Points investment method

If you can understand the first article, then the second one is even simpler, because many of the formulas and such are taught in university courses, even in humanities, it should be mandatory, right? 🥲 So everyone here should be able to understand it.

From the Law of Large Numbers, we can actually derive something else—we can look at investing this way. If the Law of Large Numbers is the static outcome, then calculus is the dynamic process.

• The Law of Large Numbers tells us where we will eventually end up, i.e., the long-term average return.
• Calculus tells us how to get there and what the path will look like, calculated through rates of change and accumulation processes.
• Formulas are the blueprint that combines these two.

For example, the Kelly Criterion, under the framework of the Law of Large Numbers, can calculate how to optimally allocate capital to maximize long-term compound growth.

f^* = \frac{p \cdot b - q}{b}

• f^* = the fraction of the current bankroll to wager
• p = the probability of winning, e.g., 0.55, or 55%.
• q = 1 - p = the probability of losing, e.g., 0.45.
• b = the odds, i.e., how much you win if you win, e.g., bet $1, win $2, net win $1, so b=1.

Its connection to the Law of Large Numbers and calculus:

• The Kelly Criterion's premise is the Law of Large Numbers. It assumes you will repeat this investment game infinitely. Only in this long-term repeated context does discussing the long-term compound growth rate maximization become meaningful. If your investment strategy doesn’t have a positive expected value p \cdot b - q > 0, the Law of Large Numbers will eventually bankrupt you, and the Kelly Criterion can’t save you.
• The derivation of the Kelly Criterion itself uses optimization theory from calculus. It solves for the maximum point of the logarithmic wealth function \mathbb{E}[\log(W)]. By taking the derivative of the expected value function and setting the derivative to zero, you can find this optimal betting fraction f^*.

The Kelly Criterion teaches investors that even with an edge p > 0.5, you shouldn’t go all-in, because f^* = 1. Overbetting, while maximizing single gains, also means a single loss can be devastating damage to your compounding process. It mathematically defines the boundary of overbetting, ensuring your capital can survive the random fluctuations of the market—short-term uncertainty—and ultimately enjoy the inevitable outcome brought by the Law of Large Numbers: long-term stable growth.

Learn to view investing as a continuously changing dynamic process.

Calculus studies change. Investing isn’t a once-a-year numbers game; it’s a process where asset prices and portfolio values change continuously. For example, derivatives measure instantaneous rates of change.

• In physics, the derivative of displacement is velocity; the derivative of velocity is acceleration.
• In investing, the derivative of your net asset curve is your instantaneous return or loss rate.
• The Law of Large Numbers describes the long-term average slope of this net asset curve. The derivative describes the slope of the tangent at every point on this curve—the daily, hourly ups and downs. In the short term, the daily fluctuations of the derivative are volatile, alternating between positive and negative, full of randomness. But in the long run, the average of these wildly fluctuating instantaneous rates of change will converge to a stable value, determined by your portfolio’s intrinsic expected return.

Inheriting a convergent state, another theory also plays a role: integration measures cumulative effects.

• In physics, integrating velocity gives total displacement.
• In investing, integrating your continuous tiny returns gives your total wealth.

This is precisely the power of compounding!

 A = P(1 + r)^t 

It is itself a discrete-time form of integration.

A more precise calculus perspective is:

A(t) = P \cdot e^{rt}

Here, e is the natural constant, and r is the continuous compound return rate. This formula is obtained by integrating the returns over infinitesimal time intervals and accumulating them.

Wealth growth isn’t a step function but a possibly very jagged continuous curve. Our goal isn’t to catch every upward derivative spike—that’s impossible. Our goal is to ensure the long-term integral value of this curve—total wealth—grows exponentially. The only reliable way to achieve this is by relying on the Law of Large Numbers, ensuring your actual annualized return r approaches your strategy’s expected return over the long term through diversification.

I personally believe that by investing in a broadly diversified asset portfolio with a long-term positive expected return p \cdot b - q > 0, such as an index fund, over a sufficiently long time horizon, my average return will inevitably converge to the intrinsic growth rate of that portfolio.

All that’s left is to formulate and execute the strategy. I won’t blindly go all-in just because I believe in the Law of Large Numbers. I’ll use the spirit of the Kelly Criterion to guide myself.

Bet in favorable games but always maintain a safety margin—cash or low-risk assets—to avoid being wiped out by short-term extreme volatility, negative derivative spikes, and thus unable to enjoy the long-term integration effect.

Learn to understand the process dynamics, accept that my portfolio’s value will fluctuate continuously like a rugged mountain path, with derivatives alternating between positive and negative. I won’t panic-sell over short-term drops or become overconfident from short-term surges. I focus on the overall direction and final height of this path—the integral result. My job is to ensure I stay on the road without getting off.

• The Law of Large Numbers is the blade, showing you the long-term direction.
• The Kelly Criterion is the hilt, telling you how fast to move, how much supplies to carry, and how to manage positions most safely and effectively on this long road.
• Calculus is the ring, helping you understand that this road will inevitably have ups and downs, rugged and smooth patches, but as long as you move toward the North Star and follow the guidance, you will surely reach the destination of wealth.

This is Xiao Kele’s Sword of the Son of Heaven.

Understanding the combination of these three, you are no longer a gambler but a true investor.