Domain: D00 — Financial Mathematics, Statistics, and Data Foundations
Family: D00-F02 — Financial Arithmetic, Time Value, and Returns
Audience: a curious beginner who may not have studied finance or programming
Delivery: plain-language lesson, hand-worked arithmetic, and a small interactive lab
The one-sentence idea
Use the natural logarithm of the ending value divided by the starting value to describe multiplicative change on an additive scale.
Why this matters in money
When returns are recorded period after period, log returns can be added to describe the total log change. The price values still multiply; the logarithm makes that multiplication easier to summarize.
What you will be able to do
- calculate a log return for positive start and end values
- explain its relationship with simple return
- recognize the positive-value boundary
What you need first
- D00-F02-A06 simple return
- D00-F01 exponents, roots, and logarithms
You do not need to program for this lesson. We use ordinary words and small numbers first. Later domains may show implementation details after the idea is comfortable.
Words to know
| Term | In simple words |
|---|---|
| Natural logarithm | The logarithm usually written ln; it asks which power of e gives a number. |
| Price ratio | Ending value divided by starting value. |
| Log return | ln of that positive price ratio. |
| Additive | Log returns can be added across periods when the value path is continuous and the same convention is used. |
Scope and simple boundary
We use positive start and end values with no intermediate cash flow. Corporate actions, zero or negative prices, and total-return series require a correctly adjusted value series first.
Definition contract
| Decision | Our simple choice | What we do not claim | Check |
|---|---|---|---|
| Inputs | Start and end must be positive. | Do not take ln(0) or ln of a negative price. | The lab labels invalid values as undefined. |
| Definition | Use ln(end/start). | Do not substitute log base 10 unless that is the named convention. | The formula names natural log. |
| Interpretation | Log return is not the same number as simple percentage return. | Do not call 0.0953 a 9.53% simple return. | The example compares both labels. |
Input contract
| Name | Kind | Unit | Empty? | Meaning |
|---|---|---|---|---|
| start | money | SAR | No | Positive starting value. |
| end | money | SAR | No | Positive ending value. |
All values in one example must use the same time period, currency, and scale unless the lesson explicitly shows a conversion. A missing or impossible input is a question to resolve, not a number to guess.
Output contract
| Output | Kind | Unit | Meaning | Safety rule |
|---|---|---|---|---|
| simple_return | percentage | % | End/start - 1, shown for comparison. | Start must be positive. |
| log_return | number | log units | Natural log of end/start. | Both values must be positive. |
The rule in everyday language
log return = ln(ending value / starting value)
Think of the rule as a sentence. Say what each number means before you put it into the sentence.
Symbol table
| Name | Meaning | Unit |
|---|---|---|
| S | positive starting value | SAR |
| E | positive ending value | SAR |
| ln | natural logarithm | unitless |
| log return | ln(E/S) | unitless |
Four small steps
- Check that both values are positive.
- Form the ending-to-starting ratio.
- Take its natural logarithm.
- If comparing with a simple return, keep both names and units visible.
Worked example (synthetic teaching numbers)
Synthetic example: a value rises from SAR 100 to SAR 110. Simple return = 110/100 - 1 = 10%. Log return = ln(1.10) = about 0.09531, often described as 9.531% in log-return units.
Independent check
The exponential reverses the log: exp(0.09531) is about 1.10. A start of zero has no valid price ratio, so the log return is undefined rather than infinitely large.
The numbers above are synthetic and author-derived for learning. They are not an observation about a named company, market, customer, or investment.
Simple playground
Open the small guided lab. Choose a number, press Step, and read the explanation under the result. The lab is designed to show one idea at a time, not to replace the lesson.
What to notice: the input, the rule, the check, and the safe interpretation are shown in that order. Open the full-size visual.
Where this is useful
- summarizing multiplicative price changes
- adding period log returns in a consistent series
- teaching why simple and log returns differ
What this does not tell you
Log returns require a positive, consistently defined value series. They do not include cash flows unless the input series already represents total wealth, and they do not promise better investment results.
This is educational content, not investment, tax, legal, accounting, or regulatory advice. A simple calculation can be correct and still be the wrong calculation for a real decision.
Historical-example decision
Not useful for this lesson. A named company would add a story but would not teach the primitive more clearly than the small synthetic numbers above. A real case would also need a verified entity, period, unit, and publication right. The lesson therefore keeps its arithmetic transparent and synthetic.
Related lessons
Evidence boundary
The plain-language definitions and measurement cautions are supported by NIST_LOG, CFA_QM. The formula wording, examples, and lab behavior are author-derived teaching choices. See the claim ledger and references for the boundary.
Optional verification implementation
You do not need code to learn this lesson. The package now includes matching Python and TypeScript verification façades, a shared worked-example fixture, and parity tests. They reproduce the lesson’s frozen rule and remain optional for nontechnical learners.
Enhancement studio: draw, compare, explain
This additive studio does not replace the beginner lesson above. It gives you two more drawings, a decision comparison, and short practice prompts so you can explain the idea without copying a formula or writing code.
Drawing 1 — name, apply, check
Read left to right: name what the data means, apply the narrow lesson rule, then use an independent check. Open the full-size concept anatomy.
Choose the right idea
| Decision | This lesson | Closest next or comparison | Why the difference matters |
|---|---|---|---|
| Main question | Use the natural logarithm of the ending value divided by the starting value to describe multiplicative change on an additive scale. | Holding-Period and Cumulative Return | Choose the question before choosing the arithmetic. |
| Safe rule | log return = ln(ending value / starting value) | Uses its own input and boundary contract. | Neighboring lessons can use the same numbers but answer different questions. |
| Required check | The exponential reverses the log: exp(0.09531) is about 1.10. A start of zero has no valid price ratio, so the log return is undefined rather than infinitely large. | Re-check its own unit, time, denominator, or schema. | A correct answer to the wrong question is still wrong. |
| Stop condition | Do not take ln(0) or ln of a negative price. | Move only when its prerequisites are satisfied. | Unknown meaning is a reason to pause, not to guess. |
Drawing 2 — common-mistake clinic
The left side states the safe interpretation; the right side shows the mistake that often produces a believable but misleading result. Open the full-size mistake comparison.
Explain it back without code
- Name it: What does the first input or observation mean?
Answer: Positive starting value. - Choose it: Which rule belongs to this question?
Answer: log return = ln(ending value / starting value) - Challenge it: What check could make you stop?
Answer: The exponential reverses the log: exp(0.09531) is about 1.10. A start of zero has no valid price ratio, so the log return is undefined rather than infinitely large.
If your explanation leaves out the unit, period, denominator, grain, or availability time that the lesson needs, it is not complete yet.
Related concepts and learning handoff
- Governed glossary: Natural logarithm, Price ratio, Log return, Additive. Browse the full financial glossary when a term is unfamiliar.
- Continue with: Holding-Period and Cumulative Return.
- Evidence boundary: all displayed numbers remain synthetic teaching data; the drawings do not claim a market observation, forecast, or investment result.
Rendered from the canonical Mermaid sources linked by this article.
Log Return — four-part map
This diagram shows the learner's path from a named input to a safe explanation.
Takeaway: the check is not an afterthought. It tells the learner whether the answer belongs to the question that was asked.
ReferencesPrimary sources and evidence notesExpand the source trail, evidence role, and limitations behind the engineering choices.
Expand the source trail, evidence role, and limitations behind the engineering choices.
Each source has a limited evidence role. The worked values, visuals, and playground controls are synthetic and author-derived.
NIST_LOG - NIST Dataplot: Exponential and Logarithmic Functions
- Organization or authors: Institutional source
- Source type: Official technical reference
- Publication or effective date: Current NIST web edition
- Version: current web edition accessed for this build
- URL: https://www.itl.nist.gov/div898/software/dataplot/expo_fun.htm
- Accessed: 2026-08-10
- Jurisdiction: general educational finance or measurement context
- Supports: Natural, base-10, and base-2 logarithm conventions and the positive-input rule.
- Limitations: The family explains the ideas by hand; it does not teach a software library.
CFA_QM - CFA Institute Quantitative Methods Study Session
- Organization or authors: CFA Institute
- Source type: Professional finance curriculum
- Publication or effective date: 2023 Level I curriculum PDF
- Version: current web edition accessed for this build
- URL: https://www.cfainstitute.org/sites/default/files/-/media/documents/study-session/2023-l1-topics-combined.pdf
- Accessed: 2026-08-10
- Jurisdiction: general educational finance or measurement context
- Supports: Why periodic rates need a stated period and why compounding is multiplicative rather than a simple sum.
- Limitations: Curriculum notation is simplified for teaching and does not replace a product's legal terms.
Author-derived and synthetic boundary
The formulas are standard classroom definitions selected for this family. The examples use small synthetic SAR amounts and percentage rates so a learner can reproduce every step by hand. They do not establish a company fact, market outcome, product quote, or investment result.
Full dependency-light reference implementations in both supported languages.
import { runTopic as runD00Topic, type D00Input, type D00Output } from "../../../../shared/typescript/d00Engine.ts";
/** Run the canonical D00-F02-A07 calculation. */
export function logReturn(input: D00Input): D00Output {
return runD00Topic("D00-F02-A07", input);
}
The embedded lab now expands to its full document height, keeping the article as the only scroll surface.