 --> 00:03:04.759 end. And why is that normalization so necessary? 00:03:05.000 --> 00:03:07.740 Because banks calculate interest at... all sorts 00:03:07.740 --> 00:03:09.919 of different frequencies, and that frequency 00:03:09.919 --> 00:03:12.860 has a direct, measurable impact on your effective 00:03:12.860 --> 00:03:15.240 return. So one bank might be compounding daily. 00:03:15.500 --> 00:03:17.580 Right, which means your interest starts earning 00:03:17.580 --> 00:03:20.280 more interest every single day. But another place 00:03:20.280 --> 00:03:22.520 might compound quarterly, which is only four 00:03:22.520 --> 00:03:24.520 times a year. And if they both just advertise, 00:03:24.560 --> 00:03:28.270 say, a 4 .5 % rate. Then just comparing that 00:03:28.270 --> 00:03:31.669 4 .5 % figure is completely misleading. The daily 00:03:31.669 --> 00:03:34.330 compounding account will always mathematically 00:03:34.330 --> 00:03:37.210 result in a higher dollar return over the course 00:03:37.210 --> 00:03:39.550 of a year. It just has to. So that's the utility 00:03:39.550 --> 00:03:42.289 right there. By normalizing every rate to what 00:03:42.289 --> 00:03:45.449 it would effectively yield over a full 365 -day 00:03:45.449 --> 00:03:48.569 period, APY lets you make a reasonable single 00:03:48.569 --> 00:03:51.129 -point comparison. It's the only way. You can 00:03:51.129 --> 00:03:53.370 genuinely put that daily compounding account 00:03:53.370 --> 00:03:55.490 up against the quarterly one. Look only at the 00:03:55.490 --> 00:03:57.770 APY figure for both. And you know for sure which 00:03:57.770 --> 00:04:00.069 one delivers the better annual return. Let's 00:04:00.069 --> 00:04:02.889 make it even simpler. Imagine Bank A advertises 00:04:02.889 --> 00:04:06.110 a 4 .5 % nominal rate compounded daily. Okay. 00:04:06.169 --> 00:04:09.550 And Bank B advertises the exact same 4 .5 % nominal 00:04:09.550 --> 00:04:11.889 rate, but they only compound semi -annually. 00:04:12.030 --> 00:04:14.129 So twice a year. Big difference in frequency. 00:04:14.349 --> 00:04:16.850 A huge difference. And when we calculate the 00:04:16.850 --> 00:04:20.430 APY, Bank A's daily compounding will give you 00:04:20.430 --> 00:04:23.829 an APY that's actually higher than 4 .5%, maybe 00:04:23.829 --> 00:04:28.750 say 4 .60%. And Bank B. Their semi -annual compounding 00:04:28.750 --> 00:04:31.209 will result in an APY that's lower. It'll be 00:04:31.209 --> 00:04:35.209 higher than 4 .5%, but maybe only 4 .5%. So the 00:04:35.209 --> 00:04:38.149 APY is what captures that vital frequency component 00:04:38.149 --> 00:04:40.529 that the nominal rate just completely ignores. 00:04:40.610 --> 00:04:43.389 It's the whole story. For a consumer, this has 00:04:43.389 --> 00:04:45.670 to be the first and most fundamental tool in 00:04:45.670 --> 00:04:48.310 the toolkit then. You look for the APY and you 00:04:48.310 --> 00:04:50.550 basically ignore the nominal rate when you're 00:04:50.550 --> 00:04:52.509 comparison shopping. That's the first rule. Now, 00:04:52.509 --> 00:04:54.389 this brings us to something that causes, I think, 00:04:54.649 --> 00:04:57.009 endless confusion because the acronyms are so 00:04:57.009 --> 00:05:00.949 close we have to talk about apy versus apr yes 00:05:00.949 --> 00:05:02.990 the essential distinction they are mathematically 00:05:02.990 --> 00:05:06.449 related but in practice their application, their 00:05:06.449 --> 00:05:08.449 directionality, it's the complete opposite. Okay, 00:05:08.470 --> 00:05:11.990 so let's clarify. APY. APY generally refers to 00:05:11.990 --> 00:05:14.470 the rate paid to a depositor. This is the return 00:05:14.470 --> 00:05:16.750 you earn on your savings account, your CD. The 00:05:16.750 --> 00:05:19.050 money flows to you from the bank. And APR, the 00:05:19.050 --> 00:05:20.970 annual percentage rate. That's the rate paid 00:05:20.970 --> 00:05:23.569 to a financial institution by a borrower. It's 00:05:23.569 --> 00:05:26.089 the cost of borrowing money, what you pay on 00:05:26.089 --> 00:05:28.790 a loan, a mortgage, or, you know, most commonly 00:05:28.790 --> 00:05:31.930 a credit card. So with APR, the money is flowing 00:05:31.930 --> 00:05:34.819 from you. The customer to the institution. Exactly. 00:05:35.000 --> 00:05:37.680 And while they both share this goal of transparency, 00:05:38.019 --> 00:05:41.060 telling you the rate over a year, APY is for 00:05:41.060 --> 00:05:44.579 earning and APR is for borrowing. And with borrowing, 00:05:44.680 --> 00:05:46.740 there are other factors, right? Like fees built 00:05:46.740 --> 00:05:48.699 into the loan structure. That's right. Which 00:05:48.699 --> 00:05:51.399 is why APR is the legally defined borrowing cost. 00:05:51.759 --> 00:05:54.560 But when we're looking at APY on deposit accounts, 00:05:54.759 --> 00:05:56.939 we have to remember a really crucial limitation 00:05:56.939 --> 00:05:59.040 that the sources point out. Yeah, what's that? 00:05:59.220 --> 00:06:02.759 The APY tells you the gross interest yield. It 00:06:02.759 --> 00:06:04.579 does not account for the possibility of account 00:06:04.579 --> 00:06:07.660 fees. Ah, that is a critical nuance. It's huge. 00:06:07.759 --> 00:06:10.839 If you deposit, say, $10 ,000 into a high -yield 00:06:10.839 --> 00:06:13.720 savings account that's advertising a 5 .00 per 00:06:13.720 --> 00:06:16.899 se or PPI. Sounds great. It does. But what if 00:06:16.899 --> 00:06:19.879 that account charges you a $10 monthly maintenance 00:06:19.879 --> 00:06:22.240 fee because you didn't meet some minimum balance 00:06:22.240 --> 00:06:25.850 rule? That's $120 over the year. Right. And that 00:06:25.850 --> 00:06:28.209 fee is subtracted from your cash balance, but 00:06:28.209 --> 00:06:31.870 it is not factored into the advertised 5 .00 00:06:31.870 --> 00:06:36.490 % APY calculation. The APY only measures the 00:06:36.490 --> 00:06:39.829 interest yield, not the net gain. So if you have 00:06:39.829 --> 00:06:42.990 two accounts with identical APYs, say... 5 % 00:06:42.990 --> 00:06:45.750 on both. But one is a zero fee account and the 00:06:45.750 --> 00:06:48.110 other has unavoidable monthly fees. The zero 00:06:48.110 --> 00:06:50.949 fee account is superior, no question. Your net 00:06:50.949 --> 00:06:53.449 return will be higher even though the APY figure 00:06:53.449 --> 00:06:55.910 is the same. The APY tells you the earning potential, 00:06:56.009 --> 00:06:57.949 but you always have to do that extra step yourself. 00:06:58.110 --> 00:07:00.189 You have to deduct any service charges or fees 00:07:00.189 --> 00:07:02.970 to figure out your true net percentage gain. 00:07:03.149 --> 00:07:05.610 Always. It's the fine print that APY doesn't 00:07:05.610 --> 00:07:07.790 capture. This brings us to where the math really 00:07:07.790 --> 00:07:09.790 meets strategy. I want to circle back to something 00:07:09.790 --> 00:07:12.100 we touched on earlier. Why do banks banks almost 00:07:12.100 --> 00:07:14.959 always choose to quote the APY and not the nominal 00:07:14.959 --> 00:07:17.259 rate when they're advertising savings products. 00:07:17.420 --> 00:07:19.680 Why not just use the simplest number? It's purely 00:07:19.680 --> 00:07:21.959 a strategic choice, and it's rooted really deeply 00:07:21.959 --> 00:07:24.620 in consumer psychology and the constant competition 00:07:24.620 --> 00:07:26.879 for funds. They need our deposits to fund their 00:07:26.879 --> 00:07:29.829 loans. They do. And to make their deposit products 00:07:29.829 --> 00:07:33.129 look as appealing as humanly possible, they leverage 00:07:33.129 --> 00:07:35.790 the math of compounding to their advantage. And 00:07:35.790 --> 00:07:37.949 the simple truth that the math dictates is that 00:07:37.949 --> 00:07:40.870 the APY will almost always be the higher, more 00:07:40.870 --> 00:07:43.529 attractive number. Almost always, unless compounding 00:07:43.529 --> 00:07:46.189 is only once a year. Because of that effect of 00:07:46.189 --> 00:07:48.949 interest earning interest, the APY represents 00:07:48.949 --> 00:07:51.389 the customer getting a numerically higher return 00:07:51.389 --> 00:07:54.230 than the nominal rate would suggest. So when 00:07:54.230 --> 00:07:56.509 a customer is looking at two banks side by side, 00:07:56.720 --> 00:07:59.220 Even a tiny difference created by compounding 00:07:59.220 --> 00:08:01.240 can provide a distinct marketing edge for the 00:08:01.240 --> 00:08:04.120 bank that's quoting the higher APY figure. Let's 00:08:04.120 --> 00:08:05.879 use that concrete example from the source material. 00:08:06.040 --> 00:08:08.660 It really puts a number on this. So imagine a 00:08:08.660 --> 00:08:10.779 certificate of deposit. Let's say the simple 00:08:10.779 --> 00:08:13.060 rate, the internal rate. which is technically 00:08:13.060 --> 00:08:17.980 the APR, is 4 .65%. OK, 4 .65%. Now, what happens 00:08:17.980 --> 00:08:20.160 if that interest is compounded, say, monthly? 00:08:20.500 --> 00:08:23.180 That's 12 compounding periods in a year. Right. 00:08:23.279 --> 00:08:25.759 And because of that monthly compounding, the 00:08:25.759 --> 00:08:28.279 institution is legally allowed to advertise that 00:08:28.279 --> 00:08:32.720 product not as 4 .65%, but as a 4 .75 % APY. 00:08:32.980 --> 00:08:36.360 That's a 10 basis point difference, a tenth of 00:08:36.360 --> 00:08:38.919 a percent. And it's derived purely from the mathematical 00:08:38.919 --> 00:08:42.429 reality of compounding 12 times a year. It might 00:08:42.429 --> 00:08:44.990 seem trivial on a single customer's thousand 00:08:44.990 --> 00:08:47.830 dollar deposit, but if you're a national bank 00:08:47.830 --> 00:08:50.769 trying to attract Billions of dollars in deposits. 00:08:51.210 --> 00:08:54.389 That 10 basis point advantage is immense. It's 00:08:54.389 --> 00:08:56.649 a huge competitive advantage. Consumers naturally 00:08:56.649 --> 00:08:58.730 gravitate toward the higher number when they're 00:08:58.730 --> 00:09:01.730 comparing two similar products. The bank is just 00:09:01.730 --> 00:09:04.450 legally stating the most favorable rate possible, 00:09:04.669 --> 00:09:06.730 the one that accounts for the maximum earning 00:09:06.730 --> 00:09:09.250 potential over a year. It creates this powerful 00:09:09.250 --> 00:09:12.570 incentive for banks to compound interest as frequently 00:09:12.570 --> 00:09:15.370 as they can, doesn't it? It does. Daily compounding 00:09:15.370 --> 00:09:17.769 is the most common today for exactly that reason. 00:09:24.540 --> 00:09:28.659 So this all reinforces why, as a depositor, you 00:09:28.659 --> 00:09:32.039 should only be comparing the APY figures, never 00:09:32.039 --> 00:09:35.259 the nominal APR. The APY is the only number that 00:09:35.259 --> 00:09:37.340 reflects the full reality of how your money is 00:09:37.340 --> 00:09:39.879 going to grow because of that frequency. And, 00:09:39.879 --> 00:09:43.149 you know, to put a fine point on it. If you ever 00:09:43.149 --> 00:09:46.070 see a bank advertising a deposit rate using the 00:09:46.070 --> 00:09:48.830 term APR, that should be an immediate red flag. 00:09:49.049 --> 00:09:51.789 Really? Why? Because it's almost certainly an 00:09:51.789 --> 00:09:54.370 attempt to be opaque. They're hiding the true 00:09:54.370 --> 00:09:57.490 effective return. If you're comparing two products, 00:09:57.629 --> 00:10:00.370 one quoting 4 .6 % APR and the other quoting 00:10:00.370 --> 00:10:04.279 4 .7 % APY. You just can't make a decision. Not 00:10:04.279 --> 00:10:06.159 until you find out the compounding frequency 00:10:06.159 --> 00:10:08.259 for that APR product and do the math yourself 00:10:08.259 --> 00:10:11.139 to find its APY. Exactly. Without that normalized 00:10:11.139 --> 00:10:13.460 figure, you are absolutely comparing apples to 00:10:13.460 --> 00:10:15.440 oranges, and you can be sure the bank knows it. 00:10:15.559 --> 00:10:17.179 All right. To really get this, we have to go 00:10:17.179 --> 00:10:19.700 beyond the marketing and get into the pure mathematics 00:10:19.700 --> 00:10:22.590 that governs this. Yes. We need to look under 00:10:22.590 --> 00:10:25.669 the hood. This calculation is universal. It applies 00:10:25.669 --> 00:10:27.409 whether we're talking about U .S. consumer banking 00:10:27.409 --> 00:10:29.649 or international finance. So let's look at the 00:10:29.649 --> 00:10:32.190 standard mathematical definition, the effective 00:10:32.190 --> 00:10:34.250 interest rate formula that's used to calculate 00:10:34.250 --> 00:10:38.370 APY globally. Okay. The formula, it might look 00:10:38.370 --> 00:10:40.490 a little complex at first glance, but it's really 00:10:40.490 --> 00:10:43.110 just a tool for annualizing any periodic rate. 00:10:43.559 --> 00:10:45.700 It's designed to calculate the actual return 00:10:45.700 --> 00:10:48.299 on an investment over one year, given the stated 00:10:48.299 --> 00:10:51.100 rate and how often it compounds. The formula 00:10:51.100 --> 00:10:56.299 is APY equals, in parentheses, 1 plus INOM divided 00:10:56.299 --> 00:10:59.399 by N. And that whole thing is raised to the power 00:10:59.399 --> 00:11:02.039 of N. And then you subtract 1. OK, let's break 00:11:02.039 --> 00:11:03.379 down those variables because that's where the 00:11:03.379 --> 00:11:06.279 story is. We'll start with INOM. INOM, that stands 00:11:06.279 --> 00:11:08.759 for the nominal interest rate. This is the basic 00:11:08.759 --> 00:11:12.080 simple rate, the 4 .65 % from our last example. 00:11:12.379 --> 00:11:14.379 And we always express it as a decimal for the 00:11:14.379 --> 00:11:17.539 math, right? So 5 % becomes 0 .05. Correct. And 00:11:17.539 --> 00:11:20.620 that nominal rate on its own is, well, it's kind 00:11:20.620 --> 00:11:22.669 of dishonest. in a world of compounding because 00:11:22.669 --> 00:11:24.929 it doesn't account for time or frequency. And 00:11:24.929 --> 00:11:27.149 that's where the variable n comes in. n is the 00:11:27.149 --> 00:11:29.389 engine of compounding. It represents the number 00:11:29.389 --> 00:11:32.110 of compounding periods per year. This is the 00:11:32.110 --> 00:11:33.850 frequency. So if your interest is calculated 00:11:33.850 --> 00:11:38.090 monthly, n is 12. Daily, n is 365. Quarterly, 00:11:38.090 --> 00:11:41.809 n is 4. And so on. So if n is the engine, the 00:11:41.809 --> 00:11:43.750 structure of the formula shows us how it works. 00:11:44.830 --> 00:11:47.409 Inside the parentheses, you're dividing inom 00:11:47.409 --> 00:11:50.360 by n. Why that step? We're converting the annual 00:11:50.360 --> 00:11:53.320 nominal rate into the periodic rate. So if your 00:11:53.320 --> 00:11:55.820 annual nominal rate is 12 % and you compound 00:11:55.820 --> 00:11:58.519 monthly, so n is 12. Then the rate for each month 00:11:58.519 --> 00:12:02.419 is 1%, 12 divided by 12. Exactly. This makes 00:12:02.419 --> 00:12:05.000 sure that only the correct fraction of the total 00:12:05.000 --> 00:12:07.480 nominal interest is applied during each specific 00:12:07.480 --> 00:12:10.000 compounding period. I see. And then we take that 00:12:10.000 --> 00:12:12.700 whole term, the 1 plus the periodic rate, and 00:12:12.700 --> 00:12:15.480 we raise it to the power of n, that exponent. 00:12:16.279 --> 00:12:18.419 That's the key to compounding, isn't it? That's 00:12:18.419 --> 00:12:21.080 the whole magic right there. The exponent n mathematically 00:12:21.080 --> 00:12:24.100 captures the idea of repeatedly reinvesting the 00:12:24.100 --> 00:12:25.860 interest you've earned over all those periods 00:12:25.860 --> 00:12:28.080 within the year. It's like a shortcut for a repetitive 00:12:28.080 --> 00:12:30.879 process. It is. It simulates that cycle. Money 00:12:30.879 --> 00:12:33.440 goes in. Interest is earned. That interest is 00:12:33.440 --> 00:12:35.399 added to the principal. And then in the next 00:12:35.399 --> 00:12:37.559 period, that slightly larger principal earns 00:12:37.559 --> 00:12:40.289 interest again. The exponent lets us calculate 00:12:40.289 --> 00:12:42.830 the final value of that entire iterative process 00:12:42.830 --> 00:12:45.669 in one single step. And then at the very end, 00:12:45.710 --> 00:12:48.590 we subtract one. We subtract one because we only 00:12:48.590 --> 00:12:50.649 want to see the percentage gain above our initial 00:12:50.649 --> 00:12:52.549 principle, which is represented by that one. 00:12:52.769 --> 00:12:56.409 So what here? The big insight, it seems really 00:12:56.409 --> 00:12:59.409 profound, it validates the entire reason we use 00:12:59.409 --> 00:13:02.990 APY. Because the math shows that the actual return, 00:13:03.230 --> 00:13:06.950 the APY, is always greater than the nominal rate. 00:13:07.279 --> 00:13:09.659 i know as long as n is greater than one as long 00:13:09.659 --> 00:13:11.419 as you compound more than once a year it's a 00:13:11.419 --> 00:13:14.220 mathematical necessity If interest is compounded 00:13:14.220 --> 00:13:16.659 more than once, APY must be higher than the nominal 00:13:16.659 --> 00:13:19.200 rate. And the more frequently you compound, the 00:13:19.200 --> 00:13:22.100 larger N is, the greater the result will be. 00:13:22.220 --> 00:13:23.519 But there's a limit to that, right? There is. 00:13:23.620 --> 00:13:25.779 And this is an important point. The increase, 00:13:25.940 --> 00:13:29.480 it slows down rapidly as N gets very large. The 00:13:29.480 --> 00:13:32.159 extra return you get from moving from, say, quarterly 00:13:32.159 --> 00:13:35.480 compounding where N is 4 to monthly where N is 00:13:35.480 --> 00:13:37.700 12. That's a much bigger jump. It's a far greater 00:13:37.700 --> 00:13:40.419 jump proportionally than the extra return you 00:13:40.419 --> 00:13:42.710 get from moving from monthly N equals. 12 to 00:13:42.710 --> 00:13:45.590 daily and equals 365. You get diminishing returns 00:13:45.590 --> 00:13:48.330 from frequency. That slowdown is really important. 00:13:48.549 --> 00:13:51.149 And it leads us right into the second big lesson 00:13:51.149 --> 00:13:53.309 from the sources here, which is the impact of 00:13:53.309 --> 00:13:56.639 the rate size itself. Yes. The effect of compounding 00:13:56.639 --> 00:14:00.120 isn't linear. Its power is hugely dependent on 00:14:00.120 --> 00:14:02.960 how large that nominal rate, INOM, is to begin 00:14:02.960 --> 00:14:05.679 with. This is such a critical realization for 00:14:05.679 --> 00:14:08.320 anyone evaluating financial products. It shows 00:14:08.320 --> 00:14:10.740 that compounding isn't always the big, scary, 00:14:10.759 --> 00:14:13.200 or wonderful thing we think it is. We have to 00:14:13.200 --> 00:14:15.659 look at the numbers to see this nonlinear effect 00:14:15.659 --> 00:14:18.080 play out. So let's start with that extreme, purely 00:14:18.080 --> 00:14:20.679 theoretical example from the sources. It's meant 00:14:20.679 --> 00:14:23.500 to illustrate the exponential potential. If you 00:14:23.500 --> 00:14:25.659 had a nominal interest rate... of 100%. Which 00:14:25.659 --> 00:14:28.500 is 1 .00 as a decimal, an insane rate, but just 00:14:28.500 --> 00:14:30.379 for the math. Right. And you compounded that 00:14:30.379 --> 00:14:33.080 frequently, let's say monthly, so n is 12. What 00:14:33.080 --> 00:14:36.019 would the APY be? The APY would jump to approximately 00:14:36.019 --> 00:14:40.220 161%. Wow. But if we let that compounding go 00:14:40.220 --> 00:14:44.080 daily, n equals 365, or even continuously, the 00:14:44.080 --> 00:14:47.899 APY approaches about 171 .8%. That is a massive 00:14:47.899 --> 00:14:49.740 difference and an additional return of more than 00:14:49.740 --> 00:14:52.500 70 percentage points. And that's purely from 00:14:52.500 --> 00:14:54.960 the frequency of compounding. At a 100 % nominal 00:14:54.960 --> 00:14:57.500 rate, the compounding effect is almost three 00:14:57.500 --> 00:14:59.399 quarters the size of the original rate itself. 00:14:59.500 --> 00:15:02.059 It's incredibly powerful. And the reason it's 00:15:02.059 --> 00:15:04.960 so startling is because when your principal is 00:15:04.960 --> 00:15:07.940 doubling in value over a year anyway, having 00:15:07.940 --> 00:15:10.279 those interest payments hit your account early 00:15:10.279 --> 00:15:13.830 and start earning their own interest. just creates 00:15:13.830 --> 00:15:16.970 this incredible feedback loop. With a huge nominal 00:15:16.970 --> 00:15:20.090 rate, every compounding event delivers a big 00:15:20.090 --> 00:15:21.870 chunk of money back to the principal, which then 00:15:21.870 --> 00:15:24.190 immediately accelerates the next return. So it 00:15:24.190 --> 00:15:25.769 shows that when the starting rate is extremely 00:15:25.769 --> 00:15:29.850 high, the frequency, the end, becomes just unbelievably 00:15:29.850 --> 00:15:33.149 powerful. It leads to true exponential growth. 00:15:33.429 --> 00:15:35.610 It's more of an intellectual exercise for deposit 00:15:35.610 --> 00:15:38.129 accounts, but it's vital for understanding high 00:15:38.129 --> 00:15:40.950 -risk investing or, on the flip side, high -interest 00:15:40.950 --> 00:15:43.480 debt. where that same compounding works against 00:15:43.480 --> 00:15:46.120 you with brutal efficiency. Let's bring it back 00:15:46.120 --> 00:15:48.419 down to earth. Let's talk about rates that actually 00:15:48.419 --> 00:15:50.980 exist in the consumer world moderate or small 00:15:50.980 --> 00:15:53.700 rates to see the practical application. Okay, 00:15:53.740 --> 00:15:56.879 let's take a nominal rate of 5 % or 0 .05. That's 00:15:56.879 --> 00:15:58.799 a very strong rate for a high -yield savings 00:15:58.799 --> 00:16:02.059 account today. And let's assume daily compounding, 00:16:02.059 --> 00:16:04.580 the industry standard. With daily compounding, 00:16:04.580 --> 00:16:07.940 that 5 % nominal rate corresponds to an APY of 00:16:07.940 --> 00:16:11.679 approximately 5 .1267%. So the difference is 00:16:11.679 --> 00:16:15.080 about 13 basis points. It's significant if you 00:16:15.080 --> 00:16:17.659 have large sums of money, but it's nothing like 00:16:17.659 --> 00:16:20.460 that 71 -point jump we saw with the 100 % rate. 00:16:20.580 --> 00:16:22.860 It's a marginal increase relative to the size 00:16:22.860 --> 00:16:24.860 of the original rate. And what if we take it 00:16:24.860 --> 00:16:27.340 down even further to a nominal rate of, say, 00:16:27.480 --> 00:16:30.450 1 %? which is pretty typical for a standard checking 00:16:30.450 --> 00:16:33.049 or savings product. With daily compounding, that 00:16:33.049 --> 00:16:36.309 1 % nominal rate corresponds to an APY of about 00:16:36.309 --> 00:16:40.029 1 .005%. That is a difference of only half a 00:16:40.029 --> 00:16:42.690 basis point. It's barely measurable in dollar 00:16:42.690 --> 00:16:45.230 terms unless you have millions and millions of 00:16:45.230 --> 00:16:47.450 dollars on deposit. And the analysis here is 00:16:47.450 --> 00:16:49.929 so crucial for a discerning consumer. If the 00:16:49.929 --> 00:16:52.429 nominal rate and its corresponding APY are small, 00:16:52.570 --> 00:16:54.940 they're very, very nearly equal. the effect of 00:16:54.940 --> 00:16:57.120 compounding becomes drastically less pronounced 00:16:57.120 --> 00:16:59.679 as the nominal rate goes down. So for you, as 00:16:59.679 --> 00:17:02.059 a consumer, comparing two low -rate accounts, 00:17:02.179 --> 00:17:06.759 say one is 0 .9 % and the other is 1 .1%, the 00:17:06.759 --> 00:17:08.640 difference in the nominal rate itself is almost 00:17:08.640 --> 00:17:10.440 certainly more important than worrying about 00:17:10.440 --> 00:17:12.799 whether one compounds daily and the other compounds 00:17:12.799 --> 00:17:15.200 monthly. So let me get this straight. If you're 00:17:15.200 --> 00:17:18.349 choosing between two banks, and one advertises 00:17:18.349 --> 00:17:22.789 1 .01 % APY compounded daily, and the other advertises 00:17:22.789 --> 00:17:26.430 1 .01 % APY compounded quarterly. The second 00:17:26.430 --> 00:17:28.769 account is definitively better. Even with the 00:17:28.769 --> 00:17:31.109 lower compounding frequency? Yes, because at 00:17:31.109 --> 00:17:33.950 such small rates, that tiny difference in the 00:17:33.950 --> 00:17:37.329 nominal rate in INOM completely outweighs the 00:17:37.329 --> 00:17:39.569 frequency of N. The critical lesson is one of 00:17:39.569 --> 00:17:42.079 scale then. You need a high nominal rate for 00:17:42.079 --> 00:17:43.980 the frequency of compounding to deliver those 00:17:43.980 --> 00:17:46.480 truly massive relative returns. That mathematical 00:17:46.480 --> 00:17:49.059 understanding is exactly what empowers you to 00:17:49.059 --> 00:17:51.519 critically evaluate an advertised rate. You now 00:17:51.519 --> 00:17:54.279 know that while a bank might splash daily compounding 00:17:54.279 --> 00:17:57.140 all over its ads as a big feature, if the rate 00:17:57.140 --> 00:17:59.480 itself is negligible, then the feature offers 00:17:59.480 --> 00:18:01.640 a negligible practical benefit to your bottom 00:18:01.640 --> 00:18:03.599 line. Okay, let's get a little deeper into the 00:18:03.599 --> 00:18:06.440 theory now. Specifically looking at what happens 00:18:06.440 --> 00:18:10.410 when that compounding variable n, becomes mathematically 00:18:10.410 --> 00:18:13.190 as large as you can possibly imagine. We're talking 00:18:13.190 --> 00:18:15.470 about continuous compounding. It's a mathematical 00:18:15.470 --> 00:18:18.150 ideal, but it has a very practical approximation 00:18:18.150 --> 00:18:21.769 in modern finance. When n approaches infinity, 00:18:22.089 --> 00:18:25.009 so compounding is happening, what, every microsecond? 00:18:25.289 --> 00:18:27.869 Theoretically, yes. When that happens, the standard 00:18:27.869 --> 00:18:30.410 formula we've been using reaches a limit. And 00:18:30.410 --> 00:18:32.650 this limit is defined by a natural constant, 00:18:32.809 --> 00:18:35.670 which gives us a much simpler approximation for 00:18:35.670 --> 00:18:39.369 very large n. And what's that formula? It's APY 00:18:39.369 --> 00:18:42.289 is approximately equal to e raised to the power 00:18:42.289 --> 00:18:44.789 of i -nom minus 1. Okay, so we're introducing 00:18:44.789 --> 00:18:47.130 the constant e, which is Euler's number. The 00:18:47.130 --> 00:18:50.910 base of natural logarithms, roughly 2 .718. Correct. 00:18:51.009 --> 00:18:53.609 And this represents the ultimate limit of compounding. 00:18:53.609 --> 00:18:56.430 You simply cannot compound interest any faster 00:18:56.430 --> 00:18:58.990 or get a higher return than what this e formula 00:18:58.990 --> 00:19:01.490 provides for a given nominal rate. It's the mathematical 00:19:01.490 --> 00:19:04.589 ceiling. It is. The formula follows the strict 00:19:04.589 --> 00:19:08.089 mathematical definition of E as a limit. And 00:19:08.089 --> 00:19:10.390 in practical finance, this approximation is used 00:19:10.390 --> 00:19:12.549 all the time because if you're compounding daily, 00:19:12.789 --> 00:19:16.150 so N is 365. Which is pretty large. It's very 00:19:16.150 --> 00:19:18.750 large. And the result you get from the standard 00:19:18.750 --> 00:19:21.589 APY formula is so incredibly close to the result 00:19:21.589 --> 00:19:23.890 from the continuous compounding formula that 00:19:23.890 --> 00:19:25.490 this approximation just becomes mathematically 00:19:25.490 --> 00:19:27.769 convenient and highly accurate for modeling. 00:19:28.009 --> 00:19:31.089 So if my bank is compounding daily, the difference 00:19:31.089 --> 00:19:34.349 in APY why between using N365 in the big formula 00:19:34.349 --> 00:19:37.369 and just using the E approximation is what? A 00:19:37.369 --> 00:19:40.029 fraction of a basis point. Typically, yes. It's 00:19:40.029 --> 00:19:42.630 often so small it rounds to the exact same figure. 00:19:43.039 --> 00:19:45.740 And it illustrates a key point. Once you get 00:19:45.740 --> 00:19:48.619 to daily compounding, you are already very, very 00:19:48.619 --> 00:19:50.839 close to the theoretical maximum possible return. 00:19:51.019 --> 00:19:52.819 There's not much more juice to squeeze. This 00:19:52.819 --> 00:19:56.019 really solidifies the math then. Daily compounding 00:19:56.019 --> 00:19:58.000 is the industry standard for deposit accounts 00:19:58.000 --> 00:20:00.960 because it maximizes that advertised APY number 00:20:00.960 --> 00:20:03.460 for marketing. Giving the bank the best edge 00:20:03.460 --> 00:20:05.839 possible while essentially reaching the mathematical 00:20:05.839 --> 00:20:09.420 ceiling of what compounding can do. This groundwork 00:20:09.420 --> 00:20:12.309 proves that APY is the essential measure. Because 00:20:12.309 --> 00:20:15.789 it quantifies and it normalizes the power of 00:20:15.789 --> 00:20:18.390 N. We've established the universal math behind 00:20:18.390 --> 00:20:20.769 APY, and we've seen the marketing incentives 00:20:20.769 --> 00:20:24.049 that flow from it. Now we need to pivot and talk 00:20:24.049 --> 00:20:26.269 about how the real world of consumer banking, 00:20:26.390 --> 00:20:28.930 especially in the U .S., imposes extremely specific 00:20:28.930 --> 00:20:31.750 rules on how that number is calculated and presented. 00:20:32.009 --> 00:20:34.829 Right. Because when math meets money, regulators 00:20:34.829 --> 00:20:36.809 inevitably step in. And what's the regulatory 00:20:36.809 --> 00:20:39.109 context here? What's the main piece of legislation 00:20:39.109 --> 00:20:41.559 we need to know about? The regulatory environment 00:20:41.559 --> 00:20:44.059 for this is crucial. In the United States, a 00:20:44.059 --> 00:20:46.200 precise method for calculating and advertising 00:20:46.200 --> 00:20:49.180 APY and the related concept of annual percentage 00:20:49.180 --> 00:20:52.559 yield earned is strictly regulated by the FDIC 00:20:52.559 --> 00:20:55.500 Truth in Savings Act. TISA. And it was passed 00:20:55.500 --> 00:20:58.119 in 1991. That's the one. Why was this needed 00:20:58.119 --> 00:21:00.160 in the first place? What was the financial world 00:21:00.160 --> 00:21:02.759 like before 1991 that made the government have 00:21:02.759 --> 00:21:05.460 to step in and standardize all this? The environment 00:21:05.460 --> 00:21:08.660 before 1991 was, um... Frankly, it was chaotic 00:21:08.660 --> 00:21:12.299 and it was ripe for consumer exploitation. Banks 00:21:12.299 --> 00:21:14.519 had incredible leeway in how they define their 00:21:14.519 --> 00:21:17.500 key terms. Like what? Well, for example, some 00:21:17.500 --> 00:21:20.079 banks would quote their interest using a 360 00:21:20.079 --> 00:21:23.700 day year instead of a 365 day year. Why would 00:21:23.700 --> 00:21:25.880 they do that? Because it artificially inflated 00:21:25.880 --> 00:21:27.940 the daily rate they claim to be paying. They 00:21:27.940 --> 00:21:30.859 divide the annual rate by a smaller number. Others 00:21:30.859 --> 00:21:33.480 used really complex, inconsistent compounding 00:21:33.480 --> 00:21:36.259 methods that made comparing two similar accounts 00:21:36.259 --> 00:21:38.799 across different banks nearly impossible. So 00:21:38.799 --> 00:21:40.720 even if a bank was offering what looked like 00:21:40.720 --> 00:21:43.339 a higher nominal rate, a smart consumer still 00:21:43.339 --> 00:21:45.500 had no clear way to know if that higher rate 00:21:45.500 --> 00:21:47.579 was just being canceled out by less frequent 00:21:47.579 --> 00:21:49.599 compounding or some kind of misleading calculation. 00:21:50.160 --> 00:21:52.380 Precisely. The lack of standardization created 00:21:52.380 --> 00:21:55.480 immense confusion for everybody. TISA was enacted 00:21:55.480 --> 00:21:58.819 specifically to promote clarity, to standardize 00:21:58.819 --> 00:22:01.440 the definitions of key terms like interest and 00:22:01.440 --> 00:22:04.220 annual percentage yield, and to prevent confusing 00:22:04.220 --> 00:22:06.819 or deceptive marketing. So it dictates exactly 00:22:06.819 --> 00:22:09.319 how banks have to communicate earnings. It makes 00:22:09.319 --> 00:22:11.099 sure that when two different banks advertise 00:22:11.099 --> 00:22:14.099 an APY, they're both using the exact same legal 00:22:14.099 --> 00:22:16.920 standard to calculate it. That context is vital. 00:22:17.079 --> 00:22:20.160 The Truth in Savings Act is essentially the regulatory 00:22:20.160 --> 00:22:22.759 answer to all those marketing strategies we just 00:22:22.759 --> 00:22:25.680 talked about. It forces the whole industry to 00:22:25.680 --> 00:22:28.480 use one standardized method. Let's look at the 00:22:28.480 --> 00:22:31.460 cornerstone of this law, then. The specific regulatory 00:22:31.460 --> 00:22:34.079 definition of APY that the board established. 00:22:34.599 --> 00:22:37.500 This definition is what sets that crucial fixed 00:22:37.500 --> 00:22:40.240 standard that every single U .S. financial institution 00:22:40.240 --> 00:22:44.079 has to follow. And the sources quote the legal 00:22:44.079 --> 00:22:47.119 mandate explicitly. It says the term annual percentage 00:22:47.119 --> 00:22:49.380 yield means the total amount of interest that 00:22:49.380 --> 00:22:52.200 would be received on a $100 deposit. Based on 00:22:52.200 --> 00:22:54.319 the annual rate of simple interest and the frequency 00:22:54.319 --> 00:22:57.819 of compounding for a 365 -day period. Expressed 00:22:57.819 --> 00:22:59.819 as a percentage calculated by a method which 00:22:59.819 --> 00:23:01.799 shall be prescribed by the board in regulations. 00:23:02.160 --> 00:23:04.880 Every single one of those bolded phrases is significant. 00:23:05.140 --> 00:23:07.880 By setting a fixed standardized comparison point 00:23:07.880 --> 00:23:11.740 $100 deposit over a 365 -day term, the regulation 00:23:11.740 --> 00:23:14.220 achieves perfect uniformity. But wait a second. 00:23:14.299 --> 00:23:18.339 If I'm planning to deposit, say, $10 ,000, why 00:23:18.339 --> 00:23:22.079 does the regulation focus on a tiny... $100 deposit. 00:23:22.599 --> 00:23:24.900 Does that change the rate? It doesn't change 00:23:24.900 --> 00:23:27.839 the rate itself, no. But it ensures that the 00:23:27.839 --> 00:23:30.059 rate calculation is consistent across the board. 00:23:30.259 --> 00:23:32.980 The $100 figure is just a benchmark principle. 00:23:33.339 --> 00:23:35.400 So it doesn't matter if I'm opening a $10 ,000 00:23:35.400 --> 00:23:38.839 savings account or a $500 CD? Doesn't matter. 00:23:38.920 --> 00:23:42.160 The advertised APY has to reflect that standardized 00:23:42.160 --> 00:23:48.109 scenario. $100 held for 365 days using that product's 00:23:48.109 --> 00:23:51.109 specific compounding rules. It takes the consumer's 00:23:51.109 --> 00:23:53.569 actual deposit amount out of the comparison metric 00:23:53.569 --> 00:23:56.549 entirely. It isolates the rate itself for a true 00:23:56.549 --> 00:23:58.670 apples -to -apples comparison. That's the goal. 00:23:58.769 --> 00:24:00.630 So it also means that whether the product being 00:24:00.630 --> 00:24:03.750 advertised is a short -term three -month CD or 00:24:03.750 --> 00:24:06.589 a long -term five -year product, the APY must 00:24:06.589 --> 00:24:09.250 always be presented as if the term were exactly 00:24:09.250 --> 00:24:13.299 one year. 365 days. It's an enforced annualization. 00:24:13.299 --> 00:24:15.619 That is the core legal mandate for transparency. 00:24:16.039 --> 00:24:19.779 It ensures that the APY truly represents an annualized 00:24:19.779 --> 00:24:22.279 comparable return, no matter what the product's 00:24:22.279 --> 00:24:24.720 actual term length is. The regulators are basically 00:24:24.720 --> 00:24:27.619 saying, if you compounded this product's rate 00:24:27.619 --> 00:24:31.519 structure for exactly 365 days on exactly $100, 00:24:32.299 --> 00:24:33.859 This is the percentage of turn you would get. 00:24:33.960 --> 00:24:36.079 And everyone has to play by that same rule. Okay. 00:24:36.220 --> 00:24:38.299 This is where we need to bridge the gap between 00:24:38.299 --> 00:24:40.299 that universal math we talked about in Section 00:24:40.299 --> 00:24:44.279 2 and this regulated world. The general formula 00:24:44.279 --> 00:24:46.940 assumes the nominal rate is already annualized, 00:24:47.000 --> 00:24:49.740 but the U .S. regulation, especially for products 00:24:49.740 --> 00:24:52.160 shorter than a year, needs a slightly different 00:24:52.160 --> 00:24:55.200 approach. It does. The TISA regulation needed 00:24:55.200 --> 00:24:57.660 a formula that could take a return earned over, 00:24:57.740 --> 00:25:01.309 say, just 90 days and accurately and legally 00:25:01.309 --> 00:25:04.849 extrapolate that return to a full 365 day figure 00:25:04.849 --> 00:25:07.750 so the board prescribed a specific formula that 00:25:07.750 --> 00:25:09.970 bakes the account's actual term right into the 00:25:09.970 --> 00:25:11.789 calculation that's right and that formula is 00:25:11.789 --> 00:25:14.390 apy equals 100 times and then in brackets you 00:25:14.390 --> 00:25:16.430 have in parentheses one plus the interest divided 00:25:16.430 --> 00:25:18.690 by the principal okay and you raise that to the 00:25:18.690 --> 00:25:21.369 power of 365 divided by the days in the term 00:25:21.369 --> 00:25:24.180 yeah and then finally you subtract one That looks 00:25:24.180 --> 00:25:26.460 similar to our general formula, but the variables 00:25:26.460 --> 00:25:29.079 are different. We're using interest and principal 00:25:29.079 --> 00:25:34.059 instead of i -nom and n. Why the change? This 00:25:34.059 --> 00:25:36.960 is the crucial mathematical difference. The Tessa 00:25:36.960 --> 00:25:40.259 formula is an annualization formula. It doesn't 00:25:40.259 --> 00:25:41.980 actually care about the number of compounding 00:25:41.980 --> 00:25:45.299 periods n. It cares about the result of the compounding 00:25:45.299 --> 00:25:48.380 over a specific time. So the bank first has to 00:25:48.380 --> 00:25:50.539 calculate the total dollar amount of interest 00:25:50.539 --> 00:25:53.460 earned during the actual term. Yes, the result 00:25:53.460 --> 00:25:55.920 of all their internal compounding events. And 00:25:55.920 --> 00:25:58.819 then they use that final dollar amount to annualize 00:25:58.819 --> 00:26:01.039 the rate according to the law. So let's define 00:26:01.039 --> 00:26:03.519 these TISA variables. Principal is the amount 00:26:03.519 --> 00:26:05.759 you deposit at the start. For the compliance 00:26:05.759 --> 00:26:08.160 test, it's that $100. Right. And then we have 00:26:08.160 --> 00:26:10.759 interest, which is the total dollar amount of 00:26:10.759 --> 00:26:12.960 interest actually earned on that principal for 00:26:12.960 --> 00:26:15.059 the term of the account. This number already 00:26:15.059 --> 00:26:17.059 has all the compounding effects baked into it. 00:26:17.140 --> 00:26:19.680 And finally, the critical factor for TESA compliance, 00:26:20.420 --> 00:26:23.599 days in term. This is the actual number of days 00:26:23.599 --> 00:26:25.480 the funds were held. So if you're calculating 00:26:25.480 --> 00:26:29.539 the APY for a 90 -day CD, days in term is 90. 00:26:30.119 --> 00:26:33.500 That fraction, 365 divided by days in term, is 00:26:33.500 --> 00:26:36.519 the annualizing factor. So if a 90 -day CD earns 00:26:36.519 --> 00:26:39.500 X dollars in interest, the formula figures out 00:26:39.500 --> 00:26:42.079 what percentage X is of your principal. And then 00:26:42.079 --> 00:26:44.920 it raises that to the power of roughly 4, 365 00:26:44.920 --> 00:26:48.700 divided by 90. It forces that short -term dollar 00:26:48.700 --> 00:26:51.220 gain to be expressed as a standardized annual 00:26:51.220 --> 00:26:54.420 rate. This level of specific regulation is fascinating. 00:26:54.660 --> 00:26:57.480 It's a formal, standardized way of saying, show 00:26:57.480 --> 00:26:59.339 us the actual money you earn during the term 00:26:59.339 --> 00:27:01.880 and we will mathematically extrapolate that real 00:27:01.880 --> 00:27:05.079 -world earning power into a 365 -day comparable 00:27:05.079 --> 00:27:07.940 percentage. This regulated approach is what institutions 00:27:07.940 --> 00:27:10.519 must use to calculate the APY they advertise 00:27:10.519 --> 00:27:12.640 to you. It ensures that whether the product is 00:27:12.640 --> 00:27:15.039 a 182 -day account or a full one -year account, 00:27:15.279 --> 00:27:17.759 the APY figure is derived from the exact same 00:27:17.759 --> 00:27:20.279 mathematical base. It guarantees consumer comparability 00:27:20.279 --> 00:27:22.309 under the law. It prevents banks from using their 00:27:22.309 --> 00:27:24.509 own little 360 -day calendar tricks like they 00:27:24.509 --> 00:27:26.549 did in the past. It does. Okay, now let's look 00:27:26.549 --> 00:27:28.930 at the practical side. The tool that you, the 00:27:28.930 --> 00:27:31.369 consumer, can actually use. The regulations don't 00:27:31.369 --> 00:27:34.069 just tell banks how to calculate APY. No, they 00:27:34.069 --> 00:27:37.289 also provide the algebraic inverse of that complex 00:27:37.289 --> 00:27:40.269 formula. And this inverse allows anyone, the 00:27:40.269 --> 00:27:43.170 institution or you, to calculate the dollar amount 00:27:43.170 --> 00:27:45.490 of interest you should earn based on the quoted 00:27:45.490 --> 00:27:48.730 regulated APY. This is the direct tool for verification. 00:27:49.440 --> 00:27:51.579 For planning. For holding your bank accountable. 00:27:51.759 --> 00:27:54.440 It is. The inverse formula is interest equals 00:27:54.440 --> 00:27:57.380 your principal times. And then in brackets you 00:27:57.380 --> 00:28:00.259 have, in parentheses, the APY divided by 100 00:28:00.259 --> 00:28:03.359 plus 1. Okay. Raised to the power of the days 00:28:03.359 --> 00:28:06.440 in the term divided by 365. And then again, subtract 00:28:06.440 --> 00:28:09.019 1. So what does this mean in real terms? This 00:28:09.019 --> 00:28:11.200 is the equation you can use to verify your earnings 00:28:11.200 --> 00:28:14.150 for any deposit amount and any term length. Think 00:28:14.150 --> 00:28:16.509 about it this way. A bank is required to quote 00:28:16.509 --> 00:28:20.809 you a specific APY. Let's say it's 5 .00%. Okay. 00:28:20.930 --> 00:28:23.410 And you open a six -month CD, which is roughly 00:28:23.410 --> 00:28:26.650 182 days, with $20 ,000. Yeah. You can plug in 00:28:26.650 --> 00:28:29.670 your principal, $20 ,000, the quoted 5 % APY, 00:28:29.670 --> 00:28:32.470 and the actual days in the term, 182. And the 00:28:32.470 --> 00:28:34.750 result of that calculation? The result will tell 00:28:34.750 --> 00:28:37.990 you, down to the penny, exactly the dollar amount 00:28:37.990 --> 00:28:39.950 of interest you should expect to see in your 00:28:39.950 --> 00:28:42.319 account at the end of that term. This transforms 00:28:42.319 --> 00:28:45.440 the APY from just some abstract, advertised percentage 00:28:45.440 --> 00:28:48.720 into a verifiable operational metric. It does. 00:28:48.839 --> 00:28:51.180 And if the dollar amount of interest you actually 00:28:51.180 --> 00:28:54.079 receive is less than what that formula says it 00:28:54.079 --> 00:28:57.720 should be, assuming no fees and no early withdrawals. 00:28:57.779 --> 00:28:59.380 Then the institution has either messed up the 00:28:59.380 --> 00:29:02.180 calculation or they have misrepresented the APY 00:29:02.180 --> 00:29:05.160 under the Truth in Savings Act. Exactly. This 00:29:05.160 --> 00:29:07.799 level of detail in the U .S. regulation just 00:29:07.799 --> 00:29:10.660 underscores the foundational importance of standardization. 00:29:10.960 --> 00:29:14.000 Without these specific TISA rules, banks could 00:29:14.000 --> 00:29:16.799 just rely on that confusing general math and 00:29:16.799 --> 00:29:19.660 use varying terms and principles, making consumer 00:29:19.660 --> 00:29:22.599 comparison useless. TISA makes sure that APY 00:29:22.599 --> 00:29:24.700 means the same thing for everyone, everywhere. 00:29:25.160 --> 00:29:27.680 It creates a regulated baseline for measuring 00:29:27.680 --> 00:29:30.140 effective return, and it empowers you with a 00:29:30.140 --> 00:29:33.240 clear tool for verification. It effectively removes 00:29:33.240 --> 00:29:36.000 that distracting debate about the frequency of 00:29:36.000 --> 00:29:38.920 compounding, that n variable, and it focuses 00:29:38.920 --> 00:29:41.680 you, the consumer, entirely on the resulting 00:29:41.680 --> 00:29:44.359 annual rate, which is the only figure that truly 00:29:44.359 --> 00:29:47.240 matters for comparison. That was a foundational 00:29:47.240 --> 00:29:50.000 deep dive into the annual percentage yield. It's 00:29:50.000 --> 00:29:52.140 a number we rely on every day, but we rarely 00:29:52.140 --> 00:29:54.799 understand the mechanics underneath. Let's quickly 00:29:54.799 --> 00:29:57.279 recap the key takeaways we solidified today. 00:29:57.660 --> 00:30:01.099 First and foremost, APY is the true normalized 00:30:01.099 --> 00:30:04.220 annualized rate. It's designed specifically for 00:30:04.220 --> 00:30:06.539 you to compare deposit accounts, and it acts 00:30:06.539 --> 00:30:08.779 as a common denominator for all these interest 00:30:08.779 --> 00:30:11.359 products, regardless of their internal compounding 00:30:11.359 --> 00:30:14.380 schedules. Second, we learned that APY is mathematically 00:30:14.380 --> 00:30:16.960 higher than the nominal APR for deposits because 00:30:16.960 --> 00:30:19.900 it accounts for compounding. But, and this was 00:30:19.900 --> 00:30:22.480 a key insight from the analysis, that difference 00:30:22.480 --> 00:30:24.640 is only really substantial when the nominal rate 00:30:24.640 --> 00:30:27.400 itself is high. For the small, real -world rates 00:30:27.400 --> 00:30:29.500 we usually see, the frequency of compounding, 00:30:29.500 --> 00:30:31.940 that N, provides a pretty negligible practical 00:30:31.940 --> 00:30:35.519 benefit. And finally, in the U .S., institutions 00:30:35.519 --> 00:30:38.519 have to use specific, standardized formulas that 00:30:38.519 --> 00:30:40.900 are rooted in the 1991 Truth in Savings Act. 00:30:41.210 --> 00:30:43.950 This regulation makes sure that the advertised 00:30:43.950 --> 00:30:47.150 APY is always based on that hypothetical $100 00:30:47.150 --> 00:30:50.670 deposit over a 365 -day period. Which ensures 00:30:50.670 --> 00:30:53.309 a consistent and legally verifiable basis for 00:30:53.309 --> 00:30:55.730 you to make comparisons. If we connect all this 00:30:55.730 --> 00:30:58.670 to the bigger picture. Understanding APY means 00:30:58.670 --> 00:31:01.190 understanding that the frequency of compounding 00:31:01.190 --> 00:31:04.170 that variable n we saw in the math can be a powerful 00:31:04.170 --> 00:31:06.769 feature, but only when it's paired with a high 00:31:06.769 --> 00:31:08.849 nominal rate. It teaches us to look past the 00:31:08.849 --> 00:31:11.509 initial percentage and ask exactly how often 00:31:11.509 --> 00:31:13.690 our money is being put back to work. And that 00:31:13.690 --> 00:31:16.160 understanding... It raises an important question, 00:31:16.259 --> 00:31:18.000 something for you to mull over now that you know 00:31:18.000 --> 00:31:20.779 the power of n. We focused heavily on APY and 00:31:20.779 --> 00:31:22.660 APR, which are the main terms in the U .S. But 00:31:22.660 --> 00:31:24.480 our source material mentioned related concepts 00:31:24.480 --> 00:31:27.059 used globally, like annual equivalent rate, or 00:31:27.059 --> 00:31:30.420 AER, and effective interest rate, EIR. So now 00:31:30.420 --> 00:31:32.180 that you're fully aware of the mathematical power 00:31:32.180 --> 00:31:35.900 of n, the compounding periods, and how that frequency 00:31:35.900 --> 00:31:38.950 dictates the effective return. Think about what 00:31:38.950 --> 00:31:41.250 other financial products might use these slightly 00:31:41.250 --> 00:31:43.609 different naming conventions. Maybe products 00:31:43.609 --> 00:31:46.490 outside of strict U .S. regulation or even complex 00:31:46.490 --> 00:31:48.869 international investing platforms. Could they 00:31:48.869 --> 00:31:52.910 use a term like AER or EIR to potentially obscure 00:31:52.910 --> 00:31:56.289 the exact frequency of compounding? And if an 00:31:56.289 --> 00:31:58.609 institution in another country only compounds, 00:31:58.789 --> 00:32:01.970 say, semi -annually but calls it an AER, how 00:32:01.970 --> 00:32:04.009 might that subtle difference in N impact your 00:32:04.009 --> 00:32:06.049 long -term returns compared to a product that 00:32:06.049 --> 00:32:08.569 you assume is compounding daily? The core principles 00:32:08.569 --> 00:32:11.750 are always identical. But the informed consumer 00:32:11.750 --> 00:32:13.869 must always seek out that underlying frequency, 00:32:14.049 --> 00:32:16.509 or at least confirm the legally mandated annualization 00:32:16.509 --> 00:32:18.730 method. That knowledge is your ultimate financial 00:32:18.730 --> 00:32:19.170 protection.