Lecture 2: Conditional Expectation Discounted Cash Flow, Section 1.2 © 2004, Lutz Kruschwitz and...
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Transcript of Lecture 2: Conditional Expectation Discounted Cash Flow, Section 1.2 © 2004, Lutz Kruschwitz and...
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Lecture 2: Conditional Expectation
Discounted Cash Flow, Section 1.2
© 2004, Lutz Kruschwitz and Andreas Löffler
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No. 2
1.2.1 Uncertainty
Uncertainty is a distinguished feature of valuation usually modelled as different states of nature with corresponding cash flows
But: to the best of our knowledge particular states of nature play no role in the valuation equations of firms, instead one uses expectation of cash flows.
( ).tFCF
[ ]tE FCF
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No. 3
1.2.1 Information
Today is certain, the future is uncertain.
Now: we always stay at time 0!
And think about the futureSimon Vouet: «Fortune-teller» (1617)
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No. 4
1.2.1 A finite example
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.
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No. 5
1.2.1 Actual and possible cash flow
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
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No. 6
1.2.1 Expectations
Let us think about cash flow paid at time
t=3, i.e. What will its
expectation be tomorrow?
This depends on the state we will have
tomorrow. Two cases are possible:
3.FCF
1 1110 or FCF 90.FCF A.N. Kolmogorov (1903-1987),
founded theory of
conditional expectation
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No. 7
1.2.1 Thinking about the future today
1
3
1
3
3
3 1
Case 1 ( 110) :
1 2 1 Expectation of FCF 193.6 96.8 145.2 133.1.
4 4 4
Case 2 ( 90) :
1 2 1 Expectation of 48.4 145.2 96.8 108.9.
4 4 4
Hence, expectation of is
133.1[ | ]
FCF
FCF
FCF
FCF
E FCF
F if the developement at 1 is up,
108.9 if the developement at 1 is down.
t
t
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No. 8
1.2.2 Conditional expectation
The expectation of FCF3 depends on the state of nature at time
t=1. Hence the expectation is conditional: conditional on the
information at time t=1 (abbreviated as |1).
A conditional expectation covers our ideas about future
thoughts.
This conditional expectation can be uncertain.
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No. 9
1.2.2 Rules
How to use conditional expectations?
We will not present proofs, but only
rules for calculation.
The following three rules will be well-known from classical expectations, two will be new.
Arithmetic textbook of
Adam Ries (1492-1559)
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No. 10
1.2.2 Rule 1: Classical expectation
At t=0 conditional expectation
and classical expectation coincide.
Or: conditional expectation generalizes classical expectation.
0|E X E X F
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No. 11
1.2.2 Rule 2: Linearity
Business as usual…
| | |t t tE a X b Y a E X b E Y F F F
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No. 12
1.2.2 Rule 3: Certain quantity
Safety first… 1| 1tE F
From this and linearity for certain quantities X,
| 1 |
1 |
t t
t
E X E X
XE
X
F F
F
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No. 13
1.2.2 Rule 4: Iterated expectation
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.
Let s t then
| | |t s sE E X E X
F F F
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No. 14
1.2.2 Rule 5: Known quanitites
We can take out from the
expectation what is known.
Or: known quantities are
like certain quantities.
If X is known at time
| |t t
t
E XY XE Y F F
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No. 15
1.2.3 Using the rules
We want to check our rules by looking at– the finite example and– an infinite example.
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No. 16
1.2.3 Finite example
3 1
3 1
Remember that we had
133.1 if up at time 1, |
108.9 if down at time 1.
From this we get
1 1| 133.1 108.9 121.
2 2
tE FCF
t
E E FCF
F
F
We want to check our rules by looking at the finite example and an infinite example.
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No. 17
1.2.3 Finite example
3 1 3 1 0
3 0
3
| | | by rule 1
| by rule 4
by rule 1
121 !
E E FCF E E FCF
E FCF
E FCF
F F F
F
And indeed
It seems purely by chance that
But it is on purpose! This will become clear later…
3 13 1
1| 1.1 ,E FCF FCF F
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No. 18
1.2.3 Infinite example
Again two factors up and
down with probability pu
and pd and 0<d<u
or
1
up,
down.t
tt
uFCFFCF
dFCF
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No. 19
1.2.3 Infinite exampleLet us evaluate the conditional expectation
where g is the expected growth rate.
If g=0 it is said that the cash flows «form a martingal». We will
later assume no growth (g=0).
1
: 1
|
,
t t u t d t
u d t
g
E FCF p uFCF p dFCF
p u p d FCF
F
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No. 20
1.2.3 Infinite example
Compare this if using only the rules
1
1
1
| | by rule 4
1 | see above
1 | by rule 2
(1 ) repeating argument.
t s t t s
t s
t s
t ss
E E FCF E E FCF
E g FCF
g E FCF
g FCF
tF F F
F
F
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No. 21
Summary
We always stay in the present. Conditional expectation
handles our knowledge of the future.
Five rules cover necessary mathematics.