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 math with your inputs

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]:

σ_p:

What this mix means (next-year story):

Choose a state count, enter probabilities & both return vectors, set w₁, then Compute.

Show math with your inputs

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).

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.

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 — First Model of Expected Return

Idea (plain English): Your expected return = a risk-free rate plus a bonus for taking market risk. The bonus is the market’s extra return (MRP) scaled by your stock’s beta (how sensitive it is to the market).

Formula:

E[R_i] = R_f + β_i · (E[R_m] − R_f)

R_f (risk-free): a Treasury yield. Use a 3-mo T-bill for short projects or a 10-yr note for long horizon work.
β (beta): “If the market moves 1%, this stock tends to move β%.” β≈1 market-like; β>1 amplifies; β<1 dampens; β<0 goes opposite.
MRP (E[R_m]−R_f): market’s extra return over the risk-free. Textbook classroom values are ~5–6% in the U.S.

Quick CAPM — compute E[R]

Tip: you can type 2 or 0.02 or 2% — all treated as 2%.

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

Beta β

E[R] (CAPM): —

MRP used: —

Try examples (teaching only; look up live betas on Yahoo/Finviz before using):

Security Market Line (SML)

Line from (β=0, R_f) to (β=1, E[R_m]). Your stock sits at (β, E[R]) on this line if CAPM holds.

What news matters for β & expected returns?

Systematic (market-wide) — affects most stocks

Fed policy changes, inflation/CPI prints, recession news
Trade wars, new tariffs, geopolitical shocks, oil supply shocks
Pandemics, financial system stress (credit/liquidity)

These push the whole market, so they drive the MRP and matter for everyone’s E[R].

Idiosyncratic (firm-specific) — diversifiable

Earnings surprises/guidance, product recalls, lawsuits
CEO/CFO news, exec comments
Customer wins/losses, factory outages

These mostly change that firm’s price, not the market’s. In large portfolios, this risk diversifies away.

Rule of thumb: Trade-war headline → systematic. CEO quote about one product → often idiosyncratic.

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%. a.Calculate the portfolio beta. (answer 1.15). b. Calculate the expected return of your portfolio (refer to CAPM calculator at https://www.jufinance.com/capm/). (answer 11.05%)

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

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

Compute 12-month correlations (how to do it)

See diversification change as ρ moves

Advice alert — Correlation & real diversification

CAPM — First Model of Expected Return

Quick CAPM — compute E[R]

Security Market Line (SML)

What news matters for β & expected returns?

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