Differential Equations - Episode 11 - Maths

You know, when you look at a a really high resolution photograph of a sprinter right in the middle of a race, you get this incredible amount of specific information. >> Oh, yeah. Like the exact muscle tension, >> right? Exactly. You see the tension in their quadriceps. You see the angle of their torso or, you know, exactly where their foot is hovering just like a millimeter above the track. >> Totally frozen. >> Yeah. It's a flawless snapshot of a single frozen moment in time. But the thing is that photograph is essentially lying to you. >> I mean, it implies a stillness that completely contradicts the reality of what's actually happening. >> Exactly. >> Because you looking at a system that is defined entirely by motion, but you're viewing it through a medium that is just well stripped all the momentum away. >> Right. It gives you a really precise location, but it completely robs you of the vector. >> Yeah. You lose the direction and the speed. I mean, from that one static image, you can't actually tell if the sprinter is accelerating toward the finish line or if they've uh just pulled a hamstring and are in the split-second process of collapsing >> or if they're just doing a slow warm-up jog, really, >> right? You have a position, but you are completely blind to what happens in the very next second. You just have no idea what the future holds. >> And mathematically speaking, for a massive portion of human history, we were basically trapped inside that photograph. >> We really were. We were operating in this world of static numbers and isolated snapshots. I mean we could measure where things were but we completely lacked the language to calculate the mechanics of the motion itself. >> So welcome to all of you listening whether you are an actuarial student who is just deep in the trenches of exam prep or uh simply someone who is insanely curious about how the hidden mechanics of reality are calculated. We are taking a massive step forward today. A really big step. >> Yeah. Think about the mathematical road we've traveled together so far in these deep dives. We started at the absolute beginning, right? Building the universe from scratch with sets, >> right? The fundamental atomic structure of logic. >> Exactly. We use sets to establish the basic parameters of existence. And then we powered that universe up. >> We introduced functions. >> Yes. We created these rigid predictable relationships between different elements. And we learned how to measure the sheer instantaneous speed of those relationships using differentiation, >> which is huge, >> right? And then we figured out how to accumulate their total weight and mass over [clears throat] time using integration. We even found the rhythm, like the literal footsteps of the logical universe, using sequences and series. >> And those are all incredibly powerful tools. I mean, you can't do modern math without them. >> Definitely. But everything you just described, differentiation, integration, functions, those are essentially just the components of an engine sitting on a workbench. >> Ah, okay. I like that analogy. >> Yeah, they are the gears, you know, the pistons, the spark plugs. But today, we are putting the engine together. We are turning the key and we're actually driving. >> I love that. Okay, let's unpack this. Today, we are diving into differential equations. And I really want to frame this for you not just as another hurdle on a syllabus or, you know, another chapter in a textbook. >> It's so much more than that. >> It really is. It is the ultimate predictive tool in all of applied mathematics. This isn't just about finding the slope of a curve. >> No, not at all. >> This is literally the written prophecy of change. It is the mathematical framework we use to calculate destiny. >> That's a bold way to put it, but it's completely accurate. And to understand how actuaries use this machinery today, like to project financial risk decades into the unseen future, we have to understand the philosophical shift that birthed it. >> Right. We have to go back. >> Yeah. We have to rewind to the late 1600s and look at Sir Isaac Newton because Newton didn't invent calculus and differential equations just for the sake of like academic publishing. >> He wasn't just trying to get tenure, >> right? He built them because the existing math of his era was completely failing him. It couldn't solve the greatest mystery of his age, >> which was the movement of the heavens. I mean, for most of human history, looking up at the night sky meant looking at this beautiful but kind of terrifying chaos. >> It was just dots of light moving in weird patterns. >> Yeah. But eventually, astronomers like Kepler managed to map out that the planets actually orbited the sun in these perfect predictable ellipses. >> Kepler did incredible work. He had the geometry down perfectly. >> He did. that he had the photograph to use our earlier analogy. >> Exactly. >> Yeah. >> Kepler could tell you what the planets were doing. He could describe the shape of the track the sprinter was running on. But the why was completely missing, >> right? The hidden engine. >> Yeah. Nobody knew the hidden engine that was forcing the planets along those specific elliptical paths. And that's where Newton comes in. He realized something that completely shattered the scientific paradigm of the 17th century. >> What was the main realization? He realized that the entire universe is governed by a continuous unbroken chain of cause and effect. >> Okay, a continuous chain. So he theorized that if you know exactly where a planet is right now at this precise millisecond and you know the mathematical rule for how the sun's gravity is pulling on it, you can calculate exactly where that planet will be 1 millisecond from now. >> Yes, you don't need a telescope to see the future anymore. You just need the mechanics of the change. That is just wild. >> It's a huge conceptual leap. He realized that the rate at which the planet's velocity changes, which is its acceleration, is directly dictated by its current position relative to the sun. >> Okay. Wait, let me make sure I have this sequence right. The position dictates the gravitational pole. >> Right. >> And then the pole dictates the acceleration. >> Exactly. And the acceleration updates the velocity, >> which in turn updates the position. It's a continuous closed feedback loop. That's exactly it. A continuous feedback loop. >> But you know, capturing a continuous loop in a mathematical formula seems like a paradox. >> How so? >> Well, standard algebra demands that an equation balance out to a static value, right? Like x= 5. But if gravity is constantly pulling and the planet is constantly moving in response to that pull, the inputs are changing infinitely fast. >> Did it ever start changing? >> Right? So instead of writing an equation that equals a static number, he wrote one where the rate of change is directly connected to the current position. But how does a mathematical feedback loop actually play out in real time? How do you write an equation that updates itself simultaneously? >> Well, the trick is you have to abandon the search for a static number entirely. >> Okay, so no more x= 5, >> right? You write an equation where the variables represent the rates of change themselves. A differential equation doesn't say position equals x. It says the change in position is equal to this function of the current position. >> Oh, I see. It is a dynamic statement of causality. >> Exactly. The change happens because of the position and then the position changes because of the change. And this is all happening simultaneously across infinitely small slices of time. >> Which brings us to the 1700s and a man who took Newton's foundation and basically built a skyscraper on top of it. >> Ah, Leon Hard Oiler. Yes, Leon Hard Oiler, a Swiss mathematician, incredibly prolific, and arguably one of the greatest analytical minds to ever exist. >> He really was. Oiler is the one who truly mastered the mechanics of these feedback loops. >> He took the theory and made it workable. >> Yeah, he developed brilliant numerical methods for taking these highly abstract differential equations and actually solving them to track real world phenomena. And the detail about Oiler that always just underscores the pure logic of this math for me is that later in his life he went completely blind. >> It is an incredible story. >> He lost his sight entirely. Yet his mathematical output actually increased. >> Which sounds impossible until you understand how he thought because a differential equation is a pure logic system of cause and effect. He didn't need to physically see geometric shapes drawn on parchment anymore. >> He could hold the dynamic feedback loop entirely in his mind. Exactly. He could calculate complex planetary orbits like the Nbody problems involving the gravitational tugof-war between the sun, the earth, and the moon completely in his head. >> Just mentally keeping track of all those moving variables. >> Yeah. He would mentally process the current positions, calculate the simultaneous rates of change, update the positions, and run thousands of mental iterations in the dark. >> That is all inspiring. taking these abstract equations and finding real world applications. He was using differential equations to track comets screaming through the solar system. Right? >> Comets, the bending of light through lenses, even the chaotic fluid dynamics of ocean currents. >> He basically proved that the mathematics of the heavens scaled perfectly down to the physical chaos of the earth. The rules of continuous change are universal. >> They absolutely are. Which brings us directly to your world listening to this today. >> Yes. Tracking a comet is great for an astronomer, but how do we translate this cosmic math into something you can actually use in everyday language? >> Well, to transition this from 17th century astrophysics to modern actuarial science, we have to look really closely at the anatomy of these equations. >> We need to see the gears, >> right? We have to clearly delineate the boundary between the math you learn in high school and the math you use to price a 30-year annuity. >> Okay, let's do it. Think about standard algebra. We need a baseline here. >> Sure. >> Algebra asks a very direct limited question. It basically asks what number when multiplied by two equals 10. >> And the answer is five. >> Right? The answer is just a static value. It doesn't tell you how that five was created or what that five is going to turn into tomorrow. It's just the photograph. >> What's fascinating here is how a differential equation asks a fundamentally different question. It assumes the environment is already in motion. >> The clock is already ticking. Exactly. And it asks what is the hidden rule governing this entire changing system? >> So the easiest way to visualize this rule in action I think is to look at a continuous savings account. This is the classic actuarial starting point. >> It's the perfect analogy, >> right? Because if you want to know how much interest you're going to earn tomorrow, you look at your balance today. Your rate of growth is permanently locked to your current state. >> Compound interest is the quintessential differential equation. The rate of change of your wealth with respect to time, let's just call it DWDT, is equal to the interest rate multiplied by your current wealth, WD. >> So if W goes up, DWTT goes up instantly. >> Yes. Which pushes W up even further, >> which pushes DWT up again. It's that locked feedback loop. >> Exactly. And because the question is fundamentally different from algebra, the solution takes a completely different form. When you solve the algebra problem, your answer is a number. But when you solve the differential equation for that savings account, >> your answer is not a number. Your answer is an entire function. >> Wait, really? The answer itself is an equation. >> Yes, it is a complete mathematical blueprint of the systems behavior. If standard math gives you a photograph, solving a differential equation gives you the entire movie. >> The entire movie. >> Yeah. It gives you a continuous curve from the moment the account was open straight through to the end of time. You can plug in any future date like t equals 10 years or t equals 50 years and the function will output the exact state of the system at that precise moment. >> Okay. So if the answer is an entire movie of the system, does that mean actuaries are essentially the movie directors or are they just the audience trying to guess the ending? >> That's a great question. Actuaries aren't the directors writing the script, >> the universe itself. >> Yeah. you know, human biology, macroeconomic forces, market psychology that writes the rules of the system. >> The universe sets the differential equation. >> Exactly. The actuary is the analyst who decodes it. They read the mathematical script beforehand so they can prepare the financial architecture for the twist ending, >> which perfectly bridges us into how actuaries actually utilize this movie to do their jobs. We can call this the actuar's burden. >> The burden of foresight really. >> Yeah. The core duty of this profession is to project the current state of the world decades into the unseen future. I mean, if you are pricing a long-term care insurance policy or reserving for a pension fund, you absolutely cannot rely on static numbers. >> You'd be bankrupt in a week. >> Exactly. A static photograph of a healthy 30-year-old gives you zero actionable data about the medical reserves you will need to liquidate when that person is 85. >> Because today's reality dictates tomorrow's risk. That is the foundational axiom of the whole field. >> Today's reality dictates tomorrow's risk. That is so crucial because actuaries look at deeply interconnected systems where one single change cascades through time. >> Right? If inflation ticks up by say 50 basis points this afternoon, it's not an isolated event. >> It changes everything. It alters the trajectory of bond yields, which alters the compound growth of retirement portfolios, which reshapes the liability profile of massive corporate pension plans 30 years from now. >> It's the butterfly effect, but with money. And that's why this predictive machinery is the absolute crowning jewel of the actuarial toolkit. >> It's the only mathematical framework rigorous enough to integrate those cascading ripples over continuous time. It allows you to step completely out of the present moment and confidently map out decades of unknown risk. Okay, here's where it gets really interesting. Let's move from the philosophy of risk to the exact specific scenarios where you, the listener, will use these equations in the field. Let's examine the actual mathematical battlegrounds. >> We should start with perhaps the most complex system of all, human mortality and morbidity. Yes, we are talking about survival models and specifically what the actuarial world relies on heavily multiple state Markoff process models. >> Right now in traditional algebra a variable is just a placeholder for an unknown quantity. But in a marov model a state is a literal condition of existence. >> Okay. So paint a picture for us. Imagine you are managing a massive block of long-term care insurance policies. Okay. You have tens of thousands of policyh holders. At any given moment, every single one of those individuals exists in one of a few highly specific state, >> right? They might be completely healthy >> or they might transition into being sick and requiring nursing care. >> They might recover and transition back to healthy. >> Exactly. Or they might transition into the terminal state of having passed away. >> Those are your states and your population is constantly flowing between them. But the analytical key here is that the flow of these populations is not mathematically random. >> Even if an individual's health feels entirely unpredictable to them, >> right? The rate at which people transition from the healthy state to the sick state over the next continuous time interval depends entirely on the volume of people currently sitting in the healthy state. >> It's the continuous savings account dynamic again, just applied to human biology instead of capital. >> The rate of change depends on the current balance. Exactly. So to model this flow across continuous time like to project how 50,000 people will migrate through these states over the next 40 years actuaries rely on a highly specialized set of differential equations. >> The famous Kaggarov forward and backward equations. >> Yes, Kurov. And for a student encountering these equations for the first time, I mean the notation alone can look like a wall of hostile Greek letters. >> Oh, it's intimidating. You've got transition probabilities, transition intensities, matrix multiplication. >> It's a lot. But underneath all that heavy notation, what exactly is the mathematical engine doing? >> Fundamentally, they are tracking the continuous rate of change of probabilities. >> Yeah, >> let's break down the mechanics. >> Let's do it. >> The Cole McGraov forward equation does exactly what it sounds like. It anchors you in the present state. Say you have a healthy 65y old today and it projects forward. >> Okay, so it's moving into the future, >> right? It integrates the transition intensities which are the continuously changing hazard rates of getting sick or dying to calculate the exact probability distribution of that person being in any specific state at time in the future. >> So it moves from a known present to a probabilistic future. >> Exactly. >> Okay. So the forward equation maps out where the population is going to end up. What does the backward equation do? The backward equation is the actuar's ultimate reserving tool. It flips the temporal perspective entirely. >> It goes in reverse. >> Yes. It fixes a specific payout requirement in the future. So let's say there's a $5,000 monthly nursing home benefit that triggers if the policy holder is in the sick state at age 85. >> Okay? So we know the future liability, >> right? The equation then works backward through continuous time applying the transition intensities in reverse to calculate the exact present value of that future liability today. >> Okay. Wait, let me push back on this for a second. >> Sure. >> If Kumagarov's equations are tracking human health, I mean, human health on an individual level is basically biological chaos. >> It definitely is. >> A perfectly healthy person could get into a catastrophic car accident tomorrow or a sick person might receive an experimental treatment and fully recover. So, how does a deterministic differential equation turn that kind of unpredictable biological chaos into a reliable, legally binding 40-year financial projection? >> It seems like it shouldn't work, >> right? Doesn't the rigid math break down the second humans start acting like humans? >> That is the exact conceptual hurdle every single actuarial student has to clear. But the math does not break down because the differential equation inside a Markoff model does not care about the individual. >> Oh, interesting. It is engineered to operate on the aggregate. >> Ah the law of large numbers. >> Yes. But applied continuously. On a micro level, health is stochastic. It is a random walk. But when you aggregate 50,000 policy holders, the variance of that random walk compresses. >> The noise cancels out. >> Exactly. The biological chaos smooths out into highly predictable continuous transition intensities. So by solving Colgrav's backward equation, an insurance company isn't trying to guess the fate of John Smith. They are calculating the mathematical expectation of the entire risk pool. >> So that so what insight here is the power of large numbers in these feedback loops. By solving that equation today, they know down to the dollar how much capital they need to hold in reserve to guarantee those claims in 2055. Yes, they are using the exact same continuous feedback loops that Oiler used to track planetary momentum but applying them to ensure a portfolio remains solvent through decades of human aging. >> That is just brilliant. But human health is just one type of unpredictability and it has a distinct mathematical advantage for an actuary, right? You eventually decays in a predictable direction. Yes, morbidity rates generally increase with age and mortality is well a certainty. The variables have biological limits, >> right? Which brings us to a massive pivot. What happens to our predictive machinery when the variable we are tracking has no biological limits? >> When it can move in any direction at any time. >> Exactly. Driven entirely by human psychology and macroeconomic pressure. We need to move from human biology to financial mathematics. We need to talk about the continuous movement of interest rates. >> This is where things get really dynamic. Yeah, >> because when you first learn about the time value of money, you basically treat the interest rate as a constant, >> right? In Exam FM, early on, a bank offers a 5% yield. You plug it into your discount factor formula and you discount your cash flows. >> It's static. But in the real world of institutional finance, pricing a 30-year bond portfolio based on a static interest rate is just financial suicide. >> Because interest rates do not sit still. They wander. They jitter continuously based on inflation data, central bank announcements, global liquidity, everything. >> They exhibit stochastic behavior. They follow a random walk over time. >> But it's not a pure random walk, is it? >> No, it's not. Because if an interest rate were purely uniformly random, it could mathematically drift up to a million% or plunge to negative 5,000%. And that obviously doesn't happen. The economic environment restricts it. >> Exactly. Interest rates are tethered. They exhibit a property known as mean reversion. >> Meaning they have this deeply ingrained mathematical tendency to pull back toward a long-term historical average. >> Right? And the classic physical analogy here is a heavy rubber band. >> Oh, the rubber band analogy is perfect because the daily volatility of the market is constantly pushing the interest rate around, right? It's stretching the rubber band >> up and down. But if rates drop too low for too long, inflationary pressures build up, capital tightens, and the rubber band violently snaps the rate back up toward the equilibrium. >> And conversely, if rates spike too high, the economy chokes, borrowing stops, the central bank intervenes, and the rubber band pulls the rate back down. >> It is a continuous tugof-war between random volatility and the magnetic pull of the mean. And to capture that exact tugof-war, actuaries use stochastic interest rate models. The foundational examples being the Vasich model and the Coxinger Ross model. Yeah. Widely known as the CIR model. >> Right. And these are built entirely on differential equations. But to capture both the randomness and the rubber band, you can't use standard calculus. >> No. Standard calculus assumes deterministic smoothness. >> You need a formula that calculates the tension of the rubber band at any given millisecond while simultaneously accounting for the random shocks. You need a sootastic differential equation. Let's look at the mechanics of the Vasich model first. Yeah, >> it defines the instantaneous change in the interest rate which we call dright. >> And it splits that change into two distinct components. >> Okay. What's the first component? >> The first component is the drift term. In the formula, it's a time B minus or T. >> That's the rubber band. B is the long-term average rate. R is the current rate. And A is the speed of reversion. Like how thick and tight the rubber band is. >> Exactly. So if the current rate or t is way below the average b that term becomes highly positive forcefully pulling the rate back up. >> Okay. So that is the deterministic part of the differential equation. But then vasiche adds the second component >> the diffusion term >> right sigma time dwt where sigma represents the volatility and dwt that is a wiener process >> also known as brownie in motion. This is the mathematical engine of randomness. It represents the unpredictable continuous wiggling of the market noise. >> Okay. So, if these models describe a random drifting combined with the magnetic pull of the average, which force usually wins out in a 30-year bond horizon? The randomness or the magnet? >> Well, if we connect this to the bigger picture, the stochastic differential equation perfectly balances these two forces over time. >> Oh, so >> over a short window, say a 3-day period, the wiener process, the random noise completely dominates the equation. the rate just jitters unpredictably. >> The magnet isn't strong enough over three days, >> right? But integrate that equation over 30 years and the drift term, the rubber band, really asserts its dominance. The math allows you to price the long-term tension of the market, generating the exact yield curve you need to value future liabilities accurately. >> But the faucet model isn't flawless, is it? >> No, it has a pretty significant quirk. Because the wiener process is normally distributed, the random shocks can theoretically push the interest rate below zero. >> The math allows for negative interest rates, >> which historically speaking was considered a major flaw in the model. >> Which is exactly why the Coxing Ross model, the CIR model was developed as an evolution. It takes the Vasich differential equation and makes one brilliant mechanical tweak. >> It multiplies the volatility term by the square root of the current interest rate. >> Why the square root? >> Think about the behavior. As the interest rate drops closer and closer to zero, the square root of that rate also approaches zero. >> Oh, I see. >> This effectively shrinks the volatility term. The closer the rate gets to zero, the weaker the random shocks become, while the upward pull of the rubber band remains strong. >> So, the differential equation mathematically builds a floor. It prevents the random walk from crossing into negative territory. >> It's an incredible piece of financial engineering. >> It really is. You are tweaking the parameters of a differential equation to alter the fundamental physics of this simulated economy. >> We're setting the rules of the universe. >> Yeah. Okay. So, we've used Marov models to map the biological decay of human populations. We've used Vasich and CIR to tether the random walk of macro interest rates, but we've protected human health in interest rates. There is one more volatile frontier that requires the most famous equation of them all. the most volatile, chaotic and relentlessly measured environment on earth, the equity markets. >> Yes, we are moving into financial economics and the valuation of derivatives, the crown jewel of finance. >> If you want to price a financial option, let's say a European call option, which gives you the right to buy a specific stock at a specific price on a specific date in the future, you are trying to price a moving target based on another moving target, >> which is incredibly hard. You cannot use arithmetic. You have used what is widely considered the most famous equation in modern quantitative finance, the black skulls formula. >> Yes, black skulls. But I want to unmask black skulls here because what many students fail to grasp when they first memorize the formula is its true nature. >> It looks so static on the page, >> right? When you see black schools printed in a study manual, it looks like this massive static algebraic formula involving cumulative normal distribution functions D1 and D2. You plug in the stock price, the strike price, the time to expiration, the risk-free rate, and the volatility, and boom, the formula spits out the fair value of the option. >> But that massive formula is just the closed form solution. >> Exactly. It is just the final photograph of the sprinter crossing the finish line. The algebraic formula is just the aftermath. At its absolute fundamental core, Black's goals is a partial differential equation. >> A partial differential equation. We have leveled up again. Meaning there isn't just one variable changing over time. There are multiple independent variables changing simultaneously. And the equation is rigorously tracking how their respective rates of change interact with each other. In the case of an equity option, its value is constantly shifting, but it is being aggressively pulled by two entirely different forces simultaneously. >> Let's break down the mechanics of those two forces. On one side, you have the passage of time, >> the ticking clock, >> right? Every single second that ticks by, the option gets closer to its expiration date. This creates time decay or what options traders call theta. Time only moves in one direction. It is a relentless, smooth, deterministic decay. The partial derivative with respect to time is easily knowable. >> But on the other side, you have the underlying stock price. >> Yes, the unpredictable volatile wiggling. >> The underlying asset is moving stochastically. It is jumping up on earnings reports, crashing on supply chain rumors, wiggling constantly. >> And this volatility affects the options value traders call the rate of change with respect to the stock price delta. And the rate of change of delta itself, Gamma. >> Exactly. So the partial differential equation of black's goals is trying to reconcile the perfectly smooth deterministic decay of time against the violently unpredictable stoastic wiggling of the stock price. >> Okay, wait. Can a partial differential equation actually tame the volatile wiggling of the stock market? Or does it just give us a very accurate measurement of how dangerous that wiggling is? How do you possibly balance those two things? You balance them by applying Ido's lema which is a cornerstone to stochastic calculus. Yeah. To construct what is known as a riskless portfolio. >> Riskless portfolio. >> Yes. Black Scholes and Merin realized that if you continuously buy and sell the underlying stock in a very specific ratio to the options you hold, a ratio dictated by the differential equation itself, the random wiggling of the stock price perfectly cancels out the random wiggling of the option value. >> Wow. So the DWT, the wiener process representing the random noise is mathematically eliminated from the equation. >> Precisely. You have hedged away the market risk entirely. >> And if the portfolio has zero market risk, economic theory dictates that it must grow at exactly the risk-free interest rate. >> That is the genius of the black skulls partial differential equation. It equates the continuous hedging of the option to the risk-free growth of cash, allowing you to solve for the exact fair value of the derivative. today. >> A great way to visualize this isn't trying to put the beast of the market in a cage because nothing tames the market, right? >> The market is driven by infinite human fear and greed. >> You can't control it. >> Instead, think of the black skulls equation like a massive continuously spinning gyroscope installed deep inside the hole of a cargo ship. >> Oh, I like that. The ocean is the market volatility, >> right? The gyroscope does not calm the ocean. The waves are going to crash. Storms are going to hit. Black swan events are going to generate massive swells. It just accepts the chaos. >> Exactly. But the differential equation inside that gyroscope is continuously calculating the roll of the ship, the angle of the waves and instantaneously adjusting the mechanical stabilizers. It mathematically offsets the chaos of the environment so that the cargo, the portfolio's core value remains perfectly level. >> And the profound nature of discovering that specific partial differential equation cannot be overstated. It didn't just earn a passing grade in a quantitative finance seminar. The conceptual framework won a Nobel Prize in economics. >> It's that huge. >> It transformed the valuation of derivatives from an esoteric guessing game into a rigorous engineering discipline. It literally catalyzed the creation of the modern global derivatives market, a system that manages hundreds of trillions of dollars in notional risk. >> That's staggering. >> The entire global architecture of financial hedging rests securely on the mathematics of continuous change. Which brings us to the ultimate synthesis of our deep dive today. What we have been exploring for the past hour is not just a collection of formulas to be memorized for an actuarial exam. >> No, it's so much bigger. >> We are talking about the mathematics of destiny. Differential equations are the precise tools that allow us to isolate the hidden rules of cause and effect operating in the present moment to financially protect the reality of tomorrow. We observe the system. Whether it is a planet trapped in gravity, a human population aging through a marov chain, a macroeconomic rubber band snapping interest rates back to the mean, or a gyroscope hedging the chaos of equity markets. >> We identify the dynamic feedback loop governing their continuous change. >> And by solving that equation, we map out the architecture of tomorrow. So if you are an actuarial student listening to this, I know it is incredibly easy to get buried in the notation, to stare at a page full of partial derivatives, wiener processes, and transition intensities and completely lose the narrative plot. >> It happens to everyone. But if you connect the grand historical visions of Newton staring at the perfect ellipses of the night sky and oiler calculating planetary momentum in the dark to your daily practical reality, it elevates the entire profession. You aren't just a financial accountant crunching static numbers. >> You are the operators of this predictive machinery. You are the engineers tasked with protecting the future. >> That is exactly right. You utilize these equations to ensure that when the future actually arrives, whether it brings a demographic shift in mortality, a massive inflationary spike in interest rates, or a catastrophic equity market crash, the financial foundations of society do not buckle. >> You use the math of destiny to ensure that the promises society makes to its citizens are actually kept. >> It's a huge responsibility. >> It is. And as we wrap up, I want to leave you with one final sobering, maybe provocative thought about the mechanics of this prophecy. We have spent an hour establishing that a differential equation gives you a perfect mathematical movie of the future driven entirely by the rules of cause and effect applied to the present state of the system. >> Right? The initial conditions, the values you plug in at time zero. >> Exactly. The equations themselves, Kaggarov, Vasich, black skulls are mathematically flawless. But if a differential equation gives you the perfect movie of the future based entirely on the conditions of today, what happens to the actuar's 30-year prophecy? If our measurement of today is off by just a fraction of a percent. >> Ah, that is the danger. The nature of a continuous feedback loop is that errors do not remain static. >> They grow. In a dynamic system, a tiny miscalculation in the initial conditions is fed back into the equation, amplified by the rate of change, and fed back in again. The error compounds exponentially over a continuous time. >> So, if the initial mortality assumption fed into the Marov model is off by just a little bit, or the estimate of current market volatility is slightly skewed, the flawless mathematics of destiny will perfectly calculate the completely wrong future. >> The math is perfect, but the starting line isn't >> right. The math of destiny is flawless, but how perfect is our starting line? So, as you push forward in your studies, as you master the rigorous equations of continuous change, never forget that your ability to project decades into the future is fundamentally constrained by your precision in observing the present moment. >> Your mastery of tomorrow is only as reliable as your understanding of right now. >> Keep pondering that. Keep exploring. Keep questioning the mechanics of the systems around you. And we will see you on the next deep dive.