When we talk about long-term investing, we often see a compounding curve extending smoothly from the lower left to the upper right.
First look at the simulator below: it automatically draws a capital path when you enter the page, and you can drag the parameters or repeatedly generate new random paths.
Under the same expected return, the greater the volatility, the more easily long-term compounded returns are dragged down. We cannot control future rises and falls, but we can try to choose instruments whose risk characteristics suit us, and through position sizing, diversification and leverage management keep the whole portfolio's volatility within a range we can bear.
Why the smooth compounding curve creates an illusion
Suppose the principal is 1 and the annualized return is 10%; after 20 years it becomes:
$$ e^{0.1 \times 20} \approx 7.39 $$
This calculation isn’t wrong, but it easily creates an illusion: that capital grows steadily along this curve, only slower early and faster later.
Real-world capital curves don’t look like this. Even with an unchanged long-term expected return, every year still experiences rises and falls. The final result depends not only on the return rate but on the order in which volatility appears, and on whether you can stay in the market all along.
How to use and read this simulator
The default parameters are a 12% expected annual return and 12% annualized volatility. You can first keep the expected annual return unchanged and only adjust the annualized volatility:
- Drop volatility to 0% — the capital path coincides with the smooth compounding curve.
- Set volatility back to 12% and click “Generate new path” repeatedly — observe how identical parameters produce different results.
- Raise volatility to 35% — observe how the path, final value and light blue region change.
The chart has three elements:
- Solid blue line: one randomly generated capital path this time.
- Grey dashed line: the smooth compounding expectation when volatility is completely ignored.
- Light blue region: the model’s 10%~90% quantile band at each time point.
The solid blue line is neither a historical backtest nor a prediction of the future. Each click of “Generate new path” is equivalent to letting the same market assumptions go through a different random process once more.
The light blue region, the mathematical model and quantile bands
The simulator assumes capital changes follow a simplified geometric Brownian motion model:
$$ S_t = S_0 \exp\left((\mu - \frac{1}{2}\sigma^2)t + \sigma W_t\right) $$
Where:
- $S_t$ is the capital after $t$ years;
- $\mu$ is the expected return rate;
- $\sigma$ is the annualized volatility;
- $W_t$ represents random disturbance accumulating over time.
The light blue region takes this model’s 10th to 90th percentile at each time point. In other words, if large numbers of results are repeatedly generated under the same parameters, then in a given specified year about 80% of results fall inside the band.
There are two easily misunderstood points here.
First, it is not a guaranteed range. An actual path may well cross the band; it is also not a stop-loss line, maximum-loss boundary or return promise.
Second, “about 80% of results fall inside it at each time point” does not equal “the whole path has an 80% probability of always staying inside”. The former describes the distribution at one time cross-section; the latter describes the entire process — two different questions.
Why the band widens over time
The scale at which random volatility accumulates over time relates to $\sqrt{t}$. The longer the time, the greater the possible path differences.
But the band in the chart appears to expand faster than $\sqrt{t}$, because volatility acts not on a fixed amount but on an ever-changing capital base. For the same 10% rise or fall:
- With principal 1, the amount change is 0.1;
- With principal 5, the amount change is 0.5;
- With principal 10, the amount change is 1.
Therefore the band widening does not mean the market will definitely become more dangerous in future. What it expresses is: time makes uncertainty accumulate continuously, and compounding applies this uncertainty to an ever-larger capital base.
Why equal average returns can still yield different final values
Volatility has another less intuitive effect: identical arithmetic average returns do not guarantee identical compound growth results.
Suppose the first year rises 50% and the second falls 50%. The two-year arithmetic average return is 0%, but the capital goes from 1 to:
$$ 1 \times 1.5 \times 0.5 = 0.75 $$
A final loss of 25%. The reason is simple: after falling 50%, a 100% rise is needed to return to the origin.
In the model the simulator uses, this effect appears in the $-\frac{1}{2}\sigma^2$ term. The higher the volatility, the larger the gap typically between the typical path’s compound growth rate and the smooth expectation. This is often called “volatility drag”.
This doesn’t say all volatility should be eliminated. Bearing risk is often precisely the source of returns. What truly needs understanding: return rates and volatility cannot be viewed separately — discussing only “how much is earned per year on average” is incomplete.
What real-world factors this model doesn't consider
To make the relationship easy to observe, this simulator makes many simplifications:
- Return and volatility rates stay constant over 20 years;
- Random changes in each time period are mutually independent;
- The market has no transaction costs, taxes or liquidity problems;
- No extreme jumps, volatility clustering or long-term regime changes are added;
- Dollar-cost averaging, withdrawals, rebalancing and personal cash flow aren’t considered.
Real markets obviously don’t meet these conditions. Therefore this tool suits explaining mechanisms, not predicting some asset’s specific price 20 years out.
1. The imagined curve
In imagination, capital rises steadily along a smooth compounding curve. Given only the annualized return and time, the endpoint seems already determined — no surprises along the way, no drawdowns to endure.
2. Real markets
Real markets don’t advance in straight lines. Even with the same long-term expected return, capital winds forward through rise after fall; the magnitude and order of volatility make the final result differ greatly.
3. After losing money, the game may be over
Large losses bring more than falling paper numbers. With leverage, losses may trigger margin calls or even liquidation; even without leverage, cash flow or psychological pressure may force selling. Once you leave the market, subsequent rebounds have nothing to do with you.
What You Can Really Control Is Your Own Portfolio
What’s most easily overlooked about the smooth compounding curve isn’t the mathematical error — it’s the human situation.
Even a strategy with positive long-term expectation may go through very deep drawdowns along the way. If position sizing, leverage or cash-flow arrangements make these fluctuations unbearable, then the theoretical long-term returns have nothing to do with you. The capital will be forced out of the market before reaching the curve’s right side.
This is also what I wanted to explain in “The Most Important Thing in Investing Is Not Losing Money”: “don’t lose money” doesn’t require every trade to profit — it means trying to avoid one unrecoverable large loss destroying the principal and depriving the compounding engine of its foundation to keep running.
So long-term investing can’t only ask:
If everything goes smoothly, how much can be earned in 20 years?
It should also ask:
On an unsmooth path, can I keep holding?
We cannot decide what return the market gives next year, nor completely eliminate risk. What can truly be actively managed: what instruments to buy, how much position to allocate to each, whether to use leverage, and what overall volatility forms when different assets are put together.
Controlling volatility isn’t simply pursuing “the lower the volatility the better” — it’s matching risk to your own tolerance. A single instrument looking good doesn’t mean it suits entering the portfolio at a very high position; a high expected return also can’t replace judgment about drawdowns and correlations.
High-Volatility Instruments: Management, Not Speculation
Futures and options are often viewed as high-volatility, high-leverage tools, but you should try to avoid using them to simply bet on price direction. A more suitable positioning: using them to discipline existing holdings — clarifying what risks the portfolio bears, within what range exposure is limited, and how much you’re willing to lose in the worst case. Futures can efficiently hedge index, commodity, interest-rate or exchange-rate risk; buying options can establish bounded or asymmetric protection for the portfolio with one definite premium.
Their value isn’t necessarily shown in how much the trade itself earns, but in how much less the whole portfolio loses when hitting adverse markets. Before using them, first ask: which part of the existing exposure does it offset? Do scale and duration match? If you can’t answer clearly, the trade is very likely not disciplining holdings but adding new speculative risk to the portfolio. Leverage, margin, time value and basis may also amplify risk in reverse.
Gold is a similar example. Viewed alone, gold itself also fluctuates and won’t be negatively correlated with other assets in all market environments; but when its driving factors aren’t fully identical to those of stocks, bonds and other assets, appropriate allocation may lower the portfolio’s concentration, volatility and tail risk. In that case, the standard for evaluating gold shouldn’t only be whether it beat stocks, but whether it improved the whole portfolio’s risk structure.
A hedging position losing on its own doesn’t mean it has no value: if the main assets rise, the insurance cost may inherently be the price the portfolio pays for stability. Conversely, staring only at futures’, options’ or gold’s own P&L easily turns risk-management tools back into directional speculation tools.
Therefore, compared with directly betting on price direction, using these tools to balance the overall position — although not guaranteeing higher absolute returns — may make long-term compound returns and risk-adjusted returns clearly better by reducing large drawdowns and forced selling. Their most valuable “return” is often letting the rest of the portfolio stay in the game more steadily.
Expected returns determine why we set out; portfolio volatility determines whether we can stay on the path. The future cannot be controlled, but positions can be managed.
This article and simulator are only for explaining the relationship between compounding and random volatility, and constitute no investment advice.