Transcript Slide 1

1
A Brief History of
Risk and Return
1-1
A Brief History of Risk and Return
• Two key observations:
1. There is a substantial reward,
on average, for bearing risk.
2. Greater risks accompany
greater returns.
1-2
Dollar & Percent Returns
• Total dollar return = the return on an
investment measured in dollars
• $ return = dividends + capital gains
• Total percent return is the return on an
investment measured as a percentage of the
original investment.
• % Return = $ return/$ invested
• The total percent return is the return for
each dollar invested.
1-3
Percent Return
Dividend Yield
Dt  1
DY 
Pt
Capital Gains
Yield
Pt 1  Pt
CGY 
Pt
% Re turn DY  CGY
Dt 1  Pt 1  Pt
% Re turn
Pt
1-4
Example: Calculating Total Dollar
and Total Percent Returns
• You invest in a stock with a share price of $25.
• After one year, the stock price per share is $35.
• Each share paid a $2 dividend.
• What was your total return?
Dividend
Capital Gain
Dollars
$2.00
$35 - $25 = $10
Percent
$2/25 = 8%
$10/25= 40 %
Total Return
$2 + $10 = $12
$12/$25 = 48%
1-5
Annualized Returns
Effective Annual Rate (EAR)
EAR  (1 HPR )  1
M
Where:
HPR = Holding Period Return
M = Number of Holding Periods per year
1-6
Annualized Returns – Example 1
You buy a stock
for $20 per
share on
January 1.
Four months
later you sell
for $22 per
share.
No dividend has
been paid yet
this year.
P0 = $20
P.33 = $22
t = “.33” since 4 months is 1/3 of a year
4-month HPR = 3 periods per year
P0.33 P 0
Holding Period Return 
P0
(HPR)
$22  $20

$20
 10%
Annualized Return
(EAR)
 ( 1  .10 ) 3  1
 33.1%
1-7
Annualized Returns – Example 2
Suppose the
P0 = $20
P2 = $28
$20 stock
HPR = 2 years (t = 2)
you bought
HPR per year = ½ (0.50)
on January 1
P2  P0

is selling for
P0
Holding Period Return
$28 two
$28  $20
(HPR)

years later
$20
No dividends
 40%
were paid in
0.50

(
1

.
40
)
1
either year.
Annualized Return
(EAR)
 18.32%
1-8
A $1 Investment in Different Types
of Portfolios, 1926—2006
1-9
Financial Market History
1-10
The Historical Record:
Total Returns on Large-Company Stocks
1-11
The Historical Record:
Total Returns on Small-Company Stocks
1-12
The Historical Record:
Total Returns on U.S. Bonds.
1-13
1-13
The Historical Record:
Total Returns on T-bills.
1-14
1-14
The Historical Record: Inflation
1-15
Historical Average Returns
• Historical Average Return = simple, or
arithmetic average.
n
His torical Ave rage Re turn
 ye arlyre turn
i 1
n
• Using the data in Table 1.1:
• Sum the returns for large-company stocks from 1926
through 2006, you get about 984 percent.
• Divide by the number of years (80) = 12.3%.
• Your best guess about the size of the return for a year
selected at random is 12.3%.
1-16
Average Annual Returns for
Five Portfolios
1-17
1-17
Average Returns: The First Lesson
• Risk-free rate:
• Rate of return on a riskless investment
• Risk premium:
• Extra return on a risky asset over the
risk-free rate
• Reward for bearing risk
• The First Lesson: There is a reward, on
average, for bearing risk.
1-18
Average Annual Risk
Premiums for Five Portfolios
1-19
Risk Premiums
• Risk is measured by the dispersion or
spread of returns
• Risk metrics:
• Variance
• Standard deviation
• The Second Lesson: The greater the
potential reward, the greater the
risk.
1-20
Return Variability Review and
Concepts
• Variance (σ2)
• Common measure of return dispersion
• Also call variability
• Standard deviation (σ)
• Square root of the variance
• Sometimes called volatility
• Same "units" as the average
1-21
Return Variability:
The Statistical Tools for Historical Returns
• Return variance: (“N" =number of returns):
 R
N
VAR(R)  σ 2 
i 1
i
R

2
N1
• Standard Deviation
SD(R)  σ  VAR(R)
1-22
Example: Calculating Historical
Variance and Standard Deviation
• Using data from Table 1.1 for large-company stocks:
(1)
(2)
Year
1926
1927
1928
1929
1930
Sum:
Return
11.14
37.13
43.31
-8.91
-25.26
57.41
Average:
11.48
(3)
(4)
(5)
Average Difference: Squared:
Return: (2) - (3) (4) x (4)
11.48
-0.34
0.12
11.48
25.65
657.82
11.48
31.83 1013.02
11.48
-20.39
415.83
11.48
-36.74 1349.97
Sum: 3436.77
Variance:
859.19
Standard Deviation:
29.31
1-23
Return Variability Review and
Concepts
• Normal distribution:
• A symmetric, bell-shaped frequency
distribution (the bell-shaped curve)
• Completely described with an average
and a standard deviation (mean and
variance)
• Does a normal distribution describe
asset returns?
1-24
Frequency Distribution of Returns on
Common Stocks, 1926—2006
1-25
Historical Returns, Standard Deviations,
and Frequency Distributions: 1926—2006
1-26
1-26
The Normal Distribution and
Large Company Stock Returns
1-27
1-27
Arithmetic Averages versus
Geometric Averages
• The arithmetic average return answers
the question: “What was your return in
an average year over a particular
period?”
• The geometric average return answers
the question: “What was your average
compound return per year over a
particular period?”
1-28
Geometric Average Return:
Formula
Equation 1.5

GAR  (1  R 1)  (1  R 2 )  ...  (1  R N)

1/N
1
Where:
Ri = return in each period
N = number of periods
1-29
Geometric Average Return


GAR   (1  Ri )
 i1

N
1/ N
1
Where:
Π = Product (like Σ for sum)
N = Number of periods in sample
Ri = Actual return in each period
1-30
Example: Calculating a
Geometric Average Return
• Using the large-company stock data from Table 1.1:
Year
1926
1927
1928
1929
1930
Percent
Return
11.14
37.13
43.31
-8.91
-25.26
One Plus
Compounded
Return
Return:
1.1114
1.1114
1.3713
1.5241
1.4331
2.1841
0.9109
1.9895
0.7474
1.4870
(1.4896)^(1/5):
1.0826
Geometric Average Return:
8.26%
1-31
Geometric Average Return
Year
% Return
1926
1927
1928
1929
1930
11.14
37.13
43.31
-8.91
-25.26
N
I/Y
PV
PMT
FV
5
CPT =
$ (1.0000)
0
$ 1.4870
$$ Invested
$
1.0000
$
1.1114
$
1.5241
$
2.1841
$
1.9895
$
1.4870
8.26%
1-32
Arithmetic Averages versus
Geometric Averages
• The arithmetic average tells you
what you earned in a typical year.
• The geometric average tells you
what you actually earned per year on
average, compounded annually.
• “Average returns” generally means
arithmetic average returns.
1-33
Geometric versus Arithmetic Averages
• For forecasting future returns:
• Arithmetic average "too high" for long forecasts
• Geometric average "too low" for short forecasts
1-34
Blume’s Formula
• Form a “T” year average return forecast
from arithmetic and geometric averages
covering “N” years, N>T.
R( T ) 
T 1
N- T
 Geometric Av erage 
 Arithmetic Av erage
N1
N-1
1-35
Check This 1.5a
Compute the Average Returns
Year
% Return
1926
1927
1928
1929
1930
10
16
-5
-8
7
4.00
Arithmetic Average
N
I/Y
PV
PMT
FV
5
CPT =
$ (1.0000)
0
$ 1.1933
$$ Invested
$
1.0000
$
1.1000
$
1.2760
$
1.2122
$
1.1152
$
1.1933
Geometric Average
3.60%
1-36
 Check This 1.5b
BLUME'S FORMULA
T 1
N- T
R( T ) 
 Geometric Average 
 Arithmetic Average
N 1
N-1
Arithmetic Average
Geometric Average
N
For T =
4.0%
3.6%
25
5
5  1
25 - 5
 3 .6 % 
 4%
25  1
25 - 1
4
20

( 3 . 6 %) 
( 4 %)
24
24
 ( 0 . 1667 )( 3 . 6 %)  ( 0 . 8333 )( 4 . 0 %)
R(5 ) 
 0 . 6001
 3 . 332
 3 . 933 %
1-37
 Check This 1.5b
BLUME'S FORMULA
R( T ) 
T 1
N- T
 Geometric Average 
 Arithmetic Average
N 1
N-1
Arithmetic Average
Geometric Average
N
For T =
4.0%
3.6%
25
10
10  1
25 - 10
 3 .6 % 
 4%
25  1
25 - 1
9
15

( 3 . 6 %) 
( 4 %)
24
24
 ( 0 . 3750 )( 3 . 6 %)  ( 0 . 6250 )( 4 . 0 %)
 1 . 35  2 . 50  3 . 85 %
R ( 10 ) 
1-38
Risk and Return
• The risk-free rate represents compensation
for the time value of money.
• First Lesson:
• If we are willing to bear risk, then we
can expect to earn a risk premium, at
least on average.
• Second Lesson:
• The more risk we are willing to bear, the
greater the expected risk premium.
1-39
Historical Risk and Return
Trade-Off
1-40
Chapter 1 End
A Brief History of
Risk and Return
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