Business Finance
(FINC 201)
Week 7
Mona Yaghoubi
Risk and Return Intro
Chapter 11 – BMM 10th ed.
Risk and Return
• So far:
• In our calculations we used the
opportunity cost of capital (rate of return).
• And we JUST said that the higher the risk,
the higher the return and the opportunity
cost of capital depends on the risk of an
investment
• This week and next week:
• We will learn how to measure risk, and
• how risk affects the cost of capital.
2
Topics Covered
• Rates of Return: A Review
• Measuring a Single Asset (today) and a
portfolio of assets (next week) Risk
• Risk and Return
• Diversification
• Portfolio
• Return
• Variance
• Firm Specific Risk vs. Market Risk
3
Rates of Return: A Review
• What does the cost of capital mean?
• It is the rate of return that investors could
expect to earn if they invested in equally
risky securities.
• It’s an opportunity cost
4
Rates of Return: A Review
The rate of return on an investment can be calculated as
follows:
𝑃𝑟𝑜𝑓𝑖𝑡
𝑅𝑒𝑡𝑢𝑟𝑛 % =
𝐼𝑛𝑣𝑒𝑠𝑡𝑚𝑒𝑛𝑡
Return =
(Amount
– Amount invested)
__ ____received
__________________
Amount invested
For example, if $1,000 is invested and $1,100 is returned
after one year, the rate of return for this investment is:
−
= 10%.
5
Rates of Return: A Review
Example
• Suppose you bought 100 shares of Wal-Mart (WMT)
one year ago today at $25 per share. Over the last
year, you received $20 in dividends (= 20 cents per
share × 100 shares). At the end of the year, the
stock sells for $30. What was your return?
• When we talk about stock the profit comes in two ways
1.
2.
Dividend
Capital gain
6
Rates of Return: A Review
Example
• Amount investment $25 × 100 = $2,500.
• At the end of the year, you have stock worth $3,000
• Capital gain =
• cash dividends of $20.
• Your total dollar gain from dividend and capital gain
was $520
• Your percentage return
for the year is
$520 = 20.8%
$2,500
7
Capital Market History
When you invest in a stock, you don’t know what
return you will earn. But by looking at the history of
security returns, you can get some idea of the return
that investors might reasonably expect from different
types of securities and of the risks that they face.
Go to https://finance.yahoo.com/
Search Tesla (TSLA), go to Historical Data tab and
download historical prices
8
Financial Markets (review from week 1)
• The Relationship Between Institutions and
Markets
• Financial markets are forums in which suppliers
of funds and demanders of funds can transact
business directly
• Transactions in short-term marketable securities
take place in the money market.
• Transactions in long-term securities take place in
the capital market
9
Market Indexes (Indices)
• Financial analysts can’t track every stock, so they rely
on market indexes to summarize the return on
different classes of securities.
• Market Index
Measure of the investment performance of the overall
market
• Dow Jones Industrial Average (The Dow)
• Index of the investment performance of a portfolio of
30 “blue-chip” stocks (e.g. Coca-Cola, IBM, Walmart)
• These 30 companies are also included in the S&P
500.
• Standard & Poor’s Composite Index (S&P 500)
• Index of the investment performance of a portfolio of
500 large US stocks
• Account for a bout 75% of the market value of the
traded stocks
10
New Zealand Exchange (NZX)
• The New Zealand Exchange (NZX) is the national
stock exchange for New Zealand and a publicly
owned company.
• On 29 March 2021, the NZX main board had a total
of 185 listed securities with a combined market
value of NZ$182 billion
• NYSE world's largest stock exchange by market
cap had a combined value of US$30.1 trillion as of
February 2018.
• NZX 50 Index is the main stock market index in
New Zealand
• It comprises the 50 biggest stocks listed in NZX
11
How an investment of $1 at the start of 1900 would
have grown by the start of 2016.
Think about risk and return!
Three portfolios:
1. A portfolio of
Treasury bills,
(short-term US
government
loans).
2. A portfolio of
longterm Treasury
bonds issued
by the U.S.
government
3. A diversified
portfolio of
common
stocks.
12
Rates of Return
Using historical Evidence to estimate today’s cost of capital
Average rates of return on Treasury bills, government bonds, and
common stocks, 1900 –2015 (in % per year)
5.3-3.8=
11.4-3.8=
Rate of return on common stocks
= interest rate on Treasury bills + risk premium
* Risk premium is expected return in excess of risk-free return
as compensation for risk. (𝑟 − 𝑟𝑓 )
13
Rates of Return on Common Stocks (1900-2015)
While common stocks have offered the highest average
returns, they have also been riskier investments
14
Measuring Risk
15
Measuring Risk
• You know that the opportunity cost of capital for
safe projects must be the rate of return offered by
safe Treasury bills,
• and you know that the opportunity cost of capital
for “average-risk” projects must be the expected
return on the market portfolio.
• But you don’t know how to estimate the cost of
capital for projects that do not fit these two simple
cases.
• Before you can do this, you need to understand
more about investment risk.
16
Measuring Risk
What we need is a
measure of how far the
returns may differ from
the average.
One way to present the
spread of possible
investment returns is by
using histograms
The bars in each histogram
show the number of years
between 1900 and 2015
that the investment’s
return fell within a specific
range.
17
Measuring Risk
• We need a numerical measure of
dispersion to quantify risk.
• The standard measures of volatility
(risk) are
• Variance
• Average value of squared deviations from
mean
• Standard Deviation
• Square root of variance
18
Variance, indicates how
spread out the returns
are from the average.
19
Measuring Risk
• To measure variance
1. Find the expected return (mean)
2. Calculate the deviation from the expected
return (mean)
3. Calculate the squared deviation
4. Variance = sum of squared deviations
weighted by probabilities
20
Measuring Risk
Step 1: Measuring the Expected Rate of Return
• The expected return is
• The rate of return expected to be realized
from an investment over a period of time,
or
• The weighted average of the possible
outcomes, where the weights are the
probabilities
21
Measuring Risk
Step 1: First we measure the Expected Rate of
Return
• If equal probabilities
n
rˆ   ri n
i 1
• If unequal
probabilities
 rˆ = Pr
1 1  P2 r2 
 Pn rn
n
  Pr
i i
i 1
22
Measuring Risk
Calculating Standard Deviation
Steps 2,3 and 4: Measures of variation around the
mean
n
2
ˆ
Variance     ( ri  r ) Pi
2
i 1
Standard deviation     2 
n
2
ˆ
(
r

r
)
Pi
 i
i 1
23
Consider the Following Investment (Asset A) and Calculate the
Variance
State of
Economy
Probability
Rate of
return (%)
Bust
0.10
28.0
Below
avg.
0.20
14.7
Avg.
0.40
0.0
Above
avg.
0.20
-10.0
Boom
0.10
-20.0
Deviation from
expected return
Squared Deviation
* probability
=1.74%
24
Consider the Following Investment and Calculate the Variance
Probability
Rate of
return (%)
Deviation from
expected return
Bust
0.10
28.0
(0.28-0.017)
0.1 0.28 − 0.017
2
Below avg.
0.20
14.7
(0.147-0.017)
0.2 0.147 − 0.017
2
Avg.
0.40
0.0
(0.00-0.017)
0.4 0.00 − 0.017
2
Above avg.
0.20
-10.0
(-0.10-0.017)
0.2 −0.10 − 0.017
2
Boom
0.10
-20.0
(-0.20-0.017)
0.1 −0.20 − 0.017
2
State of
Economy
Squared Deviation
* probability
Expected return = 1.74%
Variance= 0.018
SD= 0.018 1/2 = 0.1336 0r 13.4%
25
Topics covered in the following slides
• Comparing risk and return
• Diversification
• Portfolio
• Return
• Variance
• Firm Specific Risk vs. Market Risk
26
Comparing risk and return
Security
Expected return
Risk, σ
T-bills
8.0%
0.0%
Alta
17.4%
20.0%
AF
13.8%
18.8%
Alta
Comparing standard deviations
Prob.
T - bill
AF
Alta
0
8.0 13.8
17.4
Rate of Return (%)
The graph below shows the spread of monthly
return on two firms’ stocks
• Which firm would you choose to invest in?
• Note the expected return
Measuring Risk – review
Calculating Standard Deviation
Measures of variation around the mean
n
Variance   2   ( ri  rˆ) 2 Pi
i 1
Standard deviation     2 
n
2
ˆ
(
r

r
)
Pi
 i
i 1
30
Risk and return
• Generally, the higher the risk, the higher the
return
• We use variance and standard deviation to
measure risk
• Standard deviation (σi) measures total risk
The question is, can investors reduce the
total risk and how?
31
Yes we can, by creating a diversified
Portfolio
An investment
portfolio is any
collection or
combination of
financial assets.
32
Portfolio Risk and Return
• If we assume all investors are rational and
therefore risk averse, that investor will
ALWAYS choose to invest in portfolios rather
than in single assets.
• If an investor holds a single asset, he or she
will fully suffer the consequences of poor
performance.
• This is not the case for an investor who owns
a diversified portfolio of assets.
33
Risk & Diversification
How do we choose assets in a portfolio?
What do we mean by diversification?
• Selling umbrellas is a risky business; you may make
a killing when it rains, but you are likely to lose
your shirt in a heat wave.
• Selling ice cream is no safer; you do well in the
heat wave, but business is poor in the rain.
• Suppose, however, that you invest in both an
umbrella shop and an ice cream shop. By
diversifying your investment across the two
businesses, you make an average level of profit
come rain or shine.
34
How can we reduce risk?
• By NOT putting all our eggs in the same basket!
• Diversification
• Strategy designed to reduce risk by spreading the
portfolio across many investment
• Diversification reduces variability
• A portfolio is a grouping
of financial assets such as
stocks, bonds,
commodities and etc.
35
Portfolio
Portfolio diversification works because prices of
different stocks do not move exactly together, as is the
case of our umbrella and ice cream businesses.
36
Risk & Diversification
• Our goal is to find stocks that are negatively
correlated. When one business does well, the
other does badly.
• Unfortunately, in practice, stocks that are
negatively correlated are rare.
• The contribution of a security to the risk of a
portfolio depends on how the security’s returns
vary with the investor’s other securities in the
portfolio.
37
Risk & Diversification
How assets correlate
• Correlation (ρ ) is a statistical measure that shows
how assets move in relation to each other. (both
direction and strength of the relationship)
• It is measured on a scale of -1 to +1.
• A perfect positive correlation between assets has a
reading of +1 and means that their prices move at
exactly the same magnitude (equal percentage
moves) to the same direction.
• A perfect negative correlation has a reading of -1
and implies that assets move in opposite directions.
• A zero correlation means no relationship at all.
38
Risk of a Portfolio
Correlation
39
Risk of a Portfolio (cont.)
Even if two assets are not perfectly negatively
correlated, an investor can still realize diversification
benefits from combining them in a portfolio as
shown in the figure below.
40
Portfolio Risk and Return:
Portfolio return calculation
• The return of a portfolio is a weighted
average of the returns on the individual
assets
• where
• 𝑊𝑗 = percentage of the portfolio’s total dollar
value invested in asset j
• 𝑟𝑗 = return on asset j
41
Risk and Diversification
Two Asset Example: Gold and Auto stocks
42
Risk and Diversification
portfolio variance calculation
• Steps:
1. Create the portfolio and the return of the
portfolio in each state of economy
2. Calculate the expected return of the portfolio
3. Calculate the squared variation from the
expected return of the portfolio * the
probability of each state (Squared Deviation*
probability)
4. Calculate the sum (which will be the variance)
43
Risk and Diversification
Two Asset Example (continued)
A portfolio of 25% Gold and 75% Auto
0.75(-8)+0.25(20)=
𝑃𝑜𝑟𝑡𝑓𝑜𝑙𝑖𝑜 𝑒𝑥𝑝𝑒𝑐𝑡𝑒𝑑 𝑟𝑒𝑡𝑢𝑟𝑛 = 1/3(−1)+1/3(4.5)+1/3(8.5)= 4%
𝑃𝑜𝑟𝑡𝑓𝑜𝑙𝑖𝑜 𝑣𝑎𝑟𝑖𝑎𝑛𝑐𝑒 = 1/3 −1 − 4 2 +1/3 4.5 − 4 2 +1/3 8.5 − 4 2 = 15.6667
Standard deviation = 15.6667
0.5
= 3.9%
44
Diversification
Portfolio of two or more than two assets
Method B: Use formulas
The variance of the portfolio is the sum of the terms in all
the boxes.
σi is the standard deviation of Stock i.
Cov(Ri, Rj) is the covariance between Stock i and Stock j.
45
Portfolio Risk and Return
Method B: Use formulas
• Portfolio variance ( 2- stock portfolio)
σ2𝑝 = 𝑋12 σ12 + 𝑋22 σ22 + 2 𝑋1 𝑋2 𝐶𝑜𝑣 𝑅1 , 𝑅2
Where
•
•
•
•
𝑅𝑖 represents stock i,
𝑋𝑖 is the weight of stock i in the portfolio,
σ2𝑖 is the variance of stock i, and
𝐶𝑜𝑣 𝑅1 , 𝑅2 = ρ12 σ1 σ2
• is the covariance btw stock 1 and 2 , where ρ12 is the correlation btw
stock 1 &2
• Covariance calculations provide information on the variables
relationship (positive or negative) but cannot reveal the strength of the
connection.
46
Portfolio Risk
Formula
• Portfolio variance ( 2- stock portfolio)
σ2𝑝 = 𝑋12 σ12 + 𝑋22 σ22 + 2 𝑋1 𝑋2 ρ12 σ1 σ2
Where
•
•
•
•
•
i represents stock i,
𝑋𝑖 is the weight of stock i in the portfolio,
σ2𝑖 is the variance of stock i, and
ρ12 is the correlation btw stock 1 &2
σi is the standard deviation of stock i
47
Diversification
Alternative way to find Portfolio Variance
• Calculate Variance of portfolio made up 25% Gold
Mining stock, 75% Auto stock if the correlation btw
Gold and Auto is -0.996.
• Standard deviation of Gold (σ1 ) = 16.4
• Standard deviation of Auto (σ2 ) = 10.6
σ2𝑝 = 𝑋12 σ12 + 𝑋22 σ22 + 2 𝑋1 𝑋2 ρ12 σ1 σ2
σ2𝑝 = 0.25 2 16.4 2 + 0.75 2 10.6 2 +
2 0.25 0.75 16.4 10.6 −.996 = 15.2
σ𝑝 = 3.9%
48
Firm Specific Risk vs. Market Risk
What happens after diversification? Do we eliminate all risks?
49
Firm-Specific Risk versus Market Risk
• Market risk
• that part of a security’s risk that cannot be
eliminated by diversification because it is
associated with economic, or market factors
that systematically affect most firms
• Examples would include changes in
macroeconomic factors such interest rates,
inflation, etc.
• Also called systematic risk and undiversifiable
risk
50
Firm-Specific Risk versus Market Risk
• Firm-specific (Unique) risk
• that part of a security’s risk associated with random
outcomes generated by events, or behaviors, specific
to the firm
• eg. Management change
• It can be eliminated through proper diversification
• Also called diversifiable risk or unsystematic risk
• The unsystematic (firm-specific) risk is irrelevant
because it could easily be eliminated simply by
diversifying
51
Risk and Diversification
Adding Assets to a Portfolio
• Diversification eliminates specific risk. But there is some risk that
diversification cannot eliminate. This is called market risk.
• Risk here is measured by the variance. The total variance of a
portfolio is the sum of the variance due to the market and the
specific variance.
52
Failure to diversify
• If an investor chooses to hold a one-stock portfolio
(doesn’t diversify), would the investor be
compensated for the extra risk they bear?
• NO!
• Stand-alone (firm specific) risk is not important to a
well-diversified investor.
• Rational, risk-averse investors are concerned with σp,
which is based upon market risk.
• There can be only one price (the market return) for a
given security.
• No compensation should be earned for holding
unnecessary, diversifiable risk.
53
Big Idea
• The ONLY risk we are concerned about
is the risk that cannot be diversified
away.
• We are ONLY concerned about the risk
of a security in the context of the risk
that security adds to a well-diversified
portfolio.
54
Back-up Slides
Go through the examples at home
55
Portfolio Return
Example
Suppose you invest 60% of your portfolio in Exxon
Mobil and 40% in Coca Cola. The expected dollar
return on your Exxon Mobil stock is 10% and on
Coca Cola is 15%. The expected return on your
portfolio is:
Expected Return  (.60  10)  (.40  15)  12%
56
Portfolio Risk (example) – with formula
Suppose you invest 60% of your portfolio in Exxon Mobil and 40% in
Coca Cola. The expected dollar return on your Exxon Mobil stock is
10% and on Coca Cola is 15%. The standard deviation of their
annualized daily returns are 18% and 30%, respectively. Assume a
correlation coefficient of .20 and calculate the portfolio variance.
Portfolio Variance  x12σ 12  x 22σ 22  2( x1x 2ρ 12σ 1σ 2 )
𝑝𝑜𝑟𝑡𝑓𝑜𝑙𝑖𝑜 𝑣𝑎𝑟𝑖𝑎𝑛𝑐𝑒 =
0.60
2
∗ (0.18 2 ] + 0.40 2 ∗ 0.30 2 +
+ 2 0.6 ∗ 0.4 ∗ 0.20 ∗ 0.18 ∗ 0.30
𝑝𝑜𝑟𝑡𝑓𝑜𝑙𝑖𝑜 𝑣𝑎𝑟𝑖𝑎𝑛𝑐𝑒 =0.0313
𝑆𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝐷𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛 = 0.0313
1
2
= 17.69 %
57
Example
Consider the Following Investment Alternatives
Econ.
Prob.
T-Bill
Alta
Repo
AF
Market
Rate of return (%)
Bust
0.10
8.0
-22.0
28.0
10.0
-13.0
Below avg.
0.20
8.0
-2.0
14.7
-10.0
1.0
Avg.
0.40
8.0
20.0
0.0
7.0
15.0
Above avg.
0.20
8.0
35.0
-10.0
45.0
29.0
Boom
0.10
8.0
50.0
-20.0
30.0
43.0
Example cont. Find portfolio return
First, Find portfolio return in each state of economy
-22*0.5+28*0.5=3
Economy
Bust
Prob.
0.10
Alta
-22.0%
Repo
28.0%
Port.=
0.5(Alta) +
0.5(Repo)
3.0%
Below avg.
0.20
-2.0
14.7
6.4
Average
Above avg.
0.40
0.20
20.0
35.0
0.0
-10.0
10.0
12.5
Boom
0.10
50.0
-20.0
15.0
Example cont.
^
rp = (3.0%)0.10 + (6.4%)0.20 + (10.0%)0.40
+ (12.5%)0.20 + (15.0%)0.10 = 9.6%.
p = ((3.0 - 9.6)20.10 + (6.4 - 9.6)20.20
+(10.0 - 9.6)20.40 + (12.5 -9.6)20.20
+ (15.0 - 9.6)2(.10))1/2 = 3.3%.
Portfolio Risk and Return- Example
Method B: Use formulas
Portfolio expected return:
^
rp is a weighted average (Xi is % of
portfolio in stock i):
^
rp = 0.5(17.4%) + 0.5(1.7%) = 9.6%.