Formulae[1]

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Basic Financial Formulae 1995-2008 by Timothy R. Mayes, Ph.D. 1

Selected Financial Formulae

Purpose Formula

Basic Time Value FormulaeFuture Value of a Single Sum

Present Value of a Single Sum

Solve for N for a Single Sum

Solve for i for a Single Sum

Present Value of an Ordinary Annuity

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 aConstant Rate (g)

Future Value of an Annuity Growing at aConstant Rate (g)

Holding Period Return (single period)

FV PV 1 i+( ) N =

PV FV 1 i+( ) N

-------------------=

N

FV PV ------- ln

1 i+( )ln---------------------=

i FV PV ------- N 1 =

PV A Pmt 1 1 1 i+( ) N

i-----------------------------------=

FV A Pmt 1 i+( ) N 1

i----------------------------=

PV Ad Pmt 1 1 1 i+( ) N 1 ( )

i--------------------------------------------- Pmt +=

FV Ad Pmt 1 i+( ) N 1

i---------------------------- 1 i+( )=

PV GA Pmt 1i g ------------ 1 1 g +

1 i+------------ N

=

FV GA Pmt 1i g ------------ 1 1 g +

1 i+------------ N

1 i+( ) N =

HPR P 1 Cash Flows+

P 0----------------------------------------------- 1 =

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

Basic Security Valuation Formulae

Dividend Discount Model (AKA, the GordonModel)

Two-stage Dividend Discount Model Notes: This equation is too long for one line. g

1= Growth rate during high growth phase.

g 2 = Growth in constant growth phase after n.n = Length of high growth phase.Assume g 1 k CS and g 2 < k CS

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 OperationsVDebt , V Pref = Value of debt and preferred stock V Non-Ops Assets = Value of non-operating assets

Sustainable growth rate Note: b = retention ratio = 1 - payout ratior = return on equity

Value of a Share of Preferred Stock


Purpose Formula

HPR Reinvest 1 HPR t +( ) 1 t 1=

N

=

V CS D 0 1 g +( )

k CS g ------------------------

D 1k CS g -----------------= =

V CS D 0 1 g 1+( )

k CS g 1 -------------------------- 1

1 g 1+1 k CS +----------------- n =

D 0 1 g 1+( )n 1 g 2+( )k CS g 2

-------------------------------------------------

1 k CS +( )n

-------------------------------------------------+

V CS D 0

k CS g 2 ------------------- 1 g 2+( )

n1 n2+2

----------------- g 1 g 2 ( )+=

V CS EPS 1

k CS -------------

RE 1 ROE

k CS ------------ 1

k CS g -------------------------------------+=

V Op s FCF 1k CS g -----------------=

V CS V Ops V Debt V Pref V N on O pAs set s + =

g br =

V P Dk P -----=

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Value of a Bond on a Payment Date

Quoted Price of a Bond on a Non-PaymentDateVB,0 = Value of bond at last payment date = The fraction of the current period that has elaspsed

Basic Statistical Formulae

Arithmetic Mean (Average)

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

Expected Value (Weighted Average)

Variance

Standard Deviation

Covariance

Correlation Coefficient

Beta (Note: M is the market portfolio, and i isthe security or portfolio)


Purpose Formula

V B Pmt 1 1 1 k d +( ) N

k d -------------------------------------- FV

1 k d +( ) N

----------------------+=

V B , V B 0, 1 k d +( ) Pmt ( ) =

X 1 N ---- X t

t 1=

N

=

G 1 Rt +( )t 1=

N

N 1 =

E X ( ) t X t t 1=

N

=

X 2 t X t X ( )2

t 1=

N

=

X X 2=

X Y , t X t X ( ) Y t Y ( )[ ]t 1=

N

=

r X Y , X Y , X Y -------------=

ii M ,M

2-----------

r i M , iM M

2-----------------------= =

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Portfolio Formulae

Expected Return of a Portfolio

Variance of a 2-security Portfolio Using the covariance:

or, using the correlation coefficient:

Variance of an N-security portfolio Using the

Covariance

Standard Deviation of a Portfolio

Portfolio Beta

95% Value at Risk (Variance/Covariance

Model) Note: V p is portfolio value

Capital Market Theory Models

Capital Market Line (CML)

Capital Asset Pricing Model (CAPM) Note: This is also the equation for the SecurityMarket Line (SML)

Treynors Risk-adjusted PerformanceMeasure

Sharpes Risk-adjusted Performance Measure


Purpose Formula

E R P ( ) wi R ii 1=

N

=

P 2 w1

212 w2

222 2w1w21 2,+ +=

P 2 w1

212 w2

222 2w1w2r 1 2, 12+ +=

P 2 wiw ji j, j 1=

N

i 1= N

=

P P 2=

P wiii 1=

N

=

VaR 1.645 V p p=

E R P ( ) R f P E R M ( ) R f ( )

M --------------------------------+=

E R i( ) R f i E R M ( ) R f ( )+=

T i Ri R f

i----------------=

S i Ri R f

i----------------=

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Jensens Alpha

The Information Ratio

M2 (Modigliani & Modigliani) PerformanceMeasure

Famas Risk Decomposition Notes: Ri = Portfolio Return RM = Market Return R f = Risk-free Ratei = Portfolio BetaT = Target Beta

Brinson, Hood, and Beebower AdditiveAttribution Model

Notes:A

t= Overall Allocation Effect

St = Overall Selection EffectIt = Overall Interaction Effectw i,t = Weight of Sector i in portfolio t

bars over variables represent benchmark weights and returns.

Options and Futures Valuation Models

Black-Scholes European Call OptionValuation Model

where:


Purpose Formula

i R i R f ( ) i RM R f ( ) =

IR P R P R B R P R B -------------------=

M 2mi------- Ri R f ( ) R f +=

Risk Premium Ri R f =

Risk i RM R f ( )=Selectivity Risk Premium Risk =

Managers Risk i T ( ) RM R f ( )=Investors Risk T RM R f ( )=

DiversificationiM ------- i RM R f ( )=

Net Selectivity Selectivity Diversification =

At wi t , wi t , ( ) Ri t , Rt ( )i 1=

N

=

S t wi t , R i t , Ri t , ( )i 1=

N

=

I t wi t , wi t , ( ) Ri t , R i t , ( )i 1=

N

=

C SN d 1( ) Xert N d 2( ) =

d 1

S X --- ln r 0.5 2+( )t +

t ----------------------------------------------------=

d 2 d 1 t =

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Black-Scholes European Put OptionValuation Model (see above for d 1 and d 2)

Put-Call Parity for European Options with NoCash Flows or,

Single-period Binomial Option Pricing Modelfor Call Options ( r is the risk-free rate, u is theup factor, and d is the down factor) where,

Single-period Binomial Option Pricing Modelfor Put Options

where,

Cost of Carry Model for Pricing FuturesContracts (CC is the carrying costs as a % of the spot price)

Bond Analysis Formulae

Macaulays Duration on a Payment Date (for immunization). Note: C t is the cash flow in

period t , i is the yield to maturity

Modified Duration (for price volatility) on aPayment Date

Convexity on a Payment Date

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


Purpose Formula

P Xe rt N d 2 ( ) SN d 1 ( ) =

C P S Xe rt +=

P C Xe rt S +=

C pC u 1 p ( )C d +

1 r +( )---------------------------------------=

p r d u d ------------=

P pP u 1 p ( ) P d +

1 r +( )---------------------------------------=

p r d u d ------------=

F T 0 S 0eCC t ( )=

D

C t t ( )1 i+( )t

-----------------t 1=

N

Bond Price---------------------------=

D Mo d D

1 i+( )---------------=

C

11 i+( )2

------------------t 2

t +( )

Cf t 1 i+( )t

-----------------t 1=

N

Bond Price

---------------------------------------------------------------------=

Rt j+ n1 R i+( )

i

1 R j+( ) j

--------------------n=

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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)

Capital Budgeting Decision Formulae

Net Present Value ( NPV )

Profitability Index ( PI )

Internal Rate of Return ( IRR). Note: This is atrial and error procedure to find the i thatmakes the equality hold (i.e., what discountrate makes the NPV = 0).

Modified Internal Rate of Return ( MIRR).

Stock Market Index Construction Formulae

Price-weighted Average (e.g., DJIA) Note: The divisor (Div) at period 0 is equal tothe number of stocks in the average. It will beadjusted for stock splits or any other corporateaction that results in a non-economic changein the stock price.

Capitalization-weighted Index (e.g., S&P500)

Note: The divisor (Div) at period 0 is thedivisor that makes the initial level of the indexequal to the desired starting point. It will beadjusted for any corporate action that resultsin a change in market capitalization.


Purpose Formula

BDY FV PP FV

--------------------- 360m

---------=

BEY FV PP PP

--------------------- 365m

--------- BDY FV PP ------- 365

360---------= =

NPV Cf t

1 i+( )t -----------------

t 1=

N

IO =

PI

Cf t 1 i+( )t

-----------------t 1=

N

IO

--------------------------- NPV IO+ IO

------------------------- NPV IO------------ 1+= = =

0Cf t

1 i+( )t -----------------

t 1=

N

IO =

MIRRCf t 1 i+( ) N t ( )

t 1=

N

IO

---------------------------------------------- N 1 =

PWA t

P j j 1=

N

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

CWI t

P jQ j j 1=

N

Div t

--------------------=

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Equally-weighted Arithmetic Index (e.g.,VLA)

Note: At period 0 the index is set to somestarting value (e.g., 100). To calculate theindex for any day, multiply the average %change by the previous index level.

Equally-weighted Geometric Index (e.g.,VLG)

Note: See note above

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

Merton Value of Firm with Personal Taxes

Miscelaneous Formulae

Margin Call Trigger Price Note: IM% is the initial margin supplied,MM% is the maintenance marginrequirement, P 0 is the initial value of the

portfolio

Percentage gain to recover (% GTR) from aloss (%L)


Purpose Formula

EWAI t

EWAI t 1

P j t ,

P j t 1 ,--------------- N

j 1=

N

=

EWGI t EWGI t 1 P j t ,

P j t 1 ,---------------

j 1=

N

N =

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 ( )+[ ]=

V L

V U

S L

D+= =

V L V U tD+=

V L V U 11 t C ( ) 1 t S ( )

1 t D ( )------------------------------------- +=

P M IM% 1

MM% 1 ------------------------ P 0=

%GTR 11 %L ----------------- 1 =

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Formulae[1]