📉 Same Return, Different Wealth

Finance & Data Science
Monte Carlo Simulation · Compounding · Volatility · Python

💡 The Question

Two people invest for 10 years.

Both start with €10,000 and achieve an average annual return of 7%.

Yet one almost doubles the initial capital, while the other doesn’t.

How is that possible?

The answer is volatility.

1. The Experiment

I simulated two investment scenarios:

For Scenario B, I generated 100,000 Monte Carlo simulations using lognormal returns to reproduce the multiplicative nature of compound growth.

2. The Results

At first sight, volatility seems irrelevant: average wealth is almost identical.

But the median tells a different story.

The typical investor in the volatile scenario grows at approximately 5% instead of 7%.

Distribution of final wealth after 10 years with 7% expected annual return.

3. Why?

Investment returns compound multiplicatively.

Consider two years:

Year 1: -50%
Year 2: +100%

The arithmetic average return is:

(-50% + 100%) / 2 = +25%

But actual wealth behaves differently:

€100 × 0.50 × 2.00 = €100

The investor made 0%.

The geometric return correctly captures this:

(0.50 × 2.00)^(1/2) - 1 = 0%

The arithmetic mean describes average returns.
The geometric mean describes the growth of wealth.

4. The Volatility Drag

Under standard assumptions, the geometric growth rate can be approximated by:

Geometric Return ≈ Arithmetic Return - σ²/2

With 20% volatility:

σ²/2 = 0.20² / 2 = 2%

Therefore:

7% - 2% ≈ 5%

This difference is known as volatility drag.

The higher the volatility, the larger its impact on compound growth.

5. Mean vs Median

Why does average final wealth remain close to €19,700?

Because volatile returns generate an asymmetric distribution.

A small number of extremely successful paths pull the average upward, while the median represents the outcome of the typical simulated investor.

That’s why:

Mean   ≈ €19,658
Median ≈ €16,072

Looking only at the average can therefore hide a large difference in typical outcomes.

🎯 Key Takeaways

• The same average return does not imply the same compound growth.
• Volatility reduces the typical geometric growth rate.
• The effect is known as volatility drag.
• Mean and median wealth can diverge significantly in volatile markets.
• In long-term investing, the distribution of outcomes matters as much as the average return.

It is not only how much an investment returns that matters, but also how that return is generated.

🧰 Tools & Methods

Python · NumPy · Monte Carlo Simulation · Statistics · Compounding · Geometric Brownian Motion