Volatility Tax Cards

Saturday morning a friend forwarded his mutual fund statement. Two funds, same category, same five-year window, roughly the same average annual return. One had grown Rs 10 lakh to Rs 79 lakh. The other had grown the same Rs 10 lakh to Rs 55 lakh. He was staring at the screen. Same average. Rs 24 lakh apart.

I asked him one question. Which one swung more?

He did not know. He had never looked at volatility. Most people do not. The number on the factsheet is the average return. That number does not compound. The return you actually earn is always lower, and the gap between the two has a name nobody prints on a brochure.

The return you see on the factsheet is the arithmetic average. The return you actually earn is the geometric mean. Volatility eats the difference. Every year. Silently.

The simplest version of the trick

Start with Rs 100. Year one, it goes up 50 percent. Rs 150. Year two, it falls 50 percent. Rs 75.

The arithmetic average of those two years is zero. Up 50, down 50, average zero. Sounds like you should be back where you started.

You are not. You lost Rs 25. A quarter of your money is gone. The average told you nothing happened. The portfolio says otherwise.

This is not a rounding error. It is a structural fact about how multiplication works. A 50 percent fall needs a 100 percent rise to recover, not 50 percent. The path down and the path up are not symmetric. The wider the swing, the larger the gap between the average return and the actual return.

Try it with smaller swings. Up 20 percent then down 20 percent. Average zero. Actual loss: 4 percent. Up 10 then down 10: loss 1 percent. The swings shrink but the gap never disappears. The only swing that leaves zero drain is zero swing.

The formula that nobody puts on the factsheet

There is a clean approximation for this. The geometric return, the one you actually earn, is roughly the arithmetic mean minus half the variance.

Geometric return is approximately equal to the arithmetic return minus sigma squared divided by two.

Sigma is the annual volatility expressed as a decimal. The term sigma squared divided by two is the variance drain. It is the silent fee that volatility charges your portfolio every single year, regardless of direction.

At 10 percent volatility the drain is 0.50 percent per year. At 20 percent it is 2.00 percent. At 30 percent it is 4.50 percent. The drain does not grow in proportion. It grows with the square. Double the volatility and the drain quadruples.

What that looks like in rupees

Take five portfolios. Give every single one of them the exact same 12 percent arithmetic average return. Change nothing except the volatility of the ride.

Same 12 percent printed on the brochure. Five completely different outcomes. The small cap basket and the single stock both "averaged" 12 percent, yet one kept Rs 47 lakh and the other kept Rs 22 lakh. The gap is not in the return. It is in the ride.

That Rs 74.6 lakh the volatile stock lost to variance drain? Nobody collected it. There is no fee line for it. No fund house took it. No STT or brokerage or TER. It simply ceased to exist inside the mathematics of compounding through a bumpy path. It is the most expensive cost in investing and it does not appear on a single statement.

Two identical averages, two very different outcomes

Here is the version that makes it land. Two portfolios. Both earn 12 percent on average. Portfolio A earns a steady 12 percent every year. Portfolio B earns 42 percent in the good years and loses 18 percent in the bad years, alternating. Arithmetic average of both: exactly 12 percent.

Rs 96.5 lakh versus Rs 45.8 lakh. A gap of Rs 50.6 lakh. On the same average return. The steady path compounds at its stated rate. The bumpy path compounds at 7.91 percent, not 12, because the variance drain at 31 percent volatility chews through roughly 4 points of return every single year.

Asked myself the same question my friend was asking. How can two funds with the same average return end up this far apart? The average is not the rate. The rate is lower. And the gap between them belongs to the ride.

Why this matters more than you think

Most factsheets report the arithmetic average. "This fund averaged 15 percent over ten years." That sentence is mathematically correct and practically misleading. The return you actually earned sitting in that fund was lower. How much lower depends entirely on how bumpy the ride was.

This has a direct consequence for portfolio choice. A large cap strategy with 15 percent average return and 15 percent volatility delivers roughly 13.88 percent compounding. A small cap strategy with 18 percent average return and 28 percent volatility delivers roughly 14.08 percent compounding. The small cap brochure looks 3 points better. The actual gap is 0.2 points. Nearly identical, for a dramatically rougher ride.

The return that persuaded you was the brochure number. The return that will pay your bills is the geometric one. They are not the same number and the difference is always negative.

The honest caveat

The variance drain formula is an approximation. It works well for normally distributed returns over moderate ranges. Real returns are not perfectly normal. Fat tails exist. Skew exists. And the formula slightly overestimates the drain at very high volatilities because it is a second-order Taylor expansion, not the full expression.

More importantly, higher volatility assets genuinely do tend to earn higher arithmetic means over the very long run. Equities beat debt not just because of the brochure number but because the arithmetic mean really is higher. The variance drain does not erase the equity premium entirely. It reduces it. A small cap basket that averages 18 percent and loses 3.9 points to variance drain still compounds at 14.1 percent, which beats a bond at 7 percent. The point is not that volatile assets are bad. The point is that the gap between what you read and what you earn is real, it is permanent, and it grows with the square of the volatility.

If you are comparing two options at similar average returns, the calmer one almost certainly compounds to more. If you are accepting a wild ride, know what the toll costs before you get on.

Three plain rules

The risk profile wizard asks you how much volatility you can handle. The variance drain is the financial reason that question matters. It is not only about your sleep. It is about your money.

First. The return on the factsheet is not the return you earn. The factsheet reports the arithmetic average. You earn the geometric mean. The gap is half the variance, every year, automatically.

Second. The drain grows with the square. Double the volatility, quadruple the drain. A move from 15 percent vol to 30 percent vol does not double the cost. It multiplies it by four. From 1.12 percent drain to 4.50 percent drain per year.

Third. Compare after the drain, not before. When two strategies show similar average returns, check the volatility. Subtract sigma squared over two from each. Compare the result. That is the number your money will actually compound at. The calmer path almost always wins.

The brochure prints the ride you are promised. Volatility decides the ride you get. The gap between the two never shows up on a statement but it shows up in the corpus, twenty years later, as the lakh you cannot find.