In this lesson · 7 sections
Expected return is a statement about one period. Terminal wealth is a statement about a path. The two are connected by multiplication, and multiplication does something to a distribution that addition does not: it pulls the mean away from the middle and leaves most of the outcomes behind on the low side. A portfolio can have a perfectly respectable expected return and a typical outcome that disappoints, and nothing has gone wrong other than the reader confusing an average over possible worlds with an average over time in one world.
The previous lesson established that the geometric mean, not the arithmetic mean, is the rate that reproduces an ending balance. This one takes that from a fact about a realised series to a fact about the distribution of outcomes, and then looks at the three things that quietly reshape it: cash flows, fees and taxes on the stock of wealth rather than the flow of income.
The mean path and the typical path are different objects
Model wealth multiplicatively. Let the log return in each year be independent and normally distributed with mean m and standard deviation s. After T years the cumulative log return is normal with mean mT and variance s²T, so wealth itself is lognormal.
For a lognormal variable the median and the mean are not the same. The median wealth multiple is e^(mT). The mean wealth multiple is e^(mT + s²T/2). The ratio between them is e^(s²T/2), and it grows with both volatility and horizon.
Take m = 0.05, s = 0.18 and T = 20 years. The cumulative log return is normal with mean 0.05 × 20 = 1.00 and variance 0.18² × 20 = 0.0324 × 20 = 0.648, so its standard deviation is √0.648 = 0.8050.
Median wealth multiple: e^1.00 = 2.718. Mean wealth multiple: e^(1.00 + 0.324) = e^1.324 = 3.759. The mean is 38.3% above the median.
More useful than the ratio is the share of paths that fall short of the mean. A path beats the mean only if its cumulative log return exceeds ln(3.759) = 1.324, which is 0.324 ÷ 0.8050 = 0.402 standard deviations above its own mean. Roughly 66% of paths finish below the average outcome. That is not a defect of the market. It is what happens when you take the expectation of a convex function of a symmetric variable, and it is why the mean of a wealth distribution is a poor summary of what to expect.
Ensemble average, time average, and a bet that looks good and is not
The sharpest version of the point uses a gamble rather than a market. Each period, your entire capital is multiplied either by 1.5 or by 0.6, with equal probability.
The expected multiple per period is 0.5(1.5) + 0.5(0.6) = 1.05. Averaged across an ensemble of many simultaneous players, capital grows at 5% per period. Over 100 periods the expected wealth multiple is 1.05^100 = 131.5.
Now follow one player through time. Over 100 periods the most likely count is 50 ups and 50 downs, and the resulting multiple is (1.5 × 0.6)^50 = 0.9^50 = 0.00515. A starting stake of CHF 100 becomes about CHF 0.52. The per-period growth rate along a typical path is √0.9 − 1 = 0.94868 − 1 = −5.13%.
Both numbers are correct. The expectation is dominated by a vanishing fraction of paths that win almost every round and carry astronomical wealth; the typical path decays. When you cannot diversify across parallel copies of yourself and must instead live through the sequence, the quantity that governs your fate is the growth rate of the median, which is the geometric mean, not the arithmetic one. This is the practical content of the ergodicity distinction that Ole Peters has pressed in economics, and it is the reason position sizing exists as a discipline separate from forecasting.
Cash flows make the order of returns matter
Without deposits or withdrawals, the order of returns is irrelevant: multiplication commutes. Add a cash flow and it stops being irrelevant, because the flow changes the base that later returns are applied to.
Start with CHF 100. Two years, one return of +30% and one of −30%, and a withdrawal of CHF 20 at the end of year one.
Good year first: 100 × 1.30 = 130, withdraw 20 leaves 110, then 110 × 0.70 = 77.0.
Bad year first: 100 × 0.70 = 70, withdraw 20 leaves 50, then 50 × 1.30 = 65.0.
Same two returns, same withdrawal, CHF 12 apart. With no withdrawal both orders give 100 × 1.30 × 0.70 = 91.0.
This is sequence-of-returns risk, and its sign depends on the direction of the flow. A saver contributing steadily benefits from weak early returns, because the contributions buy more units before the recovery. A retiree drawing down is hurt by them, because units sold cheaply never participate in the rebound. The same portfolio, the same average return, opposite conclusions, determined entirely by whether money is going in or coming out.
Fees compound at exactly the same rate as returns
An annual charge levied as a percentage of assets is multiplicative, which means its effect on terminal wealth is independent of the gross return. If the fee rate is f, terminal wealth is scaled by (1 − f)^T regardless of what the market did.
With f = 1% and T = 30 years: 0.99^30 = 0.7397. The fee takes 26.0% of terminal wealth. With f = 0.20%: 0.998^30 = 0.9417, a loss of 5.8%.
One arithmetic detail worth pinning down, because it is routinely fudged. A 1% fee on a 6% gross return does not leave 5%. It leaves 1.06 × 0.99 − 1 = 4.94%. Over 30 years the gross multiple is 1.06^30 = 5.7435 and the net multiple is 1.0494^30 = 4.2483. The ratio 4.2483 ÷ 5.7435 = 0.7397 reproduces the figure above exactly, as it must. The two calculations differ only in whether you subtract the fee from the rate or apply it as a factor; only the second is correct, and the first understates the damage slightly.
Note also what the fee is charged on. It is charged on assets, not on gains. In a flat year the gross return is zero and the fee is a straight reduction of capital. Sharpe's arithmetic of active management makes the aggregate version of the point: before costs, the average actively managed franc must earn the market return, because active positions net to the market; after costs it must earn less.
Wealth taxes behave like fees, and location changes the base
Switzerland is unusual among developed economies in levying an annual tax on net wealth at cantonal and communal level, alongside income tax. The base is worldwide net assets: securities, bank balances, the tax value of property, less debts. Rates are set by canton and commune, are typically progressive, and vary by an order of magnitude across the country, which is why domicile is a live variable in Swiss financial planning in a way it is not in most jurisdictions.
Mechanically, a wealth tax is a fee with a different payee. Suppose an illustrative all-in wealth tax of 0.4% of net assets per year. Over 30 years the terminal wealth is scaled by 0.996^30 = 0.8867, a reduction of 11.3%, on top of income tax on dividends and interest. As with a fee, it is levied whether or not the portfolio made money, so in a flat or negative year it consumes principal.
Two structural features follow, and they are about where assets sit rather than what they return.
First, assets held inside the occupational pension (pillar 2) and the restricted pillar 3a are not part of the wealth tax base while they remain in the system, and the income they generate is not part of taxable income. They are taxed on withdrawal, separately from other income and at a reduced rate. Compounding inside those wrappers therefore proceeds on a gross basis in a way it does not in a taxable custody account.
Second, voluntary buy-ins into pillar 2 to fill gaps in contribution years are deductible from taxable income in the year of payment, subject to the fund's calculated shortfall, and article 79b BVG blocks a lump-sum withdrawal of benefits for three years after such a buy-in. The point here is not that a buy-in is a good idea, which depends entirely on circumstances this lesson knows nothing about, but that the compounding calculation for a Swiss resident cannot be done on gross returns alone. The wrapper changes the base, the deduction changes the effective contribution, and the lock-up changes the horizon.
Debt deducts from the wealth base, which is one reason Swiss mortgage balances behave differently from those in countries without a wealth tax. That interaction sits with the property lessons, not here.
What to take away
- Wealth is a product of returns, so its distribution is right-skewed; the mean outcome sits well above the median, and with a 5% mean log return and 18% annual log volatility over 20 years roughly two-thirds of paths finish below the average.
- The expected return governs an average across parallel worlds; the geometric mean governs the single path you actually live through, and a bet with a positive expected return can have a strongly negative growth rate.
- Order of returns is irrelevant without cash flows and decisive with them: the same +30%/−30% pair with one CHF 20 withdrawal produces CHF 77 or CHF 65 depending only on sequence.
- A percentage-of-assets fee scales terminal wealth by (1 − f)^T whatever the market does; 1% a year for 30 years removes 26.0% of the final balance.
- In Switzerland an annual wealth tax acts like an additional fee on the whole balance sheet, and pillar 2 and pillar 3a assets sit outside that base, so the location of an asset changes its compounding arithmetic, not just its tax bill.
Sources
- Ole Peters, "The ergodicity problem in economics", Nature Physics 15 (2019), on the distinction between ensemble and time averages in multiplicative processes.
- J. L. Kelly Jr., "A New Interpretation of Information Rate", Bell System Technical Journal (1956), the original treatment of growth-optimal betting.
- William F. Sharpe, "The Arithmetic of Active Management", Financial Analysts Journal (1991).
- Elroy Dimson, Paul Marsh and Mike Staunton, Triumph of the Optimists: 101 Years of Global Investment Returns (Princeton University Press, 2002), on long-horizon compounded outcomes across markets.
- Swiss Federal Tax Administration (ESTV), Steuerinformationen dossiers on the cantonal wealth tax (Vermögenssteuer) and on the taxation of occupational pension benefits.
- Federal Act on Occupational Retirement, Survivors' and Disability Pension Plans (BVG/LPP), article 79b, on voluntary buy-ins and the three-year restriction on lump-sum withdrawal.
- Swiss Federal Social Insurance Office (BSV/OFAS), statistics and guidance on pillar 2 and pillar 3a.