15.401
15.401 Finance Theory
MIT Sloan MBA Program
Andrew W. Lo
Harris & Harris Group Professor, MIT Sloan School
Lecture 12: Introduction to Risk and Return
© 2007–2008 by Andrew W. Lo
Critical Concepts
ƒ
ƒ
ƒ
ƒ
15.401
Motivation
Statistical Background
Empirical Properties of Stock Returns
Anomalies
Readings
ƒ Brealey, Myers, and Allen Chapters 7, 24.1, 24.4
Lecture 12: Intro to Risk and Return
© 2007–2008 by Andrew W. Lo
Slide 2
Motivation
15.401
NPV and Other Valuation Techniques Need Cost of Capital
ƒ Opportunity cost
ƒ Required rate of return
ƒ Risk-adjusted discount rate
ƒ Determined by “the market”
ƒ How???
Introduce Risk Into The Valuation Process
ƒ How to measure risk
ƒ How to estimate the required rate of return for a given level of risk
ƒ Related questions:
– How risky are stocks and what have their returns been historically?
– Is the stock market “efficient”?
– How can we gauge the performance of portfolio managers?
Lecture 12: Intro to Risk and Return
© 2007–2008 by Andrew W. Lo
Slide 3
Statistical Background
15.401
Terminology
Lecture 12: Intro to Risk and Return
© 2007–2008 by Andrew W. Lo
Slide 4
Statistical Background
15.401
Terminology
ƒ Mean, variance, standard deviation:
ƒ Sample estimators:
Lecture 12: Intro to Risk and Return
© 2007–2008 by Andrew W. Lo
Slide 5
Statistical Background
15.401
Other Statistics
ƒ Median
– 50th percentile (probability of 1/2 that Rt < median)
ƒ Skewness
– Is the distribution symmetric?
– Negative: big losses are more likely than big gains
– Positive: big gains are more likely than big losses
ƒ Correlation
– How closely do two variables move together?
Lecture 12: Intro to Risk and Return
© 2007–2008 by Andrew W. Lo
Slide 6
Statistical Background
15.401
Negatively Skewed Distribution
-5
-4
-3
Lecture 12: Intro to Risk and Return
-2
-1
0
1
© 2007–2008 by Andrew W. Lo
2
3
4
5
Slide 7
Statistical Background
15.401
Examples of Correlation Between Two Random Variables
4
ρ=0
3
ρ = .5
3
2
2
1
1
0
0
-4
-3
-2
-1
-4
0
1
2
3
-3
-2
-1
4
0
1
2
3
4
0
1
2
3
4
-1
-1
-2
-2
-3
-3
-4
-4
4
ρ = .8
3
ρ = –.5
3
2
2
1
1
0
0
-4
-3
-2
-1
-4
0
1
2
3
-3
-2
4
-1
-1
-1
-2
-2
-3
-3
-4
-4
Lecture 12: Intro to Risk and Return
© 2007–2008 by Andrew W. Lo
Slide 8
Statistical Background
15.401
Normal Distribution
ƒ Bell-shaped, symmetric
ƒ A model of randomness
ƒ Central Limit Theorem
Confidence Intervals
-5
-4
-3
-2
-1
0
1
2
3
4
If R is normally distributed, then …
ƒ 68% of observations fall within +/–1.00 std. deviations from mean
ƒ 90% of observations fall within +/–1.65 std. deviations from mean
ƒ 95% of observations fall within +/–1.96 std. deviations from mean
ƒ 99% of observations fall within +/–2.58 std. deviations from mean
Lecture 12: Intro to Risk and Return
© 2007–2008 by Andrew W. Lo
5
Slide 9
Statistical Background
15.401
GM Monthly Returns
0.21
0.14
0.07
0.00
-21%
-15%
Lecture 12: Intro to Risk and Return
-9%
-3%
3%
9%
© 2007–2008 by Andrew W. Lo
15%
21%
Slide 10
Empirical Properties of Stock Returns
15.401
What Characterizes U.S. Stock Returns?
ƒ How volatile are stock returns?
ƒ Are returns predictable?
ƒ How does volatility change over time?
ƒ What types of stocks have the highest returns?
What Properties Should Stock Prices Have In “Efficient” Markets?
ƒ Random, unpredictable
ƒ Prices should react quickly and correctly to news
ƒ Investors cannot earn abnormal, risk-adjusted returns (or at least it
shouldn’t be easy)
Lecture 12: Intro to Risk and Return
© 2007–2008 by Andrew W. Lo
Slide 11
Empirical Properties of Stock Returns
15.401
Predictable Price Changes
$ 80
70
60
50
40
30
20
Lecture 12: Intro to Risk and Return
© 2007–2008 by Andrew W. Lo
Slide 12
Empirical Properties of Stock Returns
15.401
Random Walks with Drift
$ 80
75
70
65
60
55
50
0
10
20
Lecture 12: Intro to Risk and Return
30
40
50
60
70
© 2007–2008 by Andrew W. Lo
80
90
100
Slide 13
Empirical Properties of Stock Returns
15.401
Four facts from history of U.S. financial markets:
1. Real interest rate has been slightly positive on average.
2. Return on more risky assets has been higher on average than return
on less risky assets.
3. Returns on risky assets can be highly correlated to each other.
4. Returns on risky assets are (usually) serially uncorrelated.
Basic Statistics, U.S., 1946 – 2001 (monthly,in percent)
Inflation
Tbill (1 yr)
Tnote (10 yr)
VW stock index
EW stock index
Motorola
Avg
Stdev
Skew
Min
Max
0.32
0.38
0.46
1.01
1.18
1.66
0.36
0.24
2.63
4.23
5.30
10.02
0.82
0.98
0.61
-0.47
-0.17
0.01
-0.84
0.03
-7.73
-22.49
-27.09
-33.49
1.85
1.34
13.31
16.56
29.92
41.67
NYSE, Amex, NASDAQ: 6,700 firms, $16.4 trillion market cap
Lecture 12: Intro to Risk and Return
© 2007–2008 by Andrew W. Lo
Slide 14
Empirical Properties of Stock Returns
15.401
Total Return of Stocks, Bonds, Bills and Inflation 1946 – 2001
10000
cpi
tbill
10 note
Dec-61
Dec-69
vw
ew
Dec-77
Dec-85
1000
100
10
1
0.1
Dec-45
Dec-53
Lecture 12: Intro to Risk and Return
© 2007–2008 by Andrew W. Lo
Dec-93
Dec-01
Slide 15
Empirical Properties of Stock Returns
15.401
Interest Rates 1953 – 2001
18%
1-yr Tbill
10-yr Tbond
15%
12%
9%
6%
3%
0%
Jun-53
Jan-63
Lecture 12: Intro to Risk and Return
Jun-72
Jan-82
© 2007–2008 by Andrew W. Lo
Jun-91
Jan-01
Slide 16
Empirical Properties of Stock Returns
15.401
Total Returns, 10-Year U.S. T-Bond, 1946 – 2001
25%
20%
15%
10%
5%
0%
-5%
-10%
-15%
-20%
-25%
Jan46
Jan51
Jan56
Lecture 12: Intro to Risk and Return
Jan61
Jan66
Jan71
Jan76
Jan81
© 2007–2008 by Andrew W. Lo
Jan86
Jan91
Jan96
Jan01
Slide 17
Empirical Properties of Stock Returns
15.401
Total Returns, U.S. Stock Market 1946 – 2001
25%
20%
15%
10%
5%
0%
-5%
-10%
-15%
-20%
-25%
Jan46
Jan51
Jan56
Lecture 12: Intro to Risk and Return
Jan61
Jan66
Jan71
Jan76
Jan81
© 2007–2008 by Andrew W. Lo
Jan86
Jan91
Jan96
Jan01
Slide 18
Empirical Properties of Stock Returns
15.401
Total Returns, Motorola 1946 – 2001
25%
20%
15%
10%
5%
0%
-5%
-10%
-15%
-20%
-25%
Lecture 12: Intro to Risk and Return
© 2007–2008 by Andrew W. Lo
Slide 19
Empirical Properties of Stock Returns
15.401
Scatterplot, VWRETD Today vs. Yesterday ,1980 – 1999
2.0%
1.0%
0.0%
-2.5%
-1.5%
-0.5%
0.5%
1.5%
2.5%
-1.0%
-2.0%
Lecture 12: Intro to Risk and Return
© 2007–2008 by Andrew W. Lo
Slide 20
Empirical Properties of Stock Returns
15.401
Scatterplot, S&P 500 This Month vs. Last Month, 1926 to 1997
20%
15%
10%
5%
0%
-25%
-15%
-5%
-5%
5%
15%
25%
-10%
-15%
-20%
Lecture 12: Intro to Risk and Return
© 2007–2008 by Andrew W. Lo
Slide 21
Empirical Properties of Stock Returns
15.401
Scatterplot GM vs. S&P 500 Monthly Returns, 1946 – 1997
25%
17%
8%
-35%
-25%
-15%
0%
-5%
5%
15%
25%
35%
-8%
-17%
-25%
Lecture 12: Intro to Risk and Return
© 2007–2008 by Andrew W. Lo
Slide 22
Empirical Properties of Stock Returns
15.401
Monthly Estimates of U.S. Stock Market Daily Volatility 1926 – 1997
6%
5%
4%
3%
2%
1%
0%
Jan 26
Jan 36
Lecture 12: Intro to Risk and Return
Jan 46
Jan 56
Jan 66
© 2007–2008 by Andrew W. Lo
Jan 76
Jan 86
Jan 96
Slide 23
Anomalies: The Size Effect, 1964 – 2004
15.401
16.0
14.0
12.0
10.0
8.0
Small
2
3
4
5
6
7
8
9
Big
Firms sorted by MARKET CAPITALIZATION
Lecture 12: Intro to Risk and Return
© 2007–2008 by Andrew W. Lo
Slide 24
Anomalies: The January Effect, 1964 – 2004
15.401
9%
8%
All Months
Januarys
7%
6%
5%
4%
3%
2%
1%
0%
Small
2
3
4
5
6
7
8
9
Big
Firm s sorted by MARKET CAPITALIZATION
Lecture 12: Intro to Risk and Return
© 2007–2008 by Andrew W. Lo
Slide 25
Anomalies: The Value Premium, 1964 – 2004
15.401
18%
16%
14%
12%
10%
8%
Low
2
3
4
5
6
7
8
9
High
Firms sorted by PRICE / BOOK EQUITY
Lecture 12: Intro to Risk and Return
© 2007–2008 by Andrew W. Lo
Slide 26
Anomalies: Momentum, 1964 – 2004
15.401
18%
15%
12%
9%
6%
3%
Low
2
3
4
5
6
7
8
9
High
Firms sorted by PAST 12-MONTH RETURN
Lecture 12: Intro to Risk and Return
© 2007–2008 by Andrew W. Lo
Slide 27
Anomalies: The Accrual Effect, 1964 – 2004
15.401
15.0
13.0
11.0
9.0
7.0
5.0
3.0
Low
2
3
4
5
6
7
8
9
High
Firms sorted by last year's OPERATING ACCRUALS
*Operating income minus operating cashflows
Lecture 12: Intro to Risk and Return
© 2007–2008 by Andrew W. Lo
Slide 28
Anomalies: IPO Returns, 1970 – 1990
15.401
Average Annual Returns, 1 – 5 Years After IPO
14.1%
16%
14.3%
13.3%
11.3%
12%
11.6%
8%
6.1%
5.0%
4%
4.0%
3.6%
Non-issuers
IPOs
1.6%
0%
1st year
2nd year
Lecture 12: Intro to Risk and Return
3rd year
4th year
© 2007–2008 by Andrew W. Lo
5th year
Slide 29
Anomalies: SEO Returns, 1970 – 1990
15.401
Average Annual Returns, 1 – 5 Years After SEO
17.7%
20%
15%
17.4%
16.2%
12.9%
12.3%
11.8%
10%
9.1%
5%
7.5%
6.6%
Non-issuers
SEOs
0.3%
0%
1st year
2nd year
Lecture 12: Intro to Risk and Return
3rd year
4th year
© 2007–2008 by Andrew W. Lo
5th year
Slide 30
Anomalies: Takeover Announcements
15.401
Cumulative average abnormal returns
Stock price of TARGET
40%
30%
20%
10%
0%
-10%
-126 -105 -84 -63
-42 -21
0
21
42
63
84
105 126 147 168 189 210 231 252
Event date relative to first bid
Successful
All
Unsuccessful
Stock Price of TARGET
Image by MIT OpenCourseWare.
Lecture 12: Intro to Risk and Return
© 2007–2008 by Andrew W. Lo
Slide 31
Anomalies: Performance of Mutual Funds
15.401
36
32
# Frequency
28
24
20
16
12
8
4
0
-3
-2
-1
0
1
2
Midpoint
Estimates of individual mutual-fund alphas 1972 to 1991. The frequency distribution
of estimated alphas for all equity mutual funds with 10-year continuous records.
Image by MIT OpenCourseWare.
Lecture 12: Intro to Risk and Return
© 2007–2008 by Andrew W. Lo
Slide 32
Key Points
15.401
Observations
ƒ
ƒ
ƒ
ƒ
ƒ
ƒ
ƒ
The average annual return on U.S. stocks from 1926 – 2004 was 11.2%.
The average risk premium was 7.8%.
Stocks are quite risky. The standard deviation of returns for the overall
market is 4.5% monthly (16.4% annually).
Individual stocks are much riskier. The average monthly standard
deviation of an individual stock is around 17% (or 50% annually).
Stocks tend to move together over time: when one stock goes up, other
stocks are likely to go up as well. The correlation is far from perfect.
Stock returns are nearly unpredictable. For example, knowing how a stock
does this month tells you very little about what will happen next month.
Market volatility changes over time. Prices are sometimes quite volatile.
The standard deviation of monthly returns varies from roughly 2% to 20%.
Financial ratios like DY and P/E ratios vary widely over time. DY hit a
maximum of 13.8% in 1932 and a minimum of 1.17% in 1999. The P/E ratio
hit a maximum of 33.4 in 1999 and a minimum of 5.3 in 1917.
Lecture 12: Intro to Risk and Return
© 2007–2008 by Andrew W. Lo
Slide 33
Key Points
15.401
Anomalies:
ƒ Size Effect: Smaller stocks typically outperform larger stocks,
especially in January.
ƒ January Effect: Returns in January tend to be abnormally high.
ƒ Value Effect: Low P/B (value) stocks typically outperform high P/B
(growth) stocks.
ƒ Momentum: Stocks with high returns over the past 12 months
typically continue to outperform stocks with low past returns.
ƒ Accruals and Issuances: Stocks with high past accruals and/or
recent stock offerings typically underperform stocks with low past
accruals and no stock offerings.
Lecture 12: Intro to Risk and Return
© 2007–2008 by Andrew W. Lo
Slide 34
Additional References
15.401
ƒ Lefevre, E., 2006, Reminiscences of a Stock Operator. New York: John Wiley & Sons.
ƒ Malkiel, B., 1996, A Random Walk Down Wall Street: Including a Life-Cycle Guide to Personal
Investing. New York: W.W. Norton.
Lecture 12: Intro to Risk and Return
© 2007–2008 by Andrew W. Lo
Slide 35
MIT OpenCourseWare
http://ocw.mit.edu
15.401 Finance Theory I
Fall 2008
For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.