ECON 372: Business Finance 2
Risk and Return in Capital Markets (part 1)
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Learning Objectives
• Identify which types of securities have historically
had the highest returns and which have been the
most volatile
• Compute the average return and volatility of
returns from a set of historical asset prices
• Understand the tradeoff between risk and return
for large portfolios versus individual stocks
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10.1 A First Look at Risk and Return
• Consider how an investment would have grown if
it were invested in each of the following from the
end of 1956 until the beginning of 2014:
–
–
–
–
S&P/TSX Composite Index: 9.3%
Standard & Poor’s 500 (S&P 500) – avg return: 9.5%
Long-term Government of Canada bonds:
Government of Canada Treasury Bills – smallest return
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10.2 Historical Risks and Returns of Stocks
• Computing Historical Returns
– Realized Returns
– Individual Investment Realized Returns
▪ The realized return from your investment in the stock from t to
t + 1 is:
𝐷𝑖𝑣𝑡+1 + 𝑃𝑡+1 − 𝑃𝑡 𝐷𝑖𝑣𝑡+1 𝑃𝑡+1 − 𝑃𝑡
=
+
𝑃𝑡
𝑃𝑡
𝑃𝑡
= 𝐷𝑖𝑣𝑖𝑑𝑒𝑛𝑑 𝑌𝑖𝑒𝑙𝑑 + 𝐶𝑎𝑝𝑖𝑡𝑎𝑙 𝐺𝑎𝑖𝑛 𝑌𝑖𝑒𝑙𝑑
𝑅𝑡+1 =
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(Eq. 10.1)
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Example 10.1: Realized Return
• Barrick Gold Corp. stock (ticker symbol: ABX.TO) paid a
dividend of $0.03 on February 24, 2017. (Note: Barrick
pays dividends in U.S. dollars [USD], but Canadian
financial websites such as Yahoo!Finance Canada
(ca.finance.yahoo.com ) report the dividends in
Canadian dollars [CAD] when one uses Barrick’s
Toronto Stock Exchange ticker symbol ABX.TO instead
of the New York Stock Exchange symbol ABX.) Suppose
you bought Barrick stock for $21.49 on December 30,
2016 and sold it immediately after the dividend was paid
for $25.59. What was your realized return from holding
the stock?
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Example 10.1: Realized Return: Plan
• We can use Equation 10.1 to calculate the
realized return. Knowing the purchase price
($21.49), the selling price ($25.59), and the
dividend ($0.03), we are ready to proceed.
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Example 10.1: Realized Return: Execute
• Using Equation 10.1, the return from December
30, 2016, until February 24, 2017, is equal to:
Divt +1 + Pt +1 − Pt 0.03 + 25.59 − 21.49
Rt +1 =
=
= 0.192, or19.2%
Pt
21.49
• This 19.2% can be broken down into the dividend
yield and the capital gain yield:
DivYield =
Divt + 1
Pt
0.03
=
= 0.001, or 0.1%
21.49
Pt +1 − Pt (25.59 − 21.49)
Capital Gain Yield =
=
= 0.191, or 19.1%
Pt
21.49
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Example 10.1: Realized Return: Evaluate
These returns include both the capital gain and the
return generated from receiving dividends. Both
dividends and capital gains contribute to the total
realized return—ignoring either one would give a
very misleading impression of Barrick’s
performance.
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10.2 Historical Risks and Returns of Stocks
• Computing Historical Returns
– Individual Investment Realized Returns
▪ For quarterly returns (or any four compounding periods that
make up an entire year) the annual realized return, Rannual, is
found by compounding:
1 + Rannual = (1 + R1 )(1 + R2 )(1 + R3 )(1 + R4 )
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(Eq.10.2)
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Example 10.2: Compounding Realized
Returns
• Suppose you purchased Barrick Gold Corp. stock
(ticker symbol: ABX.TO) on December 30, 2016,
and held it for one year, selling on January 2,
2018. What was your realized return?
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Example 10.2: Compounding Realized
Returns: Plan (2 of 2)
• We need to analyze the cash flows from holding
ABX stock for each quarter. In order to obtain the
cash flows, we must look up ABX stock price data
at the start and end of both years, as well as at
any dividend dates (see Chapter 7 and the
textbook’s website for online sources of stock
price and dividend data). From the data, we can
construct the following table to fill out our cash
flow timeline:
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Example 10.2: Compounding Realized
Returns: Plan (1 of 2)
Date
Price ($)
Dividend ($)
30-Dec-16
$21.49
24-Feb-17
$25.59
$0.03
29-May-17
$22.07
$0.03
29-Aug-17
$22.27
$0.03
29-Nov-17
$18.05
$0.03
02-Jan-18
$19.01
Source: Barrick Gold Corp. and Yahoo!Finance Canada.
Next, compute the return between each set of dates using Equation
10.1. Then determine each annual return in a way similar to that shown
in Equation 10.2 by compounding the returns for all of the periods in that
year.
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Example 10.2: Compounding Realized
Returns: Execute (1 of 3)
• In Example 10.1, we already computed the
realized return for December 30, 2016, to
February 24, 2017, as 19.2%. We continue as in
that example, using Equation 10.1 for each period
until we have a series of realized returns. For
example, from August 29, 2017, to November 29,
2017, the realized return is:
Rt +1 =
Divt + 1 + Pt +1 − Pt
Pt
0.03 + (18.05 − 22.27)
=
= − 0.188, or − 18.8%
22.27
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Example 10.2: Compounding Realized
Returns: Execute (2 of 3)
The table below includes the realized return in each
period.
Date
Price ($)
Dividend ($)
Return
30-Dec-16
$21.49
24-Feb-17
$25.59
$0.03
19.2%
29-May-17
$22.07
$0.03
−13.6%
29-Aug-17
$22.27
$0.03
1.0%
29-Nov-17
$18.05
$0.03
−18.8%
02-Jan-18
$19.01
Annual Return over 2017:
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5.3%
−11.0%
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Example 10.2: Compounding Realized
Returns: Execute (3 of 3)
• We then determine the one-year return by
compounding.
1 + R Annual = (1 + R1 )(1 + R 2 )(1 + R 3 )(1 + R 4 )(1 + R 5 )
1 + R Annual = (1.192)(0.864)(1.010)(0.812)(1.053) = 0.89
R Annual = 0.89 - 1 = -0.11 or -11%
DivYield =
Divt + 1
Pt
0.03
=
= 0.001, or 0.1%
21.49
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Example 10.2: Compounding Realized
Returns: Evaluate
• By repeating these steps, we have successfully
computed the realized annual returns for an
investor holding ABX stock over this one-year
period. From this exercise, we can see that
returns are risky: ABX lost 11% of its value in a
single year.
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10.2 Historical Risks and Returns of Stocks
• Average Annual Returns
– Average Annual Return of a Security
𝑅ሜ =
1
(𝑅1 + 𝑅2 +. . . +𝑅𝑇 )
𝑇
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(Eq. 10.3)
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10.2 Historical Risks and Returns of Stocks
• The Variance and Volatility of Returns:
– Variance: A method to measure the variability of
returns—the expected squared deviation of returns
from the mean.
𝑉𝑎𝑟 𝑅 =
1
ሜ 2 + (𝑅2 − 𝑅)
ሜ 2 +. . . +(𝑅𝑇 − 𝑅)
ሜ 2
(𝑅1 − 𝑅)
𝑇−1
(Eq. 10.4)
– Standard Deviation: A common method used to
measure the risk of a probability distribution—the
square root of the variance.
𝑆𝐷(𝑅) =
𝑉𝑎𝑟 𝑅
(Eq. 10.5)
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Example 10.3: Computing Historical
Volatility
• Using the data from Table 10.1, what is the
standard deviation of the S&P/ TSX Composite
Index returns for the years 2008 to 2017?
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Example 10.3: Computing Historical
Volatility: Plan
• With the 10 returns, compute the average return
using Equation 10.3, because it is an input to the
variance equation. Next, compute the variance
using Equation 10.4 and then take its square root
to determine the standard deviation.
2008
2009
2010
2011
2012
−33.00
%
35.05%
17.61%
−8.71%
7.19%
2013
2014
2015
2016
2017
12.99%
10.55%
−8.32%
21.08%
9.10%
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Example 10.3: Computing Historical
Volatility: Execute (1 of 2)
• The average annual return of the S&P/ TSX
Composite Index during this period as 6.35%, so
we have al the necessary inputs for the variance
calculation: Applying Eq. 10.4, we have:
1
𝑉𝐴𝑅(𝑅) =
𝑅1 − 𝑅ሜ 2 + 𝑅2 − 𝑅ሜ 2 + ⋯ + 𝑅𝑇 − 𝑅ሜ 2
𝑇−1
1
=
൫ −0.33 − 0.0635 2 + 0.3505 − 0.0635
10 − 1
= 0.0359
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2
+ ⋯ + (0.091
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Example 10.3: Computing Historical
Volatility: Execute (2 of 2)
• Alternatively, we can break the calculation of this
equation out as follows:
2008
2009
2010
2011
2012
Return
−0.3300
0.3505
0.1761
−0.0871
0.0719
Average
0.0635
0.0635
0.0635
0.0635
0.0635
Difference
−0.3935
0.2870
0.1126
−0.1506
0.0084
Squared
0.1548
0.0824
0.0127
0.0227
0.0001
2013
2014
2015
2016
2017
Return
0.1299
0.1055
−0.0832
0.2108
0.0910
Average
0.0635
0.0635
0.0635
0.0635
0.0635
Difference
0.0664
0.0420
−0.1467
0.1473
0.0275
Squared
0.0044
0.0018
0.0215
0.0217
0.0008
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Example 10.3: Computing Historical
Volatility: Execute (3 of 3)
• Summing the squared differences in the last row,
we get 0.3228. Finally, dividing by (10 − 1 = 9)
gives us 0.3228/9 = 0.0359. The standard
deviation is therefore,
SD( R) = Var ( R ) = 0.0359 = 0.1894, or 18.94%
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Example 10.3: Computing Historical
Volatility: Evaluate
• Our best estimate of the expected return for the
S&P/TSX Composite Index is its average return,
6.35%, but it is risky, with a standard deviation of
18.94%.
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Suggested Homework
•
•
•
•
Use Coca-Cola’s dividend history from here:
https://www.streetinsider.com/dividend_history.php?q=ko
And its share price history from here:
https://finance.yahoo.com/quote/KO/history?ltr=1
To calculate annualized return from holding a Coca-Cola stock from
2011 to 2021 (Jan 1, 2012 – Dec 31, 2021)
Use the annualized returns you calculated in the previous part to
calculate the average annual return and standard deviation of annual
returns for the period 2011 – 2021.
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