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The Equivalent Loan Principle and the Value of Corporate Promised Cash Flows. David C. Nachman*

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he Equivalen Loan Principle and he Value of Corporae Promied Cah Flow by David C. Nachman* Revied February, 2002 *J. Mack Robinon College of Buine, Georgia Sae Univeriy, 35 Broad Sree, Alana, GA
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he Equivalen Loan Principle and he Value of Corporae Promied Cah Flow by David C. Nachman* Revied February, 2002 *J. Mack Robinon College of Buine, Georgia Sae Univeriy, 35 Broad Sree, Alana, GA Phone ( a he Equivalen Loan Principle and he Value of Corporae Promied Cah Flow by David C. Nachman hi paper examine he valuaion principle ha aer ha he marke value of beforeax corporae promied cah flow i he preen value of he afer-ax cah flow a he afer-ax corporae borrowing rae, he financing provided by he equivalen loan for hee promied cah flow. hi valuaion principle i herefore referred o a he equivalen loan principle. he equivalen loan principle i validaed in wo cae where marke value are clear and explici calculaion are poible. One cae i when he promied cah flow are hoe of a coupon bond and he oher cae i when he promied cah flow are hoe of a fully amorized loan. In verifying ha he marke value of he before-ax promied cah flow i equal o he financing provided by he equivalen loan for he afer-ax cah flow, one need o accoun for he effecive inere charge in calculaing afer-ax cah flow. he roll of he equivalen loan principle in he valuaion of ax hield i hen explored. Here he requiie marke value are no a obviou a in he cae of loan cah flow. he roll of ax hield in expanding deb capaciy i needed o reolve an apparen conradicion in he ypical valuaion of inere ax hield and depreciaion ax hield. . Inroducion he valuaion principle ha aer ha he marke value of before-ax corporae promied cah flow i he preen value of he afer-ax cah flow a he afer-ax corporae borrowing rae ha i origin in he valuaion of leae cah flow in Myer, Dill and Bauia (976. he erm deb equivalen cah flow and afe, nominal cah flow are ued a ynonym for promied cah flow. he erm promied cah flow i ued here ince i provide he mo direc link o he required rae of reurn for valuing hee cah flow. he preen value of he afer-ax cah flow a he afer-ax corporae borrowing rae i he financing provided by wha ha become known a he equivalen loan, a borrowing plan whoe afer-ax promied deb ervice exacly duplicae he afer-ax promied cah flow in queion. 2 So he valuaion principle aer ha he marke value of before-ax corporae promied cah flow equal he financing provided by he equivalen loan. For he ake of expoiion, we will refer o hi valuaion principle a he equivalen loan principle. Ruback (986 review he lieraure on hi principle and eablihe he principle for rikle cah flow (acual promied by an arbirage argumen. Eenially, he value of he cah flow ream in queion mu equal he financing provided by he equivalen loan or here i an arbirage opporuniy. In dealing wih riky cah flow, he cae for he equivalen loan i le forceful, bu ill raighforward, relying on he value of cah flow of he ame rik a he preen value of hee cah flow a he appropriae rik adjued required rae of reurn. Brealey and Myer (2000, ecion Myer, Dill, and Bauia (976, page 799 I i ypical in he lieraure on corporae deb o alk abou value in erm of he borrowing rae ha equae he preen value of promied cah flow o marke price. For riky corporae deb, hi borrowing rae conain a rik premium for defaul rik (and any oher yemaic rik in hee cah flow. One can argue ha he equivalen loan produce promied cah flow ha, afer-axe, mach he afer-ax promied cah flow in queion. A long a he borrowing rae ued o compue he equivalen loan adju for he rik of hee promied cah flow, hen he marke value of he before-ax promied cah flow hould be he ame a he financing provided by he equivalen loan. he idea of he equivalen loan i appealing becaue i provide a leverage neural perpecive on change in capial rucure. One way o view a firm financial leverage i a he ream of afer-ax deb ervice cah flow o which he firm i commied, i aferax promied cah flow. o value a change in capial rucure, uch a a financial leae, i i imporan ha he valuaion be neural wih regard o he leverage implied by he leae. Leaing diplace deb and how much deb i diplace depend on he value of he leae. he equivalen loan preerve he leverage implied by he leae ince i duplicae he afer-ax promied leae cah flow. hi leverage neural perpecive on change in capial rucure ha been poined ou by Frank and Hodge (978 and Lewellen and Emery (980 in he conex of leaing and i exended o a pariy principle wih applicaion o bond refunding by Lewellen and Emery (980. Brealey and Myer (2000 make ue of he equivalen loan principle o value everal differen corporae promied cah flow. 3 heir reamen of he value of financial ubidie and he value of ax hield moivae he preen paper. 3 See Brealey and Myer (2000, ecion 9.5 and In applying he equivalen loan principle, one encouner wo difficulie. One difficuly arie when marke rae differ from rae ued o calculae inere paymen, a in he cae of a ubidized loan or on any loan afer marke inere rae have changed. In uch cae, one mu be careful o compue he ax hield from effecive inere charge ha allow for he fac ha he marke value of he cah flow may be differen from he bai ued o compue acual inere paymen. Failure o do o creae he illuion of arbirage opporuniie in he marke for corporae deb. We illurae hi poin in hi paper in wo imporan cae where marke value are clear, he cae of promied cah flow from a coupon bond and promied cah flow from an amorized loan. Once we have validaed he equivalen loan principle in he fir cae, we apply he principle o correcly value he ubidy of a ubidized loan. he econd difficuly arie in he valuaion of ax hield. Inere ax hield and depreciaion ax hield are like afer-ax corporae promied cah flow. A uch he value of hee ax hield, according o he equivalen loan principle hould be he preen value of he ax hield a he afer-ax co of borrowing. hi i he ypical valuaion of depreciaion ax hield. he ypical valuaion of inere ax hield, however, i he preen value of he ax hield a he before-ax co of borrowing. We reolve he apparen conradicion by noing ha he value of any ax hield ha wo componen, one i he ax aving ielf and he oher i he effec of hi ax aving on he firm deb capaciy. 4 he ypical valuaion of depreciaion ax hield aume ha hey expand he firm deb capaciy by he amoun of he equivalen loan for hee ax hield. However, 4 hi reoluion i uggeed by he analyi of depreciaion and inere ax hield and heir effec on deb capaciy in Myer, Dill, and Bauia (976. 3 hi i only he fir ep in a poenially infinie regre. Valuing hi deb capaciy expanion lead o conideraion of he value of inere ax hield. Inere ax hield alo expand he deb capaciy, which lead o more inere ax hield, which lead o more deb capaciy, which lead o... hi infinie regre can be avoided only if a ome poin more deb i no valuable. hi i a cenral iue in capial rucure heory, one we will no reolve in hi paper. he iue of deb capaciy expanion of inere ax hield i ignored in he ypical valuaion of inere ax hield. he expoiion proceed a follow. he concep of he equivalen loan and he valuaion principle ha i baed on i are preened in Secion 2. 5 In Secion 3, he equivalen loan principle i applied o value promied cah flow ariing from a coupon bond. A numerical example i given ha help illurae how o value he ubidy of a ubidized coupon bond. A an addiional validaion of he equivalen loan principle, we apply i o value he promied cah flow ariing from an amorized loan in Secion 4. In Secion 5, we ake up he valuaion of ax hield, including boh inere ax hield and depreciaion ax hield. We conclude in Secion 6. he algebra of he derivaion in he ex i given in an Appendix. 2. he Equivalen Loan he idea of he equivalen loan i formalized a follow. Aume here are period and denoe by C he afer-ax promied cah flow a he end of period. We wan o conruc a borrowing plan whoe promied afer-ax deb ervice cah flow exacly mache C, K C., 5 he promied cah flow in Secion 2 are abrac. he reader inereed in he applicaion can kip hi ecion and proceed o Secion 3 where he promied cah flow are he familiar one of a coupon bond or o Secion 4 where he promied cah flow are hoe of an amorized loan. 4 Le τ denoe he corporae ax rae and le r denoe he before-ax corporae borrowing rae. Of coure r i a marke-deermined rae for promied cah flow ha reflec he rik of defaul in he cah flow. In he cae of rikle cah flow, hi i a rikle rae of reurn, he ame rae r ued by Ruback (986 in hi argumen. 6 Le P denoe principal borrowed a he beginning of period and le A denoe he principal repaid a he end of period. If A i negaive, hen inerpre hi a addiional borrowing. he following equaion deermine he deired borrowing plan: ( τ +,, K,, ( rp A C P A P +,, K,, (2 P +. (3 0 he lef hand ide of ( give he afer-ax promied deb ervice cah flow in period, which i he um of afer-ax inere, ( τ rp, plu principal repaid A. Equaion ( expree wha equivalen mean in he erm equivalen loan. Equaion (2 i an accouning ideniy. I ae ha he principal borrowed a he beginning of period minu wha i repaid a he end of period mu equal he principal a he beginning of period +. Finally, equaion (3 i he erminal condiion ha he loan i paid off a he end of period. here are no afer-ax promied cah flow o be mached afer period. 7 ( τ From (2 and (3, i follow ha P A. From ( i hen follow ha +. Solving for P give rp P C 6 o keep hing imple, hroughou hi paper we aume ha he ax rae and inere rae are conan. See Ruback (986 ecion 3 for exenion o nonconan rae. 7 See foonoe 8 below for cah flow ream ha exend indefiniely. 5 P C + r. (4 ( τ Equaion (4 ae ha he amoun of borrowing a he beginning of period ha commi o afer-ax promied deb ervice of he afer-ax co of deb. By (2, A P P C i preciely he preen value of C a, and hence by (, ( τ rp P P C Subiuing for P from (4 and olving for P give +. P C + C ( τ r ( τ ( r (5 Equaion (5 ay ha he borrowing a he beginning of period ha uppor a commimen o afer-ax promied deb ervice cah flow ream of C, C i he preen value of hi cah flow ream a he afer-ax co of deb. By inducion on i follow ha equaion (-(3 imply ha P C, +,, ( ( τ r + K. (6 he borrowing a he beginning of period ha uppor a commimen o afer-ax promied deb ervice cah flow ream of C,, K,, i he preen value of ha cah flow ream a he afer-ax co of deb. 8 We have couched he decripion of he equivalen loan deermined by (6 a a borrowing opporuniy, bu i can alo be hough of a a lending opporuniy ha 8 o handle cah flow ream ha exend indefiniely (form an infinie equence, exend he um in (6 o infiniy o define P and ue (2 o define A P P + for each. So long a hee infinie erie (preen value are well defined (for example, if he erie ha define P for any i aboluely convergen hee definiion aify equaion ( for every. 6 provide he promied cah flow ream C,, K C afer axe on inere earned. If you lend P a he beginning of period and A i repaid a he end of period, he afer-ax cah flow in period will be C, ec. 9 principle, i he valuaion principle ha i he ubjec of hi paper, he equivalen loan Equivalen Loan Principle. P, given by (6 for, i he marke value of he beforeax verion of he afer-ax cah flow ream C, K C. he reaoning i raighforward in he cae when he before-ax verion of he cah flow ream C,, K C i rikle. If he marke value of he before-ax cah flow ream i le han P, a corporaion could buy hi before-ax cah flow ream and finance he purchae by borrowing P. Since he afer-ax cah flow ream provided by he before-ax cah flow ream purchaed i idenical o he afer-ax deb ervice of he equivalen loan, he difference beween he marke value of he cah flow ream purchaed and he financing P provided by he equivalen loan i arbirage profi. Similarly, if he marke value of he before-ax cah flow ream i more han P, a corporaion could ell hi before-ax cah flow ream and inve P of he proceed of hi ale in he equivalen loan. he afer-ax deb ervice required on he cah flow ream old i exacly produced by he lending in he equivalen loan. A a reul he difference beween he marke value of he cah flow ream old and P i arbirage profi., 9 See Brealey and Myer (2000, boom of page 565 and op of page 566 for a dicuion of he equivalen loan principle when borrowing and lending rae differ. 7 In he cae of riky promied cah flow, he equivalen loan mache he afer-ax promied cah flow and hence he marke value of he equivalen loan hould be he marke value of he before-ax promied cah flow if he borrowing rae r reflec he rik of defaul in he cah flow. I i an implici componen of he equivalen loan calculaion ha he borrowing rae r reflec he rik of defaul and any oher yemaic rik in he cah flow. 0 Since he afer-ax promied cah flow of he equivalen loan are idenical o he afer-ax promied cah flow being valued and he borrowing rae r i he required rae of reurn for uch promied cah flow, P hould be heir marke value. 3. Coupon Bond Cah Flow o make hi more concree, le conider he cae when he afer-ax promied cah flow C,, K C arie from a coupon bond. Le M denoe he principal on he bond and le I denoe he inere coupon on he bond. hen by definiion, c I M i he coupon rae on he bond. he bond promie o pay I periodically o mauriy and in addiion promie o pay M a mauriy, inere paymen period from now. Given hi, i i emping o aume ha he afer-ax cah flow of he bond are he afer-ax coupon inere ( τ I periodically o mauriy and M a mauriy. hi i in fac done in he lieraure on corporae finance, bu i correc only for bond elling a par. o ee wha can go wrong, le r denoe he yield o mauriy on he bond. By definiion of r he preen value of before-ax promied cah flow a r i he marke value of he bond 0 hi poin i emphaized explicily in he lieraure. See Myer, Dill, and Bauia (976, page 807, and Lewellen and Emery (980, page 99. See he compuaion in Brealey and Myer (2000, ecion MV I r r r ( + M + ( + r. (7 Equaion (7 expree he marke value of he bond MV in he familiar and convenien form of he preen value of he coupon annuiy plu he preen value of he principal. Aume ha he afer-ax promied cah flow from he bond are he afer-ax inere coupon ( τ I and he principal M a mauriy. Alo aume ha he beforeax corporae borrowing rae i r, he yield o mauriy on he bond. By (6, he proceed of he equivalen loan P i he preen value of afer-ax promied cah flow a he aferax verion of he before-ax corporae borrowing rae r, ˆP I ( τ r r r ( τ ( τ + ( τ ( + M ( + r ( τ. (8 he compuaion in Equaion (8 expree he preen value calculaion ˆP in he convenien form of he preen value of he afer-ax coupon annuiy plu he preen value of he principal, boh preen value uing he afer-ax borrowing rae r ( τ. hi i denoed a ˆP, and no P, becaue hi i no he financing provided by he equivalen loan unle c Appendix r. aking he difference beween (7 and (8 give (he algebra i given in he ˆ MV P M ( c r r ( ( + r + r ( τ. (9 9 For r 0 and 0 τ, i follow ha r ( τ + + r, and hence he econd erm in parenhee on he righ in (9 i poiive. I follow ha he ign of he difference (9 i he ign of he fir erm in parenhee on he righ in (9. hu i follow ha MV Pˆ ( ( 0 c ( ( r. (0 hu when we aume ha he afer-ax promied cah flow of he bond are he uual upec, he difference beween he marke value of he bond and he financing provided by he equivalen loan for he uual upec i given by (0, and hi difference i zero if and only if he bond i elling for par, i. e., if and only if c Of coure for mo corporae deb, c r. r and hi condiion i pervaive in he marke for corporae deb. From (0, i appear ha here i an arbirage opporuniy. Bu o exploi uch an opporuniy, a corporaion would have o buy he bond a marke value (a a dicoun if c c ror ell he bond a marke value (iue he bond a a premium if r. In doing o he correponding inere charge and hence he afer-ax cah flow would have o reflec he effecive-inere mehod of amorizaion of he dicoun or premium. 2 I urn ou ha he quaniy on he righ in (9 i ju he preen value a he afer-ax borrowing rae of he difference beween he ax hield from coupon inere paymen and he ax hield from inere charge calculaed uing he effecive-inere rae mehod. 3 See he Appendix for hi compuaion. he effecive-inere mehod calculae inere a he yield o mauriy on he bond baed on he carrying value of he bond, which i alered each period by he inere differenial, he difference beween he effecive inere and he coupon inere. Of coure he iniial carrying value i he marke value of he bond MV. he effec of he 2 hi wa uggeed o he auhor by Sew Myer in a privae communicaion. 0 adjumen for he inere differenial i ha he carrying value of he bond a dae i he preen value of he remaining cah flow (for +, K, from he bond a he bond yield o mauriy r. Denoe hi preen value by MV, 0,, K,. Of coure MV MV 0 and MV i given by (7 wih replaced wih. he inere charge for he fir dae i rmv rmv 0. Similarly, he inere charge for any dae i rmv, and he correponding afer-ax cah flow for he bond a dae i he acual coupon paymen I cm minu he axe aved of τrmv,, K,. For dae, he afer-ax cah flow for he bond i he afer-ax inere cm - τrmv plu he principal M. So he afer-ax cah flow for he equivalen loan are a follow. For, K,, C cm - τrmv, he acual coupon inere cm minu he ax hield on he effecive inere charge τrmv, he afer-ax inere, and C cm - τrmv + M, afer-ax inere plu principal. he equivalen loan i validaed by he fac ha MV cm τrmv + ( ( + τ r M ( + ( τ r, ( where he righ ide of ( i P, he financing provided by he equivalen loan. hu he marke value of he coupon bond i equal o he preen value of he afer-ax cah flow of he bond a he afer-ax borrowing rae for hee cah flow. hi reul i derived in he Appendix. A imple numerical example may help o illurae he compuaion of hi ecion. Aume a corporae ax rae of 35%. A year ago a firm iued a 6-year mauriy, 3 See Skouen, Sice, and Sice (2000, page , for he effecive inere rae mehod. 5% coupon noe wih par of $00 and annual coupon paymen. One year laer, he noe ha 5 year o mauriy, bu inere rae have rien o 3%. 4 Here c 5%, r 3% and 5. By (7, he marke value of he noe i MV ( ( By (8, he preen value of he afer-ax coupon inere and principal a he afer-ax co of deb i ˆP ( ( hu a indicaed in (0, MV ˆP ince in hi cae c r. he apparen arbirage here would be o purchae he noe a marke value of o finance hi purchae, borrow according o he equivalen loan for he afer-ax coupon inere and principal. Afer axe, he cah flow from he noe purchaed would exacly cover he deb ervice requiremen on hi equivalen loan. he difference i apparen arbirage profi. Noice however ha he noe purchaed for 7.86 i a a dicoun of 28.4 o par value. A a reul of hi dicoun, he effecive inere charge o income for any purchaer of he loan would be higher han he coupon inere paid, and hence he aferax cah flow would be lower. he effecive inere charge and ax hield are compued in he following able. he nex o la column in he able i he difference beween he effecive inere ax hield and he coupon inere ax hield of.35(5 4 Numerically, hi i he example a he beginning of ecion 9.5 in Brealey and Myer ( .75. he final column compue he preen value of hi differenial ax hield a he afer-ax borrowing rae of.3( Effecive Effecive Preen Carrying Inere Inere Differenial Value a ( τ Year Value Charge ax Shield ax Shield r - MV MV MV MV MV oal Preen Value 7.62 hi preen value i he difference compued in (9 above, , i. e., (9 i ju he preen value a he afer-ax borrowing rae of he difference beween he effecive inere ax hield and he coupon inere ax hield. he apparen arbirage profi of $7.62 i really he value of he incremenal ax liabiliy on he higher effecive inere income. Le change he cenario a lile. A firm i inereed in buying a machine ha co $00. he manufacurer weeen he deal by offering o finance he purchae wih a loan of $00 for five year wih annual inere paymen of 5 percen. he purchaing firm borrowing rae i 3 percen, o he loan offer i a ubidized loan. Numerically, hi cenario i he ame a he one above. Wha i he value of hi ubidy? he naural anwer i ha he value of he ubidy hould be he difference beween wha he manufacurer i paying for he firm IOU and he marke value of ha IOU, o he value of he ubidy hould be A oon a he loan i conummaed he marke value balance hee of he firm will how an increae in cah of 3 $00 and offeing increae in deb of $7.86 and an increae in he value of equiy of $ Brealey and Myer (2000, ecion 9.5, however, claim ha he NPV of he ubidized loan i he dif
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