Key Concepts and Skills
Chapter 10 Know how to calculate the return on an
investment
Know how to calculate the standard deviation of
an investment’s returns
Risk and Return: Lessons from Market History Understand the historical returns and risks on
various types of investments
Understand the importance of the normal
distribution
Understand the difference between arithmetic and
McGraw-Hill/Irwin Copyright © 2013 by The McGraw-Hill Companies, Inc. All rights reserved. geometric average returns 10-1
Chapter Outline 10.1 Returns
10.1 Returns Dollar Returns Dividends
10.2 Holding-Period Returns the sum of the cash received
10.3 Return Statistics and the change in value of the Ending
10.4 Average Stock Returns and Risk-Free Returns asset, in dollars. market value
10.5 Risk Statistics
10.6 More on Average Returns Time 0 1
Percentage Returns
10.7 The U.S. Equity Risk Premium: Historical and
International Perspectives –the sum of the cash received and the
Initial change in value of the asset, divided
10.8 2008: A Year of Financial Crisis investment by the initial investment.
10-2 10-3
Returns Returns: Example
Suppose you bought 100 shares of XYZ Co. one
Dollar Return = Dividend + Change in Market Value
year ago today at $45. Over the last year, you
dollar return received $27 in dividends (27 cents per share × 100
percentage return shares). At the end of the year, the stock sells for
beginning market value
$48. How did you do?
You invested $45 × 100 = $4,500. At the end of
dividend change in market value the year, you have stock worth $4,800 and cash
beginning market value dividends of $27. Your dollar gain was $327 =
$27 + ($4,800 – $4,500).
$327
Your percentage gain for the year is: 7.3% =
dividend yield capital gains yield $4,500
10-4 10-5
1
Returns: Example 10.2 Holding Period Return
Dollar Return: $27 The holding period return is the return
$327 gain that an investor would get when holding
$300 an investment over a period of T years,
when the return during year i is given as
Time 0 1 Ri:
Percentage Return:
HPR (1 R1 ) (1 R2 ) (1 RT ) 1
$327
-$4,500 7.3% =
$4,500
10-6 10-7
Holding Period Return: Example Historical Returns
Suppose your investment provides the A famous set of studies dealing with rates of returns
on common stocks, bonds, and Treasury bills was
following returns over a four-year conducted by Roger Ibbotson and Rex Sinquefield.
period: They present year-by-year historical rates of return
starting in 1926 for the following five important
Year Return Your holding period return types of financial instruments in the United States:
1 10%
(1 R1 ) (1 R2 ) (1 R3 ) (1 R4 ) 1 Large-company Common Stocks
2 -5% Small-company Common Stocks
3 20% (1.10) (.95) (1.20) (1.15) 1
Long-term Corporate Bonds
4 15% .4421 44.21% Long-term U.S. Government Bonds
10-8
U.S. Treasury Bills 10-9
10.3 Return Statistics Historical Returns, 1926-2011
The history of capital market returns can be Average Standard
summarized by describing the: Series Annual Return Deviation Distribution
average return Large Company Stocks 11.8% 20.3%
( R RT ) Small Company Stocks 16.5 32.5
R 1 Long-Term Corporate Bonds 6.4 8.4
T
Long-Term Government Bonds 6.1 9.8
the standard deviation of those returns
U.S. Treasury Bills 3.6 3.1
( R1 R ) 2 ( R2 R ) 2 ( RT R ) 2 Inflation 3.1 4.2
SD VAR
T 1 – 90% 0% + 90%
the frequency distribution of the returns Source: Global Financial Data (www.globalfinddata.com) copyright 2012.
10-10 10-11
2
10.4 Average Stock Returns and Risk-Free Returns Risk Premiums
The Risk Premium is the added return (over and above
the risk-free rate) resulting from bearing risk. Suppose that The Wall Street Journal announced that
One of the most significant observations of stock market the current rate for one-year Treasury bills is 2%.
data is the long-run excess of stock return over the risk- What is the expected return on the market of small-
free return. company stocks?
The average excess return from large company common
stocks for the period 1926 through 2011 was: Recall that the average excess return on small
8.2% = 11.8% – 3.6% company common stocks for the period 1926
The average excess return from small company common through 2011 was 12.9%.
stocks for the period 1926 through 2011 was:
Given a risk-free rate of 2%, we have an expected
12.9% = 16.5% – 3.6%
return on the market of small-company stocks of
The average excess return from long-term corporate bonds
for the period 1926 through 2011 was: 14.9% = 12.9% + 2%
2.8% = 6.4% – 3.6% 10-12 10-13
The Risk-Return Tradeoff 10.5 Risk Statistics
18%
Small-Company Stocks
There is no universally agreed-upon
16%
definition of risk.
Annual Return Average
14%
12% Large-Company Stocks The measures of risk that we discuss are
10%
variance and standard deviation.
The standard deviation is the standard statistical
8%
measure of the spread of a sample, and it will be
6%
T-Bonds
the measure we use most of this time.
4%
T-Bills
2%
0% 5% 10% 15% 20% 25% 30% 35%
Its interpretation is facilitated by a discussion of
the normal distribution.
Annual Return Standard Deviation
10-14 10-15
Normal Distribution Normal Distribution
A large enough sample drawn from a normal The 20.3% standard deviation we found
distribution looks like a bell-shaped curve. for large stock returns from 1926
Probability
The probability that a yearly return through 2011 can now be interpreted in
will fall within 20.3 percent of the
mean of 11.8 percent will be
the following way:
approximately 2/3.
If stock returns are approximately normally
distributed, the probability that a yearly
– 3s
– 49.1%
– 2s
– 28.8%
– 1s
– 8.5%
0
11.8%
+ 1s
32.1%
+ 2s
52.4%
+ 3s
72.7% Return on
return will fall within 20.3 percent of the
68.26%
large company common
stocks mean of 11.8% will be approximately 2/3.
95.44%
99.74% 10-16 10-17
3
Example – Return and Variance 10.6 More on Average Returns
Year Actual Average Deviation from the Squared Arithmetic average – return earned in an average
Return Return Mean Deviation period over multiple periods
1 .15 .105 .045 .002025
Geometric average – average compound return per
2 .09 .105 -.015 .000225 period over multiple periods
3 .06 .105 -.045 .002025 The geometric average will be less than the arithmetic
average unless all the returns are equal.
4 .12 .105 .015 .000225
Which is better?
Totals .00 .0045 The arithmetic average is overly optimistic for long
horizons.
Variance = .0045 / (4-1) = .0015 Standard Deviation = .03873 The geometric average is overly pessimistic for short
horizons.
10-18 10-19
Geometric Return: Example Geometric Return: Example
Recall our earlier example: Note that the geometric average is not
the same as the arithmetic average:
Year Return Geometric average return
1 10% (1 Rg ) 4 (1 R1 ) (1 R2 ) (1 R3 ) (1 R4 ) Year Return
R1 R2 R3 R4
2 -5% 1 10% Arithmetic average return
3 20% Rg 4 (1.10) (.95) (1.20) (1.15) 1 2 -5%
4
4 15% .095844 9.58% 3 20% 10% 5% 20% 15%
10%
So, our investor made an average of 9.58% per year, 4 15% 4
realizing a holding period return of 44.21%.
1.4421 (1.095844) 4
10-20 10-21
Perspectives on the Equity Risk Premium Quick Quiz
Over 1926-2011, the U.S. equity risk premium has Which of the investments discussed has had
been quite large: the highest average return and risk premium?
Earlier years (beginning in 1802) provide a smaller
estimate at 5.4% Which of the investments discussed has had
Comparable data for 1900 to 2010 put the international the highest standard deviation?
equity risk premium at an average of 6.9%, versus 7.2% in Why is the normal distribution informative?
the U.S.
Going forward, an estimate of 7% seems reasonable, What is the difference between arithmetic and
although somewhat higher or lower numbers could geometric averages?
also be considered rational
10-22 10-23
