15.401
15.401 Finance Theory
MIT Sloan MBA Program
Andrew W. Lo
Harris & Harris Group Professor, MIT Sloan School
Lecture 13–14: Risk Analytics and Portfolio Theory
© 2007–2008 by Andrew W. Lo
Critical Concepts
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ƒ
ƒ
ƒ
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15.401
Motivation
Measuring Risk and Reward
Mean-Variance Analysis
The Efficient Frontier
The Tangency Portfolio
Readings:
ƒ Brealey, Myers, and Allen Chapters 7 and 8.1
Lectures 13–14: Risk Analytics and
Portfolio Theory
© 2007–2008 by Andrew W. Lo
Slide 2
Motivation
15.401
What Is A Portfolio and Why Is It Useful?
ƒ A portfolio is simply a specific combination of securities, usually
defined by portfolio weights that sum to 1:
ƒ Portfolio weights can sum to 0 (dollar-neutral portfolios), and weights
can be positive (long positions) or negative (short positions).
ƒ Assumption: Portfolio weights summarize all relevant information.
Lectures 13–14: Risk Analytics and
Portfolio Theory
© 2007–2008 by Andrew W. Lo
Slide 3
Motivation
15.401
Example:
ƒ Your investment account of $100,000 consists of three stocks: 200
shares of stock A, 1,000 shares of stock B, and 750 shares of stock C.
Your portfolio is summarized by the following weights:
Dollar
Price/Share Investment
Portfolio
Weight
Asset
Shares
A
200
$50
$10,000
10%
B
1,000
$60
$60,000
60%
C
750
$40
$30,000
30%
$100,000
100%
Total
Lectures 13–14: Risk Analytics and
Portfolio Theory
© 2007–2008 by Andrew W. Lo
Slide 4
Motivation
15.401
Example (cont):
ƒ Your broker informs you that you only need to keep $50,000 in your
investment account to support the same portfolio of 200 shares of stock
A, 1,000 shares of stock B, and 750 shares of stock C; in other words,
you can buy these stocks on margin. You withdraw $50,000 to use for
other purposes, leaving $50,000 in the account. Your portfolio is
summarized by the following weights:
Dollar
Price/Share Investment
Portfolio
Weight
Asset
Shares
A
200
$50
$10,000
20%
B
1,000
$60
$60,000
120%
C
750
$40
$30,000
60%
Riskless Bond
−$50,000
$1
−$50,000
−100%
$50,000
100%
Total
Lectures 13–14: Risk Analytics and
Portfolio Theory
© 2007–2008 by Andrew W. Lo
Slide 5
Motivation
15.401
Example:
ƒ You decide to purchase a home that costs $500,000 by paying 20% of
the purchase price and getting a mortgage for the remaining 80% What
are your portfolio weights for this investment?
Dollar
Portfolio
Price/Share Investment Weight
Asset
Shares
Home
1
$500,000
$500,000
500%
Mortgage
1
−$400,000
−$400,000
−400%
$100,000
100%
Total
ƒ What happens to your total assets if your home price declines by 15%?
Lectures 13–14: Risk Analytics and
Portfolio Theory
© 2007–2008 by Andrew W. Lo
Slide 6
Motivation
15.401
Example:
ƒ You own 100 shares of stock A, and you have shorted 200 shares of
stock B. Your portfolio is summarized by the following weights:
Dollar
Price/Share Investment
Portfolio
Weight
Stock
Shares
A
100
$50
$5,000
???
B
−200
$25
− $5,000
???
ƒ Zero net-investment portfolios do not have portfolio weights in
percentages (because the denominator is 0)—we simply use dollar
amounts instead of portfolio weights to represent long and short
positions
Lectures 13–14: Risk Analytics and
Portfolio Theory
© 2007–2008 by Andrew W. Lo
Slide 7
Motivation
15.401
Why Not Pick The Best Stock Instead of Forming a Portfolio?
ƒ We don’t know which stock is best!
ƒ Portfolios provide diversification, reducing unnecessary risks.
ƒ Portfolios can enhance performance by focusing bets.
ƒ Portfolios can customize and manage risk/reward trade-offs.
How Do We Construct a “Good” Portfolio?
ƒ What does “good” mean?
ƒ What characteristics do we care about for a given portfolio?
ƒ Risk and reward
ƒ Investors like higher expected returns
ƒ Investors dislike risk
Lectures 13–14: Risk Analytics and
Portfolio Theory
© 2007–2008 by Andrew W. Lo
Slide 8
Measuring Risk and Reward
15.401
ƒ Reward is typically measured by return
ƒ Higher returns are better than lower returns.
ƒ But what if returns are unknown?
ƒ Assume returns are random, and consider the distribution of returns.
Several possible measures:
ƒ Mean: central tendency.
ƒ 75%: upper quartile.
ƒ 5%: losses.
5%
Mean
Lectures 13–14: Risk Analytics and
Portfolio Theory
75%
© 2007–2008 by Andrew W. Lo
Slide 9
Measuring Risk and Reward
15.401
ƒ How about risk?
ƒ Likelihood of loss (negative return).
ƒ But loss can come from positive return (e.g., short position).
ƒ A symmetric measure of dispersion is variance or standard deviation.
Variance Measures Spread:
ƒ Blue distribution is “riskier”.
ƒ Extreme outcomes more likely.
ƒ This measure is symmetric.
Lectures 13–14: Risk Analytics and
Portfolio Theory
© 2007–2008 by Andrew W. Lo
Slide 10
Measuring Risk and Reward
15.401
Assumption
ƒ Investors like high expected returns but dislike high volatility
ƒ Investors care only about the expected return and volatility of their
overall portfolio
– Not individual stocks in the portfolio
– Investors are generally assumed to be well-diversified
Key questions: How much does a stock contribute to the risk and
return of a portfolio, and how can we choose portfolio weights to
optimize the risk/reward characteristics of the overall portfolio?
Lectures 13–14: Risk Analytics and
Portfolio Theory
© 2007–2008 by Andrew W. Lo
Slide 11
Mean-Variance Analysis
15.401
Objective
ƒ Assume investors focus only on the expected return and variance (or
standard deviation) of their portfolios: higher expected return is good,
higher variance is bad
ƒ Develop a method for constructing optimal portfolios
Expected Return E[Rp]
Higher Utility
MRK
MOT
MCD
GM
T-Bills
Standard Deviation of Return SD[Rp]
Lectures 13–14: Risk Analytics and
Portfolio Theory
© 2007–2008 by Andrew W. Lo
Slide 12
Mean-Variance Analysis
15.401
Basic Properties of Mean and Variance For Individual Returns:
Basic Properties of Mean And Variance For Portfolio Returns:
Lectures 13–14: Risk Analytics and
Portfolio Theory
© 2007–2008 by Andrew W. Lo
Slide 13
Mean-Variance Analysis
15.401
Variance Is More Complicated:
Lectures 13–14: Risk Analytics and
Portfolio Theory
© 2007–2008 by Andrew W. Lo
Slide 14
Mean-Variance Analysis
15.401
Portfolio variance is the weighted sum of all the variances and
covariances:
ƒ There are n variances, and n2 − n covariances
ƒ Covariances dominate portfolio variance
ƒ Positive covariances increase portfolio variance; negative covariances
decrease portfolio variance (diversification)
Lectures 13–14: Risk Analytics and
Portfolio Theory
© 2007–2008 by Andrew W. Lo
Slide 15
Mean-Variance Analysis
15.401
Consider The Special Case of Two Assets:
ƒ As correlation increases, overall portfolio variance increases
Lectures 13–14: Risk Analytics and
Portfolio Theory
© 2007–2008 by Andrew W. Lo
Slide 16
Mean-Variance Analysis
15.401
Example: From 1946 – 2001, Motorola had an average monthly return of
1.75% and a std dev of 9.73%. GM had an average return of 1.08%
and a std dev of 6.23%. Their correlation is 0.37. How would a
portfolio of the two stocks perform?
wMot
0
0.25
0.50
0.75
1
1.25
wGM
1
0.75
0.50
0.25
0
-0.25
Lectures 13–14: Risk Analytics and
Portfolio Theory
E[RP]
1.08
1.25
1.42
1.58
1.75
1.92
var(RP)
38.8
36.2
44.6
64.1
94.6
136.3
© 2007–2008 by Andrew W. Lo
stdev(RP)
6.23
6.01
6.68
8.00
9.73
11.67
Slide 17
Mean-Variance Analysis
15.401
Mean/SD Trade-Off for Portfolios of GM and Motorola
2.1%
Motorola
Expected Return
1.7%
1.3%
GM
0.9%
0.5%
2.0%
4.0%
6.0%
8.0%
10.0%
12.0%
14.0%
Standard Deviation of Return
Lectures 13–14: Risk Analytics and
Portfolio Theory
© 2007–2008 by Andrew W. Lo
Slide 18
Mean-Variance Analysis
15.401
Example (cont): Suppose the correlation between GM and Motorola
changes. What if it equals –1.0? 0.0? 1.0?
wMot
0
0.25
0.50
0.75
1
wGM
1
0.75
0.50
0.25
0
Lectures 13–14: Risk Analytics and
Portfolio Theory
E[RP]
1.08%
1.25
1.42
1.58
1.75
Std dev of portfolio
corr = -1
corr = 0
corr = 1
6.23%
6.23%
6.23%
2.24
5.27
7.10
1.75
5.78
7.98
5.74
7.46
8.85
9.73
9.73
9.73
© 2007–2008 by Andrew W. Lo
Slide 19
Mean-Variance Analysis
15.401
Mean/SD Trade-Off for Portfolios of GM and Motorola
2.2%
Expected Return
1.8%
corr=-1
1.4%
corr=-.5
corr=0
corr=1
1.0%
0.6%
0.0%
2.0%
Lectures 13–14: Risk Analytics and
Portfolio Theory
4.0%
6.0%
8.0% 10.0% 12.0%
Standard Deviation of Return
© 2007–2008 by Andrew W. Lo
14.0%
16.0%
Slide 20
Mean-Variance Analysis
15.401
Example: In 1980, you were thinking about investing in GD. Over the
subsequent 10 years, GD had an average monthly return of 0.00%
and a std dev of 9.96%. Motorola had an average return of 1.28% and
a std dev of 9.33%. Their correlation is 0.28. How would a portfolio of
the two stocks perform?
wMot
0
0.25
0.50
0.75
1
wGD
1
0.75
0.50
0.25
0
Lectures 13–14: Risk Analytics and
Portfolio Theory
E[RP]
0.00
0.32
0.64
0.96
1.28
var(RP)
99.20
71.00
59.57
64.92
87.05
© 2007–2008 by Andrew W. Lo
stdev(RP)
9.96
8.43
7.72
8.06
9.33
Slide 21
Mean-Variance Analysis
15.401
Mean/SD Trade-Off for Portfolios of GD and Motorola
2.0%
1.5%
Expected Return
Motorola
1.0%
0.5%
0.0%
4.0%
GD
6.0%
8.0%
10.0%
12.0%
14.0%
16.0%
-0.5%
Standard Deviation of Return
-1.0%
Lectures 13–14: Risk Analytics and
Portfolio Theory
© 2007–2008 by Andrew W. Lo
Slide 22
Mean-Variance Analysis
15.401
Example: You are trying to decide how to allocate your retirement
savings between Treasury bills and the stock market. The T-Bill rate
is 0.12% monthly. You expect the stock market to have a monthly
return of 0.75% with a standard deviation of 4.25%.
wStk
0
0.33
0.67
1
wTbill
1
0.67
0.33
0
Lectures 13–14: Risk Analytics and
Portfolio Theory
E[RP]
0.12
0.33
0.54
0.75
var(RP)
0.00
1.97
8.11
18.06
© 2007–2008 by Andrew W. Lo
stdev(RP)
0.00
1.40
2.85
4.25
Slide 23
Mean-Variance Analysis
15.401
Mean/SD Trade-Off for Portfolios of T-Bills and The Stock Market
1.0%
0.8%
Expected Return
Stock Market
0.6%
50/50 portfolio
0.4%
0.2%
T-Bills
0.0%
0.0%
1.0%
Lectures 13–14: Risk Analytics and
Portfolio Theory
2.0%
3.0%
4.0%
Standard Deviation of Return
© 2007–2008 by Andrew W. Lo
5.0%
6.0%
Slide 24
Mean-Variance Analysis
15.401
Summary
Observations
ƒ E[RP] is a weighted average of stocks’ expected returns
ƒ SD(RP) is smaller if stocks’ correlation is lower. It is less than a
weighted average of the stocks’ standard deviations (unless perfect
correlation)
ƒ The graph of portfolio mean/SD is nonlinear
ƒ If we combine T-Bills with any risky stock, portfolios plot along a
straight line
Lectures 13–14: Risk Analytics and
Portfolio Theory
© 2007–2008 by Andrew W. Lo
Slide 25
Mean-Variance Analysis
15.401
The General Case:
ƒ Portfolio variance is the sum of weights times entries in the
covariance matrix
Lectures 13–14: Risk Analytics and
Portfolio Theory
© 2007–2008 by Andrew W. Lo
Slide 26
Mean-Variance Analysis
15.401
The General Case:
ƒ Covariance matrix contains n2 terms
– n terms are variances
– n2 – n terms are covariances
ƒ In a well-diversified portfolio, covariances are more important than
variances
ƒ A stock’s covariance with other stocks determines its contribution to
the portfolio’s overall variance
ƒ Investors should care more about the risk that is common to many
stocks; risks that are unique to each stock can be diversified away
Lectures 13–14: Risk Analytics and
Portfolio Theory
© 2007–2008 by Andrew W. Lo
Slide 27
Mean-Variance Analysis
15.401
Special Case:
ƒ Consider an equally weighted portfolio:
ƒ For portfolios with many stocks, the variance is determined by the
average covariance among the stocks
Lectures 13–14: Risk Analytics and
Portfolio Theory
© 2007–2008 by Andrew W. Lo
Slide 28
Mean-Variance Analysis
15.401
Example: The average stock has a monthly standard deviation of 10%
and the average correlation between stocks is 0.40. If you invest the
same amount in each stock, what is variance of the portfolio? What if
the correlation is 0.0? 1.0?
Lectures 13–14: Risk Analytics and
Portfolio Theory
© 2007–2008 by Andrew W. Lo
Slide 29
Mean-Variance Analysis
15.401
Example (cont):
12%
SD of Portfolio
10%
ρ = 1.0
8%
6%
ρ = 0.4
4%
ρ = 0.0
2%
0%
1
11
21
31
41
51
61
71
81
91
Number of Stocks
Lectures 13–14: Risk Analytics and
Portfolio Theory
© 2007–2008 by Andrew W. Lo
Slide 30
Mean-Variance Analysis
15.401
ƒ Remaining risk known as
systematic or market risk
ƒ Due to common factors
that cannot be diversified
ƒ Example: S&P 500
ƒ Other sources of
systematic risk may exist:
ƒ Credit
ƒ Liquidity
ƒ Volatility
ƒ Business Cycle
ƒ Value/Growth
ƒ Provides motivation for
linear factor models
Lectures 13–14: Risk Analytics and
Portfolio Theory
Portfolio Variance
Eventually, Diversification Benefits Reach A Limit:
Risk eliminated
by diversification
Total risk
of typical
stock
Undiversifiable or
systematic risk
Number of Stocks in Portfolio
© 2007–2008 by Andrew W. Lo
Slide 31
The Efficient Frontier
15.401
Given Portfolio Expected Returns and Variances:
How Should We Choose The Best Weights?
ƒ All feasible portfolios lie inside a bullet-shaped region, called the
minimum-variance boundary or frontier
ƒ The efficient frontier is the top half of the minimum-variance
boundary (why?)
ƒ Rational investors should select portfolios from the efficient frontier
Lectures 13–14: Risk Analytics and
Portfolio Theory
© 2007–2008 by Andrew W. Lo
Slide 32
The Efficient Frontier
15.401
2.4%
Expected Return
Efficient Frontier
1.8%
Minimum-Variance Boundary
1.2%
0.6%
0.0%
2.0%
4.0%
6.0%
8.0%
10.0%
12.0%
14.0%
16.0%
Standard Deviation of Return
Lectures 13–14: Risk Analytics and
Portfolio Theory
© 2007–2008 by Andrew W. Lo
Slide 33
The Efficient Frontier
15.401
Example: You can invest in any combination of GM, IBM, and MOT.
What portfolio would you choose?
Stock
GM
IBM
Motorola
Mean
1.08
1.32
1.75
Lectures 13–14: Risk Analytics and
Portfolio Theory
Std dev
6.23
6.34
9.73
Variance / covariance
GM
IBM
Motorola
38.80
16.13
22.43
16.13
40.21
23.99
22.43
23.99
94.63
© 2007–2008 by Andrew W. Lo
Slide 34
The Efficient Frontier
15.401
Example: You can invest in any combination of GM, IBM, and MOT.
What portfolio would you choose?
Stock
GM
IBM
Motorola
Mean
1.08
1.32
1.75
Lectures 13–14: Risk Analytics and
Portfolio Theory
Std dev
6.23
6.34
9.73
Variance / covariance
GM
IBM
Motorola
38.80
16.13
22.43
16.13
40.21
23.99
22.43
23.99
94.63
© 2007–2008 by Andrew W. Lo
Slide 35
The Efficient Frontier
15.401
Example: You can invest in any combination of GM, IBM, and MOT.
What portfolio would you choose?
Stock
GM
IBM
Motorola
Mean
1.08
1.32
1.75
Std dev
6.23
6.34
9.73
Variance / covariance
GM
IBM
Motorola
38.80
16.13
22.43
16.13
40.21
23.99
22.43
23.99
94.63
E[RP] = (wGM × 1.08) + (wIBM × 1.32) + (wMot × 1.75)
Lectures 13–14: Risk Analytics and
Portfolio Theory
© 2007–2008 by Andrew W. Lo
Slide 36
The Efficient Frontier
15.401
Example: You can invest in any combination of GM, IBM, and MOT.
What portfolio would you choose?
Stock
GM
IBM
Motorola
Mean
1.08
1.32
1.75
Std dev
6.23
6.34
9.73
Variance / covariance
GM
IBM
Motorola
38.80
16.13
22.43
16.13
40.21
23.99
22.43
23.99
94.63
E[RP] = (wGM × 1.08) + (wIBM × 1.32) + (wMot × 1.75)
var(RP) = (wGM2 × 38.80) + (wIBM × 40.21) + (wMot2 × 94.23) +
(2 × wGM × wIBM × 16.13) + (2 × wGM × wMot × 22.43) +
(2 × wIBM × wMot × 23.99)
Lectures 13–14: Risk Analytics and
Portfolio Theory
© 2007–2008 by Andrew W. Lo
Slide 37
The Efficient Frontier
15.401
Example: You can invest in any combination of GM, IBM, and MOT.
What portfolio would you choose?
Stock
GM
IBM
Motorola
Mean
1.08
1.32
1.75
Std dev
6.23
6.34
9.73
Variance / covariance
GM
IBM
Motorola
38.80
16.13
22.43
16.13
40.21
23.99
22.43
23.99
94.63
E[RP] = (wGM × 1.08) + (wIBM × 1.32) + (wMot × 1.75)
var(RP) = (wGM2 × 38.80) + (wIBM × 40.21) + (wMot2 × 94.63) +
(2 × wGM × wIBM × 16.13) + (2 × wGM × wMot × 22.43) +
(2 × wIBM × wMot × 23.99)
Lectures 13–14: Risk Analytics and
Portfolio Theory
© 2007–2008 by Andrew W. Lo
Slide 38
The Efficient Frontier
15.401
Example (cont): Feasible Portfolios
2.1%
Expected Return
1.8%
Motorola
1.5%
IBM
1.2%
GM
0.9%
0.6%
3.0%
Lectures 13–14: Risk Analytics and
Portfolio Theory
5.0%
7.0%
9.0%
Standard Deviation of Return
© 2007–2008 by Andrew W. Lo
11.0%
Slide 39
The Tangency Portfolio
15.401
The Tangency Portfolio
ƒ If there is also a riskless asset (T-Bills), all investors should
hold exactly the same stock portfolio!
ƒ All efficient portfolios are combinations of the riskless asset
and a unique portfolio of stocks, called the tangency portfolio.*
– In this case, efficient frontier becomes straight line
* Harry Markowitz, Nobel Laureate
Lectures 13–14: Risk Analytics and
Portfolio Theory
© 2007–2008 by Andrew W. Lo
Slide 40
The Tangency Portfolio
15.401
2.4%
Expected Return
1.8%
Motorola
IBM
1.2%
P
GM
0.6%
Tbill
0.0%
0.0%
2.0%
Lectures 13–14: Risk Analytics and
Portfolio Theory
4.0%
6.0%
8.0%
10.0% 12.0%
Standard Deviation of Return
© 2007–2008 by Andrew W. Lo
14.0%
16.0%
Slide 41
The Tangency Portfolio
15.401
2.4%
Expected Return
1.8%
Motorola
Tangency
portfolio
IBM
1.2%
GM
0.6%
Tbill
0.0%
0.0%
2.0%
Lectures 13–14: Risk Analytics and
Portfolio Theory
4.0%
6.0%
8.0%
10.0% 12.0%
Standard Deviation of Return
© 2007–2008 by Andrew W. Lo
14.0%
16.0%
Slide 42
The Tangency Portfolio
15.401
Sharpe ratio
A measure of a portfolio’s risk-return trade-off, equal to the
portfolio’s risk premium divided by its volatility:
Sharpe Ratio ≡
E[Rp] − rf
σp
(higher is better!)
ƒ The tangency portfolio has the highest possible Sharpe
ratio of any portfolio
ƒ Aside: Alpha is a measure of a mutual fund’s risk-adjusted
performance. The tangency portfolio also maximizes the fund’s
alpha.
Lectures 13–14: Risk Analytics and
Portfolio Theory
© 2007–2008 by Andrew W. Lo
Slide 43
The Tangency Portfolio
15.401
Slope = Sharpe Ratio (why?)
2.4%
Expected Return
1.8%
Motorola
Tangency
portfolio
IBM
1.2%
GM
0.6%
Tbill
0.0%
0.0%
2.0%
Lectures 13–14: Risk Analytics and
Portfolio Theory
4.0%
6.0%
8.0%
10.0% 12.0%
Standard Deviation of Returns
© 2007–2008 by Andrew W. Lo
14.0%
16.0%
Slide 44
Key Points
15.401
ƒ Diversification reduces risk. The standard deviation of a portfolio is
always less than the average standard deviation of the individual
stocks in the portfolio.
ƒ In diversified portfolios, covariances among stocks are more
important than individual variances. Only systematic risk matters.
ƒ Investors should try to hold portfolios on the efficient frontier.
These portfolios maximize expected return for a given level of risk.
ƒ With a riskless asset, all investors should hold the tangency
portfolio. This portfolio maximizes the trade-off between risk and
expected return.
Lectures 13–14: Risk Analytics and
Portfolio Theory
© 2007–2008 by Andrew W. Lo
Slide 45
Additional References
15.401
ƒ Bernstein, 1992, Capital Ideas. New York: Free Press.
ƒ Bodie, Z., Kane, A. and A. Marcus, 2005, Investments, 6th edition. New York: McGraw-Hill.
ƒ Brennan, T., Lo, A. and T. Nguyen, 2007, Portfolio Theory: A Review, to appear in Foundations of
Finance.
ƒ Campbell, J., Lo, A. and C. MacKinlay, 1997, The Econometrics of Financial Markets. Princeton, NJ:
Princeton University Press.
ƒ Grinold, R. and R. Kahn, 2000, Active Portfolio Management. New York: McGraw-Hill.
Lectures 13–14: Risk Analytics and
Portfolio Theory
© 2007–2008 by Andrew W. Lo
Slide 46
MIT OpenCourseWare
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15.401 Finance Theory I
Fall 2008
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