Equations

Net Present Value (NPV) Definition (3.1)

NPV = PV(Benefits) - PV(Costs)

Net Present Value of Project Cash Flows (3.2)

NPV = PV(All project cash flows)

No-Arbitrage Price of a Security (3.3)

Price(Security) = PV(All cash flows paid by the security)

Expected Return of a Risky Investment (3A.1)

Expected return = Expected gain at end of year / Initial cost

Discount Rate for Risky Cash Flows (3A.2)

rs = rf + risk premium for investment s

Future Value of a Cash Flow (4.1)

FV_n = C × (1 + r)^n

Present Value of a Cash Flow (4.2)

PV = C / (1 + r)^n

Present Value of a Cash Flow Stream (4.3)

PV = Σ (C_n / (1 + r)^n) from n=0 to N

Future Value of a Cash Flow Stream (4.4)

FV_n = PV × (1 + r)^n

Net Present Value (NPV) (4.5)

NPV = PV(Benefits) - PV(Costs)

Net Present Value of a Cash Flow Stream (4.6)

NPV = PV(Benefits) - PV(Costs)

Present Value of a Perpetuity (4.7)

PV = C / r

Present Value of an Annuity (4.9)

PV = C × (1/r) × (1 - 1/(1 + r)^N)

Present Value of an Annuity Due (4.9, alternative form)

PV = C × (1/r) × (1 - 1/(1 + r)^N) × (1 + r)

Future Value of an Annuity (4.10)

FV = C × (1/r) × ((1 + r)^N - 1)

Present Value of a Growing Perpetuity (4.11)

PV = C / (r - g)

Present Value of a Growing Annuity (4.12)

PV = C × (1/(r - g)) × (1 - ((1 + g)/(1 + r))^N)

Loan or Annuity Payment (4.14)

C = P / ((1/r) × (1 - 1/(1 + r)^N))

IRR with Two Cash Flows (4.15)

IRR = (FV/P)^(1/N) - 1

IRR of a Growing Perpetuity (4.16)

IRR = C/P + g

General Equation for Discount Rate Period Conversion (5.1)

Equivalent n-Period Discount Rate = (1 + r)^n - 1

Interest Rate per Compounding Period (5.2)

Interest Rate per Compounding Period = APR / k periods/year

Converting an APR to an EAR (5.3)

1 + EAR = (1 + APR/k)^k

Growth in Purchasing Power (5.4)

Growth in Purchasing Power = (1 + r)/(1 + i)

Real Interest Rate (5.5)

r_r = (r - i)/(1 + i) ≈ r - i

Present Value of a Risk-Free Cash Flow (5.6)

PV = Cn / (1 + rn)^n

Present Value of a Cash Flow Stream Using a Term Structure of Discount Rates (5.7)

PV = Σ (Cn / (1 + rn)^n) from n=1 to N

After-Tax Interest Rate (5.8)

r - τ × r = r(1 - τ)

Coupon Payment (6.1)

CPN = (Coupon Rate × Face Value) / Number of Coupon Payments per Year

Yield to Maturity of an n-Year Zero-Coupon Bond (6.3)

YTM_n = (FV/P)^(1/n) - 1

Risk-Free Interest Rate with Maturity n (6.4)

rn = YTMn

Yield to Maturity of a Coupon Bond (6.5)

P = CPN × (1/y) × (1 - 1/(1 + y)^N) + FV/(1 + y)^N

Price of a Coupon Bond (6.6)

P = CPN/(1 + YTM1) + CPN/(1 + YTM2)^2 + … + (CPN + FV)/(1 + YTM_n)^n

Forward Rate for Year 1 (6A.1)

f1 = YTM1

General Formula for Forward Interest Rate (6A.2)

fn = ((1 + YTMn)^n / (1 + YTM_{n-1})^(n-1)) - 1

Zero-Coupon Yield from Forward Rates (6A.3)

(1 + f1) × (1 + f2) × … × (1 + fn) = (1 + YTMn)^n

Expected Future Spot Interest Rate (6A.4)

Expected Future Spot Interest Rate = Forward Interest Rate + Risk Premium

Net Present Value of a Perpetual Cash Flow Stream (7.1)

NPV = -250 + 35/r

Net Present Value of John Star's Book Deal (7.2)

NPV = 1,000,000 - 500,000/(1 + r) - 500,000/(1 + r)^2 - 500,000/(1 + r)^3

Net Present Value of John Star's New Book Offer with Royalties (7.3)

NPV = 550,000 - 500,000/(1 + r) - 500,000/(1 + r)^2 - 500,000/(1 + r)^3 + 1,000,000/(1 + r)^4

Profitability Index (7.4)

NPV / Resource Consumed

Income Tax Calculation (8.1)

Income Tax = EBIT × τ_c

Unlevered Net Income Calculation (8.2)

Unlevered Net Income = EBIT × (1 - τ_c)

Free Cash Flow (8.6)

Free Cash Flow = (Revenues - Costs) × (1 - τc) - CapEx - ΔNWC + τc × Depreciation

Present Value of Free Cash Flow (8.7)

PV(FCFt) = FCFt / (1 + r)^t

Net Present Value (NPV) Example for HomeNet (8.8)

NPV = -19,500 + 7,000/1.12 + 9,100/1.12^2 + 9,100/1.12^3 + 9,100/1.12^4 + 2,400/1.12^5

Stock Price for a One-Year Investor (9.1)

P0 = (Div1 + P1) / (1 + rE)

Total Return of a Stock (9.2)

rE = (Div1 + P1)/P0 - 1 = Dividend Yield + Capital Gain Rate

Stock Price for a Two-Year Investor (9.3)

P0 = Div1/(1 + rE) + (Div2 + P2)/(1 + rE)^2

General Dividend-Discount Model (9.4)

P0 = Div1/(1 + rE) + Div2/(1 + rE)^2 + … + DivN/(1 + rE)^N + PN/(1 + r_E)^N

Dividend-Discount Model for Infinite Horizon (9.5)

P0 = Σ (Divn / (1 + r_E)^n) from n=1 to ∞

Constant Dividend Growth Model (9.6)

P0 = Div1 / (r_E - g)

Equity Cost of Capital with Constant Growth (9.7)

rE = (Div1 / P_0) + g

Dividend Per Share (9.8)

Divt = EPSt × Dividend Payout Rate_t

Change in Earnings (9.9)

Change in Earnings = New Investment × Return on New Investment

New Investment (9.10)

New Investment = Earnings × Retention Rate

Earnings Growth Rate (9.11)

Earnings Growth Rate = Retention Rate × Return on New Investment

Dividend Growth Rate (Sustainable Growth Rate) (9.12)

g = Retention Rate × Return on New Investment

Terminal Stock Price with Constant Long-Term Growth (9.13)

PN = Div{N+1} / (r_E - g)

Dividend-Discount Model with Constant Long-Term Growth (9.14)

P0 = Div1/(1 + rE) + Div2/(1 + rE)^2 + … + DivN/(1 + rE)^N + (Div{N+1} / (rE - g))/(1 + rE)^N

Total Payout Model (9.15)

P0 = PV(Future Total Dividends and Repurchases) / Shares Outstanding0

Free Cash Flow Estimation (9.20)

Free Cash Flow = EBIT × (1 - τ_c) - Net Investment - Increases in Net Working Capital

Enterprise Value (9.21)

V_0 = PV(Future Free Cash Flow of Firm)

Stock Price from Enterprise Value (9.22)

P0 = (V0 + Cash0 - Debt0) / Shares Outstanding_0

Terminal Enterprise Value (9.24)

VN = FCF{N+1} / (rwacc - gFCF)

Expected (Mean) Return (10.1)

E[R] = Σ (p_R × R)

Variance of the Return Distribution (10.2)

Var(R) = E[(R - E[R])^2] = Σ (p_R × (R - E[R])^2)

Standard Deviation of the Return Distribution (10.3)

SD(R) = √Var(R)

Realized Return (10.4)

R{t+1} = (Div{t+1} + P{t+1})/Pt - 1 = Dividend Yield + Capital Gain Rate

Annual Realized Return with Dividend Reinvestment (10.5)

1 + Rannual = (1 + R1) × (1 + R2) × (1 + R3) × (1 + R_4)

Average Annual Return of a Security (10.6)

\bar{R} = (1/T) × Σ R_t

Variance Estimate Using Realized Returns (10.7)

Var(R) = (1/(T-1)) × Σ (R_t - \bar{R})^2

Standard Error of the Estimate of the Expected Return (10.8)

SD(Average of Independent, Identical Risks) = SD(Individual Risk) / √Number of Observations

95% Confidence Interval for Expected Return (10.9)

Historical Average Return ± 2 × Standard Error

Excess Return (10.5, implied)

Excess Return = \bar{R} - r_f

Compound Annual Return (10.10)

Compound Annual Return = [(1 + R1) × (1 + R2) × … × (1 + R_T)]^(1/T) - 1

Cost of Capital Using CAPM (10.11)

rI = rf + βI × (E[RMkt] - r_f)

Market Risk Premium (10.12)

Market Risk Premium = E[RMkt] - rf

Portfolio Return (11.2)

RP = Σ (xi × R_i)

Expected Return of a Portfolio (11.3)

E[RP] = Σ (xi × E[R_i])

Covariance Between Returns (11.4)

Cov(Ri, Rj) = E[(Ri - E[Ri])(Rj - E[Rj])]

Estimate of Covariance from Historical Data (11.5)

Cov(Ri, Rj) = (1/(T-1)) × Σ (R{i,t} - \bar{R}i)(R{j,t} - \bar{R}j)

Correlation Between Returns (11.6)

Corr(Ri, Rj) = Cov(Ri, Rj) / (SD(Ri) × SD(Rj))

Variance of a Two-Stock Portfolio (11.8)

Var(RP) = x1^2 Var(R1) + x2^2 Var(R2) + 2 x1 x2 Cov(R1, R_2)

Variance of a Two-Stock Portfolio with Correlation (11.9)

Var(RP) = x1^2 SD(R1)^2 + x2^2 SD(R2)^2 + 2 x1 x2 Corr(R1, R2) SD(R1) SD(R_2)

Variance of a Portfolio (11.10)

Var(RP) = Cov(RP, RP) = Σ xi Cov(Ri, RP)

Variance of a Portfolio with Pairwise Covariances (11.11)

Var(RP) = Σ Σ xi xj Cov(Ri, R_j)

Variance of an Equally Weighted Portfolio (11.12)

Var(R_P) = (1/n) × Average Variance + ((n-1)/n) × Average Covariance

Volatility of a Complete Portfolio (11.15)

SD(RC) = x × SD(RP)

Sharpe Ratio (11.16)

Sharpe Ratio = (E[RP] - rf) / SD(R_P)

Beta of an Investment with a Portfolio (11.17)

βi^P = (SD(Ri) × Corr(Ri, RP)) / SD(R_P)

Required Return for an Investment (11.18)

ri = rf + βi^P × (E[RP] - r_f)

Expected Return of an Efficient Portfolio (11.19)

E[Ri] = rf + βi^P × (E[RP] - r_f)

Security Market Line (SML) (11.22)

E[Ri] = rf + βi × (E[RMkt] - r_f)

Beta with Respect to the Market Portfolio (11.23)

βi = Cov(Ri, RMkt) / Var(RMkt)

Beta of a Portfolio (11.24)

βP = Σ (xi × β_i)

Security Market Line with Differing Interest Rates (11A.1)

E[Ri] = r* + βi × (E[R_Mkt] - r*)

CAPM Equation for the Cost of Capital (12.1)

ri = rf + βi × (E[RMkt] - r_f)

Market Capitalization (12.2)

MV_i = Number of Shares Outstanding × Price per Share

Portfolio Weights in a Value-Weighted Portfolio (12.3)

xi = MVi / Σ MV_j