Lecure: Auonomous Financing and Financing Based on Marke Values I Luz Kruschwiz & Andreas Löffler Discouned Cash Flow, Secion 2.3, 2.4.1 2.4.3,
Ouline 2.3 Auonomous financing 2.4 Financing based on marke values 2.4.1 Flow o equiy Cos of equiy FTE approach 2.4.2 Toal cash flow WACC ype 1 TCF approach TCF exbook formula 2.4.3 Weighed average cos of capial WACC ype 2 WACC approach WACC exbook formula Summary,
Adjused presen value 1 Definiion 2.4 (auonomous financing): A firm is auonomously financed if is fuure amoun of deb D is cerain. Theorem 2.4 (APV): If a firm is auonomously financed, hen Proof: rivial. Ṽ l = Ṽ u + T s=+1 τr f D s 1 (1 + r f ) s. 2.3 Auonomous financing, Adjused presen value (APV)
Consan deb 2 Theorem 2.5 (consan deb, MoMi): If T = and D = cons, hen Ṽ l = Ṽ u + τd. Proof: Ṽ l = Ṽ u + = Ṽ u T s=+1 + τr f D τr f D s 1 (1 + r f ) s T s=+1 = Ṽ u + τr f D 1 r f. 1 (1 + r f ) s 2.3 Auonomous financing, Consan deb
Consan deb, again 3 An alered represenaion would be V l 0 = V u 0 + τd 0 V l 0 = V u 0 + τl 0 V l 0 (1 τl 0 )V0 l = V0 u [ u ] E FCF V0 l = (1 τl 0 )k E,u. We come back o his (or a leas a similar) equaion in he nex lecure when alking abou he Modigliani-Miller adjusmen. 2.3 Auonomous financing, Consan deb
The finie example 4 We now look a our finie example wih τ = 50%. Hence, D 0 = 100, D 1 = 100, D 2 = 50. V0 l = V0 u + τr f D 0 + τr f D 1 1 + r f (1 + r f ) 2 + τr f D 2 (1 + r f ) 3 0.5 0.1 100 0.5 0.1 100 229.75 + + 1 + 0.1 (1 + 0.1) 2 0.5 0.1 50 + (1 + 0.1) 3 240.30. This is also he value of he firm hreaened by defaul. 2.3 Auonomous financing, The finie example
Fuure firm values 5 For laer use we evaluae he fuure firm value Ṽ l 1, Ṽ1 l = Ṽ 1 u + τr f D 1 + τr f D 2 1 + r f (1 + r f ) 2 = Ṽ 1 u 0.5 0.1 100 + 1 + 0.1 { 199.88 if up, 164.74 if down. + 0.5 0.1 50 (1 + 0.1) 2 2.3 Auonomous financing, The finie example
Deb and leverage raio 6 Aenion: The corresponding deb raio is uncerain! l1 = D 1 Ṽ l 1 { 50.03 % if up, 60.03 % if down, Hence, a cerain amoun of deb implies an uncerain leverage raio! (And vice versa... ) 2.3 Auonomous financing, The finie example
The finie case wih defaul 7 Since we can sill use he APV-formula, Ṽ l 0 240.30. Anoher way of obaining his value is by evaluaing E Q [ FCF l ] and discouning i wih he riskless rae. 2.3 Auonomous financing, The finie example
The infinie example 8 Here τ = 50% and consan deb D = 100. Then V l 0 = V u 0 + = V u 0 + =0 =0 τr f D (1 + r f ) +1 τr f D 0 (1 + r f ) +1 = V u 0 + τd 0 = 550. 2.3 Auonomous financing, The infinie example
Financing based on marke values 9 Definiion 2.5 (financing based on marke values): Financing is based on marke values if deb raios l are cerain. oday amoun of deb D 0 + 1 The ime srucure of financing based on marke values ax savings τĩ+1 = The amoun of fuure deb D is uncerain! = The ax advanages from deb are uncerain as well! = APV does no apply! Insead hree differen procedures... 2.4 Financing based on marke values,
The general procedure 10 To evaluae he company 1. We sar wih appropriae cos of capial. 2. We assume ha hese cos of capial are deerminisic and apply (as usual) a corresponding valuaion formula. 3. We hen look a he connecion of hese procedures: hey are given by exbook formulas. There are hree appropriae coss of capial, hence here will be hree valuaion procedures: FTE, TCF, WACC. Noice ha defaul is no ruled ou! 2.4 Financing based on marke values,
Three valuaion procedures 11 Overview of hree procedures: procedure reference value cos of capial FTE Ẽ+ D k E,l TCF Ṽ k WACC Ṽ WACC These cos of capial are raios of corresponding cash flows o he reference values. Bu here will be an anomaly wih WACC... 2.4 Financing based on marke values,
Flow o equiy (FTE) 12 Wih FTE we are looking a he sockholders and heir cos of equiy. The cash flow o sockholders is given by FCF l +1 free cash flows ineres and repaymen Ĩ+1 R +1. Definiion 2.6 (cos of equiy): Coss of equiy are condiional expeced reurns E [Ẽ+1 + FCF l k E,l +1 Ĩ+1 R ] +1 F := 1. Ẽ 2.4.1 Flow o equiy, Cos of equiy
FTE approach 13 Theorem 2.6 (FTE): If Ẽ = T s=+1 k E,l ( are deerminisic, hen [ l E FCF 1 + k E,l s Ĩs R ] s F ) ( 1 + k E,l s 1 ). Proof: see our general valuaion heorem (Theorem 1.1). Remarks: FTE requires deerminisic cos of equiy. The heorem does no ye require financing based on marke values! The leverage raio does no appear in FTE. The knowledge of expeced repaymen is necessary. 2.4.1 Flow o equiy, FTE approach
Toal cash flow (TCF) 14 Now we are looking a he sockholders and he debholders, or he cos of equiy and deb. Definiion 2.7 (weighed average cos of capial ype 1): WACC ype 1 are condiional expeced reurns E [Ṽ l +1 + FCF l ] +1 F k := Ṽ l 1. 2.4.2 Toal cash flow, WACC ype 1
TCF approach 15 Theorem 2.7 (TCF): If k Ṽ l = T s=+1 deerminisic, hen ( 1 + k [ l E FCF ) ] s F ( ). 1 + ks 1 Proof: see our general valuaion heorem (Theorem 1.1). Remarks: TCF requires deerminisic WACC ype 1. The heorem does no ye require financing based on marke values! The leverage raio does no appear in TCF. The knowledge of expeced deb is no necessary. 2.4.2 Toal cash flow, TCF approach
2.4.2 Toal cash flow, TCF exbook formula TCF exbook formula 16 Wha is he connecion beween FTE and TCF? The answer is he exbook formula. Theorem 2.8 (TCF exbook formula): I always holds ha ) (1 l + k D l. k = k E,l
2.4.2 Toal cash flow, TCF exbook formula Proof 17 ( 1 + ( 1 + ( E,l 1 + k ( ) ( E,l 1 + k Ẽ + 1 + k D ( ) ( E,l 1 + k Ẽ + 1 + k D Ẽ + E,l k Ẽ + D + k D ) E,l k Ẽ = E [Ẽ+1 + FCF l +1 R +1 + D +1 D ] +1 Ĩ+1 F ) E,l l k Ẽ = E [Ṽ+1 + FCF l +1 R +1 D ] +1 Ĩ+1 F ) Ẽ = E ) l D = E [Ṽ ) D = ( 1 + k l [Ṽ+1 + FCF l +1 D k D +1 + FCF l ] +1 F ) Ṽ l D = Ṽ l + k Ṽ l D F ] k E,l Ẽ Ṽ l + k D D Ṽ l = k.
2.4.2 Toal cash flow, Remarks Remarks o TCF, I 18 The coss of deb are no reduced by he ax rae in he TCF exbook formula. The formula holds regardless of wheher he relevan variables are deerminisic or sochasic. In paricular: financing based on marke values is no necessary!
2.4.2 Toal cash flow, Remarks Remarks o TCF, II 19 Assume no defaul. One of wo cases possible marke-value financing If WACC ype 1 or cos of equiy are deerminisic, he oher is deerminisic as well. TCF and FTE only used simulaneously. non marke-value financing Eiher WACC ype 1 or cos of equiy has o be uncerain. TCF and FTE exclude each oher. Proof: k = k E,l (1 l) + r f l.
Weighed average cos of capial 20 We are now a sockholders and debholders again. Definiion 2.8 (weighed average cos of capial ype 2): WACC ype 2 are he condiional expeced reurns E [Ṽ l +1 + FCF u ] +1 F WACC := Ṽ l 1. Remark: These are coss of capial of a firm ha is on he one hand levered (Ṽ l ) and on he oher hand unlevered ( FCF u +1). Apples and oranges mixed here. 2.4.3 Weighed average cos of capial, WACC ype 2
WACC approach 21 Theorem 2.9 (WACC): If WACC is deerminisic, hen [ ] u T E FCF Ṽ l s F = (1 + WACC ) (1 + WACC s 1 ). s=+1 Proof: see our general valuaion heorem (Theorem 1.1) Remarks: WACC requires deerminisic WACC ype 2. The heorem above does no ye require financing based on marke values! The leverage raio does no appear in WACC. The knowledge of cash flow of an unlevered firm is necessary. 2.4.3 Weighed average cos of capial, WACC approach
WACC exbook formula 22 Wha is he connecion beween FTE and WACC? The answer is anoher exbook formula. Theorem 2.10 (WACC exbook formula): Always ) (1 l + k D (1 τ) l. WACC = k E,l 2.4.3 Weighed average cos of capial, WACC exbook formula
Proof 23 ( ) E,l 1 + k Ẽ = E [Ẽ+1 + FCF l +1 R ] +1 Ĩ+1 F ( ) E,l l 1 + k Ẽ = E [Ṽ+1 + FCF l +1 R +1 Ĩ+1 D ] +1 F ( ) E,l l 1 + k Ẽ = E [Ṽ+1 + FCF u +1 R +1 Ĩ+1 D +1 + τ(ĩ+1 D + R +1 + D ] +1 ) F ( ) E,l l 1 + k Ẽ = E [Ṽ+1 + FCF u +1 (1 + k D ) D + τ k ] D D F. 2.4.3 Weighed average cos of capial, WACC exbook formula
Proof, coninued 24 ( 1 + ( 1 + E,l k k E,l Ẽ + ) Ẽ + ( 1 + k D (1 τ) ) D = E [Ṽ l +1 + FCF u +1 F ] ) Ẽ + ( 1 + k D (1 τ) ) D = (1 + WACC ) Ṽ l E,l k Ẽ + D + k D (1 τ) D = Ṽ l + WACC Ṽ l k E,l Ẽ Ṽ l + k D (1 τ) D Ṽ l = WACC. And his was o be shown QED 2.4.3 Weighed average cos of capial, WACC exbook formula
Remarks, I 25 The coss of deb are reduced by he ax rae in he WACC exbook formula. The formula holds regardless of wheher he relevan variables are deerminisic or sochasic. In paricular: financing based on marke values is no necessary! 2.4.3 Weighed average cos of capial, Remarks
Remarks, II 26 Assume no defaul. One of wo cases possible marke-value financing If WACC ype 2 or cos of equiy are deerminisic, he oher is deerminisic as well. WACC and FTE only used simulaneously. non marke-value financing Eiher WACC ype 2 or cos of equiy has o be uncerain. WACC and FTE exclude each oher. Proof: WACC = k E,l (1 l) + r f (1 τ) l. 2.4.3 Weighed average cos of capial, Remarks
Summary 27 WACC WACC cerain TCF marke-value driven ( l cerain) FTE k cerain k E,l cerain Summary,