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Only the Logarithm Adds ── +50% Followed by -50% Is Not 0% but -25% ── Repeating +/-10% Fifty Times Each Gives -39.50%, While the Naive Sum Answers 0% ── [Paper 292]

Yuuki Yamagishi
Aug 2026 · Zenodo (CERN European Organization for Nuclear Research)

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. Scope of this paper (scope note): 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 +/-50%. 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 +/-10% fifty times each gives -39.50%. First, we check on the smallest example.+50% then -50% sums to 0% in simple returns, but is actually -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 0 (Section 2). Third, “average” names three different operations. Arithmetic mean 0%, geometric mean -13.3975%, exponentiated log mean -13.3975% (Section 3). Fourth, the gap is approximated by sigma^2/2. At +/-50% it is 12.5% against 13.3975%, an error of 7.18%; at +/-1% the ratio is 1.000025 (Section 4). Fifth, it bites over long series.+/-10% fifty times each gives -39.50%, while the naive sum answers 0% (Section 5). Sixth, annualisation is not addition either. A daily sigma=1% is 15.8745% a year, and multiplying by 252 overstates it by 15.87 times (Section 6). Returns do not add. Only the logarithm adds.+50% then -50% sums to 0% in simple returns but is actually -25%, while exponentiating the sum of logs, -0.287682, gives 0.750000 with a difference of 0. The single word “average” also names three operations──arithmetic 0%, geometric -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 +/-50% to +/-1% takes the ratio from 1.0718 to 1.000025──the shape of Ito’s correction term, appearing in the numbers. And it bites over long series──+/-10% fifty times each, equal numbers of rises and falls, gives -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. On the making of this work: The ideas and content of this work stem from the author's own considerations. Assistance from an AI (a large language model) was used for structuring, English translation, and checking the algebra. Any remaining errors or misinterpretations are solely the author's. Feedback and corrections are sincerely appreciated. ----- 「収益率」は足し算できる量に見える。本稿が問うのは、どの書き方なら足せるかである──答は、対数収益率だけである。新しい数学定理も新しい法則も主張しない。 本稿の射程(射程注記):新しい数学定理も新しい法則も主張しない──単純収益率と対数収益率の関係、幾何平均、ボラティリティ・ドラッグ、sqrt(n) 則は、いずれも標準的である。金融理論を作らない──使うのは一つの対数と、一つの平方根だけである。価格を予測しない──将来の収益率も、期待値も一切扱わない。与えられた系列の合成規則だけを扱う。裾に入らない──論文190 が重い裾と大偏差を扱った。本稿は分布の形を一切仮定せず、足し算の可否だけを問う。 sqrt(n) 則の前提を隠さない──第6節の sqrt(252) は各期の独立性を仮定している。相関があれば成り立たない。 sigma^2/2 を厳密だと言わない──第4節が示すとおり、+/-50% では 7.18% ずれる。小さい変動での近似式である。投資助言をしない──どの平均を使うべきかは問いによる。本稿はどれが何を答えるかを分けるだけである。既刊との関係:論文194 は四つの平均が一つの族であり、等号の柵が一つしかないことを示した──あちらは一時点での不等式の構造、本稿は時間方向に合成したとき、どれが正しいかであり、同じ四つの平均を材料にして問いが直交している。論文132 は伊藤の 2 が次元ではないと示した──本稿の sigma^2/2 はその補正項そのものであり、第4節の収束表がそれを数で見せる。論文190 は「稀」を対数の目盛りで測った──本稿も対数に移してはじめて足せる。論文272 は同じ「二倍」が刺激の側で 6.70 倍ひらくことを示した──本稿もどの目盛りで足すかの問題である。加えたのは対数の和を指数に戻すと実測との差が 0 になると確かめたこと、三つの「平均」を数で並べたこと、sigma^2/2 が変動の縮小とともに比 1.000025 に収束する表を作ったこと、+/-10% を 50 回ずつで -39.50% になると計算したことである。 第一に、最小の例で確かめる。+50% のあと -50% は、単純収益率の和では 0% だが、実際は -25% である(第2節)。 第二に、これが本稿の芯である。対数収益率の和 ln1.5+ln0.5=-0.287682 を指数に戻すと 0.750000──実測との差は 0 である(第2節)。 第三に、「平均」が三つの別の操作を指す。算術平均 0%、幾何平均 -13.3975%、対数平均の指数 -13.3975%(第3節)。 第四に、差は sigma^2/2 で近似できる。+/-50% では 12.5% 対 13.3975% で 7.18% の誤差だが、+/-1% では比が 1.000025 になる(第4節)。 第五に、長い系列で効く。+/-10% を 50 回ずつで -39.50%、素朴な和は 0% と答える(第5節)。 第六に、年率換算も足し算ではない。日次 sigma=1% は年率 15.8745% であり、252 倍では 15.87 倍の過大評価になる(第6節)。 収益率は足せない。足せるのは対数だけである。+50% のあと -50% は、単純収益率の和では 0% だが実際は -25%であり、対数の和 -0.287682 を指数に戻すと 0.750000 で、実測との差は 0 になる。「平均」という一語も三つの操作を指す──算術平均 0%、幾何平均 -13.3975%、そして幾何平均は対数の算術平均に他ならない(差 10^-16)。その差は sigma^2/2 で近似でき、変動を +/-50% から +/-1% に縮めると比が 1.0718 から 1.000025 になる──伊藤の補正項の形が、数の上に現れる。そして長い系列で効く──+/-10% を 50 回ずつ、上げと下げが同じ回数なのに -39.4994% であり、単純な和は 0% と答える。分けるものは一つ──その量の合成が掛け算か、足し算か。掛け算なら対数に移してから足す。ばらつきなら分散に移してから足す。移さずに足したものは、収益率でも平均でもない。 作成にあたって:本稿の着想と内容は、著者自身の考察に基づくものです。文章の構成整理や英訳、数式の確認には AI(大規模言語モデル)の助力を得ました。最終的な内容の解釈や誤りがあれば、それらはすべて著者の責に帰します。お気づきの点があれば、ご教示いただければ幸いです。

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