FIN509 • Week 6 — Chapter 13: Risk & Return

Intro — Why Risk Matters (Plain English first)

Big idea: Over time, higher expected return usually comes with higher risk. You’re paid for taking uncertainty.

Return is easy: compare prices and add dividends. If you buy at P₀, sell at P₁, and get dividend D, your period return is HPR = (P₁ − P₀ + D) / P₀.
“Risk” = loss potential + volatility: bigger, more frequent swings away from average → riskier.
Portfolios: what matters is how assets move together. Low/negative correlation reduces total risk.

Show math formulas

Expected return: E[R] = Σ p_i R_i
Variance & σ: Var = Σ p( R−E[R])², σ = √Var
Two-asset:
E[R_p] = w₁E[R₁] + w₂E[R₂]
Var = w₁²σ₁² + w₂²σ₂² + 2w₁w₂Cov
CAPM: E[R] = Rf + β( E[Rm] − Rf )

Slides (PPT)

If the embed is blocked by your LMS or browser, use the open-in-new-tab link.

Open slides ↗

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: Auto-normalize probabilities to sum to 1

Probabilities p_i Returns R_i (in %)

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) ↗

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: Auto-normalize probabilities to sum to 1

Probabilities p_i Stock 1 returns (%) Stock 2 returns (%)

Weight w₁ (w₂ auto = 1 − w₁)

w₁ (0–1) w₂ (auto)

0.50

E[R₁], σ₁:

E[R₂], σ₂:

Cov(1,2):

ρ(1,2):

E[R_p]:

Var(R_p):

σ_p:

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[R_p]: =w1_cell*ER1_cell + w2_cell*ER2_cell

Var(R_p): =w1_cell^2*Var1_cell + w2_cell^2*Var2_cell + 2*w1_cell*w2_cell*Cov_cell

σ_p: =SQRT(VarP_cell)

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).

How many stocks? Auto-normalize weights to 1

Inputs

Stock

E[R] (%)

σ (%)

1 (e.g., NVDA)

2 (e.g., TSLA)

3 (e.g., WMT)

Weights

w₁

w₂

w₃

Correlations ρ

ρ₁₂

ρ₁₃

ρ₂₃

Outputs

E[R_p]:

Var(R_p):

σ_p:

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).

Risk & Return — Why Diversify and How

Big ideas

Higher expected return usually comes with higher risk.
σ measures stand-alone total volatility.
Covariance and correlation tell us whether assets move together.
Lower correlation usually means better diversification.
Diversification reduces firm-specific risk, but not all market risk.

Teaching path

Start with one stock: expected return and σ.
Then two stocks: covariance, correlation, and portfolio σ.
Then 3–4 stocks: pairwise diversification effects.
Then beta: market-related risk.
Then CAPM: required return from beta.

Why students should care

A stock can have high stand-alone risk, but once it is placed inside a portfolio the key question becomes: how does it move with the other assets? That is why diversification matters. After diversification, the market still cares about beta, which leads to CAPM.

Useful tools for later topics

Later in finance, students can connect this chapter to hedging tools such as options, forwards, and futures.

Correlation — what it means (NVDA vs TSLA/AAPL/WMT)

Correlation (ρ) measures how two returns move together. It lives between −1 and +1. ρ ≈ +1: move together; ρ ≈ 0: unrelated; ρ ≈ −1: move opposite. In a 2-stock portfolio, lower ρ ⇒ lower risk for the same weights/volatilities.

How to get real stock data in Google Sheets

For real data, students can use either: (1) direct Google Sheets formulas or (2) the Stock Data Fetcher app.

Method 1 — Pull data directly in Google Sheets

Why use this method? It is best for learning because you can see every step: prices → returns → correlation.

In a blank Google Sheet, enter these formulas:
=GOOGLEFINANCE("NASDAQ:NVDA","close",TODAY()-370,TODAY(),"DAILY")

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

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

=GOOGLEFINANCE("NYSE:WMT","close",TODAY()-370,TODAY(),"DAILY")

=GOOGLEFINANCE("INDEXSP:.INX","close",TODAY()-370,TODAY(),"DAILY")
Next to each price series, compute returns:
=C3/C2-1
Fill down.
Then compute correlations:
=CORREL(NVDA_returns_range,TSLA_returns_range)

=CORREL(NVDA_returns_range,AAPL_returns_range)

=CORREL(NVDA_returns_range,WMT_returns_range)

=CORREL(NVDA_returns_range,SPX_returns_range)

This method is the most transparent because students see exactly where the data comes from and how the correlation is calculated.

Method 2 — Use the Stock Data Fetcher app

This app is good for collecting data quickly: Open Stock Data Fetcher ↗

Type a ticker such as NASDAQ:NVDA, NASDAQ:TSLA, NASDAQ:AAPL, or NYSE:WMT.
Choose the start date and end date.
Click Fetch Data.
The app writes the inputs into the Google Sheet and inserts the GOOGLEFINANCE formula automatically.
After the prices appear, students can compute returns and then use CORREL().

This method is faster and cleaner for students who do not want to type all formulas manually.

Compute 12-month correlations (how to do it)

Pull daily closes for past ~1 year (works for stocks & S&P 500 index):
=GOOGLEFINANCE("NVDA","close",TODAY()-370,TODAY())

=GOOGLEFINANCE("TSLA","close",TODAY()-370,TODAY())

=GOOGLEFINANCE("AAPL","close",TODAY()-370,TODAY())

=GOOGLEFINANCE("WMT","close",TODAY()-370,TODAY())

=GOOGLEFINANCE("INDEXSP:.INX","close",TODAY()-370,TODAY())
Make returns (daily is fine for correlation). Example in the cell next to the 3rd close:
=C3/C2 - 1
Fill down for each series (NVDA, TSLA, AAPL, WMT, and the S&P 500 range).
Compute correlations:
=CORREL(NVDA_returns_range, TSLA_returns_range)

=CORREL(NVDA_returns_range, AAPL_returns_range)

=CORREL(NVDA_returns_range, WMT_returns_range)

=CORREL(NVDA_returns_range, SPX_returns_range)
Want monthly instead of daily? Pull dailies, then “month-end filter” (e.g., with a helper column for EOMONTH) and compute returns from those month-end closes before CORREL.

Offline demo: load a teaching example (hypothetical values).

Pair	ρ (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

σ₁ (%) σ₂ (%) w₁

ρ (correlation)

ρ:

σ_p:

Scatter uses standardized returns to visualize shape; portfolio σ uses your σ₁, σ₂, w₁, and ρ.

Advice alert — Correlation & real diversification

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 steps below), and try our two-stock portfolio calculator to see how ρ changes portfolio risk.

Note: Correlations move over time and spike toward 1 in crises. Re-check periodically.

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[R_i] = R_f + β_i(E[R_m] − R_f)

Same idea as E[R_i] = R_f + β_i × MRP, where MRP = E[R_m] − R_f.

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.

Quick CAPM — compute E[R]

Tip: you can type 4.5 or 0.045 or 4.5%.

Risk-free R_f (%) Use: Market return MRP

Beta β

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.

Holding Period Return (HPR)

Purchase price P₀ Sell price P₁ Dividends D

HPR:

Formula: HPR = (P₁ − P₀ + D) / P₀

Homework — Week 6 (Questions with Answers Only)

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%
If investors anticipate a 7.0% risk-free rate, the market risk premium = 5.0%, beta = 1, find the return.
Answer: 12%
AAA firm has a portfolio with a value of $200,000 with the following four stocks. Calculate the beta of this portfolio.

Stock Value β

A $50,000.00 0.9500

B $50,000.00 0.8000

C $50,000.00 1.0000

D $50,000.00 1.2000

Total $200,000.00

Answer: 0.988
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
Stock A has the following returns for various states of the economy. Stock A’s expected return? Standard deviation?

State Probability Return

Recession 10% −30%

Below Average 20% −2%

Average 40% 10%

Above Average 20% 18%

Boom 10% 40%

Answer: E[R] = 8.2%, Variance = 0.02884, σ = 16.98% (calculator)
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%
An investor currently holds the following portfolio. The beta for the portfolio is?

Holding Amount Invested β

8,000 shares of Stock A $16,000 1.3

15,000 shares of Stock B $48,000 1.8

25,000 shares of Stock C $96,000 2.2

Answer: 1.99
Deleted.
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%
If you hold a portfolio made up of the following stocks, what is the beta of the portfolio?

Stock Investment Value β

A $8,000 1.5

B $10,000 1.0

C $2,000 0.5

Answer: 1.15
You own a portfolio consisting of the stocks below. The risk-free rate is 3% and market return is 10%.

Stock Percentage of Portfolio β

1 20% 1.0

2 30% 0.5

3 50% 1.6

(a) Answer: β_p = 1.15

(b) Answer: E[R_p] = 11.05%
An investor currently holds the following portfolio. Calculate the beta for the portfolio.

Holding Amount Invested β

8,000 shares of Stock A $10,000 1.5

15,000 shares of Stock B $20,000 0.8

25,000 shares of Stock C $20,000 1.2

Answer: 1.1

Stock	Value	β
A	$50,000.00	0.9500
B	$50,000.00	0.8000
C	$50,000.00	1.0000
D	$50,000.00	1.2000
Total	$200,000.00

State	Probability	Return
Recession	10%	−30%
Below Average	20%	−2%
Average	40%	10%
Above Average	20%	18%
Boom	10%	40%

Holding	Amount Invested	β
8,000 shares of Stock A	$16,000	1.3
15,000 shares of Stock B	$48,000	1.8
25,000 shares of Stock C	$96,000	2.2

Stock	Investment Value	β
A	$8,000	1.5
B	$10,000	1.0
C	$2,000	0.5

Stock	Percentage of Portfolio	β
1	20%	1.0
2	30%	0.5
3	50%	1.6

Holding	Amount Invested	β
8,000 shares of Stock A	$10,000	1.5
15,000 shares of Stock B	$20,000	0.8
25,000 shares of Stock C	$20,000	1.2

Homework Help — Videos

Quiz Help (Videos)

Part I

Open on YouTube ↗

Part II

Open on YouTube ↗

Quick Quiz (FYI only)

Click to check answers.

Intro — Why Risk Matters (Plain English first)

Slides (PPT)

Single Stock — Expected Return & Standard Deviation

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

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

Inputs

Outputs

Risk & Return — Why Diversify and How

Big ideas

Teaching path

Why students should care

Useful tools for later topics

Correlation — what it means (NVDA vs TSLA/AAPL/WMT)

How to get real stock data in Google Sheets

Compute 12-month correlations (how to do it)

See diversification change as ρ moves

Advice alert — Correlation & real diversification

CAPM & Beta — Required Return from Market Risk

Main formula

What is beta?

How to get beta

Excel / Sheets beta formulas

Quick CAPM — compute E[R]

Security Market Line (SML)

Why beta and diversification belong together

Holding Period Return (HPR)

Homework — Week 6 (Questions with Answers Only)

Homework Help — Videos

Q1 & Q5

Q2 & Q3

Q4, Q6 & Q7

Q9–End (Q9–Q12)

Quiz Help (Videos)

Part I

Part II

Quick Quiz (FYI only)

Resources