The Convexity of Stock Investing.
Why Gains Accelerate and Losses Decelerate Over Time.
Buying a stock looks symmetrical initially. Up 25%, you make 25%. Down 25%, you lose 25%. But hold long enough, and something remarkable happens: the symmetry breaks. Your losses can only compound toward zero. Your gains can compound toward infinity. This is convexity—the math of investing, rigged in your favor.
The Asymmetry of Stock Investing.
We first wrote about the asymmetry of stock investing, where it is like being long optionality, like long a call option. Asymmetry is the holy grail in finance. The initial return profile of buying a stock is symmetrical. If a stock goes up 25%, you make 25%, and vice versa, if a stock goes down 25%, you lose 25%.
Yet over the long term, it is certainly not symmetrical. The risk-reward asymmetry becomes compelling. When buying a stock, the downside is capped at 100%, but the upside is theoretically unlimited, with potential returns of 10x, 20x, or 50x. Each long stock position functions like a long call option, and a portfolio of stocks effectively resembles a giant long call option payoff.
Asymmetry is the holy grail in investing, where over a long time, in heads, you don’t lose a lot, and in tails, you win increasingly more.
Convexity versus Concavity.
I recently came across this quote by Gautam Baid, and I thought the opposites were wonderfully juxtaposed.
“Compounding is convex on the upside and concave on the downside. It means that compounding increases at an increasing rate on the upside and decreases at a decreasing rate on the downside. Positive asymmetry.”
- Gautam Baid on The Talking Billions / Excess Returns Podcast (9 Jan 2026)
But after careful thinking, I realised the first sentence was incorrect and contradictory. Compounding is not convex to the upside and concave to the downside, but is actually convex on both the upside and the downside.
Let me explain further. Let’s start with two basic concepts: convexity vs. concavity.
Convex = f” (x) > 0 = rate of change is increasing.
Concave = f” (x) < 0 = rate of change is decreasing.
On the upside, convexity curves upwards where gains increase at an increasing rate, and concavity curves downwards where gains increase at a decreasing rate.
On the downside, convex losses decrease at a decreasing rate, and concave losses decrease at an increasing rate.
Convexity has positive asymmetry: limited downside and unlimited upside, which is highly desired. Conversely, concavity has negative asymmetry, unlimited downside, and limited upside, which is not desired. We always prefer to play and structure our games to be as positively asymmetric over time, and eliminate unlimited downside.
Convexity is like a snowball rolling downhill. It starts small. But as it rolls, it picks up more snow. The bigger it gets, the more snow it picks up with each roll. Growth builds on growth.
Concavity is like inflating a balloon. The first few breaths make a big difference. But the fuller it gets, the harder you have to blow for the balloon to expand. Eventually, you’re red in the face, and it barely budges.
Why is there Convexity in Long-Term Stock Investing?
Let’s start with $100 equally invested in Stock A and B. The price of Stock A keeps rising 25% every year, and Stock B keeps declining 25% every year.
In the first year, the value of Stock A rises 25% to $125, and Stock B declines 25% to $75. The portfolio value remains unchanged at $200 with 0% returns (not good).
At the end of year 5, Stock A rose ~3X to $305, and Stock B has declined by 76% to $24. The portfolio value is $329, with a cumulative return of 64% and an annualised return of 10.5% p.a. (good).
At the end of year 10, Stock A rose ~9.3X to $931, and Stock B has declined 94% to $6. The portfolio value is $937, with a cumulative return of 368% and an annualised return of 16.7% p.a. (strong).
At the end of year 20, Stock A is an 86-bagger to $8,674, and Stock B has declined 99.7% to $0.30. The portfolio value is $8,674, with a cumulative return of 4237% and an annualised return of 20.7% p.a. (very strong).
Lessons for Buy-and-Hold Long-Term Investing?
While we acknowledge that the assumptions are highly simplistic: (1) a constant 50% batting average, (2) an equal-weighted two-stock portfolio, and (3) both stocks rising and declining at the same rate, the following lessons are instructive.
1 | Winners increasingly matter, losers increasingly don’t. If you have a winner, a little is all you need, and even if you do have a loser, it doesn’t matter. Asymmetry does not exist in discrete time periods, but shows up over time.
2 | Compounding starts slow but accelerates over time. Returns take time to show up, only if the winners are held. Being able to hold on to winners and not trim them allows one to keep growing cumulative returns exponentially and annualised returns towards steady-state rates of 20%-22%+ p.a. after 10-20 years.
3 | Winners become increasingly significant as you hit it out of the park, as portfolio concentration grows over time. The gains from the multibagger winners will keep growing, increasingly offsetting the losers’ combined losses many times over. The losers don’t matter, the winners do.
4 | Sufficient diversification is necessary to allow for this to be achieved, not being overly concentrated (<10 stocks) or being overly diversified (>100 stocks). If one is overly concentrated, and if a winner becomes overly large or runs into single-position limits, one eventually has to trim the winner. Constantly trimming your flowers, not allowing your winners to run, and watering your weeds are among the worst things one can do. Sufficient diversification by owning more stocks allows one to have a better risk appetite and more patience to hold on to winners, especially when they experience inevitable large price declines along the way.
The Convexity of Long-Term Stock Investing.
As we outlined at the beginning, while the payoff of a long-only stock is highly positively asymmetric, similar to that of a long call option, the payoff is actually convex, both to the upside and the downside, as the stock price grows over time.
Compounding is powerfully convex to the upside, and protectively convex to the downside. Compounding increases at an increasing rate on the upside and decreases at a decreasing rate on the downside.
Compound interest is the eighth wonder of the world because it is convex. Gains grow faster, and losses shrink slower. Those who understand this asymmetry earn it. Those who don’t, pay it. Find winners, hold them, and let time bend the curve in your favour.
The information contained in this article is for general informational purposes only. While every effort has been made to ensure the accuracy and reliability of the content, errors or omissions may occur. The author and publisher do not assume any responsibility or liability for any errors, inaccuracies, or omissions in the content, nor for any actions taken based on the information provided. Readers are encouraged to verify any information before relying on it and to seek professional advice as needed.
11 Jan 2026 | Eugene Ng | Vision Capital Fund | eugene.ng@visioncapitalfund.co
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Thank you for putting this together. Very insightful. I never thought about it from this angle before. Cheers.