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.
This optional follow-up shows where the probabilities in the video above come from. Follow the shaded bell curves and Excel formulas to calculate the chance of earning more than 10%, losing more than 5%, or having any negative return.
Optional learning extension: Watch if you want to understand the calculations behind the first video. Examples assume normally distributed returns; actual stock returns may differ.
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 for market risk. Start with the risk-free rate, then add compensation for systematic risk. This model provides a benchmark; it does not guarantee the return you will earn.
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.
Required return = risk-free rate + compensation for the stock’s market risk.
Expected market return − risk-free rate = expected extra return for market risk.
Then use CAPM = R𝒻 + βᵢ × MRP. If the question already gives MRP, use it directly; do not subtract R𝒻 again.
Enter R𝒻, E(Rₘ), and β to calculate the required return. For the example above, use 4%, 10%, and 1.5; the CAPM answer is 13%.
Open JUFinance.com CAPM Calculator ↗Click any symbol above to open its definition below, or click a definition heading. Use all rates for the same period.
A model that estimates the return investors require for an investment’s systematic (market) risk. Its inputs are the risk-free rate, beta, and the expected market risk premium.
Read this as “expected return on investment i.” In the CAPM equation it is the model-implied required return: the benchmark for compensation for market risk. A separate forecast of a stock’s expected return may differ from this benchmark; neither is a guaranteed realized return.
The baseline return for an investment treated as free of default risk over the relevant horizon. A Treasury rate is commonly used as a proxy. Match the horizon and currency to your analysis. Example: R𝒻 = 4%.
Beta measures sensitivity to market returns. β = 1 means market-like sensitivity; β = 1.5 means roughly 1.5 times the market’s sensitivity on average; β = 0.5 means half. Beta measures systematic risk, not total risk, and is not a guaranteed movement ratio.
The return you expect from the broad market portfolio over the period. A broad stock index is commonly used as a proxy. Example: E(Rₘ) = 10%. This is the full expected market return, before subtracting R𝒻.
MRP = E(Rₘ) − R𝒻. This is the expected extra return from taking market risk instead of holding the risk-free investment. Example: 10% − 4% = 6 percentage points. Enter 6% as the premium in the CAPM calculation.
E means “expected,” or a probability-weighted average. Parentheses E(R) and brackets E[R] mean the same thing here. The subscript i identifies the investment or stock; m identifies the market. The subscript f in R𝒻 means risk-free.
The graph of the CAPM equation: beta on the horizontal axis and required return on the vertical axis. Its intercept is R𝒻 and its slope is MRP. Compare a separately estimated expected return with the CAPM benchmark using the same horizon and assumptions.
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.
Ask two questions: What return do I expect, and what risk am I taking? Then select the tool that answers your question.
| Your situation | What to calculate | What it tells you |
|---|---|---|
| One stock | Expected return E(R) and standard deviation σ. Use HPR for a return already earned. | Expected return is the average outcome you anticipate; standard deviation measures total variability. If this is your only stock, both market risk and company-specific risk affect you. Beta alone does not capture all your risk. |
| Two stocks | Investment weights, portfolio expected return, and portfolio standard deviation using correlation ρ. | Check how the stocks move together. Correlation below +1 can reduce risk relative to the weighted average of the individual standard deviations. Two stocks do not automatically provide broad diversification. |
| Many stocks / a diversified portfolio | Portfolio beta βₚ = Σwᵢβᵢ; use standard deviation to describe remaining total portfolio variability. | Diversification reduces company-specific risk. Market risk remains; beta measures exposure to it. CAPM uses that exposure to estimate required return. |
| Is a stock’s forecast return enough for its market risk? | Find MRP, calculate CAPM required return, then compare your separate forecast expected return with that benchmark. | Above CAPM: potentially attractive / undervalued under the model. Below CAPM: potentially unattractive / overvalued. Equal: approximately fairly priced under the model. The conclusion depends on your forecasts and model assumptions. |
w is the fraction of your money in each stock; σ is its standard deviation; ρ₁₂ is the correlation between the two stocks’ returns. Portfolio risk is not simply the weighted average of the two standard deviations.
Optional enrichment: Watch if you would like to explore asset pricing models beyond CAPM. Learn what size, value, profitability, investment, and momentum mean, and follow a simple example showing why the model you choose changes measured outperformance.
Watch the optional video on YouTube ↗ · Liam narration · Captions