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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 =

N

Formula

Solve for N for a Single Sum

Solve for i for a Single Sum Present Value of an Ordinary Annuity

FV ­ 1 -----PV

1 ­ 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 i

Future Value of an Ordinary Annuity

Present Value of an Annuity Due

Future 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 P0

Basic Financial Formulae © 1995-2011 by Timothy R. Mayes, Ph.D.

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Selected Financial Formulae Purpose Formula

Holding Period Return with Reinvestment (for multiple sub-period returns)

HPR Reinvest =

t=1

1 + HPRt ­ 1

N

Basic 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 <> kCS and g2 < 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 Model

VOps = Value of Total Operations VDebt, VPref = Value of debt and preferred stock VNon-Ops Assets = Value of non-operating assets

D1 D0 1 + g V CS = ----------------------- = ---------------k CS ­ g k CS ­ g

D0 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

n

n1 + 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 ­ g

V CS

FCF 1 V Ops = ---------------k CS ­ g V CS = V Ops ­ V Debt ­ V Pref + V Non ­ OpAssets g = br D V P = ---kP

Sustainable growth rate Note: b = retention ratio = 1 - payout ratio r = return on equity Value of a Share of Preferred Stock

Basic Financial Formulae © 1995-2011 by Timothy R. Mayes, Ph.D.

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Selected Financial Formulae Purpose Value of a Bond on a Payment Date Quoted Price of a Bond on a Non-Payment Date

VB,0 = Value of bond at last payment date = The fraction of the current period that has elaspsed

Formula 1 ­ 1 1 + kd N FV V B = Pmt ------------------------------------- + --------------------kd 1 + kd N

V B = V B 0 1 + k d ­ Pmt

Basic Statistical Formulae Arithmetic Mean (Average) 1 X = --N

t=1 N

Xt

N

Geometric Mean (used for averaging returns, growth rates, etc.)

G =

N

t=1

1 + Rt ­ 1 t Xt

2 N

Expected Value (Weighted Average)

EX =

t=1 N

Variance

X =

2

t=1

t Xt ­ X

X

2

Standard Deviation Coefficient of Variation

X =

X CV = ----------EX X Y =

Covariance

t=1

t Xt ­ X Yt ­ Y

N

Correlation 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 M

Basic Financial Formulae © 1995-2011 by Timothy R. Mayes, Ph.D.

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Selected Financial Formulae Purpose Portfolio Formulae Expected Return of a Portfolio E RP = Formula

i=1

wi Ri

2 2

N

Using 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 Portfolio

2 P 2 2 2 2 2 2 2 2

=

i=1 j=1

w i w j i j

P

2

N

N

P = P =

Portfolio Beta 95% Value at Risk (Variance/Covariance Model) Note: Vp is portfolio value

i=1

wi i

N

VaR = 1.645 V p p

Capital 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 = --------------i

Basic Financial Formulae © 1995-2011 by Timothy R. Mayes, Ph.D.

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Selected Financial Formulae Purpose Jensen's Alpha The Information Ratio M2 (Modigliani & 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

N

St =

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

­ rt

Black-Scholes European Call Option Valuation Model

Basic Financial Formulae © 1995-2011 by Timothy R. Mayes, Ph.D.

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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 = ----------u­d

T F0

Single-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 Options

Cost 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 Price

t + jRn N N

Convexity on a Payment Date

The n-period forward rate given two spot rates (note that i > j, and n = i - j)

=

n

1 + Ri -------------------- ­ 1 j 1 + Rj

i

Basic Financial Formulae © 1995-2011 by Timothy R. Mayes, Ph.D.

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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 360

Capital Budgeting Decision Formulae Net Present Value (NPV) NPV =

t=1 N

-----------------t ­ IO 1 + i

Cf t

N

Cf t

Profitability Index (PI)

t=1 NPV + IO NPV PI = -------------------------- = ------------------------ = ----------- + 1 IO IO IO N

-----------------t 1 + i

Cf t

Internal 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=1

N N ­ t

Modified Internal Rate of Return (MIRR).

MIRR =

N

--------------------------------------------- ­ 1 IO

Stock 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 t

1 Pj j=

N

Basic Financial Formulae © 1995-2011 by Timothy R. Mayes, Ph.D.

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Selected Financial Formulae Purpose Capitalization-weighted Index (e.g., S&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 above

N

Formula

CWI t = -------------------Div t

j=1

Pj Qj

EWAI t = EWAI t ­ 1

j=1

--------------- N P j t ­ 1

N

N

P j t

EWGI t = EWGI t ­ 1 N

j=1

--------------P j t ­ 1

P j t

Corporate 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 Formulae

Basic Financial Formulae © 1995-2011 by Timothy R. Mayes, Ph.D.

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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) Formula

IM% ­ 1P M = ----------------------- P 0 MM% ­ 1

1 %GTR = ---------------- ­ 1 1 ­ %L

Basic Financial Formulae © 1995-2011 by Timothy R. Mayes, Ph.D.

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