WEBVTT 00:00:00.000 --> 00:00:02.279 Welcome to the Deep Dive, where we take the sources, 00:00:02.399 --> 00:00:06.139 the articles, the spreadsheets, the dense research, 00:00:06.299 --> 00:00:08.560 and really just extract the critical knowledge 00:00:08.560 --> 00:00:11.640 you need. Today, we are handing you what is, 00:00:11.640 --> 00:00:13.400 I think, the essential lens through which all 00:00:13.400 --> 00:00:15.939 financial decisions are viewed. It's really the 00:00:15.939 --> 00:00:17.760 indispensable tool. It underpins everything, 00:00:18.039 --> 00:00:19.879 right? Corporate budgeting, your mortgage payment. 00:00:20.120 --> 00:00:22.820 Everything. We are undertaking a comprehensive 00:00:22.820 --> 00:00:25.980 deep dive into present value, which you'll often 00:00:25.980 --> 00:00:28.820 see called PV, or sometimes present discounted 00:00:28.820 --> 00:00:31.859 value. Okay, so let's unpack the central dilemma 00:00:31.859 --> 00:00:34.299 that I guess drives this entire mathematical 00:00:34.299 --> 00:00:36.939 universe. It's a simple idea, but it's actually 00:00:36.939 --> 00:00:40.340 pretty profound. It asks, why is a dollar today 00:00:40.340 --> 00:00:43.390 worth more than a dollar tomorrow? And that question, 00:00:43.530 --> 00:00:46.329 that single question encapsulates the time value 00:00:46.329 --> 00:00:48.310 of money. It's not just some, you know, cute 00:00:48.310 --> 00:00:51.090 phrase. It is a fundamental economic reality. 00:00:51.329 --> 00:00:52.789 The dollar you're holding in your hand right 00:00:52.789 --> 00:00:55.789 now is inherently more valuable. It is more valuable 00:00:55.789 --> 00:00:58.609 than the promise of that exact same dollar next 00:00:58.609 --> 00:01:01.210 week or next month or, you know, a year from 00:01:01.210 --> 00:01:03.490 now. And why is that? Because the moment you 00:01:03.490 --> 00:01:06.469 possess that dollar, you gain immediate, tangible 00:01:06.469 --> 00:01:09.269 exchange value. You can spend it, of course. 00:01:09.569 --> 00:01:12.319 But more importantly in finance. You can invest 00:01:12.319 --> 00:01:15.319 it. It has the potential to start earning interest 00:01:15.319 --> 00:01:18.319 immediately. Setting it on a path to accumulate 00:01:18.319 --> 00:01:21.260 a value greater than $1 by tomorrow. Exactly. 00:01:21.680 --> 00:01:24.420 And when we talk about present value, what we're 00:01:24.420 --> 00:01:27.459 really doing is formally defining the current 00:01:27.459 --> 00:01:30.280 worth of an income stream. Or even just a single 00:01:30.280 --> 00:01:32.840 lump sum that you expect to receive sometime 00:01:32.840 --> 00:01:34.799 in the future. So it's a translation process. 00:01:34.939 --> 00:01:37.319 You're translating a future promise into an exact 00:01:37.319 --> 00:01:39.280 valuation today. You're trying to figure out 00:01:39.280 --> 00:01:42.420 its fair price right now. Precisely. And to understand 00:01:42.420 --> 00:01:44.939 PV, you really have to contrast it immediately 00:01:44.939 --> 00:01:48.340 with its opposite, future value or FV. Okay. 00:01:48.420 --> 00:01:51.939 Future value asks, if I invest $100 today at 00:01:51.939 --> 00:01:54.099 a certain interest rate, what will it accumulate 00:01:54.099 --> 00:01:57.519 to in, say, five years? It's a projection forward 00:01:57.519 --> 00:02:00.060 in time. Right. But present value flips that 00:02:00.060 --> 00:02:02.079 completely. It asks the inverse. It asks, if 00:02:02.079 --> 00:02:04.340 I am promised $100 five years from now, what 00:02:04.340 --> 00:02:07.079 is the value of that promise to me today factoring 00:02:07.079 --> 00:02:09.800 in that same interest rate? So the general rule, 00:02:10.000 --> 00:02:12.240 which we operate under for, you know, 99 percent 00:02:12.240 --> 00:02:14.719 of conventional finance. Almost always. Is that 00:02:14.719 --> 00:02:17.500 the present value will always be less than the 00:02:17.500 --> 00:02:20.599 future value. And that's because money has that 00:02:20.599 --> 00:02:23.080 interest earning potential over time. You're 00:02:23.080 --> 00:02:25.259 sort of mathematically penalized for having to 00:02:25.259 --> 00:02:27.840 wait. That's absolutely right. But, you know, 00:02:27.879 --> 00:02:30.620 the sources point out a fascinating, though very 00:02:30.620 --> 00:02:36.020 rare scenario. The exception where PV could equal 00:02:36.020 --> 00:02:39.620 or even exceed FV. And that happens with negative 00:02:39.620 --> 00:02:41.780 interest rates. Yes, negative interest rates. 00:02:41.840 --> 00:02:43.719 We've seen this crop up in certain economies, 00:02:43.840 --> 00:02:46.159 like in parts of Europe and in Japan, where central 00:02:46.159 --> 00:02:48.340 banks were pushing rates below zero. That's so 00:02:48.340 --> 00:02:50.539 counterintuitive. How does that flip the entire 00:02:50.539 --> 00:02:53.479 equation on its head? Well, think about the cost 00:02:53.479 --> 00:02:56.219 of just holding cash. If a bank charges you a 00:02:56.219 --> 00:02:58.159 fee, which is what a negative interest rate is, 00:02:58.259 --> 00:03:01.139 just to hold your money, you're actually incentivized 00:03:01.139 --> 00:03:03.259 to take that cash out today. Right. You'd rather 00:03:03.259 --> 00:03:06.699 hold it yourself or spend it. Exactly. So in 00:03:06.699 --> 00:03:09.150 that environment, The penalty is for waiting 00:03:09.150 --> 00:03:11.810 and leaving the money in the system. Therefore, 00:03:12.129 --> 00:03:15.830 the present value of $100 today, which avoids 00:03:15.830 --> 00:03:18.710 that fee, might actually be deemed more valuable 00:03:18.710 --> 00:03:21.969 than the future value of that same $100, because 00:03:21.969 --> 00:03:24.330 that future amount is going to be eroded by fees 00:03:24.330 --> 00:03:27.159 while you wait. But like you said, that's highly 00:03:27.159 --> 00:03:29.599 non -conventional. So for today, for this deep 00:03:29.599 --> 00:03:31.780 dive, we're focusing on the foundational principles 00:03:31.780 --> 00:03:34.520 of positive interest rates. We have to. So our 00:03:34.520 --> 00:03:36.780 mission then is pretty straightforward. We're 00:03:36.780 --> 00:03:40.039 going to unpack the mathematics, the complex 00:03:40.039 --> 00:03:42.780 formulas, and the really practical applications 00:03:42.780 --> 00:03:45.099 of present value. And by the time we wrap up, 00:03:45.219 --> 00:03:47.740 you will have the mathematical language to translate 00:03:47.740 --> 00:03:50.379 time into dollars. Which gives you a serious 00:03:50.379 --> 00:03:52.699 competitive advantage in understanding how loans, 00:03:52.759 --> 00:03:55.389 mortgages, annuities, and investments are truly 00:03:55.389 --> 00:03:57.349 valued. OK, let's start with the fundamental 00:03:57.349 --> 00:03:59.810 human behavior that's really driving this entire 00:03:59.810 --> 00:04:03.009 mechanism, the rational choice. Economists call 00:04:03.009 --> 00:04:05.550 this time preference, right? Exactly. Time preference. 00:04:05.810 --> 00:04:08.689 If you are offered $100 right now, no streams 00:04:08.689 --> 00:04:11.969 attached, or a guaranteed $100 one year from 00:04:11.969 --> 00:04:14.530 now, assuming a positive real interest rate, 00:04:14.830 --> 00:04:17.310 a rational person will always choose the money 00:04:17.310 --> 00:04:20.029 today. And that choice isn't just about impatience, 00:04:20.029 --> 00:04:22.110 though I'm sure that plays a small role. It's 00:04:22.110 --> 00:04:25.339 really about opportunity. That $100 today represents 00:04:25.339 --> 00:04:27.720 an immediate opportunity. What kind of opportunity? 00:04:28.019 --> 00:04:29.899 Well, I could use it to buy something I need, 00:04:29.959 --> 00:04:33.439 obviously, or more relevant to finance, I can 00:04:33.439 --> 00:04:35.240 immediately put it to work. I can put it in a 00:04:35.240 --> 00:04:37.899 savings account, a bond, any asset that starts 00:04:37.899 --> 00:04:40.680 generating a positive return instantly. That 00:04:40.680 --> 00:04:43.319 immediate access to value, that's liquidity. 00:04:43.860 --> 00:04:46.939 Precisely. If you choose the $100 today, you've 00:04:46.939 --> 00:04:48.899 sacrificed nothing. You've maximized your opportunity. 00:04:49.220 --> 00:04:52.360 But if you choose the $100 in one year, you have 00:04:52.360 --> 00:04:54.839 sacrificed the opportunity to use and to earn 00:04:54.839 --> 00:04:58.240 interest on that money for an entire 365 days. 00:04:58.500 --> 00:05:01.300 So that sacrifice of immediate access of liquidity, 00:05:01.519 --> 00:05:04.639 it has to be compensated for financially. It 00:05:04.639 --> 00:05:08.939 must be. And that compensation is interest. Interest, 00:05:08.939 --> 00:05:11.800 in its purest form, is the financial compensation 00:05:11.800 --> 00:05:15.100 required for delaying consumption. It's the premium 00:05:15.100 --> 00:05:18.100 a borrower has to pay to a lender for the privilege 00:05:18.100 --> 00:05:20.439 of using money right now. I've always found it 00:05:20.439 --> 00:05:22.480 easiest to understand when it's compared to rent. 00:05:22.699 --> 00:05:25.259 It's a perfect analogy. Think of a property owner. 00:05:25.600 --> 00:05:27.819 They give up temporary use of their property 00:05:27.819 --> 00:05:31.019 to a tenant, and in return, the tenant pays rent. 00:05:31.259 --> 00:05:33.759 The title never changes hands, but that temporary 00:05:33.759 --> 00:05:37.240 use is paid for. It's the same with money. Interest 00:05:37.240 --> 00:05:40.379 is the rent paid by a borrower to a lender. for 00:05:40.379 --> 00:05:43.360 access to money over a period of time. The lender 00:05:43.360 --> 00:05:45.779 temporarily sacrifices the money's exchange value, 00:05:45.939 --> 00:05:47.860 their ability to go out and buy something now, 00:05:48.019 --> 00:05:50.399 and they're compensated for this sacrifice with 00:05:50.399 --> 00:05:52.439 interest. So the key takeaway here is that the 00:05:52.439 --> 00:05:54.699 initial borrowed amount, the present value, is 00:05:54.699 --> 00:05:56.560 always going to be smaller than the total amount 00:05:56.560 --> 00:05:58.800 eventually paid back to the lender. It has to 00:05:58.800 --> 00:06:01.339 be. It makes perfect sense. The borrower gets 00:06:01.339 --> 00:06:03.819 the access now. The lender gets paid for waiting 00:06:03.819 --> 00:06:06.199 and for giving up their own immediate opportunities. 00:06:06.920 --> 00:06:12.009 So to... formalize this, we need terms for projecting 00:06:12.009 --> 00:06:15.730 value both forward and backward in time. We use 00:06:15.730 --> 00:06:19.009 two distinct inverse operations. First, you have 00:06:19.009 --> 00:06:21.149 capitalization. Some people call it compounding. 00:06:21.170 --> 00:06:23.889 Okay. This is the operation of taking a present 00:06:23.889 --> 00:06:26.250 value and projecting its growth into a future 00:06:26.250 --> 00:06:30.009 value. So if I have $1 ,000 today, I use capitalization 00:06:30.009 --> 00:06:32.930 to ask, how much will this $1 ,000 be worth in 00:06:32.930 --> 00:06:35.250 three years given a certain interest rate? So 00:06:35.250 --> 00:06:37.720 you're growing the money forward. Exactly. And 00:06:37.720 --> 00:06:39.639 the reverse operation, which is the whole focus 00:06:39.639 --> 00:06:42.279 of PV, is discounting. Right. We start with the 00:06:42.279 --> 00:06:44.680 future value and work our way backward. We ask, 00:06:44.779 --> 00:06:47.399 how much is that $1 ,000 I'm promised in three 00:06:47.399 --> 00:06:50.319 years actually worth to me today? In this process, 00:06:50.560 --> 00:06:53.240 we are, in a sense, peeling away the compensation, 00:06:53.620 --> 00:06:56.019 peeling away the rent that would have accumulated 00:06:56.019 --> 00:06:58.519 over those years. To find its true value in the 00:06:58.519 --> 00:07:01.660 here and now. To find its true value right now. 00:07:02.759 --> 00:07:05.759 Let's... Let's use a very specific example to 00:07:05.759 --> 00:07:08.759 really nail down this idea of equivalence. This 00:07:08.759 --> 00:07:11.420 is the point of indifference for a rational investor. 00:07:11.620 --> 00:07:15.300 Correct. Imagine an incredibly safe investment, 00:07:15.399 --> 00:07:18.699 like a high -yield savings account or a government 00:07:18.699 --> 00:07:22.199 bond, and it offers a guaranteed 5 % interest 00:07:22.199 --> 00:07:25.600 rate annually. Okay, so if I choose to receive 00:07:25.600 --> 00:07:29.199 $100 today, I can deposit it. And in one year, 00:07:29.300 --> 00:07:33.339 I'll have $105. Simple enough. Exactly. Therefore, 00:07:33.540 --> 00:07:36.740 for you, receiving $100 today is financially 00:07:36.740 --> 00:07:40.399 equivalent to receiving $105 in one year, provided, 00:07:40.459 --> 00:07:42.720 of course, that you can reliably get that 5 % 00:07:42.720 --> 00:07:45.560 return. That point of indifference is the key 00:07:45.560 --> 00:07:47.620 to all of this. Which means if someone came to 00:07:47.620 --> 00:07:50.500 me and offered me $104 next year, I would refuse 00:07:50.500 --> 00:07:53.740 it. Because I can generate $105 myself just by 00:07:53.740 --> 00:07:56.319 taking the $100 today. So I'd only accept the 00:07:56.319 --> 00:07:59.180 offer for next year if it's $105 or more. If 00:07:59.180 --> 00:08:01.759 they offer me $106, I'd probably wait. And that 00:08:01.759 --> 00:08:05.220 is the mechanism of discounting in action. That 00:08:05.220 --> 00:08:08.000 5 % rate essentially determines the minimum premium 00:08:08.000 --> 00:08:10.639 that's required to get you to delay the receipt 00:08:10.639 --> 00:08:13.540 of cash. Now, there's a critical detail from 00:08:13.540 --> 00:08:15.839 the sources we have to mention here. These base 00:08:15.839 --> 00:08:19.139 calculations, they all assume zero risk of default, 00:08:19.379 --> 00:08:21.939 right? Absolutely. That is a crucial assumption. 00:08:22.100 --> 00:08:24.339 They assume the promised payment is guaranteed 00:08:24.339 --> 00:08:27.040 or that any potential risk has already been accounted 00:08:27.040 --> 00:08:30.040 for inside the rate we're using. We'll dive much 00:08:30.040 --> 00:08:32.600 deeper into that risk adjustment later on. Okay, 00:08:32.659 --> 00:08:34.940 so since interest is the engine that drives this 00:08:34.940 --> 00:08:37.639 whole valuation process, we really need to get 00:08:37.639 --> 00:08:40.240 extremely precise about the language of interest 00:08:40.240 --> 00:08:42.600 rates. We've established it's the compensation 00:08:42.600 --> 00:08:46.039 for the time sacrifice. Interest is the additional 00:08:46.039 --> 00:08:49.039 amount gained or paid between the beginning and 00:08:49.039 --> 00:08:51.379 the end of a time period. It represents that 00:08:51.379 --> 00:08:54.679 time value of money. But the speed and the magnitude 00:08:54.679 --> 00:08:57.519 of that value growth depend entirely on the compounding 00:08:57.519 --> 00:08:59.419 period. Right. This is the length of time that 00:08:59.419 --> 00:09:02.019 has to pass before the interest is actually calculated 00:09:02.019 --> 00:09:05.299 and credited or added to the total principal 00:09:05.299 --> 00:09:07.799 amount. And I think this is where many non -specialists 00:09:07.799 --> 00:09:10.259 can get a little bit confused. If interest is 00:09:10.259 --> 00:09:13.059 compounded quarterly, it happens four times a 00:09:13.059 --> 00:09:16.919 year. If it's daily, it's 365 times a year. And 00:09:16.919 --> 00:09:19.259 crucially, the interest rate we use in our formulas, 00:09:19.519 --> 00:09:21.720 which we just call, it always has to be the rate 00:09:21.720 --> 00:09:24.419 for that specific compounding period. You absolutely 00:09:24.419 --> 00:09:26.919 have to match the rate to the period. That's 00:09:26.919 --> 00:09:29.799 a non -negotiable requirement for accuracy. And 00:09:29.799 --> 00:09:31.759 it forces us to confront the fact that we don't 00:09:31.759 --> 00:09:34.899 just have one interest rate. The sources detail 00:09:34.899 --> 00:09:38.220 seven common types used in financial modeling. 00:09:38.440 --> 00:09:40.840 And understanding the nuance between them is 00:09:40.840 --> 00:09:43.120 really the difference between true financial 00:09:43.120 --> 00:09:45.340 literacy and just, you know, plugging numbers 00:09:45.340 --> 00:09:47.440 into a formula. OK, let's start with the two 00:09:47.440 --> 00:09:49.720 fundamental ways that interest grows. First, 00:09:49.879 --> 00:09:52.179 and by far the most important for long term investing, 00:09:52.299 --> 00:09:54.730 is compound interest. This is the exponential 00:09:54.730 --> 00:09:57.309 process. It is. The interest you earn in one 00:09:57.309 --> 00:09:59.090 period is immediately added to the principal. 00:09:59.389 --> 00:10:01.809 And then the next period's interest is calculated 00:10:01.809 --> 00:10:04.230 on that new larger total. It's interest earning 00:10:04.230 --> 00:10:06.649 interest. Which leads to that characteristic 00:10:06.649 --> 00:10:10.620 accelerating upward curve of growth. It's the 00:10:10.620 --> 00:10:14.559 bedrock of retirement savings, long -term bonds, 00:10:14.679 --> 00:10:17.200 all of that. Absolutely. Now, contrast that with 00:10:17.200 --> 00:10:19.759 simple interest. Which is just additive, not 00:10:19.759 --> 00:10:22.519 exponential. Correct. The interest is only ever 00:10:22.519 --> 00:10:24.799 calculated on the original initial principal 00:10:24.799 --> 00:10:28.460 amount. The interest you earn is never reinvested 00:10:28.460 --> 00:10:31.000 or added back into the principal to earn more 00:10:31.000 --> 00:10:34.340 on its own. Where would you even see simple interest 00:10:34.340 --> 00:10:38.039 used? It seems so... It's typically for very 00:10:38.039 --> 00:10:40.080 short -term loans. Think of some short -term 00:10:40.080 --> 00:10:42.519 commercial papers or historically some basic 00:10:42.519 --> 00:10:45.539 treasury bills. If you lend $1 ,000 at 10 % simple 00:10:45.539 --> 00:10:48.320 interest for three years, you just get $100 per 00:10:48.320 --> 00:10:51.480 year. The total payback is $1 ,300. But if that 00:10:51.480 --> 00:10:53.580 was compound interest? The final amount would 00:10:53.580 --> 00:10:57.759 be $1 ,331. And that $31 difference, even on 00:10:57.759 --> 00:10:59.559 a small amount like that, really illustrates 00:10:59.559 --> 00:11:02.980 the power of compounding over time. Okay. Next 00:11:02.980 --> 00:11:05.919 up. We have concepts that are tied directly to 00:11:05.919 --> 00:11:08.000 the frequency of reporting. This is critical 00:11:08.000 --> 00:11:11.240 for comparison shopping between loans or investments. 00:11:11.519 --> 00:11:15.340 You often hear banks and lenders quoting one 00:11:15.340 --> 00:11:18.759 of two rates. Yes. And the first one, the effective 00:11:18.759 --> 00:11:22.000 interest rate, is the gold standard. This is 00:11:22.000 --> 00:11:24.960 the rate that is truly earned or paid over a 00:11:24.960 --> 00:11:27.419 full year, no matter how often it compounds. 00:11:27.580 --> 00:11:30.059 It incorporates the effect of that compounding. 00:11:30.080 --> 00:11:33.940 So if Bank A offers, say, 7 .8 % compounded monthly 00:11:33.940 --> 00:11:37.820 and Bank B offers 8 % compounded annually. The 00:11:37.820 --> 00:11:39.600 effective rate allows you to compare them on 00:11:39.600 --> 00:11:41.620 an apples to apples basis to see which one is 00:11:41.620 --> 00:11:43.639 actually the better deal. It's the true cost 00:11:43.639 --> 00:11:46.159 or the true return. And then we have the one 00:11:46.159 --> 00:11:48.259 they often put in the big font on the advertisement, 00:11:48.480 --> 00:11:51.139 the nominal annual interest rate. Right. This 00:11:51.139 --> 00:11:53.679 is often the headline rate. It's the simple annual 00:11:53.679 --> 00:11:56.279 rate of multiple interest periods, but it completely 00:11:56.279 --> 00:11:58.779 ignores the effect of compounding. And lenders 00:11:58.779 --> 00:12:00.740 love to use nominal rates because they sound 00:12:00.740 --> 00:12:03.080 lower, don't they? They do. For instance, an 00:12:03.080 --> 00:12:05.179 8 % nominal rate that's compounded quarterly 00:12:05.179 --> 00:12:07.860 actually yields an effective rate that's closer 00:12:07.860 --> 00:12:11.799 to 8 .24%. You must always, always ask for the 00:12:11.799 --> 00:12:14.220 effective rate to know the true financial impact. 00:12:14.580 --> 00:12:17.159 Okay, the fifth type is the discount rate. We 00:12:17.159 --> 00:12:19.960 use this implicitly in our PV calculation. It's 00:12:19.960 --> 00:12:22.419 conceptually the inverse of the return rate. 00:12:22.720 --> 00:12:25.399 It's the rate we use when we perform calculations 00:12:25.399 --> 00:12:29.019 backward in time. It's the rate we apply to discount 00:12:29.019 --> 00:12:31.000 the future value back to the present day. We 00:12:31.000 --> 00:12:33.039 also have theoretical limits, which are more 00:12:33.039 --> 00:12:35.340 for advanced finance, starting with continuously 00:12:35.340 --> 00:12:37.960 compounded interest. This is a beautiful mathematical 00:12:37.960 --> 00:12:40.039 concept. It really is. It's the mathematical 00:12:40.039 --> 00:12:42.539 limit of an interest rate. The compounding period 00:12:42.539 --> 00:12:45.139 approaches zero time. So instead of compounding 00:12:45.139 --> 00:12:47.769 daily or even hourly, it's just... Constantly 00:12:47.769 --> 00:12:49.690 compounding every second, every millisecond. 00:12:49.789 --> 00:12:51.870 Does any real world account actually do this? 00:12:52.110 --> 00:12:55.509 No. No real world deposit account compounds continuously. 00:12:56.289 --> 00:12:59.509 But the concept is absolutely essential in modern 00:12:59.509 --> 00:13:02.669 finance, especially for pricing financial derivatives 00:13:02.669 --> 00:13:05.509 and options contracts. The famous black schools 00:13:05.509 --> 00:13:08.429 model, for instance, relies heavily on the concept 00:13:08.429 --> 00:13:12.029 of continuous compounding. And finally, the most 00:13:12.029 --> 00:13:15.610 critical one for economic realism. The real interest 00:13:15.610 --> 00:13:18.590 rate. The real rate is essential because it accounts 00:13:18.590 --> 00:13:20.929 for the erosion of your purchasing power due 00:13:20.929 --> 00:13:23.129 to inflation. And the calculation is simple, 00:13:23.230 --> 00:13:24.730 right? Very simple. You just take the nominal 00:13:24.730 --> 00:13:26.370 interest rate and you subtract the inflation 00:13:26.370 --> 00:13:29.009 rate. If you earn 5 % interest on your investment, 00:13:29.129 --> 00:13:32.389 but inflation during that same year was 3%, your 00:13:32.389 --> 00:13:34.710 real rate of return, your actual increase in 00:13:34.710 --> 00:13:37.370 purchases power, is only 2%. And if inflation 00:13:37.370 --> 00:13:40.870 is 6%, your real rate of return is negative 1%. 00:13:40.870 --> 00:13:42.950 You've actually lost money in terms of what you 00:13:42.950 --> 00:13:44.830 can buy, even though your dollar amount went 00:13:44.830 --> 00:13:47.690 up. That breakdown clarifies the language beautifully. 00:13:48.230 --> 00:13:50.509 But it kind of circles back to that initial technical 00:13:50.509 --> 00:13:53.289 requirement. You have to modify the rate if it 00:13:53.289 --> 00:13:55.649 doesn't match the payment frequency. Why can't 00:13:55.649 --> 00:13:57.490 I just take an annual rate and divide it by 12 00:13:57.490 --> 00:14:00.110 for a monthly payment? Because if you simply 00:14:00.110 --> 00:14:02.950 divide an annual rate by 12, you are assuming 00:14:02.950 --> 00:14:06.149 simple interest for those 12 periods. But we 00:14:06.149 --> 00:14:09.070 know money almost always compounds. So if you 00:14:09.070 --> 00:14:11.590 use a monthly rate that's just the annual rate 00:14:11.590 --> 00:14:14.529 divided by 12, you will understate the final 00:14:14.529 --> 00:14:16.789 effective annual return. Because you're ignoring 00:14:16.789 --> 00:14:18.870 the interest that should have been earned on 00:14:18.870 --> 00:14:21.389 the interest from months 1 through 11. Precisely. 00:14:21.509 --> 00:14:24.139 So we need a formula. that ensures the total 00:14:24.139 --> 00:14:26.620 annual growth is equivalent, whether we calculate 00:14:26.620 --> 00:14:28.820 it once at the end of the year or four times 00:14:28.820 --> 00:14:31.539 quarterly or 12 times monthly. And there's a 00:14:31.539 --> 00:14:35.100 specific formula for that. Yes. If a bank quotes 00:14:35.100 --> 00:14:37.399 you an effective annual rate, let's call it I, 00:14:37.559 --> 00:14:40.179 and you want to calculate the equivalent nominal 00:14:40.179 --> 00:14:43.399 rate compounded quarterly, we'll call that I 00:14:43.399 --> 00:14:46.120 to the power of four. The formula enforces that 00:14:46.120 --> 00:14:48.860 equivalence. The quantity 1 plus i equals the 00:14:48.860 --> 00:14:51.879 quantity 1 plus i to the 4 divided by 4 all raised 00:14:51.879 --> 00:14:54.120 to the power of 4. Wow, okay, so the left side 00:14:54.120 --> 00:14:56.639 is the total growth factor for one year. Right, 00:14:56.720 --> 00:14:59.200 and the right side is the growth factor you get 00:14:59.200 --> 00:15:01.620 by compounding the quarterly rate, which is i 00:15:01.620 --> 00:15:04.919 to the 4 divided by 4, four separate times. You 00:15:04.919 --> 00:15:07.159 are solving for that perfect fractional rate 00:15:07.159 --> 00:15:10.720 that, when compounded quarterly, exactly matches 00:15:10.720 --> 00:15:13.220 the true annual rate I. It's a crucial adjustment. 00:15:13.460 --> 00:15:15.620 It ensures we maintain mathematical consistency 00:15:15.620 --> 00:15:18.539 between the duration of the rate and the duration 00:15:18.539 --> 00:15:20.820 of the period N. Okay, we have the engine, the 00:15:20.820 --> 00:15:22.960 interest rate, and we have the destination, which 00:15:22.960 --> 00:15:25.610 is the future payment. Now, let's look at the 00:15:25.610 --> 00:15:27.590 foundational formula for calculating the present 00:15:27.590 --> 00:15:30.309 value of a single one -time payment. This is 00:15:30.309 --> 00:15:33.070 really the cornerstone of all PV analysis. The 00:15:33.070 --> 00:15:35.710 standard formula, using compound interest, is 00:15:35.710 --> 00:15:38.669 PV equals C divided by the quantity 1 plus I 00:15:38.669 --> 00:15:41.990 raised to the power of N. Now, I know that looks 00:15:41.990 --> 00:15:43.690 intimidating, but let's just break down the variables 00:15:43.690 --> 00:15:45.710 conceptually so you can visualize what's happening. 00:15:45.809 --> 00:15:48.190 Okay, so C is the future amount of money you're 00:15:48.190 --> 00:15:50.210 discounting. It's that lump sum you expect to 00:15:50.210 --> 00:15:53.610 get. Right. And N is the number of compounding 00:15:53.610 --> 00:15:56.429 periods between today and when you receive C. 00:15:56.629 --> 00:15:59.149 And I is the effective interest rate for one 00:15:59.149 --> 00:16:00.669 of those compounding periods. And you have to 00:16:00.669 --> 00:16:02.950 express it as a decimal. Notice the denominator. 00:16:03.639 --> 00:16:06.120 That whole part, 1 plus i raised to the power 00:16:06.120 --> 00:16:09.580 of n. This denominator represents the total growth 00:16:09.580 --> 00:16:12.220 that the money would have experienced if you 00:16:12.220 --> 00:16:14.820 had invested the present value today and just 00:16:14.820 --> 00:16:17.620 let it compound for n periods. So when we divide 00:16:17.620 --> 00:16:20.440 the future amount c by this growth factor, we 00:16:20.440 --> 00:16:23.029 are, like you said before, peeling off. all the 00:16:23.029 --> 00:16:26.049 accrued interest. To find the true original starting 00:16:26.049 --> 00:16:29.129 value. That entire term, 1 plus i raised to the 00:16:29.129 --> 00:16:31.370 negative n, is often called the present value 00:16:31.370 --> 00:16:34.429 factor. It is. It's the multiplier that reduces 00:16:34.429 --> 00:16:36.690 the future cash amount into its present value. 00:16:36.909 --> 00:16:39.649 You can think of it as the erosion factor. It's 00:16:39.649 --> 00:16:42.450 the cost of waiting compounded over time. Let's 00:16:42.450 --> 00:16:44.370 make this concrete with the example from the 00:16:44.370 --> 00:16:47.129 source material. Imagine a legal settlement that 00:16:47.129 --> 00:16:50.110 promises you $1 ,000 exactly five years from 00:16:50.110 --> 00:16:52.690 now. And let's say the prevailing effective annual 00:16:52.690 --> 00:16:55.129 interest rate, the rate you could earn risk -free 00:16:55.129 --> 00:16:59.049 is 10%, or 0 .10. Okay, we just plug those numbers 00:16:59.049 --> 00:17:02.169 in. PV equals $1 ,000 divided by the quantity 00:17:02.169 --> 00:17:05.509 1 plus 0 .10, all raised to the fifth power. 00:17:05.869 --> 00:17:08.630 The denominator, 1 .10 to the fifth power, comes 00:17:08.630 --> 00:17:12.269 out to about 1 .61 over 5. So we divide 1 ,000 00:17:12.269 --> 00:17:15.150 by 1 .6105. And the calculation yields a present 00:17:15.150 --> 00:17:18.750 value of approximately $620 .92. The interpretation 00:17:18.750 --> 00:17:21.250 here is everything. At a 10 % effective annual 00:17:21.250 --> 00:17:23.250 discount rate, you should be financially indifferent. 00:17:23.569 --> 00:17:25.849 That's the key word, indifferent. Between receiving 00:17:25.849 --> 00:17:30.289 $1 ,000 in five years or receiving $620 .92 today. 00:17:30.470 --> 00:17:34.690 Because if you took that $620 .92 and you invested 00:17:34.690 --> 00:17:38.200 it today, at 10 % compounded annually, you would 00:17:38.200 --> 00:17:41.670 have exactly $1 ,000 five years from now. This 00:17:41.670 --> 00:17:45.930 $620 .92 is the fair market price for that future 00:17:45.930 --> 00:17:48.470 promise. That's a powerful decision -making tool. 00:17:48.710 --> 00:17:51.569 So if a bank offered to buy your $1 ,000 in five 00:17:51.569 --> 00:17:55.910 years promise for, say, $650 today. You should 00:17:55.910 --> 00:17:57.789 take it immediately. Because it's more than the 00:17:57.789 --> 00:18:01.190 mathematically fair price of $620 .92. Absolutely. 00:18:01.190 --> 00:18:04.009 But let's remember, this formula has a powerful 00:18:04.009 --> 00:18:07.170 dual use beyond just traditional investment discounting. 00:18:07.390 --> 00:18:10.150 We can use it to assess the erosion of purchasing 00:18:10.150 --> 00:18:12.950 power due to inflation. Okay, so how does that 00:18:12.950 --> 00:18:14.890 work? Instead of using the rate of return for 00:18:14.890 --> 00:18:17.750 I, you substitute the anticipated future inflation 00:18:17.750 --> 00:18:20.549 rate for I. This tells you what that future amount 00:18:20.549 --> 00:18:22.890 C will actually be able to buy in today's dollars. 00:18:23.190 --> 00:18:25.529 So imagine you're due to receive a pension payment 00:18:25.529 --> 00:18:28.690 of, say, $50 ,000 in 10 years. Okay, and let's 00:18:28.690 --> 00:18:31.210 assume a constant, maybe modest inflation rate 00:18:31.210 --> 00:18:34.690 of 3 % over that decade. We can calculate the 00:18:34.690 --> 00:18:36.740 current purchasing power of that payment. It 00:18:36.740 --> 00:18:40.980 would be PV equals $50 ,000 divided by 1 .03 00:18:40.980 --> 00:18:43.940 to the 10th power. Right. And that denominator 00:18:43.940 --> 00:18:47.339 is about 1 .3439. That means the purchasing power 00:18:47.339 --> 00:18:50.400 of your $50 ,000 in 10 years is only about $37 00:18:50.400 --> 00:18:54.440 ,200 in today's money. Wow. That's a substantial 00:18:54.440 --> 00:18:57.339 loss of real value just due to inflation. It's 00:18:57.339 --> 00:19:00.680 a huge loss. Yeah. The formula is literally quantifying 00:19:00.680 --> 00:19:02.920 the cost of waiting that comes from inflation. 00:19:03.200 --> 00:19:05.480 And finally, let's just revisit the simple rule 00:19:05.480 --> 00:19:08.319 of thumb here. The discount rate has an inverse 00:19:08.319 --> 00:19:11.079 relationship with the present value. The lower 00:19:11.079 --> 00:19:13.240 the rate, the higher the PV. Right. And this 00:19:13.240 --> 00:19:15.180 is where it gets really interesting because this 00:19:15.180 --> 00:19:18.200 connects the math to economic reality. You're 00:19:18.200 --> 00:19:20.240 talking about when central banks artificially 00:19:20.240 --> 00:19:23.000 suppress interest rates close to zero. What happens 00:19:23.000 --> 00:19:25.200 to asset valuation then? Well, when the discount 00:19:25.200 --> 00:19:28.420 rate is very, very low, the erosion factor, that 00:19:28.420 --> 00:19:31.059 denominator in our formula, becomes much smaller. 00:19:31.240 --> 00:19:34.059 This allows the present value of all future cash 00:19:34.059 --> 00:19:36.140 flows, whether they're from a bond or a corporate 00:19:36.140 --> 00:19:39.289 profit stream, to increase dramatically. So that's 00:19:39.289 --> 00:19:41.490 a primary driver of high asset valuations, like 00:19:41.490 --> 00:19:43.069 what we saw in the low interest rate environment 00:19:43.069 --> 00:19:46.589 of the 2010s. It's a huge driver. If the penalty 00:19:46.589 --> 00:19:49.450 for waiting is low, the future cash is almost 00:19:49.450 --> 00:19:51.890 as valuable as cash today, which pushes up current 00:19:51.890 --> 00:19:56.539 prices. Then if the Fed raises rates, that denominator 00:19:56.539 --> 00:20:00.160 grows and asset prices generally decline because 00:20:00.160 --> 00:20:02.299 the cost of waiting has suddenly increased. OK, 00:20:02.339 --> 00:20:05.059 moving from that single beautiful lump sum to 00:20:05.059 --> 00:20:07.960 a series of payments is where this math truly 00:20:07.960 --> 00:20:10.539 starts to model real world financial instruments. 00:20:10.559 --> 00:20:12.660 Things like capital budgeting for a new factory 00:20:12.660 --> 00:20:15.839 or project proposals or, of course, a mortgage. 00:20:16.019 --> 00:20:19.460 We shift now to net present value or NPV. Before 00:20:19.460 --> 00:20:21.579 we can calculate, we have to properly define 00:20:21.579 --> 00:20:24.240 what a cash flow is. It's simply the movement 00:20:24.240 --> 00:20:26.740 of money either in or out of a specific entity 00:20:26.740 --> 00:20:29.200 at the end of a specific period. And we differentiate 00:20:29.200 --> 00:20:31.779 them using signs, positive and negative. Right. 00:20:32.019 --> 00:20:34.660 Cash flows you receive are positive. Cash flows 00:20:34.660 --> 00:20:37.279 you pay out, like for maintenance, taxes, or 00:20:37.279 --> 00:20:39.279 an initial investment, are negative. The cash 00:20:39.279 --> 00:20:41.480 flow for any single period just represents the 00:20:41.480 --> 00:20:43.700 net change in money for that specific time frame. 00:20:44.079 --> 00:20:46.640 The crucial step here is understanding the NPV 00:20:46.640 --> 00:20:49.819 calculation process. Because PV is additive, 00:20:50.140 --> 00:20:53.140 you can add them up. Calculating NPV involves 00:20:53.140 --> 00:20:55.740 discounting each individual cash flow using the 00:20:55.740 --> 00:20:58.480 right PV factor for its own time period. And 00:20:58.480 --> 00:21:00.559 then you sum up all those individual discounted 00:21:00.559 --> 00:21:02.279 values. You can't take a shortcut. There's no 00:21:02.279 --> 00:21:05.259 averaging. And the resulting NTV figure is the 00:21:05.259 --> 00:21:08.539 absolute current. net economic value, that whole 00:21:08.539 --> 00:21:11.299 sequence of transactions. This is the gone -ago 00:21:11.299 --> 00:21:14.339 number in business, isn't it? It is. If the NPV 00:21:14.339 --> 00:21:16.799 is positive, the project is theoretically profitable 00:21:16.799 --> 00:21:19.900 at the discount rate you chose. If the NPV is 00:21:19.900 --> 00:21:22.059 negative, the project actually destroys value. 00:21:22.420 --> 00:21:24.019 Okay, let's work through the complex example 00:21:24.019 --> 00:21:26.420 provided in our source material to really visualize 00:21:26.420 --> 00:21:30.279 this sum of individual PVs. Imagine a small project 00:21:30.279 --> 00:21:34.500 discounted at a 5 % rate, so as is 0 .05, and 00:21:34.500 --> 00:21:36.859 it's spread over three years. Okay. So in period 00:21:36.859 --> 00:21:40.079 one, you receive a positive $100. In period two, 00:21:40.140 --> 00:21:42.000 you have to pay out $50, so that's a negative 00:21:42.000 --> 00:21:43.980 cash flow. And in period three, you receive another 00:21:43.980 --> 00:21:47.619 $35. So we have to calculate three separate PVs. 00:21:47.779 --> 00:21:50.980 For the $100 received in period 1, we just discount 00:21:50.980 --> 00:21:54.960 it back one period. PV1 equals 100 divided by 00:21:54.960 --> 00:21:58.720 1 .05 to the first power. Which is $95 .24. That's 00:21:58.720 --> 00:22:00.319 the present value of that first receipt. For 00:22:00.319 --> 00:22:02.859 the negative $50 paid out in period 2, we have 00:22:02.859 --> 00:22:06.420 to discount it back two periods. So PV2 is negative 00:22:06.420 --> 00:22:09.819 50 divided by 1 .05 squared. And since it's a 00:22:09.819 --> 00:22:11.680 negative cash flow, the present value is also 00:22:11.680 --> 00:22:14.700 negative, negative $45 .35. That's the present 00:22:14.700 --> 00:22:17.200 cost of that future expense. And for the $35 00:22:17.200 --> 00:22:19.079 received in period three, we discount it back 00:22:19.079 --> 00:22:23.740 three periods. PV3 is 35 divided by 1 .05 cubed. 00:22:24.160 --> 00:22:28.279 Which gives us $30 .23. So the final step is 00:22:28.279 --> 00:22:30.519 to calculate the net present value by just summing 00:22:30.519 --> 00:22:36.119 them up. 95 .24 minus 45 .35 plus 30 .23. The 00:22:36.119 --> 00:22:39.720 result is a positive NTV of $80 .12. And that 00:22:39.720 --> 00:22:43.099 $80 .12 is the current value of that entire sequence 00:22:43.099 --> 00:22:45.460 of positive and negative transactions. So if 00:22:45.460 --> 00:22:47.359 the initial cost to launch this whole project 00:22:47.359 --> 00:22:49.880 was, let's say, $70. It would be a profitable 00:22:49.880 --> 00:22:53.099 investment because your net present value is 00:22:53.099 --> 00:22:56.819 a positive $10 .12. But if the initial cost was 00:22:56.819 --> 00:22:59.299 $90, the net present value would be negative 00:22:59.299 --> 00:23:03.380 $9 .88. And the rational decision is to reject 00:23:03.380 --> 00:23:06.319 the project. That looks pretty straightforward 00:23:06.319 --> 00:23:09.000 when the interest rate is constant. But in real 00:23:09.000 --> 00:23:11.079 world finance, cash flow streams can introduce 00:23:11.079 --> 00:23:13.880 several complexities. What happens if the cash 00:23:13.880 --> 00:23:16.559 flows are non -consecutive? Maybe I get a payment 00:23:16.559 --> 00:23:18.839 in year one, skip years two and three, and get 00:23:18.839 --> 00:23:21.039 another payment in year four. The key is just 00:23:21.039 --> 00:23:23.849 remembering that the exponent n reflects the 00:23:23.849 --> 00:23:26.009 true number of compounding periods that have 00:23:26.009 --> 00:23:29.049 elapsed between today time zero and the receipt 00:23:29.049 --> 00:23:31.009 of the cash flow. So if you skip a year, you 00:23:31.009 --> 00:23:32.890 don't change the rate, but your exponent just 00:23:32.890 --> 00:23:34.930 jumps accordingly. Exactly. If the first payment 00:23:34.930 --> 00:23:37.569 occurs at the end of period one, so n is one, 00:23:37.730 --> 00:23:39.349 and the next payment occurs at the end of period 00:23:39.349 --> 00:23:41.650 five, then for that second calculation, n is 00:23:41.650 --> 00:23:44.029 five. You simply use the correct power for each. 00:23:44.519 --> 00:23:46.819 And is an index of time, not an index of the 00:23:46.819 --> 00:23:49.240 number of payments. OK, here's a crucial technical 00:23:49.240 --> 00:23:51.880 detail that analysts run into, especially in 00:23:51.880 --> 00:23:55.000 volatile markets. What happens if the risk changes 00:23:55.000 --> 00:23:57.900 or the interest rate itself changes midstream? 00:23:58.099 --> 00:24:01.019 This requires something called sequential discounting. 00:24:01.059 --> 00:24:04.299 And it's a necessary adjustment because you absolutely 00:24:04.299 --> 00:24:07.799 must use the discount rate that accurately reflects 00:24:07.799 --> 00:24:10.599 the risk and opportunity cost. in that specific 00:24:10.599 --> 00:24:13.039 timeframe. You can just use a blended average 00:24:13.039 --> 00:24:15.259 rate. You have to discount the flow using the 00:24:15.259 --> 00:24:17.500 rate that's appropriate for the period it actually 00:24:17.500 --> 00:24:19.920 occurs in. You do. Let's illustrate that with 00:24:19.920 --> 00:24:21.960 a two -period example. Say you have a cash flow 00:24:21.960 --> 00:24:25.720 of $100 in period one at a 5 % rate, and then 00:24:25.720 --> 00:24:28.619 a cash flow of $200 in period two, but the rate 00:24:28.619 --> 00:24:31.519 has now jumped to 10%. Maybe due to a recession 00:24:31.519 --> 00:24:34.160 or just higher perceived risk. The first cash 00:24:34.160 --> 00:24:37.420 flow, the $100, is simple. It's just Y100 divided 00:24:37.420 --> 00:24:40.720 by 1 .05 to the first power, which we already 00:24:40.720 --> 00:24:45.660 know is $95 .24. Right. But for the $200 in period 00:24:45.660 --> 00:24:49.140 two, we have to chain the discounts. We first 00:24:49.140 --> 00:24:52.000 discount the $200 using the 10 % rate for the 00:24:52.000 --> 00:24:54.920 start of period two. Then... We have to take 00:24:54.920 --> 00:24:57.519 that resulting value and discount it back to 00:24:57.519 --> 00:25:00.140 the present using the 5 % rate that was in effect 00:25:00.140 --> 00:25:02.420 during period one. So the calculation visually 00:25:02.420 --> 00:25:05.799 becomes $200 divided by the product of 1 .10 00:25:05.799 --> 00:25:08.640 to the first power and 1 .05 to the first power. 00:25:08.740 --> 00:25:11.240 You're dividing by both discount factors. And 00:25:11.240 --> 00:25:13.339 the calculation for that second cash flow results 00:25:13.339 --> 00:25:17.680 in about $173 .16. So summing those gives us 00:25:17.680 --> 00:25:22.119 an MPV of 95 .24 plus 173 .1 cents, which is 00:25:22.119 --> 00:25:26.200 $268. And this shows that even if interest rates 00:25:26.200 --> 00:25:28.799 change, the fundamental principle is maintained. 00:25:29.119 --> 00:25:31.559 Every single cash flow is reduced by the cumulative 00:25:31.559 --> 00:25:33.740 discount factors between its receipt date and 00:25:33.740 --> 00:25:36.140 today. It ensures the prevailing cost of capital 00:25:36.140 --> 00:25:38.140 for every single period is properly accounted 00:25:38.140 --> 00:25:40.000 for. All right. So once we get our heads around 00:25:40.000 --> 00:25:41.960 discounting these streams of cash flows, we can 00:25:41.960 --> 00:25:43.960 start looking at specialized financial instruments 00:25:43.960 --> 00:25:46.200 where the cash flows are really structured and 00:25:46.200 --> 00:25:48.839 highly predictable. And the first and most common 00:25:48.839 --> 00:25:51.579 of these structured flows is the annuity. An 00:25:51.579 --> 00:25:54.500 annuity is simply any financial arrangement that 00:25:54.500 --> 00:25:57.559 stipulates a structured payment schedule. Crucially, 00:25:57.720 --> 00:25:59.859 the payments are the same amount and they happen 00:25:59.859 --> 00:26:02.480 at regular time intervals. And annuities are 00:26:02.480 --> 00:26:04.980 everywhere. They're used to model bonds, equipment 00:26:04.980 --> 00:26:08.059 leases, retirement payout schedules, even certain 00:26:08.059 --> 00:26:10.539 calculated depreciation charges that are used 00:26:10.539 --> 00:26:13.299 in accounting. When we calculate their present 00:26:13.299 --> 00:26:15.480 value, though, we have to distinguish between 00:26:15.480 --> 00:26:18.880 two critical types, the annuity immediate and 00:26:18.880 --> 00:26:21.619 the annuity due. And this isn't just some small 00:26:21.619 --> 00:26:23.900 academic distinction. It actually changes the 00:26:23.900 --> 00:26:26.799 value. It does. For an annuity immediate, payments 00:26:26.799 --> 00:26:29.240 occur at the end of each period. Think of a standard 00:26:29.240 --> 00:26:31.440 home mortgage payment or a bond coupon payment. 00:26:32.019 --> 00:26:34.339 You live there or you benefit from the loan for 00:26:34.339 --> 00:26:36.279 the month and then you pay at the end. Okay. 00:26:36.359 --> 00:26:38.960 And an annuity due? For an annuity due, payments 00:26:38.960 --> 00:26:41.200 occur at the beginning of each period. You'll 00:26:41.200 --> 00:26:42.859 sometimes hear this referred to as happening 00:26:42.859 --> 00:26:46.000 at time zero. Think of paying your rent or an 00:26:46.000 --> 00:26:48.200 insurance premium. or contributing to a savings 00:26:48.200 --> 00:26:50.759 plan. You pay up front for the use of the asset 00:26:50.759 --> 00:26:53.640 or the service for that period. And mathematically, 00:26:54.059 --> 00:26:56.299 the annuity due is always more valuable than 00:26:56.299 --> 00:26:57.980 the annuity immediate simply because the money 00:26:57.980 --> 00:27:00.880 is received earlier. Exactly. The present value 00:27:00.880 --> 00:27:03.660 of an annuity due is just the PV of annuity immediate 00:27:03.660 --> 00:27:07.130 multiplied by the quantity 1 plus i. It has one 00:27:07.130 --> 00:27:09.789 more full period to earn interest. And the fundamental 00:27:09.789 --> 00:27:12.809 structure of the formula for both types is a 00:27:12.809 --> 00:27:16.069 closed form calculation, right? It saves you 00:27:16.069 --> 00:27:18.809 from calculating dozens or hundreds of individual 00:27:18.809 --> 00:27:21.230 cash flows separately. That's right. It's a summation 00:27:21.230 --> 00:27:23.789 of a geometric series, a very efficient shortcut. 00:27:24.220 --> 00:27:25.680 Now, most of us aren't walking around with a 00:27:25.680 --> 00:27:28.299 finance degree or a scientific calculator, and 00:27:28.299 --> 00:27:30.299 those closed -form annuity formulas can look 00:27:30.299 --> 00:27:32.440 pretty dense. This is where we can introduce 00:27:32.440 --> 00:27:35.059 a truly practical nugget of knowledge for the 00:27:35.059 --> 00:27:38.099 listener, the annuity approximation rule. Ah, 00:27:38.119 --> 00:27:40.319 yes. This approximation is a brilliant shortcut. 00:27:40.500 --> 00:27:43.660 It's designed for rapid mental arithmetic to 00:27:43.660 --> 00:27:46.819 quickly gauge the payment amount, C, that's required 00:27:46.819 --> 00:27:49.359 for an annuity or a loan. And the formula C is 00:27:49.359 --> 00:27:51.420 approximately equal to the present value times 00:27:51.420 --> 00:27:54.059 the quantity of 1 over N plus 2 thirds times 00:27:54.059 --> 00:27:57.220 I. Right. Where C is the periodic payment, PV 00:27:57.220 --> 00:27:59.960 is the loan amount, N is the number of payments, 00:28:00.119 --> 00:28:02.740 and I is the interest rate per period. Let's 00:28:02.740 --> 00:28:04.859 illustrate the incredible utility of this with 00:28:04.859 --> 00:28:08.299 a tough example. You take a $10 ,000 loan paid 00:28:08.299 --> 00:28:11.059 annually over N equals 10 years at a pretty high 00:28:11.059 --> 00:28:15.230 15 % interest rate. So I is 0 .15. Okay, we can 00:28:15.230 --> 00:28:17.470 do the approximation mentally. We need to calculate 00:28:17.470 --> 00:28:20.329 the term in the parentheses first. 1 tenth plus 00:28:20.329 --> 00:28:24.380 2 thirds of 0 .15. One -tenth is just 0 .1. And 00:28:24.380 --> 00:28:27.240 two -thirds of 0 .15 is also 0 .1. So the term 00:28:27.240 --> 00:28:30.460 is just 0 .1 plus 0 .1, which is 0 .2. Then we 00:28:30.460 --> 00:28:33.339 multiply the loan principle by that 0 .2, 10 00:28:33.339 --> 00:28:35.720 ,000 times 0 .2. That gives us an approximate 00:28:35.720 --> 00:28:38.759 annual payment of $2 ,000. And the true payment, 00:28:38.819 --> 00:28:40.700 if you were to calculate it using the complex 00:28:40.700 --> 00:28:43.680 geometric series formula, is actually 1 ,993 00:28:43.680 --> 00:28:46.319 dollars. That is remarkably close. It's accurate 00:28:46.319 --> 00:28:49.910 to within 0 .35%. And the sources concern this 00:28:49.910 --> 00:28:52.190 approximation is accurate to within plus or minus 00:28:52.190 --> 00:28:55.390 6%, even for interest rates up to 20%. It's intended 00:28:55.390 --> 00:28:57.849 for rough, quick analysis, but its accuracy really 00:28:57.849 --> 00:28:59.670 speaks volumes about the mathematical structure 00:28:59.670 --> 00:29:01.730 of the annuity. It lets you quickly assess a 00:29:01.730 --> 00:29:04.029 loan offer and spot a potentially misleading 00:29:04.029 --> 00:29:06.910 or bad deal without ever having to open a spreadsheet. 00:29:07.250 --> 00:29:10.009 Precisely. Now, moving beyond finite payments, 00:29:10.230 --> 00:29:12.349 let's talk about perpetuities. These are payments 00:29:12.349 --> 00:29:14.890 that theoretically go on forever. A perpetuity 00:29:14.890 --> 00:29:18.450 refers to a fixed periodic payment that is receivable 00:29:18.450 --> 00:29:20.970 indefinitely. I imagine they're pretty rare. 00:29:21.470 --> 00:29:23.829 They are rare. Governments sometimes issue them, 00:29:23.910 --> 00:29:26.750 and they're used in valuation to model certain 00:29:26.750 --> 00:29:29.630 very long -lived assets or endowments. And the 00:29:29.630 --> 00:29:32.089 calculation is remarkably simple. It comes from 00:29:32.089 --> 00:29:35.069 taking the limit of the annuity formula as the 00:29:35.069 --> 00:29:37.930 number of periods, N, approaches infinity. It 00:29:37.930 --> 00:29:40.890 simplifies beautifully, too. PV equals C divided 00:29:40.890 --> 00:29:43.470 by I. So the present value is just the periodic 00:29:43.470 --> 00:29:45.609 payment divided by the interest rate. Why is 00:29:45.609 --> 00:29:48.609 it so simple? Because when payments go on forever... 00:29:48.759 --> 00:29:51.000 The later payments are so heavily discounted 00:29:51.000 --> 00:29:53.759 that their present value approaches zero. All 00:29:53.759 --> 00:29:56.099 the real value is concentrated in the early years. 00:29:56.180 --> 00:29:58.500 Okay, give me an example. If an endowment promises 00:29:58.500 --> 00:30:02.279 to pay you $500 every year, forever, and the 00:30:02.279 --> 00:30:05.759 prevailing discount rate is 5%, The present value 00:30:05.759 --> 00:30:10.220 is just 500 divided by 0 .05, which is $10 ,000. 00:30:10.579 --> 00:30:14.160 Ah, because if you invest $10 ,000 today at 5%, 00:30:14.160 --> 00:30:17.900 it generates $500 per year forever without ever 00:30:17.900 --> 00:30:20.079 touching the principal. That's the logic. Okay, 00:30:20.140 --> 00:30:22.519 and finally, we arrive at one of the most practical 00:30:22.519 --> 00:30:26.039 and complex applications of PV, the valuation 00:30:26.039 --> 00:30:29.700 of a bond. A bond is fundamentally the intersection 00:30:29.700 --> 00:30:32.720 of an annuity and a lump sum. That's a great 00:30:32.720 --> 00:30:35.579 way to put it. A bond is an interest -earning 00:30:35.579 --> 00:30:38.099 debt security with a face value, which we call 00:30:38.099 --> 00:30:41.539 F, a fixed coupon rate, R, and a maturity date. 00:30:41.640 --> 00:30:44.079 The bondholder receives regular coupon payments, 00:30:44.279 --> 00:30:46.480 typically semi -annual, and they're calculated 00:30:46.480 --> 00:30:49.400 as the face value times the coupon rate. So the 00:30:49.400 --> 00:30:51.400 present value, which is the purchase price of 00:30:51.400 --> 00:30:53.680 that bond, has to be calculated as the sum of 00:30:53.680 --> 00:30:56.059 two distinct discounted values. That's right. 00:30:56.359 --> 00:30:57.880 First, you have the stream of coupon payments. 00:30:58.019 --> 00:30:59.980 Since there are equal amounts received at regular 00:30:59.980 --> 00:31:02.039 intervals, you calculate them as the PV of an 00:31:02.039 --> 00:31:04.559 annuity. The final return of the face value, 00:31:04.819 --> 00:31:08.720 F, at maturity. Since this is a single, one -time 00:31:08.720 --> 00:31:11.279 payment, you calculate it as the PV of a lump 00:31:11.279 --> 00:31:13.920 sum. You add the two resulting figures together 00:31:13.920 --> 00:31:16.359 to get the bond's fair market value today. And 00:31:16.359 --> 00:31:18.779 the relationship between the bond's fixed coupon 00:31:18.779 --> 00:31:21.500 rate, R, and the variable market interest rate, 00:31:21.599 --> 00:31:24.980 I, is everything. That relationship determines 00:31:24.980 --> 00:31:27.839 whether the bond trades at par, a discount, or 00:31:27.839 --> 00:31:30.500 a premium. This is the critical financial dynamic. 00:31:30.940 --> 00:31:34.039 The market rate, I, is the discount rate we use 00:31:34.039 --> 00:31:36.460 in our PV formula. Okay, so what if the coupon 00:31:36.460 --> 00:31:39.380 rate happens to equal the market rate? Then the 00:31:39.380 --> 00:31:43.359 bond is sold at par, meaning its PV, its price, 00:31:43.500 --> 00:31:46.180 is equal to its base value. But if the coupon 00:31:46.180 --> 00:31:48.599 rate is less than the market rate, the bond's 00:31:48.599 --> 00:31:50.500 fixed payments are less attractive than what 00:31:50.500 --> 00:31:52.480 the market is currently offering. So to make 00:31:52.480 --> 00:31:54.819 the bond's yield comparable to the market, it 00:31:54.819 --> 00:31:57.359 has to be sold at a discount. The purchase price, 00:31:57.480 --> 00:31:59.759 the PV, is less than the face value. Exactly. 00:31:59.799 --> 00:32:01.940 And conversely, if the coupon rate is greater 00:32:01.940 --> 00:32:04.380 than the market rate, the bond is paying a higher 00:32:04.380 --> 00:32:07.180 fixed return than new market instruments. It's 00:32:07.180 --> 00:32:09.559 highly desirable. So investors are willing to 00:32:09.559 --> 00:32:12.079 pay a price greater than the face value it's 00:32:12.079 --> 00:32:15.369 sold. And that's it. So if interest rates in 00:32:15.369 --> 00:32:18.130 the market suddenly spike, all existing bonds 00:32:18.130 --> 00:32:21.069 drop in value because their fixed coupon rate 00:32:21.069 --> 00:32:23.410 is now too low compared to the new market rate, 00:32:23.549 --> 00:32:27.230 forcing their PV down to compensate. It's pure 00:32:27.230 --> 00:32:30.150 present value math driving the daily volatility 00:32:30.150 --> 00:32:32.859 of the bond market. We've established the math 00:32:32.859 --> 00:32:36.099 and the specialized formulas. Now, let's look 00:32:36.099 --> 00:32:38.880 at how PV is used to navigate risk, to assess 00:32:38.880 --> 00:32:41.640 uncertainty, and ultimately to make investment 00:32:41.640 --> 00:32:44.019 decisions. Before we do, let's quickly confirm 00:32:44.019 --> 00:32:46.059 the bedrock technical foundations that allow 00:32:46.059 --> 00:32:48.240 all this complex math to work so consistently. 00:32:48.559 --> 00:32:51.059 First is the principle of additivity. Right. 00:32:51.140 --> 00:32:54.559 The PV of a bundle of cash flows is, simply put, 00:32:54.700 --> 00:32:56.960 the sum of the individual PVs. We saw that in 00:32:56.960 --> 00:32:59.920 our NPD example. You can add, subtract, and manipulate 00:32:59.920 --> 00:33:02.720 these discounted values. which is why the NPV 00:33:02.720 --> 00:33:04.680 process works so cleanly. And second, and this 00:33:04.680 --> 00:33:06.420 is probably the most important part, are the 00:33:06.420 --> 00:33:08.359 underlying assumptions that are built into the 00:33:08.359 --> 00:33:10.220 discount rate. What are those? The calculation 00:33:10.220 --> 00:33:13.920 requires that the two major risks, price inflation 00:33:13.920 --> 00:33:17.839 and the risk of default, are already fully incorporated 00:33:17.839 --> 00:33:20.140 into the interest rate you're using. you're either 00:33:20.140 --> 00:33:22.539 using the real interest rate to account for purchasing 00:33:22.539 --> 00:33:25.000 power erosion. Or you're adding a risk premium 00:33:25.000 --> 00:33:27.180 to the rate to account for the possibility of 00:33:27.180 --> 00:33:29.200 non -payment. Exactly. If you use a rate that 00:33:29.200 --> 00:33:31.799 isn't properly risk -adjusted, your valuation 00:33:31.799 --> 00:33:34.369 will be wildly inaccurate. Now, before we get 00:33:34.369 --> 00:33:36.130 into the practical decision making, you mentioned 00:33:36.130 --> 00:33:38.309 something interesting for the curious listener. 00:33:38.430 --> 00:33:41.549 For continuous payments, the PV is actually mathematically 00:33:41.549 --> 00:33:44.410 related to the Laplace transform. It is, yes. 00:33:44.589 --> 00:33:47.210 For cash flows that are modeled as a continuous 00:33:47.210 --> 00:33:49.990 stream rather than discrete annual or quarterly 00:33:49.990 --> 00:33:53.190 payments, the math shifts into continuous calculus. 00:33:53.769 --> 00:33:56.869 And the discounting factor is exactly analogous 00:33:56.869 --> 00:33:59.369 to the inverse operation that's performed by 00:33:59.369 --> 00:34:01.940 the Laplace transform. It just goes to show that 00:34:01.940 --> 00:34:04.519 this concept of discounting future value is a 00:34:04.519 --> 00:34:07.000 universal mathematical concept that scales from 00:34:07.000 --> 00:34:10.059 basic algebra right up to advanced physics. That's 00:34:10.059 --> 00:34:12.460 fascinating. But let's focus on the reality of 00:34:12.460 --> 00:34:15.400 forecasting the future. Practitioners use PV, 00:34:15.599 --> 00:34:17.960 but the future is always uncertain. Our sources 00:34:17.960 --> 00:34:21.619 highlight two distinct flavors of the present 00:34:21.619 --> 00:34:23.840 value approach to handle this uncertainty. Right. 00:34:23.920 --> 00:34:25.780 The first is what's called the traditional PV 00:34:25.780 --> 00:34:28.559 approach. This is the method most often taught 00:34:28.559 --> 00:34:31.820 in introductory courses. You use a single best 00:34:31.820 --> 00:34:34.679 guess set of estimated cash flows and you discount 00:34:34.679 --> 00:34:37.179 them using a single aggregated risk adjusted 00:34:37.179 --> 00:34:39.679 interest rate. This rate, often called the cost 00:34:39.679 --> 00:34:41.800 of capital, is supposed to compensate for everything. 00:34:41.980 --> 00:34:45.159 Market risk, credit risk, inflation risk. Right. 00:34:45.260 --> 00:34:47.840 It's simple, it's fast, but it's totally reliant 00:34:47.840 --> 00:34:50.280 on getting that one single discount rate perfectly 00:34:50.280 --> 00:34:53.050 right. The second is the expected PV approach. 00:34:53.489 --> 00:34:56.070 This sounds more complex. It is far more complex, 00:34:56.090 --> 00:34:59.230 but also more robust. Instead of relying on one 00:34:59.230 --> 00:35:02.030 single guess for the future, the analyst models 00:35:02.030 --> 00:35:05.150 multiple cash flow scenarios, say a best case, 00:35:05.289 --> 00:35:07.849 a worst case, and a most likely case, and they 00:35:07.849 --> 00:35:10.050 assign a specific probability to each one of 00:35:10.050 --> 00:35:11.809 those scenarios. And then they calculate the 00:35:11.809 --> 00:35:15.469 NPV for each one. They do. And crucially, they 00:35:15.469 --> 00:35:17.769 use a credit -adjusted risk -free rate in this 00:35:17.769 --> 00:35:21.639 process. By calculating the expected value, which 00:35:21.639 --> 00:35:24.559 is the PV, multiplied by its probability for 00:35:24.559 --> 00:35:26.960 each outcome, they separate the probability of 00:35:26.960 --> 00:35:29.480 the outcome from the time value penalty. Then 00:35:29.480 --> 00:35:31.659 they add up the results to get a weighted average 00:35:31.659 --> 00:35:35.420 expected PV. Exactly. This approach is superior 00:35:35.420 --> 00:35:38.400 for high -risk, volatile projects, where the 00:35:38.400 --> 00:35:40.579 variability of the cash flow is the main uncertainty 00:35:40.579 --> 00:35:42.800 rather than the stability of the required rate 00:35:42.800 --> 00:35:45.079 of return. The choice between these two approaches 00:35:45.079 --> 00:35:47.719 really just highlights that valuation is rarely 00:35:47.719 --> 00:35:50.739 an absolute truth. It's more of a sophisticated, 00:35:51.019 --> 00:35:53.219 structured estimate based on our assumptions 00:35:53.219 --> 00:35:55.300 about what the future holds. And this brings 00:35:55.300 --> 00:35:57.599 us to the most critical decision in the entire 00:35:57.599 --> 00:36:00.639 PV process, choosing the interest rate itself. 00:36:00.739 --> 00:36:02.719 You can't just use the rate your bank offers. 00:36:02.780 --> 00:36:04.280 You have to start with the risk -free interest 00:36:04.280 --> 00:36:06.380 rate. And the risk -free rate is the return you 00:36:06.380 --> 00:36:08.119 could get on the safest investment possible. 00:36:08.760 --> 00:36:10.639 Usually this is modeled on short -term government 00:36:10.639 --> 00:36:13.840 debt like treasury bills. It serves as the absolute 00:36:13.840 --> 00:36:16.679 minimum benchmark for any investment you're considering. 00:36:17.039 --> 00:36:20.340 If a financial project, even one that seems guaranteed, 00:36:20.739 --> 00:36:23.800 doesn't offer a rate of return that equals or 00:36:23.800 --> 00:36:26.679 exceeds this minimum risk -free rate. You should 00:36:26.679 --> 00:36:29.380 just reject it. You should. You would be better 00:36:29.380 --> 00:36:31.360 off simply putting your capital into the risk 00:36:31.360 --> 00:36:33.980 -free asset instead. And if the project involves 00:36:33.980 --> 00:36:36.860 any uncertainty, which most do, you must then 00:36:36.860 --> 00:36:39.300 incorporate the risk premium. The risk premium 00:36:39.300 --> 00:36:41.559 is the additional percentage that's added to 00:36:41.559 --> 00:36:44.059 the risk -free rate to compensate the investor 00:36:44.059 --> 00:36:45.960 for taking on that uncertainty. So how do you 00:36:45.960 --> 00:36:48.539 determine the size of that premium? You compare. 00:36:48.820 --> 00:36:51.199 You look at the required rates of return for 00:36:51.199 --> 00:36:53.860 other projects or assets with similar levels 00:36:53.860 --> 00:36:56.610 of risk out on the market. If a publicly traded 00:36:56.610 --> 00:36:59.190 company with a similar risk profile demands an 00:36:59.190 --> 00:37:02.210 8 % return from its investors, then your private 00:37:02.210 --> 00:37:05.070 project with that same risk level should also 00:37:05.070 --> 00:37:07.969 be discounted at an 8 % rate. This ensures your 00:37:07.969 --> 00:37:10.070 discount rate accounts for both the time value 00:37:10.070 --> 00:37:12.269 of money and the likelihood of actually getting 00:37:12.269 --> 00:37:14.769 those future payments. It's critical. So how 00:37:14.769 --> 00:37:17.510 do investors use PV to choose between competing 00:37:17.510 --> 00:37:19.670 projects that seem to offer similar returns? 00:37:19.949 --> 00:37:22.530 They calculate the PV for all competing projects 00:37:22.530 --> 00:37:26.050 using the same appropriate risk -adjusted interest 00:37:26.050 --> 00:37:29.590 rate. And the golden rule is elegantly simple. 00:37:30.070 --> 00:37:32.510 The project with the smallest present value, 00:37:32.690 --> 00:37:35.500 meaning the least initial outlay required, is 00:37:35.500 --> 00:37:37.500 the most efficient choice. Because it offers 00:37:37.500 --> 00:37:39.980 the same level of future return for less capital 00:37:39.980 --> 00:37:42.880 invested today. And that is the bedrock of capital 00:37:42.880 --> 00:37:45.000 deployment decisions in corporations everywhere. 00:37:45.219 --> 00:37:47.800 To wrap up our detailed look at valuation, let's 00:37:47.800 --> 00:37:49.320 go back in time and talk about a traditional 00:37:49.320 --> 00:37:52.079 method of valuation that was used long before 00:37:52.079 --> 00:37:54.219 complex spreadsheets and geometric series were 00:37:54.219 --> 00:37:56.860 common. It's called year's purchase. Year's purchase 00:37:56.860 --> 00:37:59.000 was the traditional method, especially in real 00:37:59.000 --> 00:38:01.519 estate and land management, for valuing a future 00:38:01.519 --> 00:38:04.289 income stream as a present capital sum. Instead 00:38:04.289 --> 00:38:06.269 of complex formulas, you would simply multiply 00:38:06.269 --> 00:38:08.929 the average expected annual cash flow by a multiple 00:38:08.929 --> 00:38:11.469 known as years' purchase. Let's use the sources 00:38:11.469 --> 00:38:14.269 example. Valuing a property that's leased at 00:38:14.269 --> 00:38:17.570 a fixed rent of $10 ,000 per annum, and the standard 00:38:17.570 --> 00:38:19.989 practice in that area dictates using 20 years' 00:38:20.090 --> 00:38:23.170 purchase. So the valuation is simply 20 times 00:38:23.170 --> 00:38:26.050 $10 ,000, resulting in a present capital sum 00:38:26.050 --> 00:38:29.269 of $200 ,000. Okay, so what's the fascinating 00:38:29.269 --> 00:38:32.090 aha moment here? The aha moment is that this 00:38:32.090 --> 00:38:34.730 simple historical rule mathematically equates 00:38:34.730 --> 00:38:37.630 exactly to discounting that income stream in 00:38:37.630 --> 00:38:40.989 perpetuity at a 5 % rate. Oh, wow. Remember the 00:38:40.989 --> 00:38:44.409 perpetuity formula? PV equals C divided by I. 00:38:44.510 --> 00:38:48.449 10 ,000 divided by .05 is 200 ,000. The factor 00:38:48.449 --> 00:38:50.630 of 20 years purchase is simply the reciprocal 00:38:50.630 --> 00:38:53.929 of the discount rate. 1 divided by .05 is 20. 00:38:54.380 --> 00:38:56.699 That is incredible. It shows that the fundamental 00:38:56.699 --> 00:38:59.159 economic intuition of discounting future cash 00:38:59.159 --> 00:39:01.739 flows was already in place centuries ago, even 00:39:01.739 --> 00:39:04.079 if the math was expressed differently. And we 00:39:04.079 --> 00:39:07.460 have hard historical evidence of its use. The 00:39:07.460 --> 00:39:09.619 English crown utilized this exact methodology 00:39:09.619 --> 00:39:12.880 extensively in the early 16th century, when they 00:39:12.880 --> 00:39:15.159 were setting resale prices for manors and lands 00:39:15.159 --> 00:39:17.699 seized during the dissolution of the monasteries. 00:39:17.900 --> 00:39:20.679 They had established a standard usage of 20 years' 00:39:20.780 --> 00:39:23.269 purchase. They were performing present value 00:39:23.269 --> 00:39:25.929 calculations using intuitive multiples, proving 00:39:25.929 --> 00:39:28.710 the cost of waiting is a core financial concept 00:39:28.710 --> 00:39:31.349 that never changes. That brings us to the end 00:39:31.349 --> 00:39:34.090 of our deep dive into the dollar dilemma. To 00:39:34.090 --> 00:39:36.389 quickly recap, present value is the essential 00:39:36.389 --> 00:39:38.530 tool for comparing financial promises across 00:39:38.530 --> 00:39:41.110 time, and it's defined by three core inputs. 00:39:41.389 --> 00:39:43.980 The future amount. the time until you get it, 00:39:44.039 --> 00:39:46.519 and the risk -adjusted discount rate. And mastering 00:39:46.519 --> 00:39:48.739 these calculations, from discounting a single 00:39:48.739 --> 00:39:51.059 lump sum to valuing complex instruments like 00:39:51.059 --> 00:39:53.440 annuities and bonds, allows you to look past 00:39:53.440 --> 00:39:55.840 the superficial headline promises. You can now 00:39:55.840 --> 00:39:58.039 determine the intrinsic value of financial instruments 00:39:58.039 --> 00:40:00.519 today. You can translate future hope into current 00:40:00.519 --> 00:40:03.059 dollars. So what does this all mean for you, 00:40:03.099 --> 00:40:05.730 the listener? We explored the two high -level 00:40:05.730 --> 00:40:07.849 approaches to handling financial uncertainty, 00:40:08.170 --> 00:40:11.389 the traditional PV, which uses a single aggregated 00:40:11.389 --> 00:40:14.710 discount rate, and the expected PV, which actively 00:40:14.710 --> 00:40:17.610 maps out multiple probabilistic futures. And 00:40:17.610 --> 00:40:20.690 that distinction forces us to ask a final provocative 00:40:20.690 --> 00:40:22.969 question that really builds on all the knowledge 00:40:22.969 --> 00:40:25.269 we have discussed today. When you're valuing 00:40:25.269 --> 00:40:27.230 an investment that you truly care about, say 00:40:27.230 --> 00:40:29.750 you're funding a startup or buying a long -term 00:40:29.750 --> 00:40:32.429 commercial property, should you rely on a single? 00:40:32.860 --> 00:40:35.719 black box discount rate that attempts to aggregate 00:40:35.719 --> 00:40:39.440 all market credit and operational risk into one 00:40:39.440 --> 00:40:42.960 single number? Or is it more prudent, even if 00:40:42.960 --> 00:40:45.320 it's more time consuming, to actively map out 00:40:45.320 --> 00:40:47.940 multiple probabilistic futures, assigning weights 00:40:47.940 --> 00:40:50.380 to different potential outcomes before you estimate 00:40:50.380 --> 00:40:52.300 its current worth? Because the choice between 00:40:52.300 --> 00:40:54.340 those two methods determines how much variability 00:40:54.340 --> 00:40:56.820 and risk you are truly willing to accept in your 00:40:56.820 --> 00:40:57.980 final valuation figure.