Lecture: Conditional Expectation

Transcription

1 Lecture: Conditional Expectation Lutz Kruschwitz & Andreas Löffler Discounted Cash Flow, Section 1.2,

2 Outline 1.2 Conditional expectation Uncertainty and information Rules Application of the rules,

3 Uncertainty 1 Uncertainty is a distinguished feature of valuation usually modelled as different future states of nature ω with corresponding cash flows FCF t (ω). But: to the best of our knowledge particular states of nature play no role in the valuation equations [ ] of firms, instead one uses expectations E FCF t of cash flows. 1.2 Conditional expectation, Uncertainty and information

4 Information 2 Today is certain, the future is uncertain. Now: we always stay at time 0! And think about the future Simon Vouet: Fortune teller (1617) FCF s 0 t s today future later future 1.2 Conditional expectation, Uncertainty and information

5 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. t = 0 t = 1 t = 2 t = 3 time 1.2 Conditional expectation, Uncertainty and information

6 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 ( ), failed fortune-teller 1.2 Conditional expectation, Uncertainty and information

7 Expectations 5 Let us think about cash flow paid at time t = 3, i.e. FCF 3. What will its expectation be tomorrow? A.N. Kolmogorov ( ), This depends on the state we will have tomorrow. Two cases are possible: FCF 1 = 110 or FCF 1 = 90. founded theory of conditional expectation 1.2 Conditional expectation, Uncertainty and information

8 Thinking about the future today 6 Case 1 ( FCF 1 = 110) = Expectation of FCF 3 = = Case 2 ( FCF 1 = 90) = Expectation of FCF 3 = = Hence, expectation of FCF 3 is [ ] E FCF 3 F 1 = { if the development at t = 1 is up, if the development at t = 1 is down. 1.2 Conditional expectation, Uncertainty and information

9 Conditional expectation 7 The expectation of FCF 3 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 F 1 ). A conditional expectation covers our ideas about future thoughts. This conditional expectation can be uncertain. 1.2 Conditional expectation, Uncertainty and information

10 Rules 8 How to use conditional expectations? We will not present proofs, but only rules for calculation. Arithmetic textbook of The first three rules will be well-known from classical expectations, two will be new. Adam Ries ( ) 1.2 Conditional expectation, Rules

11 Rule 1: Classical expectation 9 ] ] E [ X F0 = E [ X At t = 0 conditional expectation and classical expectation coincide. Or: conditional expectation generalizes classical expectation. 1.2 Conditional expectation, Rules

12 Rule 2: Linearity 10 [ E a X ] ] ] + bỹ F t = a E [ X Ft + b E [Ỹ Ft Business as usual Conditional expectation, Rules

13 Rule 3: Certain quantity 11 E [1 F t ] = 1 Safety first... From this and linearity for certain quantities X, E [X F t ] = E [X 1 F t ] = X E [1 F t ] = X 1.2 Conditional expectation, Rules

14 Rule 4: Iterated expectation 12 Let s t then [ ] ] ] E E [ X Ft F s = E [ X Fs 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, Rules

15 Rule 5: Known or realized quantities 13 If X is known at time t ] E [ X Ỹ F t = X ] E [Ỹ Ft We can take out from the expectation what is known. Or: known quantities are like certain quantities. 1.2 Conditional expectation, Rules

16 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 From this we get [ ] E FCF 3 F 1 = { if up at time t = 1, if down at time t = 1. [ [ ]] E E FCF 3 F 1 = = Conditional expectation, Application of the rules

17 Finite example 15 And indeed [ [ ]] E E FCF 3 F 1 [ [ ] ] = E E FCF 3 F 1 F 0 [ ] = E FCF 3 F 0 [ ] = E FCF 3 = 121! by rule 1 by rule 4 by rule 1 It seems purely by chance that [ ] E FCF 3 F 1 = FCF 1, but it is on purpose! This will become clear later Conditional expectation, Application of the rules

18 Infinite example 16 u 2g FCF t gfcf t u g FCF t ud g FCF t Again two factors up and down with probability p u and p d and 0 < d < u d g FCF t or d 2g FCF t gfcf t+1 =( ug FCF t up, dg FCF t down. t t+1 t+2 time 1.2 Conditional expectation, Application of the rules

19 Infinite example 17 Let us evaluate the conditional expectation [ ] E FCF t+1 F t = p u u FCF t + p d d FCF t where g is the expected growth rate. = (p u u + p d d) FCF }{{} t, :=1+g 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, Application of the rules

20 Infinite example 18 Compare this if using only the rules [ [ ]] E E FCF t F s [ [ ] ] = E E FCF t F t 1 F s = E [(1 + g) FCF ] t 1 F s [ ] = (1 + g) E FCF t 1 F s by rule 4 see above by rule 2 = (1 + g) t s FCF s repeating argument. 1.2 Conditional expectation, Application of the rules

21 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, Application of the rules