Lecture 2: Risk and Financial Crises

Economics 252, Spring 2011Prof. Robert Shiller, Yale University

Return

Expected Value, Mean, Average

∑ ∞

====

1)()(

iix xxxprobxE iµ

∫∞

∞−

== xdxxfxE x )()( µ

∑=

=n

ii nxx

1/

nn

iixx /1

1

)()G( ∏=

=

Variance and Standard Deviation

nxxsn

iix /)( 2

1

2 −= ∑=

2

1))((prob)var( xi

n

ii xxxx µ−== ∑

=

Covariance

nyyxxyxn

i/))((),cov(

1−−= ∑

=

Correlation

• A scaled measure of how much two variables move together

• -1 ≤ ρ ≤ 1

)/(),cov( yxssyx=ρ

Variance of Sum

Stock Market Level, 2000-2010, 2000=100

Apple, Inc. and S&P 500 Monthly Adjusted Price First Decade of 2000s, 2000=100

Apple, Inc. and S&P 500 Monthly Returns, First Decade of 2000s

Scatter, Apple vs S&P 500

Same Scatter with Regression Line

Norm al Distribution w ith Zero M ean

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

-15 -10 -5 0 5 10 15

Return (x)

f(x) Standard Dev. = 3

Standard Dev. = 1

Norm al Versus Fat Tailed Distributions

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

-15 -10 -5 0 5 10 15

Return x

f(x) Normal Distribution

Cauchy Distribution