The annualized return looks great. How much is actually left?

Start with 100. The first year it rises 50%; the second year it falls 50%. Add the two percentages and divide by two, and you get zero: an arithmetic average of zero.

The money itself did something else.

100 → 150 → 75.

The second year's drop was taken off 150, not off the original 100, so a "50% loss" cost 75. Over the two years you are down 25%.

If you want a single annual number that describes those two years, ask what figure, applied twice in a row, takes 100 to 75. The answer is about −13.40%. That is the compound annualized return: it flattens the whole holding period onto one yearly scale so different periods can be compared. The real path is still 100, up to 150, down to 75.

And if money moves in and out of the account during the period, each chunk has its own clock, in which case the numbers should be measured with the money-weighted method.

A bigger teaching example: 100,000 yuan, growing 6% a year before fees, held for twenty years, every gain reinvested, no tax. Two accounts hold identical assets, both charge their fee at year-end, on the balance after growth. One takes 0.5% a year; the other takes 1.5%.

Year one is the same for both: the balance grows to 106,000. Then the split happens. The cheap account pays out 530 and keeps 105,470. The expensive one pays out 1,590 and keeps 104,410. From then on, each year's growth starts from those different balances, and the gap between them grows too.

Run it forward twenty years:

The two fee structures end up about 53,100 apart. And a fee costs more than the amount taken out that year. It also removes whatever growth that amount would have produced in every remaining year.

When reading a specific fund, find what the fee is charged on, when it is charged, and how, then check whether the advertised returns already sit net of it. Amounts already deducted live inside the NAV performance you see; fees you pay out of your own account are a separate line item with a separate effect.

Once the balance has grown, compare it against prices. Keep the one-year example: 10,000 goes in, grows 6%, a 1% fee comes off at year-end, no tax.

10,000 × 1.06 × 0.99 = 10,494.

That is a gain of 494, or 4.94% on the money. Now suppose the basket of goods and services you actually buy, the one that cost 10,000 at the start, costs 10,300 a year later. Then 10,494 buys only about 1.88% more of the basket than you started with.

Measured in start-of-year prices, that is 10,494 ÷ 1.03 ≈ 10,188.35. Most of the 494 went toward paying higher prices for the same things; the genuine gain in buying power is about 188.35 in start prices.

This comparison only works if the basket stays fixed. If your own spending rose, that might mean you bought more, upgraded, or simply reorganized your life, none of which is inflation. For general price movement, the consumer price index over the same period is the usual reference point; for a personal goal, look at the specific categories you actually spend on.

Now a single holding can be written up as a complete record: what went in, which ups and downs it went through, what fees and taxes came out, what is left, and how much of the spending you had planned it can support.

If the money is earmarked for a date years from now, note when it will actually be needed. If you have to reach for it early, that chunk's holding period ends at the withdrawal, and whatever the book value does afterward belongs to the money still sitting in the investment. Your own timeline of when you need cash should go into the return calculation from the start.

The formulas for compounding, fees, and real returns live in compounding, fees, and real purchasing power. Then fold the results into the three ledgers, income, net wealth, and available cash: money, income, and wealth.