`Selected Financial Formulae Purpose Basic Time Value Formulae Future Value of a Single Sum Present Value of a Single Sum FV = PV  1 + i  N FV PV = ----------------- 1 + iN FV ln  ------   PV N = -------------------ln  1 + i  i =NFormulaSolve for N for a Single SumSolve for i for a Single Sum Present Value of an Ordinary AnnuityFV ­ 1 -----PV1 ­ 1   1 + i N PV A = Pmt ---------------------------------i  1 + i N ­ 1 FV A = Pmt --------------------------i 1 ­ 1   1 + i   N ­ 1PV Ad = Pmt -------------------------------------------- + Pmt iFuture Value of an Ordinary AnnuityPresent Value of an Annuity DueFuture Value of an Annuity Due Present Value of an Annuity Growing at a Constant Rate (g) Future Value of an Annuity Growing at a Constant Rate (g) Holding Period Return (single period) 1 + i N ­ 1 FV Ad = Pmt ---------------------------  1 + i  i Pmt 1 1+g N PV GA = ------------  1 ­  -----------    1 + i  i­g Pmt 1 1+g N N FV GA = ------------  1 ­  -----------    1 + i   1 + i   i­g P 1 +  Cash Flows HPR = ---------------------------------------------- ­ 1 P0Basic Financial Formulae © 1995-2011 by Timothy R. Mayes, Ph.D.1Selected Financial Formulae Purpose FormulaHolding Period Return with Reinvestment (for multiple sub-period returns)HPR Reinvest =t=1  1 + HPRt  ­ 1NBasic Security Valuation Formulae Dividend Discount Model (AKA, the Gordon Model) Two-stage Dividend Discount Model Notes: This equation is too long for one line. g1 = Growth rate during high growth phase. g2 = Growth in constant growth phase after n. n = Length of high growth phase. Assume g1 &lt;&gt; kCS and g2 &lt; kCS Three-stage Dividend Discount Model Notes: n1 = Length of high growth phase. n2 = Periods until constant growth phase. n2 = n1 + length of transistion phase. Earnings Model Constant Growth FCF Valuation ModelVOps = Value of Total Operations VDebt, VPref = Value of debt and preferred stock VNon-Ops Assets = Value of non-operating assetsD1 D0  1 + g  V CS = ----------------------- = ---------------k CS ­ g k CS ­ gD0  1 + g1  1 + g1 n V CS = -------------------------- 1 ­  ----------------- +  1 + k CS k CS ­ g 1 D0  1 + g1   1 + g2  -----------------------------------------------k CS ­ g 2 -----------------------------------------------n  1 + k CS nn1 + n2 D0 V CS = -------------------  1 + g 2  + ----------------  g 1 ­ g 2  k CS ­ g 2 2 ROE RE 1  ----------- ­ 1  k CS  EPS 1 = ------------ + -----------------------------------k CS k CS ­ gV CSFCF 1 V Ops = ---------------k CS ­ g V CS = V Ops ­ V Debt ­ V Pref + V Non ­ OpAssets g = br D V P = ---kPSustainable growth rate Note: b = retention ratio = 1 - payout ratio r = return on equity Value of a Share of Preferred StockBasic Financial Formulae © 1995-2011 by Timothy R. Mayes, Ph.D.2Selected Financial Formulae Purpose Value of a Bond on a Payment Date Quoted Price of a Bond on a Non-Payment DateVB,0 = Value of bond at last payment date  = The fraction of the current period that has elaspsedFormula 1 ­ 1   1 + kd N FV V B = Pmt ------------------------------------- + --------------------kd  1 + kd NV B  = V B 0  1 + k d  ­   Pmt Basic Statistical Formulae Arithmetic Mean (Average) 1 X = --Nt=1 N XtNGeometric Mean (used for averaging returns, growth rates, etc.)G =Nt=1  1 + Rt  ­ 1  t Xt2 NExpected Value (Weighted Average)EX =t=1 NVarianceX =2t=1 t  Xt ­ X X2Standard Deviation Coefficient of VariationX =X CV = ----------EX  X Y =Covariancet=1  t  Xt ­ X   Yt ­ Y  NCorrelation Coefficient Beta (Note: M is the market portfolio, and i is the security or portfolio) X Y r X Y = -----------X Y r i M  i  M  i M  i = ---------- = ---------------------2 2 M MBasic Financial Formulae © 1995-2011 by Timothy R. Mayes, Ph.D.3Selected Financial Formulae Purpose Portfolio Formulae Expected Return of a Portfolio E  RP  = Formulai=1 wi Ri2 2NUsing the covariance: Variance of a 2-security Portfolio  P = w 1  1 + w 2  2 + 2w 1 w 2  1 2 or, using the correlation coefficient:  P = w 1  1 + w 2  2 + 2w 1 w 2 r 1 2  1  2 Variance of an N-security portfolio Using the Covariance Standard Deviation of a Portfolio2 P 2 2 2 2 2 2 2 2=i=1 j=1  w i w j  i jP2NNP = P =Portfolio Beta 95% Value at Risk (Variance/Covariance Model)  Note: Vp is portfolio valuei=1 wi iNVaR = 1.645  V p   pCapital Market Theory Models Capital Market Line (CML) Capital Asset Pricing Model (CAPM) Note: This is also the equation for the Security Market Line (SML) Treynor's Risk-adjusted Performance Measure Sharpe's Risk-adjusted Performance Measure  E  RM  ­ Rf  E  R P  = R f +  P ------------------------------M E  Ri  = Rf + i  E  RM  ­ Rf  Ri ­ Rf T i = --------------i Ri ­ Rf S i = --------------iBasic Financial Formulae © 1995-2011 by Timothy R. Mayes, Ph.D.4Selected Financial Formulae Purpose Jensen's Alpha The Information Ratio M2 (Modigliani &amp; Modigliani) Performance Measure Fama's Risk Decomposition Notes: Ri = Portfolio Return RM = Market Return Rf = Risk-free Rate i = Portfolio Beta T = Target Beta Formula i =  Ri ­ Rf  ­ i  RM ­ Rf  RP ­ RB IR P = ----------------- RP ­ RB m 2 M =  ------  R i ­ R f  + R f  i  Risk Premium = R i ­ R f Risk =  i  R M ­ R f  Selectivity = Risk Premium ­ Risk Managers Risk =   i ­  T   R M ­ R f  Investors Risk =  T  R M ­ R f  i Diversification =  ------ ­  i  R M ­ R f   M  Net Selectivity = Selectivity ­ Diversification Brinson, Hood, and Beebower Additive Attribution Model Notes: At = Overall Allocation Effect St = Overall Selection Effect It = Overall Interaction Effect wi,t = Weight of Sector i in portfolio t bars over variables represent benchmark weights and returns. At =i=1 N  w i t ­ w i t   R i t ­ R t NSt =i=1 N wi t  Ri t ­ Ri t It =i=1  wi t ­ wi t   Ri t ­ Ri t Options and Futures Valuation Models C = SN  d 1  ­ Xe N  d 2  where: S 2 ln  --  +  r + 0.5 t  X d 1 = -------------------------------------------------- t d2 = d1 ­  t­ rtBlack-Scholes European Call Option Valuation ModelBasic Financial Formulae © 1995-2011 by Timothy R. Mayes, Ph.D.5Selected Financial Formulae Purpose Black-Scholes European Put Option Valuation Model (see above for d1 and d2) Put-Call Parity for European Options with No Cash Flows Formula P = Xe ­ rt N  ­ d 2  ­ SN  ­ d 1  C = P + S ­ Xe ­ rt or, P = C + Xe ­ rt ­ S pC u +  1 ­ p C d C = --------------------------------------1 + r where, r­d p = ----------u­d pP u +  1 ­ p P d P = -------------------------------------1 + r where, r­d p = ----------u­dT F0Single-period Binomial Option Pricing Model for Call Options (r is the risk-free rate, u is the up factor, and d is the down factor)Single-period Binomial Option Pricing Model for Put OptionsCost of Carry Model for Pricing Futures Contracts (CC is the carrying costs as a % of the spot price)= S 0 e CC  t Bond Analysis Formulae Macaulay's Duration on a Payment Date (for immunization). Note: Ct is the cash flow in period t, i is the yield to maturity Modified Duration (for price volatility) on a Payment Date Ct  t  ---------------  1 + i -t t=1 D = -------------------------Bond Price D D Mod = --------------1 + i Cf t 1 -----------------   t 2 + t  --------------- 1 + i 2 t = 1  1 + i t C = -------------------------------------------------------------------Bond Pricet + jRn N NConvexity on a Payment DateThe n-period forward rate given two spot rates (note that i &gt; j, and n = i - j)=n 1 + Ri  -------------------- ­ 1 j  1 + Rj iBasic Financial Formulae © 1995-2011 by Timothy R. Mayes, Ph.D.6Selected Financial Formulae Purpose Bank Discount Yield for discount securities (FV = face value, PP = purchase price, m = periods per year) Bond Equivalent Yield for discount securities (see definitions for BDY) Formula FV ­ PP 360 BDY = --------------------  -------FV m FV ­ PP 365 FV 365 BEY = --------------------  -------- = BDY  ------  -------PP m PP 360Capital Budgeting Decision Formulae Net Present Value (NPV) NPV =t=1 N -----------------t ­ IO 1 + iCf tNCf tProfitability Index (PI)t=1 NPV + IO NPV PI = -------------------------- = ------------------------ = ----------- + 1 IO IO IO N -----------------t 1 + iCf tInternal Rate of Return (IRR). Note: This is a trial and error procedure to find the i that makes the equality hold (i.e., what discount rate makes the NPV = 0).0 =t=1 -----------------t ­ IO 1 + i  Cft  1 + i  t=1N N ­ tModified Internal Rate of Return (MIRR).MIRR =N--------------------------------------------- ­ 1 IOStock Market Index Construction Formulae Price-weighted Average (e.g., DJIA) Note: The divisor (Div) at period 0 is equal to the number of stocks in the average. It will be adjusted for stock splits or any other corporate action that results in a non-economic change in the stock price.PWA t = ------------Div t1 Pj j=NBasic Financial Formulae © 1995-2011 by Timothy R. Mayes, Ph.D.7Selected Financial Formulae Purpose Capitalization-weighted Index (e.g., S&amp;P 500) Note: The divisor (Div) at period 0 is the divisor that makes the initial level of the index equal to the desired starting point. It will be adjusted for any corporate action that results in a change in market capitalization. Equally-weighted Arithmetic Index (e.g., VLA) Note: At period 0 the index is set to some starting value (e.g., 100). To calculate the index for any day, multiply the average % change by the previous index level. Equally-weighted Geometric Index (e.g., VLG) Note: See note aboveNFormulaCWI t = -------------------Div tj=1 Pj QjEWAI t = EWAI t ­ 1 j=1  ---------------  N  P j t ­ 1NNP j tEWGI t = EWGI t ­ 1  Nj=1 --------------P j t ­ 1P j tCorporate Financial Formulae Net Operating Profit After Taxes (NOPAT) Net Operating Working Capital (NOWC) Operating Capital (Op. Cap.) Free Cash Flow (FCF) Economic Value Added (EVA) Beta of a Leveraged Firm MM Value of Firm, No Corporate Taxes MM Value of Firm With Corporate Taxes Miller Value of Firm with Personal Taxes NOPAT = EBIT  1 ­ t  NOWC = Op. C.A. ­ Op. C.L. Op. Cap. = NOWC + NFA FCF = NOPAT ­ Net Investment in Op. Cap. EVA = NOPAT ­  Op. Cap.  Cost of Cap.  L = U  1 +  1 ­ t   D  S   VL = VU = SL + D V L = V U + tD  1 ­ tC   1 ­ tS  V L = V U + 1 ­ ------------------------------------ D  1 ­ tD Miscelaneous FormulaeBasic Financial Formulae © 1995-2011 by Timothy R. Mayes, Ph.D.8Selected Financial Formulae Purpose Margin Call Trigger Price Note: IM% is the initial margin supplied, MM% is the maintenance margin requirement, P0 is the initial value of the portfolio Percentage gain to recover (% GTR) from a loss (%L) FormulaIM% ­ 1P M = -----------------------  P 0 MM% ­ 11 %GTR = ---------------- ­ 1 1 ­ %LBasic Financial Formulae © 1995-2011 by Timothy R. Mayes, Ph.D.9`

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