Finance, with the arithmetic left in.

Compounding

Returns earning returns. The curve does almost nothing, then almost everything.

Everyone knows compounding is powerful. Almost nobody internalises where the power sits. It is not spread evenly across the years — it is stacked violently at the end. That's why the boring middle years feel like failure, and why quitting at year 12 is the single most expensive thing you can do.

The practical consequence: time in the market is not one input among several. It is the input. A 25-year-old saving $200 a month will likely beat a 40-year-old saving $600 a month, and no amount of clever stock-picking closes that gap.

Savings rate

The share of income you don't spend. The only variable that really moves your date.

This is the one that surprises people. Your savings rate does two things at once: it fills the portfolio faster and it shrinks the finish line, because the finish line is a multiple of your spending. Spend less and you need less. It works from both ends, which is why it dominates investment returns.

Starting from zero, at a 5% real return, needing 25× annual expenses:

Read the jump from 10% to 20%: it doesn't halve the wait, it cuts it by 15 years. Note also what's not in that table — your income. A high earner saving 10% retires later than a modest earner saving 50%. The rate is what matters, not the salary.

The honest caveat: this assumes a constant income, a constant real return, and no life. Nobody's path is this smooth. It's a compass, not a schedule.

Your FI number & the 4% rule

The portfolio that covers your spending forever. Roughly 25× what you spend in a year.

The 4% rule came out of research into historical US portfolios: withdraw 4% of the starting balance, adjust for inflation each year, and you'd have survived most 30-year windows. Invert it and you get the target: divide by 4%, or multiply spending by 25.

Where it breaks: 4% is a US-centric backtest, not a law. It assumed a 30-year retirement — retire at 40 and you need the money to last 50+. It ignores taxes and fees. And it fails hardest in exactly the scenario below.

Real vs nominal returns

Nominal is the number on the statement. Real is what it buys.

Inflation is the opponent nobody plans against, because the loss never shows up as a line item. Note the maths is division, not subtraction — 7% minus 3% is not quite 4%.

The rule of 72

Divide 72 by the return to get the doubling time. Mental arithmetic, accurate enough.

Useful because it lets you sanity-check a claim in your head, in a conversation, without a spreadsheet.

Close enough to be worth memorising, and it works in reverse: if someone promises to double your money in three years, they're implying a ~24% annual return. Now you know what to ask them.

Sequence of returns risk

The order of returns is irrelevant while saving, and brutal once withdrawing.

This is the most under-taught idea in retirement planning. Two people can earn the identical average return over the identical decade and end up in different worlds — purely because of what happened first. Bad years early force you to sell more shares to fund the same spending, and those shares aren't there for the recovery.

Expense ratios

The annual % a fund takes. The only fee you can control with certainty.

A fee looks trivial because it's quoted as a decimal. It isn't trivial, because it compounds against you for the same thirty years your money compounds for you.

Dollar-cost averaging vs lump sum

Investing gradually vs all at once. One is mathematically better; the other is survivable.

If markets rise more often than they fall, lump sum wins on average — time in the market beats waiting. The evidence generally favours it. But this is a case where the maths and the human diverge:

• Lump sum maximises expected return and maximises regret if you buy the week before a crash.
• DCA gives up some expected return to buy an insurance policy against your own behaviour.

The right answer is whichever one you'll actually stick with. A theoretically optimal plan you abandon in month four returns less than a mediocre plan you keep for thirty years. Note also that most people never face the question — if you're investing from a salary, you're dollar-cost averaging by default.

Rebalancing

Selling what won, buying what lost, to hold your target allocation.

It feels wrong every single time, which is the point — it's a rule that forces you to sell high and buy low when your instincts scream the opposite.

The P/E ratio

Price divided by earnings per share. What you pay for $1 of annual profit.

A P/E alone tells you almost nothing — cheap is often cheap for a reason. The useful move is to flip it. One divided by the P/E gives the earnings yield, which puts a stock on the same footing as a bond and makes the comparison concrete.

At 35×, you're accepting a 2.86% earnings yield today because you believe those earnings grow fast. That's the actual bet. The multiple just hides it.

Free cash flow vs net income

Net income is an opinion. Free cash flow is closer to a fact.

Net income is the output of accounting choices — depreciation schedules, write-offs, revenue recognition. Free cash flow is operating cash minus capital spending: money that actually arrived and stayed. It's what pays dividends, buybacks and debt.

When the two diverge for years, ask why. Cash flow far above net income can be healthy (heavy depreciation on assets bought long ago). Net income far above cash flow, persistently, is the classic warning sign.

Buybacks and EPS

Fewer shares, same profit, higher earnings per share. Growth without growing.

This is the mechanic behind a huge share of reported EPS growth, and most people never see it.

Buybacks are an investment decision, and like any investment the return depends on the price paid. Retiring shares at 10× earnings is excellent. Doing it at 40× destroys value while still making the EPS line look good — which is exactly why some management teams do it.

Shareholder yield

Dividends plus net buybacks, over market cap. The full picture of cash returned.

Dividend yield alone badly understates modern capital return, because buybacks have largely replaced dividends as the way US companies hand cash back. A company with a 1% dividend can be returning 9%.

Payout ratio

The share of earnings paid as dividends. A safety gauge, not a yield gauge.

The question a payout ratio answers isn't "how much am I paid" — it's "how likely is this to survive a bad year, and is there room to grow it?"

• Under ~40%: comfortable. Room to raise, room to absorb a downturn.
• 60–80%: mature. The dividend is roughly the whole story.
• Over 100%: paying out more than it earns. Funded by debt or asset sales. Ask how long that can continue.

A 7% yield with a 110% payout ratio is not a 7% yield. It's a countdown to a cut, and the market usually knows — which is why the yield is 7% in the first place.