Overview formulas EMF

FORMULAS EMF
Part 1
Event Studies
𝐴𝑅#$ = 𝑅#$ − 𝑁𝑅#$
𝐴𝑅#$ ~ 𝑁(0, 𝜎 . )
0 12
• Mean adjusted: 𝑁𝑅# = $614 𝑅#$ , doesn’t take into account market changes.
12 3 14 50
• Market adjusted: 𝑁𝑅#$ = 𝑅7$ , assumes all stocks move same way as market.
• Market model: 𝑁𝑅#$ = 𝛼# + 𝛽# 𝑅;$
o 𝛼# = 𝑅# − 𝛽# 𝑅;
<=> ?@ ,?A
o 𝛽# =
BCD ?A
𝟏 𝑻𝟐
§ 𝐶𝒐𝒗(𝑹𝒊 , 𝑹𝒎 ) = (𝑹𝒊𝒕 − 𝑹𝒊 )(𝑹𝑴𝒕 − 𝑹𝒎 )
𝑻𝟐 3 𝑻𝟏 𝒕6𝑻𝟏
𝟏 𝑻𝟐
§ 𝑽𝒂𝒓(𝑹𝒎 ) = (𝑹𝑴𝒕 − 𝑹𝒎 )𝟐
𝑻𝟐 3 𝑻𝟏 𝒕6𝑻𝟏
𝟏 𝑻𝟐
§ 𝑹𝒊 = 𝒕6𝑻𝟏 𝑹𝒊𝒕
𝑻𝟐 3 𝑻𝟏 5𝟏
• CAPM: 𝑁𝑅#$ = 𝑅S$ + 𝛽# (𝑅7$ − 𝑅S$ ), beta estimated using excess returns estimation period.
o Restriction on α imposed: 𝛼# = 𝑅S$ (1 − 𝛽# )

0 U
𝐴𝐴𝑅# = #60 𝐴𝑅#$
U


𝐻W : 𝐸(𝐴𝑅#$ ) = 0
\\?]
If ARs are independent, identically, normally distributed: 𝑇𝑆0 = 𝑁 ~ 𝑁(0,1), 𝜎 . is variance of AR.
^


𝐴𝐴𝑅$
𝑇𝑆0 = 𝑁 ~ 𝑁(0,1)
𝑠
𝟏 𝑵
• 𝒔𝒕 = 𝒊6𝟏 (𝑨𝑹𝒊𝒕 − 𝑨𝑨𝑹𝒕 )𝟐 , estimator of variance
𝑵3𝟏
• Has student-t distribution with N-1 degrees of freedom if ARs are independent, identically,
normally distributed.

$2
𝐶𝐴𝑅# = #6$4 𝐴𝑅#$
0 U
𝐶𝐴𝐴𝑅 = 𝐶𝐴𝑅#
U #60
<\\?]
o 𝑇𝑆. = 𝑁 ~ 𝑁(0,1)
c]
𝟏 𝑵
§ 𝒔𝒕 = 𝒊6𝟏 (𝑪𝑨𝑹𝒊 − 𝑪𝑨𝑨𝑹)𝟐
𝑵3𝟏


Crude dependence adjustment: estimated standard error of AAR from time series (not recommended):
0 12
• 𝑠= $614 (𝐴𝐴𝑅$ − 𝐴𝑅 ∗ ).
12 3 14
o AR*: overall average of AR over estimation period
\\?]
• 𝑇𝑆f = ~ 𝑁(0,1) à no 𝑁
c