Chapter 7

Introduction to Risk and

Return
Definitions

I The return of a stock:

I r= Dividend + Final Price − Initial Price
Initial Price

I When you buy a stock, you know the initial price, but you
don’t know what it will be the final price after a period.
I Stock returns are random variables.
I Many possible scenarios that lead to different returns.

I The firm isn’t a black box: you can learn about the possible
scenarios and how likely they are. You may:
I Analyze the firm thoroughly to predict the future.
I Learn from the past to predict the future (easy way).
Definitions

I The return of a stock:

I r= Dividend + Final Price − Initial Price
Initial Price

I When you buy a stock, you know the initial price, but you
don’t know what it will be the final price after a period.
I Stock returns are random variables.
I Many possible scenarios that lead to different returns.

I The firm isn’t a black box: you can learn about the possible
scenarios and how likely they are. You may:
I Analyze the firm thoroughly to predict the future.
I Learn from the past to predict the future (easy way).
The Expected Return
Learning from the past to predict the future...

I We can calculate the average of past returns to form an idea

of what, more or less, we can expect to get.
I Expected return (ER): the average of the returns of all
possible scenarios; the true average of what you can get.
I Whatever the return may be, it will be around ER.
I But you can’t calculate the true average, ER, unless you know
every possible scenario. It requires more than a thorough
analysis of the firm.

I The average of past returns ≈ expected return.

I Average past returns provide a good approximation of what
you can expect to get as long as the firm and the possible
scenarios have not changed much.
Risk Measures

What is risk?

I Multiple definitions and measures. Examples:

I The risk of losing money.
I Worst case scenario.
I How scattered/disperse/volatile they are.

I Our measure: How spread out the returns are around the
expected return. How much they vary around the ER.
I Popular: Variance (VAR) and standard deviation (SD).
Measuring Risk: Variance & Standard Deviation
I Variance: is the average squared deviation from the expected
return. It’s a measure of dispersion.
I Excel function: “VAR.P”.

I Standard Deviation: The square root of variance.

I Excel function: “STDEV.P”.
I It’s the average difference between returns and the expected
return. It’s a measure of dispersion.

Notice that:
I You never observe the true SD and VAR.

I Sample SD and VAR ≈ true SD and VAR (as long as the firm
and the possible scenarios have not changed much).
Measuring Risk: Variance & Standard Deviation

Exercise Set 1:
1. Firm A had -12 percent, 24 percent, and 18 percent rates of
return during the last three years; calculate ER, VAR, and SD.
If you invest in firm A, what return do you expect to get in a
year, more or less?

2. Firm B had 1 percent, 11 percent, and 18 percent rates of

return during the last three years; calculate ER, VAR, and SD.
If you invest in firm B, what is the average difference between
the return you actually get and what you, more or less,
expected to get?

3. With which firm you can form better predictions of the return
you will get (i.e., more certainty)?
Measuring Risk: Variance & Standard Deviation
Exercise Set 2:
1. Calculate the monthly ER and SD of Alphabet Inc. (GOOGL)
using end-of-month adjusted close prices from February 2011
to February 2021.

2. Calculate the monthly ER and SD of Microsoft Corp. (MSFT)

using end-of-month adjusted close prices from February 2011
to February 2021.

3. Calculate the monthly ER and SD of Apple Inc. (AAPL) using

end-of-month adjusted close prices from February 2011 to
February 2021.

4. Calculate the monthly ER and SD of Amazon, Inc. (AMZN)

using end-of-month adjusted close prices from February 2011
to February 2021.
Review and Next Step

Review:
I Expected return.

I Risk.

I How do we measure “risk”?

Next step:

1. How can we decrease the risk?

Review and Next Step

Review:
I Expected return.

I Risk.

I How do we measure “risk”?

Next step:

1. How can we decrease the risk? Portfolio diversification.

Portfolio

I Portfolio: to invest in a number financial assets such as

stocks.
I Portfolio return, rp : weighted sum of the stock returns:

rp = wS1 × rS1 + wS2 × rS2 + wS3 × rS3 + ...

I rS1 , rS2 , rS3 , ... : the returns of each individual stock in your
portfolio.
I wS1 , wS2 , wS3 , ... : the weights of each individual stock in your
portfolio (the proportions of money invested in each stock).
I e.g. wS3 = Money invested in stock S3
Total money invested in the portfolio
.

I Excel function: SUMPRODUCT(weights, returns)

Portfolio Diversification - Real Data

Exercise Set 3:

1. Calculate the monthly between February 2011 to February

2021 of an equal-weights portfolio of Google, Microsoft,
Apple, and Amazon.

2. Compare the ER of the portfolio to the average of the ERs of

the stocks. What do you observe?
• ERp = wS1 × ERS1 + wS2 × ERS2 + wS3 × ERS3 + ...

3. Compare the SD of the portfolio to the average of the SDs of

the stocks. What do you observe?
• SDp < wS1 × SDS1 + wS2 × SDS2 + wS3 × SDS3 + ...
Portfolio Diversification - Real Data

Exercise Set 3:

1. Calculate the monthly between February 2011 to February

2021 of an equal-weights portfolio of Google, Microsoft,
Apple, and Amazon.

2. Compare the ER of the portfolio to the average of the ERs of

the stocks. What do you observe?
I ERp = wS1 × ERS1 + wS2 × ERS2 + wS3 × ERS3 + ...

3. Compare the SD of the portfolio to the average of the SDs of

the stocks. What do you observe?
• SDp < wS1 × SDS1 + wS2 × SDS2 + wS3 × SDS3 + ...
Portfolio Diversification - Real Data

Exercise Set 3:

1. Calculate the monthly between February 2011 to February

2021 of an equal-weights portfolio of Google, Microsoft,
Apple, and Amazon.

2. Compare the ER of the portfolio to the average of the ERs of

the stocks. What do you observe?
I ERp = wS1 × ERS1 + wS2 × ERS2 + wS3 × ERS3 + ...

3. Compare the SD of the portfolio to the average of the SDs of

the stocks. What do you observe?
I SDp < wS1 × SDS1 + wS2 × SDS2 + wS3 × SDS3 + ...
Portfolio Diversification - Lessons & Next steps

Huge takeaway:

I Diversification reduces risk without decreasing your

average return.

Next steps:

1. What is the best way to diversify?

2. How/Why does it work?

Measuring Portfolio Returns - Example & Exercises

Exercise Set 4: Use ’DATA 2’ tab in the excel file “Portfolio

Diversification-DUMMY” to complete the following exercises:

1. Calculate the ER and SD of an equal-weights portfolio of

stock A.

2. Calculate the ER and SD of an equal-weights portfolio of

stocks A and B.

3. Calculate the ER and SD of an equal-weights portfolio of

stocks A, B, and C.

4. Calculate the ER and SD of an equal-weights portfolio of

stocks A, B, C, and D.

5. Calculate the ER and SD of an equal-weights portfolio of

stocks A, B, C, D, and E.
Portfolio Diversification

I Let’s plot ERs and the SDs of the above portfolios (next
slide).
I On the x-axis, we indicate the number of stocks in the
portfolios, and on the y-axis, the ERs and SDs.

I 1st huge conclusion:

I The more stocks you include in your portfolio, the more
diversification you achieve.

I Diversification has a limit. You will not eliminate all the risk by
including infinite stocks in your portfolio
Portfolio Diversification

Portfolio Diversification
3.00%

2.50%

2.00%

1.50%

1.00%

0.50%

0.00%
1 2 3 4 5
Number of Stocks in Portfolio

Avg Return Risk (s.d.)

Portfolio Diversification - Exercises

I What types of stocks should you include in your portfolio to

achieve more diversification?

Exercise Set 5:
1. Calculate the ER and SD of an equal-weights portfolio of
stocks A, B, C, D, and E, using the data in “Data 1” tab.

2. Calculate the ER and SD of an equal-weights portfolio of

stocks A, B, C, D, and E, using the data in “Data 3” tab.

3. Calculate the ER and SD of an equal-weights portfolio of

stocks A, and B using the data in “Data 4” tab.
Portfolio Diversification

I Let’s put the portfolios’ expected returns and standard

deviations of the previous exercise in a plot (next slide).
I What do you observe?
I Are we mixing more stocks in Data 4 portfolio?
I Are Data 3 and 4 stocks less “risky” than Data 2 stocks?

I Let’s just plot Stock A and Stock B returns during the last
three years for each data set (“Data 1, 2, 3, and 4” data).
I What do you observe?
Portfolio Diversification

Portfolio Diversification
3.00%

2.50%

2.00%

1.50%

1.00%

0.50%

0.00%
Data 1 Data 2 Data 3 Data 4

Risk (s.d.) Expected R.

Portfolio Diversification
Stock A & B Returns - Data 1 Stock A & B Returns - Data 2
8% 8.0%

6% 6.0%

4% 4.0%

2% 2.0%

0% 0.0%

-2% -2.0%

-4% -4.0%

-6% -6.0%

-8% -8.0%
Stock A Stock B Stock A Stock B

Stock A & B Returns - Data 3 Stock A & B Returns - Data 4

8.0% 8.0%

6.0% 6.0%

4.0% 4.0%

2.0% 2.0%

0.0% 0.0%

-2.0% -2.0%

-4.0% -4.0%

-6.0% -6.0%

-8.0% -8.0%
Stock A Stock B Stock A Stock B
Measuring Co-movements: Covariance

I 2nd huge conclusion:

I The less stock returns co-move, the higher diversification
you achieve.

How can we measure the co-movement of two stocks?

1. Correlation (COR): it measures the co-movement of two
stocks. Excel function: “Correl”.
I Always between -1 and 1:
I corr = 1 : they move in perfect synchronicity.
I 0 < corr < 1 : they tend to move in the same direction.
I corr = 0 : they move in unrelated ways.
I −1 < corr < 0: they tend to move in opposite directions
I corr = −1 : they move in perfect opposite synchronicity.
Measuring Co-movements: Covariance

2. Covariance (COV): it also measures the co-movement of two

stocks. Excel function: “Covariance.P”.
I if positive ⇒ they tend to move in the same direction.
I if negative ⇒ they tend to move in opposite directions.
I if zero ⇒ they move in unrelated ways.

What is the relationship between covariance and correlation?

I COV (rA , rB ) = COR(rA , rB ) × SD(rA ) × SD(rB ).
I CO-VARIANCE is a mix of COrrelation and VARIANCE.
I Of the part of stock returns that co-move, it measures the
variance (volatility) of those co-movements.
Covariance & Correlation - Exercises

I Let’s calculate the COV and COR of Stocks A and B in the

“Data 2” tab.

Exercise Set 6:

1. Calculate COV and COR of Stocks A and B in “Data 1”.

2. Calculate COV and COR of Stocks A and B in “Data 3”.

3. Calculate COV and COR of Stocks A and B in “Data 4”.

4. What do you observe?

Diversification - Intuition
What is the best way to diversify?
I Diversification is maximized when we combine firms with
low correlation (negative correlation better).

Intuition:
I Diversification reduces the risk of a portfolio because the
prices of different firms do not move exactly together.
I Sometimes when one stock has a positive deviation from the
expected return, another stock has a negative one.
I If we invest in such stocks, the positive deviation will cancel
out with the negative deviation.
I Therefore, the portfolio return will not deviate much from the
expected portfolio return, and volatility will be low.
Portfolio Diversification - Real Data
Exercise Set 7:
1. Calculate the ER and SDs of Google (GOOGL), T-Mobile US,
Inc. (TMUS), SPDR Gold Trust (GLD), and Hormel Foods
Corporation (HRL).

2. Compare them to the SDs of Google, Microsoft, Apple, and

Amazon.

3. Calculate the ER and SD of an equal-weights portfolio of

GOOGL, TMUS, GLD, and HRL.

4. Compare the SD of the above portfolio to the SD of the

portfolio of Google, Microsoft, Apple, and Amazon.

5. Explain your results.

Portfolio Diversification - Real Data
Exercise Set 8:
1. Complete the following correlation matrix:
GOOGL MSFT AAPL AMZN
GOOGL
MSFT
AAPL
AMZN

2. Complete the following correlation matrix:

GOOGL TMUS GLD HRL
GOOGL MSFT AAPL AMZN
TMUS
GLD
HRL