FIN509 • Chapter 13: Risk, Return & Portfolio Management

★ Optional Excel ICE — Up to 100 Additional Chapter 13 Homework Points

MBA Edition · Fall 2026 · Interactive learning, real-data examples, graduate portfolio analytics, and practical investment decisions.
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FIN509 · CHAPTER 13 STUDY GUIDE

How to use this chapter

The four required topics below will be tested on the quiz and the final exam. Work through them in order. Try each calculation by hand, check it with the calculator, then use the three practice quizzes to check your understanding.

  1. One stock · Probability, return, and risk Required

    Use a table of outcomes and probabilities to calculate expected return, variance, and standard deviation. Standard deviation measures total volatility.

  2. Two stocks · Portfolio return, risk, and correlation Required

    Use weights to find expected portfolio return. Use weights, stock volatilities, and correlation to calculate portfolio variance and standard deviation. Explain why lower correlation can reduce portfolio risk.

  3. Systematic and unsystematic risk Required

    Distinguish market-wide risk from company-specific risk, also called idiosyncratic risk. Explain why spreading your holdings across companies and industries reduces company-specific risk while leaving market risk.

  4. Beta and CAPM · Required return Required

    Explain what beta measures. Know the risk-free rate, expected market return, and market risk premium. Calculate the market risk premium as expected market return minus the risk-free rate, then calculate required return as the risk-free rate plus beta times the market risk premium.

Optional Enrichment and extra practice

Explore these after the required topics. They extend the lesson beyond the quiz and final exam scope.

Use Professor Foley’s JUFinance.com Calculators

Try the calculation yourself, then use the matching calculator to check your work. Keep track of your inputs and explain what the answer means.

Expected Return & Risk

One stock: calculate expected return and standard deviation.

Open Expected Return & Risk Calculator ↗

Holding Period Return

Calculate return from purchase price, sale price, and dividends.

Open Holding Period Return Calculator ↗

Two-Stock Portfolio

Explore weights, correlation, portfolio return, and risk.

Open Two-Stock Portfolio Calculator ↗

CAPM Required Return

Use the risk-free rate, expected market return, and beta.

Open CAPM Required Return Calculator ↗

Why study risk and return?

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.

What you use this for

  • Compare investments using numbers, not guesses.
  • Estimate expected return before investing.
  • Measure volatility (risk) with standard deviation.
  • Reduce risk through diversification.
  • Use beta and CAPM for required return in a diversified portfolio.

Flow of this chapter

1) One stock: expected return + standard deviation 2) Two stocks: correlation + diversification 3) (Optional) Three stocks: see the pattern 4) Many stocks: shift to beta 5) CAPM + SML: required return for systematic risk

What is return?

Return is the gain (or loss) from an investment over a period. It can come from price change and cash income (such as dividends).

Key return formulas
Dollar Return = Ending Value − Beginning Value + Cash Income
Holding Period Return (HPR) = (P1 − P0 + D1) / P0
One stock · Expected return
E(R) = n∑i = 1 pi × Ri
∑ means add all outcomes. Multiply each probability pi by its return Ri, then add. Probabilities sum to 1.

Notes: HPR is a realized return for one period. Expected return uses probabilities across possible outcomes.

Quick HPR example

Buy at $50, end at $54, dividend $1.

HPR = (54 − 50 + 1) / 50 = 5 / 50 = 0.10 = 10%

What is risk?

In this chapter, risk means uncertainty (variability) of returns. If returns swing a lot, risk is higher.

How we measure risk early in the chapter
One stock · Variance
Var(R) = n∑i = 1 pi × [Ri − E(R)]2
For each outcome: subtract expected return, square the difference, multiply by its probability, and add.
One stock · Standard deviation
σ = √Var(R)
Take the square root of variance to put risk back in the same units as return.
  • Variance uses squared deviations from expected return.
  • Standard deviation is easier to interpret because it is in return units (%).

Notes: Convert percentages to decimals before calculations (10% = 0.10).

Visual lesson · 1 minute 36 seconds

Why Is Standard Deviation a Measure of Risk?

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.

Watch on YouTube ↗

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.

OPTIONAL · Excel follow-up · 6 minutes 7 seconds

Calculate Return Probabilities in Excel: NORM.DIST & Bell Curves

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.

Watch optional lesson on YouTube ↗

Topic 1 · One Stock: Expected Return & Standard Deviation Required

First, we use a textbook probability table (Apple teaching example). For optional practice, use the real-data method: use November 2021–October 2026 month-end prices → monthly returns → mean and standard deviation in Excel.

One Stock vs. Two Stocks: Return, Risk & Correlation

Start here: Apple’s one-stock probability example, then Apple and Moderna monthly data, portfolio return, risk, and correlation.

Watch on YouTube

Textbook example (Apple): states of the economy
State of the EconomyProbabilityApple Return (Example)
Recession10%-30%
Below Average20%-2%
Average40%10%
Above Average20%18%
Boom10%40%

Notes: This is a teaching probability model (not historical Apple returns).

Textbook Step 2: Compute expected return for Apple (show the math)
Apple · Expected return
E(RA) = n∑i = 1 pi × RA,i
Probability × return for every economic state; then add the results.
State p R p × R
Recession0.10-0.30-0.0300
Below Average0.20-0.02-0.0040
Average0.400.100.0400
Above Average0.200.180.0360
Boom0.100.400.0400
Total = E[RA]0.0820 = 8.20%

This matches the Apple textbook expected return shown on this page: 8.20%.

Textbook Step 3: Compute Apple variance + standard deviation (show the math)
Apple · Variance
Var(RA) = n∑i = 1 pi × [RA,i − E(RA)]2
Use the probability-weighted squared deviations shown in the table below.

Use E[RA] = 0.0820 (decimal).

State R R − E[RA] (R − E[RA])² p × (R − E[RA])²
Recession-0.30-0.3820.1459240.0145924
Below Average-0.02-0.1020.0104040.0020808
Average0.100.0180.0003240.0001296
Above Average0.180.0980.0096040.0019208
Boom0.400.3180.1011240.0101124
Variance = Var(RA)0.028836
Apple · Standard deviation
σA = √0.028836 = 0.1698 = 16.98%
Variance is in decimal return squared; its square root is a decimal return, then format as a percentage.

Apple textbook results (match check)

Expected return: 8.20%

Risk (stdev): 16.98%

What this shows

  • Expected return comes from the weighted average (p×R).
  • Risk comes from weighted squared deviations from the mean.
  • Next: compare this textbook method to the real-data method below.

Check after you hand-solve

JU Return / Risk Calculator
Real data (from Excel): Apple (AAPL) — monthly returns → mean + stdev Optional
Download FIN509 Chapter 13 Class Data (Excel)

If the download doesn’t start, right-click the button → “Save link as…”

Excel formulas

Monthly return (copy down): =(B6-B5)/B5 Mean monthly return: =AVERAGE(D6:D64) Monthly standard deviation (risk): =STDEV.S(D6:D64)

AAPL results (from embedded Excel data)

Mean (monthly): —

Stdev (monthly): —

One-SD range: —

Interpretation

  • Mean = historical average monthly return (estimate).
  • Stdev = historical monthly volatility (risk).
View AAPL monthly data (60 rows)
DateAAPL PriceAAPL Monthly Return

First row has no return (no prior month).

Real data (from Excel): Moderna (MRNA) — monthly returns → mean + stdev Optional
Download FIN509 Chapter 13 Class Data (Excel)

If the download doesn’t start, right-click the button → “Save link as…”

Excel formulas

Monthly return (copy down): =(C6-C5)/C5 Mean monthly return: =AVERAGE(E6:E64) Monthly standard deviation (risk): =STDEV.S(E6:E64)

MRNA results (from embedded Excel data)

Mean (monthly): —

Stdev (monthly): —

One-SD range: —

Interpretation

  • MRNA has higher volatility than AAPL in this sample.
  • That’s why correlation + diversification matters next.
View MRNA monthly data (60 rows)
DateMRNA PriceMRNA Monthly Return

First row has no return (no prior month).

Mini-calculator (textbook: expected return + stdev from your own state table)

Paste rows as: probability, return (prob in %, return in %). Example: 10, -30

JU Return / Risk Calculator

Results

Expected return: —

Variance: —

Standard deviation: —

Variance is shown in decimal units (for example, 0.028836).

FIN509 application · Investment committee case

Apply core risk, return, diversification, and CAPM concepts to a practical investment decision.

Single Stock — Expected Return & Standard Deviation

Want a quick help? Try our expected return calculator here: Expected Return Calculator ↗
Describe your next-year scenarios as “probability → return (%)”. Example (5 states): “10% chance of −30%, 20% chance of −2%, 40% chance of 10%, 20% chance of 18%, 10% chance of 40%.”
State count:
E[R]:
Var:
σ:
What this means (next-year story):
Choose a state count, enter probabilities & returns (%), then Compute.
Show full math work (step by step)
We first convert returns from % to decimals, then compute E[R], Var(R), and σ = √Var(R).
Run “Compute” to generate the steps.
Excel & quick tools:
E[R]: =SUMPRODUCT(p_rng, r_rng/100)
Var: =SUMPRODUCT(p_rng,(r_rng/100 - E_R_cell)^2) σ: =SQRT(Var_cell)
Want diversification? Try your two-stock mix here: Two-Stock Portfolio (A & B) ↗

Holding Period Return (HPR)

HPR:
Formula: HPR = (P₁ − P₀ + D) / P₀

Topic 2 · Covariance, Correlation & Two-Asset Diversification Required

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) = 0.0550.

Watch First · Covariance & Correlation — Apple vs. Moderna

Learn why lower correlation can reduce portfolio risk. Includes a hypothetical Apple–Moderna example, sample covariance and correlation calculations, Excel COVARIANCE.S and CORREL, a scatterplot, and the two-stock portfolio risk formula.

Watch on YouTube ↗ Narration by Thomas · For FIN301 & FIN509

Note: The video uses illustrative returns, not historical Apple or Moderna returns. The historical-return calculations elsewhere in this lesson are separate.

Step 1: Excel functions (mean, stdev, correlation)
AAPL mean monthly return: =AVERAGE(D6:D64) AAPL monthly stdev: =STDEV.S(D6:D64) MRNA mean monthly return: =AVERAGE(E6:E64) MRNA monthly stdev: =STDEV.S(E6:E64) Correlation (AAPL, MRNA): =CORREL(D6:D64,E6:E64) → 0.0550
Input (monthly)AAPLMRNAExcel function
Mean return——=AVERAGE(returns)
Stdev——=STDEV.S(returns)
Correlation (AAPL, MRNA)0.0550=CORREL(D6:D64,E6:E64)
Step 2: Two-stock portfolio formulas
Two stocks · Expected return
E(Rp) = wAE(RA) + wME(RM)
Multiply each stock’s expected return by its portfolio weight and add. Here wA + wM = 1.
Two stocks · Covariance
Cov(A,M) = ρA,M × σA × σM
ρ (rho) is correlation. A = Apple; M = Moderna. Covariance describes how their returns move together.
Two stocks · Portfolio variance
Var(Rp) = wA2σA2 + wM2σM2 + 2wAwMCov(A,M)
The first two terms use each stock’s variance. The last term includes how the two stocks move together.
Two stocks · Portfolio standard deviation
σp = √wA2σA2 + wM2σM2 + 2wAwMρA,MσAσM
Equivalently, σp = √Var(Rp). Calculate portfolio variance first, then take its square root.
Example (real data): 60% Apple + 40% Moderna (monthly)

Result (computed from the embedded data)

Portfolio mean (monthly): —

Portfolio stdev (monthly): —

Covariance: —

Interpretation

  • Weights + correlation determine how much diversification helps.
  • Even with correlation (0.0550), portfolio volatility can be lower than the risky stock alone.
JU 2-Stock Portfolio Calculator
Mini-calculator (Apple + Moderna, monthly)

2-Stock Results

Weight in Moderna: —

Portfolio mean: —

Covariance (A,M): —

Portfolio variance: —

Portfolio stdev: —

FIN509 application · Investment committee case

Apply core risk, return, diversification, and CAPM concepts to a practical investment decision.

Two-Stock Portfolio — E[R], σ, Covariance & Correlation

Want a quick help? Try our two-stock portfolio calculator here: Two-Stock Portfolio Calculator ↗
Use the same next-year states (probabilities) and enter each stock’s return (%) in those states. Lower correlation ⇒ stronger diversification.
State count:
0.50
E[R₁], σ₁:
E[R₂], σ₂:
Cov(1,2):
ρ(1,2):
E[Rp]:
Var(Rp):
σp:
What this mix means (next-year story):
Choose a state count, enter probabilities & both return vectors, set w₁, then Compute.
Show full math work (step by step)
We compute each stock’s expected return and variance first, then covariance, correlation, portfolio expected return, portfolio variance, and portfolio standard deviation.
Run “Compute” to generate the steps.
Excel formulas:
E[R₁]: =SUMPRODUCT(p_rng, r1_rng/100)
E[R₂]: =SUMPRODUCT(p_rng, r2_rng/100)
Var(R₁): =SUMPRODUCT(p_rng,(r1_rng/100-ER1_cell)^2)
Var(R₂): =SUMPRODUCT(p_rng,(r2_rng/100-ER2_cell)^2)
Cov(1,2): =SUMPRODUCT(p_rng,(r1_rng/100-ER1_cell)*(r2_rng/100-ER2_cell))
ρ(1,2): =Cov_cell/(SD1_cell*SD2_cell)
E[Rp]: =w1_cell*ER1_cell + w2_cell*ER2_cell
Var(Rp): =w1_cell^2*Var1_cell + w2_cell^2*Var2_cell + 2*w1_cell*w2_cell*Cov_cell
σp: =SQRT(VarP_cell)

Optional Activity · Stock Data and Correlation

Use Google Sheets to see whether NVIDIA (NVDA) and Tesla (TSLA) tend to move together. Start with the short video, then try the steps if you want extra practice.

Open the optional video, Google Sheets steps, and correlation demo

Watch · From stock prices to correlation

Watch on YouTube ↗

1. Get historical closing prices

Both methods use Google Sheets’ GOOGLEFINANCE function. A historical close is the stock’s closing price for a trading day. The function returns two columns, date and close.

Stock Data Fetcher app

The app is a shortcut for inserting the Google Finance formula. Enter NASDAQ:NVDA, choose the start and end dates, and click Fetch Data. Once the date and close columns appear in the sheet, repeat for NASDAQ:TSLA using the same dates.

Open Stock Data Fetcher ↗
Enter the formula directly in Google Sheets

In a blank sheet, enter the first formula in A1 and the second in D1. Leave room underneath for the results to fill in.

=GOOGLEFINANCE("NASDAQ:NVDA","close",TODAY()-370,TODAY(),"DAILY") =GOOGLEFINANCE("NASDAQ:TSLA","close",TODAY()-370,TODAY(),"DAILY")

For more practice, try NASDAQ:AAPL, NASDAQ:WMT, or INDEXSP:.INX for the S&P 500. Use the shared date range you can verify for both series. If a comparison has fewer common dates, report the period you actually used.

Why use Sheets here? Yahoo Finance lets you look up historical prices. Its historical CSV download requires a Gold subscription where export is available. Sheets brings the data into cells where you can calculate directly. Yahoo Finance Help ↗ · Google Sheets Help ↗

2. Calculate daily returns

On a new tab, put matching trading dates in column A, NVDA closes in B, and TSLA closes in C. Sort the common dates oldest to newest. Use consecutive trading days so each return covers the same one-day interval for both stocks.

With headers in row 1 and the first prices in row 2, enter =B3/B2-1 in D3 and =C3/C2-1 in E3. Fill down and format the returns as percentages. Leave row 2’s return cells blank. A close that rises from $100 to $102 has a +2% price return.

3. Calculate and interpret correlation

Use =CORREL(D3:D5,E3:E5) for the video’s three-return example. For your own data, replace row 5 with the last row containing both returns. The video uses hypothetical prices. A live comparison uses about a year of daily returns.

Correlation (ρ) ranges from −1 to +1. Near +1 means returns tend to move together. Near 0 means little linear relationship. Near −1 means they tend to move in opposite directions. For the same weights and individual stock volatilities, lower correlation lowers the risk of a two-stock portfolio with positive weights in both stocks.

Try the offline example

Offline demo: load a teaching example (hypothetical values).
Pairρ (12-mo)
NVDA vs TSLA—
NVDA vs AAPL—
NVDA vs WMT—
NVDA vs S&P 500 (INDEXSP:.INX)—
Tip: Click “Import from Two-Stock states” below to estimate ρ from your state table above.

See diversification change as ρ moves

ρ:
σp:
Scatter uses standardized returns to visualize shape; portfolio σ uses your σ₁, σ₂, w₁, and ρ.
Optional · Correlation and diversification takeaways

If you’re picking just a handful of stocks, always check pairwise correlations.

  • Prefer low or negative ρ between picks — that’s where diversification actually reduces portfolio σ.
  • Reality check: In the U.S. market, most stocks are positively correlated (they share market/news shocks). So five tech names ≠ diversified.
  • Broaden the mix: Add international stocks, and consider other asset classes (Treasuries, investment-grade bonds, real assets). Cross-market exposure usually lowers average correlations.
  • How to do it fast: Use Sheets to compute 12-month return correlations (see the optional activity above), and try our two-stock portfolio calculator to see how ρ changes portfolio risk.

Note: Correlations can change over time and may rise during market stress.

Topic 3 · Multi-Asset Portfolio Risk & Allocation Optional

Optional, but useful: adding a third stock can reduce firm-specific risk further.

Three-Stock Portfolios: Why Correlation Matters

Next: extend the two-stock portfolio to three stocks, see why each pair’s correlation matters, and briefly explore four stocks.

Watch on YouTube

Three-stock formulas (expanded)
E[Rp] = w1E[R1] + w2E[R2] + w3E[R3]
Var(Rp) = w1²σ1² + w2²σ2² + w3²σ3² + 2w1w2Cov(1,2) + 2w1w3Cov(1,3) + 2w2w3Cov(2,3)
Cov(i,j) = Corr(i,j) × σi × σj
σp = √Var(Rp)

Bridge to CAPM: from 3 stocks to many 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.

  • General rule (textbook idea): once you have a well-chosen portfolio with roughly 20–25+ stocks, most diversifiable (idiosyncratic) risk is already reduced.
  • Important: this does not eliminate systematic risk (market risk).
  • At that point, standard deviation still measures total volatility, but for pricing/required return we focus more on beta.
  • Why? CAPM says investors are compensated for systematic risk, not for firm-specific risk that can be diversified away.

Notes: This is why the chapter shifts from standard deviation (σ) to beta (β).

FIN509 application · Investment committee case

Apply core risk, return, diversification, and CAPM concepts to a practical investment decision.

3–4 Stock Portfolio (Pairwise View — no Σ, full expansion)

Treat a portfolio of 3 or 4 stocks as a collection of pairs.
3 stocks → 3 pairs: (1,2), (1,3), (2,3).
4 stocks → 6 pairs: (1,2), (1,3), (1,4), (2,3), (2,4), (3,4).
Portfolio inputs
Stocks
StockE[R] (%)σ (%)Weight
1 · NVDA
2 · TSLA
3 · WMT
4 · JNJ
Pairwise correlations

Outputs

E[Rp]:
Var(Rp):
σp:
Pairs and their variance contributions (2·wᵢ·wⱼ·ρᵢⱼ·σᵢ·σⱼ):
What’s going on?
  • Expected return (no Σ):
  • Variance (no Σ, fully expanded):
Show full equation with your numbers

As N grows, pair count explodes. For intuition on expected returns with many assets, use CAPM below. For risk with many assets, use matrix tools/Excel (or course calculators).

Topic 4 · Beta, CAPM, the SML & Market Risk Required

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.)

Beta, CAPM & the Security Market Line Explained

Finally: use beta and the CAPM equation to find required return, then see how the same equation becomes the SML.

Watch on YouTube

CAPM is a model
CAPM
MODEL

CAPM: Capital Asset Pricing Model

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.

Open JUFinance.com CAPM Calculator

Systematic vs. Idiosyncratic Risk: What Can Diversification Fix?

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.

Watch on YouTube ↗

Diversification graph: as number of stocks increases (why beta matters later) Required

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).

What is idiosyncratic risk?

Risk specific to one company (or a small group of companies). This can be diversified away.

  • CEO scandal at one firm
  • Factory fire at one company
  • Product recall for one brand
  • Bad earnings surprise for one stock

What is systematic risk?

Market-wide risk that affects many or most stocks. This cannot be diversified away.

  • U.S. recession risk
  • Fed rate shocks / interest-rate changes
  • Inflation surprises
  • Broad market selloff
Location analogy (U.S. → Florida → Jacksonville)

Think of risk layers like geography:

  • U.S.-wide shock (national recession, Fed policy shock) → affects most firms across states. This is like systematic risk.
  • Florida-specific shock (state policy change, hurricane season impact) → affects many Florida firms more than other states. This may be partly diversifiable if your portfolio is nationally diversified.
  • Jacksonville-specific shock (local employer issue, local disruption) → more local and narrower. This is more like idiosyncratic/local risk and is easier to diversify away in a broad portfolio.

Notes: CAPM focuses on the broad market component (systematic risk), which is why beta is used.

Diversification Risk Curve

The curve below shows total portfolio risk falling as the number of stocks increases, then leveling off at systematic risk.

Where can you find beta? (Apple / Walmart / Moderna)

Apple (AAPL) beta

Yahoo Finance: AAPL

Beta (used here): 1.11

Walmart (WMT) beta

Yahoo Finance: WMT

Beta (used here): 0.67

Moderna (MRNA) beta

Yahoo Finance: MRNA

Beta (used here): 1.32

Note: betas vary across sources and time windows. For any assignment, use the same source for all stocks.

CAPM formula & market risk premium — click a term for its meaning

1. CAPM: required return for market risk

= + × [ − ]

Required return = risk-free rate + compensation for the stock’s market risk.

Example: 4% + 1.5 × (10% − 4%) = 13%

2. Market risk premium: calculate this first

= −

Expected market return − risk-free rate = expected extra return for market risk.

Example: 10% − 4% = 6% market risk premium

Then use CAPM = R𝒻 + βᵢ × MRP. If the question already gives MRP, use it directly; do not subtract R𝒻 again.

Now try it: Professor Foley’s CAPM Calculator

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 ↗

CAPM terms: click to show / hide each definition

Click any symbol above to open its definition below, or click a definition heading. Use all rates for the same period.

CAPM — Capital Asset Pricing Model

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.

E(Rᵢ) — CAPM required return

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.

R𝒻 — Risk-free rate

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%.

βᵢ — Beta of investment i

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.

E(Rₘ) — Expected market return

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 — Market risk premium

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.

βᵢ × MRP — Stock’s risk premium

Multiply the market risk premium by the stock’s beta to estimate that stock’s required compensation for market risk. Example: 1.5 × 6% = 9%. Add R𝒻 to get the full CAPM required return: 4% + 9% = 13%.

E( ), i, and m — Read the notation

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.

SML — Security Market Line

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.

CAPM + SML mini-calculator (Apple + Walmart + Moderna on the SML)
JUFinance.com CAPM Calculator

CAPM / SML Results

Market risk premium (MRP): —

Apple CAPM return: —

Walmart CAPM return: —

Moderna CAPM return: —

Custom stock CAPM return: —

Custom stock SML message: —

SML line Apple Walmart Moderna Custom exp.

FIN509 application · Investment committee case

Apply core risk, return, diversification, and CAPM concepts to a practical investment decision.

CAPM & Beta — Required Return from Market Risk

Idea: CAPM says a stock’s expected or required return depends on the risk-free rate plus compensation for market risk.

Main formula

E[Ri] = Rf + βi(E[Rm] − Rf)
Same idea as E[Ri] = Rf + βi × MRP, where MRP = E[Rm] − Rf.

What is beta?

  • β = 1: stock tends to move with the market.
  • β > 1: stock tends to move more than the market.
  • β < 1: stock tends to move less than the market.
  • β < 0: stock tends to move opposite the market.

Beta measures systematic risk, not total risk.

How to get beta

  • Yahoo Finance: Statistics → Beta (5Y Monthly).
  • Finviz: search the ticker and look for the Beta field.
  • Excel / Google Sheets: estimate beta from historical returns using SLOPE.
Beta = slope of (stock returns) on (market returns)

Excel / Sheets beta formulas

Beta using slope: =SLOPE(stock_returns_range, market_returns_range)
Same beta with covariance/variance: =COVARIANCE.S(stock_returns_range,market_returns_range)/VAR.S(market_returns_range)
Portfolio beta: =SUMPRODUCT(weight_rng, beta_rng)
In SLOPE(y,x), stock returns are y and market returns are x.
Open the CAPM Calculator ↗

Quick CAPM — compute E[R]

Tip: you can type 4.5 or 0.045 or 4.5%.
E[R] (CAPM): —
MRP used: —
Plug-in result:
Run “Compute” to show the substitution step.
Show CAPM math work (step by step)
Run “Compute” to generate the steps.
Try examples (teaching only; look up live betas on Yahoo or Finviz before using):

Security Market Line (SML)

The SML starts at (β=0, Rf) and passes through (β=1, E[Rm]).

Why beta and diversification belong together

  • σ measures total stand-alone risk.
  • Beta measures market-related risk.
  • Diversification reduces firm-specific risk, but not market risk.
  • That is why higher-beta stocks usually require higher expected returns.

FIN509 application · Investment committee case

★ Optional Excel ICE (November 10) · Up to 100 Additional Chapter 13 Homework Points · Click to expand
FIN509 · Week 6 · Optional Extra CreditNovember 10, 2026

Optional Extra Credit · Excel Risk & Return ICE

Use the supplied AAPL, WMT, NVDA, and S&P 500 return data. Follow the Excel walkthrough, reproduce the results and Security Market Line (SML), and submit your completed workbook for optional extra credit.

Open FIN509 Chapter 13 Excel ICE Instructions → Download Excel Data ↓

FIN509 MBA · Shared Excel workbook with a FIN509 Chapter 13 solutions video.

Follow the FIN509 step-by-step solutions for the 59 matched monthly returns from the September 2021–August 2026 price data. Watch on YouTube ↗

Complete the exercise in three steps

  1. Open the instructions and download the Excel workbook.
  2. Follow the video and reproduce each result in your own spreadsheet.
  3. Check your final answers, save your workbook, and submit it on Blackboard.
OPTIONAL ICE — UP TO 100 ADDITIONAL POINTS FOR CHAPTER 13 HOMEWORK

You can earn up to 100 additional points added to your Chapter 13 homework grade by completing this optional in-class exercise (ICE). Work through the supplied data in Excel, show your formulas and SML graph, and submit your own finished workbook on Blackboard.

MBA interpretation: Briefly interpret the correlations, compare the three stock betas, discuss diversification and systematic risk, and explain what the SML implies for expected returns.

Final Excel checklist (click to expand)

These boxes are a personal review aid; check your own Excel file before submitting.

Homework — Week 6 (Questions with Answers Only)

  1. AAA firm’s stock has a 0.25 possibility to make 30.00% return, a 0.50 chance to make 12% return, and a 0.25 possibility to make −18% return. Calculate expected rate of return.
    Answer: 9%
  2. If investors anticipate a 7.0% risk-free rate, the market risk premium = 5.0%, beta = 1, find the return.
    Answer: 12%
  3. AAA firm has a portfolio with a value of $200,000 with the following four stocks. Calculate the beta of this portfolio.
    StockValueβ
    A$50,000.000.9500
    B$50,000.000.8000
    C$50,000.001.0000
    D$50,000.001.2000
    Total$200,000.00
    Answer: 0.988
  4. A portfolio with a value of $40,000,000 has a beta = 1. Risk-free rate = 4.25%, market risk premium = 6.00%. An additional $60,000,000 will be included in the portfolio. After that, the expected return should be 13%. Find the average beta of the new stocks to achieve the goal.
    Answer: 1.76
  5. Stock A has the following returns for various states of the economy. Stock A’s expected return? Standard deviation?
    StateProbabilityReturn
    Recession10%−30%
    Below Average20%−2%
    Average40%10%
    Above Average20%18%
    Boom10%40%
    Answer: E[R] = 8.2%, Variance = 0.02884, σ = 16.98% (calculator)
  6. Collectibles Corp. has a beta of 2.5 and a standard deviation of returns of 20%. The return on the market portfolio is 15% and the risk-free rate is 4%. What is the risk premium on the market?
    Answer: 11%
  7. An investor currently holds the following portfolio. The beta for the portfolio is?
    HoldingAmount Investedβ
    8,000 shares of Stock A$16,0001.3
    15,000 shares of Stock B$48,0001.8
    25,000 shares of Stock C$96,0002.2
    Answer: 1.99
  8. Deleted.
  9. Assume that you have $165,000 invested in a stock that is returning 11.50%, $85,000 invested in a stock that is returning 22.75%, and $235,000 invested in a stock that is returning 10.25%. What is the expected return of your portfolio?
    Answer: 12.87%
  10. If you hold a portfolio made up of the following stocks, what is the beta of the portfolio?
    StockInvestment Valueβ
    A$8,0001.5
    B$10,0001.0
    C$2,0000.5
    Answer: 1.15
  11. You own a portfolio consisting of the stocks below. The risk-free rate is 3% and market return is 10%.
    StockPercentage of Portfolioβ
    120%1.0
    230%0.5
    350%1.6
    (a) Answer: βp = 1.15
    (b) Answer: E[Rp] = 11.05%
  12. An investor currently holds the following portfolio. Calculate the beta for the portfolio.
    HoldingAmount Investedβ
    8,000 shares of Stock A$10,0001.5
    15,000 shares of Stock B$20,0000.8
    25,000 shares of Stock C$20,0001.2
    Answer: 1.1

Previous Homework & Quiz Help — YouTube Links

Q1 & Q5

Watch older video on YouTube ↗

Q2 & Q3

Watch older video on YouTube ↗

Q4, Q6 & Q7

Watch older video on YouTube ↗

Q9–End (Q9–Q12)

Watch older video on YouTube ↗

Quiz Help (Videos)

Part I

Watch older video on YouTube ↗

Part II

Watch older video on YouTube ↗

FIN509 Homework & Practice · Chapter 13

Show your steps. Use Excel functions where appropriate: SUMPRODUCT, STDEV/STDEV.S, CORREL, SLOPE.

Homework questions (official list with provided answers)
  1. Stock A has the following returns for various states of the economy. What is Stock A’s expected return?
    State of the EconomyProbabilityStock A’s Return
    Recession10%-30%
    Below Average20%-2%
    Average40%10%
    Above Average20%18%
    Boom10%40%

    Answer: 8.2%

  2. Joe purchased 800 shares of Robotics Stock at $3 per share on 1/1/19 and sold the shares on 12/31/19 for $3.45. Robotics stock has a beta of 1.9, the risk-free rate of return is 4%, and the market risk premium is 9%. What is Joe’s holding period return?

    Answer: 15%

  3. You own a portfolio with the following expected returns given various states of the economy. What is the overall portfolio expected return?
    State of EconomyProbability of StateRate of Return if State Occurs
    Boom27%14%
    Normal70%8%
    Recession3%-11%

    Answer: 9.05%

  4. The prices for Electric Circuit Corporation for the first quarter of 2019 are given below. The price on January 1, 2019 was $130. Find the holding period return for an investor who purchased the stock on January 1, 2019 and sold it on the last day of March 2019.
    Month EndPrice
    January$125.00
    February$138.50
    March$132.75

    Answer: 2.12%

  5. Collectibles Corp. has a beta of 2.5 and a standard deviation of returns of 20%. The return on the market portfolio is 15% and the risk-free rate is 4%. What is the risk premium on the market?

    Answer: 11%

  6. An investor currently holds the following portfolio. What is the beta for the portfolio?
    HoldingAmount InvestedBeta
    Stock A (8,000 shares)$16,0001.3
    Stock B (15,000 shares)$48,0001.8
    Stock C (25,000 shares)$96,0002.2

    Answer: 1.99

  7. Assume that you have $165,000 invested in a stock returning 11.50%, $85,000 invested in a stock returning 22.75%, and $235,000 invested in a stock returning 10.25%. What is the expected return of your portfolio?

    Answer: 13%

  8. If you hold a portfolio made up of the following stocks, what is the beta of the portfolio?
    StockInvestment ValueBeta
    Stock A$8,0001.5
    Stock B$10,0001.0
    Stock C$2,0000.5

    Answer: 1.15

  9. The risk-free rate of return is 3.9% and the market risk premium (Rm − Rf) is 6.2%. What is the expected rate of return on a stock with a beta of 1.21?

    Answer: 11.4%

  10. You own a portfolio consisting of the stocks below.
    StockPercentage of PortfolioBeta
    120%1.0
    230%0.5
    350%1.6

    The risk-free rate is 3% and the market return is 10%.

    • (a) Calculate the portfolio beta. Answer: 1.15
    • (b) Calculate the expected return of your portfolio. Answer: 11.05%
  11. Compute the holding period return for Jazman and Solomon for period 1 through 3 (bought in period 1 and sold in period 3). Show the holding period returns for each company.
    PeriodJazmanSolomon
    1$10$20
    2$12$25
    3$15$15

    Answer: 50%, -25%

  12. Calculate expected return.
    State of the EconomyProbability% Return (Cash Flow / Investment Cost)
    Economic Recession30%5%
    Strong and Moderate Economic Growth70%15%

    Answer: 12%

  13. Calculate the expected returns of the following cases, respectively.
    1. Invest $10,000 in Treasury bill with guaranteed return of 4%. Answer: 4%
    2. Invest $10,000 in Apple. 50% possibility to earn 20% return and 50% possibility to lose 10% of investment. Answer: 5%
    3. Invest $10,000 in Wal-Mart. 50% possibility to earn 5% return and 50% possibility to earn 0% return. Answer: 2.5%
  14. Rank the risk of the following cases from the least risky to the most risky.
    1. Invest $10,000 in Treasury bill with guaranteed return of 4%.
    2. Invest $10,000 in Apple. 50% possibility to earn 20% return and 50% possibility to lose 10% of investment.
    3. Invest $10,000 in Wal-Mart. 50% possibility to earn 5% return and 50% possibility to earn 0% return.

    Answer: 1, 3, 2

  15. An investor currently holds the following portfolio. Calculate the beta for the portfolio.
    HoldingAmount InvestedBeta
    Stock A (8,000 shares)$10,0001.5
    Stock B (15,000 shares)$20,0000.8
    Stock C (25,000 shares)$20,0001.2

    Answer: 1.1

The additional FIN301-derived exercises below are optional foundation practice; verify graded FIN509 assignments and due dates in Blackboard.

★ NEW · Featured homework walkthrough · Natasha voice

Chapter 13 Homework: All 15 Problems Explained — Natasha

New homework walkthrough with Natasha’s narration: work through all 15 questions, including expected returns, holding period returns, portfolio beta, CAPM, Excel formulas, and graphs. Pause to practice each question.

Watch Natasha’s 15-question homework walkthrough on YouTube ↗

★ Check your understanding

Chapter 13 Practice Quizzes

Try three true/false practice quizzes. Each answer gives you immediate feedback.

Quiz 1

Correlation and diversification

Open Quiz 1 ↗

Quiz 2

CAPM, beta, and the Security Market Line

Open Quiz 2 ↗

Quiz 3

Systematic and idiosyncratic risk

Open Quiz 3 ↗
★ Check your calculations

JUFinance Calculators

Show your formula and steps first, then use these tools to check your answers.

Expected Return & Risk

Probability and return tables, including homework questions 1, 3, and 12.

Open calculator ↗

Holding Period Return

Purchase price, dividends, and sale price for questions 2, 4, and 11.

Open calculator ↗

CAPM

Risk-free rate, market return, and beta for questions 9 and 10.

Open calculator ↗

Two-Stock Portfolio

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.

FIN509 application · Investment committee case

Resources

Appendix: Risk and Return in Real Life ★ Optional

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.

Diversification: Why One Stock Isn’t Enough

See how combining investments can change portfolio risk.

Watch on YouTube

Beta, CAPM, and Undervalued Stocks

Apply required return to a possible stock choice.

Watch on YouTube
★ Real data lesson

Where Does Stock Beta Come From? Optional

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.

Watch the beta lesson on YouTube ↗

Chapter 13 Summary: What should I do with these tools?

MBA takeaway: evaluate both expected returns and covariance, distinguish diversifiable risk from market risk, and defend portfolio decisions using CAPM, factor exposures, risk-adjusted performance, and scenario analysis.

Ask two questions: What return do I expect, and what risk am I taking? Then select the tool that answers your question.

Your situationWhat to calculateWhat it tells you
One stockExpected 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 stocksInvestment 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 portfolioPortfolio 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.
Two-stock formulas — click to review
E(Rₚ) = w₁E(R₁) + w₂E(R₂), where w₁ + w₂ = 1
σₚ = √[w₁²σ₁² + w₂²σ₂² + 2w₁w₂σ₁σ₂ρ₁₂]

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.

CAPM: follow these three steps

  1. Calculate market risk premium: MRP = E(Rₘ) − R𝒻. Example: 10% − 4% = 6%.
  2. Calculate required return: R𝒻 + βᵢ × MRP. For β = 1.5: 4% + 1.5 × 6% = 13%.
  3. Compare: if your independent forecast is 15%, it exceeds the 13% CAPM benchmark by 2 percentage points. If it is 11%, it falls short by 2 percentage points. Actual returns may differ.
Remember: Standard deviation measures total risk. Correlation helps explain diversification. Beta measures market risk. CAPM estimates the required return for that market risk.

Review CAPM and its term definitions ↑

Topic 6 · Factor Models Beyond CAPM (Graduate) Optional

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

FIN509 application · Investment committee case

Apply core risk, return, diversification, and CAPM concepts to a practical investment decision.

MBA Application · Investment Committee Case Optional

Case: You advise a $10 million endowment. The board is evaluating a concentrated growth-stock portfolio versus a broad diversified portfolio. The growth strategy has higher assumed return but also greater volatility and beta. A recent market downturn has exposed liquidity needs and unstable correlations.

  1. Use the foundational two-stock portfolio calculator to compare two representative risky assets. Explain how correlation changes total portfolio volatility and why diversification matters.
  2. Evaluate two managers using the . Clearly state which benchmark and estimation period you assume.
  3. Run a −30% market stress test. Describe the effect of time-varying beta and correlation breakdown.
  4. Use the the CAPM model to explain whether a reported alpha is potentially compensation for systematic factor risk.
  5. Write a 200–300-word investment committee recommendation with assumptions, constraints, risks and a clearly justified allocation. There is no single correct answer.
Suggested evaluation guide (for self-study)

30% correct return/risk calculations; 25% diversification and correlation reasoning; 20% performance and factor interpretation; 15% stress-testing limitations; 10% communication and suitability. This exercise is supplemental and does not change the existing FIN509 homework requirements.

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