Use these to verify hand calculations after you finish your own work.
Every investment decision is a tradeoff: you want a higher return, but higher return usually comes with higher risk. This chapter gives you a structured way to measure both.
Return is the gain (or loss) from an investment over a period. It can come from price change and cash income (such as dividends).
Notes: HPR is a realized return for one period. Expected return uses probabilities across possible outcomes.
Buy at $50, end at $54, dividend $1.
In this chapter, risk means uncertainty (variability) of returns. If returns swing a lot, risk is higher.
Notes: Convert percentages to decimals before calculations (10% = 0.10).
Keep expected return at 6% and compare narrow, wide, and zero-spread outcomes. See how the chance of a gain above 10%, a loss below −5%, or any loss changes.
The bell curves illustrate a normal distribution. Actual stock returns may not follow that shape. Standard deviation measures variation on both the gain and loss sides.
First, we use a textbook probability table (Apple teaching example). Then, we show a real-data method: use 5-year monthly prices → monthly returns → mean and standard deviation in Excel.
Start here: Apple’s one-stock probability example, then Apple and Moderna monthly data, portfolio return, risk, and correlation.
| State of the Economy | Probability | Apple Return (Example) |
|---|---|---|
| Recession | 10% | -30% |
| Below Average | 20% | -2% |
| Average | 40% | 10% |
| Above Average | 20% | 18% |
| Boom | 10% | 40% |
Notes: This is a teaching probability model (not historical Apple returns).
| State | p | R | p × R |
|---|---|---|---|
| Recession | 0.10 | -0.30 | -0.0300 |
| Below Average | 0.20 | -0.02 | -0.0040 |
| Average | 0.40 | 0.10 | 0.0400 |
| Above Average | 0.20 | 0.18 | 0.0360 |
| Boom | 0.10 | 0.40 | 0.0400 |
| Total = E[RA] | 0.0820 = 8.20% | ||
This matches the Apple textbook expected return shown on this page: 8.20%.
Use E[RA] = 0.0820 (decimal).
| State | R | R − E[RA] | (R − E[RA])² | p × (R − E[RA])² |
|---|---|---|---|---|
| Recession | -0.30 | -0.382 | 0.145924 | 0.0145924 |
| Below Average | -0.02 | -0.102 | 0.010404 | 0.0020808 |
| Average | 0.10 | 0.018 | 0.000324 | 0.0001296 |
| Above Average | 0.18 | 0.098 | 0.009604 | 0.0019208 |
| Boom | 0.40 | 0.318 | 0.101124 | 0.0101124 |
| Variance = Var(RA) | 0.028836 | |||
Expected return: 8.20%
Risk (stdev): 16.98%
If the download doesn’t start, right-click the button → “Save link as…”
Mean (monthly): —
Stdev (monthly): —
One-SD range: —
| Date | AAPL Price | AAPL Monthly Return |
|---|
First row has no return (no prior month).
If the download doesn’t start, right-click the button → “Save link as…”
Mean (monthly): —
Stdev (monthly): —
One-SD range: —
| Date | MRNA Price | MRNA Monthly Return |
|---|
First row has no return (no prior month).
Paste rows as: probability, return (prob in %, return in %). Example: 10, -30
Expected return: —
Variance: —
Standard deviation: —
Variance is shown in decimal units (for example, 0.028836).
Now we move from one stock to a two-stock portfolio. Use the same monthly return series and compute mean, stdev, and correlation in Excel, then compute portfolio risk. Correlation (AAPL, MRNA) = 0.1889.
| Input (monthly) | AAPL | MRNA | Excel function |
|---|---|---|---|
| Mean return | — | — | =AVERAGE(returns) |
| Stdev | — | — | =STDEV.S(returns) |
| Correlation (AAPL, MRNA) | 0.1889 | =CORREL(C3:C62, F3:F62) | |
Portfolio mean (monthly): —
Portfolio stdev (monthly): —
Covariance: —
Weight in Moderna: —
Portfolio mean: —
Covariance (A,M): —
Portfolio variance: —
Portfolio stdev: —
Optional, but useful: adding a third stock can reduce firm-specific risk further.
Next: extend the two-stock portfolio to three stocks, see why each pair’s correlation matters, and briefly explore four stocks.
We can keep adding stocks to this portfolio (4 stocks, 5 stocks, ...). As the portfolio becomes broader across industries, idiosyncratic (firm-specific) risk keeps falling.
Notes: This is why the chapter shifts from standard deviation (σ) to beta (β).
CAPM uses beta (β) to measure systematic risk. Here we keep Apple + Walmart + Moderna on the SML. (Given class inputs: Apple β=1.11, Walmart β=0.67, Moderna β=1.32.)
Finally: use beta and the CAPM equation to find required return, then see how the same equation becomes the SML.
Use beta to estimate the required return.
Watch five local or company-specific shocks and five marketwide shocks. See why diversification reduces one kind of risk but leaves exposure to the market.
As you add more stocks, idiosyncratic risk (firm-specific risk) falls. After a broad portfolio (often around 20–25+ stocks), most of that risk is reduced. What remains is mainly systematic risk (market-wide risk).
Risk specific to one company (or a small group of companies). This can be diversified away.
Market-wide risk that affects many or most stocks. This cannot be diversified away.
Think of risk layers like geography:
Notes: CAPM focuses on the broad market component (systematic risk), which is why beta is used.
The curve below shows total portfolio risk falling as the number of stocks increases, then leveling off at systematic risk.
Note: betas vary across sources and time windows. For any assignment, use the same source for all stocks.
MRP is the “extra return” for taking market risk. Beta tells you how much market risk a stock has.
Market risk premium (MRP): —
Apple CAPM return: —
Walmart CAPM return: —
Moderna CAPM return: —
Custom stock CAPM return: —
Custom stock SML message: —
Show your steps. Use Excel functions where appropriate:
SUMPRODUCT, STDEV/STDEV.S, CORREL, SLOPE.
| State of the Economy | Probability | Stock A’s Return |
|---|---|---|
| Recession | 10% | -30% |
| Below Average | 20% | -2% |
| Average | 40% | 10% |
| Above Average | 20% | 18% |
| Boom | 10% | 40% |
Answer: 8.2%
Answer: 15%
| State of Economy | Probability of State | Rate of Return if State Occurs |
|---|---|---|
| Boom | 27% | 14% |
| Normal | 70% | 8% |
| Recession | 3% | -11% |
Answer: 9.05%
| Month End | Price |
|---|---|
| January | $125.00 |
| February | $138.50 |
| March | $132.75 |
Answer: 2.12%
Answer: 11%
| Holding | Amount Invested | Beta |
|---|---|---|
| Stock A (8,000 shares) | $16,000 | 1.3 |
| Stock B (15,000 shares) | $48,000 | 1.8 |
| Stock C (25,000 shares) | $96,000 | 2.2 |
Answer: 1.99
Answer: 13%
| Stock | Investment Value | Beta |
|---|---|---|
| Stock A | $8,000 | 1.5 |
| Stock B | $10,000 | 1.0 |
| Stock C | $2,000 | 0.5 |
Answer: 1.15
Answer: 11.4%
| Stock | Percentage of Portfolio | Beta |
|---|---|---|
| 1 | 20% | 1.0 |
| 2 | 30% | 0.5 |
| 3 | 50% | 1.6 |
The risk-free rate is 3% and the market return is 10%.
| Period | Jazman | Solomon |
|---|---|---|
| 1 | $10 | $20 |
| 2 | $12 | $25 |
| 3 | $15 | $15 |
Answer: 50%, -25%
| State of the Economy | Probability | % Return (Cash Flow / Investment Cost) |
|---|---|---|
| Economic Recession | 30% | 5% |
| Strong and Moderate Economic Growth | 70% | 15% |
Answer: 12%
Answer: 1, 3, 2
| Holding | Amount Invested | Beta |
|---|---|---|
| Stock A (8,000 shares) | $10,000 | 1.5 |
| Stock B (15,000 shares) | $20,000 | 0.8 |
| Stock C (25,000 shares) | $20,000 | 1.2 |
Answer: 1.1
Follow the calculations for expected return, holding period return, portfolio beta, and CAPM. Pause the video to try each problem yourself.
Work through the three short quizzes after reviewing the homework solutions.
Show your formula and steps first, then use these tools to check your answers.
Probability and return tables, including homework questions 1, 3, and 12.
Open calculator ↗Purchase price, dividends, and sale price for questions 2, 4, and 11.
Open calculator ↗Explore portfolio return and risk with two stocks in the chapter lesson.
Open calculator ↗For three-stock portfolio beta, multiply each investment weight by its beta and add the results.
Chapter 6 study help
Ask about return, risk, diversification, beta, CAPM, or a homework question.
I will use the prepared lesson answers and show you where to review the idea.
Ask Maggie uses prepared Chapter 6 answers and will not guess when a question does not match.
These videos are not required. I will introduce the ideas briefly in class. Watch them if you would like to see how diversification, beta, and CAPM can help you think through investment choices.
See how combining investments can change portfolio risk.
Apply required return to a possible stock choice.
Use real Apple, Moderna, and S&P 500 monthly prices to calculate returns and beta in Excel. The video also explains why beta differs from total risk, why a fuller estimate uses about 60 months, and where the Treasury rate enters CAPM.