WEBVTT 00:00:00.000 --> 00:00:02.580 okay let's unpack this with a question a question 00:00:02.580 --> 00:00:06.980 so so basic so intuitive that its answer really 00:00:06.980 --> 00:00:10.740 forms the bedrock of well all of modern finance 00:00:10.740 --> 00:00:14.220 evaluation and investment strategy so imagine 00:00:14.220 --> 00:00:16.699 you have a choice It's a guaranteed, totally 00:00:16.699 --> 00:00:19.660 risk -free choice. Would you prefer to have $50 00:00:19.660 --> 00:00:22.100 ,000 deposited into your account this morning 00:00:22.100 --> 00:00:24.920 right now, or would you prefer to receive that 00:00:24.920 --> 00:00:28.039 exact same $50 ,000 one year from today? Well, 00:00:28.059 --> 00:00:29.920 unless you know something profoundly disturbing 00:00:29.920 --> 00:00:32.560 about the future stability of the economy, the 00:00:32.560 --> 00:00:36.320 answer for any rational actor is immediate possession. 00:00:36.539 --> 00:00:38.979 You take the money now. Right. Always now. And 00:00:38.979 --> 00:00:41.979 the why behind that choice, that gut feeling, 00:00:42.119 --> 00:00:44.020 that intuitive preference for money now, that 00:00:44.020 --> 00:00:45.960 is the subject of our deep dive today. We are 00:00:45.960 --> 00:00:48.600 plunging headfirst into the time value of money 00:00:48.600 --> 00:00:51.759 or TVM. TVM is the foundational principle that 00:00:51.759 --> 00:00:55.219 formalizes this common sense observation. It's 00:00:55.219 --> 00:00:57.399 the engine driving interest rates, bond pricing, 00:00:57.600 --> 00:01:00.020 stock valuation. Mortgages. Nearly every mortgage 00:01:00.020 --> 00:01:02.340 calculation on the planet. Exactly. Our sources 00:01:02.340 --> 00:01:05.219 for this deep dive are, I have to say, surprisingly 00:01:05.219 --> 00:01:08.650 layered. We're going to start. with some ancient 00:01:08.650 --> 00:01:11.969 legal and philosophical roots, which is fascinating. 00:01:12.530 --> 00:01:15.209 Then we'll move into the necessary algebra that 00:01:15.209 --> 00:01:18.049 underpins compound interest. And then for those 00:01:18.049 --> 00:01:20.030 of you who really want to see behind the curtain, 00:01:20.150 --> 00:01:22.989 we're going to push right into the advanced mathematics. 00:01:23.090 --> 00:01:26.040 We're talking. differential equations, integral 00:01:26.040 --> 00:01:28.859 calculus, the stuff financial engineers use to 00:01:28.859 --> 00:01:31.599 model risk dynamically. So our mission is pretty 00:01:31.599 --> 00:01:34.040 simple. We want you to walk away from this understanding, 00:01:34.219 --> 00:01:36.400 not just that a dollar today is worth more than 00:01:36.400 --> 00:01:39.340 a dollar tomorrow, but how we prove that mathematically 00:01:39.340 --> 00:01:42.159 and maybe more importantly, why that principle 00:01:42.159 --> 00:01:45.319 is the constant star in the volatile galaxy of 00:01:45.319 --> 00:01:47.620 global finance. Exactly. So the core concept 00:01:47.620 --> 00:01:49.900 summary, the big takeaway, if you leave with 00:01:49.900 --> 00:01:52.459 nothing else. TVM states that there is normally 00:01:52.459 --> 00:01:54.659 a greater benefit. to receiving a sum of money 00:01:54.659 --> 00:01:57.959 now rather than an identical sum later. And the 00:01:57.959 --> 00:02:00.260 reason is simple. Money you have today can be 00:02:00.260 --> 00:02:03.200 put to work. It can be invested or saved to earn 00:02:03.200 --> 00:02:05.379 a positive rate of return. Producing more money 00:02:05.379 --> 00:02:08.000 tomorrow. And this is the vital connection to 00:02:08.000 --> 00:02:10.439 your daily life. This isn't just theory. It's 00:02:10.439 --> 00:02:13.159 the lens through which you have to weigh the 00:02:13.159 --> 00:02:15.659 opportunity costs of spending versus saving or 00:02:15.659 --> 00:02:18.439 investing. Right. When you save money in a CD 00:02:18.439 --> 00:02:21.740 or when you lend capital to someone, the interest 00:02:21.740 --> 00:02:25.210 you are paid is fundamentally compensation. It's 00:02:25.210 --> 00:02:28.370 payment for the lender's loss of the use of that 00:02:28.370 --> 00:02:30.870 money during that time period. It's the cost 00:02:30.870 --> 00:02:33.009 of deferring gratification. And that cost is 00:02:33.009 --> 00:02:35.229 measurable. We could put a number on it. Okay. 00:02:35.330 --> 00:02:37.409 So let's start with part one, those historical 00:02:37.409 --> 00:02:40.969 roots. To truly get why TVM matters, we have 00:02:40.969 --> 00:02:42.870 to look past the spreadsheets for a minute and 00:02:42.870 --> 00:02:46.629 recognize that this concept is rooted in fundamental 00:02:46.629 --> 00:02:49.330 human psychology. It's about decision making. 00:02:50.069 --> 00:02:52.349 TVM is often seen as a direct implication of 00:02:52.349 --> 00:02:55.189 what economists call time preference. Time preference. 00:02:55.250 --> 00:02:57.229 It sounds a little bit like a psychology term, 00:02:57.310 --> 00:02:59.530 but it's an economic term describing what the 00:02:59.530 --> 00:03:01.689 preference for present consumption over future 00:03:01.689 --> 00:03:04.210 consumption. That's it. Exactly. We are, for 00:03:04.210 --> 00:03:06.550 the most part, wired to value satisfaction right 00:03:06.550 --> 00:03:09.669 now versus waiting for a similar or even a slightly 00:03:09.669 --> 00:03:11.889 greater satisfaction later. I want the cookie 00:03:11.889 --> 00:03:14.909 now, not two cookies tomorrow. Pretty much. This 00:03:14.909 --> 00:03:16.889 preference for the present is nearly universal. 00:03:17.740 --> 00:03:20.419 If you give someone a choice between an immediate 00:03:20.419 --> 00:03:23.360 small reward and a slightly larger delayed reward, 00:03:23.699 --> 00:03:26.879 most people and certainly most economic models 00:03:26.879 --> 00:03:29.379 will assume the immediate option is favored. 00:03:29.960 --> 00:03:33.620 This, this inherent human impatience has a monetary 00:03:33.620 --> 00:03:36.400 cost. And that cost translates directly into 00:03:36.400 --> 00:03:39.180 the interest rate. So if I'm asking you to suppress 00:03:39.180 --> 00:03:42.000 your natural preference for spending your money 00:03:42.000 --> 00:03:44.759 today, I have to offer you some form of compensation. 00:03:45.610 --> 00:03:47.750 And that compensation has to cover more than 00:03:47.750 --> 00:03:50.490 just that psychological cost of waiting. This 00:03:50.490 --> 00:03:53.289 brings us to the required rate of return. An 00:03:53.289 --> 00:03:55.849 investor or a lender is willing to forego spending 00:03:55.849 --> 00:03:58.669 money now only if they expect a favorable net 00:03:58.669 --> 00:04:01.250 return. And that return has to be high enough 00:04:01.250 --> 00:04:03.590 to satisfy three core requirements. OK, what's 00:04:03.590 --> 00:04:05.810 the first one? First, as we said, overcoming 00:04:05.810 --> 00:04:08.069 that natural time preference, the pure cost of 00:04:08.069 --> 00:04:10.370 waiting. The impatience factor. The impatience 00:04:10.370 --> 00:04:13.889 factor. Second, and this is absolutely critical 00:04:13.889 --> 00:04:17.250 in modern economies. offsetting inflation. If 00:04:17.250 --> 00:04:20.490 you loan me $100 today, and in a year inflation 00:04:20.490 --> 00:04:23.029 has eroded the purchasing power of that $100 00:04:23.029 --> 00:04:27.089 by, say, 3%, you are already behind unless I 00:04:27.089 --> 00:04:29.250 pay you at least 3 % interest. Right, because 00:04:29.250 --> 00:04:32.050 my $100 buys fewer groceries next year, so just 00:04:32.050 --> 00:04:34.149 getting my $100 back means I've actually lost 00:04:34.149 --> 00:04:36.670 purchasing power. Exactly. Your capital must 00:04:36.670 --> 00:04:39.589 at least maintain its purchasing power. And thirdly, 00:04:39.610 --> 00:04:42.810 in most non -guaranteed scenarios, you have to 00:04:42.810 --> 00:04:45.889 be compensated for risk. The higher the probability 00:04:45.889 --> 00:04:48.569 that I won't pay you back, the higher the required 00:04:48.569 --> 00:04:51.290 rate of return must be to justify giving me the 00:04:51.290 --> 00:04:53.089 money in the first place. So it's time preference, 00:04:53.329 --> 00:04:56.189 inflation and risk. That combination, the real 00:04:56.189 --> 00:04:58.529 risk -free rate, the inflation premium and the 00:04:58.529 --> 00:05:00.889 risk premium sets the required rate of return. 00:05:01.449 --> 00:05:04.250 That rate in turn is the fundamental driver of 00:05:04.250 --> 00:05:05.910 all the discount rates we are going to discuss 00:05:05.910 --> 00:05:08.949 today. It serves as the hurdle rate. The hurdle 00:05:08.949 --> 00:05:11.610 you have to clear. The minimum expected return 00:05:11.610 --> 00:05:14.170 necessary to make an investment worthwhile. If 00:05:14.170 --> 00:05:16.550 the potential return falls below that required 00:05:16.550 --> 00:05:18.629 rate, you just keep the money in your pocket. 00:05:18.930 --> 00:05:21.810 And while we associate this kind of rigorous 00:05:21.810 --> 00:05:24.550 financial modeling with modern banks and hedge 00:05:24.550 --> 00:05:27.709 funds, our sources highlight this truly fascinating 00:05:27.709 --> 00:05:30.670 historical thread that shows this thinking is 00:05:30.670 --> 00:05:33.449 ancient. The concept was explicitly recognized 00:05:33.449 --> 00:05:36.790 and formalized in law in the Talmud around 500 00:05:36.790 --> 00:05:39.529 CE. Yes, it's a powerful illustration of the 00:05:39.529 --> 00:05:42.810 practicality of ancient law. The Babylonian Talmud. 00:05:43.079 --> 00:05:45.899 specifically in Tractate MACOS 3A, addresses 00:05:45.899 --> 00:05:49.000 a very specific legal situation concerning false 00:05:49.000 --> 00:05:51.620 witnesses in a loan contract. Okay, let's unpack 00:05:51.620 --> 00:05:53.680 this scenario. It sounds like a legal drama, 00:05:53.740 --> 00:05:56.459 but with high finance at its core. It really 00:05:56.459 --> 00:05:59.100 is. So imagine a lender provides a sum of money. 00:05:59.199 --> 00:06:02.040 The contract states the borrower must repay the 00:06:02.040 --> 00:06:05.060 loan in 10 years, a long -term loan. Okay. Now 00:06:05.060 --> 00:06:07.600 imagine two false witnesses come forward and 00:06:07.600 --> 00:06:09.439 they swear in court that the loan term was only 00:06:09.439 --> 00:06:11.889 30 days. They're lying, obviously, in order to 00:06:11.889 --> 00:06:14.069 force an immediate repayment. But then they get 00:06:14.069 --> 00:06:16.509 caught. So the court has to figure out the penalty. 00:06:16.709 --> 00:06:19.589 It can't just be a simple fine for lying because 00:06:19.589 --> 00:06:22.610 the damage they tried to inflict was purely financial. 00:06:23.029 --> 00:06:25.410 Precisely. The court needed to calculate the 00:06:25.410 --> 00:06:27.709 financial loss that the lender would have suffered 00:06:27.709 --> 00:06:30.670 if the perjury had succeeded. If the lender were 00:06:30.670 --> 00:06:33.769 forced to accept repayment in 30 days, rather 00:06:33.769 --> 00:06:36.269 than waiting the contract to 10 years, what have 00:06:36.269 --> 00:06:38.250 they lost? Well, they would have been denied 00:06:38.250 --> 00:06:41.209 the use of that capital for... Nine years and 00:06:41.209 --> 00:06:44.189 11 months. Exactly. The value of that obligation, 00:06:44.490 --> 00:06:47.269 that debt, changes dramatically based on the 00:06:47.269 --> 00:06:50.889 time horizon. A 30 -day debt obligation is vastly 00:06:50.889 --> 00:06:53.449 more valuable to the creditor than a 10 -year 00:06:53.449 --> 00:06:55.829 debt obligation, even though the principal repayment 00:06:55.829 --> 00:06:58.129 amount is identical. Why more valuable? Break 00:06:58.129 --> 00:07:00.230 that down. Because the lender gets their capital 00:07:00.230 --> 00:07:02.129 back faster. Right. That means they can immediately 00:07:02.129 --> 00:07:03.870 turn around and reinvest it. They're getting 00:07:03.870 --> 00:07:06.939 the opportunity cost sooner. By lying... The 00:07:06.939 --> 00:07:08.899 witnesses were trying to force the borrower to 00:07:08.899 --> 00:07:12.519 surrender a less valuable long -term asset. Which 00:07:12.519 --> 00:07:14.339 is the 10 -year term. For a much more valuable 00:07:14.339 --> 00:07:17.639 short -term asset, the 30 -day term. So what 00:07:17.639 --> 00:07:19.600 did the court rule? How did they quantify the 00:07:19.600 --> 00:07:22.319 punishment? The court ruled that the false witnesses 00:07:22.319 --> 00:07:24.839 had to pay the difference in the valuation. They 00:07:24.839 --> 00:07:26.540 had to figure out what the market value of a 00:07:26.540 --> 00:07:29.319 10 -year loan was versus a 30 -day loan. And 00:07:29.319 --> 00:07:32.459 the witnesses had to pay that premium. That means 00:07:32.459 --> 00:07:36.269 1 ,500 years ago. Judges were performing a calculation 00:07:36.269 --> 00:07:39.470 that is functionally identical to a modern present 00:07:39.470 --> 00:07:42.589 value calculation to assess damages. That is 00:07:42.589 --> 00:07:45.350 just breathtaking. It proves the core intellectual 00:07:45.350 --> 00:07:48.790 framework, that time is an explicit factor in 00:07:48.790 --> 00:07:51.850 monetary valuation, is embedded deep in the history 00:07:51.850 --> 00:07:54.829 of commerce and law. They knew the value of an 00:07:54.829 --> 00:07:56.810 asset wasn't just the number on the page, but 00:07:56.810 --> 00:07:59.050 the use and the reinvestment potential of that 00:07:59.050 --> 00:08:02.470 sum over time. Right. And this framework didn't 00:08:02.470 --> 00:08:05.430 just stop there. As global trade exploded during 00:08:05.430 --> 00:08:07.689 the Renaissance, the need for more rigorous, 00:08:07.850 --> 00:08:10.069 universally accepted valuation systems became 00:08:10.069 --> 00:08:12.170 critical. And this is where the School of Salamanca 00:08:12.170 --> 00:08:14.449 comes in. Yes. The notion was later described 00:08:14.449 --> 00:08:16.750 in detail by the Spanish theologian and economist 00:08:16.750 --> 00:08:19.709 Martín de Azpilcueta of the School of Salamanca 00:08:19.709 --> 00:08:21.889 in the 16th century. So what was his contribution? 00:08:22.550 --> 00:08:24.850 Azpilcueta focused heavily on the concepts of 00:08:24.850 --> 00:08:27.629 scarcity and demand. He noted that money is worth 00:08:27.629 --> 00:08:31.009 more when and where it is scarce. This tied directly 00:08:31.009 --> 00:08:34.200 into time. Money available today, when resources 00:08:34.200 --> 00:08:36.799 might be scarce or immediate opportunities exist, 00:08:37.080 --> 00:08:39.279 is inherently worth more than money promised 00:08:39.279 --> 00:08:42.700 for a future unknown environment. So he was formalizing 00:08:42.700 --> 00:08:45.299 the opportunity cost argument. He was. And this 00:08:45.299 --> 00:08:47.179 intellectual development really paved the way 00:08:47.179 --> 00:08:50.159 for the formal algebraic tools that we use today. 00:08:50.399 --> 00:08:53.340 The need for precise calculation in a rapidly 00:08:53.340 --> 00:08:56.240 globalizing world demanded that mathematics replace 00:08:56.240 --> 00:08:58.600 legal precedent. Which is a perfect transition 00:08:58.600 --> 00:09:01.539 to part two, the algebra of time. So we move 00:09:01.539 --> 00:09:03.500 from the philosophy of time preference in ancient 00:09:03.500 --> 00:09:06.259 law to the rigor of modern financial equations. 00:09:06.639 --> 00:09:09.379 Every single TVM calculation, whether it's simple 00:09:09.379 --> 00:09:11.799 or incredibly complex, revolves around finding 00:09:11.799 --> 00:09:13.899 the net value of cash flows at different points 00:09:13.899 --> 00:09:16.460 in time. And it all relies on four key variables. 00:09:16.700 --> 00:09:18.659 The universal language of finance, you called 00:09:18.659 --> 00:09:21.399 it. It is. You must have. One. The balance, which 00:09:21.399 --> 00:09:23.759 is either your starting lump sum, the present 00:09:23.759 --> 00:09:26.259 value, or your ending lump sum, the future value. 00:09:26.360 --> 00:09:30.220 Okay, PV or FE. Two, the periodic rate, usually 00:09:30.220 --> 00:09:33.539 written as I or R. This is your rate of interest 00:09:33.539 --> 00:09:35.679 or your discount rate or your rate of return 00:09:35.679 --> 00:09:38.399 per period. The growth engine. Exactly. Three, 00:09:38.559 --> 00:09:41.779 the number of periods, N or T. The number of 00:09:41.779 --> 00:09:44.059 compounding or discounting intervals, years, 00:09:44.299 --> 00:09:47.059 months, quarters. How long the money works for. 00:09:47.279 --> 00:09:50.820 And four. Cash flows, often written as A or PMT. 00:09:51.279 --> 00:09:54.120 This represents a series of payments or contributions 00:09:54.120 --> 00:09:56.820 or withdrawals. Okay, let's use that simple example 00:09:56.820 --> 00:09:58.519 again, but this time let's label the variables. 00:09:58.639 --> 00:10:00.879 If you invest 100 pounds. That's your present 00:10:00.879 --> 00:10:03.759 value, your PV. For one year. That's N equals 00:10:03.759 --> 00:10:07.360 1. Earning 5 % interest. I equals .05. It will 00:10:07.360 --> 00:10:10.000 be worth 105 after one year. And that's your 00:10:10.000 --> 00:10:12.740 future value, your FV. That extra five pounds 00:10:12.740 --> 00:10:14.940 is the compensation for deferring your consumption 00:10:14.940 --> 00:10:17.200 for one year. So that simple scenario reveals 00:10:17.200 --> 00:10:19.679 the core operation. We're either compounding 00:10:19.679 --> 00:10:22.860 forward to find a future value or the reverse. 00:10:23.039 --> 00:10:24.940 Or discounting backward to find a present value. 00:10:25.139 --> 00:10:27.419 Let's focus on that discounting process because 00:10:27.419 --> 00:10:30.620 this is where valuation really happens. The entire 00:10:30.620 --> 00:10:33.870 principle allows us to value an asset. say, a 00:10:33.870 --> 00:10:36.149 company or a building, based on the expected 00:10:36.149 --> 00:10:38.169 stream of future income it's going to generate. 00:10:38.309 --> 00:10:40.549 That's right. We take those expected future incomes, 00:10:40.730 --> 00:10:43.529 we discount each one back to the present, and 00:10:43.529 --> 00:10:45.710 then we sum them all up. This is the heart of 00:10:45.710 --> 00:10:49.129 discounted cash flow analysis, or DCF. So DCF 00:10:49.129 --> 00:10:51.610 turns a whole series of future promises into 00:10:51.610 --> 00:10:54.850 a single concrete number today. A lump sum present 00:10:54.850 --> 00:10:57.049 value of the entire stream. It tells you what 00:10:57.049 --> 00:10:58.929 it's worth right now. Okay, let's look at the 00:10:58.929 --> 00:11:01.330 absolute kernel of the mathematics, the core 00:11:01.330 --> 00:11:03.870 formula for calculating the present value, the 00:11:03.870 --> 00:11:07.409 PV, of a single future value, FV. This is the 00:11:07.409 --> 00:11:09.710 simplest form. This formula shows how a future 00:11:09.710 --> 00:11:13.029 sum, FV, to be received in one year is discounted 00:11:13.029 --> 00:11:15.909 back to today using an interest rate, R. It's 00:11:15.909 --> 00:11:19.590 simply PV equals FV divided by 1 plus R. Okay, 00:11:19.649 --> 00:11:22.110 that makes perfect sense. The denominator, that 00:11:22.110 --> 00:11:24.779 1 plus R, is the growth factor. To find what 00:11:24.779 --> 00:11:26.879 a future amount is worth today, you have to divide 00:11:26.879 --> 00:11:29.080 by that growth factor. You have to undo the growth. 00:11:29.200 --> 00:11:31.940 Now, when we generalize this for multiple periods, 00:11:32.059 --> 00:11:34.879 n, the power of compounding comes into play. 00:11:35.019 --> 00:11:37.220 If you're discounting a sum to be received in 00:11:37.220 --> 00:11:40.340 10 years, you have to undo the compounding that 00:11:40.340 --> 00:11:42.639 happened over all 10 of those periods. You're 00:11:42.639 --> 00:11:45.419 not just dividing by 1 plus r once. You're dividing 00:11:45.419 --> 00:11:48.500 by it 10 times. The foundational equation for 00:11:48.500 --> 00:11:51.960 the future value, fv, of a present sum shows 00:11:51.960 --> 00:11:55.919 this compounding in action. It's FV equals PV 00:11:55.919 --> 00:11:59.379 times 1 plus i to the power of n. And the inverse 00:11:59.379 --> 00:12:02.220 of that equation gives us the generalized present 00:12:02.220 --> 00:12:06.080 value PV formula for a single lump sum received 00:12:06.080 --> 00:12:09.059 n periods in the future. Which is PV equals FV 00:12:09.059 --> 00:12:12.360 divided by 1 plus i to the power of n. That exponent, 00:12:12.539 --> 00:12:14.659 that n, that's where the magic happens, isn't 00:12:14.659 --> 00:12:16.600 it? That is where compounding lives. It's what 00:12:16.600 --> 00:12:18.759 differentiates simple interest from compound 00:12:18.759 --> 00:12:21.259 interest. 100%. Simple interest is just calculated 00:12:21.259 --> 00:12:22.980 on the original principle. Compound interest 00:12:22.980 --> 00:12:25.200 is calculated on the principle and on all the 00:12:25.200 --> 00:12:27.500 previously accumulated interest. It's interest 00:12:27.500 --> 00:12:29.480 earning its own interest. Can we use a quick 00:12:29.480 --> 00:12:31.940 example? Let's say $10 ,000 at 10 % for three 00:12:31.940 --> 00:12:34.299 years. Sure. With simple interest, it's easy. 00:12:34.879 --> 00:12:38.039 10 % of $10 ,000 is $1 ,000. Over three years, 00:12:38.100 --> 00:12:40.700 that's $3 ,000 in interest. Your total is $13 00:12:40.700 --> 00:12:44.899 ,000. Okay. Now with compound interest. Year 00:12:44.899 --> 00:12:47.159 one is the same. You earn $1 ,000. Your balance 00:12:47.159 --> 00:12:50.779 is now $11 ,000. But in year two, you earn 10 00:12:50.779 --> 00:12:54.559 % on $11 ,000. That's $1 ,100. Your balance is 00:12:54.559 --> 00:12:58.419 now $12 ,100. So you made an extra $100 on last 00:12:58.419 --> 00:13:00.919 year's interest. Exactly. And in year three, 00:13:01.059 --> 00:13:04.179 you earned 10 % on $12 ,100, which is $1 ,210. 00:13:04.860 --> 00:13:09.879 Your final total is $13 ,310. That extra $310 00:13:09.879 --> 00:13:13.159 is the power of compounding. And that gap gets 00:13:13.159 --> 00:13:15.700 exponentially wider over time. It's not just 00:13:15.700 --> 00:13:17.799 about interest earning interest. It's about geometric 00:13:17.799 --> 00:13:20.720 growth. When you look at long -term savings or 00:13:20.720 --> 00:13:23.529 debt. The difference between n equals 10 and 00:13:23.529 --> 00:13:26.850 n equals 30 is absolutely staggering because 00:13:26.850 --> 00:13:29.230 the growth rate is applied not additively but 00:13:29.230 --> 00:13:32.470 multiplicatively every single period. Now let's 00:13:32.470 --> 00:13:34.730 talk about the single most critical variable 00:13:34.730 --> 00:13:38.269 in all of this, the discount rate, that R. We 00:13:38.269 --> 00:13:40.590 said earlier that an incorrect rate makes the 00:13:40.590 --> 00:13:43.730 results meaningless. Why is it so hard to determine 00:13:43.730 --> 00:13:47.159 the right discount rate? What goes into it? Well, 00:13:47.220 --> 00:13:49.100 the discount rate is that hurdle rate we discussed. 00:13:49.720 --> 00:13:53.480 But in practice, it's a subjective dynamic estimate 00:13:53.480 --> 00:13:56.679 of the appropriate cost of capital for a specific 00:13:56.679 --> 00:13:59.159 investment. It's not a single number you can 00:13:59.159 --> 00:14:01.639 just look up. So it's not just the current interest 00:14:01.639 --> 00:14:03.860 rate at the bank? Not at all. For a corporation 00:14:03.860 --> 00:14:06.460 valuing a new project, for example, they might 00:14:06.460 --> 00:14:07.980 use something called the weighted average cost 00:14:07.980 --> 00:14:11.340 of capital, or WACC. That blends the cost of 00:14:11.340 --> 00:14:13.389 their debt. and the cost of their equity. And 00:14:13.389 --> 00:14:15.629 for a stock analyst? They might use the capital 00:14:15.629 --> 00:14:18.730 asset pricing model, the CAPM, to determine the 00:14:18.730 --> 00:14:20.970 required rate of return for that specific stock 00:14:20.970 --> 00:14:23.490 based on its market risk relative to the whole 00:14:23.490 --> 00:14:25.529 market. So it's a measure of the risk inherent 00:14:25.529 --> 00:14:27.990 in the cash flows you're valuing, the opportunity 00:14:27.990 --> 00:14:30.029 cost available elsewhere, and that time premium 00:14:30.029 --> 00:14:33.190 we talked about. Precisely. If you are valuing 00:14:33.190 --> 00:14:35.309 a guaranteed cash flow from a U .S. Treasury 00:14:35.309 --> 00:14:39.029 bill, your R is going to be very low. If you're 00:14:39.029 --> 00:14:41.840 valuing the projected cash flow from a volatile 00:14:41.840 --> 00:14:44.360 tech startup, ours is going to be extremely high, 00:14:44.440 --> 00:14:47.860 maybe 25 % or 30%. Because the risk of failure 00:14:47.860 --> 00:14:50.620 is so much higher. Exactly. The higher the perceived 00:14:50.620 --> 00:14:53.179 risk or the higher the available return in alternative 00:14:53.179 --> 00:14:55.759 investments, the higher your discount rate. And 00:14:55.759 --> 00:14:57.980 a higher discount rate means? A lower present 00:14:57.980 --> 00:15:00.820 value for those future dollars. If I can easily 00:15:00.820 --> 00:15:03.559 earn 20 % of my money somewhere else, I am not 00:15:03.559 --> 00:15:05.519 going to pay much today for a dollar that only 00:15:05.519 --> 00:15:08.200 arrives in five years. That future dollar gets 00:15:08.200 --> 00:15:10.700 discounted much more steeply because my opportunity 00:15:10.700 --> 00:15:14.059 cost is enormous. So using the wrong rate, say, 00:15:14.179 --> 00:15:16.519 using 5 % when the actual cost of your capital 00:15:16.519 --> 00:15:19.139 is 10%, could lead to a massive overvaluation. 00:15:19.480 --> 00:15:21.120 You'd think a project is worth way more than 00:15:21.120 --> 00:15:23.940 it is. And you could lose a lot of money. Conversely, 00:15:23.940 --> 00:15:25.860 if you use a rate that is too high, you might 00:15:25.860 --> 00:15:27.899 incorrectly reject a very profitable investment 00:15:27.899 --> 00:15:30.559 opportunity. The discount rate is the absolute 00:15:30.559 --> 00:15:33.360 linchpin of valuation integrity. Okay, let's 00:15:33.360 --> 00:15:35.340 move from these single lump sums to the reality 00:15:35.340 --> 00:15:38.740 of business valuations, which often involve multiple 00:15:38.740 --> 00:15:41.539 distinct cash flows happening at different points 00:15:41.539 --> 00:15:44.679 in time. Right. To handle this, we use the cumulative 00:15:44.679 --> 00:15:47.419 present value formula. It simply generalizes 00:15:47.419 --> 00:15:50.960 that basic PV concept to sum up multiple cash 00:15:50.960 --> 00:15:54.519 flows, let's call them FV sub T, occurring at 00:15:54.519 --> 00:15:58.000 different times, T. So if a project pays me $1 00:15:58.000 --> 00:16:01.159 ,000 in year one, $2 ,000 in year two, and $5 00:16:01.159 --> 00:16:04.039 ,000 in year three, three. I can't use one simple 00:16:04.039 --> 00:16:06.179 formula. You can't. You have to treat each future 00:16:06.179 --> 00:16:08.779 cash flow as its own unique lump sum problem. 00:16:09.100 --> 00:16:11.220 You discount the thousand from year one back 00:16:11.220 --> 00:16:13.679 one period. You discount the two thousand from 00:16:13.679 --> 00:16:16.059 year two back two periods. You discount the five 00:16:16.059 --> 00:16:18.019 thousand from year three back three periods. 00:16:18.200 --> 00:16:20.159 And then you add those three resulting present 00:16:20.159 --> 00:16:22.940 values together. The formula with the big sigma 00:16:22.940 --> 00:16:25.120 for summation looks intimidating, but that's 00:16:25.120 --> 00:16:27.179 all it's doing. It's just a systematic way of 00:16:27.179 --> 00:16:29.299 doing that three step process for however many 00:16:29.299 --> 00:16:32.720 periods you have. Exactly. The formula collapses 00:16:32.720 --> 00:16:35.720 all those disparate future promises into a single 00:16:35.720 --> 00:16:38.879 comprehensive value today. It provides clarity. 00:16:39.279 --> 00:16:42.139 You can look at a proposal for, say, an infrastructure 00:16:42.139 --> 00:16:44.279 project that has variable income projections 00:16:44.279 --> 00:16:47.519 over 50 years and boil that entire complex schedule 00:16:47.519 --> 00:16:50.159 down to one number. This is what that project 00:16:50.159 --> 00:16:52.679 is worth today, given our hurdle rate. Before 00:16:52.679 --> 00:16:55.440 we move on, we have to revisit a crucial detail 00:16:55.440 --> 00:16:57.659 you mentioned, which is the cause of so many 00:16:57.659 --> 00:17:01.200 financial errors. The requirement that the interest 00:17:01.200 --> 00:17:05.440 rate I must always match the period N. This is 00:17:05.440 --> 00:17:07.200 non -negotiable and is where people get tripped 00:17:07.200 --> 00:17:09.579 up all the time. If you are dealing with an annual 00:17:09.579 --> 00:17:11.339 interest rate, but your cash flows are happening 00:17:11.339 --> 00:17:14.019 monthly, you must divide that annual rate by 00:17:14.019 --> 00:17:16.420 12. And then you have to adjust the periods too, 00:17:16.519 --> 00:17:18.099 right? And then you multiply the number of periods, 00:17:18.160 --> 00:17:20.359 N, by 12. If you have a five -year loan with 00:17:20.359 --> 00:17:23.660 monthly payments, N isn't five, it's 60. A mortgage 00:17:23.660 --> 00:17:25.599 is the classic example. The bank quotes you an 00:17:25.599 --> 00:17:28.039 annual percentage rate, an APR, but you make 00:17:28.039 --> 00:17:30.740 payments monthly. Right. If you try to calculate 00:17:30.740 --> 00:17:32.779 the future value of your monthly savings contribution 00:17:32.779 --> 00:17:35.700 using the annual rate as I and the number of 00:17:35.700 --> 00:17:38.240 years as N, your calculation will be dramatically 00:17:38.240 --> 00:17:41.559 wrong. Why? Because you fail to account for the 00:17:41.559 --> 00:17:43.779 fact that you're earning interest and compounding 00:17:43.779 --> 00:17:46.160 every single month, not just once at the end 00:17:46.160 --> 00:17:48.259 of the year. That monthly compounding is powerful. 00:17:48.640 --> 00:17:51.220 It is. Let's say you have a 12 % annual rate. 00:17:51.500 --> 00:17:55.599 If you incorrectly use I equals 0 .12 and N equals 00:17:55.599 --> 00:17:59.200 one year, your growth factor is 1 .1. But if 00:17:59.200 --> 00:18:02.779 you correctly use I equals 0 .01, which is 12 00:18:02.779 --> 00:18:06.339 % divided by 12, and N equals 12 periods, the 00:18:06.339 --> 00:18:09.299 factor is 1 .01 to the power. hour of 12. Which 00:18:09.299 --> 00:18:12.539 it comes out to. About 1 .1268. That 0 .68 % 00:18:12.539 --> 00:18:14.960 difference might sound small, but over a 30 -year 00:18:14.960 --> 00:18:18.480 $500 ,000 mortgage, that error becomes tens of 00:18:18.480 --> 00:18:21.259 thousands of dollars. Always, always match the 00:18:21.259 --> 00:18:23.200 rate to the period. Okay, that's a great practical 00:18:23.200 --> 00:18:25.519 point. Let's move into part three, applying TVM 00:18:25.519 --> 00:18:27.539 to these recurring cash streams, starting with 00:18:27.539 --> 00:18:29.819 annuities. Moving from single long sums to the 00:18:29.819 --> 00:18:32.119 reality of recurring periodic payments brings 00:18:32.119 --> 00:18:34.079 us to the most practical and frequently used 00:18:34.079 --> 00:18:38.150 application of TVM. An annuity. An annuity. It's 00:18:38.150 --> 00:18:41.250 the structural basis for practically every recurring 00:18:41.250 --> 00:18:43.910 financial obligation or asset you can think of. 00:18:44.130 --> 00:18:47.890 Leases, rent payments, installment loans, retirement 00:18:47.890 --> 00:18:50.750 withdrawals. Formally, it's a series of equal 00:18:50.750 --> 00:18:52.849 payments of receipts that occur at evenly spaced 00:18:52.849 --> 00:18:55.750 intervals. The equal payment and evenly spaced 00:18:55.750 --> 00:18:58.430 interval constraints are key, right? That's what 00:18:58.430 --> 00:19:00.690 lets us simplify the math. That's what allows 00:19:00.690 --> 00:19:04.049 the general cumulative PV formula to be simplified 00:19:04.049 --> 00:19:07.789 into the much cleaner annuity formulas. We need 00:19:07.789 --> 00:19:09.849 to distinguish between the two fundamental types 00:19:09.849 --> 00:19:12.410 of annuities. It all comes down to timing. It 00:19:12.410 --> 00:19:14.950 does. The ordinary annuity assumes payments occur 00:19:14.950 --> 00:19:17.490 at the end of each period. Think of a standard 00:19:17.490 --> 00:19:20.450 car loan or home mortgage payment. It's due 30 00:19:20.450 --> 00:19:23.450 days after the loan is disbursed. Most standard 00:19:23.450 --> 00:19:25.829 financial models just assume an ordinary annuity 00:19:25.829 --> 00:19:28.130 unless you specify otherwise. And the other type 00:19:28.130 --> 00:19:31.390 is the annuity due. The annuity due assumes payments 00:19:31.390 --> 00:19:33.549 or receipts occur at the beginning of each period. 00:19:34.079 --> 00:19:36.299 The classic example is an apartment rental payment, 00:19:36.460 --> 00:19:37.980 which is usually due on the first day of the 00:19:37.980 --> 00:19:39.880 month before you use the space for that month. 00:19:40.059 --> 00:19:42.119 So why does that timing distinction matter so 00:19:42.119 --> 00:19:45.759 much? Mathematically and financially, a payment 00:19:45.759 --> 00:19:48.200 you receive at the beginning of a period has 00:19:48.200 --> 00:19:51.880 one extra compounding period to earn interest 00:19:51.880 --> 00:19:54.680 compared to a payment received at the end. It's 00:19:54.680 --> 00:19:57.160 that simple. So if you're calculating the present 00:19:57.160 --> 00:20:00.299 value of an annuity due, it will always be higher 00:20:00.299 --> 00:20:03.579 than the PV of an ordinary annuity. Always. Because 00:20:03.579 --> 00:20:06.019 you receive that first payment immediately, it's 00:20:06.019 --> 00:20:08.579 not discounted at all, and every subsequent payment 00:20:08.579 --> 00:20:11.059 starts earning interest one period sooner. So 00:20:11.059 --> 00:20:13.359 if I'm the one receiving payments, I want an 00:20:13.359 --> 00:20:15.380 annuity due. If I'm the one making payments, 00:20:15.559 --> 00:20:18.380 I want an ordinary annuity. You've got it. The 00:20:18.380 --> 00:20:20.680 calculation adjustment is incredibly simple. 00:20:20.799 --> 00:20:24.220 To get the PV or the FV of an annuity due, you 00:20:24.220 --> 00:20:26.299 just calculate the ordinary annuity value and 00:20:26.299 --> 00:20:29.529 then multiply it by 1 plus i. That simple multiplication 00:20:29.529 --> 00:20:32.230 accounts for that one extra period of compounding 00:20:32.230 --> 00:20:35.069 across the entire stream of payments. Okay, let's 00:20:35.069 --> 00:20:37.450 focus on the present value of an annuity, PVA. 00:20:37.710 --> 00:20:40.549 This tells us the current lump sum cash value 00:20:40.549 --> 00:20:43.369 of that stream of fixed future payments. Right. 00:20:43.589 --> 00:20:46.930 The PVA formula, while it looks complex, is really 00:20:46.930 --> 00:20:49.750 just a mathematical shortcut. Instead of running 00:20:49.750 --> 00:20:52.950 the cumulative PD formula from Part 2 over and 00:20:52.950 --> 00:20:55.609 over again and adding them all up, the PVA formula 00:20:55.609 --> 00:20:58.150 uses the characteristics of a geometric series 00:20:58.150 --> 00:21:01.670 to collapse that long calculation into one single 00:21:01.670 --> 00:21:03.910 expression. And this is what loan servicers use 00:21:03.910 --> 00:21:06.069 to figure out how much principal is left on my 00:21:06.069 --> 00:21:09.130 car loan or my mortgage at any given time. Exactly. 00:21:09.150 --> 00:21:12.670 The future value of an annuity, FEA, is just 00:21:12.670 --> 00:21:15.049 as important, especially for our listeners thinking 00:21:15.049 --> 00:21:17.740 about long -term savings or retirement. This 00:21:17.740 --> 00:21:20.180 one is about building wealth, not valuing a debt. 00:21:20.420 --> 00:21:23.519 Yes. The FBA calculates the accumulated value 00:21:23.519 --> 00:21:26.519 of a stream of regular contributions after end 00:21:26.519 --> 00:21:28.779 periods, assuming those payments are consistently 00:21:28.779 --> 00:21:31.259 invested at a rate Asian. This is the power of 00:21:31.259 --> 00:21:33.579 dollar cost averaging and compounding over decades. 00:21:34.079 --> 00:21:36.740 For most people, this is the most powerful formula 00:21:36.740 --> 00:21:39.319 they'll ever encounter because it shows the true 00:21:39.319 --> 00:21:42.019 wealth building potential of regular discipline 00:21:42.019 --> 00:21:44.559 saving. But the real world is rarely static. 00:21:45.099 --> 00:21:46.819 Payments aren't always fixed. What if they're 00:21:46.819 --> 00:21:49.059 designed to increase over time? This brings us 00:21:49.059 --> 00:21:51.220 to the growing annuity. This is a scenario that 00:21:51.220 --> 00:21:54.200 perfectly matches reality. For instance, when 00:21:54.200 --> 00:21:56.339 you're planning for a long term pension, you 00:21:56.339 --> 00:21:58.019 should probably assume your annual contributions 00:21:58.019 --> 00:22:00.480 will increase by a certain percentage as your 00:22:00.480 --> 00:22:03.119 salary increases over your career. Or in real 00:22:03.119 --> 00:22:05.079 estate, a commercial lease might have an annual 00:22:05.079 --> 00:22:07.740 contractual rent escalator indexed to inflation 00:22:07.740 --> 00:22:11.240 or some fixed rate. Perfect example. So in a 00:22:11.240 --> 00:22:14.029 growing annuity. Each successive cash flow grows 00:22:14.029 --> 00:22:18.029 by a factor of 1 plus g, where g is that expected 00:22:18.029 --> 00:22:20.329 growth rate. And this must make the math a lot 00:22:20.329 --> 00:22:22.670 harder. It adds a significant layer of mathematical 00:22:22.670 --> 00:22:26.049 complexity. It forces the analyst to use two 00:22:26.049 --> 00:22:28.109 different specialized formulas for the present 00:22:28.109 --> 00:22:30.450 value, depending on whether the growth rate, 00:22:30.529 --> 00:22:33.630 g, is equal to the interest rate, i, or not. 00:22:34.119 --> 00:22:36.480 Most standard financial calculators can't handle 00:22:36.480 --> 00:22:38.599 this. It requires a deeper understanding of the 00:22:38.599 --> 00:22:40.880 underlying math. It really highlights that the 00:22:40.880 --> 00:22:43.519 moment you introduce variability, you leave simple 00:22:43.519 --> 00:22:45.980 spreadsheet arithmetic behind, and you enter 00:22:45.980 --> 00:22:49.059 the domain of true financial modeling. Now let's 00:22:49.059 --> 00:22:51.220 go beyond the finite and discuss the theoretical 00:22:51.220 --> 00:22:54.539 ultimate expression of TVM, the concept of forever, 00:22:54.759 --> 00:22:58.039 the perpetuity. A perpetuity is defined as an 00:22:58.039 --> 00:23:00.880 infinite and constant stream of identical cash 00:23:00.880 --> 00:23:04.400 flows. This seems like a paradox How can you 00:23:04.400 --> 00:23:06.339 possibly calculate the current value of something 00:23:06.339 --> 00:23:08.539 that lasts forever? Wouldn't it be infinite? 00:23:08.819 --> 00:23:11.440 It seems like it should be But the key is that 00:23:11.440 --> 00:23:14.640 the math simplifies dramatically, precisely because 00:23:14.640 --> 00:23:17.740 we introduce infinity. We derive the perpetuity 00:23:17.740 --> 00:23:21.380 formula by taking that PVA formula and allowing 00:23:21.380 --> 00:23:24.039 the number of periods, n, to approach infinity. 00:23:24.299 --> 00:23:25.839 Okay. Talk us through the mathematical insight 00:23:25.839 --> 00:23:28.279 there. In the PVA formula, there's a term, 1 00:23:28.279 --> 00:23:31.779 divided by 1 plus i to the power of n. That represents 00:23:31.779 --> 00:23:34.079 the discounting of the final payment. Right. 00:23:34.140 --> 00:23:36.819 And as n, the number of periods, gets larger 00:23:36.819 --> 00:23:39.940 and larger approaching infinity, that denominator... 00:23:40.190 --> 00:23:43.210 1 plus i to the n grows infinitely large. And 00:23:43.210 --> 00:23:45.809 when you divide 1 by an infinitely large number, 00:23:46.009 --> 00:23:48.849 the result gets closer and closer to? Zero. The 00:23:48.849 --> 00:23:51.589 present value of any payment made in the extremely 00:23:51.589 --> 00:23:54.630 distant future essentially vanishes. It becomes 00:23:54.630 --> 00:23:56.630 mathematically irrelevant to its value today. 00:23:56.849 --> 00:23:58.829 Ah, so the value of a payment I'm supposed to 00:23:58.829 --> 00:24:02.009 receive 500 years from now is so heavily discounted 00:24:02.009 --> 00:24:04.230 that it's basically worthless today. We don't 00:24:04.230 --> 00:24:06.970 even need to account for it. Precisely. The effect 00:24:06.970 --> 00:24:09.730 of discounting overcomes the infinitude of time. 00:24:10.059 --> 00:24:12.920 This simplification causes major parts of the 00:24:12.920 --> 00:24:16.140 PVA formula to cancel out, leaving us with this 00:24:16.140 --> 00:24:18.480 elegantly simple calculation for the present 00:24:18.480 --> 00:24:21.700 value of a perpetuity, PVP. And what is that 00:24:21.700 --> 00:24:24.740 formula? It's just PV equals A divided by I, 00:24:24.819 --> 00:24:26.980 the payment amount divided by the interest rate. 00:24:27.079 --> 00:24:29.779 That's wonderfully clean. So if you want to receive 00:24:29.779 --> 00:24:33.140 a fixed payment of, say, $1 ,000 a year indefinitely, 00:24:33.240 --> 00:24:36.160 and the interest rate is 5%, You just divide 00:24:36.160 --> 00:24:39.859 $1 ,000 by 0 .05. And you get $20 ,000. That 00:24:39.859 --> 00:24:42.099 formula tells you the exact lump sum you need 00:24:42.099 --> 00:24:45.420 to invest today at rate I to generate that stream 00:24:45.420 --> 00:24:47.579 of returns forever without ever touching the 00:24:47.579 --> 00:24:50.039 principal. Where is this used in the real world? 00:24:50.220 --> 00:24:52.880 It's used to value specific assets, like preferred 00:24:52.880 --> 00:24:55.339 stock, which often pays a set dividend indefinitely, 00:24:55.480 --> 00:24:57.980 or certain old government bonds called consuls. 00:24:58.119 --> 00:25:00.079 But just as with annuities, we have to account 00:25:00.079 --> 00:25:02.779 for growth. And this leads to the very powerful 00:25:02.779 --> 00:25:05.869 growing perpetuity. This is a perpetual annuity 00:25:05.869 --> 00:25:08.809 where the payment grows at a fixed rate, gee, 00:25:08.990 --> 00:25:12.769 forever. This formula is absolutely crucial because 00:25:12.769 --> 00:25:16.549 it attempts to model businesses that are expected 00:25:16.549 --> 00:25:19.769 to exist indefinitely and generate perpetually 00:25:19.769 --> 00:25:22.970 increasing cash flows, maybe slightly outpacing 00:25:22.970 --> 00:25:25.190 inflation. There's a critical condition for this 00:25:25.190 --> 00:25:27.529 formula to work, though, right? For it to give 00:25:27.529 --> 00:25:30.150 a rational economic result. Yes. And this is 00:25:30.150 --> 00:25:33.259 non -negotiable. The growth rate, G, must be 00:25:33.259 --> 00:25:36.160 strictly less than the discount rate, I. What 00:25:36.160 --> 00:25:38.299 happens if it's not? What if G is bigger than 00:25:38.299 --> 00:25:40.819 I? If the expected growth rate of the payments 00:25:40.819 --> 00:25:43.359 were equal to or higher than your required discount 00:25:43.359 --> 00:25:45.980 rate, the resulting present value would be infinite 00:25:45.980 --> 00:25:48.839 or negative. It would mean the asset is too valuable 00:25:48.839 --> 00:25:51.319 to quantify a mathematical impossibility under 00:25:51.319 --> 00:25:53.940 stable economic conditions. It basically means 00:25:53.940 --> 00:25:55.640 the payments are growing faster than you can 00:25:55.640 --> 00:25:57.819 discount them. OK, so assuming that rational 00:25:57.819 --> 00:26:01.299 condition G is less than I holds, the resulting 00:26:01.299 --> 00:26:03.960 formula is perhaps the single most famous valuation 00:26:03.960 --> 00:26:07.339 equation in equity markets. It is. It's PV equals 00:26:07.339 --> 00:26:10.640 A divided by I minus G. And that is the Gordon 00:26:10.640 --> 00:26:12.740 growth model. Also known as the dividend discount 00:26:12.740 --> 00:26:14.720 model. Let's spend a moment on this one because 00:26:14.720 --> 00:26:17.240 this is the perfect illustration of TVM moving 00:26:17.240 --> 00:26:20.259 from theory into practical high -stakes valuation. 00:26:20.720 --> 00:26:24.559 How exactly is it used to value a stock? Okay, 00:26:24.599 --> 00:26:28.160 so in stock valuation, the A represents the company's 00:26:28.160 --> 00:26:30.180 expected dividend payment one year from now. 00:26:30.339 --> 00:26:33.539 The I is the investor's required rate of return 00:26:33.539 --> 00:26:36.299 for that stock, which is based on its risk. And 00:26:36.299 --> 00:26:39.440 the G is the expected long -term stable growth 00:26:39.440 --> 00:26:41.819 rate of those dividends into perpetuity. And 00:26:41.819 --> 00:26:43.660 the results of that simple division tells you? 00:26:43.779 --> 00:26:46.039 The intrinsic value of the stock. It's the maximum 00:26:46.039 --> 00:26:47.859 price you should be willing to pay for that stock 00:26:47.859 --> 00:26:50.759 today, based on those assumptions. It's the intellectual 00:26:50.759 --> 00:26:53.059 framework for a huge amount of equity research. 00:26:53.400 --> 00:26:55.460 Well, what makes it so challenging to use in 00:26:55.460 --> 00:26:57.900 practice? It seems so simple. It's powerful, 00:26:58.079 --> 00:27:00.900 but it's highly sensitive to its inputs. First, 00:27:01.000 --> 00:27:03.319 that required rate of return, I, is just an estimate 00:27:03.319 --> 00:27:06.299 based on risk. Second, estimating a sustainable 00:27:06.299 --> 00:27:09.319 perpetual growth rate, G, is notoriously difficult. 00:27:10.170 --> 00:27:12.230 Analysts often anchor this to the long -term 00:27:12.230 --> 00:27:14.670 expected rate of inflation or GDP growth, but 00:27:14.670 --> 00:27:18.190 even a tiny error, say 2 % versus 2 .5%, gets 00:27:18.190 --> 00:27:20.589 magnified because G is in the denominator. And 00:27:20.589 --> 00:27:23.009 third, I assume it only works for certain kinds 00:27:23.009 --> 00:27:26.049 of companies. Absolutely. It assumes the company 00:27:26.049 --> 00:27:28.490 will exist forever and pay a stable, growing 00:27:28.490 --> 00:27:31.549 dividend. That immediately excludes most high 00:27:31.549 --> 00:27:33.569 -growth tech companies that are reinvesting all 00:27:33.569 --> 00:27:36.329 of their earnings and don't pay a dividend. It's 00:27:36.329 --> 00:27:39.130 best suited for a stable, mature... Dividend 00:27:39.130 --> 00:27:41.190 -paying companies like utilities or consumer 00:27:41.190 --> 00:27:44.049 staples. Powerful, but you have to be very careful 00:27:44.049 --> 00:27:47.170 with your assumptions. You do. Now let's synthesize 00:27:47.170 --> 00:27:49.069 all these applications with a classic example. 00:27:49.869 --> 00:27:52.650 Coupon bond pricing. This is where we combine 00:27:52.650 --> 00:27:55.089 the two major calculation types we've just covered. 00:27:55.309 --> 00:27:57.069 So when you buy a coupon bond, you're essentially 00:27:57.069 --> 00:27:59.509 buying two distinct financial assets bundled 00:27:59.509 --> 00:28:01.869 into one security. That's a perfect way to put 00:28:01.869 --> 00:28:04.269 it. A coupon bond provides two types of cash 00:28:04.269 --> 00:28:07.190 flows over its life. First, you get the stream 00:28:07.190 --> 00:28:09.930 of periodic coupon payments. These are fixed 00:28:09.930 --> 00:28:12.349 regular payments. So that part is just an annuity. 00:28:12.569 --> 00:28:14.609 Precisely. That part is treated as an ordinary 00:28:14.609 --> 00:28:17.349 annuity. You calculate the present value of that 00:28:17.349 --> 00:28:20.190 entire stream of coupons using the PVA formula. 00:28:20.490 --> 00:28:23.549 And second, at the very end, when the bond matures, 00:28:23.650 --> 00:28:27.170 the investor receives the large lump sum return 00:28:27.170 --> 00:28:30.009 of capital. the face value or par value of the 00:28:30.009 --> 00:28:33.289 bond. And that single large final payment is 00:28:33.289 --> 00:28:37.130 treated as a simple future value lump sum. You 00:28:37.130 --> 00:28:39.549 discount it back to the present using the PV 00:28:39.549 --> 00:28:42.670 formula from Part 2. So to get the bond's current 00:28:42.670 --> 00:28:45.910 price, its market value, you calculate the PV 00:28:45.910 --> 00:28:48.049 of the annuity of coupons, you calculate the 00:28:48.049 --> 00:28:50.170 PV of the final lump sum, and you just add them 00:28:50.170 --> 00:28:52.589 together. That's all there is to it. It highlights 00:28:52.589 --> 00:28:55.779 the elegance of TVM. No matter how complex a 00:28:55.779 --> 00:28:58.200 financial instrument seems, if you can just break 00:28:58.200 --> 00:29:00.200 down its cash flows into periodic streams and 00:29:00.200 --> 00:29:02.960 lump sums, you can determine its current market 00:29:02.960 --> 00:29:05.680 value. That synthesis is really the professional 00:29:05.680 --> 00:29:07.480 application of everything we've covered so far. 00:29:07.619 --> 00:29:09.579 But now we transition from the application of 00:29:09.579 --> 00:29:11.960 algebra to the rigorous tools of calculus. Yes, 00:29:11.960 --> 00:29:14.079 which are essential for modeling risk and dynamic 00:29:14.079 --> 00:29:16.480 market conditions. Okay, part four, advanced 00:29:16.480 --> 00:29:19.259 mathematical framing. We are now putting on the 00:29:19.259 --> 00:29:21.440 financial engineering hat, moving beyond fixed 00:29:21.440 --> 00:29:23.839 periodic calculations into the realm of constant 00:29:23.839 --> 00:29:27.599 dynamic change. This starts with continuous compounding. 00:29:27.640 --> 00:29:30.200 Most people are familiar with discrete compounding, 00:29:30.220 --> 00:29:34.140 daily, monthly, or quarterly. Continuous compounding 00:29:34.140 --> 00:29:36.660 is the theoretical limit of that. It's what happens 00:29:36.660 --> 00:29:39.380 when the rate is compounded infinitely many times 00:29:39.380 --> 00:29:41.960 over a given period. It's the ultimate smoothing 00:29:41.960 --> 00:29:44.950 of the growth curve. But if the difference between 00:29:44.950 --> 00:29:46.990 annual and daily compounding is already pretty 00:29:46.990 --> 00:29:49.890 small, why do we bother with this conceptual 00:29:49.890 --> 00:29:53.730 leap to infinite continuous compounding? The 00:29:53.730 --> 00:29:56.170 primary reason is honestly mathematical convenience. 00:29:56.670 --> 00:29:59.750 When you use discrete compounding, the resulting 00:29:59.750 --> 00:30:02.140 formulas are chunky. They have exponents that 00:30:02.140 --> 00:30:04.700 are difficult to manipulate using calculus. By 00:30:04.700 --> 00:30:06.359 converting rates into the continuous compound 00:30:06.359 --> 00:30:09.259 interest rate equivalent, we get a smooth, differentiable 00:30:09.259 --> 00:30:11.200 function. And that makes it easier to model. 00:30:11.339 --> 00:30:13.039 Much easier to work with when you're modeling 00:30:13.039 --> 00:30:15.420 complex derivatives or developing hedging strategies. 00:30:15.720 --> 00:30:17.980 And this is where the base of the natural logarithm 00:30:17.980 --> 00:30:20.460 E, that famous transcendental number, makes its 00:30:20.460 --> 00:30:23.400 appearance. Exactly. The present value of a future 00:30:23.400 --> 00:30:26.559 payment, FV, at timeskeet, using the continuously 00:30:26.559 --> 00:30:29.519 compounded rate R, is given by a new formula. 00:30:30.160 --> 00:30:33.279 PV equals FE times E to the power of negative 00:30:33.279 --> 00:30:36.339 art. That expression, E to the negative art, 00:30:36.619 --> 00:30:40.039 is known as the continuous discount factor. It 00:30:40.039 --> 00:30:42.859 replaces the discrete discount factor of 1 over 00:30:42.859 --> 00:30:46.400 1 plus R to the T. Right. And while computationally 00:30:46.400 --> 00:30:49.079 it's very close to daily compounding, its real 00:30:49.079 --> 00:30:51.920 value lies in how neatly it integrates into higher 00:30:51.920 --> 00:30:54.779 -level mathematical structures. It allows analysts 00:30:54.779 --> 00:30:57.279 to perform instantaneous calculations of sensitivity 00:30:57.279 --> 00:31:00.440 and change. And this framework is essential when 00:31:00.440 --> 00:31:02.819 we consider the reality of varying discount rates. 00:31:03.359 --> 00:31:05.980 As we established, the discount rate R should 00:31:05.980 --> 00:31:08.140 ideally account for changing market conditions 00:31:08.140 --> 00:31:11.000 and risk over time. It's not a constant. Right. 00:31:11.079 --> 00:31:13.259 If your required rate of return is not constant, 00:31:13.380 --> 00:31:16.019 if you expect low risk and a 5 % return for the 00:31:16.019 --> 00:31:18.400 first two years, but then high risk and a 15 00:31:18.400 --> 00:31:21.279 % return for the following three years, you can't 00:31:21.279 --> 00:31:23.339 just plug a single R into the simple formula. 00:31:23.539 --> 00:31:25.319 So what do you do? Instead of a constant rate 00:31:25.319 --> 00:31:28.430 R, we use a time -dependent function R, T. To 00:31:28.430 --> 00:31:30.470 calculate the present value of a sum under this 00:31:30.470 --> 00:31:32.750 varying rate environment, we have to integrate 00:31:32.750 --> 00:31:34.930 the continuously compounded rate over the entire 00:31:34.930 --> 00:31:38.690 period. OK, so integration in this context is 00:31:38.690 --> 00:31:41.029 the mathematical tool for summing up the effect 00:31:41.029 --> 00:31:44.069 of an infinite number of instantaneous discount 00:31:44.069 --> 00:31:46.450 factors along that timeline. You got it. The 00:31:46.450 --> 00:31:48.789 resulting formula involves the exponential of 00:31:48.789 --> 00:31:51.170 a negative integral, which looks terrifying. 00:31:51.230 --> 00:31:54.390 But conceptually, all it's saying is calculate 00:31:54.390 --> 00:31:57.210 the total cumulative discount factor generated 00:31:57.210 --> 00:32:01.230 by the changing rate from today times zero to 00:32:01.230 --> 00:32:04.309 the maturity time. It's powerful because it allows 00:32:04.309 --> 00:32:06.940 for dynamic valuation. If the market suddenly 00:32:06.940 --> 00:32:09.279 changes its risk assessment, the function already 00:32:09.279 --> 00:32:12.240 updates and the system can instantly recalculate 00:32:12.240 --> 00:32:14.660 the new present value. It does. And the source 00:32:14.660 --> 00:32:16.680 material provides the continuous equivalence 00:32:16.680 --> 00:32:19.920 for all our part three concepts, annuities, perpetuities, 00:32:19.980 --> 00:32:22.640 growing annuities. This just shows that the conceptual 00:32:22.640 --> 00:32:25.279 structure of finance is robust. We're simply 00:32:25.279 --> 00:32:27.319 swapping the mathematical language from algebra 00:32:27.319 --> 00:32:29.859 to calculus for more precision and flexibility. 00:32:30.160 --> 00:32:32.400 But the highest level of conceptual sophistication 00:32:32.400 --> 00:32:34.880 is framing TVM through differential equations. 00:32:35.339 --> 00:32:38.119 This is the ultimate conceptual leap in modern 00:32:38.119 --> 00:32:40.619 finance. It really is. So let's clarify what 00:32:40.619 --> 00:32:43.720 we mean by a conceptual leap here. In parts two 00:32:43.720 --> 00:32:46.000 and three, we were always calculating a single 00:32:46.000 --> 00:32:49.319 number, the PV today. What does the differential 00:32:49.319 --> 00:32:52.660 equation approach calculate instead? It shifts 00:32:52.660 --> 00:32:56.059 the focus from a single static snapshot, a number, 00:32:56.160 --> 00:32:59.599 to a dynamic function. When we use ordinary differential 00:32:59.599 --> 00:33:04.400 equations, or ODEs, we are calculating VT. which 00:33:04.400 --> 00:33:06.799 is the present value of the asset at any point 00:33:06.799 --> 00:33:10.000 in the future, t, not just at time zero. So we're 00:33:10.000 --> 00:33:12.519 modeling how the value evolves or decays over 00:33:12.519 --> 00:33:15.059 time. This makes sense because the value of an 00:33:15.059 --> 00:33:17.279 asset that doesn't generate income for five years 00:33:17.279 --> 00:33:19.660 is constantly increasing in value toward its 00:33:19.660 --> 00:33:22.380 future cash flow as the discounting time shrinks. 00:33:22.799 --> 00:33:25.440 Correct. The core statement of TVM, that value 00:33:25.440 --> 00:33:28.019 decreases as time passes, assuming no cash flows 00:33:28.019 --> 00:33:30.160 occur, is formally defined by something called 00:33:30.160 --> 00:33:33.019 a linear differential operator. That minus sign 00:33:33.019 --> 00:33:35.019 on the partial derivative with respect to time 00:33:35.019 --> 00:33:37.400 mathematically encodes that decrease in value. 00:33:37.819 --> 00:33:40.099 The other term represents the loss of value due 00:33:40.099 --> 00:33:42.200 to the foregone opportunity cost of the investment. 00:33:42.420 --> 00:33:45.240 So if the value function, Vt, were to change, 00:33:45.480 --> 00:33:48.380 say, because of an unexpected payment, the differential 00:33:48.380 --> 00:33:50.859 equation would immediately model that shock to 00:33:50.859 --> 00:33:53.279 the system. And that allows analysts to study 00:33:53.279 --> 00:33:55.480 the sensitivity and the evolution of the asset's 00:33:55.480 --> 00:33:59.519 value dynamically, in real time. And the standard 00:33:59.519 --> 00:34:01.960 tool used in physics and math to solve these 00:34:01.960 --> 00:34:05.519 ODEs is called Green's functions. That sounds 00:34:05.519 --> 00:34:08.429 dense. Let's find an analogy. What does the Green's 00:34:08.429 --> 00:34:10.690 function represent in financial terms? Think 00:34:10.690 --> 00:34:13.630 of it as the atomic structure of cash flow. In 00:34:13.630 --> 00:34:16.269 finance, the Green's function for TVM is defined 00:34:16.269 --> 00:34:19.150 as the value of an asset that pays exactly $1 00:34:19.150 --> 00:34:21.869 at a single instantaneous point in time. This 00:34:21.869 --> 00:34:24.670 instantaneous, isolated cash flow is modeled 00:34:24.670 --> 00:34:27.409 mathematically as a direct delta function. which 00:34:27.409 --> 00:34:29.230 is just a fancy way of saying a single spike 00:34:29.230 --> 00:34:31.110 at a specific point on the timeline. So it's 00:34:31.110 --> 00:34:33.690 the basic unit of value, the value today of $1 00:34:33.690 --> 00:34:36.489 paid tomorrow and nothing else. Exactly. The 00:34:36.489 --> 00:34:38.750 greens function gives you the value of that single 00:34:38.750 --> 00:34:42.210 instant event. And the profound so what here 00:34:42.210 --> 00:34:45.510 is that any complex financial instrument. Any 00:34:45.510 --> 00:34:48.070 combination of coupons, maturities, fees, or 00:34:48.070 --> 00:34:50.750 dividends can be modeled as an infinite series 00:34:50.750 --> 00:34:53.530 of these instantaneous cash flows. And since 00:34:53.530 --> 00:34:55.969 greens functions are the solution to the differential 00:34:55.969 --> 00:34:59.170 equation for that one instantaneous event. By 00:34:59.170 --> 00:35:01.090 integrating all those greens functions together, 00:35:01.269 --> 00:35:03.849 you can build the value of the whole complex 00:35:03.849 --> 00:35:06.340 cash flow stream. This is the foundation of the 00:35:06.340 --> 00:35:08.940 really complex stuff. This sophisticated methodology, 00:35:09.340 --> 00:35:11.460 using differential equations and Green's functions, 00:35:11.659 --> 00:35:14.420 is the very foundation upon which complex derivatives 00:35:14.420 --> 00:35:16.820 pricing models are built. When you hear about 00:35:16.820 --> 00:35:19.099 the black skulls formula or models for interest 00:35:19.099 --> 00:35:21.579 rate swaps, you are hearing about mathematical 00:35:21.579 --> 00:35:23.599 structures that are ultimately solving differential 00:35:23.599 --> 00:35:27.320 equations, where TVM is the core operator. Wow. 00:35:27.780 --> 00:35:30.940 We started with a judge in 500 CE determining 00:35:30.940 --> 00:35:33.400 the financial difference between a 30 -day debt 00:35:33.400 --> 00:35:36.480 and a 10 -year debt. And we have ended this dive 00:35:36.480 --> 00:35:39.059 with the instantaneous discounting of future 00:35:39.059 --> 00:35:42.360 cash flows using calculus to price billion -dollar 00:35:42.360 --> 00:35:46.139 derivatives. That conceptual path is truly incredible. 00:35:46.380 --> 00:35:48.900 The logic never changed. We just got better tools 00:35:48.900 --> 00:35:51.199 to measure it. Okay, let's wrap this up. We've 00:35:51.199 --> 00:35:52.840 covered a tremendous amount of ground today, 00:35:53.000 --> 00:35:55.500 tracking the concept of time and money from its 00:35:55.500 --> 00:35:58.059 roots in ancient law all the way to its application 00:35:58.059 --> 00:36:00.400 in Wall Street valuation models. And the core 00:36:00.400 --> 00:36:03.239 logic, as I said, never changed. We simply formalized 00:36:03.239 --> 00:36:05.599 it. We established that the critical variables 00:36:05.599 --> 00:36:08.059 remain constant across all levels of complexity. 00:36:08.420 --> 00:36:11.760 Present value, PV, future value, FVA, the rate, 00:36:11.780 --> 00:36:15.010 I or R, and the time, N or T. Whether you're 00:36:15.010 --> 00:36:17.230 calculating a simple savings goal using basic 00:36:17.230 --> 00:36:20.530 algebra or dynamically modeling asset risk using 00:36:20.530 --> 00:36:22.949 differential equations, you are fundamentally 00:36:22.949 --> 00:36:25.429 just asking how those four variables interact 00:36:25.429 --> 00:36:28.460 under compounding and discounting. And now you 00:36:28.460 --> 00:36:31.300 have the conceptual toolkit. You understand the 00:36:31.300 --> 00:36:34.260 necessity of time preference, the crucial role 00:36:34.260 --> 00:36:37.300 of determining the correct discount rate, the 00:36:37.300 --> 00:36:39.639 practical application of annuities and mortgages 00:36:39.639 --> 00:36:42.519 and bonds, and why financial engineers turn to 00:36:42.519 --> 00:36:45.300 calculus when they need maximum precision and 00:36:45.300 --> 00:36:47.909 flexibility in dynamic markets. every time you 00:36:47.909 --> 00:36:49.989 open a savings account or consider paying down 00:36:49.989 --> 00:36:52.530 a mortgage early or evaluate a business investment 00:36:52.530 --> 00:36:55.869 you are engaging in a tvm calculation you are 00:36:55.869 --> 00:36:58.210 measuring the opportunity cost of having or not 00:36:58.210 --> 00:37:00.730 having that capital today versus tomorrow you 00:37:00.730 --> 00:37:03.670 are so here's a final provocative thought something 00:37:03.670 --> 00:37:05.750 that builds directly on the deepest assumption 00:37:05.750 --> 00:37:08.289 of everything we've talked about today the entire 00:37:08.289 --> 00:37:11.570 structure of tvm all the formulas all the compounding 00:37:11.570 --> 00:37:14.570 all the valuation hinges on the certainty that 00:37:14.570 --> 00:37:16.349 money today can be invested to earn a sustained 00:37:16.619 --> 00:37:18.699 positive rate of return. Right. That assumption 00:37:18.699 --> 00:37:21.780 dictates that the discount factor, 1 over 1 plus 00:37:21.780 --> 00:37:24.909 r to the n, must always be less than 1. which 00:37:24.909 --> 00:37:27.409 means future money is worth less. But imagine 00:37:27.409 --> 00:37:30.349 a sustained, guaranteed economic environment 00:37:30.349 --> 00:37:33.289 where the expected rate of return on all capital 00:37:33.289 --> 00:37:36.030 was structurally negative, not just temporarily 00:37:36.030 --> 00:37:39.190 negative in inflation -adjusted terms, but fundamentally 00:37:39.190 --> 00:37:42.090 numerically negative. So money actively shrinks 00:37:42.090 --> 00:37:44.590 just by holding it. Exactly. If that happened, 00:37:44.650 --> 00:37:47.230 the value of a dollar today would be mathematically 00:37:47.230 --> 00:37:49.829 less than the value of the same dollar in the 00:37:49.829 --> 00:37:53.070 future. the fundamental time preference for present 00:37:53.070 --> 00:37:55.269 consumption would be economically penalized. 00:37:55.389 --> 00:37:58.190 It would completely reverse the logic. A dollar 00:37:58.190 --> 00:38:00.389 tomorrow would be worth more than a dollar today. 00:38:00.670 --> 00:38:04.050 If capital actively decayed in value just sitting 00:38:04.050 --> 00:38:06.730 in the bank, what would that force us to do with 00:38:06.730 --> 00:38:09.230 our resources? Would that accelerate consumption 00:38:09.230 --> 00:38:12.090 to prevent decay? Or would it fundamentally destroy 00:38:12.090 --> 00:38:14.230 the incentive to save and invest for the future? 00:38:14.429 --> 00:38:16.909 It completely reverses the gravitational pull 00:38:16.909 --> 00:38:20.119 of time on money. And the mathematics of TVM 00:38:20.119 --> 00:38:23.380 is the only tool robust enough to map what that 00:38:23.380 --> 00:38:26.500 strange new economy would look like. It's a profound 00:38:26.500 --> 00:38:28.860 question to ponder on your next deep dive.