Lecture: Corporate Income Tax

Transcription

1 Lectre: Corporate Income Tax Ltz Krschwitz & Andreas Löffler Disconted Cash Flow, Section 2.1,

2 Otline 2.1 Unlevered firms Similar companies Notation Valation eqation Weak atoregressive cash flows Independent and ncorrelated increments Gordon Shapiro Discont rates Example The finite case The infinite case,

3 2.1 Unlevered firms, The nlevered firm 1 Companies are indebted, i.e. levered. Why shold we consider nlevered firms, i.e. firms withot debt? Valation reqires knowledge of cash flows = from bsiness plans, annal balance sheets etc. taxes = from tax law cost of capital = from similar companies. What is a similar company?

4 Similar companies 2 Companies are similar with respect to bsiness risk financial risk (= different leverage ratio). We eliminate the financial risk by determining the cost of capital of an nlevered firm ( nlevering ) and then of a levered firm ( relevering ). Unlevered firm nlevering relevering Firm A Firm B 2.1 Unlevered firms, Similar companies

5 Notation Unlevered firms, Notation levered firm index l nlevered firm index free cash flows (after taxes) vale cost of capital FCF t Ṽ t k E, t h E gfcf e i t+1 = + V t+1 F t Def 1 ev t

6 Vale of nlevered firm 4 We have, analogosly to Chapter 1 (even with the same proof!) Theorem 2.1 (vale of nlevered firm): With deterministic cost of capital [ ] T E FCF Ṽt s F t = ( ) ( ). s=t kt E, 1 + k E, s Valation eqation,

7 An important assmption 5 Aim: We want to show that for the nlevered firm, costs of capital (expected retrns) are discont rates as well! Means: An assmption is necessary, which was already sed by Feltham/Christensen, Feltham/Ohlson, also everyday bsiness in statistics: atoregressive cash flows. Remark: This assmption will concern the stochastic strctre of the nlevered cash flows Weak atoregressive cash flows,

8 Independence vs. ncorrelation 6 Atoregressive cash flows, also called AR(1): FCF t+1 = (1 + g) FCF t + ε t+1. In finance the sal assmption is: noise terms ε t are pairwise independent. Bt here we only assme that the noise terms are pairwise ncorrelated which is less restrictive: X, Ỹ are ncorrelated if independent if for all fnctions f and g ] [ ) )] Cov [ X, Ỹ = 0 Cov f ( X, g (Ỹ = Weak atoregressive cash flows, Independent and ncorrelated increments

9 A formlation sing conditional expectations 7 Frthermore, or growth rate g t can be time-dependent that is why we speak of weak atoregression. Assmption 2.1 (weak atoregression): There are growth rates (real nmbers!) g t sch that for the cash flows of the nlevered firm [ ] E FCF t+1 F t = (1 + g t ) FCF t. Is this the same as definition AR(1) above? Yes, and this will be shown sing or rles! Weak atoregressive cash flows, Independent and ncorrelated increments

10 Noise and weak atoregression 8 We will show now 1. Noise has no expectation E [ ε t ] = Noise terms are ncorrelated Cov [ ε s, ε t ] = 0 if s t where noise is defined by ε t := FCF t+1 (1 + g t ) FCF t Weak atoregressive cash flows, Independent and ncorrelated increments

11 Proof (1) 9 Noise terms have no expectation: E [ ε t ] = E [ ε t F 0 ] by rle 1 = E [E [ ε t F t ] F 0 ] by rle 4 [ [ = E E FCF t+1 (1 + g t ) FCF ] ] t F t F 0 by definition [ [ ] = E E FCF t+1 F t E [(1 + g t ) FCF ] ] t F t F 0 by rle 2 [ [ ] = E E FCF t+1 F t (1 + g t ) FCF ] t F 0 by rle 5 = E [(1 + g t ) FCF t (1 + g t ) FCF ] t F 0 by assmption = 0 QED Weak atoregressive cash flows, Independent and ncorrelated increments

12 Proof (2) 10 Noise terms are ncorrelated: Cov [ ε s, ε t ] = E [ ε t ε s ] E [ ε t ] E [ ε s ] }{{} =0 = E [ ε t ε s ] = E [ ε t ε s F 0 ] by rle 1 = E [E [ ε t ε s F s ] F 0 ] by rle 4 [ ] = E ε s E [ ε t F s ] F 0 by rle 4 }{{} will now shown to be 0 = 0 QED by definition of covariance Weak atoregressive cash flows, Independent and ncorrelated increments

13 Proof (2) contined 11 [ E [ ε t F s ] = E FCF t (1 + g t 1 ) FCF ] t 1 F s [ [ FCF = E = E E [ E t (1 + g t 1 ) FCF ] ] t 1 F t 1 F s [ ] FCF t F t 1 (1 + g t 1 ) FCF ] t 1 F s rle 4 rle 2 and 5 = 0 assmption Weak atoregressive cash flows, Independent and ncorrelated increments

14 Implications 12 What follows from weak atoregressive cash flows? Two important theorems: 1. There is a deterministic dividend price ratio. 2. The cost of capital of the nlevered firm may be sed as a discont rate Weak atoregressive cash flows, Independent and ncorrelated increments

15 First conclsion: Gordon Shapiro 13 Theorem 2.2 (Williams, Gordon Shapiro, Feltham/Ohlson): If costs of capital are deterministic and cash flows are weak atoregressive, then Ṽ t = FCF t d t holds for a deterministic dividend price ratio d t. (Or first mltiple!) Weak atoregressive cash flows, Gordon Shapiro

16 Proof of Theorem First notice that [ ] [ [ ] ] E FCF s F t = E E FCF s F s 1 F t = E [(1 + g s 1 ) FCF ] s 1 F t [ ] = (1 + g s 1 ) E FCF s 1 F t [ ] = (1 + g s 1 ) (1 + g t+1 ) E FCF t F t by rle 4 by assmption 2.1 by rle 4 contined = (1 + g s 1 ) (1 + g t+1 ) FCF t by rle Weak atoregressive cash flows, Gordon Shapiro

17 Proof of theorem 2.2 (contined) 15 Ṽ t = T s=t+1 ( 1 + k E, t [ E FCF ) ] s F t ( 1 + k E, s 1 T (1 + g s 1 (1 + g t+1 = ( ) ( ) FCF s=t kt E, 1 + k E, t s 1 }{{} :=1/dt ) from Theorem 1.1 see slide above = FCF t d t QED Weak atoregressive cash flows, Gordon Shapiro

18 Second conclsion: discont rates 16 Aim: to show that costs of capital are discont rates! First let s precisely define discont rates. Notice that discont rates will depend on the cash flow we want to vale ( FCF s ), on the point in time where we determine this vale (index t) and on the actal time period (index r) we are disconting. t r FCF s We se the notation κ t s r for disconting from r + 1 to r Weak atoregressive cash flows, Discont rates

19 Definition of discont rates 17 Definition 2.2 (discont rates): Real nmbers are called discont rates of the cash flow FCF t if they satisfy [ ] [ ] E Q FCF s F t E FCF s F t (1 + r f ) s t }{{} vale = (1 + κ t s t ) (1 + κ t s s 1 ). Interpretation of lhs: vale, follows from fndamental theorem. Interpretation of rhs: the way we se discont rates Weak atoregressive cash flows, Discont rates

20 Costs of capital and discont rates 18 Now, finally, or second implication from weak atocorrelated cash flows. Theorem 2.3 (eqivalence of valation concepts): If costs of capital are deterministic and cash flows are weak atoregressive, then E Q [ FCF s F t ] (1 + r f ) s t = ( 1 + k E, t [ E FCF ) ] s F t ( 1 + k E, s 1 ) or: costs of capital are discont rates! Weak atoregressive cash flows, Discont rates

21 This reslt is not trivial at all! Weak atoregressive cash flows, Discont rates Meaning of Theorem Notice that sms are eqal [ ] T E Q FCF s F t (1 + r s=t+1 f ) s t = }{{} fndamental theorem Ṽ t = T [ ] E FCF s F t s=t+1 (1 + kt E, }{{} theorem = 10 = ) (1 + k E, s 1 ). Theorem 2.3 tells s that smmands are eqal as well [ ] [ ] E Q FCF s F t E FCF s F t (1 + r f ) s t = (1 + kt E, ) (1 + k E, s 1 ). 4 3 and 6 7

22 Proof of Theorem We skip the proof! The shining of the proof Weak atoregressive cash flows, Discont rates

23 The finite example 21 We assme that k E, = 20%. The expectations are [ ] [ ] [ ] E FCF 1 = 100, E FCF 2 = 110, E FCF 3 = 121, k E, = 20 %. The vale of the firm is given by [ ] [ ] [ ] E FCF Ṽ0 1 E FCF = (1 + k E, ) + 2 E FCF (1 + k E, ) (1 + k E, ) 3 = ( ) ( ) Example, The finite case

24 Determining Ṽ 1 22 Althogh not clear yet why necessary, we want to determine the market vale at t = 1: [ ] { 121 p, [ ] { p, E FCF 2 F 1 = E FCF 99 down. 3 F 1 = down. hence ev 1 () [g 1+k E, + E FCF = E [g FCF 2 ()] = (1+0.2) ev 1 (d) () ] (1+k E, ) 2 e V 1 = 8 < : p, down Example, The finite case

25 Determining Q 23 Let r f = 10%. Another additional reslt is the determination of the risk-netral probability Q. De to theorem 2.3 ( costs of capital are discont rates ) we have [ ] [ ] E Q FCF 3 F 2 E FCF 3 F 2 = 1 + r f 1 + k E,. Assme that state ω occrred at time t = 2. Then this eqation translates to Example, The finite case

26 Determining Q (contined) 24 E Q[g FCF 3 F 2] g = Q 3 ( ω) FCF 3 ( ω)+q 3 (d ω) g FCF 3 (d ω) 1+r f 1+r f g = P 3 ( ω) FCF 3 ( ω)+p 3 (d ω) FCF g 3 (d ω) [g 1+k E, = E FCF Also, the conditional Q-probabilities add to one: Q 3 ( ω) + Q 3 (d ω) = 1. This is a 2 2-system that can be solved! 3 F 2] 1+k E, Example, The finite case

27 Q in the finite example t = 0 t = 1 t = 2 t = 3 time Example, The finite case

28 The infinite case 26 Let k E, = 20%, then the vale of the nlevered firm is V 0 = Ẽ[ FCF 1] k E, = 500. Let r f = 10%. Then, as above Q can be determined: Q t+1 ( ω) = regardless of t and ω. 1+r f 1+k E, d d, Q t+1 (d ω) = 1+r f 1+k E, d Remark: The factors and d cannot be chosen arbitrarily in the infinite case if the cost of capital k E, is given, becase d < 1 + r f 1 + k E, < mst hold in order to ensre positive Q-probabilities Example, The infinite case

29 Smmary 27 We will consider nlevered and levered firms. Cash flows of the nlevered firm are weak atoregressive, i.e. noise terms are ncorrelated. The costs of capital of nlevered firm are discont rates. The mltiple dividend price ratio is deterministic Example, The infinite case