Educational YouTube channel · money math
The world of money runs on math.
Every dollar you save, borrow or invest follows a formula. MoneyMathician works it out on screen — compound interest, the rule of 72, APR vs APY, what inflation does to a dollar, how a 1% fee compounds, and why average returns mislead.
What this channel does
One piece of money math per video, shown rather than asserted. The point is not that you take our word for it — it is that you can follow the sum and check it yourself.
Derived, not stated
The formula appears out of the picture, with its variables named and the decisive step worked through. No result drops out of nowhere.
Plain dollars, round rates
Examples use amounts you can put into a calculator. Every figure on screen comes from the same model that draws the animation, so the picture and the number cannot drift apart.
No tickers, no advice
This channel explains the sums; you make the decisions. No stock picks, no products, no market news — the math is the same next year.
Who it’s for
You have money in an account or a fund, you have run into a number somewhere — “7% on average”, “1% in fees”, “the 4% rule” — and you would like to know why it holds, or why it doesn’t. No background beyond school arithmetic is assumed.
You’ll feel at home here if…
- you would rather understand a rule of thumb than memorise it
- you want to see where a number comes from before you trust it
- you like a formula better once someone has drawn it
- you are saving, paying off a loan, or looking at a pension projection
What you won’t find
- stock picks, tickers, or “what you should do with your money”
- market news, rate decisions, or commentary that expires
- get-rich framing, or a secret the pros supposedly hide
- numbers without the calculation that produced them
Episodes
Every episode stands on its own. Watching them in order builds the toolkit faster.
Simple vs Compound Interest: Why the Curve Bends
Same $10,000, same 5%: year one earns $500, year thirty earns $2,058. We draw the line and the curve side by side, take the growth apart, and pull A = P(1 + r)ⁿ out of the picture.
Rule of 72 Explained: Where the 72 Comes From
Divide 72 by your rate and you get the years until your money doubles — 9 years at 8%. But 72 is not a magic number: it is really 69.3, nudged up. We climb the staircase, pull n = ln 2 / ln(1 + r) out of the picture, and show which number actually fits.
APR vs APY: The Compounding Fine Print
Two banks, the same 5% in the window — and after one year the jars are not equal. The rate they multiply by is the APR; the rate your jar actually reaches is the APY. Cut the year into pieces and every piece that lands early starts earning too: APY = (1 + r ÷ m)^m − 1. Twelve dollars and sixty-seven cents after one year, almost sixteen hundred after thirty.
Real vs Nominal Return: What Inflation Takes
Same jar as last time: 5% on the sign, 5.13% more after a year — but the price tag on the shelf climbed too. Put the jar on the basket scale and it buys only about 2% more. Nominal is the raw number on the statement; real is the same jar with inflation taken out: real = (1 + n) ÷ (1 + p) − 1. After thirty years the statement says $44,812 and the shelf says $18,462.
Time Value of Money: What $1,030 Is Worth Today
Your sister can pay you back $1,000 tonight or $1,030 next year. If your jar pays 3%, those two offers are worth exactly the same today — a dead tie. So the answer comes from the three things that make money now worth more than money later, and each one gets a price on that one question: opportunity cost puts her offer $19.05 short, inflation says the waiting earns nothing real, and one chance in twenty that it never comes puts it $50 short. The tool underneath is present value: PV = FV ÷ (1 + r)ⁿ — you divide, you never subtract.
Annuities: The Present Value of an Income Stream
A letter says you won $1,200,000, paid as $120,000 a year for ten years, or take the cash option today. Ten times $120,000 is $1.2 million, so the cash should be $1.2 million too — except a check ten years from now is not a check this morning. Bring each one back at 5% and the row is worth $926,608, over a quarter of a million short. The short way is shorter still: a row that never ends is worth a pile that pays it out forever, $120,000 ÷ 0.05 = $2,400,000, and ten years is simply forever with the tail cut off. That subtraction is the annuity formula every textbook prints, PV = P/r × [1 − 1/(1+r)ⁿ], and it lands on the same $926,608.19 to the cent.
The Asymmetry of Losses: The Gain You Need to Recover
Max's account is down by half, so it needs half back, right? Say it held $10,000 and dropped to $5,000: half back is $2,500, and the app shows $7,500 — a quarter short, and nobody took it. Put the loss and the gain on a seesaw in dollars and the plank only levels when $5,000 comes back; on $5,000 that is 100%. Read the rule straight off the plank: the loss side holds L of the whole, the gain side g of what remains, level means they weigh the same, so g = L / (1 − L). A tenth down needs 11.1% back, a half needs 100%, nine tenths needs 900%. And at 7% a year, doubling takes about ten years and growing tenfold takes thirty-four.
Investment Fees Over Time: What 1% Really Costs You
One line on the form says: annual fee, 1%. A penny on the dollar, so on $1,000 that is $10 a year, and $300 over thirty years. That sum is off by more than six times. The fee is 1% of everything in the account, not of the gain: in year one it takes $10.70 of the $70 your $1,000 earned. It grows as the account grows: on $10,000 it is $107 in year one and $569 in year thirty, about $8,400 in all. And every dollar that leaves as a fee would have kept growing: on $100,000 the account ends $198,147 short of the same account with no fee, while the fees paid add up to $83,557. The rule that falls out: what you keep is one minus the fee, to the power of the years. With a 1% fee for thirty years that is 74%, whatever the market returns and whatever you started with.
Why Average Returns Mislead: Arithmetic vs Geometric Mean
Say a fund goes up 60%, then down 40%. Add the two and divide by two: plus 10% a year, so $1,000 should have become $1,210. The statement says $960. A return does not add to your money, it multiplies it: 1.6 × 0.6 = 0.96, and the one steady rate that lands on the same balance, the geometric mean, is about minus 2% a year. Draw the two years as a rectangle and both averages appear at once: adding the returns keeps the perimeter (a square of 1.1, the $1,210), the geometric mean keeps the area (your $960), and the empty corner is half the swing, squared. On 98 years of the S&P 500 with dividends (1928–2025, A. Damodaran, NYU Stern) the average year was +11.9% while the money grew 10.0% a year: $100 became about $1.16 million, not the $5.86 million the average describes.
In the works
- Volatility drag
- Time-weighted vs money-weighted return
Come for one formula, stay for the rest.
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