Returns: arithmetic, geometric and log

A return looks like the least controversial number in finance. It is a ratio of two prices, and everyone agrees on the prices. Yet more disputes about performance come from the definition of the return than from any disagreement about what the market did. Two people looking at the same price series, the same dividends and the same account statement can honestly report different numbers, and both can be right, because they have answered different questions.

This lesson sets out the three definitions that matter — simple, geometric and logarithmic — and, more importantly, the operations each one is safe for. The recurring failure is not ignorance of the formulas. It is using a return that is additive in one dimension as though it were additive in the other.

Three definitions, and the question each answers

Take a price that goes from P₀ to P₁ over one period. The simple return (also called the arithmetic or discrete return) is

R = P₁/P₀ − 1

This is the fraction of your capital you gained. It is the natural unit for anything that involves money at a point in time: how much you made, how much a position contributed, how a portfolio's return relates to its holdings.

The log return (also called the continuously compounded return) is

r = ln(P₁/P₀) = ln(1 + R)

This is the exponent that turns P₀ into P₁. It is the natural unit for anything that involves stacking periods end to end.

The geometric mean return is not a third kind of return but a way of averaging. Over T periods with simple returns R₁, …, R_T, it is the constant rate that would have produced the same terminal wealth:

G = [(1 + R₁)(1 + R₂) … (1 + R_T)]^(1/T) − 1

Simple and log returns convert cleanly into each other: r = ln(1 + R) and R = e^r − 1. For small moves they are nearly the same. A simple return of 1.00% is a log return of 0.995%. A simple return of 5.00% is a log return of 4.879%. The gap opens up fast at large magnitudes: a simple return of +100% is a log return of 69.31%, and a simple return of −50% is a log return of −69.31%. That symmetry is the whole appeal of logs. Doubling and halving are equal and opposite in log space, and grossly unequal in simple space.

The gap between the average and what you actually kept

Start with 100. The price rises 20%, then falls 20%.

Period 1: 100 × 1.20 = 120. Period 2: 120 × 0.80 = 96.

The arithmetic mean of the two simple returns is (0.20 − 0.20)/2 = 0. The terminal wealth is 96, a cumulative return of −4.00%. The geometric mean is √0.96 − 1 = 0.97980 − 1 = −2.020% per period.

Nothing here is a paradox. The arithmetic mean answers "what was the expected one-period return, drawn at random from this sample?" The geometric mean answers "what constant rate reproduces the ending balance?" Only the second is a claim about wealth. Anyone quoting an arithmetic average of a volatile series as though it described growth is overstating the result, and the overstatement grows with volatility.

The size of the gap is not arbitrary. To a good approximation,

G ≈ R̄ − σ²/2

where R̄ is the arithmetic mean and σ the standard deviation of the simple returns. In the example, the two observations +0.20 and −0.20 have mean 0 and population standard deviation 0.20, so σ²/2 = 0.04/2 = 0.02. The approximation predicts −2.00% against the exact −2.020%. This term is sometimes called volatility drag, and the mechanism is arithmetic rather than behavioural: a proportional loss requires a larger proportional gain to reverse, because the gain is applied to a smaller base. The companion lesson on compounding takes this further into the distribution of terminal wealth.

Now do the same example in logs. ln(1.20) = 0.182322 and ln(0.80) = −0.223144. They sum to −0.040822, and e^(−0.040822) = 0.9600. The sum of log returns over time is exactly the log of the cumulative return, with no approximation. The mean log return is −0.020411, and e^(−0.020411) = 0.97980, which is precisely the geometric mean found above. Averaging log returns and converting back always gives the geometric mean.

Where log returns break

Log returns are additive across time and are not additive across assets. This is the single most common technical error in return arithmetic.

A portfolio holds 50% in asset A and 50% in asset B, rebalanced at the start of the period. A returns +20% and B returns −20%. The portfolio's simple return is the weighted average of the simple returns: 0.5(0.20) + 0.5(−0.20) = 0. The portfolio ended flat, which is obviously correct: half the money grew by a fifth, half shrank by a fifth.

If instead you average the log returns, you get 0.5(0.182322) + 0.5(−0.223144) = −0.020411, implying −2.02%. That is the wrong answer to this question. It is the right answer to a different one: it is what a single asset would have compounded to over two periods with those returns in sequence.

The rule to carry: combine across assets in simple returns, combine across time in log returns. Convert at the boundary. A portfolio's log return is the log of one plus its weighted-average simple return, not the weighted average of the constituents' log returns.

Log returns have a second property worth naming. Because the log of a product is the sum of logs, and sums of many roughly independent variables tend toward the normal distribution, log returns are the variable most statistical machinery is built around. That is why prices are commonly modelled as lognormal: it keeps prices positive and makes the multiplicative structure additive. Whether the normal assumption survives contact with real data is the subject of the lessons on distributions and on volatility. It does not survive well.

Price return is not the return

A price series alone understates what a shareholder earned, because it omits dividends. A total return series reinvests each distribution at the price on the ex-date. Over decades the difference is not cosmetic; for a mature, high-payout market it can be the majority of the cumulative gain.

For a Swiss-resident investor there is a further wedge between the index's total return and the one that reaches the bank account. Switzerland levies a federal withholding tax (Verrechnungssteuer) of 35% on dividends paid by Swiss companies and on interest from Swiss bank accounts and bonds. The paying company remits 35% to the Federal Tax Administration and the investor receives 65%. For a Swiss-resident individual, this is a security deposit rather than a final tax: declaring the gross income on the cantonal tax return reclaims the full 35%, credited against the tax bill or refunded. The catch is that the reclaim depends on correct declaration, and it takes time. Between the ex-date and the refund, the investor is financing the state. A total return index assumes gross reinvestment on the ex-date. An investor reinvesting 65% and receiving the balance a year later has, in effect, a small timing drag on compounding.

The picture changes for foreign dividends, where relief runs through the double taxation treaty network. Switzerland's treaties typically reduce foreign withholding on portfolio dividends to 15%, and the residual is claimed back through the Swiss return as a lump-sum tax credit rather than refunded by the source country. Anyone comparing a fund's reported return to a gross index should know which of these frictions the fund actually bore.

Currency: multiply, do not add

A Swiss investor holding a foreign asset earns two returns, and they compound rather than add. If the asset returns R_loc in its local currency and the foreign currency moves R_fx against the franc, the return in francs is

R_CHF = (1 + R_loc)(1 + R_fx) − 1

Suppose a US-listed holding returns +10.0% in dollars while the dollar depreciates 5.0% against the franc. The return in francs is 1.10 × 0.95 − 1 = 1.0450 − 1 = +4.50%, not the +5.0% a subtraction would suggest. The missing half-point is the cross term R_loc × R_fx = 0.10 × (−0.05) = −0.005.

In logs it is a clean sum: ln(1.10) = 0.095310 and ln(0.95) = −0.051293, totalling 0.044017, and e^0.044017 = 1.0450. This is the clearest single illustration of why both conventions earn their place.

Time-weighted and money-weighted are different questions

One last distinction, because it decides whose skill is being measured. A time-weighted return chains the sub-period returns between every external cash flow, so deposits and withdrawals have no effect. It measures the performance of the strategy. A money-weighted return is the internal rate of return on the actual cash flow schedule. It measures the performance of the investor, including the timing of contributions.

They diverge whenever money arrives before a bad stretch or after a good one. A manager who cannot control client flows is judged on the time-weighted number, which is why the Global Investment Performance Standards are built around it. A person saving into a pillar 3a account, who chooses when to pay in, should look at the money-weighted number, because the timing was theirs.

What to take away

• Simple returns are additive across assets at a point in time; log returns are additive across time for a single asset. Mixing the two is the most common source of wrong return arithmetic.
• The arithmetic mean of a volatile series always exceeds its geometric mean, by roughly σ²/2, and only the geometric mean corresponds to terminal wealth.
• Converting between conventions is exact: r = ln(1 + R) and R = e^r − 1. They differ negligibly for daily moves and materially for annual ones.
• A foreign asset's return in francs is the product of the local return and the currency return, not their sum; the cross term matters once either leg is large.
• Swiss withholding tax of 35% on domestic dividends is reclaimable by declaring the gross income, but the delay between deduction and refund is a genuine, if small, drag on compounding that no index captures.

Sources

• John Y. Campbell, Andrew W. Lo and A. Craig MacKinlay, The Econometrics of Financial Markets (Princeton University Press, 1997), Chapter 1, on the definition and statistical properties of simple and continuously compounded returns.
• Zvi Bodie, Alex Kane and Alan J. Marcus, Investments, chapters on rates of return, arithmetic versus geometric averages, and time-weighted versus dollar-weighted performance.
• Attilio Meucci, Risk and Asset Allocation (Springer, 2005), on linear versus compounded returns and the aggregation properties of each.
• Elroy Dimson, Paul Marsh and Mike Staunton, Triumph of the Optimists: 101 Years of Global Investment Returns (Princeton University Press, 2002), on the contribution of reinvested income to long-run returns.
• CFA Institute, Global Investment Performance Standards (GIPS), on required return calculation methodology for reported composite performance.
• Swiss Federal Tax Administration (Eidgenössische Steuerverwaltung), guidance on Verrechnungssteuer: deduction, declaration and refund procedure for Swiss-resident individuals.
• Swiss National Bank, exchange rate and financial market statistics, for franc series used in currency-adjusted return calculations.