Chapter 21

Stocks, Bonds, Mutual Funds, and Investment Mathematics

Dividends, yields, total return, bond prices, and fund costs

Investing apps show prices, yields, and percentages all day. This chapter teaches the math behind those numbers so you can read them accurately. Every company, fund, price, and rate here is hypothetical. Nothing in this chapter is investment advice, and no return is ever guaranteed.

21.1

Stocks: Dividends, Yield, and Total Return

Learning objectives

After this section you will be able to:

Calculate the cost of a stock purchase and annual dividend income.
Find dividend yield and the price-to-earnings ratio.
Compute total return, including both price change and dividends.

Real-life situation

A first brokerage account

A 19-year-old in Philadelphia buys 50 shares of a hypothetical company, Riverside Goods, at $40 per share. The company pays a $0.30 dividend every quarter. A year later the share price is $44.

Item	Value
Shares	50
Purchase price	$40.00
Quarterly dividend per share	$0.30
Price one year later	$44.00
Earnings per share (EPS)	$2.50

Figure 21.1 Hypothetical stock position after one year

The concept

A share of stock is a small piece of ownership in a company. The cost of a purchase is the number of shares times the price per share, plus any trading fee. Many apps now charge no commission, but always check.

Some companies pay part of their profits to shareholders as dividends, often quarterly. Dividend yield compares the yearly dividend to the current share price. Dividends are not promised; a company can cut or stop them.

Total return measures everything you earned: the change in price plus any dividends received, divided by what you paid. A stock can rise in price and still disappoint, or fall in price while dividends soften the loss.

The price-to-earnings (P/E) ratio divides the share price by earnings per share. It shows how many dollars investors pay for each $1 of yearly profit. A high or low P/E is a starting question, not a buy or sell signal.

Stocks: Dividends, Yield, and Total Return: solution map

1
Identify the question
Using Figure 21.1, find the purchase cost (no fee), the year's dividend income, and the dividend yield at the $40 purchase price.
2
Calculation 1
Cost: 50 × $40 = $2,000.
3
Calculation 2
Annual dividend per share: 4 × $0.30 = $1.20.
4
Calculation 3
Dividend income: 50 × $1.20 = $60.
5
Calculation 4
Yield: $1.20 ÷ $40 = 0.03 = 3%.
6
Interpret and check
Cost $2,000; dividends $60; yield 3%

Figure 21.1.V Stocks: Dividends, Yield, and Total Return: solution map. Hypothetical teaching data.

Money Words

Plain-language words you will use in this lesson — learn them before we reach the formulas.

stock
A share of ownership in a company.
bond
A loan to a government or company that pays interest and repays the face amount at maturity.
mutual fund
A pool of money from many investors used to buy a mix of stocks, bonds, or both.

Key terms

Share: One unit of ownership in a company.
Dividend: A payment of company profits to shareholders.
Dividend yield: Annual dividend per share ÷ price per share.
Total return: Price change plus income, as a percent of the amount invested.
Earnings per share (EPS): A company's yearly profit divided by its number of shares.

Formulas & rules

Cost of purchase
Cost = shares × price per share + fees
Dividend yield
Yield = annual dividend per share ÷ price × 100%
Total return
Total return = (ending price − starting price + dividends) ÷ starting price × 100%
P/E ratio
P/E = price per share ÷ earnings per share

Worked examples

Example 1Cost, dividends, and yield

Using Figure 21.1, find the purchase cost (no fee), the year's dividend income, and the dividend yield at the $40 purchase price.

Solution

1Cost: 50 × $40 = $2,000.
2Annual dividend per share: 4 × $0.30 = $1.20.
3Dividend income: 50 × $1.20 = $60.
4Yield: $1.20 ÷ $40 = 0.03 = 3%.

Answer: Cost $2,000; dividends $60; yield 3%

Example 2Total return

Find the price return and total return for one share held for the year.

Solution

1Price change: $44 − $40 = $4.
2Price return: $4 ÷ $40 = 10%.
3Add dividends: $4 + $1.20 = $5.20 per share.
4Total return: $5.20 ÷ $40 = 0.13 = 13%.
5For 50 shares: 50 × $5.20 = $260 total gain.

Answer: Price return 10%; total return 13% ($260 on the position)

Example 3When the price falls

Another investor buys 20 shares at $25, receives $0.50 per share in dividends, and the price drops to $21. Find the total return.

Solution

1Cost: 20 × $25 = $500.
2Price change for the position: 20 × ($21 − $25) = −$80.
3Dividends: 20 × $0.50 = $10.
4Total return: (−80 + 10) ÷ 500 = −70 ÷ 500 = −14%.

Answer: −14%; the dividends reduced, but did not erase, the loss

Practice exercises 21.1

Find the cost of 30 shares at $60 with a $0 fee.
A stock priced at $50 pays $0.50 per quarter. Find its dividend yield.
You buy at $15, receive $0.40 in dividends, and the price rises to $18. Find total return to the nearest hundredth of a percent.
A share costs $48 and EPS is $3. Find the P/E ratio and explain it.
Using Figure 21.1's EPS of $2.50, find the P/E at the $40 and $44 prices.
A $1,000 position loses 20% in price and pays $15 in dividends. What is it worth including dividends?

Section 21.1 recap

Yield = annual dividend ÷ price; P/E = price ÷ EPS.
Total return = (price change + dividends) ÷ starting price.
Dividends and price gains are never guaranteed.

Answer key with explanations: Section 21.1

Practice exercises

30 × $60 = $1,800.
Annual dividend = 4 × $0.50 = $2.00; yield = 2 ÷ 50 = 4%.
(18 − 15 + 0.40) ÷ 15 = 3.40 ÷ 15 ≈ 22.67%.
48 ÷ 3 = 16. Investors are paying $16 for each $1 of yearly earnings.
40 ÷ 2.50 = 16; 44 ÷ 2.50 = 17.6.
Price value 1,000 × 0.80 = $800; plus $15 = $815, a total return of −18.5%.

Recap quiz

4%: 1 ÷ 25 = 0.04 = 4%.
Price change and dividends: Total return adds income to the price change.
15: 30 ÷ 2 = 15.

21.2

Bonds: Coupon Payments, Par, and Current Yield

Learning objectives

After this section you will be able to:

Calculate annual and semiannual coupon payments.
Convert bond price quotes to dollars.
Find current yield and explain why bond prices move opposite to interest rates.

Real-life situation

A city bond quote

A hypothetical city issues bonds with a $1,000 face value and a 5% coupon, paid twice a year. Months later, interest rates for new bonds have risen, and the bond is quoted at 96.

City of Example 5% bond

Face (par) value: $1,000
Coupon rate: 5.00% annual, paid semiannually
Quote: 96.00 (percent of par)
Years to maturity: 10
All terms hypothetical

Figure 21.2 Hypothetical bond quote

The concept

A bond is a loan from an investor to a government or company. The issuer promises to pay interest, called the coupon, and to repay the face value, called par, at maturity.

The coupon is figured on par, not on the price you pay. A 5% coupon on a $1,000 bond pays $50 a year, or $25 every six months if paid semiannually.

Bond prices are quoted as a percent of par. A quote of 96 means 96% of $1,000, or $960. A bond below par trades at a discount; above par, at a premium.

When new bonds pay higher rates, older bonds with lower coupons become less attractive, so their prices fall. When rates drop, older bonds' prices rise. Current yield, the annual coupon divided by the price, reflects this. Bonds can also default, so higher yields usually come with more risk.

Read a bond quote

1
Par and quote
$1,000 par; quote 96.00 → $960 price
2
Annual coupon
5% × $1,000 = $50; $25 semiannually
3
Current yield
$50 ÷ $960 ≈ 5.21%; not yield to maturity

Figure 21.2.V Read a bond quote. Hypothetical teaching data.

Money Words

Plain-language words you will use in this lesson — learn them before we reach the formulas.

dividend
A share of a company's profit paid to its stockholders.
yield
The income an investment pays, shown as a percent of its price.
capital gain
Profit made by selling an investment for more than it cost.

Key terms

Bond: A loan to an issuer that pays interest and returns the face value at maturity.
Par (face) value: The amount repaid at maturity, often $1,000.
Coupon rate: The yearly interest rate paid on par value.
Discount / premium: Price below par / price above par.
Current yield: Annual coupon ÷ current price.

Formulas & rules

Annual coupon
Coupon = par × coupon rate
Price from quote
Price = quote ÷ 100 × par
Current yield
Current yield = annual coupon ÷ price × 100%

Worked examples

Example 1Coupons and current yield at a discount

For the bond in Figure 21.2, find the annual coupon, each semiannual payment, the price, and the current yield.

Solution

1Annual coupon: $1,000 × 0.05 = $50.
2Semiannual payment: $50 ÷ 2 = $25.
3Price: 96 ÷ 100 × $1,000 = $960.
4Current yield: $50 ÷ $960 ≈ 0.05208 ≈ 5.21%.

Answer: $50 per year ($25 twice); price $960; current yield ≈ 5.21%

Example 2A premium bond

Rates later fall and the same bond is quoted at 104. Find the price and current yield.

Solution

1Price: 104 ÷ 100 × $1,000 = $1,040.
2The coupon does not change: $50.
3Current yield: $50 ÷ $1,040 ≈ 0.04808 ≈ 4.81%.

Answer: Price $1,040; current yield ≈ 4.81%, below the 5% coupon

Practice exercises 21.2

A $1,000 bond has a 4% coupon paid semiannually. Find each payment.
That 4% bond is priced at $950. Find its current yield.
What do 10 bonds quoted at 98 cost (par $1,000, no fees)?
A 6% bond with $1,000 par value is priced at $1,020. Find the current yield.
How much coupon income does one 5% $1,000 bond pay over 3 years?
Interest rates on new bonds rise. What usually happens to the price of an existing bond, and why?

Section 21.2 recap

Coupons are based on par; prices are quoted as a percent of par.
Current yield = annual coupon ÷ price. Prices and rates move in opposite directions.

Answer key with explanations: Section 21.2

Practice exercises

Annual 1,000 × 0.04 = $40; semiannual $40 ÷ 2 = $20.
40 ÷ 950 ≈ 4.21%.
Each: 0.98 × 1,000 = $980; 10 × 980 = $9,800.
Annual coupon 1,000 × 0.06 = $60; 60 ÷ 1,020 ≈ 5.88%.
$50 × 3 = $150.
It falls, because its fixed coupon is less attractive than new bonds' higher payments.

Recap quiz

Fall: Older, lower coupons become less attractive.
$1,020: 1.02 × 1,000 = $1,020.
Annual coupon ÷ price: It compares income to what you pay now.

21.3

Mutual Funds and ETFs: NAV, Expenses, and Diversification

Learning objectives

After this section you will be able to:

Calculate net asset value (NAV) per share and shares purchased.
Find the yearly cost of an expense ratio and a sales load.
Compute a weighted portfolio return to see the effect of diversification.

Real-life situation

Comparing two index funds

A student has $10,000 saved from summer jobs and a small creator side income. Two hypothetical funds track the same broad market. One charges 0.50% a year; the other charges 0.05%.

Fund	Expense ratio	Yearly cost on $10,000
Fund A	0.50%	$50.00
Fund B	0.05%	$5.00

Figure 21.3 Two hypothetical funds holding the same stocks

The concept

A mutual fund or exchange-traded fund (ETF) pools many investors' money to buy many stocks or bonds at once. Owning one share gives a small slice of everything the fund holds.

Net asset value (NAV) per share is the fund's total assets minus its liabilities, divided by the number of shares. Mutual funds are bought at NAV once per day; ETFs trade all day at market prices close to NAV.

Funds charge a yearly expense ratio, taken out of the fund's assets. Some mutual funds also charge a sales load when you buy. Because costs are certain and returns are not, small differences in fees add up over many years.

Diversification means spreading money across many investments so one bad result hurts less. A portfolio's return is the weighted average of its parts. Diversification reduces risk, but it cannot eliminate losses when the whole market falls.

Mutual Funds and ETFs: NAV, Expenses, and Diversification: solution map

1
Identify the question
A fund holds $52,000,000 in assets, owes $2,000,000, and has 2,500,000 shares. Find NAV, then the number of shares $1,500 buys.
2
Calculation 1
Net assets: 52,000,000 − 2,000,000 = $50,000,000.
3
Calculation 2
NAV: 50,000,000 ÷ 2,500,000 = $20.00.
4
Calculation 3
Shares bought: $1,500 ÷ $20 = 75 shares.
5
Interpret and check
NAV $20.00; 75 shares

Figure 21.3.V Mutual Funds and ETFs: NAV, Expenses, and Diversification: solution map. Hypothetical teaching data.

Money Words

Plain-language words you will use in this lesson — learn them before we reach the formulas.

diversification
Spreading money across different investments to reduce risk.
portfolio
The full collection of investments a person or fund owns.
risk
The chance that an investment or decision will turn out worse than expected.

Key terms

Mutual fund / ETF: A pooled investment that holds many securities.
NAV: Net asset value per share: (assets − liabilities) ÷ shares.
Expense ratio: The yearly fee as a percent of the money invested.
Sales load: A commission charged by some funds when shares are bought.
Diversification: Spreading money across many investments to reduce risk.

Formulas & rules

NAV
NAV = (total assets − liabilities) ÷ shares outstanding
Expense cost
Yearly cost = amount invested × expense ratio
Weighted return
Portfolio return = w₁r₁ + w₂r₂ + …
Weights are decimals that add to 1.

Worked examples

Example 1Finding NAV and shares bought

A fund holds $52,000,000 in assets, owes $2,000,000, and has 2,500,000 shares. Find NAV, then the number of shares $1,500 buys.

Solution

1Net assets: 52,000,000 − 2,000,000 = $50,000,000.
2NAV: 50,000,000 ÷ 2,500,000 = $20.00.
3Shares bought: $1,500 ÷ $20 = 75 shares.

Answer: NAV $20.00; 75 shares

Example 2The cost of fees

Using Figure 21.3, find each fund's first-year cost on $10,000 and the difference.

Solution

1Fund A: 10,000 × 0.0050 = $50.
2Fund B: 10,000 × 0.0005 = $5.
3Difference: $50 − $5 = $45 in the first year, before any growth.

Answer: Fund A costs $45 more per year for the same holdings

Example 3A diversified portfolio

A portfolio is 60% in a stock fund that returned 8% and 40% in a bond fund that returned −2%. Find the portfolio return.

Solution

1Stock part: 0.60 × 8% = 4.8%.
2Bond part: 0.40 × (−2%) = −0.8%.
3Total: 4.8% − 0.8% = 4.0%.

Answer: 4.0%

Practice exercises 21.3

A fund has $30,000,000 in assets, $1,000,000 in liabilities, and 1,450,000 shares. Find the NAV.
Find the yearly expense on $5,000 at a 0.75% expense ratio.
How many shares does $1,000 buy at a NAV of $25?
A portfolio is 50% at +6%, 30% at +2%, and 20% at −5%. Find its return.
A fund charges a 5% front-end load. How much of a $2,000 purchase is actually invested?
Why can't diversification guarantee you won't lose money?

Section 21.3 recap

NAV = (assets − liabilities) ÷ shares. Fees are certain; returns are not.
Portfolio return is a weighted average. Diversification lowers risk but cannot remove it.

Answer key with explanations: Section 21.3

Practice exercises

(30,000,000 − 1,000,000) ÷ 1,450,000 = 29,000,000 ÷ 1,450,000 = $20.00.
5,000 × 0.0075 = $37.50.
1,000 ÷ 25 = 40 shares.
0.5(6) + 0.3(2) + 0.2(−5) = 3 + 0.6 − 1 = 2.6%.
Load 2,000 × 0.05 = $100; invested 2,000 − 100 = $1,900.
When most investments fall together, as in a market-wide drop, a spread-out portfolio still loses value; diversification only reduces the damage from any single holding.

Recap quiz

$20: 20,000 × 0.001 = $20.
(Assets − liabilities) ÷ shares: Subtract liabilities before dividing by shares.
5%: 0.5 × 10 + 0.5 × 0 = 5%.

21.4

Return on Investment (ROI) and Annualized Return

Learning objectives

After this section you will be able to:

Calculate net gain and return on investment (ROI) for a purchase, project, or investment.
Include every cost and fee before computing ROI, and interpret negative ROI.
Convert a multi-year ROI to a simple and a compound annual rate so different time periods can be compared fairly.

Real-life situation

Is the camera kit worth it?

A 17-year-old content creator buys a hypothetical $600 camera and lighting kit. Over the next year, better-looking videos bring in $900 of extra sponsorship pay, and she spends $60 on editing software for the new footage. Was the kit a good investment?

Item	Amount
Cost of kit (investment)	$600
Extra sponsorship pay	$900
Extra costs (software)	$60
Time period	1 year

Figure 21.4 Hypothetical one-year results of the camera kit

The concept

Return on investment (ROI) answers one question: for every dollar I put in, how much did I gain or lose? It works for stocks, a business purchase, a side hustle, or equipment.

First find the net gain: everything the investment brought in, minus every cost tied to it, including the original investment, fees, shipping, and extra expenses. Leaving out a cost makes ROI look better than it really was.

ROI is the net gain divided by the amount invested, written as a percent. A negative ROI means you lost money. ROI says nothing about risk or time on its own, so a large ROI earned over many years can be worse than a smaller one earned in one year.

To compare periods fairly, convert to a yearly rate. The simple annual rate divides ROI by the number of years. The compound annual rate, (1 + ROI)^(1/years) − 1, is the steady yearly growth that produces the same result and is the fairer comparison.

Return on Investment (ROI) and Annualized Return: solution map

1
Identify the question
Using Figure 21.4, find the net gain and ROI of the camera kit.
2
Calculation 1
Net gain: $900 − $600 − $60 = $240.
3
Calculation 2
ROI: $240 ÷ $600 = 0.40.
4
Calculation 3
As a percent: 0.40 × 100% = 40%.
5
Interpret and check
Net gain $240; ROI 40%

Figure 21.4.V Return on Investment (ROI) and Annualized Return: solution map. Hypothetical teaching data.

Money Words

Plain-language words you will use in this lesson — learn them before we reach the formulas.

ROI
Return on investment — the gain compared to what you put in, as a percent: (gain ÷ cost) × 100.
return
The money or gain an investment produces, often shown as a percent.
net profit
The final profit after every cost is paid — sometimes called the bottom line.

Key terms

Return on investment (ROI): Net gain ÷ amount invested, as a percent.
Net gain: Total money returned minus all costs, including the original investment.
Annualized return: A multi-year return restated as a rate per year.

Formulas & rules

Net gain
Net gain = total returned − amount invested − other costs
ROI
ROI = net gain ÷ amount invested × 100%
Simple annual ROI
Simple annual ROI = ROI ÷ years
Compound annual rate
Annual rate = (1 + ROI)^(1/years) − 1

Worked examples

Example 1ROI on equipment

Using Figure 21.4, find the net gain and ROI of the camera kit.

Solution

1Net gain: $900 − $600 − $60 = $240.
2ROI: $240 ÷ $600 = 0.40.
3As a percent: 0.40 × 100% = 40%.

Answer: Net gain $240; ROI 40%

Example 2Comparing different time periods

Option A turns $2,000 into $2,300 in 1 year. Option B turns $5,000 into $5,600 over 2 years. Compare their ROI and their yearly rates.

Solution

1A: net gain $2,300 − $2,000 = $300; ROI = 300 ÷ 2,000 = 15% for one year.
2B: net gain $5,600 − $5,000 = $600; ROI = 600 ÷ 5,000 = 12% over two years.
3B simple annual: 12% ÷ 2 = 6% per year.
4B compound annual: (1.12)^(1/2) − 1 = 1.0583 − 1 ≈ 5.83% per year.
5B earned more dollars, but A grew faster each year.

Answer: A: 15% per year. B: 12% total, about 5.83% per year compounded

Example 3When fees make ROI negative

A reseller buys 4 pairs of sneakers at $180 each and sells them for $200 each. The selling app keeps a 10% fee and shipping costs $12 per pair. Find the ROI.

Solution

1Invested: 4 × $180 = $720.
2Sales: 4 × $200 = $800. Fee: 0.10 × $800 = $80. Shipping: 4 × $12 = $48.
3Net gain: $800 − $720 − $80 − $48 = −$48.
4ROI: −48 ÷ 720 ≈ −0.0667 = −6.67%.

Answer: About −6.67%; a $20 markup per pair became a loss after fees and shipping

Practice exercises 21.4

You spend $250 on supplies for a bake sale and take in $400. Find the ROI.
A $1,200 investment is worth $1,080 a year later. Find the ROI.
An $800 investment earns an ROI of 25%. Find the net gain and the ending value.
$3,000 grows to $3,630 over 2 years. Find the ROI, the simple annual ROI, and the compound annual rate.
Project A gains $90 on $300. Project B gains $200 on $1,000. Which has the higher ROI, and which earns more dollars?
Why can't you compare a 30% ROI earned over 3 years directly with a 15% ROI earned in 1 year?

Section 21.4 recap

ROI = net gain ÷ amount invested; subtract every cost first.
Negative ROI means a loss. ROI alone ignores time and risk.
Compare different time periods with an annual rate: (1 + ROI)^(1/years) − 1.

Answer key with explanations: Section 21.4

Practice exercises

Net gain 400 − 250 = $150; ROI = 150 ÷ 250 = 60%.
Net gain 1,080 − 1,200 = −$120; ROI = −120 ÷ 1,200 = −10%.
Gain 0.25 × 800 = $200; ending value 800 + 200 = $1,000.
Gain $630; ROI 630 ÷ 3,000 = 21%; simple 21% ÷ 2 = 10.5% per year; compound (1.21)^(1/2) − 1 = 1.1 − 1 = 10% per year.
A: 90 ÷ 300 = 30%. B: 200 ÷ 1,000 = 20%. A has the higher ROI; B earns more dollars ($200 vs. $90).
ROI ignores time. 30% over 3 years is (1.30)^(1/3) − 1 ≈ 9.14% per year compounded, which is lower than 15% per year.

Recap quiz

25%: Net gain $50; 50 ÷ 200 = 25%.
The purchase price plus every fee and extra cost: Leaving out any cost overstates ROI.
5%: 20% ÷ 4 = 5% per year.

21.5

Stock Market: Own It, Trade It, Understand It

Learning objectives

After this section you will be able to:

Explain what a share represents, how companies raise money in the primary market, and how shares change hands in the secondary market.
Calculate market value, dollar and percent gains or losses, and separate realized from unrealized results.
Include the bid-ask spread and fees to find what really hit your account, and read a basic candlestick table and simple moving average without treating them as predictions.

Real-life situation

What's the play? Before you risk your money, learn how the money moves

A 16-year-old opens a custodial brokerage account with a parent and watches a fictional company, Northside Robotics. The app shows a price, a chart, a bid, and an ask. Before buying anything, she writes down five days of the chart's numbers. Every company, price, fee, and return in this lesson is fictional. This is education, not investment advice.

Day	Open	High	Low	Close	Volume
Mon	$24.00	$25.10	$23.80	$24.90	1.2M
Tue	$24.90	$25.60	$24.50	$25.40	1.5M
Wed	$25.40	$25.50	$24.40	$24.70	2.1M
Thu	$24.70	$26.00	$24.60	$25.80	1.8M
Fri	$25.80	$26.40	$25.70	$26.20	1.4M

Figure 21.5 Chart reading lab: five fictional trading days of Northside Robotics (open, high, low, close, volume)

The concept

Think like an owner. A share is a small piece of ownership in a business. Companies sell new shares in the primary market (for example, an initial public offering) to raise money to grow. After that, investors buy and sell those shares to each other in the secondary market through brokers on exchanges, where market makers stand ready to buy and sell. When you buy on an app, your money usually goes to another investor, not to the company.

Market capitalization is price per share times shares outstanding: the market's price tag on the whole company. Volume is how many shares traded. Liquidity is how easily you can buy or sell without moving the price; heavily traded stocks usually have tighter spreads.

Here's the catch. Every quote has two prices. The bid is the highest price buyers will pay right now; the ask is the lowest price sellers will accept. You usually buy at the ask and sell at the bid, so the bid-ask spread is a real cost even when an app advertises zero commissions. A market order fills right away at the best available price, which can be worse than you expected when prices jump. A limit order sets the most you will pay (or the least you will accept); it protects your price but may never fill.

Realized vs. unrealized. A gain on a stock you still own is unrealized: it can shrink or disappear tomorrow. It becomes realized only when you sell. Total return adds dividends to the price change. Volatility measures how much a price swings; higher volatility means bigger possible gains and bigger possible losses.

Don't get fooled by the chart. Each candlestick shows a day's open, high, low, and close. A green candle closed above its open; a red candle closed below it. Support and resistance are price levels where buying or selling showed up before; they are observations, not guarantees. A simple moving average (SMA) is the average of recent closes; an exponential moving average (EMA) weights recent days more. Both lag price because they are built from the past, so a 20-day SMA reacts faster than a 50-day SMA, and neither predicts the future.

Investing, trading, and gambling are different. Investors buy ownership and usually hold for years, letting compounding and dividends work. Traders buy and sell over days or minutes, so costs repeat on every trade. Gambling is betting without an edge on outcomes you cannot control. An ETF or index fund holds many companies in one share, spreading risk; it can still fall in value.

Friction waterfall: what really hit your account

Gross profit (10 × $2)$ 20

Spread cost (10 × $0.10)$ 1

Sale fee$ 2

Net profit$ 17

Bars share a zero baseline.

Figure 21.5.V Friction waterfall: what really hit your account. Hypothetical teaching data.

Money Words

Plain-language words you will use in this lesson — learn them before we reach the formulas.

bid-ask spread
The gap between what buyers will pay (bid) and what sellers will accept (ask); a hidden cost on every trade.
limit order
An order to buy or sell only at a price you choose or better; it protects your price but may not fill.
liquidity
How quickly and easily something can be bought or sold for cash without changing its price much.

Key terms

Primary market: Where a company sells new shares to raise money.
Secondary market: Where investors buy and sell existing shares with each other.
Market capitalization: Price per share × shares outstanding.
Bid / ask: The best price buyers will pay / the best price sellers will accept right now.
Market order / limit order: Fill now at the best available price / fill only at your price or better.
Unrealized gain: A paper gain on shares you still own; it becomes realized only when you sell.
Simple moving average (SMA): The average of the last n closing prices.
Risk-to-reward ratio: Potential gain ÷ potential loss for a planned trade.

Formulas & rules

Market value
Market value = shares × current price
Percent gain or loss
% change = (current value − cost) ÷ cost × 100%
Spread cost
Spread cost = spread per share × shares
Net trading profit
Net profit = sale proceeds − purchase cost − spread cost − fees
Dividend yield
Yield = annual dividend per share ÷ price × 100%
Simple moving average
SMA = sum of last n closes ÷ n
Market capitalization
Market cap = price per share × shares outstanding

Worked examples

Example 1You own a piece of the company

Run the numbers: a student buys 5 shares of a fictional company at $40 each. Three months later the price is $48. Find the original investment, current market value, dollar gain, and percent return. Then redo it if the price drops to $32 instead.

Solution

1Original investment: 5 × $40 = $200.
2Market value at $48: 5 × $48 = $240.
3Dollar gain: $240 − $200 = $40.
4Percent return: $40 ÷ $200 = 0.20 = 20%.
5Here's the catch: if the price drops to $32, value is 5 × $32 = $160, a change of $160 − $200 = −$40, or −20%.
6What really hit? Nothing yet. Both results are unrealized until she sells, and the price can keep moving either way.

Answer: $200 invested; at $48 it is worth $240 (+$40, +20%); at $32 it is worth $160 (−$40, −20%). Both are unrealized.

Example 2The trade looked better on the screen

A trader buys 10 shares at $25 and sells them at $27. The bid-ask spread costs a combined $0.10 per share on entry and exit, and there is a $2 fee on the sale. Find the gross profit, the net profit, and the net percent return.

Solution

1Purchase cost: 10 × $25 = $250. Sale proceeds: 10 × $27 = $270.
2Gross profit: $270 − $250 = $20.
3Here's the catch: spread cost 10 × $0.10 = $1, plus the $2 fee, is $3 of friction.
4What really hit your account: $20 − $3 = $17.
5Net return: $17 ÷ $250 = 0.068 = 6.8%, not the 8% the screen suggested.
6Go or No-Go? Still a gain, but friction took 15% of the profit ($3 of $20). A zero-commission app still had a spread cost.

Answer: Gross $20 (8%); net $17 (6.8%) after $3 of spread and fees

Example 3Investing versus trading

Two students each start with $500 for one year. Assumptions (hypothetical, not a forecast): the investor buys an index ETF once, pays no fee, and it gains 8% for the year. The trader makes 24 trades whose gains total $30 before costs, but pays $1.50 in spread and fees per trade. Compare the ending values.

Solution

1Investor: 500 × 1.08 = $540, a gain of $40 (8%).
2Trader's costs: 24 × $1.50 = $36.
3Trader's net: $30 − $36 = −$6, so the account ends at $494 (−1.2%).
4Here's the catch: the trader picked more winners than losers, yet costs repeated on every trade.
5This does not mean investing always wins. If the ETF had fallen 10% that year, the investor would end at $450. Results depend on costs, risk, timing, and consistency.

Answer: Investor $540 (+8%); trader $494 (−1.2%) under these stated assumptions

Example 4The dividend decision

A student owns 20 shares of a fictional company priced at $25 that pays an annual dividend of $0.50 per share. Find the dividend income and dividend yield. Then explain what happens to the share price when the dividend is paid.

Solution

1Dividend income: 20 × $0.50 = $10 per year.
2Dividend yield: $0.50 ÷ $25 = 0.02 = 2%.
3Here's the catch: the cash comes out of the company, so the price typically drops by about the dividend on the ex-dividend date: $25 − $0.50 = $24.50.
4Value check: 20 × $24.50 = $490 in shares, plus $10 cash = $500, the same $500 as before.
5Think like an owner: a dividend moves value from the company to you; wealth grows only if the business keeps earning. Dividends can be cut at any time.

Answer: $10 per year; yield 2%; the payment by itself does not create extra wealth

Practice exercises 21.5

Go or No-Go? You buy 4 shares at $10.00 and sell at $10.50. The spread costs a combined $0.10 per share and the app charges a $2 fee on the sale. Find the net result and explain your decision.
Go or No-Go? A setup buys at $50 with a plan to exit at $46 if wrong and $51.50 if right. Find the risk, the reward, and the risk-to-reward ratio, and explain.
Long term: $1,000 grows at a hypothetical 6% per year for 10 years versus 3% per year. Find both ending values and the difference. What assumption matters most?
Stock or ETF? You put $1,000 into one company that falls 40%. Compare with $1,000 split equally across 50 companies in an ETF where that same company falls 40% and the rest are flat.
You buy 30 shares at $20, receive $0.80 per share in dividends over the year, and the price ends at $19. Find the total return in dollars and percent.
No trade is a decision: a fictional $8 stock jumps before the open and shows a bid of $7.80 and an ask of $8.20. What percent of the price is the spread, and why might waiting be reasonable?
Using Figure 21.5, find the 5-day simple moving average of the closes, and say whether Friday's close is above or below it.
Level up: a fictional company has 2,000,000 shares outstanding at $26.20. Find its market capitalization. Talk your math: explain why a $26 stock is not automatically 'cheaper' than a $260 stock.

Section 21.5 recap

A share is ownership. Market value = shares × price; % change = (value − cost) ÷ cost.
Gains are unrealized until you sell. Spreads and fees come out of every trade, even 'zero-commission' ones.
Charts and moving averages describe the past, not the future. Diversification lowers risk but never removes it. Choosing not to trade can be the smart move.

Answer key with explanations: Section 21.5

Practice exercises

Gross 4 × $0.50 = $2. Costs 4 × $0.10 = $0.40 plus $2 = $2.40. Net $2 − $2.40 = −$0.40. A gain on the screen became a loss after costs; No-Go at this size.
Risk $50 − $46 = $4. Reward $51.50 − $50 = $1.50. Reward ÷ risk = 1.5 ÷ 4 = 0.375. Even if it often works, one loss erases more than two wins; most would call it No-Go.
1,000 × 1.06^10 ≈ $1,790.85; 1,000 × 1.03^10 ≈ $1,343.92; difference ≈ $446.93. The steady return rate is an assumption; real returns vary year to year and can be negative.
Single stock: 1,000 × 0.60 = $600 (−$400). ETF: each holding is $20; 20 × 0.40 = $8 loss, so $992. Diversification limited the damage, though a broad market drop would hit both.
Cost 30 × $20 = $600. Price change 30 × (−$1) = −$30. Dividends 30 × $0.80 = $24. Total −$30 + $24 = −$6; −6 ÷ 600 = −1%.
Spread $8.20 − $7.80 = $0.40; 0.40 ÷ 8 = 5%. The stock must rise 5% just to break even on a quick round trip, and a market order could fill worse. Waiting or a limit order protects your money.
Sum 24.90 + 25.40 + 24.70 + 25.80 + 26.20 = 127.00; 127 ÷ 5 = $25.40. Friday's $26.20 is above it. That describes the past week; it does not predict next week.
2,000,000 × $26.20 = $52,400,000. Price per share alone says nothing about value; you must compare the whole company (market cap) to what it earns.

Recap quiz

Unrealized: It becomes realized only when you sell; it can still change.
$0.10 per share: Buy at the ask, sell at the bid: $20.05 − $19.95 = $0.10.
It is calculated from past closes: It averages the last 50 closes, so it reacts after price moves.

Chapter 21 Organizer and Review

Chapter Organizer

Every section at a glance: key formulas, key terms, and the main idea.

21.1 Stocks: Dividends, Yield, and Total Return
Key formulas
- Cost of purchase: Cost = shares × price per share + fees
- Dividend yield: Yield = annual dividend per share ÷ price × 100%
- Total return: Total return = (ending price − starting price + dividends) ÷ starting price × 100%
- P/E ratio: P/E = price per share ÷ earnings per share
Key terms
Share, Dividend, Dividend yield, Total return, Earnings per share (EPS)
Remember
Yield = annual dividend ÷ price; P/E = price ÷ EPS.
21.2 Bonds: Coupon Payments, Par, and Current Yield
Key formulas
- Annual coupon: Coupon = par × coupon rate
- Price from quote: Price = quote ÷ 100 × par
- Current yield: Current yield = annual coupon ÷ price × 100%
Key terms
Bond, Par (face) value, Coupon rate, Discount / premium, Current yield
Remember
Coupons are based on par; prices are quoted as a percent of par.
21.3 Mutual Funds and ETFs: NAV, Expenses, and Diversification
Key formulas
- NAV: NAV = (total assets − liabilities) ÷ shares outstanding
- Expense cost: Yearly cost = amount invested × expense ratio
- Weighted return: Portfolio return = w₁r₁ + w₂r₂ + …
Key terms
Mutual fund / ETF, NAV, Expense ratio, Sales load, Diversification
Remember
NAV = (assets − liabilities) ÷ shares. Fees are certain; returns are not.
21.4 Return on Investment (ROI) and Annualized Return
Key formulas
- Net gain: Net gain = total returned − amount invested − other costs
- ROI: ROI = net gain ÷ amount invested × 100%
- Simple annual ROI: Simple annual ROI = ROI ÷ years
- Compound annual rate: Annual rate = (1 + ROI)^(1/years) − 1
Key terms
Return on investment (ROI), Net gain, Annualized return
Remember
ROI = net gain ÷ amount invested; subtract every cost first.
21.5 Stock Market: Own It, Trade It, Understand It
Key formulas
- Market value: Market value = shares × current price
- Percent gain or loss: % change = (current value − cost) ÷ cost × 100%
- Spread cost: Spread cost = spread per share × shares
- Net trading profit: Net profit = sale proceeds − purchase cost − spread cost − fees
- Dividend yield: Yield = annual dividend per share ÷ price × 100%
- Simple moving average: SMA = sum of last n closes ÷ n
- Market capitalization: Market cap = price per share × shares outstanding
Key terms
Primary market, Secondary market, Market capitalization, Bid / ask, Market order / limit order, Unrealized gain, Simple moving average (SMA), Risk-to-reward ratio
Remember
A share is ownership. Market value = shares × price; % change = (value − cost) ÷ cost.

Cumulative Review

Mixed practice from this chapter and the two chapters before it.

Section 21.1
A $1,000 position loses 20% in price and pays $15 in dividends. What is it worth including dividends?
Show answer
Price value 1,000 × 0.80 = $800; plus $15 = $815, a total return of −18.5%.
Section 21.2
Interest rates on new bonds rise. What usually happens to the price of an existing bond, and why?
Show answer
It falls, because its fixed coupon is less attractive than new bonds' higher payments.
Section 21.3
Why can't diversification guarantee you won't lose money?
Show answer
When most investments fall together, as in a market-wide drop, a spread-out portfolio still loses value; diversification only reduces the damage from any single holding.
Section 21.4
Why can't you compare a 30% ROI earned over 3 years directly with a 15% ROI earned in 1 year?
Show answer
ROI ignores time. 30% over 3 years is (1.30)^(1/3) − 1 ≈ 9.14% per year compounded, which is lower than 15% per year.
Section 21.5
Level up: a fictional company has 2,000,000 shares outstanding at $26.20. Find its market capitalization. Talk your math: explain why a $26 stock is not automatically 'cheaper' than a $260 stock.
Show answer
2,000,000 × $26.20 = $52,400,000. Price per share alone says nothing about value; you must compare the whole company (market cap) to what it earns.
Review of Section 19.3
A hypothetical business earns $18,000 revenue and has $6,000 allowed expenses. Find taxable profit.
Show answer
Taxable profit = $18,000 − $6,000 = $12,000.00.
Review of Section 20.3
A 2% chance of a $5,000 loss otherwise means no loss. Find expected loss.
Show answer
0.02 × $5,000 + 0.98 × $0 = $100.00.

All prices, stores, people, and accounts in this chapter are hypothetical and for teaching only. Product references are generic categories; no brand endorses or is endorsed by this textbook.

Chapter 20 Chapter 22