WEBVTT 00:00:00.000 --> 00:00:02.439 Welcome to the Deep Dive, where we take the sources 00:00:02.439 --> 00:00:05.240 you share with us, the deep research, the dense 00:00:05.240 --> 00:00:07.660 articles, the complicated spreadsheets, and cut 00:00:07.660 --> 00:00:09.300 straight through to the knowledge that truly 00:00:09.300 --> 00:00:12.000 matters. Today, we are tackling a concept that, 00:00:12.039 --> 00:00:14.460 depending on your perspective, is either the 00:00:14.460 --> 00:00:16.620 most powerful creator of generational wealth 00:00:16.620 --> 00:00:20.559 or, well, The silent accelerating mechanism of 00:00:20.559 --> 00:00:22.679 suffocating debt. That's right. We're diving 00:00:22.679 --> 00:00:25.160 into compound interest. We are. Our listeners 00:00:25.160 --> 00:00:27.300 shared a fascinating stack of sources detailing 00:00:27.300 --> 00:00:30.019 the mechanics, the mathematics, and the surprisingly 00:00:30.019 --> 00:00:33.280 long, often controversial history of this concept, 00:00:33.500 --> 00:00:36.320 a concept you often hear dubbed the eighth wonder 00:00:36.320 --> 00:00:39.259 of the world. And our mission today is, I think, 00:00:39.299 --> 00:00:41.649 pretty ambitious. I'd say so. We're going to 00:00:41.649 --> 00:00:44.429 take these dry mathematical constants, this detailed 00:00:44.429 --> 00:00:47.310 historical context and some frankly intimidating 00:00:47.310 --> 00:00:50.369 formulas and distill them into absolutely essential. 00:00:51.000 --> 00:00:53.439 actionable knowledge. So that by the end of this, 00:00:53.479 --> 00:00:55.780 you understand not just what compound interest 00:00:55.780 --> 00:00:58.679 is, but really how it operates. Exactly. How 00:00:58.679 --> 00:01:00.899 it operates from, say, ancient Babylonian accounting 00:01:00.899 --> 00:01:03.159 all the way up to the complex pricing of financial 00:01:03.159 --> 00:01:05.519 derivatives today. And we really need to drive 00:01:05.519 --> 00:01:09.040 home that how it operates part because compound 00:01:09.040 --> 00:01:12.879 interest is incredibly powerful because of its 00:01:12.879 --> 00:01:15.980 simplicity. At its core, it's just interest earned 00:01:15.980 --> 00:01:18.219 not only on the initial money you put down, the 00:01:18.219 --> 00:01:21.219 principal, but also on all the interest that 00:01:21.219 --> 00:01:23.280 is already accumulated. So the interest itself 00:01:23.280 --> 00:01:26.040 starts earning interest. It's like your money 00:01:26.040 --> 00:01:28.519 is constantly giving birth to more money, and 00:01:28.519 --> 00:01:30.439 then those babies also start producing cash. 00:01:31.159 --> 00:01:33.359 Precisely. That compounding effect is the source 00:01:33.359 --> 00:01:36.120 of its massive power. It's what drives that exponential 00:01:36.120 --> 00:01:38.739 growth in investments over decades. And it's 00:01:38.739 --> 00:01:40.780 also what makes the total repayment on, say, 00:01:40.859 --> 00:01:43.659 a 30 -year loan so much larger than the original 00:01:43.659 --> 00:01:45.500 amount you borrowed. It's the exact same engine, 00:01:45.540 --> 00:01:47.739 just running in reverse. Okay, let's unpack this 00:01:47.739 --> 00:01:49.879 with the core key distinction right at the very 00:01:49.879 --> 00:01:52.799 top. When people talk about interest, they often 00:01:52.799 --> 00:01:56.359 mix up two fundamental concepts, simple interest 00:01:56.359 --> 00:01:59.959 and compound interest. What separates these two 00:01:59.959 --> 00:02:02.620 financially? The defining difference is what 00:02:02.620 --> 00:02:05.700 base the interest is calculated on in the next 00:02:05.700 --> 00:02:08.560 period. With simple interest, the interest you 00:02:08.560 --> 00:02:11.560 earned in previous periods is never added back 00:02:11.560 --> 00:02:14.460 to the principal for the next calculation. So 00:02:14.460 --> 00:02:16.379 you're always earning interest based only on 00:02:16.379 --> 00:02:18.379 that original amount you started with? Always. 00:02:18.460 --> 00:02:20.620 It never changes. Let's run a quick example. 00:02:20.840 --> 00:02:24.400 Say I deposit $1 ,000 in an account that pays 00:02:24.400 --> 00:02:27.729 10 % simple annual interest. After year one, 00:02:27.949 --> 00:02:32.150 I have $1 ,100. After year two, the interest 00:02:32.150 --> 00:02:34.430 is calculated again on the original $1 ,000, 00:02:34.629 --> 00:02:37.550 so I earn another $100, bringing me to $1 ,200. 00:02:37.810 --> 00:02:40.169 That's spot on. The calculation always starts 00:02:40.169 --> 00:02:44.389 fresh from that initial $1 ,000 base. Now, with 00:02:44.389 --> 00:02:46.469 common interest, that previously accumulated 00:02:46.469 --> 00:02:50.439 interest is retained or reinvested. and added 00:02:50.439 --> 00:02:52.520 to the principal. So it forms a bigger base for 00:02:52.520 --> 00:02:54.599 the next calculation. A bigger base, exactly. 00:02:54.919 --> 00:02:57.780 Okay, using the same example. $1 ,000 at 10 % 00:02:57.780 --> 00:03:01.699 compound interest. Year one, still $1 ,100. Same 00:03:01.699 --> 00:03:05.360 result. But in year two, the 10 % is calculated 00:03:05.360 --> 00:03:09.060 on $1 ,100, not the original $1 ,000. So I earn 00:03:09.060 --> 00:03:12.240 $110 in the second year, bringing my total to 00:03:12.240 --> 00:03:16.360 $1 ,210. And while that extra $10 seems, you 00:03:16.360 --> 00:03:19.310 know, negligible over two years. It's nothing. 00:03:19.389 --> 00:03:22.629 Right. But imagine extending that period to 40 00:03:22.629 --> 00:03:25.590 years. That small difference over a single year 00:03:25.590 --> 00:03:28.569 becomes seismic over the long haul. That's the 00:03:28.569 --> 00:03:30.349 fundamental difference that dictates whether 00:03:30.349 --> 00:03:34.789 your portfolio is coasting or rocketing. That's 00:03:34.789 --> 00:03:37.050 the one. Speaking of rocketing, let's move into 00:03:37.050 --> 00:03:39.169 the vocabulary that defines the speed of that 00:03:39.169 --> 00:03:42.699 rocket. The source material highlights one variable 00:03:42.699 --> 00:03:45.520 above all else that dictates the velocity of 00:03:45.520 --> 00:03:47.479 compounding. And that's the compounding frequency. 00:03:47.800 --> 00:03:49.479 The compounding frequency. This is where the 00:03:49.479 --> 00:03:51.699 simple definition meets the practical mechanics 00:03:51.699 --> 00:03:54.620 of finance. We use the variable n to define it. 00:03:54.719 --> 00:03:57.680 It is simply the number of times per unit of 00:03:57.680 --> 00:04:00.479 time, which is almost always a year. that the 00:04:00.479 --> 00:04:02.379 interest you earned is officially capitalized. 00:04:02.659 --> 00:04:04.659 Capitalized meaning added back to the principal. 00:04:04.919 --> 00:04:07.340 Added back to the principal balance, yes. So 00:04:07.340 --> 00:04:09.759 if a bank advertises that they compound interest 00:04:09.759 --> 00:04:13.479 yearly, our N is 1. If it's half yearly, N is 00:04:13.479 --> 00:04:17.360 2. But wait, if I have two products, both offering 00:04:17.360 --> 00:04:21.240 a 5 % annual rate, but one compounds yearly and 00:04:21.240 --> 00:04:23.819 the other compounds quarterly, so N equals 4, 00:04:24.160 --> 00:04:27.360 why does the quarterly option yield more money? 00:04:27.959 --> 00:04:30.579 That's the critical insight. The more often N 00:04:30.579 --> 00:04:32.819 increases, the more chances the accrued interest 00:04:32.819 --> 00:04:34.959 has to start earning interest on itself within 00:04:34.959 --> 00:04:37.339 that single year. Even though the annual rate 00:04:37.339 --> 00:04:39.540 is the same, the quarterly compounding means 00:04:39.540 --> 00:04:41.319 your money starts growing on a slightly bigger 00:04:41.319 --> 00:04:44.079 base three extra times before the year is out. 00:04:44.259 --> 00:04:46.000 I see. It's about leveraging that exponential 00:04:46.000 --> 00:04:49.000 power sooner and more often. Exactly. And we 00:04:49.000 --> 00:04:51.800 see concrete examples of this across all financial 00:04:51.800 --> 00:04:54.600 products. We have yearly, half yearly, quarterly, 00:04:54.800 --> 00:04:56.879 and then monthly where N is 12, which is probably 00:04:56.879 --> 00:04:58.839 the most. common for consumer debt and savings 00:04:58.839 --> 00:05:00.860 account right like credit cards credit cards 00:05:00.860 --> 00:05:04.720 mortgages savings but it doesn't stop there interest 00:05:04.720 --> 00:05:07.879 can be compounded weekly so n is 52 or daily 00:05:07.879 --> 00:05:11.560 where n is 365 and then ultimately we reach the 00:05:11.560 --> 00:05:13.839 mathematical limit of continuous compounding 00:05:13.839 --> 00:05:15.699 where n goes to infinity we'll get to that later 00:05:15.699 --> 00:05:17.899 let's focus on that common monthly example because 00:05:17.899 --> 00:05:20.060 that's what most people interact with if i have 00:05:20.060 --> 00:05:23.120 a 12 annual nominal rate but the interest is 00:05:23.120 --> 00:05:26.459 capitalized monthly, the frequency is 12. But 00:05:26.459 --> 00:05:28.519 the period for calculation changes, doesn't it? 00:05:28.560 --> 00:05:30.959 It does. You're dividing that nominal annual 00:05:30.959 --> 00:05:33.500 rate, the R, by the frequency, which is 12. So 00:05:33.500 --> 00:05:36.819 12 % divided by 12 is a 1 % monthly rate. You 00:05:36.819 --> 00:05:39.800 then apply that 1 % rate 12 times over the year, 00:05:39.879 --> 00:05:41.980 but each time on a slightly larger principle. 00:05:42.259 --> 00:05:44.720 This is what generates the slightly higher effective 00:05:44.720 --> 00:05:47.240 return compared to that stated 12 % nominal rate. 00:05:47.360 --> 00:05:49.980 This brings us directly to a key actionable takeaway 00:05:49.980 --> 00:05:52.920 for our listener. If every product uses a different 00:05:52.920 --> 00:05:55.879 compounding frequency, one compounds daily, one 00:05:55.879 --> 00:05:58.459 quarterly, one semi -annually, how is a consumer 00:05:58.459 --> 00:06:00.839 supposed to compare them fairly? How do you figure 00:06:00.839 --> 00:06:03.240 out which one is genuinely the better deal? That 00:06:03.240 --> 00:06:05.600 confusion is exactly the problem that the annual 00:06:05.600 --> 00:06:09.319 equivalent rate, or AER, solves. AER. Right. 00:06:09.439 --> 00:06:12.000 This standardized metric is also known as the 00:06:12.000 --> 00:06:15.600 effective annual percentage rate. or eapr or 00:06:15.600 --> 00:06:19.480 sometimes annual percentage yield apy its purpose 00:06:19.480 --> 00:06:22.439 is pure standardization so it cuts through the 00:06:22.439 --> 00:06:24.360 noise it strips away the compounding frequency 00:06:24.360 --> 00:06:27.279 the daily versus the quarterly to give you a 00:06:27.279 --> 00:06:30.199 single comparable figure that represents the 00:06:30.199 --> 00:06:33.199 true cost or true yield of the money. So if I'm 00:06:33.199 --> 00:06:35.639 looking at two different savings accounts, and 00:06:35.639 --> 00:06:38.939 one offers a 5 % nominal rate compounded daily, 00:06:39.079 --> 00:06:42.939 and the other offers 5 .1 % compounded semi -annually, 00:06:43.079 --> 00:06:45.319 the only number that really tells me which one 00:06:45.319 --> 00:06:48.069 is better is the AER. Is that right? That is 00:06:48.069 --> 00:06:50.810 the single most critical, actionable piece of 00:06:50.810 --> 00:06:52.829 knowledge here. If you only remember one thing 00:06:52.829 --> 00:06:54.790 about comparing financial products, it's the 00:06:54.790 --> 00:06:58.009 AER. Ignore the nominal rate. That's just marketing. 00:06:58.189 --> 00:07:00.129 The headline number. It's the headline. Pay attention 00:07:00.129 --> 00:07:02.649 to the AER because that tells you the real yield 00:07:02.649 --> 00:07:04.769 after the effects of compounding have been factored 00:07:04.769 --> 00:07:07.009 in for one full year. I appreciate that emphasis. 00:07:07.290 --> 00:07:10.170 The sources also point out that the AER is often 00:07:10.170 --> 00:07:12.870 even more useful because it includes charges 00:07:12.870 --> 00:07:15.980 beyond just a pure interest. That inclusion is 00:07:15.980 --> 00:07:18.540 a consumer safeguard. And it's often mandated 00:07:18.540 --> 00:07:21.800 by regulatory bodies in many countries. The definition 00:07:21.800 --> 00:07:25.160 of AER is the total accumulated interest plus 00:07:25.160 --> 00:07:28.060 certain charges like taxes and fees that would 00:07:28.060 --> 00:07:30.740 be payable or earned up to the end of one full 00:07:30.740 --> 00:07:32.920 year. Divided by the original principle. Right. 00:07:33.079 --> 00:07:35.420 This gives you the clearest possible picture 00:07:35.420 --> 00:07:39.040 of the real annual cost or gain. If a bank uses 00:07:39.040 --> 00:07:41.279 daily compounding to make their nominal rate 00:07:41.279 --> 00:07:44.000 look attractive, the AER will pull back that 00:07:44.000 --> 00:07:46.259 curtain and show the effective rate after those 00:07:46.259 --> 00:07:50.829 300. It is so easy to fall into the trap of thinking 00:07:50.829 --> 00:07:53.250 compound interest is a modern financial invention, 00:07:53.529 --> 00:07:55.430 maybe something cooked up in the 20th century. 00:07:55.470 --> 00:07:57.209 Right, like it's a product of Wall Street or 00:07:57.209 --> 00:07:59.850 something. Exactly. But the sources we reviewed 00:07:59.850 --> 00:08:02.350 reveal its pedigree is incredibly long, stretching 00:08:02.350 --> 00:08:05.529 way back into antiquity. Okay, let's unpack this 00:08:05.529 --> 00:08:07.990 history. Where do the roots of compounding truly 00:08:07.990 --> 00:08:10.560 lie? The practice is definitely not new. It was 00:08:10.560 --> 00:08:13.259 known to ancient civilizations, and we have remarkable 00:08:13.259 --> 00:08:16.560 physical evidence. The sources cite a clay tablet 00:08:16.560 --> 00:08:19.360 recovered from Babylon. A clay tablet? A clay 00:08:19.360 --> 00:08:22.100 tablet, yeah. It dates back to the period between 00:08:22.100 --> 00:08:27.040 2000 and 1700 BC. Wow. Over 4 ,000 years ago. 00:08:27.079 --> 00:08:29.720 What does this tablet show? It appears to contain 00:08:29.720 --> 00:08:32.100 a problem involving the calculation of compound 00:08:32.100 --> 00:08:35.679 interest. While the context is, you know, economic, 00:08:35.899 --> 00:08:38.759 the mathematical understanding of how a sum grows 00:08:38.759 --> 00:08:41.580 exponentially over time was clearly present in 00:08:41.580 --> 00:08:43.960 the Mesopotamian era. So financial planning for 00:08:43.960 --> 00:08:46.240 long term growth or debt management has been 00:08:46.240 --> 00:08:48.539 around as long as recorded history. It seems 00:08:48.539 --> 00:08:51.039 so. It's a powerful perspective. And yet for 00:08:51.039 --> 00:08:52.879 much of the subsequent history, particularly 00:08:52.879 --> 00:08:55.720 leading up to the medieval period, compound interest 00:08:55.720 --> 00:08:58.600 wasn't just a standard financial tool. it was 00:08:58.600 --> 00:09:00.700 something severely frowned upon. Oh, absolutely. 00:09:00.980 --> 00:09:03.620 Morally condemned, especially by Roman law. Why? 00:09:03.759 --> 00:09:06.200 What was the controversy? Well, in ancient and 00:09:06.200 --> 00:09:08.500 medieval societies, charging any interest was 00:09:08.500 --> 00:09:11.320 often viewed as usury and condemned by religious 00:09:11.320 --> 00:09:14.240 and moral authorities. But charging interest 00:09:14.240 --> 00:09:16.860 on interest compound interest was regarded as 00:09:16.860 --> 00:09:20.240 the worst kind of usury. The worst kind? They 00:09:20.240 --> 00:09:23.840 even had a special name for it. Anaticism. Why 00:09:23.840 --> 00:09:26.019 the special condemnation for compounding? What 00:09:26.019 --> 00:09:28.740 made it so much worse? It was seen as unnatural. 00:09:29.159 --> 00:09:31.340 I mean, simple interest was sometimes tolerated 00:09:31.340 --> 00:09:33.899 as compensation for the risk of lending your 00:09:33.899 --> 00:09:37.639 principal. But earning interest was viewed as 00:09:37.639 --> 00:09:40.629 profiting from time itself. Or generating money 00:09:40.629 --> 00:09:42.289 from nothing. Generating money from nothing, 00:09:42.350 --> 00:09:44.950 which was considered deeply exploitative. Roman 00:09:44.950 --> 00:09:47.029 law and the common law that followed in many 00:09:47.029 --> 00:09:49.470 countries severely restricted or outlawed the 00:09:49.470 --> 00:09:51.570 practice for centuries because of this moral 00:09:51.570 --> 00:09:53.889 judgment. That context is so important, showing 00:09:53.889 --> 00:09:56.029 that finance is always intertwined with morality. 00:09:56.629 --> 00:09:59.110 Yet commerce and the necessity of long -term 00:09:59.110 --> 00:10:01.669 trade eventually forced a mathematical reckoning, 00:10:01.730 --> 00:10:04.610 right? It had to. So when do we see the first 00:10:04.610 --> 00:10:07.570 real written analysis of how to calculate this 00:10:07.570 --> 00:10:09.960 forbidden growth? That begins to pop up in the 00:10:09.960 --> 00:10:13.299 medieval era. We see early written analysis surface 00:10:13.299 --> 00:10:15.200 through the records of practicing merchants. 00:10:15.559 --> 00:10:18.879 A key figure here is Francesco Balducci Pigolotti. 00:10:18.980 --> 00:10:22.720 A Florentine merchant. Right. His 1340 book Crática 00:10:22.720 --> 00:10:25.779 de la Mercatura contains some of the earliest 00:10:25.779 --> 00:10:28.620 detailed calculations. And what kind of calculations 00:10:28.620 --> 00:10:30.639 are we talking about? He provided a practical 00:10:30.639 --> 00:10:33.500 ready -reckoner table. It detailed the compound 00:10:33.500 --> 00:10:36.440 interest accrued on 100 lire for various common 00:10:36.440 --> 00:10:39.620 mercantile rates, specifically from 1 % up to 00:10:39.620 --> 00:10:43.340 8%, calculated for periods up to 20 years. So 00:10:43.340 --> 00:10:45.500 this wasn't abstract math. This was an essential 00:10:45.500 --> 00:10:48.379 tool for merchants to quickly estimate the value 00:10:48.379 --> 00:10:51.179 of long -term investments or obligations. Despite 00:10:51.179 --> 00:10:53.539 the legal restrictions, yes, it was a business 00:10:53.539 --> 00:10:55.779 tool. And shortly after that, we get one of the 00:10:55.779 --> 00:10:58.179 most famous rules in all of finance, one that 00:10:58.179 --> 00:11:00.360 speaks directly to the power of compounding and 00:11:00.360 --> 00:11:03.120 time, and one that every investor still relies 00:11:03.120 --> 00:11:06.360 on today. The rule of 72. The rule of 72. That 00:11:06.360 --> 00:11:08.700 rule was presented in Luca Pacioli's foundational 00:11:08.700 --> 00:11:11.399 work, the Summa De Arithmetica, which was published 00:11:11.399 --> 00:11:15.360 in 1494. The rule of 72 is perhaps the most elegant 00:11:15.360 --> 00:11:18.080 mathematical approximation in finance. Tell us 00:11:18.080 --> 00:11:22.139 how it works and why 72. It's designed for rapid 00:11:22.139 --> 00:11:24.720 mental computation. The rule states that the 00:11:24.720 --> 00:11:27.100 approximate number of years required for an investment 00:11:27.100 --> 00:11:30.200 at compound interest to double its value is found 00:11:30.200 --> 00:11:32.779 by simply dividing the annual interest rate as 00:11:32.779 --> 00:11:36.070 a whole number into 72. So let's use a couple 00:11:36.070 --> 00:11:38.990 of scenarios. If I achieve a modest 4 % return 00:11:38.990 --> 00:11:42.070 annually, how long until my money doubles? You 00:11:42.070 --> 00:11:45.750 divide 72 by 4, which gives you 18 years. And 00:11:45.750 --> 00:11:47.970 if I'm a riskier investor and manage to achieve 00:11:47.970 --> 00:11:51.490 a 9 % return? 72 divided by 9 is 8 years. It 00:11:51.490 --> 00:11:54.250 links time, rate, and growth very elegantly. 00:11:54.269 --> 00:11:56.549 And importantly, it gets the job done quickly. 00:11:56.789 --> 00:11:59.470 It's a fantastic mental shortcut. It is. And 00:11:59.470 --> 00:12:01.629 while 72 is the most common number, the sources 00:12:01.629 --> 00:12:04.529 note that mathematically 69 .3 is closer to the... 00:12:04.519 --> 00:12:06.759 theoretical ideal, especially for continuous 00:12:06.759 --> 00:12:09.539 compounding. But 72 is easier to divide. Exactly. 00:12:09.840 --> 00:12:12.659 72 has more convenient factors. It divides cleanly 00:12:12.659 --> 00:12:16.600 by 2, 3, 4, 6, 8, 9, 12, etc., making it superior 00:12:16.600 --> 00:12:19.679 for quick mental estimation. The rule of 72 essentially 00:12:19.679 --> 00:12:21.960 gives us a quick way to conceptualize that exponential 00:12:21.960 --> 00:12:24.419 curve. It shows us how sensitive the doubling 00:12:24.419 --> 00:12:27.440 time is to the rate. It does. But the move towards 00:12:27.440 --> 00:12:30.379 truly modern, accurate calculation came in the 00:12:30.379 --> 00:12:33.580 17th century. And here we have to highlight Richard 00:12:33.580 --> 00:12:36.700 Witt's Arithmetical Questions, published in 1613. 00:12:37.000 --> 00:12:39.519 Why was Witt's work considered such a landmark 00:12:39.519 --> 00:12:42.840 change? Because prior writers like Passioli only 00:12:42.840 --> 00:12:45.759 treated compound interests briefly within broader 00:12:45.759 --> 00:12:48.759 mathematical treatises. Witt's work was wholly 00:12:48.759 --> 00:12:51.379 and exclusively devoted to the subject. The whole 00:12:51.379 --> 00:12:53.960 book was just on this. The entire thing. He was 00:12:53.960 --> 00:12:56.759 a London mathematical practitioner, and his book 00:12:56.759 --> 00:12:59.639 provided immense clarity, accuracy, and depth 00:12:59.639 --> 00:13:02.129 of insight. He wasn't just giving - tables, he 00:13:02.129 --> 00:13:04.190 was providing the methodology. And the detail 00:13:04.190 --> 00:13:07.049 was extreme for the time. It really was. He provided 00:13:07.049 --> 00:13:10.929 124 worked examples and included D2L tables based 00:13:10.929 --> 00:13:13.350 on the 10 % maximum allowable interest rate in 00:13:13.350 --> 00:13:16.070 London at that time. This transformed compounding 00:13:16.070 --> 00:13:19.169 from an approximation or a side note into a fully 00:13:19.169 --> 00:13:21.909 formalized, rigorous field of mathematics. Okay, 00:13:21.950 --> 00:13:23.870 here's where it gets really interesting. connecting 00:13:23.870 --> 00:13:26.809 this very practical financial problem to pure 00:13:26.809 --> 00:13:29.009 foundational mathematics. This is one of the 00:13:29.009 --> 00:13:31.350 biggest aha moments in the history of mathematics 00:13:31.350 --> 00:13:33.889 and finance. It wasn't just practitioners and 00:13:33.889 --> 00:13:36.129 merchants pushing the boundaries. It was pure 00:13:36.129 --> 00:13:38.990 mathematicians looking at the limits of compounding. 00:13:39.029 --> 00:13:41.490 And the pursuit of compound interest led directly 00:13:41.490 --> 00:13:44.549 to a monumental discovery, the mathematical constant 00:13:44.549 --> 00:13:47.590 E. The number E. Yes. Jacob Bernoulli discovered 00:13:47.590 --> 00:13:51.129 this constant in 1683 while exploring a question 00:13:51.129 --> 00:13:53.820 about compound interest. specifically he was 00:13:53.820 --> 00:13:56.700 asking what happens when you let the compounding 00:13:56.700 --> 00:13:59.919 frequency our variable increase toward infinity. 00:14:00.100 --> 00:14:03.500 Exactly. He asked, if I offer a 100 % annual 00:14:03.500 --> 00:14:05.320 rate and compound it more and more frequently, 00:14:05.539 --> 00:14:09.100 monthly, daily, hourly, continuously, what is 00:14:09.100 --> 00:14:11.879 the maximum possible accumulated value? And it 00:14:11.879 --> 00:14:14.159 doesn't just go to infinity. It doesn't. He found 00:14:14.159 --> 00:14:16.379 that instead of growing infinitely large, the 00:14:16.379 --> 00:14:18.700 total value approached a finite limit, which 00:14:18.700 --> 00:14:22.299 we now know as E, approximately 2 .71828. That 00:14:22.299 --> 00:14:25.289 connection. From Jacob Bernoulli studying a specific 00:14:25.289 --> 00:14:28.029 financial problem, the maximum possible return 00:14:28.029 --> 00:14:30.370 on an investment, to discovering the constant 00:14:30.370 --> 00:14:32.750 that governs all exponential growth across physics, 00:14:32.809 --> 00:14:35.009 biology, and engineering. That's just phenomenal. 00:14:35.289 --> 00:14:37.509 It is. The engine of wealth is literally governed 00:14:37.509 --> 00:14:40.330 by a constant of nature. It perfectly illustrates 00:14:40.330 --> 00:14:43.429 how practical financial problems often drive 00:14:43.429 --> 00:14:45.929 these theoretical breakthroughs that reshape 00:14:45.929 --> 00:14:48.509 our understanding of the universe. And that practice 00:14:48.509 --> 00:14:50.909 of trying to tame the math continued centuries 00:14:50.909 --> 00:14:53.710 later. The sources note that in the 19th century, 00:14:53.850 --> 00:14:56.110 Persian merchants employed a simplified mathematical 00:14:56.110 --> 00:14:59.250 technique. They used a modified linear Taylor 00:14:59.250 --> 00:15:01.889 approximation to calculate monthly payments. 00:15:02.149 --> 00:15:05.450 A Taylor approximation sounds like advanced university 00:15:05.450 --> 00:15:08.320 calculus. Why would a merchant be using that? 00:15:08.440 --> 00:15:11.240 Well, they weren't using the full, complex calculus. 00:15:11.379 --> 00:15:13.860 They used a slightly modified, simplified, linear 00:15:13.860 --> 00:15:16.279 version derived from the Taylor series expansion. 00:15:16.720 --> 00:15:18.840 The point was to create a method that allowed 00:15:18.840 --> 00:15:22.139 for easy, accurate mental computation of complex 00:15:22.139 --> 00:15:25.179 loan terms. So without needing pen, paper, or 00:15:25.179 --> 00:15:28.379 complex tables. Exactly. It shows this long human 00:15:28.379 --> 00:15:31.279 tradition, spanning from Babylon to 19th century 00:15:31.279 --> 00:15:34.059 Persia, of trying to tame the exponential nature 00:15:34.059 --> 00:15:37.210 of compounding for everyday commerce. We've established 00:15:37.210 --> 00:15:39.389 the core variables and traced the concept through 00:15:39.389 --> 00:15:41.950 4 ,000 years of history. Now let's see how that 00:15:41.950 --> 00:15:44.350 compounding frequency, n, is manifest across 00:15:44.350 --> 00:15:47.250 modern financial tools. Let's start with large 00:15:47.250 --> 00:15:50.409 standardized instruments like corporate and government 00:15:50.409 --> 00:15:53.809 bonds. Okay. These large instruments operate 00:15:53.809 --> 00:15:56.929 on a very explicit standardized schedule globally. 00:15:57.389 --> 00:16:00.070 The standard approach is that the interest payments, 00:16:00.350 --> 00:16:03.190 often called coupon payments, are typically payable 00:16:03.190 --> 00:16:06.389 twice yearly. Semi -annually. So the compounding 00:16:06.389 --> 00:16:09.929 frequency, n, is fixed at 2. Yes. The calculation 00:16:09.929 --> 00:16:12.049 is straightforward. The interest paid every six 00:16:12.049 --> 00:16:14.269 months is the disclosed nominal rate divided 00:16:14.269 --> 00:16:17.250 by 2, multiplied by the principal amount. But 00:16:17.250 --> 00:16:19.450 the key takeaway for the investor is understanding 00:16:19.450 --> 00:16:21.889 the effective rate, right? Absolutely. Because 00:16:21.889 --> 00:16:23.830 you receive that first CUCON payment halfway 00:16:23.830 --> 00:16:26.190 through the year, and if you immediately reinvest 00:16:26.190 --> 00:16:28.149 it, it starts earning interest for the remaining 00:16:28.149 --> 00:16:30.649 six months. So the yearly compounded rate... 00:16:30.940 --> 00:16:33.559 the aer is always slightly higher than the nominal 00:16:33.559 --> 00:16:35.980 rate stated on the bond assuming you're reinvesting 00:16:35.980 --> 00:16:38.539 those coupons that's a subtle but absolutely 00:16:38.539 --> 00:16:41.379 critical point for you the learner if you are 00:16:41.379 --> 00:16:44.100 comparing two bonds always calculate the effective 00:16:44.100 --> 00:16:47.539 annual rate based on that semi -annual compounding 00:16:47.539 --> 00:16:50.440 to get the true yield. Never mistake the headline 00:16:50.440 --> 00:16:53.000 rate for the actual rate of return if you are 00:16:53.000 --> 00:16:55.399 actively reinvesting. Now let's move to consumer 00:16:55.399 --> 00:16:58.360 finance, specifically mortgages, which provide 00:16:58.360 --> 00:17:00.600 a fascinating contrast depending on where you 00:17:00.600 --> 00:17:03.460 live. Our sources draw a sharp distinction between 00:17:03.460 --> 00:17:06.400 Canadian and U .S. mortgages. This is a perfect 00:17:06.400 --> 00:17:09.259 illustration of how legislation dictates financial 00:17:09.259 --> 00:17:12.660 structure. In Canada, mortgage loans are generally 00:17:12.660 --> 00:17:15.599 mandated by law to be compounded semi -annually. 00:17:15.720 --> 00:17:19.130 So N equals 2. By law. By law. It's a fixed, 00:17:19.250 --> 00:17:21.990 regulated feature of the loan structure, even 00:17:21.990 --> 00:17:23.890 though the borrower is typically making monthly 00:17:23.890 --> 00:17:26.769 or even biweekly payments. So the Canadian lender 00:17:26.769 --> 00:17:29.230 figures out the total interest owed for that 00:17:29.230 --> 00:17:31.650 half -year period and then breaks that down into 00:17:31.650 --> 00:17:33.690 the monthly payment schedule. Precisely. The 00:17:33.690 --> 00:17:35.670 interest accrues semi -annually, even though 00:17:35.670 --> 00:17:37.349 payments are more frequent. Now contrast that 00:17:37.349 --> 00:17:39.910 with U .S. mortgages. These use an amortizing 00:17:39.910 --> 00:17:42.150 loan structure, which handles interest in a fundamentally 00:17:42.150 --> 00:17:44.910 different way. What does amortizing mean in this 00:17:44.910 --> 00:17:48.160 context? It means the entire process is governed 00:17:48.160 --> 00:17:50.920 by an amortization schedule. Think of Canadian 00:17:50.920 --> 00:17:54.000 compounding like a semi -annual accounting day 00:17:54.000 --> 00:17:56.740 where all the interest for that period gets calculated. 00:17:57.019 --> 00:18:00.539 Got it. U .S. amortization means you clear that 00:18:00.539 --> 00:18:03.700 interest debt every single month. Interest is 00:18:03.700 --> 00:18:05.799 calculated monthly on the remaining principal 00:18:05.799 --> 00:18:09.140 balance, but critically, that interest generated 00:18:09.140 --> 00:18:11.579 is paid off immediately by the monthly payment. 00:18:11.759 --> 00:18:14.240 So the interest never gets added back to the 00:18:14.240 --> 00:18:16.880 principal to compound further because I'm paying 00:18:16.880 --> 00:18:19.160 it down every month. The principal only reduces 00:18:19.160 --> 00:18:21.579 with whatever is left of my payment after the 00:18:21.579 --> 00:18:23.880 interest for that month has been covered. That's 00:18:23.880 --> 00:18:26.339 the key difference. The monthly calculation happens, 00:18:26.400 --> 00:18:28.519 but the compounding effect on the principal is 00:18:28.519 --> 00:18:30.940 mitigated by the rapid forced payment of the 00:18:30.940 --> 00:18:33.759 accrued interest. It keeps the exponential curve 00:18:33.759 --> 00:18:36.240 of the debt from accelerating. That's a great 00:18:36.240 --> 00:18:37.960 way to summarize the difference. Now let's move 00:18:37.960 --> 00:18:39.920 to the mathematical extreme we touched on earlier, 00:18:40.160 --> 00:18:42.940 continuous compounding. This is where we revisit 00:18:42.940 --> 00:18:46.490 Jacob Bernoulli and the constant E. Continuous 00:18:46.490 --> 00:18:48.630 compounding is the mathematical limit achieved 00:18:48.630 --> 00:18:51.170 when the number of compounding periods per year, 00:18:51.329 --> 00:18:54.329 our n, increases without limit approaching infinity. 00:18:54.710 --> 00:18:57.309 So interest is being calculated and added back 00:18:57.309 --> 00:18:59.730 to the principle every microsecond, constantly. 00:18:59.990 --> 00:19:02.430 That's the idea. Why do we care about this theoretical 00:19:02.430 --> 00:19:05.200 limit? I mean, it seems irrelevant to a homeowner 00:19:05.200 --> 00:19:08.099 or a typical saver. While it is irrelevant for 00:19:08.099 --> 00:19:10.200 a savings account, it is absolutely essential 00:19:10.200 --> 00:19:13.039 for the dynamic pricing of complex instruments, 00:19:13.339 --> 00:19:15.920 especially financial derivatives like options 00:19:15.920 --> 00:19:18.759 and futures. Okay, so why is continuous time 00:19:18.759 --> 00:19:21.500 preferred in those advanced markets? It's a natural 00:19:21.500 --> 00:19:23.720 consequence of the type of mathematics required 00:19:23.720 --> 00:19:26.400 to model assets whose prices change constantly. 00:19:26.619 --> 00:19:28.539 When you get into higher -level mathematical 00:19:28.539 --> 00:19:31.440 finance, specifically something called Ito calculus, 00:19:32.430 --> 00:19:34.329 Modeling these rapid continuous price movements 00:19:34.329 --> 00:19:36.950 is mathematically simpler if you assume continuous 00:19:36.950 --> 00:19:39.890 time rather than discrete jumps. Ah, so rather 00:19:39.890 --> 00:19:42.529 than trying to model quarterly or daily compounding. 00:19:42.589 --> 00:19:45.940 Right. If the asset price is changing every millisecond, 00:19:46.019 --> 00:19:48.559 it's easier to model the growth rate as continuous. 00:19:48.859 --> 00:19:51.460 I see. Continuous compounding simplifies the 00:19:51.460 --> 00:19:53.839 math needed to value risk and expected returns. 00:19:54.200 --> 00:19:56.920 The constant E allows the models to assume a 00:19:56.920 --> 00:19:59.920 continuous stream of small changes. And as we 00:19:59.920 --> 00:20:02.039 mentioned, mathematically, we know the absolute 00:20:02.039 --> 00:20:04.500 ceiling for this growth. The upper limit. The 00:20:04.500 --> 00:20:07.039 effective annual rate under continuous compounding 00:20:07.039 --> 00:20:10.200 approaches the upper limit defined by e to the 00:20:10.200 --> 00:20:13.819 power of r minus 1. No matter how fast you compound, 00:20:14.059 --> 00:20:16.670 you can never exceed that rate. We've seen how 00:20:16.670 --> 00:20:20.130 merchants tamed this math using approximations. 00:20:20.410 --> 00:20:23.029 Now let's transition into the quantitative core 00:20:23.029 --> 00:20:26.049 of our sources, the actual exact formulas that 00:20:26.049 --> 00:20:28.329 underpin those ancient tricks and govern modern 00:20:28.329 --> 00:20:30.250 finance. We're going to try and demystify these. 00:20:30.529 --> 00:20:32.670 This is where we define the power of exponential 00:20:32.670 --> 00:20:35.809 growth explicitly. We begin with the periodic 00:20:35.809 --> 00:20:37.930 compounding formula, often called the growth 00:20:37.930 --> 00:20:40.529 engine formula. This is the general equation 00:20:40.529 --> 00:20:43.230 that calculates the total accumulated value A. 00:20:43.430 --> 00:20:44.990 Give us the formula and then we'll break down 00:20:44.990 --> 00:20:47.269 every... piece by what it does for us the total 00:20:47.269 --> 00:20:49.650 accumulated value a which is the final amount 00:20:49.650 --> 00:20:52.890 is defined by a equals p times in parentheses 00:20:52.890 --> 00:20:56.109 one plus r over n all raised to the power of 00:20:56.109 --> 00:20:58.549 t times n okay that looks a little intimidating 00:20:58.549 --> 00:21:02.150 but let's go piece by piece p is easy The initial 00:21:02.150 --> 00:21:04.869 money invested, the principal. Correct. R is 00:21:04.869 --> 00:21:06.769 the nominal annual interest rate, always expressed 00:21:06.769 --> 00:21:11.190 as a decimal, so 5 % is 0 .05. N is our friend, 00:21:11.369 --> 00:21:13.589 the compounding frequency, how often per year. 00:21:13.829 --> 00:21:16.349 And a key is the overall length of time, usually 00:21:16.349 --> 00:21:18.650 measured in years. So that inner part, the 1 00:21:18.650 --> 00:21:21.750 plus rn, is effectively the growth rate per compounding 00:21:21.750 --> 00:21:24.119 period. That's it. So if I invest for 30 years 00:21:24.119 --> 00:21:27.680 at 8 % compounded quarterly, so NZO4, I'm dividing 00:21:27.680 --> 00:21:30.759 that 8 % annual rate into four periods. So every 00:21:30.759 --> 00:21:33.859 three months, my money grows by 1 plus 0 .08 00:21:33.859 --> 00:21:37.720 over 4, or 1 .02. Exactly. You are multiplying 00:21:37.720 --> 00:21:40.519 your investment by a factor of 1 .02 every single 00:21:40.519 --> 00:21:43.460 quarter. And the exponent, TN, is the total number 00:21:43.460 --> 00:21:45.579 of compounding periods over the entire life of 00:21:45.579 --> 00:21:47.420 the investment. So in that 30 -year quarterly 00:21:47.420 --> 00:21:51.039 example, the exponent is 30 times 4 of 120. 120. 00:21:52.170 --> 00:21:54.410 Massive exponent is the key lever, isn't it? 00:21:54.430 --> 00:21:57.170 That is the exponential power in action. That 00:21:57.170 --> 00:21:59.509 is why time matters more than the rate over the 00:21:59.509 --> 00:22:01.990 long run. It is the exponential multiplier. And 00:22:01.990 --> 00:22:04.549 remember, the total interest generated, I, is 00:22:04.549 --> 00:22:07.289 simply the final amount, A, minus the initial 00:22:07.289 --> 00:22:09.970 principle, P. Now, you mentioned earlier that 00:22:09.970 --> 00:22:13.170 mathematicians often simplify this by dropping 00:22:13.170 --> 00:22:16.109 the principle P. Why do they do that? And what 00:22:16.109 --> 00:22:18.069 is the resulting formula called? We call that 00:22:18.069 --> 00:22:22.950 the accumulation function, $8T. Why drop P? Because 00:22:22.950 --> 00:22:25.839 the principle is just a scaling factor. Dropping 00:22:25.839 --> 00:22:28.380 it allows mathematicians to create a standardized 00:22:28.380 --> 00:22:32.359 model showing exactly what $1 grows to after 00:22:32.359 --> 00:22:35.339 time t. So it lets you compare growth rates universally, 00:22:35.640 --> 00:22:37.779 regardless whether you started with $1 or $1 00:22:37.779 --> 00:22:40.140 million. Exactly. The accumulation function is 00:22:40.140 --> 00:22:42.579 simply that same formula without the PA of t 00:22:42.579 --> 00:22:46.180 equals 1 plus r over n, all to the power of tn. 00:22:46.339 --> 00:22:48.619 It tells you the factor by which any money will 00:22:48.619 --> 00:22:50.980 grow. That makes perfect sense. Let's now revisit 00:22:50.980 --> 00:22:53.980 the highest boundary condition. Continuous compounding, 00:22:53.980 --> 00:22:56.500 where n approaches infinity. Right, which we 00:22:56.500 --> 00:22:58.859 established is critical for options pricing. 00:22:59.380 --> 00:23:02.019 Continuous compounding shifts the formula completely 00:23:02.019 --> 00:23:05.559 away from discrete countable periods and into 00:23:05.559 --> 00:23:08.660 the realm of the natural logarithm base e. And 00:23:08.660 --> 00:23:11.799 the formula for the amount after time t with 00:23:11.799 --> 00:23:14.599 continuous compounding becomes strikingly simple, 00:23:14.779 --> 00:23:17.069 doesn't it? Yes, the complexity of the discrete 00:23:17.069 --> 00:23:20.009 formula melts away, and we are left with p of 00:23:20.009 --> 00:23:23.150 t equals p naught times e to the power of ort. 00:23:23.289 --> 00:23:25.509 So p naught is your initial principle, r is the 00:23:25.509 --> 00:23:28.369 annual rate, and t is the time in years. And 00:23:28.369 --> 00:23:30.769 that simplicity is why it is preferred for modeling 00:23:30.769 --> 00:23:32.869 things that change dynamically and continuously. 00:23:33.369 --> 00:23:36.410 When time never stops compounding, growth is 00:23:36.410 --> 00:23:38.829 governed by nature's constant. This leads us 00:23:38.829 --> 00:23:41.450 to a concept that is mathematically most abstract, 00:23:41.589 --> 00:23:44.410 but conceptually vital. The force of interest, 00:23:44.670 --> 00:23:46.910 represented by the Greek letter delta. The force 00:23:46.910 --> 00:23:49.309 of interest, or delta, is what we call the continuous 00:23:49.309 --> 00:23:52.329 rate when n tends to infinity. For the learner, 00:23:52.470 --> 00:23:54.390 don't get intimidated by the calculus notation. 00:23:54.730 --> 00:23:57.009 Think of delta as the instantaneous rate of growth. 00:23:57.250 --> 00:23:59.529 The purest, most unadulterated measure of how 00:23:59.529 --> 00:24:01.509 fast your money is compounding at any single 00:24:01.509 --> 00:24:03.769 moment. It's like a speed limit sign on the highway 00:24:03.769 --> 00:24:06.410 of finance, exactly. It's not an average rate 00:24:06.410 --> 00:24:08.809 over a year. It's describing the rate of change 00:24:08.809 --> 00:24:11.329 at every single instant in time. So what's the 00:24:11.329 --> 00:24:14.970 relationship between this delta and our standard 00:24:14.970 --> 00:24:18.930 compound rate? When the annual rate r is constant, 00:24:19.089 --> 00:24:21.710 delta is also constant, and delta is simply the 00:24:21.710 --> 00:24:26.109 natural log of 1 plus r. So delta ln 1 plus r. 00:24:26.329 --> 00:24:29.170 This tells you the specific continuous rate required 00:24:29.170 --> 00:24:32.430 to mathematically match the discrete annual rate. 00:24:32.549 --> 00:24:34.869 That's critical. It gives us a way to translate 00:24:34.869 --> 00:24:38.839 discrete... real world compounding into the continuous 00:24:38.839 --> 00:24:41.099 theoretical world. And the comparative context 00:24:41.099 --> 00:24:43.440 is illuminating. The force of interest is always 00:24:43.440 --> 00:24:45.480 less than the annual effective interest rate, 00:24:45.539 --> 00:24:47.859 the AER, because the AER includes the total effect 00:24:47.859 --> 00:24:50.079 of the discrete compounding jumps while Delti 00:24:50.079 --> 00:24:52.140 sort of smooths it out. Right. And the sources 00:24:52.140 --> 00:24:54.720 bring this highly theoretical concept back to 00:24:54.720 --> 00:24:57.000 real world economics, showing it's used not just 00:24:57.000 --> 00:24:59.079 for derivative pricing, but for modeling inflation. 00:24:59.440 --> 00:25:01.799 Using something called Studeley's formula. Yes, 00:25:01.900 --> 00:25:04.799 which uses a time varying force of interest to 00:25:04.799 --> 00:25:08.309 model inflation. The full formula is quite complex, 00:25:08.509 --> 00:25:11.170 but it shows the practical utility of these continuous 00:25:11.170 --> 00:25:14.410 growth models. They give actuaries and economists 00:25:14.410 --> 00:25:17.009 the tools to project long -term economic forces 00:25:17.009 --> 00:25:20.309 accurately. Okay, that is heavy math, but the 00:25:20.309 --> 00:25:23.829 context is critical. Before we move on, we need 00:25:23.829 --> 00:25:25.789 to address the necessity of the modern financial 00:25:25.789 --> 00:25:29.109 market. Converting interest rates between compounding 00:25:29.109 --> 00:25:31.950 bases. This is a practical everyday necessity 00:25:31.950 --> 00:25:34.690 for financial professionals. If you have an interest 00:25:34.690 --> 00:25:37.549 rate quoted at 6 % compounded quarterly, so N 00:25:37.549 --> 00:25:40.069 equals 4, and you want to compare it to an equivalent 00:25:40.069 --> 00:25:42.950 rate that compounds monthly, where N equals 12. 00:25:43.210 --> 00:25:44.990 You have to convert one to the other. You have 00:25:44.990 --> 00:25:46.710 to, otherwise you're comparing apples and oranges. 00:25:46.990 --> 00:25:49.289 You need to ensure they yield the exact same 00:25:49.289 --> 00:25:52.410 total accumulated value over a year. So what's 00:25:52.410 --> 00:25:54.529 the conversion formula for that? The general 00:25:54.529 --> 00:25:57.890 conversion formula is a bit of a mouthful, but 00:25:57.890 --> 00:26:00.369 what it's doing is simply forcing the growth 00:26:00.369 --> 00:26:03.130 factor of the first rate to be equal to the growth 00:26:03.130 --> 00:26:05.470 factor of the second, ensuring they are truly 00:26:05.470 --> 00:26:07.890 equivalent. And if we want to convert a discrete 00:26:07.890 --> 00:26:11.089 rate, say our monthly compounding rate, directly 00:26:11.089 --> 00:26:13.890 to the continuous basis, the force of interest. 00:26:14.089 --> 00:26:16.470 We use a formula derived directly from the definition 00:26:16.470 --> 00:26:21.869 of delta. Delta equals n. NN1 plus RNLA. This 00:26:21.869 --> 00:26:24.490 allows any discrete rate to be seamlessly expressed 00:26:24.490 --> 00:26:27.109 in terms of its continuously compounded equivalent. 00:26:27.369 --> 00:26:29.809 We have established the engine and demystified 00:26:29.809 --> 00:26:32.109 the formulas. Now let's apply this power to the 00:26:32.109 --> 00:26:34.509 financial tools that affect almost everyone. 00:26:34.920 --> 00:26:37.559 Amortized loans like mortgages and long -term 00:26:37.559 --> 00:26:39.900 savings plans. We'll start with monthly amortized 00:26:39.900 --> 00:26:42.119 loan payments. We established that calculating 00:26:42.119 --> 00:26:44.640 the smooth monthly payment is a fundamental application 00:26:44.640 --> 00:26:46.980 of compounding mathematics. This is the smooth 00:26:46.980 --> 00:26:49.440 payment calculator. It's the formula that spits 00:26:49.440 --> 00:26:51.759 out that magic number. What you owe every month 00:26:51.759 --> 00:26:54.160 for the life of the loan. It is a critical piece 00:26:54.160 --> 00:26:56.339 of financial engineering. The exact formula used 00:26:56.339 --> 00:26:58.839 to calculate the smooth, constant monthly payment, 00:26:58.960 --> 00:27:02.619 which we'll call C, is C equals R times P divided 00:27:02.619 --> 00:27:05.240 by, in parenthesis, 1 minus the reciprocal of 00:27:05.240 --> 00:27:08.500 1 plus R to the power of N. Let's clarify the 00:27:08.500 --> 00:27:10.619 variables here as they're a bit different. P 00:27:10.619 --> 00:27:12.900 is still the principal, the amount borrowed. 00:27:13.380 --> 00:27:16.599 What are R and N in this specific context? Crucially, 00:27:16.779 --> 00:27:19.579 R is the monthly interest rate. You must take 00:27:19.579 --> 00:27:21.440 the nominal annual rate and divide it by 12. 00:27:21.680 --> 00:27:26.680 So a 6 % annual rate means r equals 0 .005. And 00:27:26.680 --> 00:27:29.299 n. And n is the total number of payment periods. 00:27:29.559 --> 00:27:32.259 Not the frequency, but the total count. For a 00:27:32.259 --> 00:27:35.200 typical 30 -year monthly loan, n is 360. The 00:27:35.200 --> 00:27:37.279 denominator looks very much like a present value 00:27:37.279 --> 00:27:39.890 calculation. It's ensuring that if you took every 00:27:39.890 --> 00:27:42.670 single one of those 360 payments, discounted 00:27:42.670 --> 00:27:44.609 them back to today, they would perfectly equal 00:27:44.609 --> 00:27:47.430 the original principal P. That is precisely what 00:27:47.430 --> 00:27:49.589 the formula is doing. It is relating the present 00:27:49.589 --> 00:27:51.589 value of an annuity, that fixed stream of future 00:27:51.589 --> 00:27:54.309 payments, to the lump sum borrowed today. Now, 00:27:54.329 --> 00:27:56.789 that formula provides the exact smooth payment. 00:27:56.970 --> 00:27:58.789 But let's go back to the ingenuity we discussed 00:27:58.789 --> 00:28:01.549 earlier, the power of approximation. The sources 00:28:01.549 --> 00:28:03.910 detailed a high -accuracy, simplified formula 00:28:03.910 --> 00:28:06.529 used historically. This is a fantastic example 00:28:06.529 --> 00:28:09.490 of mathematics meeting necessity. For typical 00:28:09.490 --> 00:28:12.730 interest rates, where the monthly rate r is tiny 00:28:12.730 --> 00:28:15.690 compared to 1, we can use a simplification based 00:28:15.690 --> 00:28:18.190 on Taylor approximation. Where the natural log 00:28:18.190 --> 00:28:21.109 of 1 plus r is approximately equal to r. That's 00:28:21.109 --> 00:28:22.930 the mathematical trick, and it's what allowed 00:28:22.930 --> 00:28:25.289 those 19th century Persian merchants to do this 00:28:25.289 --> 00:28:27.809 in their heads. The source suggests defining 00:28:27.809 --> 00:28:30.329 two auxiliary variables to make this approximation 00:28:30.329 --> 00:28:33.670 cleaner. First, you define a term called CNOT, 00:28:33.730 --> 00:28:36.730 which is P divided by N. So what is that? This 00:28:36.730 --> 00:28:38.849 is the theoretical monthly payment if the loan 00:28:38.849 --> 00:28:41.230 had zero interest. It is just the principal divided 00:28:41.230 --> 00:28:43.250 equally over the payment periods. That gives 00:28:43.250 --> 00:28:45.230 us a baseline payment we can easily calculate 00:28:45.230 --> 00:28:47.589 in our heads. Okay, so let's walk through the 00:28:47.589 --> 00:28:49.950 specific concrete example provided in the sources 00:28:49.950 --> 00:28:52.109 to show just how accurate this historical method 00:28:52.109 --> 00:28:54.750 is. The example is a substantial common scenario. 00:28:55.009 --> 00:28:58.130 Right. A $120 ,000 mortgage with a term of 30 00:28:58.130 --> 00:29:01.750 years and a nominal rate of 4 .5%. payable monthly 00:29:01.750 --> 00:29:05.769 so 30 years means n is 360 periods and r is the 00:29:05.769 --> 00:29:08.509 monthly rate so 0 .045 divided by 12 which is 00:29:08.509 --> 00:29:12.730 0 .00375 first that zero interest payment c naught 00:29:12.730 --> 00:29:16.289 that simple division 120 000 divided by 360 which 00:29:16.289 --> 00:29:20.490 is 333 .33 okay that's our baseline right then 00:29:20.490 --> 00:29:22.910 we use a simplified approximation formula provided 00:29:22.910 --> 00:29:25.589 in the lars you calculate a factor let's call 00:29:25.589 --> 00:29:28.730 it x which is one half times n times r for this 00:29:28.730 --> 00:29:32.779 loan x comes out to And the formula. The practical 00:29:32.779 --> 00:29:35.460 folk method is that the payment, C, is approximately 00:29:35.460 --> 00:29:38.440 C naught times the quantity 1 plus X plus 1 third 00:29:38.440 --> 00:29:40.480 of X squared. So we plug in those numbers. C 00:29:40.480 --> 00:29:44.619 naught is 333 .33 and X is .675. And when you 00:29:44.619 --> 00:29:46.819 run that calculation, the approximate payment 00:29:46.819 --> 00:29:50.240 comes out to be about $608 .96. And the actual 00:29:50.240 --> 00:29:53.160 exact payment calculated using the complex formula 00:29:53.160 --> 00:29:55.940 we saw earlier. The exact payment amount is $608 00:29:55.940 --> 00:30:01.119 .02. Wow. 608 .96 versus 608 .02, that approximation 00:30:01.119 --> 00:30:03.579 is off by less than a dollar. About 94 cents, 00:30:03.779 --> 00:30:05.819 an error of about a sixth of a percent. That 00:30:05.819 --> 00:30:08.039 shows the immense practical power of simplifying 00:30:08.039 --> 00:30:10.599 calculus for mental computation, especially in 00:30:10.599 --> 00:30:12.940 an era without ubiquitous computers. It speaks 00:30:12.940 --> 00:30:15.279 volumes about the human need to tame complex 00:30:15.279 --> 00:30:18.099 financial math for everyday transactions. Now, 00:30:18.099 --> 00:30:21.460 let's pivot from debt back to growth savings 00:30:21.460 --> 00:30:23.660 and investment plans, where you have an initial 00:30:23.660 --> 00:30:26.779 lump sum P and you make regular recurring deposits. 00:30:26.890 --> 00:30:29.529 This is what most people are doing with a retirement 00:30:29.529 --> 00:30:31.670 account. Right. So in this scenario, we have 00:30:31.670 --> 00:30:34.650 two components growing simultaneously. The original 00:30:34.650 --> 00:30:36.890 principal grows via the standard compounding 00:30:36.890 --> 00:30:39.089 formula. And the stream of monthly deposits are 00:30:39.089 --> 00:30:41.670 essentially a series of small deferred principles. 00:30:42.349 --> 00:30:44.990 Each one of those contributions is a new investment 00:30:44.990 --> 00:30:47.250 that starts compounding for a shorter duration 00:30:47.250 --> 00:30:49.920 than the original lump sum. So the calculation 00:30:49.920 --> 00:30:52.359 has to sum up the future value of the original 00:30:52.359 --> 00:30:55.660 money plus the future value of every single subsequent 00:30:55.660 --> 00:30:58.519 monthly deposit. Precisely. The formula relies 00:30:58.519 --> 00:31:00.740 on recognizing that the sequence of recurring 00:31:00.740 --> 00:31:04.299 deposits forms a geometric series. We sum up 00:31:04.299 --> 00:31:06.480 the future value of every single monthly deposit 00:31:06.480 --> 00:31:09.740 made over the total time t. And the total accumulated 00:31:09.740 --> 00:31:12.500 value, which we'll call p prime, combines both 00:31:12.500 --> 00:31:14.519 the compounding series and the initial principle. 00:31:14.990 --> 00:31:17.789 It does. Assuming r is the monthly rate and t 00:31:17.789 --> 00:31:20.289 is the time in months, the formula has two parts. 00:31:20.470 --> 00:31:23.390 The first part is m times the quantity, 1 plus 00:31:23.390 --> 00:31:26.430 r, to the t minus 1, all divided by r. That's 00:31:26.430 --> 00:31:28.250 the sum of the deposits. And the second part? 00:31:28.369 --> 00:31:30.769 The second part is just our familiar growth engine 00:31:30.769 --> 00:31:33.190 formula for the initial lump sum, so p times 00:31:33.190 --> 00:31:35.750 1 plus r to the power of t. You add them together. 00:31:36.279 --> 00:31:38.539 The ability to use this closed -form solution 00:31:38.539 --> 00:31:42.779 simplifies what would otherwise be 360 separate 00:31:42.779 --> 00:31:45.420 calculations if we were doing a 30 -year monthly 00:31:45.420 --> 00:31:47.720 calculation manually. It gives us an immediate, 00:31:47.839 --> 00:31:50.799 exact answer for the future value of a constantly 00:31:50.799 --> 00:31:53.740 funded savings plan. And that wraps up our deep 00:31:53.740 --> 00:31:56.609 dive into the engine of wealth and debt. We started 00:31:56.609 --> 00:31:58.809 with a simple definition, immediately established 00:31:58.809 --> 00:32:01.130 that critical difference between simple and compound 00:32:01.130 --> 00:32:03.430 interest. And then we jumped into the real mechanics 00:32:03.430 --> 00:32:06.450 compounding frequency, N, and the standardized 00:32:06.450 --> 00:32:09.529 yardstick, the annual equivalent rate, or AER, 00:32:09.609 --> 00:32:11.549 which we established is the only number you should 00:32:11.549 --> 00:32:14.130 trust when comparing financial products. We covered 00:32:14.130 --> 00:32:16.849 four millennia of history, tracing the concept 00:32:16.849 --> 00:32:19.170 from ancient Babylonian clay tablets through 00:32:19.170 --> 00:32:21.950 the condemnation of compounding as usury in Roman 00:32:21.950 --> 00:32:25.089 law, all the way to Passioli's elegant rule of 00:32:25.089 --> 00:32:28.450 72. And crucially, we saw that Jacob Bernoulli's 00:32:28.450 --> 00:32:32.150 study of compound interest in 1683 led directly 00:32:32.150 --> 00:32:34.269 to the discovery of the fundamental constant 00:32:34.269 --> 00:32:38.039 E. the bedrock of exponential growth. We analyzed 00:32:38.039 --> 00:32:40.259 the financial landscape, comparing the regulated 00:32:40.259 --> 00:32:42.799 semiannual compounding of corporate bonds and 00:32:42.799 --> 00:32:45.759 Canadian mortgages with the actively amortizing 00:32:45.759 --> 00:32:48.500 monthly interest structure of U .S. loans. And 00:32:48.500 --> 00:32:51.420 we dove into the complex mathematics, demystifying 00:32:51.420 --> 00:32:53.940 the periodic compounding formula, the role of 00:32:53.940 --> 00:32:56.880 E in continuous compounding, and the highly technical 00:32:56.880 --> 00:32:59.319 concept of the force of interest used for valuing 00:32:59.319 --> 00:33:01.380 derivatives and modeling inflation. Finally, 00:33:01.480 --> 00:33:03.720 we applied those principles to practical formulas. 00:33:04.519 --> 00:33:06.680 exact calculation for smooth monthly payments 00:33:06.680 --> 00:33:09.900 on loans and the highly accurate historical approximations 00:33:09.900 --> 00:33:13.039 that prove the ingenuity of finance. The scope 00:33:13.039 --> 00:33:16.440 of compound interest is truly enormous. It dictates 00:33:16.440 --> 00:33:19.000 the timeline for retirement, the total cost of 00:33:19.000 --> 00:33:21.200 a home and the profitability of global financial 00:33:21.200 --> 00:33:24.339 instruments. It is the concept that separates 00:33:24.339 --> 00:33:27.390 just saving money from truly investing. And we 00:33:27.390 --> 00:33:29.349 noted that continuous compounding is not just 00:33:29.349 --> 00:33:31.750 theoretical. It's crucial for pricing advanced, 00:33:31.950 --> 00:33:34.789 rapidly moving financial instruments. The force 00:33:34.789 --> 00:33:37.230 of interest is a measure of this continuous growth, 00:33:37.430 --> 00:33:40.430 and we can model highly complex economic phenomena 00:33:40.430 --> 00:33:43.170 like inflation using it. This raises an important 00:33:43.170 --> 00:33:45.490 question for you, the learner, to consider as 00:33:45.490 --> 00:33:48.410 you manage your own portfolio. If mathematically 00:33:48.410 --> 00:33:51.470 continuous is the theoretical limit of compounding, 00:33:51.490 --> 00:33:54.069 if money is growing every microsecond, what does 00:33:54.069 --> 00:33:56.470 the real -world utility of continuous compounding 00:33:56.470 --> 00:33:59.089 tell us about the ideal frequency of reviewing 00:33:59.089 --> 00:34:01.269 or rebalancing your own long -term investment 00:34:01.269 --> 00:34:18.809 strategy? deep dive. We hope this has equipped 00:34:18.809 --> 00:34:20.809 you with the knowledge to not just participate 00:34:20.809 --> 00:34:23.230 in the financial world, but to truly understand 00:34:23.230 --> 00:34:26.050 the mechanics, the history and the powerful math 00:34:26.050 --> 00:34:27.670 that underlies it all. Until next time.