How to use Examplify
If you are new to Examplify or have had problems with Examplify, attend one of
Examplify briefing sessions arranged by CIT
https://wiki.nus.edu.sg/display/DA/Common+Briefing+Sessions
NOTE: I highly recommend that you attend one of the sessions in Round 1
(before our midterm)
If none of the above options is possible:
https://wiki.nus.edu.sg/display/DA/Getting+Started
- Go through the briefing slides
• Examplify Assessment - https://wiki.nus.edu.sg/x/daBJCw
• Or watch the briefing video - https://wiki.nus.edu.sg/x/tgg_EQ
- Take a few practice exams
- Write to citbox25@nus.edu.sg if you run into any technical issues
You are responsible for familiarizing yourself
with Examplify before our midterm
1
FIN 2704/2704X
Week 4
Risk and Return part 1
2
Learning objectives
• Understanding investment return
• Understand the difference between nominal returns and
real returns
• Be able to estimate expected return
• Know how to calculate geometric and arithmetic return
• Understand what investment risk is and what the risk and
return trade-off is
• Understand how risk is measured and calculated
• Understand what risk aversion is and how it relates to risk
premium
• Understand the effects of diversification on portfolio risk
and return
3
A First Look at Risk and Return
We begin our look at risk and return by looking at
historical risk (volatility) and return experienced by various
investments.
Suppose your great-grandparents had invested $1 on
your behalf in 1925 and instructed their broker to reinvest
any dividends and/or interest earned in the account until
the beginning of 2019.
Consider how an investment would have grown if it were
wholly invested in any one of the following investments:
4
A First Look at Risk and Return
1. Large Company stocks (S&P 500): A portfolio, constructed by
Standard and Poor’s, comprising 90 U.S. stocks up to 1957 and 500
U.S. stocks after that. The firms represented are leaders in their
respective industries and are among the largest firms, in terms of
market capitalization (share price times the number of shares in the
hands of the shareholders), traded on U.S. markets.
2. Small Company stocks: A portfolio of stocks of U.S. firms whose
market capitalizations are in the bottom 10% of all stocks traded on the
NYSE. (As stocks’ market values change, this portfolio is updated so it
always consists of the smallest 10% of stocks.)
3. Long-term Government Bonds: A portfolio of long-term, U.S.
Treasury bonds with maturities of approximately 20 years.
4. Treasury Bills: An investment in three-month U.S. Treasury Bills
(reinvested as the bills mature).
5
A First Look at Risk and Return
A $1 investment from 1925 to
2019 in the following portfolios:
•
•
•
•
Small-company stocks
Large-company stocks
Long-term government bonds
Treasury bills
The inflation line is shown as a
reference point.
Note that the investments that
performed the best in the longrun also had the greatest
fluctuations from year to year.
6
Return
7
What Are Investment Returns?
Investment returns measure the financial results of an
investment
Returns may be historical or prospective (expected)
Any period’s returns can be expressed in:
• Dollar terms:
= Amount received (End of Period) – Amount invested (Beginning of Period)
• Percentage terms:
= Amount received (End of Period) – Amount invested (Beginning of Period)
Amount invested (Beginning of Period)
8
Example: PepsiCo Returns
When an investor buys a stock or a bond, their return
comes in 2 forms:
1. Any _________ or interest payment (income) received, and
2. A ________ gain or a ________ loss (due to change in price)
Suppose you bought PepsiCo stocks at $43 a share.
By the end of the year, the value of each share has risen
to $49, giving you a capital gain of $(49 – $43) = $6.
In addition, PepsiCo paid a dividend of $0.56 a share.
Total dollar return = Dividend income + Capital gain/loss
= $6.56
9
Percentage Return for the Period
The percentage return is the sum of
1
Dividend
Dividend Yield =
Initial Share Price
2
Capital Gain
Capital Gains Yield =
Initial Share Price
, and
𝐓𝐨𝐭𝐚𝐥 𝐏𝐞𝐫𝐜𝐞𝐧𝐭𝐚𝐠𝐞 𝐑𝐞𝐭𝐮𝐫𝐧
= 𝐃𝐢𝐯𝐢𝐝𝐞𝐧𝐝 𝐘𝐢𝐞𝐥𝐝 + 𝐂𝐚𝐩𝐢𝐭𝐚𝐥 𝐆𝐚𝐢𝐧𝐬 𝐘𝐢𝐞𝐥𝐝
10
Example: PepsiCo Percentage Return
0.56
Dividend Yield =
= 0.013 or 1.3%
43
6
Capital Gains Yield =
= 0.14 or 14.0%
43
𝐷𝑖𝑣𝑖𝑑𝑒𝑛𝑑 + 𝐶𝑎𝑝𝑖𝑡𝑎𝑙 𝐺𝑎𝑖𝑛𝑠
Total Percentage Return =
𝐼𝑛𝑖𝑡𝑖𝑎𝑙 𝑆ℎ𝑎𝑟𝑒 𝑃𝑟𝑖𝑐𝑒
=
!.#$%$
&'
= 0.153 or 15.3%
11
What’s The Impact Of Inflation?
What we have calculated earlier is a _________ return. This
refers to actual $ received versus actual $ invested.
But given inflation, am I able to buy the same, less or more
goods with the money I receive at the end as the amount I
could have bought with the $ investment at the time I made
the investment?
What was my return in terms of this change in purchasing
power over time?
The ______ rate of return tells you how much more you will
be able to buy with your money at the end of the period.
12
What’s The Impact Of Inflation?
To convert from a nominal to a real rate of return:
1 + 𝑛𝑜𝑚𝑖𝑛𝑎𝑙 𝑟𝑒𝑡𝑢𝑟𝑛
1 + 𝑟𝑒𝑎𝑙 𝑟𝑒𝑡𝑢𝑟𝑛 =
1 + 𝑖𝑛𝑓𝑙𝑎𝑡𝑖𝑜𝑛 𝑟𝑎𝑡𝑒
A common approximation for the real rate of return is:
𝑟𝑒𝑎𝑙 𝑟𝑒𝑡𝑢𝑟𝑛 ≈ 𝑛𝑜𝑚𝑖𝑛𝑎𝑙 𝑟𝑒𝑡𝑢𝑟𝑛 − 𝑒𝑥𝑝𝑒𝑐𝑡𝑒𝑑 𝑖𝑛𝑓𝑙𝑎𝑡𝑖𝑜𝑛
13
Example: PepsiCo
Let’s go back to the PepsiCo. % return example. If the
inflation for the year was 2.8 percent, what was the real rate
of return on a share of PepsiCo. stock?
1 + 𝑛𝑜𝑚𝑖𝑛𝑎𝑙 𝑟𝑒𝑡𝑢𝑟𝑛
1 + 𝑟𝑒𝑎𝑙 𝑟𝑒𝑡𝑢𝑟𝑛 =
1 + 𝑖𝑛𝑓𝑙𝑎𝑡𝑖𝑜𝑛 𝑟𝑎𝑡𝑒
1 + 0.153
1 + 𝑟𝑒𝑎𝑙 𝑟𝑒𝑡𝑢𝑟𝑛 =
= 1.1216
1 + 0.028
𝑟𝑒𝑎𝑙 𝑟𝑒𝑡𝑢𝑟𝑛 = 0.1216 = 𝟏𝟐. 𝟏𝟔%
14
Which Stocks To Invest In?
• First, you need to estimate the stock’s expected returns.
• ____________ returns are returns that take into account
uncertainties that are present in different scenarios.
• With perfect quantifiable information, an investor must
estimate the different return scenarios possible and the
probability of each return scenario.
15
How Do You Calculate Expected Return?
If we know the possible return scenarios and their
probabilities
𝑺
Expected Return:
𝒓! = $ 𝑷𝒔 𝒓𝒔
𝒔"𝟏
• Ps = Probability of possible return
• rs = Possible return
Note: there are n possible returns
16
Example: Expected Return Using Probabilities
Probability
of Return
Possible
Return
• 30% probability of a 10% return
0.3
10%
• 10% probability of a -10% return
0.1
-10%
• 60% probability of a 25% return
0.6
25%
An asset has a
Total = 1.0
What is the expected return?
𝑟̂ = (0.30)(10%) + (0.10)(-10%) + (0.60)(25%) = 17%
17
How to calculate average return?
If we do not have probabilities of future scenarios, but we have
historical returns instead, average return can be calculated in 2 ways:
1. Geometric average return also called Geometric mean
𝑟̅ =
1 + 𝑟! 1 + 𝑟" 1 + 𝑟# ⋯ 1 + 𝑟$
!
$
−1
where 𝑟! are the actual nominal returns in year 1, year 2, etc.
2. Arithmetic average return also called Arithmetic mean
&
𝑟̅ =
! 𝑟#
#$%
𝑛
18
Geometric vs. Arithmetic
If an investor buys an asset at time 0 and holds it till time T:
1. The geometric mean is what the investor actually earned
per year on average compounded annually
– Also known as the mean holding period return or average
compound return earned per year over a multi-year period.
2. The arithmetic mean is what the investor earned in a
typical/average year.
–
Also known as ex-post average return, observed average
return or historical average return.
19
Example: Geometric mean
Suppose you bought an investment and held it for 4 years.
It provided the following returns over the 4-year period:
Year
Return
1
10%
2
-5%
3
20%
4
15%
𝐇𝐨𝐥𝐝𝐢𝐧𝐠 𝐏𝐞𝐫𝐢𝐨𝐝 𝐑𝐞𝐭𝐮𝐫𝐧 𝐇𝐏𝐑
= 1 + 𝑟% 1 + 𝑟' 1 + 𝑟( 1 + 𝑟) − 1
= 1.10 0.95 1.20 1.15 − 1
= 0.4421 = 𝟒𝟒. 𝟐𝟏%
𝐆𝐞𝐨𝐦𝐞𝐭𝐫𝐢𝐜 𝐀𝐯𝐞𝐫𝐚𝐠𝐞 𝐑𝐞𝐭𝐮𝐫𝐧
%
= 1 + 𝑟% 1 + 𝑟' 1 + 𝑟( 1 + 𝑟) ) − 1
= 1 + 0.1 1 − 0.05 1 + 0.2 1 + 0.15
= 0.09584 = 𝟗. 𝟓𝟖%
%
)
−1
So, you realized an average annual compound return of 9.58% on
your money over 4 years or, equivalently, a 4-year holding period
return (HPR) of 44.21%.
20
Example: Arithmetic Mean
Note the difference between geometric average and
arithmetic average.
Year
Return
1
10%
2
-5%
3
20%
4
15%
𝐀𝐫𝐢𝐭𝐡𝐦𝐞𝐭𝐢𝐜 𝐀𝐯𝐞𝐫𝐚𝐠𝐞 𝐑𝐞𝐭𝐮𝐫𝐧
𝑟% + 𝑟' + 𝑟( + 𝑟)
=
𝑛
10% − 5% + 20% + 15%
=
4
= 𝟏𝟎%
10% is the average return in a typical year.
21
Is this a Good Investment?
So, let’s say we’ve estimated an expected return:
• How does one determine whether the return is adequate?
– By comparing to a benchmark
• What should the benchmark be?
– It is the required rate of return on the investment
• What determines the required return?
– Required return depends on the risk of the investment
22
Risk
23
What is Risk?
• Risk is the uncertainty associated with future possible
outcomes
• We are concerned with investment risk
• Investment risk refers to the potential for your investment
return to fluctuate in value – go up or down – from period
to period
• All investments carry some risk, but some carry more risk
than others.
– Generally, we will see that if you want higher returns, you
need to be comfortable with accepting a higher level of risk
24
Probability Distribution
1. Which stock is riskier?
The greater the chance of
a return far below or above
the expected return, the
greater the risk à Stock Y
is riskier than Stock X.
Stock X
Probability
2. Which stock would
investors generally prefer?
Stock Y
-20
0
15
Rate of return (%)
50
Risk Averse ➔ _________
vs
Risk Neutral ➔ _________
vs
Risk Lover ➔ _________
25
How Do We Measure Uncertainty?
How do we measure this variability or volatility in
investment returns?
We calculate the variance or standard deviation of the
possible returns.
If we know the future possible returns and the probability of
each possibility:
𝑺
𝝈=
J 𝑷𝒔 𝒓𝒔 − 𝒓M
𝒔&𝟏
𝟐
Expected return
26
Variance and Standard Deviation
If we only have historical past data, then variance and
standard deviation may be estimated using the realized rate
of return at any time t
n
Estimated 𝜎 =
∑( r − r )
t
2
Avg
t=1
n −1
– t ranges from 1 to n
– n is the number of historical observations
The average
annual return
(arithmetic mean)
over the last n years
27
Example – Estimating Std Deviation from Historical Data
Year
Actual
Return
Average
Return
Deviation from
the Mean
Squared
Deviation
1
0.15
0.105
0.045
0.002025
2
0.09
0.105
-0.015
0.000225
3
0.06
0.105
-0.045
0.002025
4
0.12
0.105
0.015
0.000225
Total
0.42
Variance = 0.0045 / (4-1) = 0.0015
0.0045
Standard Deviation = 0.03873
Stand-Alone Risk
Standard deviation measures ______ risk, also called the
stand-alone risk of an investment.
• Stand-alone risk is the risk an investor would face if
he/she held only this one asset
• The larger the standard deviation, the higher the probability
that actual returns will be far away from the expected return
– Thus standard deviation measures dispersion around an
expected value
29
The Normal Distribution
The graph below is based on historical
annual return and standard deviation for
a portfolio of large-firm common stocks.
There is a 99%
probability that the annual
return will be between
–48.8% and 73.6%.
The average annual
return is 12.4%
There is a 68%
probability that the annual
return will be between
–8.0% and 32.8%.
There is a 95%
probability that the annual
return will be between
–28.4% and 53.2%.
1𝜎 = 32.8% − 12.4%
= 20.4%
30
Risk & Return: Comprehensive Example
Economy
Probability
Recession
Assets
T-Bill
Alta
Repo
Am F.
10%
8.0%
–22.0%
28.0%
10.0%
Below Avg
20%
8.0%
–2.0%
14.7%
–10.0%
Average
40%
8.0%
20.0%
0.0%
7.0%
Above Avg
20%
8.0%
35.0%
–10.0%
45.0%
Boom
10%
8.0%
50.0%
–20.0%
30.0%
Calculate the expected returns and standard deviations
of the above investment alternatives.
31
Comprehensive example: Calculation for Alta
Economy
Probability
Alta
Recession
10%
–22.0%
Below Avg
20%
–2.0%
Average
40%
20.0%
Above Avg
20%
35.0%
Boom
10%
50.0%
𝑺
M = J 𝑷𝒔 𝒓 𝒔
Expected Return 𝒓
𝒔&𝟏
𝑟"#$%
̂
= 0.1 −22% + 0.2 −2% + 0.4 20% + 0.2 35% + 0.1 50% = 𝟏𝟕. 𝟒%
Comprehensive example: Calculation for Alta
Economy
Probability
Alta
Recession
10%
–22.0%
Below Avg
20%
–2.0%
Average
40%
20.0%
Above Avg
20%
35.0%
Boom
10%
50.0%
𝑺
Standard Deviation of Return 𝝈𝑨𝒍𝒕𝒂 = ! 𝑷𝒔 𝒓𝒔 − 𝒓N
𝟐
𝒔$𝟏
𝜎"#$% =
0.1(−22% − 17.4%)& +0.2(−2% − 17.4%)&
+0.4(20% − 17.4%)& +0.2(35% − 17.4%)& +0.1(50% − 17.4%)&
= 401.44 = 𝟐𝟎. 𝟎%
Comprehensive Example:
Expected Returns & Standard Deviations
Investment
𝒓M
𝝈
17.4%
20.0%
Market
15.0
15.3
Am. F.
13.8
18.8
Alta
T-Bill
8.0
0
Repo Men
1.7
13.4
Looking at these assets, we do not see a consistent trade-off between
expected return and the total risk measure 𝜎 of the individual stocks.
Coefficient of Variation
A widely used measure of relative variability is the
Coefficient of Variation (CV).
CV is a standardized measure of dispersion about the
expected value, that shows the risk per unit of return.
Standard Deviation
CV =
Expected Rate of Return
or
Standard Deviation 𝜎
CV =
=
Mean
𝑟̂
35
Illustrating CV As A Measure Of Relative Risk
Probability
Stock A
Stock B
Rate of Return (%)
𝜎N = 𝜎O
But _________ is riskier than __________ because of a larger
probability of losses (negative returns). In other words, the same
amount of risk (as measured by 𝜎) but for lower expected return.
36
An Example of CV
Stock X and Stock Y have widely
differing rates of return and
standard deviations of return.
If you compare standard
deviations, Stock X seems to be
less risky than Stock Y.
Stock X
Stock Y
Expected
Return
7%
12%
Standard
Deviation
5%
7%
Comparing CVs, the result is different:
• CV of Stock X =
• CV of Stock Y =
2%
4%
4%
= 0.714
%'%
= 0.583
• Stock X has greater relative variability or higher risk per unit of
expected return than Stock Y à more risky.
37
Comprehensive Example:
Coefficient of Variation
Security
Expected Return
𝝈
CV
Alta
17.4%
20.0%
1.1
Market
15.0
15.3
1.0
Am F
13.8
18.8
1.4
T-Bill
8.0
0
0
Repo Men
1.7
13.4
7.9
Lesson from Capital Market History
• When examining portfolios of assets (as per our first example of
4 possible historical portfolios), we find:
– There is a reward for bearing risk
– The greater the potential reward, the greater the risk
– This is called the risk-return trade-off
• Investors face a trade-off between _____ and _______________
• Historical data confirm our intuition that assets with lower
degrees of risk provide lower returns on average than do those of
higher risk.
• Thus, in general, low levels of uncertainty (low risk) are
associated with low potential returns, whereas high levels of
uncertainty (high risk) are associated with high potential returns.
39
Historical Returns, Standard Deviations, and
Frequency Distributions: 1926 - 2000
40
Example: High Return Low Risk a Red Flag
ST, Feb 2016
ST, May 2015
ST, Feb 2015
- 70 investors in Singapore
were crying foul over tree
investments gone wrong.
Some of them had
invested as much as
$60,000.
- $60 million ponzi scam had
been running for 15 years.
- 250 people with approximately
$35 million in investments were
conned by Suisse International’s
gold buyback scheme.
- Investors would put money
into buying saplings or
semi-matured agarwood
trees with promises of up
to 700% after 6-7 years.
- This scam involved investing
money with a “renowned” agent
to buy distressed properties in
prime districts and selling them
to a pool of buyers in China.
- Duped more than 100 investors
with promises of 10% to 48%
over a period of four to eight
months
- Under this scheme, investors
would invest in physical gold
with the promise of a return of
about 2% every month.
Investors thought they could
earn over 20% per annum
without taking much risk.
41
Risk Aversion
Risk aversion: assumes investors dislike risk and
therefore require ______ rates of return to encourage
them to hold riskier securities
– The “extra” return earned for taking on risk is referred to
as risk premium
– An asset’s risk premium is equal to its required return
less the rate of return available on treasury securities
(i.e., the risk-free asset)
42
Risk Premiums
The risk premium is the return over and above the risk-free rate
But what risk-free rate benchmark should we use?
1. Treasury Bills: In most developed markets, where the
government can be viewed as default-free, Treasury Bills
(maturity of 1 year or less) are considered to be risk-free.
2. Treasury Bonds: However, for analysis of longer-term
projects or company valuation, where the long-term risk
premium is investor’s main concern, many argue that the
risk-free rate should be the long-term (10Y) government
bond rate.
43
Historical Risk Premiums
• The historic risk premium is defined as the excess return
over the risk-free rate*.
• For the sample 1926–2000, the historical risk premiums are:
§ Large stocks: 13.0 – 3.9 = 9.1%
§ Small stocks: 17.3 – 3.9 = 13.4%
§ Long-term corporate bonds: 6.0 – 3.9 = 2.1%
* For this module, we will define US Treasury Bills as a riskless asset and
hence we will use its returns (3.9%) as the risk-free rate.
44
Portfolio
45
Portfolio: Alta & Repo
Assume you invest an equal amount of $50,000 each in Alta and Repo.
Economy
Probability
Recession
Assets
Alta
Repo
10%
–22.0%
28.0%
Below Avg
20%
–2.0%
14.7%
Average
40%
20.0%
0.0%
Above Avg
20%
35.0%
–10.0%
Boom
10%
50.0%
–20.0%
Calculate 𝑟K̂ and 𝜎K for the portfolio.
Expected return on a portfolio
Portfolio risk
Portfolio Expected Return
• It is the weighted average of the expected returns of the
individual stocks that make up the portfolio
7
𝑟,̂ = e 𝑤4 𝑟4̂
456
𝑤# = the fraction of the portfolio5 s dollar value invested in stock i
Note that the 𝑤# ’s must add up to 1
• Example:
$50,000
$50,000
𝑟'̂ =
17.4% +
1.7% = 9.6%
$50,000 + $50,000
$50,000 + $50,000
Note that 𝑟,̂ is between 𝑟-./0
̂
and 𝑟12,3
̂
.
47
Expected Portfolio Return: Alternative Method
Here we look at the portfolio’s return in each scenario and the
probability of that scenario occurring.
Economy
Prob.
Recession
Assets
Alta
Repo
Portfolio
10%
–22.0%
28.0%
0.5 −22% + 0.5 28.0% = 3.0%
Below Avg
20%
–2.0%
14.7%
0.5 −2% + 0.5 14.7% = 6.4%
Average
40%
20.0%
0.0%
0.5 20% + 0.5 0% = 10.0%
Above Avg
20%
35.0%
–10.0%
0.5(35%) + 0.5 −10% = 12.5%
Boom
10%
50.0%
–20.0%
0.5 50% + 0.5 −20% = 15.0%
𝒓M 𝒑 = 0.1 3.0% + 0.2 6.4% + 0.4 10.0%
+0.2 12.5% + 0.1 15.0% = 𝟗. 𝟔%
48
What about the portfolio’s risk?
Applying the standard deviation & CV formulae:
é 0.10 (3.0 - 9.6)
ê+ 0.20 (6.4 - 9.6) 2
ê
s p = ê+ 0.40 (10.0 - 9.6) 2
ê+ 0.20 (12.5 - 9.6) 2
ê
2
+
0.10
(15.0
9.6)
êë
2
ù
ú
ú
ú
ú
ú
úû
1
2
= 3.3%
3.3%
CVp =
= 0.34
9.6%
The standard deviation of the portfolio can be derived in the same
way as for an individual asset. We use the portfolio’s scenario
expected returns and the probability of those scenarios occurring.
49
Closer look at portfolio risk
Note that the standard deviation of the portfolio, 𝜎, , at 3.3%,
is much lower than
– The standard deviation of either Alta (20%) or Repo (13.4%).
– The average of Alta’s and Repo’s standard deviations (16.7%).
The portfolio provides a return that is the average of the
returns of both stocks, at 9.6%, but the portfolio has a much
lower risk as measured by the standard deviation of the
portfolio, at 3.3%.
How does this happen?
50
Recall the return distribution of Alta & Repo
Economy
Prob.
Recession
Assets
Alta
Repo
Portfolio
10%
–22.0%
28.0%
0.5 −22% + 0.5 28.0% = 3.0%
Below Avg
20%
–2.0%
14.7%
0.5 −2% + 0.5 14.7% = 6.4%
Average
40%
20.0%
0.0%
0.5 20% + 0.5 0% = 10.0%
Above Avg
20%
35.0%
–10.0%
0.5(35%) + 0.5 −10% = 12.5%
Boom
10%
50.0%
–20.0%
0.5 50% + 0.5 −20% = 15.0%
In states of the economy where Alta performs poorly (Recession,
Below Avg.), Repo performs well.
In states of the economy where Alta performs well (Above Avg.,
Boom), Repo performs poorly.
The returns of Alta and Repo display a negative relationship.
51
Covariance
When looking at the risk of a portfolio of assets, it is important to recognize
and consider the ________ between the individual stocks with one another.
This leads us to the concept of covariance; that is, how the performance of
two assets “move” or “do not move” together.
The covariance of the annual rates of return of 2 different investments is
used to measure how the two assets’ rates of return vary together over
the same time period. For n periods of measured returns of stock X and
stock Y, the covariance of X and Y is found as follows:
𝐶𝑜𝑣𝑎𝑟𝑖𝑎𝑛𝑐𝑒 = 𝜎!" = 𝜌!" 𝜎! 𝜎"
where ρXY = the correlation coefficient between stock X and stock Y
(
and
𝜎!" = K 𝑝& × 𝑟),& − 𝑟)̂ × 𝑟+,& − 𝑟+̂
&'#
and
𝜎!" =
𝑟!# − 𝑟!̅ 𝑟"# − 𝑟"̅ + 𝑟!$ − 𝑟!̅ 𝑟"$ − 𝑟"̅ + ⋯ 𝑟!% − 𝑟!̅ 𝑟"% − 𝑟"̅
𝑛−1
52
Correlation Coefficient
The correlation coefficient between two stocks (X and Y), denoted by 𝝆𝑿𝒀 ,
measures the extent to which two securities X and Y move together
ρXY = 0
ρXY = -1
In general
+1 ³ 𝝆𝑿𝒀 ³ -1
ρXY = 0
ρXY = +1
Perfectly
positively
correlated
Perfectly
negatively
correlated
The variance (and therefore standard deviation) of a portfolio depends on the
correlation coefficients between the assets included in the portfolio.
Note: the correlation coefficient standardizes the units of covariance measure.
53
The Volatility of a Portfolio
• Generally, investors care less about an individual
company’s specific return and more about how the
company’s returns contribute to their portfolio’s overall
return. As well, they care about their portfolio’s overall risk.
• When we combine stocks into a portfolio, some of the
stock’s risk is eliminated through diversification. The amount
of risk that will remain in the portfolio depends upon the
degree to which the stocks included in the portfolio share
common risk (i.e., their correlation).
The volatility of a portfolio is the total risk of the portfolio, as
measured by the portfolio standard deviation.
Returns for Stocks & Portfolios
When will stock returns be highly correlated with each other?
Stock returns will tend to move together if they are affected similarly by the
same economic events. Thus, stocks in the same industry tend to have
more highly correlated returns than stocks in different industries.
Diversification Key Points
• Portfolio diversification involves investment in several different
asset classes or sectors that are not perfectly positively
correlated with each other.
• Diversification is not just holding a lot of assets.
– E.g., if you own 50 technology stocks, you are not well-diversified.
However, if you own 50 stocks that span 20 different industries,
then you are clearly much better diversified.
• Diversification can substantially reduce the variability of returns
without an equivalent reduction in expected returns.
– This happens because worse-than-expected returns from one asset
are offset by better-than-expected returns from another
• However, there is a minimum level of risk that cannot be
diversified away and that is the systematic portion – we will look
further into this next week!
Overall Summary
Return:
• What is a return on investment?
• Real return vs. Nominal return
• How to do you calculate the expected return of an investment?
• Geometric vs. Arithmetic return
Risk:
• Investment risk measurement: variance & standard deviation
• Use Coefficient of Variation (CV) to compare the risk and return of different
investment options
• Low risk investments are associated with low potential returns, whereas high risk
investments are associated with high potential returns
• By assumption, investors are risk averse. Hence, they require “extra” return for
risky assets compared to risk-free assets
Portfolio:
• How to calculate portfolio return & risk
• Diversification can substantially reduce the variability of returns without an
equivalent reduction in expected returns.
57
Covariance and
Correlation Coefficient Example
Week 4 Additional Materials
Please review these slides on your own.
You are responsible for all the material
58
Example 1: Covariance & Correlation Coefficient
A security analyst has prepared the following probability
distribution of the possible returns on 2 stocks:
CompuGraphics Inc (CGI) and Data Switch Corp (DSC).
Probability
Return on
CGI
Return on
DSC
0.30
10%
40%
0.50
14%
16%
0.20
20%
20%
59
Example 1 : Covariance & Correlation Coefficient
For CGI, the expected return is:
<
𝑟678
̂ = ! 𝑝9 𝑟678,9
9:;
= 0.3 10% + 0.5 14% + 0.2 20% = 𝟏𝟒%
For DSC, the expected return is:
<
𝑟>?6
̂
= ! 𝑝9 𝑟>?6,9
9:;
= 0.3 40% + 0.5 16% + 0.2 20% = 𝟐𝟒%
60
Example 1: Covariance & Correlation Coefficient
For CGI, the standard deviation of returns is:
0
𝜎*+, =
O 𝑃- 𝑟*+, − 𝑟*+,
̂
&
-./
= 0.3 10% − 14%
&
+ 0.5 14% − 14%
&
+ 0.2 20% − 14%
&
= 𝟑. 𝟒𝟔%
&
= 𝟏𝟎. 𝟓𝟖%
For DSC, the standard deviation of returns is:
0
𝜎12* =
O 𝑃- 𝑟12* − 𝑟12*
̂
&
-./
= 0.3 40% − 24%
&
+ 0.5 16% − 24%
&
+ 0.2 20% − 24%
61
Example 1: Covariance & Correlation Coefficient
Recall that probabilities of potential returns are known, so we
use the corresponding formula
Probability
0.30
0.50
0.20
Return on
CGI
10%
14%
20%
Return on
DSC
40%
16%
20%
<
𝜎678,>?6 = ! 𝑝9 𝑟678,9 − 𝑟678
̂
𝑟>?6,9 − 𝑟>?6
̂
9:;
= 0.3 10% − 14% 40% − 24%
+0.5 14% − 14% 16% − 24%
+0.2 20% − 14% 20% − 24% = −𝟎. 𝟎𝟎𝟐𝟒
62
Example 1: Covariance & Correlation Coefficient
The correlation coefficient between CGI and DSC is :
𝜌PQR,TUP
𝜎PQR,TUP
=
𝜎PQR 𝜎TUP
−0.0024
=
3.46%×10.58%
= −𝟎. 𝟔𝟓𝟓
63
Example 2: Covariance & Correlation Coefficient
A security analyst has collected the following historical
returns on 2 stocks:
Abacus Inc (ABC) and Definition Corp (DEF).
Year
Return on
ABC
Return on
DEF
1
10%
40%
2
14%
16%
3
20%
20%
64
Example 2 : Covariance & Correlation Coefficient
For ABC, the mean return is:
$
𝑟̅ 𝐴𝐵𝐶 =
B 𝑟!
!"#
3
=
10% + 14% + 20%
= 14.7%
3
For DEF, the mean return is:
$
𝑟̅ 𝐷𝐸𝐹 =
B 𝑟!
!"#
3
40% + 16% + 20%
=
= 25.3%
3
65
Example 2: Covariance & Correlation Coefficient
For ABC, the standard deviation of returns is:
n
∑( r − r )
t
𝜎./0 =
=
2
Avg
t=1
n −1
(10% − 14.7%)$ +(14% − 14.7%)$ +(20% − 14.7%)$
= 𝟓. 𝟎𝟑%
2
For DEF, the standard deviation of returns is:
n
𝜎134 =
=
∑(
rt − rAvg
t=1
)
2
n −1
(40% − 25.3%)$ +(16% − 25.3%)$ +(20% − 25.3%)$
= 𝟏𝟐. 𝟖𝟔%
2
66
Example 2: Covariance & Correlation Coefficient
Recall that the historical returns are known, so we use the
corresponding formula
𝜎%&'()* =
=
Year
Return on
ABC
Return on
DEF
1
10%
40%
2
14%
16%
3
20%
20%
𝑟%&'# − 𝑟%&'
̅
𝑟()*# − 𝑟()*
̅
+ 𝑟%&'+ − 𝑟%&'
̅
𝑟()*+ − 𝑟()*
̅
+ 𝑟%&'$ − 𝑟%&'
̅
𝑟()*$ − 𝑟()*
̅
3−1
10% − 14.7% 40% − 25.3% + 14% − 14.7% 16% − 25.3% + 20% − 14.7% 20% − 25.3%
2
= −𝟎. 𝟎𝟎𝟒𝟓
67
Example 2: Covariance & Correlation Coefficient
The correlation coefficient between ABC and DEF is :
𝜌-]P,T^_
𝜎-]PT^_
=
𝜎-]P 𝜎T^_
−0.0045
=
5.03%×12.86%
= −𝟎. 𝟕𝟎
68