Real and Complex Numbers - Episode 12 - Maths

I want you to imagine uh just for a moment the architecture of a massive insurance giant, >> right? >> We are talking about like an institution that underwrites the risks of millions of lives, uh properties, entire global supply chains. >> Yeah. The scale is just massive. >> It really is. And the professionals tasked with safeguarding all of this, the actuaries, they operate in a realm that is completely governed by the cold, hard realities of finance. >> Absolutely. I mean, they're tasked with quantifying the likelihood of a catastrophe, right? Or modeling the trajectory of global inflation. They have to ensure that decades from now when a policyholder actually needs their pension or their death benefit, the capital is just, you know, there. >> It has to be there. There's no room for error. >> Exactly. So, this profession deals exclusively in the tangible. We're talking real money, real demographic shifts, real mortality tables, and yet the ultimate mathematical machinery that allows these professionals to aggregate that risk to stabilize our financial future. It relies really heavily on a concept that a 17th century philosopher basically dismissed as completely utterly quote unquote imaginary. It's I mean it highlights this really profound irony in applied mathematics you know because you have this highly regulated intensely pragmatic profession right it's dedicated to grounding our economy in reality predicting the exact capital reserves required to keep civilization ticking >> right keeping the lights on. >> Exactly. But the secret to executing those calculations at scale, it involves mapping real world data into a mathematical dimension that for literally centuries was considered a hallucination. >> A literal hallucination. That's so wild to me. >> Yeah. The tools actuaries use to model reality are like inextricably linked to numbers that just do not physically exist on the standard number line. >> Which brings us to the mission of our deep dive today. And we are tailoring this exploration specifically for you, the listener. >> Yeah. This one is for the math nerds. Oh, 100%. So whether you are an actuarial student currently just buried in preparations for your upper level exams or maybe you're a seasoned credentialed actuary prepping for like a really complex riskmanagement committee >> or honestly even if you're just someone who is insanely curious about the hidden mathematical architecture of reality. >> Yes. Because our goal today is to demystify real and complex numbers. We're going to trace their evolution from uh basically a concept that caused violent onlogical panic in the ancient world, >> which is my favorite part of the history. Honestly, >> it's so dramatic. We'll go through the era where they were considered mental torture by Renaissance algebraists all the way to their modern application as the computational engine behind actuarial science >> because the history of numbers is essentially uh the history of humanity being dragged often kicking and screaming into higher levels of abstraction. Right? And we are going to look at the mechanics of why these abstractions were completely necessary. We won't just be doing a history lesson, right? We're going to break down the actual mechanisms >> like characteristic functions. >> Yeah, exactly. Characteristic functions, the complex roots of auto reggressive models just to understand exactly how stepping out of the real number line solves problems that are otherwise mathematically impossible to solve. >> So let's start that journey by defining the boundaries of what humans historically consider to be real. >> Okay, let's do it. And to do that, we actually have to look back to ancient Greece, specifically to the Pythagoreans because they had constructed this worldview. It was this really intense hybrid of philosophy, religion, and mathematics. And it was incredibly rigid. >> Very rigid. Yeah. The Figurian philosophy was built entirely on this premise of commensurability. >> Commenserability. Okay. Break that down for us. >> So they basically believed that the universe was constructed upon perfect harmony. >> Okay. which meant that any continuous magnitude, so any length, weight or period of time could ultimately be expressed as a ratio of integers >> like a fraction. >> Exactly like a fraction. In their view, if you had two distinct lengths, you could always eventually find some smaller common unit of measure that fit into both lengths a perfect whole number of times. >> Right? Okay. >> And that is the strict definition of a rational number, right? One integer divided by another. I mean, it's a highly intuitive way to look at the world. Honestly, if I have a piece of wood that is roughly a yard long and another that is roughly half a yard, I can just keep breaking down my measuring stick into inches or half inches or 16 of an inch. >> Yeah. You just keep going smaller, >> right? Until I find a discrete unit that measures both pieces of wood. It really implies a universe made of fundamental indivisible building blocks. >> It does. It feels very clean and organized. But the Pythagoreans famously ran into a geometric wall, right? When they analyzed their own most basic shapes. >> Yeah. The crisis occurred when they attempted to find the ratio between the diagonal of a square and its side. >> Okay. So, picture a square, >> right? Picture a square where each side is exactly one unit long. The Pythagorean theorem, which they obviously loved, dictates that the square of the diagonal is equal to 1 2 + 1 2. >> So 1 + 1 is 2. the diagonal squar equals 2. >> Exactly. Therefore, the actual length of the diagonal is the square root of two. >> And the assumption going into this was that the square root of two just had to be a fraction, right? Some whole number divided by some other whole number. >> They completely assumed it had to be. It was foundational to their religion. But when they applied a method of proof by contradiction, which was likely based on the properties of even and odd numbers, they discovered this massive paradox. >> Walk us through that paradox because it's fascinating. Okay, so if you assume the square root of two is a rational fraction in its absolute simplest form, let's just call it a over b. >> Okay, a over b. >> If you do the algebra, you eventually mathematically prove that both a and b must be even numbers. >> Wait, but if both the top and the bottom are even numbers, >> then the fraction wasn't in its simplest form to begin with. >> Oh, because you could just divide them both by two. >> Exactly. You'd have to keep dividing forever. So it's like an infinite loop which means the initial assumption that the square root of two can be expressed as a neat little ratio of integers must be completely false. >> It's totally false. It cannot be written as a fraction if you try to write it out as a decimal. It just goes on infinitely. It's like 1.41421356 and it just keeps going without ever settling into a repeating pattern. >> It's like a glitch in a video game. >> That is a great way to put it. like finding a pixel on your screen that refuses to just be a solid color. You know, it just cycles infinitely through random noise. >> Yeah. And this is the birth of the irrational number. And you have to understand for them it wasn't just a fun mathematical curiosity. It was a localized cosmological crisis >> because it broke their religion >> completely. If the diagonal of a square cannot be measured by the same discrete units as its sides, it means the entire concept of commensurability is just false. It implies that space is not discreet but continuously dense in a way that fractions cannot capture. >> Wow. And the legend of course is that the Pythagoreans were so disturbed by this discovery, this absolute proof that their fundamental operating system of the universe had a fatal flaw that they took the mathematician who discovered it, this guy Hippacis, and threw him overboard at sea to drown. >> Yeah. They supposedly murdered him for math. Now, while the historical veracity of the actual murder is definitely debated by historians, the underlying panic was very real, >> right? Why does infinity or just the inability to neatly package a number into a fraction inherently terrify humans so much that it leads to like legendary murder? >> But I think it's because it fundamentally challenges our common sense, right? We want the universe to be understandable. We want it to be a clean, predictable grid. The discovery of irrational numbers forced mathematicians to eventually expand their concept of reality. >> It forced them to grow up mathematically speaking. >> Yeah. It demanded the formalization of the real number line. The real numbers represent a continuous unbroken continuum. To map physical space properly, you need the integers. Sure, you need the rational fractions to fill in the gaps between the integers, but then you need an uncountably infinite number of irrational numbers just to fill in the gaps between the fractions. >> It's just gaps all the way down. So, the real number line becomes the ultimate measurable tape. It covers every possible measurable magnitude in the physical universe, >> everything. >> Whether you are dealing with a portfolio's total financial exposure, a mortality rate for a certain demographic, or even just the exact temperature of your server room, right? That value lives somewhere on that one-dimensional real number line. >> It has a specific address on that line. >> But the mathematical universe refused to stay confined to a single dimension. >> It sure did. >> And to see how that line was forced to expand into a whole plane, we have to jump forward. We're leaving ancient Greece and going to 16th century Italy. >> Yes, the Renaissance. We move to the era of Geraldo Cardano. >> Cardano. Such a great character. >> He really is. He was a highly influential mathematician, a physician, and a very avid gambler. >> A good combination for a mathematician >> definitely. And unlike the Pythagoreans who were totally focused on geometry and shapes, Cardano was operating in the realm of algebra. He was specifically trying to find general solutions for cubic equations. >> Okay. Cubic equations. Refresh my memory. That's where the highest exponent is a three, right? >> Correct. Like x cub + px plus q= z. >> Got it. And finding the roots of equations is like a fundamental part of mathematical modeling. You are essentially asking at what exact values of x does this complex polomial curve cross the zero line? >> Right? Where does it hit the floor? >> Yeah. >> But Cardano ran into this bizarre algebraic roadblock that was fundamentally different from the Pythagorean problem we just talked about. >> How so? >> Well, the Pythagoreans found a number that was just hard to write down, right? But it clearly existed in reality. You could physically draw the diagonal of a square. You could see it. >> Yeah, it's right there on the page. >> But Cardano was using a newly developed algebraic formula to solve these cubic equations. And the formula occasionally forced him to calculate the square root of a negative number. >> Okay, let's pause and break down why that completely breaks the rules of basic arithmetic as they understood it at the time. >> Sure. >> When you square a real number, you just multiply it by itself. And because of the rules of signs, squaring a positive number gives you a positive result. Like 3 * 3 is nine, right? >> But squaring a negative number also yields a positive result. -3 * -3 is also positive 9. >> Exactly. The physical interpretation of multiplication simply didn't allow for a negative square. If you're looking for say the square root of -15, you are searching for a value that when multiplied by itself produces -15, >> which is impossible, >> completely impossible. No such value exists anywhere on the continuous, infinitely dense real number line we just established. >> It's like following your GPS, right? It's like your GPS telling you to drive north, but then the next step is somehow asking you to drive blue. >> Drive blue. Yeah. >> It's an instruction that the universe just doesn't have a steering wheel for. That's a perfect analogy. Yeah. >> And that is exactly why Cardano referred to these numbers as fictitious. >> Yeah. >> He literally described the process of working with them as mental torture. >> Mental torture. >> Yeah. And later on, Renee Deart, the famous philosopher, he actually coined the term imaginary numbers strictly as a derogatory insult. >> Wait, it was meant as an insult. >> Oh, totally. >> Yeah. He believed that if an algebraic formula spit out a negative square root, it was simply the math's way of telling you that the geometric problem you were trying to solve was stupid and impossible. >> So, if the math demands something completely impossible in reality, why didn't brilliant minds like Cardano just declare the equation broken or just say it's unsolvable and walk away? Why did they let it quote unquote torture them? >> Because they couldn't walk away. >> Yeah. >> This is where the history of algebra takes a really fascinating turn. It's a phenomenon known as caus irreducibilis or the irreducible case. Mathematicians notice something deeply unsettling about these cubic equations. A cubic equation just by its geometric nature can cross the zero line three times. >> Right? It weaves up and down. >> Right? So it can have three perfectly normal real world roots. >> Meaning there are three tangible points on the real number line that solve the equation perfectly. >> Correct? But here is the catch. When mathematicians applied their algebraic formulas to find those three perfectly real roots, the formula would require them to use the square root of a negative number as an intermediate step. >> Wait, really? So, the final answer is a normal number, but the middle of the equation has imaginary numbers. >> Yes, you have to go through the imaginary to get to the real. >> That is wild. The math essentially forces you to walk through a non-existent imaginary hallway just to get to a perfectly real room on the other side. >> Exactly. Like you can see the real roots on a graph, you know they are there, but the algebra refuses to let you calculate them unless you accept the mental torture of negative square roots. >> And this was the pivotal realization. They couldn't just dismiss imaginary numbers as indicators of an impossible problem anymore because the problems were absolutely possible. The roots were real, right? >> The imaginary numbers were an unavoidable bridge. The internal logic of the algea was perfectly sound. It was humanity's interpretation of the number line that was too narrow. >> So, the math was functioning correctly the whole time, but humans were basically trying to force a multi-dimensional concept onto a one-dimensional tape measure. >> That's exactly what was happening. >> So, how did we eventually figure out where these imaginary numbers actually live? If they aren't on the tape measure, where are they? >> Well, the geometric intuition for complex numbers evolved over a pretty long period, but it really culminated with the work of Carl Friedrich Gaus and Jean Robera Gandant. Okay. Gaus, heavy hitter. >> Oh, the heaviest. They provided the visualization that finally gave these abstract algebraic ghosts a thinkable space to inhabit. >> Let's visualize this for the listener because I feel like this is the conceptual leap that completely alters how we model data later on. >> Okay. So, picture the real number line again. It's a horizontal axis, right? Extending infinitely to the left with negative numbers and infinitely to the right with positive numbers. >> Got it? Left and right. zero is sitting right in the middle. >> Now, what does it mean to geometrically multiply a real number by negative 1? >> Um, I mean, you just make it negative, >> right? But geometrically on the line, it means you take a value, say positive five on the right, and you flip it 180° around the zero point. So, it lands on neg five on the left. >> Okay. I see. Multiplication by negative 1 is 180° rotation. >> Exactly. Now mathematicians define the imaginary unit which we denote as I as the square root of1 right >> therefore i * i equals1 >> okay tracking >> so if multiplying by negative 1 is a full 180°ree rotation then multiplying by i twice results in 180° rotation >> oh mathematically implies that multiplying by i just once must be a 90° rotation >> exactly or well to avoid the word exactly that is the precise geometric mechanism If you take the real number one and you multiply it by I, you rotate it 90° off the horizontal axis, you're now pointing straight up. You've literally created a vertical axis. >> So by simply defining the square root of -1 as a 90° rotation, Gaus and his peers transform the one-dimensional number line into a two-dimensional grid. >> Yes, the horizontal axis is the real domain. The vertical axis is the imaginary domain. And together they form the complex plane. So, humanity basically upgraded from a 2D sidescrolling video game where you can only move left or right to an open world map where you can explore entirely new directions. >> Yeah. A complex number therefore is no longer a torturous abstract concept. It is simply a two-dimensional coordinate on this open world map. >> Right? >> It has a real component which tells you how far left or right to go and an imaginary component which tells you how far up or down to go. For example, the complex number 3 + 4 I is just a point sitting three units to the right and four units up. >> It's just a location. It fundamentally changes the geometry of arithmetic. And it's so beautiful how something deemed an insult by Daycart became this explorable geometric space. >> It is beautiful. And furthermore, it allowed mathematicians to utilize polar coordinates. >> What does that mean? So instead of describing a complex number by its horizontal and vertical grid position like a city block, you can describe it by its direct distance from the center which is its magnitude and its angle of rotation from the real axis. >> Okay. Like a radar sweep >> exactly like a radar. And this was formalized beautifully by Oiler's formula Leon Hard Oiler. His formula states that Eis * an angle theta equals the cosine of theta plus I * the s of theta. >> Okay. E to the I theta= cosine theta plus I sin theta. This equation is like famously known for being incredibly elegant in math circles. But for someone applying math in the real world, what is it actually doing? >> It acts as the ultimate bridge between algebra and geometry. It basically proves that exponential growth when you apply it in the imaginary dimension manifests as rotation. >> Exponential growth becomes rotation. >> Yes. When you multiply complex numbers, you are essentially multiplying their magnitudes and adding their angles of rotation. This ability to mathematically encode rotation, frequency, and oscillation into simple exponents is the foundation of modern signal processing, quantum mechanics, and crucially for our audience, advanced probability theory, >> which brings us to the absolute core of our deep dive today. The history is fascinating, but it's really just a prerequisite to understand the tool, right? We have established this two-dimensional complex plane. But why does an actuarial student or, you know, a chief risk officer at a reinsurance firm care about a 90° rotation off the real number line, >> right? Why does it matter for the bottom line? >> Yeah, actuaries deal with hard dollars, mortality tables, property damage. How does an quote unquote imaginary access help price a real world insurance premium? Well, it comes down to a strategy of mathematical translation. Actuaries routinely encounter problems in probability and statistics that are computationally intractable in the real domain. >> Meaning, they're just too hard to solve normally. >> Too hard is an understatement. The standard arithmetic of the real number line becomes so deeply entangled that calculating an exact answer is practically impossible. >> So, actuaries are basically sci-fi dimensional travelers. >> What do you mean? Like when the math gets too heavy in our dimension, we use the complex plane as a sort of secret tunnel. We can't climb the mountain of data. So we phase out of physical reality, walk through the mountain in the complex plane and phase back into the real world on the other side. >> That is honestly that is exactly what they do. Think of it as mapping a problem. You take a highly convoluted realworld probability problem and you use a mathematical transformation to beam it or map it into the complex plane. Okay, >> the rules of algebra are completely different in the complex plane. >> Specifically, operations like convolution, which are a nightmare in the real world, become simple multiplication in the complex world. >> Wow. >> So, you solve the problem easily in that complex space and then you apply an inverse transformation to map the solution back down to the real world. You bypass the impassible mountain by traveling through the complex dimension. >> Okay, let's apply this directly to the actuarial syllabus. The most visceral example of the secret tunnel in action is in the construction of aggregate claims distribution. >> Yes, >> this is a foundational concept in risk management. >> It is the absolute bedrock of property and casualty pricing and capital reserving. >> Mhm. >> The aggregate loss model which is usually denoted as S, represents the total amount of money an insurance company will have to pay out over a given period, say one year. So to model S, the actuary has to account for two entirely separate sources of randomness. Right? >> Correct. >> First you have the frequency out of a portfolio of let's say 100,000 policies. How many claims will actually occur? You don't know. It's a random variable. Let's call it N, >> right? And N is typically modeled using a discrete probability distribution. So something like a pson distribution or a negative binomial distribution depending on the variance of the main frequency you expect, >> right? And then you have the severity. When a claim does occur, how much is it actually going to cost? That is the second random variable. Let's call it X. >> And X is modeled using a continuous probability distribution that is usually heavily skewed to the right >> like a logn normal or a gamma distribution. >> Exactly. Or a Pareto distribution to account for the possibility of rare massive payouts like a total loss on a skyscraper. >> Okay. So the aggregate loss S is the sum of X1 plus X2 plus X3 all the way up to N. You are adding up a random number of random variables. >> A random number of random variables. >> If you want to find the overall probability distribution of S, which you absolutely need to do to figure out your 99th percentile value at risk and set your capital reserves, how do you do it using standard real world math? >> Well, you have to perform a mathematical operation known as convolution. >> And convolution is the nightmare. >> It is the nightmare. Let's simplify and just look at adding two continuous random variables together, X and Y. To find the probability density function of their sum, you cannot just add their individual probabilities together. >> Why not? >> Because there are infinite ways they could add up to your target number. You have to integrate over every possible combination of X and Y that equals your target sum. >> Let's visualize the calculus there. If I want to know the probability that total claims equal exactly $1 million, I have to calculate the probability that claim X was exactly $1 and claim Y was $999,999. Right? >> Then I have to calculate the probability that X was $2 and Y was $999,998. And I have to do that for every single combination. Yes, you must calculate the product of the probability of X taking a specific value lowercase X and the probability of Y taking the value of the target sum minus X and then you integrate that product across the entire domain of possible values. >> The formula for the convolution of two probability density functions f and g is the integral from negative infinity to positive infinity of f of x * g of z minus x with respect to x. >> And that is just for two claims. In reality, n the total number of claims could be 10,000. >> Exactly. Which means you would have to compute a 10,000fold iterated integral. >> A 10,000fold integral. >> Analytically, this is impossible for almost all combinations of frequency and severity distributions used in practice. There is simply no closed form algebraic solution. And historically, before we had massive computational power, even numerical approximations of these integrals were devastatingly slow and incredibly prone to error. So this is the impossible mountain. Sticking to the real number line, convoluting these distributions is mathematically exhausting if not impossible. >> Enter the bypass. The characteristic function. >> The characteristic function. Okay, break this down for us. >> The characteristic function is one of the most powerful tools in an actuary's theoretical arsenal. It is essentially the forier transform of a probability distribution. The definition of a characteristic function of a random variable x which we denote by fi subx of t is the expected value of e raised to the power of i * t * x. >> Okay, let's dissect that formula. E of e to the itx. We have the expected value operator which means we are looking for an average over the whole distribution. We have x which is a random real world claim amount. We have t which is a real number representing frequency and right there in the exponent multiplied by a real world data is i the imaginary unit. >> Exactly. By inserting it into the exponent we are utilizing oilers's formula which we discussed earlier. >> The rotation thing. >> Yes, we are mapping the real world probability distribution onto the complex plane. We are expressing the distribution not as a fixed curve on a flat graph but as a sum of continuously rotating complex vectors. >> Now I know some actuarial students listening might be wondering why use the characteristic function instead of the moment generating function the mgf because the mgf does not use imaginary numbers. The mgf is just the expected value of e to the tx not the ad involved. >> That is a crucial distinction. The moment generating function only works if the integral actually converges >> meaning it doesn't blow up to infinity. >> Right? But for heavy tail distributions like the paro distribution which actuaries frequently use to model liability claims or natural catastrophe claims, the expected value of e to the tx explodes to infinity. The mgf simply does not exist for these heavy tail distributions >> because e to a positive power grows exponentially and a heavy tail distribution doesn't die off fast enough to contain that massive growth. >> Precisely. the integral diverges. But, and this is the magic trick, when you add the imaginary unit i to the exponent to create the characteristic function, Oiler's formula dictates that e to the itx becomes cosine of tx plus i * s of tx. >> Okay. >> Both the s and cosine functions are bounded. They simply oscillate between positive 1 and negative 1. >> Oh, they act as mathematical limiter. >> They do. Yeah. Because the complex exponential is banded by a magnitude of one, the integral for the characteristic function is absolutely convergent for every legitimate probability distribution. >> Every single one. >> Every single one. No matter how wild, infinite or heavy tailed your real world claims data is the characteristic function will always exist safely mapped onto the complex plane. >> So we have successfully beamed our messy heavytailed claims data into the imaginary dimension. We have completely bypassed the divergence issue. >> Yes. >> But how does this actually solve the nightmare of convolution? How does this help us add up 10,000 random claims without doing 10,000 integrals? >> This is where the mathematical translation pays off via the convolution theorem. >> The convolution theorem. >> In the real domain, the probability density function of the sum of independent random variables is the convolution of their individual density functions. As we said, a mathematical nightmare, right? But in the forier domain in the complex plane the characteristic function of the sum of independent random variables is simply the product of their individual characteristic functions. >> Wait, let me make sure that lands for the listener. Convolution, the horrific impassible iterated integral is entirely erased. >> Erased >> in the complex dimension, if you want to combine 10,000 random claims, you just take the characteristic function of a single claim and you multiply it by itself 10,000 times. Yep. You just raise it to the power of n. >> It's like it's like trying to untangle a thousand different knotted headphones in the real world. That's convolution. But instead of doing that, you beam them into the complex dimension where they instantly turn into a neat ball of string and then you just beam it back as perfectly coiled wire. >> That is exactly it. The complex plane translates convolution into pointwise multiplication. You calculate the expected frequency using probability generating functions. You calculate the complex severity using the characteristic function and you just multiply them together. You now have the characteristic function of the total aggregate loss S. >> Okay, so you have the exact answer, >> but it's currently sitting in the complex plane. It's a rotating vector. We need real dollars to put on our balance sheet, >> right? To get the real dollars, you apply the inverse forier transform >> which does what? You integrate the complex aggregate characteristic function multiplied by e to the negative itx over the real line. >> This mathematical process pulls the answer back out of the complex plane and deposited firmly back on the real number line yielding the exact probability density function of your aggregate claims. >> And in modern practice, actuaries use the fast for transform or FFT algorithms to execute this numerically on computers in like milliseconds, right? literally milliseconds. >> So you are essentially using the geometric properties of a dimension that Deart called a hallucination to dictate exactly how many hundreds of millions of dollars an insurer needs to hold in liquid assets to survive a catastrophic hurricane season. >> It really is the ultimate validation of abstract mathematics. >> The imaginary math literally guarantees the real survival of the company. >> It does. And while aggregate claims distribution is a cornerstone of property and casualty work, complex numbers play an equally critical role in the other major pillar of actuarial science, >> which is predicting macroeconomic variables over time. >> Okay, this transitions us from the chaotic daily microevents of individual claims to the macroevents of the global economy. Because actuaries are heavily involved in asset liability management, >> very heavily. >> You cannot appropriately price a whole life insurance policy or determine the funding status of a defined benefit pension plan if you do not have a robust model predicting inflation, interest rates, and equity returns decades into the future. >> Right? You need a long-term view. And to forecast these variables, actuaries and financial engineers utilize time series analysis. >> Time series analysis. A time series is simply a sequence of data points indexed in time order. The goal is to build a mathematical model that explains the underlying structure of the past data so you can simulate its future path. >> And the most common family of models used for this are ARMA models, right? Auto reggressive integrated moving average models. Let's break down just the auto reggressive part, the AR component. >> Sure. An auto reggressive model assumes that the current value of a variable is a linear combination of its own past values plus a random shock. >> Okay. Give me an example. >> For example, an AR1 model AR1 states that inflation today is equal to some coefficient multiplied by inflation last month plus a random white noise error term representing unpredictable current events. >> Okay, so AR1 looks back one month, >> right? And an AR2 model would look at the past two months and so on. >> So it's basically predicting the future by looking in the rearview mirror, assuming the momentum of the past carries forward. >> Exactly. But before an actuary can take one of these models and simulate 30 years of interest rates to stress test a billion dollar portfolio, they have to ensure the model itself is mathematically stable. >> And the technical term for this stability is stationerity, right? >> Yes. For a time series model to be useful for long-term forecasting, it must be weakly stationary. This means that the statistical properties of the process, specifically its mean, its variance, and its autocoarian structure, do not change over time. What happens if the model's not stationary? >> The model becomes divergent. It falls apart. If the variance grows indefinitely over time, the confidence intervals of your predictions just explode, rendering the forecast completely useless. >> Right? Or the model might exhibit explosive behavior where a single random economic shock today doesn't fade away like it should, but instead amplifies, pushing the simulated interest rate to infinity or driving it down to an impossible negative vacuum. >> So relying on a non-stationary model is like setting your autopilot base on a compass that spins increasingly out of control the further you fly. It's a massive systemic risk. >> It's a huge risk. So, how does an actuary analyze a complex equation like a high order AR model with multiple lag coefficients and definitively prove whether it is stationary or explosive? Like, how do they know? >> We return once again to finding the roots of an equation. >> Back to the roots. >> Back to the roots. To analyze the stability of an auto reggressive model, actuaries use something called a lag operator, which is often denoted by the letter B. The lag operator simply shifts the time index back one period. So, B * X subtals X subt. Okay, it steps you back in time. By applying this operator, you can rewrite the entire auto reggressive time series equation as a polomial function of this lag operator B. >> Yes, you extract what is called the characteristic equation of the model. >> Okay. >> For an AR2 model where the current value depends on the past two values, the characteristic equation would be a quadratic polomial like 1 - 5 1 Z - 5 2 Z ^2 equals 0. where the fi terms are the coefficients from your real world data and z represents our variable >> and to check for stability we have to solve for z we have to find the roots of this polomial >> correct and just like cardano trying to solve his cubic equations back in the renaissance when you solve this characteristic equation the roots you find are very frequently complex numbers >> here we are again >> here we are because they're are complex numbers we cannot simply look at whether they are positive or negative to determine stability we must plot them on the complex plane We take the roots of our economic model and we plot them as coordinates using their real and imaginary components. >> This brings us to one of the most critical visual diagnostics in time series analysis. We have our complex plane, our vertical and horizontal axis. And on this map we draw a very specific boundary. >> We draw the unit circle. >> The unit circle. >> This is a circle perfectly centered on the origin zero with a radius of exactly one. So it crosses the real horizontal axis at positive 1 and negative 1 and it crosses the imaginary vertical axis at positive i and negative i. >> Exactly. The fundamental theorem of stationerity for auto reggressive models states that the process is stationary if and only if all the roots of its characteristic equation lie strictly outside the unit circle in the complex plane. >> Wait, let's unpack the mechanics of that. Why does a root sitting outside a circle on an imaginary grid guarantee that inflation won't mathematically explode in our real world simulation? What is the actual mathematical connection there? >> It has to do with the inversion of the polomial in the impulse response function. When you express the current value of the time series in terms of all past random shocks which requires dividing by that characteristic polomial, you have to use a tailaylor series expansion. Okay. So you're expanding the polomial into an infinite series to see how past shocks carry forward into the future. >> Yes. The coefficients of that infinite series represent the impulse response. They tell you how much an economic shock from say 5 years ago is still affecting the economy today. Right. >> For this series to be stable, the effect of past shocks must decay over time. They must eventually fade out to zero. >> The economy has to have a finite memory. It has to forget old shock. >> Exactly. Mathematically for a power series to converge and decay the absolute value the magnitude of the roots of the characteristic polinomial must be greater than one. >> Okay. >> If the magnitude is greater than one, it means the root sits outside the unit circle on the complex plane. When you invert a number whose magnitude is greater than one, you get a fraction. >> Right? Like 1 / two is 1/2. >> Yes. And as you raise that fraction to higher and higher powers as time goes on, it shrinks closer and closer to zero. 1/2 becomes 1/4 becomes 1/8. The shock decays. The model stabilizes. >> Exactly. Conversely, what if a root sits inside the unit circle? Its magnitude is less than one. Say 1/2. When you invert it, you get a number greater than one. You get two. >> Oh, I see. >> As time marches forward, that number raised to a power grows exponentially. 2 4 8 16. An economic shock today amplifies infinitely. The variance explodes. The model is non-stationary. >> Wow. And what if the root sits exactly on the unit circle? Say the root is exactly one. >> That is the famous unit root scenario which you test for using mechanisms like the Dicky Fuller test. >> I've heard of that. >> A process with a unit route is a random walk. >> Yeah. >> The variance of a random walk is equal to time at multiplied by the variance of the shock. Therefore, as time goes to infinity, the variance goes to infinity. It never decays. It is not stationary. So, we are basically looking at a radar screen, the complex plane. And as long as the blips, which are the mathematical roots, stay safely outside the danger zone, the unit circle, our realworld economy model won't blow up. >> That's it. When an actuarial analyst is staring at their statistical software output, plotting the roots of their IRMA model on a graph, they're literally looking at a complex plane. They're looking to see if those mathematical blips are safely outside the radius of the unit circle. And if they are, >> if they are, the actuary can confidently simulate the macroeconomic environment, price the long-term liabilities, and assure the regulators that the systemic risk is contained. >> How reassuring is it for an actuary to literally be able to draw a circle on an imaginary map and know that because of that circle, real people's pensions and investments are safe from collapsing to zero. >> It demonstrates that the architecture of financial stability is inextricably linked to the geometry of the complex plane. Without the ability to map auto reggressive polomials into the complex dimension to verify stationerity, long-term financial modeling would be little more than dangerous guesswork. >> It is truly staggering when you zoom out and look at the trajectory we have covered today. I mean, we started two millennia ago with the Pythagoreans who violently rejected the concept of infinite fractions because they threatened the notion of a perfectly discrete countable universe, >> right? A universe that made simple sense. and we witnessed the expansion of the real number line to include the irrational chaos of continuous space. >> We then explored the algebraic frontier of the Renaissance where mathematicians like Cardano were forced to utilize negative square roots dismissed as fictitious mental torture because the algebra demanded an invisible bridge to find perfectly real answers. And then we saw Gaus and Oiler take those invisible ghosts and give them geographic reality by adding a perpendicular axis creating the two-dimensional complex plane and formalizing the mathematics of rotation via Oilers's formula. >> And ultimately we dissected how that abstract geometry serves as the computational engine for modern actuarial science from utilizing for transforms and characteristic functions to bypass the impossible calculus of convolution for aggregate claims. >> The knotted headphones, >> right? the knotted headphones to plotting the roots of auto reggressive polomials against the unit circle to guarantee macroeconomic stationerity. >> The actuaries listening to this are using a map that Deart called an insult to guarantee that real people's pensions and life savings do not collapse into ruin. The real numbers allow us to measure the reality we can see, but complex numbers give us the analytical leverage to step entirely outside of observable reality, solve the fundamentally unsolvable, and return with the exact parameters needed to protect our future. >> As we conclude, I want to leave you with a thought regarding the nature of paradigm shifts in applied science. We have seen how imaginary numbers, a concept deemed mathematically illegal and philosophically absurd in the 16th century, are now a baseline requirement for managing humanity's most complex financial architectures. >> It makes you wonder what we are currently dismissing. >> Consider the theoretical frameworks being debated today in physics, in quantum computing, in advanced topology. What mathematical concept currently being dismissed by pragmatists as an unphysical glitch or a purely theoretical hallucination will be the absolute foundational requirement for modeling the risks of the 22nd century? >> That is the perfect question to ponder the next time you are plotting a time series or running an FFT on a claims distribution. The glitch isn't always an error, right? Sometimes it is just the shadow of a dimension we haven't fully learned how to map yet. Keep interrogating the formulas in front of you. Understand the mechanisms beneath the models. And until next time, keep diving deep.