WEBVTT 00:00:00.000 --> 00:00:02.279 Okay, let's unpack this. Welcome to the deep 00:00:02.279 --> 00:00:06.179 dive, where we take concepts that sound overwhelmingly 00:00:06.179 --> 00:00:09.060 complicated, the kind of topic that makes your 00:00:09.060 --> 00:00:11.800 eyes glaze over in a textbook, and turn them 00:00:11.800 --> 00:00:14.140 into insights that fundamentally change how you 00:00:14.140 --> 00:00:16.769 view your own finances. Today, we are undertaking 00:00:16.769 --> 00:00:19.969 a really comprehensive deep dive into the architecture 00:00:19.969 --> 00:00:22.789 and the mathematics of annuities. Annuity. For 00:00:22.789 --> 00:00:25.510 a lot of people, annuities live in this sort 00:00:25.510 --> 00:00:28.829 of fuzzy world of retirement products sold by 00:00:28.829 --> 00:00:31.730 insurance agents. They sound complex, but at 00:00:31.730 --> 00:00:33.649 their heart, they are probably the most essential 00:00:33.649 --> 00:00:36.590 financial building block. That's right. When 00:00:36.590 --> 00:00:39.009 you strip away the sales pitch, the core idea 00:00:39.009 --> 00:00:41.829 is stunningly simple. Our source material gives 00:00:41.829 --> 00:00:45.149 us a concise and powerful definition. What's 00:00:45.149 --> 00:00:46.829 that? An annuity is just a series of uniform 00:00:46.829 --> 00:00:49.170 payments or receipts made at perfectly equal 00:00:49.170 --> 00:00:51.509 time intervals, usually over a specified term. 00:00:51.750 --> 00:00:54.689 And that's our mission today. Exactly. The mission 00:00:54.689 --> 00:00:56.950 is to quickly and thoroughly understand the foundational 00:00:56.950 --> 00:01:00.090 types, the exact valuation methods, and the practical 00:01:00.090 --> 00:01:02.630 calculations that underpin everything from your 00:01:02.630 --> 00:01:05.310 home loan to your retirement security. To immediately 00:01:05.310 --> 00:01:09.269 ground this abstract math in your daily life. 00:01:09.629 --> 00:01:11.650 Just think about the routine cash flows you already 00:01:11.650 --> 00:01:14.150 handle. So many of them are annuities. They really 00:01:14.150 --> 00:01:16.609 are. The consistent deposits you make into your 00:01:16.609 --> 00:01:19.430 savings account week after week or maybe month 00:01:19.430 --> 00:01:22.609 after month, that's an annuity. Your inescapable 00:01:22.609 --> 00:01:26.969 level monthly home mortgage or car loan payment. 00:01:27.250 --> 00:01:29.609 That is absolutely an annuity. The annual or 00:01:29.609 --> 00:01:32.129 monthly premiums you send to your insurance provider. 00:01:32.450 --> 00:01:34.909 Also an annuity. And, of course, the classic 00:01:34.909 --> 00:01:37.969 retirement scenario, regular pension payments 00:01:37.969 --> 00:01:40.980 coming to you. The consistency of these parents, 00:01:41.120 --> 00:01:44.299 that regularity, is what allows us to apply some 00:01:44.299 --> 00:01:47.000 really rigorous mathematics. So why does that 00:01:47.000 --> 00:01:49.099 regularity matter so much? Because it lets us 00:01:49.099 --> 00:01:51.579 apply the time value of money, but in a highly 00:01:51.579 --> 00:01:54.200 efficient sort of summarized way. Instead of 00:01:54.200 --> 00:01:57.739 calculating the present value of, say, 360 separate 00:01:57.739 --> 00:02:00.680 mortgage payments, we can use a single factor 00:02:00.680 --> 00:02:03.159 to calculate it all at once. Ah, so it's a shortcut. 00:02:03.420 --> 00:02:06.680 It's a massive shortcut. The value of an annuity 00:02:06.680 --> 00:02:09.860 is always expressed as either a present value, 00:02:10.139 --> 00:02:12.580 which is the lump sum those future payments are 00:02:12.580 --> 00:02:15.379 worth today, or a future value, which is their 00:02:15.379 --> 00:02:21.039 total accumulated worth down the line. to either 00:02:21.039 --> 00:02:23.500 discount for the present value or accumulate 00:02:23.500 --> 00:02:26.520 for the future value. Precisely. And the interesting 00:02:26.520 --> 00:02:29.319 real -world context here is that while the math 00:02:29.319 --> 00:02:32.159 is pure finance, the contracts themselves are, 00:02:32.259 --> 00:02:34.680 well, they're overwhelmingly issued by life insurance 00:02:34.680 --> 00:02:37.199 companies. Why them specifically? They're the 00:02:37.199 --> 00:02:40.099 entities best suited to manage the risks inherent 00:02:40.099 --> 00:02:42.460 in guaranteeing that regular income, particularly 00:02:42.460 --> 00:02:44.639 for things like retirement or survivor benefits. 00:02:44.979 --> 00:02:46.979 Right. And as we'll get into, that insurance 00:02:46.979 --> 00:02:49.680 context introduces the most mathematically common 00:02:49.740 --> 00:02:52.500 complex, and frankly, fascinating element, mortality 00:02:52.500 --> 00:02:55.740 risk. It does. But you can't build the roof until 00:02:55.740 --> 00:02:58.439 you have the foundation. So we absolutely must 00:02:58.439 --> 00:03:01.000 first master the structure of the simple guaranteed 00:03:01.000 --> 00:03:04.180 annuity. That sets the stage beautifully. Now, 00:03:04.199 --> 00:03:06.139 before we look at a single formula, we have to 00:03:06.139 --> 00:03:07.939 grasp how these things are categorized. Yes, 00:03:07.979 --> 00:03:10.379 this is crucial. Because the way an annuity is 00:03:10.379 --> 00:03:12.719 classified determines exactly which formula you 00:03:12.719 --> 00:03:15.680 pull off the shelf. If you mix up the type, the 00:03:15.680 --> 00:03:18.120 calculation is worthless. Our source material 00:03:18.120 --> 00:03:21.699 breaks annuities down across four distinct but 00:03:21.699 --> 00:03:24.300 often intertwined dimensions. We need to walk 00:03:24.300 --> 00:03:26.780 through each one because they really define the 00:03:26.780 --> 00:03:28.960 structure of the payments. Let's start with what 00:03:28.960 --> 00:03:30.979 seems to be the most critical distinction for 00:03:30.979 --> 00:03:33.379 valuation, something that changes the math right 00:03:33.379 --> 00:03:36.699 away, the timing of payments. Yes. This gives 00:03:36.699 --> 00:03:39.060 us the annuity immediate versus the annuity due. 00:03:39.259 --> 00:03:41.919 Here's where we move beyond simple definitions 00:03:41.919 --> 00:03:46.439 and into... Financial architecture. The standard 00:03:46.439 --> 00:03:48.900 default structure, the one used for nearly all 00:03:48.900 --> 00:03:51.479 loan repayment schedules, is the annuity immediate, 00:03:51.719 --> 00:03:53.560 or you might hear it called an ordinary annuity. 00:03:53.740 --> 00:03:56.400 Okay. And the core rule here is what? Payments 00:03:56.400 --> 00:03:58.560 are made at the end of each period. At the end. 00:03:58.800 --> 00:04:01.719 And the reason for that timing is really deeply 00:04:01.719 --> 00:04:04.159 ingrained in lending practices, isn't it? It 00:04:04.159 --> 00:04:06.759 reflects how interest accrues. Exactly. When 00:04:06.759 --> 00:04:08.680 you take out a loan, you use the money for the 00:04:08.680 --> 00:04:10.729 first month. Interest accrues on that principal 00:04:10.729 --> 00:04:13.669 over those, say, 30 days. So when you make your 00:04:13.669 --> 00:04:15.930 payment at the end of the month. You are covering 00:04:15.930 --> 00:04:18.310 the interest that accrued during the preceding 00:04:18.310 --> 00:04:20.709 period, plus a little bit of principal reduction. 00:04:21.170 --> 00:04:23.889 The payments are always in arrears. Your home 00:04:23.889 --> 00:04:26.629 mortgage is the perfect example. Right. Now, 00:04:26.709 --> 00:04:29.649 if that's the default, why do we need the alternative, 00:04:29.949 --> 00:04:32.629 the annuity due? The annuity due just flips the 00:04:32.629 --> 00:04:34.850 timing. Payments are made at the beginning of 00:04:34.850 --> 00:04:37.589 each period. They are paid in advance. That small 00:04:37.589 --> 00:04:40.410 shift. changes the math entirely, as we'll see. 00:04:40.509 --> 00:04:42.350 It does because that first payment is made at 00:04:42.350 --> 00:04:45.550 time zero. It gets a whole extra period to compound 00:04:45.550 --> 00:04:48.250 interest compared to a payment made a month later. 00:04:48.470 --> 00:04:50.410 And the examples here are things where you pay 00:04:50.410 --> 00:04:53.889 up front for the right to use an asset or receive 00:04:53.889 --> 00:04:56.870 a service. Exactly. Rent is the most common example. 00:04:57.069 --> 00:04:59.290 You pay on the first of the month to use the 00:04:59.290 --> 00:05:01.410 apartment for that month. Insurance premiums 00:05:01.410 --> 00:05:03.589 too. Insurance premiums operate the same way. 00:05:03.949 --> 00:05:06.129 You pay in advance to cover the risk during the 00:05:06.129 --> 00:05:08.790 upcoming period. So payment at the end for obligations 00:05:08.790 --> 00:05:11.209 like loans, that's immediate. And payment at 00:05:11.209 --> 00:05:13.329 the beginning for rights or services like rent 00:05:13.329 --> 00:05:16.709 or insurance, that's due. That is a powerful 00:05:16.709 --> 00:05:19.629 and necessary mental shortcut you need when you 00:05:19.629 --> 00:05:21.790 start tackling the math. Okay, let's move to 00:05:21.790 --> 00:05:24.370 the second dimension, contingency of payments. 00:05:24.649 --> 00:05:26.629 This one defines the duration of the payment 00:05:26.629 --> 00:05:29.170 stream, specifically whether the term is certain 00:05:29.170 --> 00:05:32.370 or if it's based on human survival. So first 00:05:32.370 --> 00:05:34.639 we have the annuity certain. which I assume is 00:05:34.639 --> 00:05:36.759 the financially straightforward version. It is. 00:05:36.800 --> 00:05:40.819 This pays over a fixed known period. The number 00:05:40.819 --> 00:05:43.959 of payments, we call it N, is set in stone in 00:05:43.959 --> 00:05:47.319 the contract, say, 10 years or 360 months. And 00:05:47.319 --> 00:05:49.180 that payment schedule will be completed no matter 00:05:49.180 --> 00:05:51.839 what. Regardless of whether the individual receiving 00:05:51.839 --> 00:05:55.180 the payments lives or dies, the uncertainty is 00:05:55.180 --> 00:05:57.959 removed entirely. But once we introduce human 00:05:57.959 --> 00:06:00.680 life, we get the life annuity. And here, the 00:06:00.680 --> 00:06:03.300 payment is contingent upon survival. It pays 00:06:03.300 --> 00:06:05.920 only while one or more specified lives survive. 00:06:06.319 --> 00:06:08.300 The number of payments is therefore uncertain 00:06:08.300 --> 00:06:10.459 because it's entirely dependent on longevity. 00:06:10.839 --> 00:06:13.319 This is the cornerstone of pensions that provide 00:06:13.319 --> 00:06:16.139 income for life. Exactly. Payments cease the 00:06:16.139 --> 00:06:18.259 moment the annuitant passes away. I find this 00:06:18.259 --> 00:06:20.360 distinction so fascinating because it highlights 00:06:20.360 --> 00:06:23.180 the fundamental difference between pure finance 00:06:23.180 --> 00:06:28.889 and actuarial science. How so? Finance deals 00:06:28.889 --> 00:06:31.569 with known cash flows, while actuarial science 00:06:31.569 --> 00:06:34.029 has to deal with expected cash flows. Absolutely. 00:06:34.050 --> 00:06:36.129 And of course, the industry has created a hybrid, 00:06:36.370 --> 00:06:40.269 the certain and life annuity. Which is also known 00:06:40.269 --> 00:06:42.649 as a life annuity with period certain. Right. 00:06:43.019 --> 00:06:44.779 It sounds like a hedge against dying too soon. 00:06:44.860 --> 00:06:47.399 That's precisely what it is. It guarantees payments 00:06:47.399 --> 00:06:50.920 for a minimum term, let's say 10 years, even 00:06:50.920 --> 00:06:53.240 if the annuitant dies early. And then after that? 00:06:53.399 --> 00:06:55.519 And thereafter, payments continue for as long 00:06:55.519 --> 00:06:57.439 as the annuitant is alive. So if the annuitant 00:06:57.439 --> 00:07:00.019 dies in year three? The beneficiaries still receive 00:07:00.019 --> 00:07:02.420 payments for the remaining seven years of that 00:07:02.420 --> 00:07:04.920 guaranteed term. It offers a balance between 00:07:04.920 --> 00:07:07.420 the risk management of a life annuity and the 00:07:07.420 --> 00:07:10.000 security of a fixed term guarantee. Okay, our 00:07:10.000 --> 00:07:12.180 third classification. This one looks at how the 00:07:12.180 --> 00:07:14.839 investment return is calculated. Variability 00:07:14.839 --> 00:07:17.439 of payments. This is crucial for long -term planning 00:07:17.439 --> 00:07:20.199 because it dictates your risk profile. The safest 00:07:20.199 --> 00:07:23.329 option is the fixed annuity. Fixed. So the insurer 00:07:23.329 --> 00:07:26.250 just declares a fixed interest rate. Yes. And 00:07:26.250 --> 00:07:29.250 that rate provides a guaranteed minimum return 00:07:29.250 --> 00:07:32.290 on the underlying account value. It's all about 00:07:32.290 --> 00:07:34.649 stability and predictability. You know exactly 00:07:34.649 --> 00:07:36.930 what you'll get. Then you jump onto the risk 00:07:36.930 --> 00:07:39.029 and reward spectrum with the variable annuity. 00:07:39.189 --> 00:07:42.189 This is where you, the consumer, take on the 00:07:42.189 --> 00:07:45.639 investment risk. Your premiums are invested directly 00:07:45.639 --> 00:07:48.079 in what are called subaccounts, which are often 00:07:48.079 --> 00:07:50.300 structured just like mutual funds. So your contract 00:07:50.300 --> 00:07:52.980 value and your future income payments can fluctuate. 00:07:53.060 --> 00:07:55.220 They fluctuate directly with the underlying investment 00:07:55.220 --> 00:07:57.850 performance. High market returns mean higher 00:07:57.850 --> 00:08:01.050 payments. A market crash means your income stream 00:08:01.050 --> 00:08:04.589 can shrink. So the insurer is managing the mortality 00:08:04.589 --> 00:08:07.250 risk, but the policyholder is managing the market 00:08:07.250 --> 00:08:10.009 risk. Correct. And then sitting right in the 00:08:10.009 --> 00:08:11.970 middle, attempting to offer the best of both 00:08:11.970 --> 00:08:14.589 worlds is this specialized structure known as 00:08:14.589 --> 00:08:17.350 the equity indexed annuity. Right. These are 00:08:17.350 --> 00:08:19.230 often marketed very aggressively because they 00:08:19.230 --> 00:08:21.750 seem to promise market upside participation without 00:08:21.750 --> 00:08:24.670 the market downside risk. How does the structure 00:08:24.670 --> 00:08:26.939 actually manage that balancing act? Well, they 00:08:26.939 --> 00:08:28.959 credit interest based partly on the performance 00:08:28.959 --> 00:08:31.660 of a specified market index like the S &P 500. 00:08:31.920 --> 00:08:35.440 But, and this is the key, they impose contractual 00:08:35.440 --> 00:08:37.820 restrictions to limit the risk. What are those 00:08:37.820 --> 00:08:40.759 key rules? The two big ones are the cap and the 00:08:40.759 --> 00:08:43.820 participation rate. The cap is the maximum rate 00:08:43.820 --> 00:08:46.120 you can earn in any given year, even if the index 00:08:46.120 --> 00:08:49.639 skyrockets by, say, 20%. And the participation 00:08:49.639 --> 00:08:52.299 rate. A participation rate dictates what percentage 00:08:52.299 --> 00:08:55.320 of the index's gain you actually receive. So 00:08:55.320 --> 00:08:57.330 if the index... gains 10 % and your participation 00:08:57.330 --> 00:09:01.009 rate is 70%, you only get credited with 7%. So 00:09:01.009 --> 00:09:03.970 you trade unlimited upside potential for a layer 00:09:03.970 --> 00:09:06.490 of protection. Exactly. And that protection is 00:09:06.490 --> 00:09:09.049 the third crucial feature. They always include 00:09:09.049 --> 00:09:11.809 a minimum guaranteed return. If the market index 00:09:11.809 --> 00:09:15.549 drops 30%, you earn zero from the index, but 00:09:15.549 --> 00:09:17.889 your contract value won't fall below that minimum 00:09:17.889 --> 00:09:20.009 guarantee. It protects your principal. It's a 00:09:20.009 --> 00:09:22.210 very complex risk transfer mechanism. It is. 00:09:22.570 --> 00:09:25.149 Finally, the fourth classification, deferral 00:09:25.149 --> 00:09:27.129 of payments, which just adjusts the timing of 00:09:27.129 --> 00:09:29.250 when the income starts. This is simply about 00:09:29.250 --> 00:09:31.610 whether you need the money now or later. We start 00:09:31.610 --> 00:09:34.389 with the immediate annuity. Right. Often called 00:09:34.389 --> 00:09:38.250 a SPIA, a single premium immediate annuity. And 00:09:38.250 --> 00:09:40.330 how does that work? This is purchased with a 00:09:40.330 --> 00:09:42.850 lump sum and the income payments begin shortly 00:09:42.850 --> 00:09:46.210 thereafter, typically within one year. This is 00:09:46.210 --> 00:09:48.450 the product of choice for someone who has just 00:09:48.450 --> 00:09:51.590 retired, has a large sum of cash and wants to 00:09:51.590 --> 00:09:54.289 immediately convert that capital into a steady, 00:09:54.370 --> 00:09:57.330 reliable income stream. And its counterpart is 00:09:57.330 --> 00:09:59.830 the deferred annuity. The deferred version is 00:09:59.830 --> 00:10:02.769 the accumulation phase product. It starts income 00:10:02.769 --> 00:10:05.570 payments only after a specified deferral period, 00:10:05.730 --> 00:10:08.250 which might last for decades. And during that 00:10:08.250 --> 00:10:10.809 deferral period? The capital grows tax deferred, 00:10:10.809 --> 00:10:13.289 earning interest or investment returns, until 00:10:13.289 --> 00:10:15.850 the policyholder decides to annuitize and begin 00:10:15.850 --> 00:10:18.440 taking income. And it's important to synthesize 00:10:18.440 --> 00:10:20.779 these classifications. Any of the variability 00:10:20.779 --> 00:10:24.320 types, fixed, variable, or indexed, can be written 00:10:24.320 --> 00:10:26.559 as either an immediate or deferred contract, 00:10:26.919 --> 00:10:29.000 right? That's right. The structure of the investment 00:10:29.000 --> 00:10:31.340 return is separate from the timing of the payout 00:10:31.340 --> 00:10:34.019 phase. So if you're 35 and planning for retirement, 00:10:34.379 --> 00:10:36.720 you might buy a variable deferred annuity, hoping 00:10:36.720 --> 00:10:39.399 for maximum accumulation growth over 30 years 00:10:39.399 --> 00:10:41.419 before starting the income stream. And if you're 00:10:41.419 --> 00:10:45.139 65, Ready to retire. You might buy a fixed immediate 00:10:45.139 --> 00:10:48.120 annuity for immediate guaranteed payments. We've 00:10:48.120 --> 00:10:50.320 built the scaffolding. We understand the four 00:10:50.320 --> 00:10:52.840 defining characteristics. Now we can transition 00:10:52.840 --> 00:10:55.679 to the actual mechanics. How do we take a stream 00:10:55.679 --> 00:10:58.960 of 300 guaranteed payments and accurately summarize 00:10:58.960 --> 00:11:01.799 their value today or tomorrow? We are now focusing 00:11:01.799 --> 00:11:04.139 entirely on the annuity certain where the term 00:11:04.139 --> 00:11:07.240 N is known. And the valuation process is, you 00:11:07.240 --> 00:11:10.120 said, elegant? It is. It's purely the summary 00:11:10.120 --> 00:11:12.860 of that stream of predetermined cash flows into 00:11:12.860 --> 00:11:15.700 a single value. It's the ultimate time -saving 00:11:15.700 --> 00:11:17.960 calculation. And this math hinges on knowing 00:11:17.960 --> 00:11:21.139 two things. The effective interest rate per period, 00:11:21.279 --> 00:11:23.700 which we call a dollars, and the number of total 00:11:23.700 --> 00:11:26.240 payments, none. Let's start with the standard 00:11:26.240 --> 00:11:28.399 structure. Annuity immediate, so payments are 00:11:28.399 --> 00:11:30.860 at the end. When we value these, we use specific 00:11:30.860 --> 00:11:33.539 factors, which are essentially mathematical shortcuts. 00:11:34.000 --> 00:11:37.169 So you multiply the periodic payment, R. by one 00:11:37.169 --> 00:11:39.809 of these factors to get the total value. Exactly. 00:11:40.230 --> 00:11:43.690 First up is the present value factor. Instead 00:11:43.690 --> 00:11:46.029 of reading out the complex algebraic notation, 00:11:46.370 --> 00:11:49.250 let's just focus on what this factor does. Good 00:11:49.250 --> 00:11:51.750 idea. We can just call it the PV factor. Its 00:11:51.750 --> 00:11:54.309 purpose is to discount every single future unit 00:11:54.309 --> 00:11:57.919 payment back to time zero. If you had 60 monthly 00:11:57.919 --> 00:12:00.600 payments, this factor saves you the painful task 00:12:00.600 --> 00:12:03.120 of discounting payment one by one month, payment 00:12:03.120 --> 00:12:05.379 two by two months, and so on. All the way to 00:12:05.379 --> 00:12:08.620 payment 60. Right. It collapses the entire liability 00:12:08.620 --> 00:12:12.559 or asset into a single lump sum today. Conceptually, 00:12:12.679 --> 00:12:15.019 the formula is built using the math of a geometric 00:12:15.019 --> 00:12:17.460 series. You're just summing up the present value 00:12:17.460 --> 00:12:19.779 of dental payments, each discounted by an increasing 00:12:19.779 --> 00:12:22.500 power. Yes. And that underlying structure is 00:12:22.500 --> 00:12:24.720 what allows it to simplify so beautifully into 00:12:24.720 --> 00:12:26.860 that closed form equation. When you multiply 00:12:26.860 --> 00:12:29.600 the PV factor by your recurring payment, R, you 00:12:29.600 --> 00:12:31.740 get the total present value. Okay, now let's 00:12:31.740 --> 00:12:33.919 look at the other side of the timeline, the future 00:12:33.919 --> 00:12:36.500 value factor. This factor accumulates the payments 00:12:36.500 --> 00:12:39.000 forward. It tells you the total cash value, all 00:12:39.000 --> 00:12:40.940 the payments, plus all the compounded interest 00:12:40.940 --> 00:12:43.879 immediately after the very last payment is made. 00:12:44.019 --> 00:12:45.460 This is what you'd care about if you're trying 00:12:45.460 --> 00:12:47.649 to hit a savings goal. Precisely. And what's 00:12:47.649 --> 00:12:50.309 crucial is that these two factors are inherently 00:12:50.309 --> 00:12:52.610 linked. They are just two sides of the same coin 00:12:52.610 --> 00:12:55.370 separated by time. Meaning if you know the present 00:12:55.370 --> 00:12:58.049 value. If you know the present value of an annuity, 00:12:58.169 --> 00:13:01.330 you can calculate the future value simply by 00:13:01.330 --> 00:13:04.169 accumulating that PV forward over non -billy 00:13:04.169 --> 00:13:07.190 periods at the given interest rate. The two values 00:13:07.190 --> 00:13:09.450 must be consistent with the time value of money. 00:13:09.610 --> 00:13:11.789 Let's make this real. The source material gives 00:13:11.789 --> 00:13:14.190 a great example involving a five -year annuity. 00:13:14.289 --> 00:13:17.389 Yes, a nominal annual interest rate of 12 % and 00:13:17.389 --> 00:13:20.149 monthly payments of $100. We need to find the 00:13:20.149 --> 00:13:22.700 present value. Okay, this is a fantastic teaching 00:13:22.700 --> 00:13:24.779 moment because it requires the mandatory first 00:13:24.779 --> 00:13:27.080 step in annuity math. Rate adjustment. Rate adjustment. 00:13:27.220 --> 00:13:30.259 We have a nominal annual rate of 12%, compounded 00:13:30.259 --> 00:13:32.679 monthly. Since payments are monthly, we must 00:13:32.679 --> 00:13:35.620 use the effective monthly rate. So 12 % divided 00:13:35.620 --> 00:13:38.080 by 12 months equals an effective periodic rate, 00:13:38.200 --> 00:13:42.159 $1 of 1%, or 0 .01. And the term adjustment. 00:13:42.419 --> 00:13:44.799 Five years of monthly payments mean $1 equals 00:13:44.799 --> 00:13:47.960 $60 of periods. So we are calculating the value 00:13:47.960 --> 00:13:51.639 of 60 future $100 payments at a 1 % monthly rate. 00:13:51.779 --> 00:13:54.779 We take the $100 payment and multiply it by the 00:13:54.779 --> 00:13:58.940 PV factor for 60 periods at 1%. And running those 00:13:58.940 --> 00:14:01.139 60 monthly payments through the PV factor formula, 00:14:01.360 --> 00:14:05.720 we find that the factor is roughly 44 .955. Meaning 00:14:05.720 --> 00:14:07.960 that if you multiply that factor by the $100 00:14:07.960 --> 00:14:11.740 monthly payment, the total present value is approximately... 00:14:12.139 --> 00:14:16.220 $4 ,495 .50. And think about what that number 00:14:16.220 --> 00:14:21.360 really means. Go on. $4 ,495 .50 today is the 00:14:21.360 --> 00:14:24.100 exact financial equivalent of receiving $100 00:14:24.100 --> 00:14:26.320 at the end of every month for the next five years, 00:14:26.399 --> 00:14:28.620 assuming you can earn a 12 % nominal rate on 00:14:28.620 --> 00:14:30.240 your money elsewhere. It gives you immediate 00:14:30.240 --> 00:14:32.460 clarity on the current worth of a future income 00:14:32.460 --> 00:14:35.200 stream. Exactly. We just calculated the standard 00:14:35.200 --> 00:14:37.580 annuity immediate. Now we look at the minor structural 00:14:37.580 --> 00:14:39.720 change that introduces a significant financial 00:14:39.720 --> 00:14:42.059 difference. the annuity due formulas. Right, 00:14:42.159 --> 00:14:43.580 where payments happen at the beginning of the 00:14:43.580 --> 00:14:45.379 period. The good news is we don't have to start 00:14:45.379 --> 00:14:47.840 from scratch. Not at all. The mathematics of 00:14:47.840 --> 00:14:49.980 the annuity due simply builds upon the annuity 00:14:49.980 --> 00:14:52.500 immediate, thanks to that one extra period of 00:14:52.500 --> 00:14:54.879 interest. Because every payment is made one period 00:14:54.879 --> 00:14:58.000 earlier, it has one more compounding cycle. Precisely. 00:14:58.350 --> 00:15:01.389 This means the present value factor for an annuity 00:15:01.389 --> 00:15:04.350 due is simply the annuity immediate factor multiplied 00:15:04.350 --> 00:15:07.230 by one plus I. And that one plus I factor might 00:15:07.230 --> 00:15:09.629 look tiny, especially if a wallet is small. But 00:15:09.629 --> 00:15:11.629 over a long period of savings or investment, 00:15:11.850 --> 00:15:14.269 that small factor represents the difference between 00:15:14.269 --> 00:15:16.629 constantly playing catch up and getting ahead. 00:15:17.080 --> 00:15:19.679 It absolutely encapsulates the exponential power 00:15:19.679 --> 00:15:22.100 of early investment, and the same relationship 00:15:22.100 --> 00:15:25.100 holds for the future value factors. The annuity 00:15:25.100 --> 00:15:27.059 due future value factor is the annuity immediate 00:15:27.059 --> 00:15:30.240 future value factor, also multiplied by 1 plus 00:15:30.240 --> 00:15:32.659 i. Let's apply this to a future value example, 00:15:32.820 --> 00:15:34.580 which is typically used for calculating retirement 00:15:34.580 --> 00:15:37.039 savings goals. We are finding the accumulated 00:15:37.039 --> 00:15:39.580 value of a 7 -year annuity due with a nominal 00:15:39.580 --> 00:15:42.500 annual interest rate of 9 % and monthly payments 00:15:42.500 --> 00:15:46.919 of $100. Again, we normalize the inputs. 12 months 00:15:46.919 --> 00:15:50.840 gives us an effective monthly rate of 1 ,00075. 00:15:51.220 --> 00:15:54.080 And the term is 7 years times 12 months, so $1 00:15:54.080 --> 00:15:57.279 .84 for payments. So we're solving for the accumulated 00:15:57.279 --> 00:16:00.019 value of 84 payments of $100 made at the beginning 00:16:00.019 --> 00:16:02.799 of each month compounded at 0 .75 % monthly. 00:16:03.080 --> 00:16:05.399 The source calculation shows that the annuity 00:16:05.399 --> 00:16:07.519 due future value factor for these parameters 00:16:07.519 --> 00:16:12.970 is approximately 117 .3001. If we multiply that 00:16:12.970 --> 00:16:15.450 factor by the $100 payment, the final accumulated 00:16:15.450 --> 00:16:20.230 value is approximately $11 ,730 .01. And that 00:16:20.230 --> 00:16:22.730 total represents the full value of all 84 deposits. 00:16:22.850 --> 00:16:25.970 Plus, all the compounded interest earned measured 00:16:25.970 --> 00:16:28.750 immediately after that 84th and final deposit 00:16:28.750 --> 00:16:31.070 hits the account. We must now address perhaps 00:16:31.070 --> 00:16:33.809 the most abstract yet foundational concept in 00:16:33.809 --> 00:16:36.789 finance theory. The perpetuity. This is literally 00:16:36.789 --> 00:16:39.870 the annuity that never, ever ends. Payments continue 00:16:39.870 --> 00:16:42.149 indefinitely. While they're rare for individual 00:16:42.149 --> 00:16:44.269 consumer products, the mathematical structure 00:16:44.269 --> 00:16:46.909 is vital for valuing long -lived assets like 00:16:46.909 --> 00:16:49.210 certain stocks or government bonds. The fundamental 00:16:49.210 --> 00:16:51.570 question is exactly what you asked earlier. If 00:16:51.570 --> 00:16:54.129 the stream of payments goes on forever, how can 00:16:54.129 --> 00:16:56.590 we possibly assign a finite value to it today? 00:16:56.830 --> 00:16:59.049 And the answer lies in the discount factor. The 00:16:59.049 --> 00:17:01.169 further out a payment is, the more heavily it 00:17:01.169 --> 00:17:04.019 is discounted back to time zero. Eventually, 00:17:04.099 --> 00:17:06.059 the present value of a payment 10 ,000 years 00:17:06.059 --> 00:17:08.599 from now is, well, it's infinitesimally small. 00:17:08.779 --> 00:17:11.380 So we use the limit function. Exactly. The present 00:17:11.380 --> 00:17:13.980 value of a level perpetuity is obtained by taking 00:17:13.980 --> 00:17:16.839 the limit of the annuity immediate PV factor 00:17:16.839 --> 00:17:21.180 as the term $1 tends to infinity. Because the 00:17:21.180 --> 00:17:23.279 interest rate of dollars is positive, that time 00:17:23.279 --> 00:17:25.720 factor we saw in the formula, 1 plus i to the 00:17:25.720 --> 00:17:28.319 power of negative n, it completely disappears 00:17:28.319 --> 00:17:31.380 as n approaches infinity. And the formula collapses 00:17:31.380 --> 00:17:34.740 into something remarkably clean. The closed form 00:17:34.740 --> 00:17:36.700 for the present value of an annuity immediate 00:17:36.700 --> 00:17:40.019 perpetuity with payments of R is simply R divided 00:17:40.019 --> 00:17:43.660 by I. R over I. That simplicity is staggering. 00:17:43.880 --> 00:17:46.539 It is. It allows financial analysts to quickly 00:17:46.539 --> 00:17:48.720 estimate the fundamental value of a permanent 00:17:48.720 --> 00:17:51.859 income source. To ground this historically, look 00:17:51.859 --> 00:17:53.599 at something like the British Consul's perpetual 00:17:53.599 --> 00:17:56.039 bonds issued by the British government, sometimes 00:17:56.039 --> 00:17:58.680 centuries ago. They paid interest forever. Forever, 00:17:58.799 --> 00:18:01.440 with no maturity date. Their market price was 00:18:01.440 --> 00:18:03.519 a reflection of the discounted value of that 00:18:03.519 --> 00:18:06.420 infinite stream of payments. It's R over I in 00:18:06.420 --> 00:18:08.980 action. And what about the annuity due perpetuity 00:18:08.980 --> 00:18:11.059 where the payment is made immediately? Since 00:18:11.059 --> 00:18:13.400 the payment is made at time zero, it earns that 00:18:13.400 --> 00:18:17.400 first period of interest. Its PV factor is 1 00:18:17.400 --> 00:18:19.839 over D, where D is the effective rate of discount. 00:18:20.200 --> 00:18:23.500 But crucially, the PV of the annuity due perpetuity 00:18:23.500 --> 00:18:27.519 is logically 1 plus I times the PV of the annuity 00:18:27.519 --> 00:18:29.900 immediate perpetuity. The relationship holds, 00:18:30.099 --> 00:18:33.119 even with infinity. We've mastered the core math 00:18:33.119 --> 00:18:36.420 of guaranteed cash flows. Now we pivot to the 00:18:36.420 --> 00:18:38.660 two most important applications of these formulas 00:18:38.660 --> 00:18:41.619 in the real world. the architecture of loans, 00:18:41.819 --> 00:18:44.279 and the management of survival risk. Starting 00:18:44.279 --> 00:18:47.220 with amortization calculations. This is a structure 00:18:47.220 --> 00:18:49.380 that defines virtually every mortgage, student 00:18:49.380 --> 00:18:51.920 loan, or auto loan you might encounter. When 00:18:51.920 --> 00:18:53.980 an annuity is used to repay a loan with level 00:18:53.980 --> 00:18:56.400 payments, it is universally treated as an annuity 00:18:56.400 --> 00:18:58.640 immediate. Right, because as we established, 00:18:58.859 --> 00:19:00.960 each payment is made in arrears covering the 00:19:00.960 --> 00:19:03.019 interest accrued since the last payment. So the 00:19:03.019 --> 00:19:04.839 entire mathematical goal is to make sure the 00:19:04.839 --> 00:19:07.319 present value of all those payments exactly equals 00:19:07.319 --> 00:19:10.049 the initial loan principle. Exactly. The genius 00:19:10.049 --> 00:19:12.329 of the amortization formula is that it calculates 00:19:12.329 --> 00:19:15.470 the precise level payment R required to hit that 00:19:15.470 --> 00:19:17.630 zero balance target on the very last payment. 00:19:17.809 --> 00:19:20.750 And that level payment R is calculated by taking 00:19:20.750 --> 00:19:23.910 the principal P and dividing it by the PV factor 00:19:23.910 --> 00:19:26.210 for the loan term. That's right. The formula 00:19:26.210 --> 00:19:28.690 looks a bit intimidating, but it's simply reversing 00:19:28.690 --> 00:19:30.849 the present value calculation we already discussed. 00:19:31.130 --> 00:19:33.490 It dictates your monthly financial commitment 00:19:33.490 --> 00:19:36.829 for decades. Once we know R, the question quickly 00:19:36.829 --> 00:19:40.809 becomes... how much of that debt is left? We 00:19:40.809 --> 00:19:42.930 need to calculate the outstanding loan balance 00:19:42.930 --> 00:19:47.849 after, say, no payments. And banks and borrowers 00:19:47.849 --> 00:19:50.430 use two methods to do this, both of which yield 00:19:50.430 --> 00:19:53.710 the exact same result. The first is the retrospective 00:19:53.710 --> 00:19:55.809 method, which looks backward from the loan's 00:19:55.809 --> 00:19:58.309 starting date. Retrospective. So you start with 00:19:58.309 --> 00:20:00.690 the original principle, you accumulate it forward 00:20:00.690 --> 00:20:03.269 with interest as if you made no payments, and 00:20:03.269 --> 00:20:05.230 then you subtract the accumulated value of the 00:20:05.230 --> 00:20:07.460 payments you did make. It's an accounting method, 00:20:07.640 --> 00:20:10.400 really, used to verify the total interest charged 00:20:10.400 --> 00:20:12.819 and payments received up to that date. But there's 00:20:12.819 --> 00:20:15.319 an alternative. Yes, and often mathematically 00:20:15.319 --> 00:20:17.960 cleaner for planning is the prospective method. 00:20:18.339 --> 00:20:20.220 This one looks forward from the current point 00:20:20.220 --> 00:20:22.500 in time. So you just treat the outstanding balance 00:20:22.500 --> 00:20:24.980 as the present value of the remaining payments. 00:20:25.380 --> 00:20:27.779 You calculate it by taking the level payment 00:20:27.779 --> 00:20:30.880 R and multiplying it by the PV factor for the 00:20:30.880 --> 00:20:32.980 remaining payments. This method is favored for 00:20:32.980 --> 00:20:35.539 financial planning because it's quicker and directly 00:20:35.539 --> 00:20:38.339 tied to the future obligation. If you were a 00:20:38.339 --> 00:20:40.440 lender selling the loan to another institution, 00:20:40.839 --> 00:20:43.059 you'd calculate its value using the prospective 00:20:43.059 --> 00:20:45.539 method. Let's apply the source example to see 00:20:45.539 --> 00:20:47.720 this equivalence in action. We have a loan principal 00:20:47.720 --> 00:20:50.200 P of $1 ,000, a pretty high rate of temporary. 00:20:50.279 --> 00:20:53.160 and a short term of three annual payments. First, 00:20:53.420 --> 00:20:56.200 calculating the level payment R. Using the amortization 00:20:56.200 --> 00:20:58.619 formula, we find the annual payment required 00:20:58.619 --> 00:21:03.019 is approximately $402 .11. That payment ensures 00:21:03.019 --> 00:21:05.359 the debt is precisely paid off in three years. 00:21:05.640 --> 00:21:07.519 Now let's find the outstanding balance after 00:21:07.519 --> 00:21:10.019 one payment, B1, using the simpler perspective 00:21:10.019 --> 00:21:12.240 method. Okay, so we need the present value of 00:21:12.240 --> 00:21:15.099 the remaining two payments at 10%. The PV factor 00:21:15.099 --> 00:21:18.619 for two periods at 10 % is approximately 1 .7355. 00:21:19.500 --> 00:21:21.819 Multiplying the payment of 402 .01 by that factor 00:21:21.819 --> 00:21:23.980 gives us an outstanding balance after one year 00:21:23.980 --> 00:21:28.440 of approximately $697 .89. And if you use the 00:21:28.440 --> 00:21:30.680 retrospective method, accumulating the thousand 00:21:30.680 --> 00:21:33.539 for one year and subtracting the future value 00:21:33.539 --> 00:21:36.200 of that first payment, you would get the identical 00:21:36.200 --> 00:21:38.319 number. The financial mathematics is robust. 00:21:38.619 --> 00:21:41.400 A look backward and a look forward converge on 00:21:41.400 --> 00:21:44.319 the same current debt level. Exactly. And just 00:21:44.319 --> 00:21:47.240 as a quick footnote. Even though most loans are 00:21:47.240 --> 00:21:49.880 annuity immediate, if you encountered an unusual 00:21:49.880 --> 00:21:51.900 loan where the payment was due at the beginning, 00:21:52.099 --> 00:21:55.200 you would simply use the annuity due PV factors. 00:21:55.460 --> 00:21:58.039 The methodology of amortization doesn't change, 00:21:58.200 --> 00:22:01.339 just the specific factor you use. Now we transition 00:22:01.339 --> 00:22:04.660 from pure finance, where N is known, to actuarial 00:22:04.660 --> 00:22:06.799 science, where N is dependent on human existence. 00:22:07.140 --> 00:22:10.019 Life annuities introduce mortality into the equation. 00:22:10.279 --> 00:22:13.000 This fundamentally changes the risk profile and 00:22:13.000 --> 00:22:15.599 the math. This is the crucial intersection of 00:22:15.599 --> 00:22:17.759 insurance and mathematics. So we're dealing with 00:22:17.759 --> 00:22:19.880 payments that are conditional upon the annuitant 00:22:19.880 --> 00:22:22.900 surviving to receive them. Which means we can't 00:22:22.900 --> 00:22:24.740 use the simple annuity certain formulas anymore 00:22:24.740 --> 00:22:26.940 because we just don't know the term. We have 00:22:26.940 --> 00:22:29.240 to use expected values. We have to calculate 00:22:29.240 --> 00:22:32.160 the actuarial present value or APV. Exactly. 00:22:32.670 --> 00:22:35.710 The APV is the expected value of the discounted 00:22:35.710 --> 00:22:37.950 payment stream. It's not the guaranteed value, 00:22:38.130 --> 00:22:40.769 but the statistically probable value derived 00:22:40.769 --> 00:22:43.329 by pooling large amounts of population data. 00:22:43.490 --> 00:22:45.849 So what's the core tool we need from the actuarial 00:22:45.849 --> 00:22:48.609 side? We need the probability of survival, which 00:22:48.609 --> 00:22:51.549 we denote as TPX. This is the probability that 00:22:51.549 --> 00:22:54.410 a person currently aged X will survive for at 00:22:54.410 --> 00:22:57.880 least more periods. Actuaries derive this from 00:22:57.880 --> 00:23:00.259 massive amounts of data compiled into mortality 00:23:00.259 --> 00:23:03.059 tables. Right, which track how large groups of 00:23:03.059 --> 00:23:06.319 people pass away over time. So if I'm a 60 -year 00:23:06.319 --> 00:23:09.750 -old Ajax buying a life annuity. the insurer 00:23:09.750 --> 00:23:11.710 needs to know the probability that I'll be alive 00:23:11.710 --> 00:23:14.670 at age 70. Right, where T equals 10. And for 00:23:14.670 --> 00:23:17.150 every single payment, that probability is factored 00:23:17.150 --> 00:23:20.329 in. The APV calculation is a summation that runs 00:23:20.329 --> 00:23:22.930 from the first possible payment all the way to 00:23:22.930 --> 00:23:25.250 the maximum possible human lifespan. So you're 00:23:25.250 --> 00:23:26.809 summing up the present value of every dollar. 00:23:27.230 --> 00:23:29.009 but weighted by the chance that the person will 00:23:29.009 --> 00:23:31.009 still be around to collect it. And this framework 00:23:31.009 --> 00:23:33.130 is what allows the life insurance company to 00:23:33.130 --> 00:23:35.710 manage longevity risk, the risk that you live 00:23:35.710 --> 00:23:37.410 much longer than average and they have to keep 00:23:37.410 --> 00:23:39.509 paying you. They can't predict your individual 00:23:39.509 --> 00:23:42.789 lifespan. No, but by pooling thousands of lives 00:23:42.789 --> 00:23:45.789 together, they can reliably predict the average 00:23:45.789 --> 00:23:48.869 payout duration for the entire pool. So if we 00:23:48.869 --> 00:23:51.150 look at the standard actuarial notation for a 00:23:51.150 --> 00:23:53.549 whole life annuity immediate of one per year 00:23:53.549 --> 00:23:56.789 on a life -aged X, it's written as a sub -X. 00:23:57.130 --> 00:24:00.630 And that factor is the sum of all future discounted 00:24:00.630 --> 00:24:03.289 one payments, each one weighted by the probability 00:24:03.289 --> 00:24:06.250 of survival. And if it's a whole life annuity 00:24:06.250 --> 00:24:08.890 due, it's a double dot X and the summation just 00:24:08.890 --> 00:24:11.349 starts at time T0 to account for that immediate 00:24:11.349 --> 00:24:13.910 first payment. Right. And this math is the reason 00:24:13.910 --> 00:24:16.470 life annuities offer such unique financial certainty. 00:24:16.990 --> 00:24:19.529 You give up control of your principal, but in 00:24:19.529 --> 00:24:22.549 return, the insurer assumes the longevity risk, 00:24:22.829 --> 00:24:25.190 guaranteeing you an income stream you cannot 00:24:25.190 --> 00:24:28.130 outlive. It's a powerful tradeoff and it's rooted 00:24:28.130 --> 00:24:30.750 entirely in the math of expected values. For 00:24:30.750 --> 00:24:33.569 our final section, we pivot to the crucial inverse. 00:24:33.769 --> 00:24:36.690 calculation. Solving for the payment R. Right. 00:24:36.750 --> 00:24:39.289 In practical financial planning, you rarely start 00:24:39.289 --> 00:24:41.190 knowing the payment amount. You usually start 00:24:41.190 --> 00:24:43.089 with the goal. You know the lump sum you have 00:24:43.089 --> 00:24:45.650 today or the lump sum you need tomorrow. And 00:24:45.650 --> 00:24:47.849 you need to determine the required periodic deposit 00:24:47.849 --> 00:24:50.390 or withdrawal. We'll focus on the annuity due 00:24:50.390 --> 00:24:52.549 scenario here as it's often used for planned 00:24:52.549 --> 00:24:55.049 withdrawals from a retirement fund or consistent 00:24:55.049 --> 00:24:57.569 savings contributions. Let's start with present 00:24:57.569 --> 00:25:00.329 value to payment, annuity due. This is the income 00:25:00.329 --> 00:25:02.910 generation scenario. Yeah. You have a starting 00:25:02.910 --> 00:25:05.650 lump sum, A, and you want to know the maximum 00:25:05.650 --> 00:25:08.670 level payment, R, you can withdraw over a fixed 00:25:08.670 --> 00:25:11.650 term. Since we know that the present value, A, 00:25:11.849 --> 00:25:14.569 equals the payment, R, multiplied by the annuity 00:25:14.569 --> 00:25:16.849 due factor, we just divide the lump sum by that 00:25:16.849 --> 00:25:19.430 factor, R equal A divided by the A double dot 00:25:19.430 --> 00:25:21.250 factor. Okay, let's work through example one. 00:25:21.450 --> 00:25:24.769 You have A. $70 ,000, and you plan to withdraw 00:25:24.769 --> 00:25:27.130 annual payments over three years, assuming an 00:25:27.130 --> 00:25:29.450 effective interest rate of 15%. First, we need 00:25:29.450 --> 00:25:31.369 to calculate the annuity due factor for three 00:25:31.369 --> 00:25:34.009 years at 15%. This factor, a double dot angle 00:25:34.009 --> 00:25:37.880 3 at 15%, is approximately 2 .63. So we divide 00:25:37.880 --> 00:25:41.019 the $70 ,000 principal by the factor 2 .63. Which 00:25:41.019 --> 00:25:43.299 requires a level annual payment R of approximately 00:25:43.299 --> 00:25:48.220 $26 ,659 .47. That is the precise amount you 00:25:48.220 --> 00:25:49.740 can withdraw at the beginning of each of the 00:25:49.740 --> 00:25:51.599 next three years before the principal is exhausted. 00:25:51.980 --> 00:25:54.819 Example 2 gets more complex by introducing quarterly 00:25:54.819 --> 00:25:57.559 compounding. We have a present value A equals 00:25:57.559 --> 00:26:01.039 $250 ,700 with quarterly payments for eight years 00:26:01.039 --> 00:26:04.420 and a nominal annual rate of 5 % compounded quarterly. 00:26:04.619 --> 00:26:06.700 We normalize first. Eight years of quarterly 00:26:06.700 --> 00:26:10.140 payments means N equals 32 total payments. The 00:26:10.140 --> 00:26:13.859 rate is 5 % divided by 4, so I is 1 .25 % per 00:26:13.859 --> 00:26:15.960 quarter. So we need the annuity due factor for 00:26:15.960 --> 00:26:19.779 32 periods at 1 .25%. That factor, a double dot 00:26:19.779 --> 00:26:22.859 angle 32 at 1 .25%, is calculated to be about 00:26:22.859 --> 00:26:25.799 26 .57. Dividing the starting principal, $250 00:26:25.799 --> 00:26:29.859 ,700, by that factor of 26 .57. Yields the required 00:26:29.859 --> 00:26:32.519 quarterly payment. R. That level quarterly payment 00:26:32.519 --> 00:26:36.519 is approximately $9 ,435 .71. This is the kind 00:26:36.519 --> 00:26:38.299 of calculation that lets financial institutions 00:26:38.299 --> 00:26:40.299 instantly determine a sustainable withdrawal 00:26:40.299 --> 00:26:42.799 rate from a trust fund. Now for the final mathematical 00:26:42.799 --> 00:26:45.480 application. Yeah. Accumulated value to payment. 00:26:45.680 --> 00:26:47.940 Annuity due. This is the savings solution. Right. 00:26:47.980 --> 00:26:50.079 You know the future value S you want to reach 00:26:50.079 --> 00:26:51.619 and you need to calculate the minimum payment. 00:26:51.740 --> 00:26:53.960 R required to hit that target. So we use the 00:26:53.960 --> 00:26:56.000 accumulated value factor. Since the accumulated 00:26:56.000 --> 00:26:58.599 value S is R multiplied by the future value factor, 00:26:58.740 --> 00:27:01.549 we divide the target S by that factor. R equals 00:27:01.549 --> 00:27:04.410 S divided by the S double dot factor. Example 00:27:04.410 --> 00:27:07.650 3 involves monthly payments. We want to reach 00:27:07.650 --> 00:27:10.369 an accumulated savings goal of S if it's $55 00:27:10.369 --> 00:27:13.930 ,000 in three years. The nominal rate is 15 % 00:27:13.930 --> 00:27:15.930 compounded monthly. We adjust the parameters 00:27:15.930 --> 00:27:18.750 one last time. 15 % divided by 12 gives us a 00:27:18.750 --> 00:27:22.269 monthly rate I of 1 .25%. The turn is three years 00:27:22.269 --> 00:27:25.970 times 12 months, so N is 36 payments. We need 00:27:25.970 --> 00:27:28.170 the annuity due accumulated value factor for 00:27:28.170 --> 00:27:32.460 36 periods at 1. Plugging in those numbers, the 00:27:32.460 --> 00:27:36.000 factor S double dot angle 36 at 1 .25 % is approximately 00:27:36.000 --> 00:27:39.960 45 .68. And if we take our $55 ,000 target and 00:27:39.960 --> 00:27:42.799 divide it by that accumulation factor of 45 .68, 00:27:43.019 --> 00:27:45.720 we get the required monthly deposit. That precise 00:27:45.720 --> 00:27:47.740 monthly deposit made at the beginning of each 00:27:47.740 --> 00:27:51.019 month for three years is approximately $12 ,204 00:27:51.019 --> 00:27:54.559 .04. This is the hard financial discipline required 00:27:54.559 --> 00:27:57.299 to meet that specific savings objective. It proves 00:27:57.299 --> 00:27:59.480 the feasibility and the cost of the goal instantly. 00:28:00.170 --> 00:28:01.769 So what does this all mean for you, the well 00:28:01.769 --> 00:28:03.529 -informed listener? We started with the definition 00:28:03.529 --> 00:28:05.769 of a regular payment stream and ended with the 00:28:05.769 --> 00:28:07.630 powerful formulas that structure billions of 00:28:07.630 --> 00:28:09.730 dollars in finance daily. The biggest takeaway 00:28:09.730 --> 00:28:11.970 is really twofold, and it's centered on those 00:28:11.970 --> 00:28:15.490 key structural distinctions. First, always remember 00:28:15.490 --> 00:28:17.849 the difference between annuity immediate and 00:28:17.849 --> 00:28:21.130 annuity due. Immediate is for loans. Due is for 00:28:21.130 --> 00:28:24.539 savings and rent. Exactly. Immediate is payments 00:28:24.539 --> 00:28:27.339 in arrears. Due is payments in advance. That 00:28:27.339 --> 00:28:30.460 timing difference introduces that crucial one 00:28:30.460 --> 00:28:33.240 plus I factor into all calculations, which is 00:28:33.240 --> 00:28:36.259 so vital over long compounding periods. And second, 00:28:36.359 --> 00:28:38.599 the distinction between the certain and the uncertain. 00:28:38.839 --> 00:28:41.640 Annuity certain use is known. Fixed terms making 00:28:41.640 --> 00:28:44.539 the calculation straightforward. Life annuities 00:28:44.539 --> 00:28:47.279 introduce actuarial science, replacing the fixed 00:28:47.279 --> 00:28:49.839 term N with the probabilistic factor of survival, 00:28:50.140 --> 00:28:53.279 TPX, allowing insurers to pool and manage the 00:28:53.279 --> 00:28:56.079 risk of longevity. These mathematical tools are 00:28:56.079 --> 00:28:58.440 indispensable because they allow us to rationally 00:28:58.440 --> 00:29:01.180 equate a single lump sum today, whether it's 00:29:01.180 --> 00:29:03.640 a mortgage principal or a retirement nest egg, 00:29:03.740 --> 00:29:06.410 with a long stream of future cash flows. And 00:29:06.410 --> 00:29:08.329 before we wrap up the financial discussion, let's 00:29:08.329 --> 00:29:10.589 briefly touch on the global context. We mentioned 00:29:10.589 --> 00:29:12.589 that while the mathematics is universal, the 00:29:12.589 --> 00:29:14.509 products are governed by national regulations. 00:29:14.529 --> 00:29:17.250 Right. Our source material notes that annuities 00:29:17.250 --> 00:29:19.369 operate under different legal regimes globally, 00:29:19.630 --> 00:29:23.299 American, European and Swiss law. And while PV 00:29:23.299 --> 00:29:27.039 equals R over I holds everywhere, the rules governing 00:29:27.039 --> 00:29:29.500 how these products are issued, what guarantees 00:29:29.500 --> 00:29:32.559 must be offered, the tax implications, the solvency 00:29:32.559 --> 00:29:35.180 requirements. They're all highly specific to 00:29:35.180 --> 00:29:37.559 those jurisdictions. It means an index -linked 00:29:37.559 --> 00:29:39.460 annuity offered in the U .S. will have different 00:29:39.460 --> 00:29:41.460 consumer protections and tax treatment than a 00:29:41.460 --> 00:29:44.099 similar product offered in Switzerland. The financial 00:29:44.099 --> 00:29:47.559 product is truly a blend of math and law. And 00:29:47.559 --> 00:29:49.359 finally, let's return to the idea that allows 00:29:49.359 --> 00:29:52.319 us to connect the abstract mathematics back to 00:29:52.319 --> 00:29:54.720 the economy at large. Yeah. The perpetuity. We 00:29:54.720 --> 00:29:56.680 established that the present value of an infinite 00:29:56.680 --> 00:29:59.700 stream of payments, R, is simply R divided by 00:29:59.700 --> 00:30:02.259 the interest rate, I. And this simple ratio means 00:30:02.259 --> 00:30:04.920 that the higher the interest rate, I, the smaller 00:30:04.920 --> 00:30:07.339 the present value becomes. If you could earn 00:30:07.339 --> 00:30:10.119 1 ,000 % interest, then an infinite stream of 00:30:10.119 --> 00:30:12.579 $100 payments would be worth almost nothing today, 00:30:12.700 --> 00:30:14.799 mathematically speaking. It raises a profound 00:30:14.799 --> 00:30:17.559 question about economic perspective. The valuation 00:30:17.559 --> 00:30:20.339 of a perpetuity demonstrates just how heavily 00:30:20.339 --> 00:30:23.279 we discount future income based on current opportunity 00:30:23.279 --> 00:30:26.039 cost. When money is expensive, when interest 00:30:26.039 --> 00:30:28.779 rates are high, we place a lower present value 00:30:28.779 --> 00:30:31.299 on tomorrow's certainty. Conversely, in a low 00:30:31.299 --> 00:30:33.660 rate environment, the value of that future cash 00:30:33.660 --> 00:30:36.480 flow stream skyrockets. The interest rate is 00:30:36.480 --> 00:30:38.779 not just a number on a spreadsheet. It's a measure 00:30:38.779 --> 00:30:41.769 of how much we currently value the future. It's 00:30:41.769 --> 00:30:43.990 a fascinating thought to mull over the next time 00:30:43.990 --> 00:30:45.990 you hear a central bank discussing rate hikes. 00:30:46.569 --> 00:30:48.990 We hope you enjoyed this deep dive into the mathematics 00:30:48.990 --> 00:30:49.769 of annuities.