Lecture 2: Conditional Expectation

Discounted Cash Flow, Section 1.2

© 2004, Lutz Kruschwitz and Andreas Löffler

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

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)

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.

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

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

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

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.

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)

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

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

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

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

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

No. 15

1.2.3 Using the rules

We want to check our rules by looking at– the finite example and– an infinite example.

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.

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

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

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

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

No. 21

Summary

We always stay in the present. Conditional expectation

handles our knowledge of the future.

Five rules cover necessary mathematics.