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