1. Lecture 2: Conditional Expectation Discounted Cash Flow, Section 1.2 © 2004, Lutz Kruschwitz and Andreas Löffler

2. 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.

3. No. 3 1.2.1 Information Today is certain, the future is uncertain. Now: we always stay at time 0! And think about the future Simon Vouet: «Fortune-teller» (1617)

4. 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.

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

6. 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: A.N. Kolmogorov (1903-1987), founded theory of conditional expectation

7. No. 7 1.2.1 Thinking about the future today

8. No. 8 1.2.2 Conditional expectation 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 |  1 ). A conditional expectation covers our ideas about future thoughts. This conditional expectation can be uncertain.

9. 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)

10. No. 10 1.2.2 Rule 1: Classical expectation At t=0 conditional expectation and classical expectation coincide. Or: conditional expectation generalizes classical expectation.

11. No. 11 1.2.2 Rule 2: Linearity Business as usual…

12. No. 12 1.2.2 Rule 3: Certain quantity Safety first… From this and linearity for certain quantities X,

13. 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.

14. 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.

15. No. 15 1.2.3 Using the rules We want to check our rules by looking at –the finite example and –an infinite example.

16. No. 16 1.2.3 Finite example We want to check our rules by looking at the finite example and an infinite example.

17. No. 17 1.2.3 Finite example And indeed It seems purely by chance that But it is on purpose! This will become clear later…

18. No. 18 1.2.3 Infinite example Again two factors up and down with probability p u and p d and 0<d<u or

19. No. 19 1.2.3 Infinite example Let 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).

20. No. 20 1.2.3 Infinite example Compare this if using only the rules

21. No. 21 Summary We always stay in the present. Conditional expectation handles our knowledge of the future. Five rules cover necessary mathematics.