MAT-144 · Mathematical Reasoning Topic 07 · Taxes & Stocks
Lesson 04 · Owning a piece of the business

Stocks: shares, dividends, return.

A stock is a tiny ownership stake in a company. Its price moves up and down, and along the way the company may pay you dividends. The return on your investment is the sum of both pieces.

01Shares + price 02Dividends 03Return = gain + dividends
▸ THE HOOK

When a company sells shares of itself to the public, every buyer becomes a partial owner. Apple has about 15 billion shares outstanding; owning 100 of them makes you a 15-billionth owner of the company, entitled to your proportional sliver of its profits and a vote at its annual meeting. Most individual investors will never come close to a controlling stake in anything, but the math of ownership scales linearly: 100 shares is 100 times what 1 share is, and the return on a thousand-dollar investment scales accordingly.

Two things can happen between the day you buy a stock and the day you sell it. The price can move — up if the market thinks the company is worth more than it did when you bought, down if it thinks less. And the company may pay dividends — regular (usually quarterly) cash payments to shareholders, funded out of the company's profits. Your total return on the investment is the sum of both effects.

This lesson develops the share-price math, the dividend math, and the combined return-on-investment calculation. Lesson 5 will do the same for bonds (which are loans, not ownership), and Lesson 6 ties the two together with the question of how to mix them in a real portfolio.

Total return = capital gain + dividends.

A share is a unit of ownership in a corporation. Multiply the share count by the current price per share to get the dollar value of the position: 100 shares at $50 per share = $5,000.

Two sources of return are possible. The capital gain is the change in share price multiplied by the number of shares owned: if the price rises from $50 to $58, the capital gain on 100 shares is ($58 − $50) × 100 = $800. A capital gain is unrealized while you still hold the shares; it becomes realized (and triggers capital-gains tax) when you sell.

Dividends are regular cash payments the company makes to each shareholder, usually quarterly, on a per-share basis. If the company pays $0.50 per share each quarter for a year, the annual dividend per share is $2.00, and 100 shares receive $200 in dividends.

The total return on the year is the sum: capital gain + dividends. Divide by the original amount invested to get the return on investment (ROI) as a percent: ROI = total return / amount invested.
TOTAL RETURN = CAPITAL GAIN + DIVIDENDS $5,000 invested · one year later PURCHASE 100 shares @ $50 = $5,000 one year CAPITAL GAIN ($58 − $50) × 100 = +$800 price moved $50 → $58 per share DIVIDENDS $2 × 100 = +$200 company paid $2 per share in dividends total return $1,000 RETURN ON INVESTMENT $1,000 / $5,000 = 20%

A year of stock ownership decomposed. $5,000 buys 100 shares at $50. A year later the price is $58 (capital gain $800) and the company has paid $2/share in dividends ($200). Total return $1,000 on $5,000 = 20% ROI.

▸ DEFINITION

A share is a unit of corporate ownership. The capital gain on a position is (sale price − purchase price) × shares, and the dividend income is dividend per share × shares. The total return is the sum; the return on investment (ROI) is total return divided by amount invested, expressed as a percent.

Words you'll see on every stock quote

  • Share A single unit of ownership in a company. A typical public company has hundreds of millions or billions of shares outstanding; owning 100 is a small but meaningful (and tradable) position.
  • Share price The dollar amount one share trades for. Determined moment-to-moment by buyers and sellers in the open market. The share-price ticker on financial news websites is the most-watched number in finance.
  • Dividend A cash payment made by the company to shareholders, usually quarterly, on a per-share basis. Companies that pay dividends are typically more mature; younger, growth-focused companies often pay none and reinvest profits internally.
  • Capital gain (or loss) The change in the value of the position due to share-price movement, not including dividends. Realized when you sell the shares; until then, the gain is unrealized and not yet taxed.
  • Return on investment (ROI) Total return divided by amount invested, expressed as a percent. ROI lets you compare investments of very different sizes on the same scale: 20% on $5,000 and 20% on $5,000,000 are the same rate of return, even though the dollar gains differ by a thousandfold.

$5,000 in XYZ stock: one year of return.

Decompose a single year of stock ownership into its two return components, then combine for the total and ROI.

"You buy 100 shares of XYZ Corp at $50 per share. One year later, the share price has risen to $58, and during the year XYZ paid $0.50 per share each quarter in dividends. Compute the capital gain, the total dividend income, the total return, and the return on investment."

1

Compute the amount invested.

amount invested = 100 shares × $50 = $5,000
→ purchase day
2

Compute the capital gain.

Price moved from $50 to $58 — a rise of $8 per share. Multiply by share count:

capital gain = ($58 − $50) × 100 = $800
→ price-movement piece
3

Compute the dividend income.

$0.50 per share per quarter, four quarters in a year, equals $2.00 per share annually. Multiply by share count:

dividends = $2.00 × 100 = $200
→ cash-payments piece
4

Sum for total return.

total return = capital gain + dividends = $800 + $200 = $1,000
→ both pieces combined
5

Compute ROI.

Divide total return by amount invested:

ROI = $1,000 / $5,000 = 0.20 = 20%

A 20% one-year return is excellent, well above the long-run S&P 500 average of about 10% per year. The decomposition reveals that 80% of the return came from price movement and 20% from dividends — a typical mix for a dividend-paying stock in a good year.

→ the comparable rate

Three investments. Compute the return.

Same recipe. Some scenarios involve losses; watch the signs.
PROBLEM 01 ☆ ☆   warm-up · price-only gain

You bought 50 shares of ABC at $30 per share. A year later, the price is $36. ABC paid no dividends. What is your ROI?

ROI % =
PROBLEM 02 ★ ★ ☆   gain plus dividends

You bought 200 shares of DEF at $25. A year later the price is $28, and DEF paid $1.50 per share in dividends over the year. What is your total return in dollars?

return = $
PROBLEM 03 ★ ★ ★   the loss case

You bought 40 shares of GHI at $75. A year later the price has fallen to $68, but GHI paid $2.50 per share in dividends. What is your ROI (to one decimal place)?

ROI % =

Three fast questions before you move on.

Tap an answer. Feedback shows up immediately.

Q1. What does owning a share of a company entitle you to?

Why B? A share is an ownership stake. The company is not obligated to pay you anything — no guaranteed return, no fixed interest. A and C describe bonds (Lesson 5), which are loans rather than ownership. D is false for any stock; no return is guaranteed.

Q2. Your total return on a stock investment is the sum of which two components?

Why C? Two sources: the change in price (capital gain, which can be negative for a loss) and the dividends received during the holding period. D is the formula for amount invested, not return.

Q3. If a stock's price stays exactly the same from purchase to sale but the company pays $1.20 in dividends per share, is the investor's total return positive, negative, or zero?

Why C? Even when the price doesn't move, the dividends are still a real return to the shareholder. The total return is positive, equal to the dividend income. D is technically true that the dollar amount depends on share count, but the sign of the return is positive regardless.
▸ UP NEXT — LESSON 05

Owning vs lending.

A stock is a piece of the business itself. Owning shares ties your fortunes to the company's performance: when it does well, the price rises and dividends grow; when it does poorly, both fall, and there is no guarantee of recovery. Stocks have historically delivered higher long-run returns than safer alternatives (about 10% annually for the broad U.S. stock market since World War II), but with substantial year-to-year volatility — some years up 30%, others down 30%.

The math of this lesson scales transparently. 100 shares or 100,000 shares, $5,000 or $5,000,000 invested — the ROI percentage is the same as long as the per-share story is the same. That linearity is why ROI is the standard yardstick for comparing investments of different sizes.

Next: Lesson 5 introduces bonds, the lending counterpart of stocks. Where a stockholder owns part of the company and shares in its uncertain future, a bondholder has loaned the company money in exchange for predictable interest payments and a fixed return of principal at maturity. The math is different, and so is the risk profile.

Continue to Lesson 05

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