Lecture: Conditional Expectation
Lutz Kruschwitz & Andreas Löffler
Discounted Cash Flow, Section 1.2
,
Outline
1.2 Conditional expectation
1.2.1 Uncertainty and information
1.2.2 Rules
1.2.3 Application of the rules
,
Uncertainty
1
Uncertainty is a distinguished feature of valuation
usually modelled as different future states of
g t (ω).
nature ω with corresponding cash flows FCF
But: to the best of our knowledge particular states
of nature play no role in the valuation equations
h
i of
g t of
firms, instead one uses expectations E FCF
cash flows.
1.2 Conditional expectation,
1.2.1 Uncertainty and information
Information
2
Today is certain, the future is uncertain.
Now: we always stay at time 0!
And think about the future
gs
FCF
Simon Vouet: ‘Fortune–teller’ (1617)
1.2 Conditional expectation,
0
today
t
future
1.2.1 Uncertainty and information
s
later future
A finite example
3
193.6
132
110
96.8
110
90
145.2
88
48.4
t=0
t=1
1.2 Conditional expectation,
t=2
t=3
1.2.1 Uncertainty and information
There are three points in
time in the future.
Different cash flow
realizations can be
observed.
The movements up and
down along the path
occur with probability
0.5.
time
Actual and possible cash flow
4
What happens if actual cash flow at time t = 1 is
neither 90 nor 110 (for example, 100)?
Our model proved to be wrong. . .
Nostradamus (1503–1566),
failed fortune-teller
1.2 Conditional expectation,
1.2.1 Uncertainty and information
Expectations
5
Let us think about cash flow paid at time
g 3 . What will its
t = 3, i.e. FCF
expectation be tomorrow?
A.N. Kolmogorov (1903–1987),
This depends on the state we will have
tomorrow. Two cases are possible:
g 1 = 110 or FCF
g 1 = 90.
FCF
founded theory of
conditional expectation
1.2 Conditional expectation,
1.2.1 Uncertainty and information
Thinking about the future today
6
g 1 = 110)
Case 1 (FCF
g 3 = 1 ×193.6+ 2 ×96.8+ 1 ×145.2 = 133.1.
=⇒ Expectation of FCF
4
4
4
g 1 = 90)
Case 2 (FCF
g 3 = 1 ×96.8+ 2 ×145.2+ 1 ×48.4 = 108.9.
=⇒ Expectation of FCF
4
4
4
g 3 is
Hence, expectation of FCF
(
h
i
g 3 |F1 = 133.1 if the development at t = 1 is up,
E FCF
108.9 if the development at t = 1 is down.
1.2 Conditional expectation,
1.2.1 Uncertainty and information
Conditional expectation
7
g 3 depends on the state of nature at time
The expectation of FCF
t = 1. Hence, the expectation is conditional: conditional on the
information at time t = 1 (abbreviated as |F1 ).
A conditional expectation covers our ideas about future thoughts.
This conditional expectation can be uncertain.
1.2 Conditional expectation,
1.2.1 Uncertainty and information
Rules
8
How to use conditional expectations? We
will not present proofs, but only rules for
calculation.
The first three rules will be well-known
from classical expectations, two will be
new.
Arithmetic textbook of
Adam Ries (1492–1559)
1.2 Conditional expectation,
1.2.2 Rules
Rule 1: Classical expectation
9
At t = 0 conditional expectation and
classical expectation coincide.
i
h i
h
e
e |F0 = E X
E X
1.2 Conditional expectation,
Or: conditional expectation generalizes
classical expectation.
1.2.2 Rules
Rule 2: Linearity
10
i
i
h
h
i
h
e |Ft
e |Ft + b E Y
e + bY
e |Ft = a E X
E aX
1.2 Conditional expectation,
1.2.2 Rules
Business as usual . . .
Rule 3: Certain quantity
E [1|Ft ] = 1
11
Safety first. . .
From this and linearity for certain quantities X ,
E [X |Ft ] = E [X 1|Ft ]
= X E [1|Ft ]
=X
1.2 Conditional expectation,
1.2.2 Rules
Rule 4: Iterated expectation
Let s ≤ t then
i
i
i
h
h h
e |Fs
e |Ft |Fs = E X
E E X
12
When we think today about
what we will know tomorrow
about the day after tomorrow,
we will only know what we
today already believe to know
tomorrow.
1.2 Conditional expectation,
1.2.2 Rules
Rule 5: Known or realized quantities
If X is known at time t
i
i
h
h
e |Ft
eY
e |Ft = X
eE Y
E X
1.2 Conditional expectation,
1.2.2 Rules
13
We can take out from the
expectation what is known.
Or: known quantities are
like certain quantities.
Using the rules
14
We want to check our rules by looking at the finite example and an
infinite example. We start with the finite example:
Remember that we had
i
g 3 |F1 =
E FCF
h
(
133.1 if up at time t = 1 ,
108.9 if down at time t = 1 .
From this we get
h h
ii
g 3 |F1 = 1 × 133.1 + 1 × 108.9 = 121.
E E FCF
2
2
1.2 Conditional expectation,
1.2.3 Application of the rules
Finite example
15
And indeed
i
i
ii
h h
h h
g 3 |F1 |F0
g 3 |F1 = E E FCF
E E FCF
i
h
g 3 |F0
= E FCF
i
h
g3
= E FCF
= 121 !
It seems purely by chance that
h
i
g 3 |F1 = 1.13−1 × FCF
g1,
E FCF
but it is on purpose! This will become clear later. . .
1.2 Conditional expectation,
1.2.3 Application of the rules
by rule 1
by rule 4
by rule 1
Infinite example
16
g
u 2 FCF t
g
Again two factors up and
down with probability pu
and pd and 0 < d < u
u FCF t
g
g
FCF t
ud FCF t
g
or
d FCF t
g
FCF t+1 =
gt
d 2 FCF
t
t+1
1.2 Conditional expectation,
t+2
time
1.2.3 Application of the rules
( g
u FCF t
gt
d FCF
up,
down.
Infinite example
17
Let us evaluate the conditional expectation
i
h
g t + pd d FCF
gt
g t+1 |Ft = pu u FCF
E FCF
gt,
= (pu u + pd d) FCF
|
{z
}
:=1+g
where g is the expected growth rate.
If g = 0 it is said that the cash flows ‘form a martingal’. In the
infinite example we will later assume no growth (g = 0).
1.2 Conditional expectation,
1.2.3 Application of the rules
Infinite example
18
Compare this if using only the rules
i
i
ii
h h
h h
g t |Ft−1 |Fs
g t |Fs = E E FCF
E E FCF
h
i
g t−1 |Fs
= E (1 + g )FCF
h
i
g t−1 |Fs
= (1 + g ) E FCF
gs
= (1 + g )t−s FCF
1.2 Conditional expectation,
1.2.3 Application of the rules
by rule 4
see above
by rule 2
repeating argument.
Summary
19
We always stay in the present. Conditional expectation handles our
knowledge of the future.
Five rules cover the necessary mathematics.
1.2 Conditional expectation,
1.2.3 Application of the rules