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Compounding, Fees, and Real Purchasing Power

To see what an account has actually done over the years, you need three answers: how the money grew, what the fees ate, and what the ending balance could buy. Follow the same account across the same period, and the three calculations slot together cleanly.

Period by period, then one average

Compounding occurs when retained returns take part in the next period’s growth. If there are no deposits or withdrawals, the end balance is simply:

End balance = start balance × (1 + first period's return) × (1 + second period's return) × ...

If every year happens to deliver the same return r, the formula collapses to start × (1 + r)^n. Real years don't cooperate that way, so usually you multiply each year's factor separately. When dividends are reinvested, use the total-return figures that include them; when you pocket the dividends, treat that as a withdrawal.

To turn the whole history into a single annual number:

Compound annual return = (end balance ÷ start balance)^(1/n) − 1

Here n is the holding period in years, so eighteen months is 1.5 years. Returns stay in the account, with no external contributions or withdrawals. The compound annual rate is the constant rate that would have produced the same result over the same stretch.

This is why averaging percentages misleads. Grow 100 yuan by 50%, then let it fall 50%, and you land on 75 yuan. The arithmetic average of the two returns is zero, the compound rate is about −13.4%. Both statements are true of different things, but only the second one describes what happened to your money. For accounts with money coming and going, the note fund NAV and personal investment returns covers recording each deposit and withdrawal with its date.

Fees: count them where they actually come out

A fee charged this year shrinks this year's balance, and it also shrinks every future year that compounds off that smaller balance. In the simplified setup for this note, you earn return r first, then at year-end a fee at rate f is taken from the year-end balance, with taxes set to zero. Rates in the formula use decimal form, so a 1% fee means f = 0.01:

One year's growth factor = (1 + r) × (1 − f)

Repeat that for n years with no additions or withdrawals, and the end balance is start × [(1 + r) × (1 − f)]^n. Real products define their own fee bases, frequencies, and amounts, so the formula is a teaching aid rather than a product spec. A full comparison of fee levels over a 20-year horizon sits in what remains after annual returns, fees, and inflation.

One trap worth naming: management and custody fees are typically levied out of the fund's own assets, so they're already baked into the net value you look up. Calculate returns from that post-fee NAV and don't deduct the fee again. Subscription, redemption, and account fees are different; record any charges paid separately in the actual cash flows when they occur. Every fee gets exactly one home.

Convert back to what the money buys

Suppose the after-fee, after-tax nominal return for the year is r, and the basket of goods and services you actually compare against rose by π:

Real return = (1 + r) ÷ (1 + π) − 1

Walk through the arithmetic once. Put in 10,000 yuan. It earns 6%, then loses 1% to fees, no taxes. That's 10,000 × 1.06 = 10,600, then × 0.99 = 10,494. Nominally you're up 4.94%. But the same basket cost 3% more this year, so expressed in last year's purchasing power the balance is 10,494 ÷ 1.03 ≈ 10,188.35 yuan. The real gain is about 1.88%, which measures the increase in this account’s purchasing power against the specified basket.

Over several years, multiply each year's price factor together and divide the end balance by that cumulative price change. If the household changed what it buys, or how well it buys it, don't lump that into inflation; track quantity and quality changes separately from price changes. Picking the right price index for that job is handled in price measures and household purchasing power.

Further reading