Abstract
A “rate of return” looks like a quantity that can be added. This paper asks which way of writing it can——the answer is the log return alone. No new mathematical theorem and no new law is claimed.
First, we check on the smallest example. $+50\%$ then $-50\%$ sums to $0\%$ in simple returns, but is actually $\mathbf{-25\%}$ (Section 2).
Second, this is the core of the paper. Exponentiating the sum of log returns $\ln1.5+\ln0.5=-0.287682$ gives $0.750000$——the difference from the measured value is $\mathbf{0}$ (Section 2).
Third, “average” names three different operations. Arithmetic mean $0\%$, geometric mean $\mathbf{-13.3975\%}$, exponentiated log mean $-13.3975\%$ (Section 3).
Fourth, the gap is approximated by $\sigma^2/2$. At $\pm50\%$ it is $12.5\%$ against $13.3975\%$, an error of $7.18\%$; at $\pm1\%$ the ratio is $\mathbf{1.000025}$ (Section 4).
Fifth, it bites over long series. $\pm10\%$ fifty times each gives $\mathbf{-39.50\%}$, while the naive sum answers $0\%$ (Section 5).
Sixth, annualisation is not addition either. A daily $\sigma=1\%$ is $\mathbf{15.8745\%}$ a year, and multiplying by $252$ overstates it by $\mathbf{15.87}$ times (Section 6).
Scope (what is not claimed)
No new mathematical theorem and no new law is claimed——the relation between simple and log returns, the geometric mean, volatility drag, and the $\sqrt{n}$ rule are all standard. We do not build financial theory——all we use is one logarithm and one square root. We do not forecast prices——future returns and expectations are not treated at all. Only the composition rule for a given series is treated. We do not enter the tails——Paper 190 treated heavy tails and large deviations. This paper assumes no distributional shape and asks only whether addition is permitted. We do not hide the premise of the $\sqrt{n}$ rule——the $\sqrt{252}$ of Section 6 assumes independence between periods. With correlation it fails. We do not say $\sigma^2/2$ is exact——as Section 4 shows, it is off by $7.18\%$ at $\pm50\%$. It is an approximation for small fluctuations. We give no investment advice——which mean to use depends on the question. This paper only separates which one answers what. Relation to earlier papers: Paper 194 showed that the four means are one family with only one fence of equality——that paper concerns the structure of the inequality at one instant; this one concerns which is right when composing along time, the same four means as material with orthogonal questions. Paper 132 showed that the $2$ in Ito’s lemma is not a dimension——the $\sigma^2/2$ here is that correction term itself, and the convergence table of Section 4 shows it numerically. Paper 190 measured “rare” on a logarithmic scale——here too one can add only after moving to the logarithm. Paper 272 showed that one and the same “twice” opens by $6.70$ on the stimulus side——this paper is likewise about on which scale one adds. What is added is confirming that exponentiating the sum of logs leaves a difference of $0$, lining up the three “averages” numerically, building a table in which the ratio to $\sigma^2/2$ converges to $1.000025$ as the fluctuation shrinks, and computing that $\pm10\%$ fifty times each gives $-39.50\%$.
1. Notation and the Question
Setting: When a price moves from $P_0$ to $P_1$ there are two ways to write it.
| Name | Definition | Range |
|---|---|---|
| Simple return | $r=P_1/P_0-1$ | $[-1,\infty)$ |
| Log return | $\ell=\ln(P_1/P_0)$ | $(-\infty,\infty)$ |
Question: Joining two periods, which of them adds?
Observation 1.1 Prices compose by multiplication——$P_2/P_0=(P_1/P_0)(P_2/P_1)$. Take a logarithm and multiplication becomes addition——the answer is already written in the form. This paper confirms it numerically.
2. The Core: The Sum of Logs Leaves a Difference of $0$
Check 2.1: $+50\%$ then $-50\%$.
| Method | Computation | Result |
|---|---|---|
| Sum of simple returns | $+0.5000+(-0.5000)$ | $0.00\%$ [wrong] |
| Actual factor | $1.5\times0.5$ | $0.7500=\mathbf{-25.00\%}$ |
| Sum of log returns | $+0.405465+(-0.693147)$ | $-0.287682$ |
| Exponentiated | $e^{-0.287682}$ | $\mathbf{0.750000}$ [right] |
| Difference from measured | $\mathbf{0}$ |
Observation 2.2 This is the core of the paper. Exponentiating the sum of logs leaves a difference of $\mathbf{0}$ (not even machine rounding)——not close, but the same.
Observation 2.3 The sum of simple returns misses by $25$ percentage points. The difference between $0\%$ and $-25\%$ is not an error but a different operation——something that should have been multiplied was added.
Observation 2.4 Note that log returns are asymmetric. $+50\%$ is $\ln1.5=+0.405$; $-50\%$ is $\ln0.5=-0.693$——the same “$50\%$” differs by $1.71$ times in logarithm. The downside is heavier because $-100\%$ corresponds to $-\infty$.
3. “Average” Names Three Different Operations
Check 3.1: For the same two periods we compute three “averages.”
| Average | Computation | Value |
|---|---|---|
| Arithmetic (simple returns) | $(0.5-0.5)/2$ | $\mathbf{0.0000\%}$ |
| Geometric (actual growth) | $\sqrt{1.5\times0.5}-1$ | $\mathbf{-13.3975\%}$ |
| Mean of log returns | $(-0.287682)/2=-0.143841$ | |
| Exponentiated | $e^{-0.143841}-1$ | $\mathbf{-13.3975\%}$ |
| Difference of the last two | $1.11\times10^{-16}$ |
Observation 3.2 The geometric mean and the exponentiated log mean are the same. The difference is $10^{-16}$, machine rounding——the geometric mean is the arithmetic mean of the logarithms.
Observation 3.3 Only the arithmetic mean answers $0\%$. The gap is $\mathbf{13.3975}$ percentage points——and both are called “the average return”.
Observation 3.4 Paper 194 showed that the four means are one family with only one fence of equality. That concerns the structure of the inequality at one instant——this paper asks which returns the right answer when composed along time. The same four means as material, with orthogonal questions.
Observation 3.5 Separator——which is right depends on the question. For “the expectation of the next single period,” the arithmetic mean; for “what it came to after $n$ periods,” the geometric mean——both correct, answering different questions.
4. The Gap Is Approximated by $\sigma^2/2$, Exactly in the Small-Fluctuation Limit
Setting: The gap between arithmetic and geometric means is called “volatility drag” and is approximated by $\sigma^2/2$.
Check 4.1: We check at $\pm50\%$.
| Quantity | Value |
|---|---|
| Standard deviation of simple returns $\sigma$ | $0.500000$ |
| $\sigma^2/2$ | $0.125000=12.5000\%$ |
| Measured drag (arithmetic $-$ geometric) | $0.133975=\mathbf{13.3975\%}$ |
| Ratio | $\mathbf{1.0718}$ (the approximation errs by $7.18\%$) |
Check 4.2: With the fluctuation set to $\pm x$, we watch the ratio.
| $\pm x$ | Measured drag | $\sigma^2/2=x^2/2$ | Ratio |
|---|---|---|---|
| $0.500$ | $0.13397460$ | $0.12500000$ | $1.071797$ |
| $0.200$ | $0.02020410$ | $0.02000000$ | $1.010205$ |
| $0.100$ | $0.00501256$ | $0.00500000$ | $1.002513$ |
| $0.050$ | $0.00125078$ | $0.00125000$ | $1.000626$ |
| $0.020$ | $0.00020002$ | $0.00020000$ | $1.000100$ |
| $\mathbf{0.010}$ | $0.00005000$ | $0.00005000$ | $\mathbf{1.000025}$ |
Observation 4.3 Shrinking $x$ by fifty takes the ratio from $1.0718$ to $\mathbf{1.000025}$. $\sigma^2/2$ becomes exact in the small-fluctuation limit——this is the shape of Ito’s correction term.
Observation 4.4 Paper 132 showed that the $2$ in Ito’s lemma is not a dimension. This table shows numerically where that $2$ appears——the $2$ in $x^2/2$ comes from the second-order term of a Taylor expansion and has nothing to do with the dimension of space.
Observation 4.5 A point to write honestly——at $\pm50\%$, $\sigma^2/2$ is off by $7.18\%$. Applied to a series containing crashes it systematically understates——one must not forget it is an approximation.
5. It Bites over Long Series
Check 5.1: $+10\%$ and $-10\%$, fifty times each, $100$ periods in all.
| Method | Result |
|---|---|
| Sum of simple returns | $0\%$ [wrong] |
| Actual factor | $\mathbf{0.60500607}=\mathbf{-39.4994\%}$ |
| Sum of logs $50(\ln1.1+\ln0.9)$ | $-0.50251679$ |
| Exponentiated | $\mathbf{0.60500607}$ [right] |
| Geometric mean per period | $-0.501256\%$ |
| $\sigma^2/2$ (per period) | $0.500000\%$ |
Observation 5.2 Over $100$ periods it is $\mathbf{-39.50\%}$. The sum of simple returns answers $0\%$——$39.5$ percentage points of loss vanish entirely from view.
Observation 5.3 The per-period geometric mean $-0.501256\%$ agrees with $\sigma^2/2=0.500000\%$ to three digits. $\pm10\%$ is small, so the approximation works well——exactly as the table of Section 4 predicted.
Observation 5.4 And this $-39.5\%$ results from equal numbers of rises and falls. The intuition that “even odds cancel out” misses by $39.5$ percentage points——the fluctuation itself is a cost.
6. Annualisation Is Not Addition Either
Setting: If the periods are independent, variances add. So standard deviations go as $\sqrt{n}$.
Check 6.1: From a daily $\sigma=1\%$.
| Horizon | Trading days $n$ | $\sigma\sqrt{n}$ |
|---|---|---|
| Week | $5$ | $2.2361\%$ |
| Month | $21$ | $4.5826\%$ |
| Year | $252$ | $\mathbf{15.8745\%}$ |
Observation 6.2 The annual figure is $\mathbf{15.8745\%}$, not $252\%$. Naively multiplying by $252$ gives $252\%$, an overstatement by $\mathbf{15.8745}$ times——because $252/\sqrt{252}=\sqrt{252}$.
Observation 6.3 Here too the question is what adds. What adds is the variance, not the standard deviation——for returns the logarithm adds; for dispersion the variance adds.
Observation 6.4 A point to write honestly——the $\sqrt{n}$ rule assumes independence between periods. With correlation it fails——independence is a premise, not an observed fact.
Observation 6.5 Separator——what must be moved for addition to become permitted? The scale.
| Quantity | Composition rule | The writing that adds |
|---|---|---|
| Return | multiplication | logarithm |
| Dispersion | sum of squares | variance |
Observation 6.6 In neither case does the original quantity add. Move to the scale that adds, add there, and come back——the same procedure by which Paper 190 measured “rare” logarithmically.
7. Roll Call and Closing
- $+50\%$ then $-50\%$ is $-25\%$ (Check 2.1): The simple sum says $0\%$.
- The sum of logs leaves difference $0$ (Check 2.1): Not even rounding.
- $\ln1.5$ against $\ln0.5$ is $1.71$ (Observation 2.4): The downside is heavier.
- Geometric mean $-13.3975\%$ (Check 3.1): The arithmetic mean is $0\%$.
- Geometric $=$ exponentiated log mean (Observation 3.2): Difference $1.11\times10^{-16}$.
- $\sigma^2/2$ errs by $7.18\%$ at $\pm50\%$ (Check 4.1): An approximation.
- Ratio $1.000025$ at $\pm1\%$ (Check 4.2): The shape of Ito’s correction.
- $-39.4994\%$ over $100$ periods (Check 5.1): The sum answers $0\%$.
- $-0.501256\%$ per period (Check 5.1): Three digits with $\sigma^2/2=0.5\%$.
- Annual $15.8745\%$ (Check 6.1): $252$ overstates by $15.87$ times.
Closing: Returns do not add. Only the logarithm adds. $+50\%$ then $-50\%$ sums to $0\%$ in simple returns but is actually $\mathbf{-25\%}$, while exponentiating the sum of logs, $-0.287682$, gives $0.750000$ with a difference of $\mathbf{0}$. The single word “average” also names three operations——arithmetic $0\%$, geometric $\mathbf{-13.3975\%}$, and the geometric mean is nothing but the arithmetic mean of the logarithms (difference $10^{-16}$). That gap is approximated by $\sigma^2/2$, and shrinking the fluctuation from $\pm50\%$ to $\pm1\%$ takes the ratio from $1.0718$ to $\mathbf{1.000025}$——the shape of Ito’s correction term, appearing in the numbers. And it bites over long series——$\pm10\%$ fifty times each, equal numbers of rises and falls, gives $\mathbf{-39.4994\%}$, and the naive sum answers $0\%$. One thing separates them——whether that quantity composes by multiplication or by addition. If by multiplication, move to logarithms and add there. If dispersion, move to variances and add there. What is added without moving is neither a return nor an average.
References
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- Fernholz, R. and Shay, B. (1982). Stochastic Portfolio Theory and Stock Market Equilibrium. The Journal of Finance, 37(2).
- Campbell, J. Y., Lo, A. W. and MacKinlay, A. C. (1997). The Econometrics of Financial Markets. Princeton: Princeton University Press.
- Hull, J. C. (2018). Options, Futures, and Other Derivatives. New York: Pearson.
- Kelly, J. L. (1956). A New Interpretation of Information Rate. Bell System Technical Journal, 35(4).
- Yamagishi, Y. (2026). The $2$ in Itô's Lemma Is Not a Dimension [Paper 132]. Zenodo. 10.5281/zenodo.22080358
- Yamagishi, Y. (2026). “Rare” Is Measured on a Logarithmic Scale [Paper 190]. Zenodo. 10.5281/zenodo.22106182
- Yamagishi, Y. (2026). The Four Means Are One Family, and There Is Only One Equality Fence [Paper 194]. Zenodo. 10.5281/zenodo.22106809
- Yamagishi, Y. (2026). One and the Same “Twice as Strong” Opens by a Factor of 6.70 on the Stimulus Side [Paper 272]. Zenodo. 10.5281/zenodo.22169965
Reference
Yamagishi, Y. (2026). Only the Logarithm Adds [Paper 292]. YOI: https://www.yamagishiyuuki.com/yoi/292
DOI: 10.5281/zenodo.22171519
ORCID: 0009-0006-8587-3450
PDF (English) · PDF (Japanese) · Date proof of the English PDF (OpenTimestamps .ots) · SHA256SUMS
The record page for this paper (in Japanese, with the PDFs in each language and the date proof): YOI 292. This paper is a preprint and has not been peer reviewed.