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Every guide we have published, newest first. Each one shows its arithmetic and states its assumptions, so you can check the conclusion with your own numbers.

24 guides

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Free Cash Flow vs Net Income: Why They Differ

Both numbers get called profit, and they can differ by 50 percent or more in the same year. The four wedges that drive them apart, a full worked example, and what each red flag actually indicates.

A person at a dim desk studying a large screen showing a long rising and falling line chart
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Recency Bias: Why Yesterday's Returns Mislead You

Recency bias makes the last two years of market history feel like the next ten. Work through the arithmetic of buying high and selling low, see the base-rate data on forecasts, and build a checklist that survives both bull and bear markets.

A mechanical balance scale with coins on one side and a folded document on the other
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The Price-to-Book Ratio: What It Measures and When

Price-to-book compares a stock price to the accounting value of what the company owns. Here is the formula, two fully worked examples, why P/B suits banks and breaks on software companies, and where it fits among valuation metrics.

A financial newspaper-style layout of blurred stock charts on a desk beside a calculator and reading glasses
8 min read

Stock Buybacks, Explained With Simple Math

When a company buys back its own shares, your slice of it grows — but only if the price was fair. The EPS math of buybacks, when they help, when they waste money, and where to find them in filings.

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9 min read

Changing Jobs? The Math of Your 401(k) Options

Leave it, roll it to an IRA, move it to the new plan, or cash it out — the four 401(k) options compared with actual numbers, including what cashing out a $60,000 balance really costs.

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Target-Date Funds: How They Actually Work

The one-fund retirement default explained: what a glide path does, what target-date fees cost over 30 years with exact math, and the blind spots worth knowing.

A calculator, a pen, and a notebook with hand-drawn simple charts on a wooden desk in soft daylight, a cup of tea nearby
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Dividend Payout Ratio: Is the Dividend Safe?

A high yield is only half the story. How the payout ratio — from earnings and from free cash flow — separates sustainable dividends from future cut candidates.

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8 min read

Rebalancing Bands: The Math of When to Rebalance

Rebalance on a schedule or on a threshold? A worked $100,000 example shows what a 5-percentage-point band actually triggers, how much the discipline was worth in one bull-then-bust sequence, and why frequency alone answers the wrong question.

A stethoscope resting on a financial document next to a calculator on a clean desk, bright natural light, realistic photo style
8 min read

The HSA Triple Tax Advantage, Explained With Math

No other account deducts money on the way in, shelters growth, and waives tax on the way out. A fully worked 30-year example shows what the three breaks are worth together — and the one mistake that throws most of the value away.

An older couple sitting together at a sunlit dining table looking at a laptop, financial documents and a notebook spread out before them
11 min read

Sequence-of-Returns Risk: Why the Same Average Can End Two Ways

Two portfolios can earn the same average return and end with very different balances, because the order of those returns decides how much a withdrawal costs. This guide runs the same sequence two ways and shows how much the ordering alone is worth.

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12 min read

Reading a Cash Flow Statement: Where the Money Actually Went

Profit is an accounting opinion; cash is a fact. This walkthrough builds a cash flow statement line by line from a fictional company, shows why a profitable business can still run short of cash, and sets out the three tests worth running before trusting a reported profit.

A person reviewing printed account statements at a kitchen table with a pen and a calculator nearby
11 min read

Capital Gains Tax When You Sell: The Arithmetic of a Sale

A gain is not taxed when the price rises, only when the position is sold — and the rate applied depends on how long it was held. This walkthrough covers cost basis, holding periods, loss offsets, and the after-tax arithmetic of a single sale.

A printed fee table and a pair of reading glasses on a wooden desk beside a calculator in soft window light
11 min read

What an Expense Ratio Actually Costs You: The Arithmetic

A 1% fee looks like a rounding error beside a 6% return. Run both through 30 years of compounding and the fee removes roughly 24% of the ending balance. Here is the arithmetic, worked in full.

An open annual report lying flat on a desk with a printed balance sheet visible beside a pen and a coffee cup
11 min read

How to Read a Balance Sheet: Assets, Liabilities, Equity

A balance sheet is a snapshot rather than a story: what a company owns, what it owes, and what is left for shareholders. A worked example, the ratios that fall out of it, and what it cannot tell you.

Coins arranged in neat columns beside a printed share certificate and a fountain pen on a dark wooden table
12 min read

Dividend Yield and Total Return: What You Actually Get

A dividend is not extra money on top of the share price — the price adjusts by roughly the amount paid. How yield, payout ratio and total return fit together, worked out step by step.

A printed interest rate chart and a coffee cup on a desk beside a computer screen showing a yield curve in daylight
11 min read

Why Bond Prices Fall When Interest Rates Rise

The most counterintuitive line in finance, explained with a bond you can price on a calculator. Includes the full discounting arithmetic, a duration shortcut, and why bond funds behave differently from bonds held to maturity.

An analyst comparing printed financial tables with a pencil and a hand calculator on a cluttered wooden desk
11 min read

What the P/E Ratio Actually Tells You, and What It Hides

The price-to-earnings ratio is the most quoted number in investing and one of the least understood. Here is the arithmetic behind it, one fully worked example, and the four situations where it quietly stops meaning anything.

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12 min read

Roth vs Traditional: The Tax Math That Decides It

The Roth versus traditional question is not about which account is better. It is about the tax rate you face now versus the one you face later. Here is the arithmetic that turns a debate into a single comparison.

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11 min read

The Behavior Gap: Why Investor Returns Lag Fund Returns

A fund can earn 8% a year while the people who own it earn 6%. The difference is called the behavior gap, and it comes from when money arrives rather than what it buys. Here is the arithmetic behind it, and the specific habits that close it.

An open annual report with printed financial tables on a desk beside a calculator and a cup of coffee in warm afternoon light
11 min read

How to Read an Income Statement, Line by Line (With a Worked Example)

Revenue, gross profit, operating income, net income, EPS — an income statement is one subtraction chain read top to bottom. Here is what each line measures, which ratios matter, and the three places the numbers can be framed to mislead.

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11 min read

Asset Allocation: What Your Stock and Bond Split Actually Does

The stock/bond split is the one decision that changes your results more than any fund pick. Here is the arithmetic of expected return, the arithmetic of a bad year, and a fully worked rebalancing example — with the assumptions stated at every step.

A financial newspaper and reading glasses resting on a wooden desk beside a window with soft morning light
10 min read

How to Start Index Fund Investing: A Plain-Arithmetic Walkthrough

Index investing is popular because the arithmetic is hard to argue with, not because it is exciting. This walkthrough covers what an index fund actually is, what fees do over decades, and the exact steps from brokerage account to first purchase.

Stock charts and green and red figures glowing on a trading terminal screen in a dimly lit room
9 min read

How to Read a Stock Quote: Every Number on the Screen, Explained

A stock quote packs a dozen numbers onto one screen and most people ignore half of them. Here is what price, bid-ask spread, volume, market cap, P/E, dividend yield and the 52-week range each actually tell you.

A calculator, coins stacked in growing columns and a printed savings schedule on a clean desk
10 min read

Dollar-Cost Averaging vs. Lump Sum: What the Numbers Actually Show

You have money to invest and one decision to make first: all at once, or in installments? The historical data favors lump sum more often than intuition suggests — but the reasons people still choose DCA are rational, and here is the arithmetic on both sides.