Use these to verify hand calculations after you finish your own work.
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
Expected Return = E[R] = Σ (p × R)
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
Variance: Var(R) = Σ { p × [R − E(R)]² }
Standard deviation: σ = √Var(R)
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).
Topic 1: One stock (textbook Apple example) → Real data (Apple + Moderna)
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.
Textbook example (Apple): states of the economy
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).
Textbook Step 2: Compute expected return for Apple (show the math)
E[RA] = Σ (p × R)
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%.
Textbook Step 3: Compute Apple variance + standard deviation (show the math)
Var(RA) = Σ { p × [R − E(RA)]² }
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
σA = √0.028836 = 0.1698 = 16.98%
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.
Variance is shown in decimal units (for example, 0.028836).
Topic 2: Two-stock portfolio — Apple + Moderna (using Excel data)
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.
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).
Topic 4: Beta, CAPM, and the Security Market Line (SML)
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.)
Diversification graph: as number of stocks increases (why beta matters later)
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)
Chapter 6 Homework (due with the second midterm exam)
Show your steps. Use Excel functions where appropriate:
SUMPRODUCT, STDEV/STDEV.S, CORREL, SLOPE.
Homework questions (official list with provided answers)
Stock A has the following returns for various states of the economy. What is Stock A’s expected return?
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%
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%
You own a portfolio with the following expected returns given various states of the economy. What is the overall portfolio expected return?
State of Economy
Probability of State
Rate of Return if State Occurs
Boom
27%
14%
Normal
70%
8%
Recession
3%
-11%
Answer: 9.05%
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 End
Price
January
$125.00
February
$138.50
March
$132.75
Answer: 2.12%
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. What is the beta for the portfolio?
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
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%
If you hold a portfolio made up of the following stocks, what is the beta of the portfolio?
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
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%
You own a portfolio consisting of the stocks below.
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%.
(a) Calculate the portfolio beta. Answer: 1.15
(b) Calculate the expected return of your portfolio. Answer: 11.05%
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.
Period
Jazman
Solomon
1
$10
$20
2
$12
$25
3
$15
$15
Answer: 50%, -25%
Calculate expected return.
State of the Economy
Probability
% Return (Cash Flow / Investment Cost)
Economic Recession
30%
5%
Strong and Moderate Economic Growth
70%
15%
Answer: 12%
Calculate the expected returns of the following cases, respectively.
Invest $10,000 in Treasury bill with guaranteed return of 4%. Answer: 4%
Invest $10,000 in Apple. 50% possibility to earn 20% return and 50% possibility to lose 10% of investment. Answer: 5%
Invest $10,000 in Wal-Mart. 50% possibility to earn 5% return and 50% possibility to earn 0% return. Answer: 2.5%
Rank the risk of the following cases from the least risky to the most risky.
Invest $10,000 in Treasury bill with guaranteed return of 4%.
Invest $10,000 in Apple. 50% possibility to earn 20% return and 50% possibility to lose 10% of investment.
Invest $10,000 in Wal-Mart. 50% possibility to earn 5% return and 50% possibility to earn 0% return.
Answer: 1, 3, 2
An investor currently holds the following portfolio. Calculate the beta for the portfolio.