Selected Financial Formulae. Basic Time Value Formulae PV A FV A. FV Ad
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1 Basc Tme Value e Fuure Value of a Sngle Sum PV( + Presen Value of a Sngle Sum PV ( + Solve for for a Sngle Sum ln PV ln( + Solve for for a Sngle Sum PV Presen Value of an Ordnary Annuy PV A Pm ( + Fuure Value of an Ordnary Annuy A Pm ( Presen Value of an Annuy ue PV Ad Pm ( + ( + Pm Fuure Value of an Annuy ue Ad Pm ( ( + Presen Value of an Annuy Growng a a Consan Rae (g Fuure Value of an Annuy Growng a a Consan Rae (g Pm PV GA g + g Pm GA g + g ( + + Holdng Perod Reurn (sngle perod HPR P + Cash Flows P 0 Basc Fnancal e by Tmohy R. Mayes, Ph..
2 Holdng Perod Reurn wh Renvesmen (for mulple sub-perod reurns HPR Renves ( + HPR Basc Secury Valuaon e vdend scoun Model (AKA, he Gordon Model V 0 ( + g CS g g Two-sage vdend scoun Model oes: Ths equaon s oo long for one lne. g Growh rae durng hgh growh phase. g Growh n consan growh phase afer n. n Lengh of hgh growh phase. Assume g <> and g < Three-sage vdend scoun Model oes: n Lengh of hgh growh phase. n Perods unl consan growh phase. n n + lengh of ransson phase. Earnngs Model Consan Growh FCF Valuaon Model V Ops Value of Toal Operaons V eb, V Pref Value of deb and preferred sock V on-ops Asses Value of non-operang asses V CS 0 ( + g + g n g ( + g n ( + g g ( + n 0 n V CS n ( + g g ( g g RE ROE EPS V k CS CS g FCF V Ops WACC g V CS V Ops V eb V Pref + V on OpAsses Susanable growh rae oe: b reenon rao - payou rao r reurn on equy g br Value of a Share of Preferred Sock V P ---- k P Basc Fnancal e by Tmohy R. Mayes, Ph..
3 Value of a Bond on a Paymen ae V B Pm ( + k d k d ( + k d Quoed Prce of a Bond on a on-paymen ae V B,0 Value of bond a las paymen dae α The fracon of he curren perod ha has elaspsed V B, α V B 0 Basc Sascal e, ( + k d α α( Pm Arhmec Mean (Average X --- X Geomerc Mean (used for averagng reurns, growh raes, ec. G ( + R Expeced Value (Weghed Average EX ( ρ X Varance σ X ρ ( X X Sandard evaon σ X σ X Coeffcen of Varaon CV σ X EX ( Covarance σ XY, [ ρ ( X X ( Y Y ] Correlaon Coeffcen σ r XY, XY, σ X σ Y Bea (oe: M s he marke porfolo, and s he secury or porfolo β σ , M σ M r M, σ σ M σ M Basc Fnancal e by Tmohy R. Mayes, Ph.. 3
4 Porfolo e Expeced Reurn of a Porfolo Varance of a -secury Porfolo ER ( P w R Usng he covarance: + + w w σ, or, usng he correlaon coeffcen: σ P σ P w σ w σ w σ + + w σ w w r, σ σ Varance of an -secury porfolo Usng he Covarance σ P j w w j σ j, Sandard evaon of a Porfolo Mnmum Varance Porfolo Weghs oe: V s he varance/covarance marx and e s a vecor of s of lengh number of secures. σ P σ P W V e MnVar e T V e Porfolo Bea β P w β 95% Value a Rsk (paramerc Model oe: V p s porfolo value VaR.645 V p σ p Capal Marke Theory Models Capal Marke Lne (CML Capal Asse Prcng Model (CAPM oe: Ths s also he equaon for he Secury Marke Lne (SML Treynor s Rsk-adjused Performance Measure ( ER ( M ER ( P R f + σ P σ M ER ( R f + β ( ER ( M R f R T β Basc Fnancal e by Tmohy R. Mayes, Ph.. 4
5 Sharpe s Rsk-adjused Performance Measure Jensen s Alpha The Informaon Rao R S σ α ( R β ( R M R IR P R B P σ RP R B M (Modglan & Modglan Performance Measure M σ m σ ( R R f + R f Fama s Rsk ecomposon oes: R Porfolo Reurn R M Marke Reurn R f Rsk-free Rae β Porfolo Bea β T Targe Bea Brnson, Hood, and Beebower Addve Arbuon Model oes: A Overall Allocaon Effec S Overall Selecon Effec I Overall Ineracon Effec w, Wegh of Secor n porfolo bars over varables represen benchmark weghs and reurns. Rsk Premum R Rsk β ( R M Selecvy Rsk Premum Rsk Managers Rsk ( β β T ( R M Invesors Rsk β T ( R M versfcaon σ β ( RM σ M e Selecvy Selecvy versfcaon A S I w, w, ( ( R, R w, R, R, ( ( w, w, R, R, ( Basc Fnancal e by Tmohy R. Mayes, Ph.. 5
6 Opons and Fuures Valuaon Models Black-Scholes European Call Opon Valuaon Model C S( d Xe r where: d d ( S ln-- + ( r + 0.5σ X σ d d σ Black-Scholes European Pu Opon Valuaon Model (see above for d and d P ( d S( d Xe r Pu-Call Pary for European Opons wh o Cash Flows Sngle-perod Bnomal Opon Prcng Model for Call Opons (r s he rsk-free rae, u s he up facor, and d s he down facor Sngle-perod Bnomal Opon Prcng Model for Pu Opons Cos of Carry Model for Prcng Fuures Conracs (CC s he carryng coss as a % of he spo prce C P+ S Xe r or, S P C+ Xe r pc C u + ( pc d ( + r where, p r d u d pp P u + ( pp d ( + r where, p r d u d ( TF0 S 0 e CC Bond Analyss e Macaulay s uraon on a Paymen ae (for mmunzaon. oe: C s he cash flow n perod, s he yeld o maury C ( ( Bond Prce Modfed uraon (for prce volaly on a Paymen ae Mod ( + Basc Fnancal e by Tmohy R. Mayes, Ph.. 6
7 Convexy on a Paymen ae C Cf ( + ( ( Bond Prce The n-perod forward rae gven wo spo raes (noe ha > j, and n - j j R + n ( + R ( + R j j n Bank scoun Yeld for dscoun secures ( face value, PP purchase prce, m perods per year BY PP m Bond Equvalen Yeld for dscoun secures (see defnons for BY BEY PP BY PP m PP 360 Capal Budgeng ecson e e Presen Value (PV PV Cf IO ( + Profably Index (PI PI Cf ( PV IO IO IO PV IO Inernal Rae of Reurn (IRR. oe: Ths s a ral and error procedure o fnd he ha makes he equaly hold (.e., wha dscoun rae makes he PV 0. 0 Cf IO ( + Modfed Inernal Rae of Reurn (MIRR. MIRR Cf ( + ( IO Sock Marke Index Consrucon e Basc Fnancal e by Tmohy R. Mayes, Ph.. 7
8 Prce-weghed Average (e.g., JIA oe: The dvsor (v a perod 0 s equal o he number of socks n he average. I wll be adjused for sock spls or any oher corporae acon ha resuls n a non-economc change n he sock prce. PWA P j j v Capalzaon-weghed Index (e.g., S&P 500 oe: The dvsor (v a perod 0 s he dvsor ha makes he nal level of he ndex equal o he desred sarng pon. I wll be adjused for any corporae acon ha resuls n a change n marke capalzaon. CWI j P j Q j v Equally-weghed Arhmec Index (e.g., VLA oe: A perod 0 he ndex s se o some sarng value (e.g., 00. To calculae he ndex for any day, mulply he average % change by he prevous ndex level. Equally-weghed Geomerc Index (e.g., VLG oe: See noe above EWAI EWAI EWGI EWGI P j, P j j, P j, P j j, Corporae Fnancal e e Operang Prof Afer Taxes (OPAT e Operang Workng Capal (OWC Operang Capal (Op. Cap. Free Cash Flow (FCF Economc Value Added (EVA Bea of a Leveraged Frm MM Value of Frm, o Corporae Taxes MM Value of Frm Wh Corporae Taxes OPAT EBIT( OWC Op. C.A. Op. C.L. Op. Cap. OWC + FA FCF OPAT e Invesmen n Op. Cap. EVA OPAT ( Op. Cap. Cos of Cap. β L β U [ + ( ( S ] V L V U S L + V L V U + Basc Fnancal e by Tmohy R. Mayes, Ph.. 8
9 Mller Value of Frm wh Personal Taxes V L V U + ( C ( S ( Mscelaneous e Margn Call Trgger Prce oe: IM% s he nal margn suppled, MM% s he manenance margn requremen, P 0 s he nal value of he porfolo IM% P M P MM% 0 Percenage gan o recover (% GTR from a loss (%L %GTR %L Basc Fnancal e by Tmohy R. Mayes, Ph.. 9
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