International Investment 2005-2006

Professor André Farber

Solvay Business School

Université Libre de Bruxelles

Invest 2006 -1 |2April 18, 2023

Academic contributions to investment process

• 1952 Markowitz: Portfolio Selection

• 2 dimensions: expected returns and risk

• Returns: normally distributed random variables

• Crucial role of covariances (correlation coefficients)

• 1965 Fama, Samuelson: Efficient Market Hypothesis

• Stock prices are unpredictable, move randomly

• 1964-1966 Sharpe, Lintner, Mossin: Capital Asset Pricing Model

• Expected return function of systematic risk ()

• 1973-1974 Black, Scholes, Merton: Option Pricing Model

• Pricing in a risk neutral world

Source: Bernstein, P. Capital Ideas, Free Press 1992

Invest 2006 -1 |3April 18, 2023

From Gown to Town

• 1971 Wells Fargo launches first index fund.

• 1980 Portfolio insurance introduced by Leland and Rubinstein

• Big problem in 1987

• 1994 Creation of Long Term Capital Management

• 1998: LTCM rescued after losing $4 billions

Invest 2006 -1 |4April 18, 2023

A 1 slide review of Finance (no formula..yet)

Expected returnof portfolio

Standarddeviation of

portfolio’s return.

Risk-freerate (Rf )

4

M.5

..

Capital market line

.X

Y

Which portfolio to choose?

Invest 2006 -1 |5April 18, 2023

Standard & Poor 500

S&P 500 Daily 1996-2003 StDev = 1.28% (n=1,850)

0

50

100

150

200

250

300

350

400

-10.

0%-9

.0%

-8.0

%-7

.0%

-6.0

%-5

.0%

-4.0

%-3

.0%

-2.0

%-1

.0%

0.0%

1.0%

2.0%

3.0%

4.0%

5.0%

6.0%

7.0%

8.0%

9.0%

10.0

%

Return (daily)

Fre

qu

ency

Invest 2006 -1 |6April 18, 2023

Microsoft

Microsoft Daily 1996-2003 StDev=2.58% (n=1,850)

0

20

40

60

80

100

120

140

160

180

200

-10.

0%

-9.5

%

-9.0

%

-8.5

%

-8.0

%

-7.5

%

-7.0

%

-6.5

%

-6.0

%

-5.5

%

-5.0

%

-4.5

%

-4.0

%

-3.5

%

-3.0

%

-2.5

%

-2.0

%

-1.5

%

-1.0

%

-0.5

%

0.0%

0.5%

1.0%

1.5%

2.0%

2.5%

3.0%

3.5%

4.0%

4.5%

5.0%

5.5%

6.0%

6.5%

7.0%

7.5%

8.0%

8.5%

9.0%

9.5%

10.0

%

Invest 2006 -1 |7April 18, 2023

Normal distribution illustrated

Normal distribution

0.0000

0.0050

0.0100

0.0150

0.0200

0.0250

68.26%

95.44%

Standard deviation from mean

Invest 2006 -1 |8April 18, 2023

Normal distribution – technical (and useful) details

The normal distribution is identified by two parameters: the expected value (mean) and the standard deviation. If R is a random return, we write:

( , )R N R

For the standard normal distribution, the expectation is zero and the standard deviation is equal to 1.0

The cumulative normal distribution, denoted gives the probability that the random variable will be less than or equal to x.

( ; , )N x R

( ; , ) Pr( )N x R R x

1

2

3

4

5

6

A B C D E F G HNormal distribution

Parameters Mean 13.0%StDev 20.0%

Proba of return < 2.0% 0.291 D5. =NORMDI ST(C5,E2,E3,TRUE)

Proba of return > 2.0% 0.709

In Excel, use NORMDIST(Value,Mean,StandardDeviation,TRUE)

Example:

Invest 2006 -1 |9April 18, 2023

Normal distribution – more details

The probability that a normal variate will take on a value in the range [a,b] is:

Pr( ) ( ; , ) ( ; , )a R b N b R N b R

Confidence interval: the range within which the return will fall with probability 1-α (the confidence level – α is the probability of error)

Pr( ) 1

Pr( )2

Pr( )2

a R b

R a

R b

In Excel, use NORMINV(p,Mean,StandardDeviation)

8

9

10

11

12

13

14

A B C D E F GValue of portf olio $250.00

Normal distribution Expected Return 13.0%

StDev 20.0% What probability of error do you accept? 5.0%

Return Value

Max 52.20% $380.50 Proba above 2.50%

Min -26.20% $184.50 Proba below 2.50%

Invest 2006 -1 |10April 18, 2023

Value at Risk (VaR)

Value at Risk is a measure of the maximum loss than can experienced over a period of time with a x% probability of exceeding this amount.

1234

5

6

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12

13

14

A B C D E F G HValue at Risk (VaR) "How bad can things get?"

What is the loss level such as we are X% certain that it will not be exceeded?

VaR is the loss level that will not be exceeded with a certain level of probability

Value of portf olio $100.00

Normal distribution Expected Return 13.0%

StDev 20.0% What confidence level do you choose? 95%

The return with a 95% proba of being above is -19.9% E11. =NORMI NV(E12,E6,E7)

The probability of being below this level is 5%

The value of the portf olio would be $80.10

The loss would be $19.90 =VaR

Invest 2006 -1 |11April 18, 2023

Risk premium on a risky asset

• The excess return earned by investing in a risky asset as opposed to a risk-free asset

•

• U.S.Treasury bills, which are a short-term, default-free asset, will be used a the proxy for a risk-free asset.

• The ex post (after the fact) or realized risk premium is calculated by substracting the average risk-free return from the average risk return.

• Risk-free return = return on 1-year Treasury bills

• Risk premium = Average excess return on a risky asset

Invest 2006 -1 |12April 18, 2023

Historical Returns, 1926-2002

Source: © Stocks, Bonds, Bills, and Inflation 2003 Yearbook™, Ibbotson Associates, Inc., Chicago (annually updates work by Roger G. Ibbotson and Rex A. Sinquefield). All rights reserved.

– 90% + 90%0%

Average Standard Series Annual Return Deviation Distribution

Large Company Stocks 12.2% 20.5%

Small Company Stocks 16.9 33.2

Long-Term Corporate Bonds 6.2 8.7

Long-Term Government Bonds 5.8 9.4

U.S. Treasury Bills 3.8 3.2

Inflation 3.1 4.4

Invest 2006 -1 |13April 18, 2023

Is the U.S a special case?

Table 1 Summary Statistics for Annual Real Equity Returns: 16 Markets anda World Index, 1900-2002

CountryGeometric

MeanArithmetic

MeanStandard Deviation Autocorrelation

Belgium 1.8 4.0 22.1 0.23Italy 2.1 6.2 29.4 0.03

Germany 2.8 8.1 32.4 -0.17France 3.1 5.5 22.7 0.19Spain 3.2 5.4 22.0 0.33Japan 4.1 8.8 30.2 0.20

Switzerland 4.1 5.9 19.8 0.20Ireland 4.3 6.6 22.2 -0.04

Denmark 4.6 6.2 20.1 -0.14Netherlands 5.0 7.0 21.5 0.09

United Kingdom 5.2 7.1 20.2 -0.05World 5.4 6.8 17.2 0.13

Canada 5.9 7.2 16.9 0.17United States 6.3 8.3 20.3 0.01South Africa 6.7 8.9 22.6 0.04

Sweden 7.3 9.5 22.7 0.13Australia 7.4 8.9 17.8 -0.02

Average( ex world) 4.6 7.1 22.7 0.08

Source: Dimson, Marsh and Staunton, Irrational Optimism, Financial Analysts Journal, 60, 1 (2004) pp.15-25

Invest 2006 -1 |14April 18, 2023

Market Risk Premium: The Very Long Run

1802-1870 1871-1925 1926-1999 1802-2002

Common Stock 6.8 8.5 12.2 9.7

Treasury Bills 5.4 4.1 3.8 4.3

Risk premium 1.4 4.4 8.4 5.4

Source: Ross, Westerfield, Jaffee (2005) Table 9A.1

The equity premium puzzle:

Was the 20th century an anomaly?

Invest 2006 -1 |15April 18, 2023

Siegel on the Equity Risk Premium

Table 1. Historical Real Stock and Bond Returns and the Equity Premium

Period Comp. Arith. Comp. Arith. Comp. Arith. Comp. Arith. Comp. Arith.A. Long period to present1802-2004 6.82% 8.38% 3.51% 3.88% 2.84% 3.02% 3.31% 4.50% 3.98% 5.36%1871-2004 6.71 8.43 2.85 3.24 1.68 1.79 3.86 5.18 5.03 6.64

B. Major subperiods1802-1870 7.02% 8.28% 4.78% 5.11% 5.12% 5.40% 2.24% 3.17% 1.90% 2.87%1871-1925 6.62 7.92 3.73 3.93 3.16 3.27 2.89 3.99 4.46 4.651926-2004 6.78 8.78 2.25 2.77 0.69 0.75 4.53 6.01 6.09 8.02

C. Post-World War II full sample, bull markets and bear markets1946-2004 6.83% 8.38% 1.44% 2.04% 0.56% 0.62% 5.39% 6.35% 6.27% 7.77%1946-1965 10.02 11.39 -1.19 -0.95 -0.84 -0.75 11.21 12.34 10.86 12.141966-1981 -0.36 1.38 -4.17 -3.86 -0.15 -0.13 3.81 5.24 -0.21 1.511982-1999 13.62 14.30 8.4 9.28 2.91 2.92 5.22 5.03 10.71 11.381982-2004 9.47 10.64 8.01 8.74 2.31 2.33 1.46 1.90 7.16 8.32Note: "Comp" stands for "compound"; "Arith." stands for "arithmetic."Source: Siegel, J., Perspectives on the Equity Risk Premium,Financial Analyst Journal; Nov/Dec 2005; 61, 6

BillsReal returns Stock Return minus Return on:

Stocks Bonds Bills Bonds

Invest 2006 -1 |16April 18, 2023

And now the formulas: 2 assets portfolio

• Expected return

• Risk

• More formulas:

1 1 2 2PR X R X R

2 2 2 2 21 1 2 2 1 2 122P X X X X

12 1 2 12 1 2( , )Cov R R

2 2 21 1 1 2 12 2 1 12 2 2[ ] [ ]P X X X X X X

21 1 2 2

1 1 2 2

( , ) ( , )P P P

P P

X Cov R R X Cov R R

X X

Invest 2006 -1 |17April 18, 2023

Formulas using matrix algebra

21 21 RXRXRP

122122

22

21

21

2 2 XXXXP

2

1

2221

1221

212

X

XXXP

Expected return:

Variance:

),(~ PPP RNR Returns: normal distribution

211212

XXP '2

Invest 2006 -1 |18April 18, 2023

Choosing portfolios from many stocks

• Porfolio composition :

• (X1, X2, ... , Xi, ... , XN)

• X1 + X2 + ... + Xi + ... + XN = 1

• Expected return:

• Risk:

• Note:

• N terms for variances, N(N-1) terms for covariances

• Covariances dominate

NNP RXRXRXR ...2211

2 2 2P j j i j ij i j ij

j i j i i j

j jPj

X X X X X

X

Invest 2006 -1 |19April 18, 2023

Calculation in Excel

1 1 1

1

1

1 .. ..

1 ..

.. 1j j j j

N N N N

X R

X R

X R

21 1 1

21

21

j N

j j jN

N jN N

Step 1. Compute covariance matrix

1P jP NP Step 2. Compute covariances of individual securities with portfolio

jP k jkk

X

Step 3. Compute expected return P j jj

R X R

Step 4. Compute variance 2P j jP

j

X

Useful Excel trick: use SUMPRODUCT

Invest 2006 -1 |20April 18, 2023

Using matrices

NX

X

X ...1

NR

R

R ...

1

NNN

N

...

.........

...

1

111

XX

RXR

P

P

'

'2

Invest 2006 -1 |21April 18, 2023

Example

• Consider the risk of an equally weighted portfolio of N "identical«  stocks:

• Equally weighted:

• Variance of portfolio:

• If we increase the number of securities ?:

• Variance of portfolio:

NX j

1

cov)1

1(1 22

NNP

NP cov2

cov),(,, jijj RRCovRR

Invest 2006 -1 |22April 18, 2023

Diversification

Risk Reduction of Equally Weighted Portfolios

0.00%

5.00%

10.00%

15.00%

20.00%

25.00%

30.00%

35.00%

# stocks in portfolio

Po

rtfo

lio

sta

nd

ard

de

via

tio

n

Market risk

Unique risk

Invest 2006 -1 |23April 18, 2023

Conclusion

• 1. Diversification pays - adding securities to the portfolio decreases risk. This is because securities are not perfectly positively correlated

• 2. There is a limit to the benefit of diversification : the risk of the portfolio can't be less than the average covariance (cov) between the stocks

• The variance of a security's return can be broken down in the following way:

• The proper definition of the risk of an individual security in a portfolio M is the covariance of the security with the portfolio:

Total risk of individual security

Portfolio risk

Unsystematic or diversifiable risk

Invest 2006 -1 |24April 18, 2023

Mean-Variance Frontier Calculation: brute force

Mean variance portfolio: i j

ijjw

iw

Pn

wwwMin 2

,..,2

,1

jj

jjj

w

Eew

1

s.t.

Matrix notations:

w

VwwMin '

Eew '

1w’u=

Invest 2006 -1 |25April 18, 2023

Some math…

)'1(2)'(2'),,( 2121 uwewEVwwwL

01'

0'

021

wu

Ewe

ueVw

UEX 12

11

1''

''1

21

1

12

11

uVueVu

EuVeeVe

2111 ' ' ' ABCDuVuCeVeBuVeA

D

EABD

AEC

2

1

Ehgw

Lagrange:

FOC:

Define:

eVD

AuV

D

Bh

uVD

AeV

D

Cg

11

11

Invest 2006 -1 |26April 18, 2023

Interpretation

1g+h

0H

E

The frontier can be spanned by two frontier returns

Minimum variance portfolio MVPA/C

Ehgw

C

1

Beta

Prof. André FarberSOLVAY BUSINESS SCHOOLUNIVERSITÉ LIBRE DE BRUXELLES

Invest 2006 -1 |28April 18, 2023

Measuring the risk of an individual asset

• The measure of risk of an individual asset in a portfolio has to incorporate the impact of diversification.

• The standard deviation is not an correct measure for the risk of an individual security in a portfolio.

• The risk of an individual is its systematic risk or market risk, the risk that can not be eliminated through diversification.

• Remember: the optimal portfolio is the market portfolio.

• The risk of an individual asset is measured by beta.

• The definition of beta is:

22 )(

),(

M

iM

M

Mii

R

RRCov

Invest 2006 -1 |29April 18, 2023

Beta

• Several interpretations of beta are possible:

• (1) Beta is the responsiveness coefficient of Ri to the market

• (2) Beta is the relative contribution of stock i to the variance of the market portfolio

• (3) Beta indicates whether the risk of the portfolio will increase or decrease if the weight of i in the portfolio is slightly modified

Invest 2006 -1 |30April 18, 2023

Beta as a slope

15, 25

15, 15

-5, -5

-5, -15

-10, -17.5

20, 27.5

-20

-15

-10

-5

0

5

10

15

20

25

30

-15 -10 -5 0 5 10 15 20 25

Return on market

Ret

urn

on

ass

et

Slope = Beta = 1.5

Invest 2006 -1 |31April 18, 2023

A measure of systematic risk : beta

• Consider the following linear model

• Rt Realized return on a security during period t

A constant : a return that the stock will realize in any period

• RMt Realized return on the market as a whole during period t

A measure of the response of the return on the security to the return on the market

• ut A return specific to the security for period t (idosyncratic return or unsystematic return)- a random variable with mean 0

• Partition of yearly return into:

– Market related part ß RMt

– Company specific part + ut

tMtt uRR

Invest 2006 -1 |32April 18, 2023

Beta - illustration

• Suppose Rt = 2% + 1.2 RMt + ut

• If RMt = 10%

• The expected return on the security given the return on the market

• E[Rt |RMt] = 2% + 1.2 x 10% = 14%

• If Rt = 17%, ut = 17%-14% = 3%

Invest 2006 -1 |33April 18, 2023

Measuring Beta

• Data: past returns for the security and for the market

• Do linear regression : slope of regression = estimated beta

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A B C D E F G H IBeta Calculation - monthly data

Market A B

Mean 2.08% 0.00% 4.55% D3. =AVERAGE(D12:D23)

StDev 5.36% 4.33% 10.46% D4. =STDEV(D12:D23)

Correl 78.19% 71.54% D5. =CORREL(D12:D23,$B$12:$B$23)

R² 61.13% 51.18% D6. =D5 2̂

Beta 1 0.63 1.40 D7. =SLOPE(D12:D23,$B$12:$B$23)

I ntercept 0 -1.32% 1.64% D8. =I NTERCEPT(D12:D23,$B$12:$B$23)

Data

Date Rm RA RB

1 5.68% 0.81% 20.43%

2 -4.07% -4.46% -7.03%

3 3.77% -1.85% -10.14%

4 5.22% -1.94% 6.91%

5 4.25% 3.49% 4.65%

6 0.98% 3.44% 7.64%

7 1.09% -4.27% 8.41%

8 -6.50% -2.70% -1.25%

9 -4.19% -4.29% -11.19%

10 5.07% 3.75% 13.18%

11 13.08% 9.71% 19.22%

12 0.62% -1.67% 3.77%

Invest 2006 -1 |34April 18, 2023

Decomposing of the variance of a portfolio

• How much does each asset contribute to the risk of a portfolio?

• The variance of the portfolio with 2 risky assets

• can be written as

• The variance of the portfolio is the weighted average of the covariances of each individual asset with the portfolio.

22222 2 BBABBAAAP XXXX

BPBAPA

BBABABABBAAA

BBABBAABBAAAP

XX

XXXXXX

XXXXXX

)()(

)()(22

22222

Invest 2006 -1 |35April 18, 2023

Example

Exp.Return Sigma VarianceRiskless rate 5 0 0A 15 20 400B 20 30 900Correlation 0

Prop. Variance-covarianceA 0.50 400 0B 0.50 0 900

Cov(Ri,Rp) 200.00 450.00X 0.50 0.50

Variance 325.00St.dev. 18.03Exp.Ret. Rp 17.50

Invest 2006 -1 |36April 18, 2023

Beta and the decomposition of the variance

• The variance of the market portfolio can be expressed as:

• To calculate the contribution of each security to the overall risk, divide each term by the variance of the portfolio

nMniMiMMM XXXX ......22112

1......

1......

2211

2222

221

1

nMniMiMM

M

nMn

M

iMi

M

M

M

M

XXXX

or

XXXX

Invest 2006 -1 |37April 18, 2023

Marginal contribution to risk: some math

• Consider portfolio M. What happens if the fraction invested in stock I changes?

• Consider a fraction X invested in stock i

• Take first derivative with respect to X for X = 0

• Risk of portfolio increase if and only if:

• The marginal contribution of stock i to the risk is

22222 )1(2)1( iiMMP XXXX

)(2 2

0

2

MiM

X

P

dX

d

2MiM

iM

Invest 2006 -1 |38April 18, 2023

Marginal contribution to risk: illustration

0.00

5.00

10.00

15.00

20.00

25.00

30.00

35.00

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1

Fraction in B

Ris

k o

f p

ort

folio

Cor = 0 Cor = 0.25 Cor = 0.50 Cor = 0.75 Cor = 1.0

Invest 2006 -1 |39April 18, 2023

Beta and marginal contribution to risk

• Increase (sightly) the weight of i:

• The risk of the portfolio increases if:

• The risk of the portfolio is unchanged if:

• The risk of the portfolio decreases if:

12

2 M

iMiMMiM

12

2 M

iMiMMiM

12

2 M

iMiMMiM

Invest 2006 -1 |40April 18, 2023

Inside beta

• Remember the relationship between the correlation coefficient and the covariance:

• Beta can be written as:

• Two determinants of beta

– the correlation of the security return with the market

– the volatility of the security relative to the volatility of the market

Mi

iMiM

M

iiM

M

iMiM

2

Invest 2006 -1 |41April 18, 2023

Properties of beta

• Two importants properties of beta to remember

• (1) The weighted average beta across all securities is 1

• (2) The beta of a portfolio is the weighted average beta of the securities

1......2211 nMniMiMM XXXX

nMnPiMiPMPMPP XXXX ......2211

Invest 2006 -1 |42April 18, 2023

Modeling choices under uncertainty

• We need to specify how an investor will choose.

– Economist use a utility function: a number associated with each possible choice (here each possible portfolio)

• First a few word about a very general specification.

• You have €100 to invest. You face 2 possible portfolios

Futures values (Proba)

Portfolio 1: 90 (0.50) 120 (0.50)

Portfolio 2: 50 (0.50) 200 (0.50)

Which portfolio would you choose?

Invest 2006 -1 |43April 18, 2023

Expected utility

• General formulation: choice based on Expected utility(Future Wealth)

• a weighted average of utilities of future wealth

E(u) = p1u(W1) + p2u(W2) + p3u(W3) + … + pnu(Wn)

• Utility function u(W)

– an increasing function of W (more wealth is preferred)

– shape captures attitude toward risk

• constant marginal utility u’ (linear) : risk neutrality

• decreasing marginal utility (concave): risk aversion

Wealth Wealth

Utility Utility

Risk neutrality

Risk aversion

Invest 2006 -1 |44April 18, 2023

Back to our example

• Lisa is risk neutral: u(W) = W• John is risk averse: u(W) = ln(W)• What will they choose?

– Lisa• Expected utility portfolio 1 = 0.5090 + 0.50120 = 105• Expected utility portfolio 2 = 0.5050 + 0.50200 = 125

• Lisa would choose portfolio 2– John

• Expected utility portfolio 1 = 0.50ln(90) + 0.50ln(120) = 4.64• Expected utility portfolio 2 = 0.50ln(50) + 0.50ln(200) = 4.60

• John would choose portfolio 1

Invest 2006 -1 |45April 18, 2023

Why different choices?

• Lisa is risk neutral, only expected value matters

• John is risk averse:

– He will always prefer a sure value over a risky one with the same expected value

– The greater expected value of portfolio 2 is not sufficient to compensate for the additional risk

Invest 2006 -1 |46April 18, 2023

Mean-variance utility

• In a more general setting, a pretty good approximation of the expected utility of a portfolio can be obtained with the following formulation

• It combines into one number the expected return and the risk of the portfolio

• The degree of risk aversion is captured by a

25.0 PP aRU

Expected return

Standard deviation

U

A

B

Invest 2006 -1 |47April 18, 2023

Using mean-variance utility

• Back to our example:

• Lisa : a = 0 (she is risk neutral)

• John : a = 4 (he is risk averse)

• Portfolio 1: Expected return = 5% Standard deviation = 12.75%

• Portfolio 2: Expected return = 25% Standard deviation = 79.06%

• Utilities Lisa John

• Portfolio 1 .05 .05 - 0.5 4 .1275² = .0325

• Portfolio 2 .25 .25 - 0.5 4 .7906² = -1.00

• Choice 2 1

Invest 2006 -1 |48April 18, 2023

Portfolio Selection & Risk Aversion

E(r)

Efficientfrontier ofrisky assets

Morerisk-averseinvestor

U’’’ U’’ U’

Q

PS

St. Dev

Lessrisk-averseinvestor

Invest 2006 -1 |49April 18, 2023

Finding the optimal risky asset allocation

• Risk-free asset : RF

• Risky portfolio : Expected return = R Standard deviation = • Invest fraction X in the risky portfolio

• Choose X to maximize U

• Optimal allocation :

• (Proof on demand)

• The fraction invested in the risky portfolio is decreasing with risk aversion (the higher risk aversion, the lower the fraction invested in the risky portfolio)

2a

RRX F

Invest 2006 -1 |50April 18, 2023

Asset Allocation and Risk Aversion

Expected return

Standard deviation

Optimal asset allocation

Optimal risky portfolio

U

RF

Efficient frontier

Invest 2006 -1 |51April 18, 2023

Risk aversion: a crude estimate

• Let ’s start from the historical for the US 1926-1996

Arithmetic Standard

Mean Deviation

Large company stocks 12.5% 20.4%

US Treasury Bills 3.8% 3.3%

Historical risk premium 8.7%

• Set X=1 (average stock holding in equilibrium)

• Warning: the debate on the expected risk premium is still on

09.2)²204.0(

087.0a

Invest 2006 -1 |52April 18, 2023

Historical risk premium: long term perspective

Real Historical Standard Sharpe

Equity Premium Deviation ratio

1872-1999 5.73 13.01 0.32

1872-1949 4.10 19.52 0.30

1950-1999 8.28 16.65 0.49

Fama, French « The Equity Premium » University of Chicago, WP 522, July 2000

What happened ?