Numbers have 3 companions: When is it from? What is it for? And what is it compared to?
A number missing any one of these isn't really a number yet.
Deciphering Amounts, Percentages, and Statements
How much can you trust numbers?
Learn to compare figures for interest rates, fees, and prices, and where to find them in statements and contracts.
Time to read the text: about 8 min
The review questions are optional and not included in this time.
What you can read: the course opening and contents are free; the full text is for those who have purchased
After purchase, you can read the full text once your access starts.
How to Read Numbers in 3 Minutes
1See it
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Nominal is the amount on paper. Real is how much you can actually buy with it today.
You find the real value by dividing by the price level.
If prices rise 2% every year, 1,000,000 yen will only be worth 820,000 yen in 10 years.
At 3% a year, it would be worth 740,000 yen.
If interest rates go from 1% to 2%, we say they 'rose by 1 percentage point'.
Use 'points' for the difference, and '%' for the ratio.
Compound interest is interest on your interest. Take 1,000,000 yen at 3% a year for 10 years.
1,344,000 yen
With simple interest, it would be 1,300,000 yen.
Even a 1% annual fee will eat up 9.6% of your principal in 10 years.
It's deducted at the same rate every year, regardless of how your investment performs.
When looking at a graph, check the vertical axis, the time period, if it's nominal or real, and the source.
Trust the trend only after you've checked these four things.
Read contracts in this order: total amount, time period, cancellation policy, fees, and who you're dealing with.
This helps you grasp the big picture before getting lost in a sea of clauses.
For your Nenkin Teikibin (pension statement), if you find any errors in your record, use the response form to report them.
The estimated benefit amount is a nominal figure.
2In numbers
9.6%
Percentage of principal lost to a 1% annual fee over 10 years
820,000yen's worth
What 1,000,000 yen can buy in 10 years (if prices rise 2% annually)
3What would you do?
Someone at the bank tells you, 'The annual fee is only 1%.' If you hold the investment for 20 years, how much of your principal will disappear in fees?
Pick one
About 10%
Actually, it's more than that. Annual fees accumulate using the same formula as compound interest. Over 20 years, it comes to 18.2%, or about 20%.
About 20%
That's right. The formula is 1 - (1 - 0.01)^20, which equals 18.2%. Even a rate that seems small gets bigger over many years.
About 40%
It's not quite that much. It's 18.2% over 20 years, and 26.0% over 30 years. Still, that's more than 1/4 of your principal.
The real test is this: can you add up all the fees—for buying, holding, and selling—and figure out a single annual percentage? If you can do that, you've got it.
4Read more
Suppose a document says "1,000,000 yen in ten years." If prices rise by 2% a year, that 1,000,000 yen will buy what 820,000 yen buys today (100 ÷ 1.02 to the tenth power). At 3% a year, it is 740,000 yen's worth. The amount on paper is the same, but its meaning changes with what it is measured against.
When reading a number, check its three companions: when the number is from (point in time), what it counts (unit), and what it is relative to (denominator).
Article V of the 1858 Treaty of Amity and Commerce between the United States and Japan provided that foreign coin would pass for the same weight of Japanese coin of the same description. But the Ichibugin was a silver coin counted by its face value, not its weight. What happened when the world of weight and the world of face value were mixed can be read in Section 2 and on the routes.
Today's household finances have the same two worlds: the nominal pension amount and the real pension amount divided by prices; the headline interest rate and the rate after fees.
In this course you will acquire the habit of attaching the three companions to a number, and an order for reading the documents you receive. All formulas are written in the text.
What you will understand after this course
- Can distinguish between nominal and real values, and can personally calculate the erosion of real value using the inflation rate.
- Can verify the denominator of a ratio and question the original number behind terms like "double," "half," and "%".
- Can place simple and compound interest, fees, and interest rates into a single formula, and can calculate the difference after ten years using one's own numbers.
More about this course
Course number: P2 — How much can you trust numbers?
Supervised by: Mitsuru Hayama
Beyond the text: 6 exercises · 2 coins · 3 readings How reading times are calculated
You will also understand
- Has a method for reading statements, contracts, and official documents, and can state where to seek clarification for unfamiliar sections.
You can start this course without reading any other course.
If you want to know more about checking sources (optional)
- Is It Really True? Checking History
We re-examine the origins of money and famous anecdotes, comparing them with historical records and actual artifacts.
Assessment
6 exercises in this course (4 check, 0 compare, 2 written). Progress is recorded by self-assessment.
The 8 chapters of this course
- Numbers Have Three Accompaniments — Point in Time, Unit, and DenominatorTime point, unit, denominator. A number that lacks even one of these three is not yet a number.
- Nominal and Real — The Lesson of the IchibuginNominal is the paper value; real is how much you can buy now. Nominal ÷ (1 + inflation rate)^years.
- The denominator of a ratio — What is the basis for "double"?State the denominator and period for ratios. Differences are in points, ratios are in %.
- Simple and Compound Interest — Putting a Price on TimeSimple interest is on the principal only; compound interest is also on the interest. Principal × (1 + annual rate) ^ number of years. Even a small difference in rates leads to a large difference in the amount over the years.
- Fees are an invisible priceFees compound. 1 − (1 − rate)^years. Add the fees for buying, holding, and exiting into a single rate.
- Questioning the graph — Axes, period, units, sourceA graph is a picture. Trust its direction only after checking four things: the vertical axis, the time period, whether it is nominal or real, and the source.
- The order for reading statements and contracts — total amount, period, cancellation, fees, counterpartyTotal amount, period, cancellation, fees, counterparty. Reading in this order helps you understand the overall weight before getting lost in the details.
- Reading official documents — Using a pension notice as an exampleWhat kind of document is it? Which format applies? Are the values nominal or real? Are there any errors? Where can it be verified? The procedure is the same, regardless of the country.
Readings (3)
- Before the beginning Overview: the world the 18th century BC Babylonia
A world where numbers always had a unit: "X grams of silver". Numbers without units did not exist. - The black ships and the outflow of gold Japan 1858 Shimoda, the American consulate
A sentence from a treaty that mixed the world of weight with the world of face value. The first lesson on nominal versus real value. - When temples were banks Japan the 7th–13th centuries The granaries of the provinces
The interest rate on a temple loan. An example of the price of time exceeding the principal within a year.
Read next
Household Finances, Housing, and Retirement Funds →What you learn next: Be able to summarize one's annual income and expenses on a single sheet, using the same categories as official statistics.
Time to read the text: about 11 min ／ What you can read: the course opening and contents are free; the full text is for those who have purchased
Price
After purchase, the full courses, the illustrated books, every country’s story and the numismatics examination are open to you.
See the price →