Why Short-Term Investing is Called Gambling: A Mathematical Explanation Based on Geometric Brownian Motion

Updates

• 2024-11-03: Published a stock price and New NISA asset management simulator built with Next.js!

• 2025-06-15: Updated the app! (The link above has been updated)

https://x.com/gosrum/status/1934082793228935599

This is not designed for mobile screens, so a PC or tablet environment is recommended.

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Introduction

Who is this article for?

• People interested in asset management
• People who want to know why short-term investment is often called gambling
• People interested in stock price fluctuations and New NISA investment simulations that consider volatility (risk)

Content

In this article, we will simulate stock price fluctuations based on finance theory. Specifically, we will conduct analysis and discussion considering the following properties:

• Engaging in probabilistic discussions that consider risk, rather than just deterministic discussions based only on returns.
• Performing theoretical analysis rather than Monte Carlo-style simulations.

By understanding this article, you will be able to mathematically explain the following:

• The reason why short-term investment is said to be gambling
• Why the Sharpe Ratio is important

Prerequisites for this article

Regardless of its validity, this article theoretically analyzes what properties stock prices (market averages) possess assuming they follow the Black-Scholes model. Please note that this analysis itself is not new; I am writing it down as a memorandum to strengthen my own "holding power" (conviction) in stock investing.

My Investment Policy

My Investment Policy

I basically believe in the Geometric Brownian Motion mentioned above and invest in All Country (ACWI) or S&P 500 indices as my core, with satellite investments in NASDAQ100, high-dividend ETFs, bonds, etc.

A good property of Geometric Brownian Motion is that it naturally incorporates the characteristic that stock prices always remain above 0, as they can be expressed exponentially. Furthermore, looking at history, it is a fact that long-term stock price trends have increased exponentially, and I personally recognize it as a simple yet somewhat reliable model.

Of course, it is important to note that the past does not guarantee the future.

As seen from this analysis, by holding stocks for a long period, the probability that stock investing will pay off increases (as long as capitalist society and population growth continue). Therefore, as soon as I have surplus funds, I buy and hold investment trusts or ETFs such as All Country stocks and try not to sell them as much as possible to minimize costs like taxes and fees. If I do sell, I intend to sell only what is necessary when it is necessary, and until then, I plan to keep buying. I hope to introduce exit strategies and such on another occasion.

By the way, my investment policy is heavily influenced by "S&P500 Saikyo Densetsu" (The Legend of S&P500 being the strongest).

Formulation of Financial Model

I will formulate the financial model based on Imoz Financial Theory^[1].

Black–Scholes Model

The Black–Scholes model is a model of a securities market where two types of securities exist: one non-dividend-paying stock and one type of bond. Furthermore, it assumes that continuous trading is possible and the market is a perfect market.

Let $S_t$ be the stock price at time $t$. We assume that the stock price follows the following stochastic differential equation.

$$\begin{align} {\displaystyle dS_{t}=\bar{\mu} S_{t}dt + \sigma S_{t}dB_{t}} \end{align}$$

Here, $B_t$ is a standard Wiener process, $\sigma$ and $\bar{\mu}$ are constants, where $\sigma$ is the volatility (risk) and $\bar{\mu}$ is the drift (arithmetic mean return). Equation (1) implies that stock price movements are represented by Geometric Brownian Motion.

Ito's Lemma

When a stochastic process ${X_t}$ follows the stochastic differential equation

$$dX_t = f(t)dt + g(t)dB_t$$

and assuming $h(t, x)$ is twice continuously differentiable with respect to $t$ and $x$, then

$$\begin{align} dh(t, X_t) = h_t(t,X_t)dt+ h_x(t,X_t) \left[f(t)dt + g(t)dB_t\right] +\frac{1}{2} h_{xx}(t,X_t) g(t)^2 dt \end{align}$$

holds true^[2]. Here, the definitions of $h_t(t,x),h_x(t,x),h_{xx}(t,x)$ are as follows:

$$h_t(t,x) =\frac{\partial h(t,x)}{\partial t},\quad h_x(t,x) =\frac{\partial h(t,x)}{\partial x},\quad h_{xx}(t,x) =\frac{\partial^2 h(t,x)}{\partial x^2}$$

Transformation of Geometric Brownian Motion

We define $X_t, f(t), g(t), h(t, x)$ as follows:

$$X_t=S_t,\quad f(t)=\bar{\mu}S_t,\quad g(t)=\sigma S_t,\quad h(t,x) = \ln x$$

Then,

$$h_t(t,x) = 0,\quad h_x(t,x) = \frac{1}{x}, \quad h_{xx}(t,x) = -\frac{1}{x^2}$$

From this, the stochastic differential equation (1) representing Geometric Brownian Motion can be transformed into the following stochastic differential equation using Ito's Lemma (2):

$$\begin{align} d\ln S_t &= \left(\bar{\mu}-\frac{\sigma^2}{2}\right)dt + \sigma dB_t\nonumber\\ &= \mu dt + \sigma dB_t \end{align}$$

Here, $\displaystyle \mu=\bar{\mu}- \frac{\sigma^2}{2}$ is called the geometric mean return. Also, $dB_t$ represents the increment of Brownian motion (Wiener process).

Difference Between Arithmetic and Geometric Mean Returns

Assume that the actual stock return for the $i$-th year was $\mu_i$. This includes elements of volatility (noise).

• Definition of arithmetic mean return $\bar{\mu}$:

$$\bar{\mu} = \frac{1}{n}\sum_i \mu_i \nonumber$$

• Definition of geometric mean return $\mu$:

$$\begin{align} (1+{\mu})^n&=\prod_i(1+\mu_i)\nonumber\\ \mu&=\left[\prod_i(1+\mu_i)\right]^{\frac{1}{n}}-1\nonumber\\ &\approx \bar{\mu}-\frac{\sigma^2}{2} \nonumber \end{align}$$

When the term "average return" is used over a long period, geometric mean return is generally used more frequently.

Analytical Solution of Geometric Brownian Motion

When the initial value of the stock at the initial time $t_0=0$ is $S_0$, the solution to differential equation (3) is as follows:

$$\begin{align} S_{t}&=S_0\exp\left[\left(\bar{\mu}- \frac{\sigma^2}{2}\right)t+\sigma B_t\right] \end{align}$$

Since the Wiener process follows $B_t\sim\mathcal{N}(0,t)$, Equation (4) can be expressed as follows:

$$\begin{align} \ln \frac{S_{t}}{S_0}&\sim \mathcal{N}\left(\left(\bar{\mu}- \frac{\sigma^2}{2}\right)t,\sigma^2{t} \right)\nonumber\\ &=\mathcal{N}\left(\mu t,\sigma^2{t} \right) \end{align}$$

Equation (5) represents the probability distribution satisfied by a financial instrument following Geometric Brownian Motion.

Mathematical Considerations of the Analytical Solution

From the analytical expression (5), assuming that the stock price (market average) fluctuates according to the Black-Scholes model, the following interpretations can be made:

• On average, the stock price grows exponentially according to the geometric mean return $\mu$.
• However, when the investment period $t$ is short, $\mu t \ll \sigma\sqrt{t}$ holds, making volatility (noise) dominant. This is the reason why short-term investment is often called gambling.
• However, if the investment period $t_{eq}$ satisfies the following, there is approximately an 84%^[3] probability of not falling below the principal.

$$\begin{align} \mu t - \sigma\sqrt{t}&=0 \nonumber\\ \Leftrightarrow t_{eq}(-\sigma) &= \frac{\sigma^2}{\mu^2}=\frac{1}{s_r^2} \end{align}$$

Here, $s_r$ represents the ratio of geometric mean return to volatility (the so-called Sharpe ratio).

• Equation (6) shows that the investment period $t_{eq}(-\sigma)$ until recovering the principal when the stock price swings down by $-\sigma$ is inversely proportional to the square of the Sharpe ratio.
• Furthermore, the investment period $t_{\rm min}(-\sigma)$ when the stock price reaches its minimum when swinging down by $-\sigma$ can be estimated as follows:

$$\begin{align} \frac{d}{dt}\left(\mu t - \sigma\sqrt{t}\right) = 0\nonumber\\ \Leftrightarrow t_{\rm min}(-\sigma) &= \frac{\sigma^2}{4\mu^2}=\frac{1}{4s_r^2} \end{align}$$

Where is the boundary between short-term and long-term investment?

As mentioned in the previous section, as long as the stock price continues its Geometric Brownian Motion, we've seen that short-term investment is close to gambling. So, how long should one hold to consider it long-term investment?

There is no definitive answer, but as a clear indicator, this article considers the timing when the expected value and the standard deviation coincide as the boundary between short-term and long-term investment. The rationale is that after this timing, the expected value becomes dominant against fluctuations within $\pm1\sigma$. And this timing coincides with the value of $t_{eq}(-\sigma)$ derived in the previous section.

In summary, the boundary between short-term and long-term investment can be regarded as the inverse of the square of the Sharpe ratio. From this, it is clear that to invest with reduced elements of luck in the shortest possible time, one only needs to consider maximizing the Sharpe ratio. Of course, since the expected value of investment returns depends only on the geometric mean return, which one to prioritize depends on individual values.

One point of caution: some might think that for bank deposits, since the volatility is $0$, the Sharpe ratio is infinite, but this is a dangerous way of thinking. This is because this analysis does not take the inflation rate into account. If the deposit interest rate does not exceed the inflation rate, one must not forget that bank deposits become an investment where losing is guaranteed, with a Sharpe ratio of negative infinity.

Tips: Median and Expected Value

• Recall that for Geometric Brownian Motion, the asset growth rate follows a normal distribution $\mathcal{N}\left(\mu t, \sigma\sqrt{t}\right)$ when viewed in logarithmic terms.

• In a normal distribution, the expected value, median, and mode all coincide.

  • Expected value (log space): $\mu t$
  • Median (log space): $\mu t$
  • Mode (log space): $\mu t$

• On the other hand, when we transform the above normal distribution back to the original random variable (by taking the exponent), the expected value, median, and mode no longer coincide.

  • Expected value: $\displaystyle\exp\left[\left(\mu + \frac{\sigma^2}{2}\right)t\right]$
  • Median: $\displaystyle\exp\left(\mu t\right)$
  • Mode: $\displaystyle\exp\left[\left(\mu - {\sigma^2}\right)t\right]$

The table summarizing these is as follows:

Item	$\ln S_t/S_0$	$S_t/S_0$
Expected Value	$\mu t$	$\displaystyle\exp\left[\left(\mu + \frac{\sigma^2}{2}\right)t\right]$
Median	$\mu t$	$\displaystyle\exp\left(\mu t\right)$
Mode	$\mu t$	$\displaystyle\exp\left[\left(\mu - {\sigma^2}\right)t\right]$

Simulation

*Note: Here, performance data as of May 5, 2024, is used.

In the Case of S&P 500 (1992-2024)

• Evaluation is conducted using S&P 500 historical performance (1992-2024) in US dollars.
• The results are as follows. For reference, the performance of the ACWI index (1987-2023), a global stock-linked index in US dollars, is also listed.

Parameter	Symbol	S&P 500	ACWI	Conservative Assumption
Geometric Mean Return	$\mu$	10.25%	7.99%	5.0%
Risk	$\sigma$	14.81%	15.22%	20.0%
Sharpe Ratio	$s_r$	0.65	0.51	0.25
Period until principal recovery during $-\sigma$ downturn (Estimate) ★ Retention period that can be considered long-term investment	$t_{eq}(-\sigma)$	2.4 years	3.8 years	16.0 years
Timing of asset bottom during $-\sigma$ downturn (Estimate)	$t_{\rm min}(-\sigma)$	0.6 years	1.0 years	4.0 years

[Update] I created a web app for an asset management simulator

Below, I show graphs of simulation results, but I have also created an app where you can run investment simulations with your preferred parameters! Please give it a try.

By the way, the meaning of confidence interval XX% is as follows:

• The range within which the value will fall with XX% probability for the input parameters (geometric mean return $\mu$, risk $\sigma$), assuming that the stock price follows Geometric Brownian Motion.

This is not designed for mobile screens, so a PC or tablet environment is recommended.

Simulation Results

Stock Price Multiplier

Case	Stock Price
S&P 500 1992-2024	$\mu=10.25$%, $\sigma=14.81$%, $s_r=0.65$
ACWI 1987-2023	$\mu=7.99$%, $\sigma=15.22$%, $s_r=0.51$
Conservative Assumption	$\mu=5.00$%, $\sigma=20.00$%, $s_r=0.25$

Looking at this, you can see how incredible the growth rate of the past S&P 500 index was. Although we are looking at it in US dollars, the Sharpe ratio is also quite high, and the investment efficiency is extremely good.

• "seed = 23" is an example of stock price fluctuation generated using random numbers.

New NISA Asset Amount

Here, we simulate how much assets will increase relative to the monthly investment amount in the New NISA, assuming stock price fluctuations based on the conservative assumptions mentioned above.

While simulations considering only returns (the median in this article: yellow dashed line) are often seen on blogs and YouTube, I rarely see simulation results that also consider volatility, so I believe this graph is quite valuable.

By the way, we assume the investment starts from Year 0 here.

Monthly Investment Amount	New NISA Asset Amount Only $\mu=5.00$%, $\sigma=20.00$%, $s_r=0.25$
300,000 yen
150,000 yen
100,000 yen
50,000 yen

Well, it might be cliché, but what I want to convey here is that if you want to maximize your assets in the long term, you'll want to fill up your New NISA allowance as early as possible.

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Summary

In this article, assuming that the market average of stocks follows the Black–Scholes model, we analytically derived the probability distribution satisfied by the stock price as a random variable using Ito's Lemma.

From the form of the analytical solution, it is evident that the logarithm of the stock price multiplier has the following characteristics:

• Mean: Geometric Mean Return × Elapsed Time ($\mu t$)
• Standard Deviation: Risk (Volatility) × Elapsed Time to the power of 1/2 ($\sigma \sqrt{t}$)

In other words, when the holding period of stocks (market average) is short, fluctuations due to volatility are dominant. This is the reason why short-term investment is often called gambling.

Bank deposits are the most liquid assets, but at the same time, we must not forget that as long as the inflation rate exceeds the deposit interest rate, bank deposits become an investment product where losing is guaranteed. I hope to continue building assets while taking appropriate risks.

Thank you for reading this far. I look forward to seeing you again in the next post.

Reference Articles

• Imoz Financial Theory

脚注

1. Black–Scholes Equation

2. On a Formula Concerning Stochastic Differentials

3. The probability of staying within $1\sigma$ is 68%, but of the remaining 32%, the probability of a downward swing is only 16%.