Formula Cheat Sheet — SnapFi by Netmark

Single Sum (Lump Sum)

FV of Lump Sum (Annual Compounding)

FV = PV \times (1 + k)^n

Solves for: FV

FV of Lump Sum (Periodic Compounding)

FV = PV \times \left(1 + \frac{k}{m}\right)^{n \times m}

Solves for: FV

FV of Lump Sum (Continuous Compounding)

FV = PV \times e^{k \times n}

Solves for: FV

PV of Lump Sum (Annual Compounding)

PV = FV \times (1 + k)^{-n}

Solves for: PV

PV of Lump Sum (Periodic Compounding)

PV = FV \times \left(1 + \frac{k}{m}\right)^{-(n \times m)}

Solves for: PV

PV of Lump Sum (Continuous Compounding)

PV = FV \times e^{-(k \times n)}

Solves for: PV

Simple Interest

I = PV \times k \times n

Solves for: SimpleInterest

Discount Factor

DF = \frac{1}{(1 + k)^n}

Solves for: DF

Compound Annual Growth Rate (CAGR)

CAGR = \left(\frac{V_{\text{end}}}{V_{\text{begin}}}\right)^{1/n} - 1

Solves for: CAGR

Rule of 72 (Doubling Time)

n_{\text{double}} \approx \frac{72}{100 \, k}

Solves for: DoublingYears

Ordinary Annuity

FV of Ordinary Annuity (Annual Compounding)

FVA = PMT \times \frac{(1 + k)^n - 1}{k}

Solves for: FVA

FV of Ordinary Annuity (Periodic Compounding)

FVA = PMT \times \frac{\left(1 + \frac{k}{m}\right)^{n \times m} - 1}{\frac{k}{m}}

Solves for: FVA

FV of Ordinary Annuity (Continuous Compounding)

FVA = PMT \times \frac{e^{k \times n} - 1}{e^{k} - 1}

Solves for: FVA

PV of Ordinary Annuity (Annual Compounding)

PVA = PMT \times \frac{1 - (1 + k)^{-n}}{k}

Solves for: PVA

PV of Ordinary Annuity (Periodic Compounding)

PVA = PMT \times \frac{1 - \left(1 + \frac{k}{m}\right)^{-(n \times m)}}{\frac{k}{m}}

Solves for: PVA

PV of Ordinary Annuity (Continuous Compounding)

PVA = PMT \times \frac{1 - e^{-(k \times n)}}{e^{k} - 1}

Solves for: PVA

Payment from PV (Annual Compounding)

PMT = PV \times \frac{k}{1 - (1 + k)^{-n}}

Solves for: PMT

Payment from PV (Periodic Compounding)

PMT = PV \times \frac{\frac{k}{m}}{1 - \left(1 + \frac{k}{m}\right)^{-(n \times m)}}

Solves for: PMT

Payment from FV (Annual Compounding)

PMT = FV \times \frac{k}{(1 + k)^n - 1}

Solves for: PMT

Payment from FV (Periodic Compounding)

PMT = FV \times \frac{\frac{k}{m}}{\left(1 + \frac{k}{m}\right)^{n \times m} - 1}

Solves for: PMT

Payment from PV (Continuous Compounding)

PMT = PV \times \frac{e^{k} - 1}{1 - e^{-k \times n}}

Solves for: PMT

Payment from FV (Continuous Compounding)

PMT = FV \times \frac{e^{k} - 1}{e^{k \times n} - 1}

Solves for: PMT

Annuity Due

FV of Annuity Due (Annual Compounding)

FVA = PMT \times \frac{(1 + k)^n - 1}{k} \times (1 + k)

Solves for: FVA

FV of Annuity Due (Periodic Compounding)

FVA = PMT \times \frac{\left(1 + \frac{k}{m}\right)^{n \times m} - 1}{\frac{k}{m}} \times \left(1 + \frac{k}{m}\right)

Solves for: FVA

PV of Annuity Due (Annual Compounding)

PVA = PMT \times \frac{1 - (1 + k)^{-n}}{k} \times (1 + k)

Solves for: PVA

PV of Annuity Due (Periodic Compounding)

PVA = PMT \times \frac{1 - \left(1 + \frac{k}{m}\right)^{-(n \times m)}}{\frac{k}{m}} \times \left(1 + \frac{k}{m}\right)

Solves for: PVA

Payment from PV (Annual Compounding)

PMT = PV \times \frac{k}{1 - (1 + k)^{-n}} \times \frac{1}{1 + k}

Solves for: PMT

Payment from PV (Periodic Compounding)

PMT = PV \times \frac{\frac{k}{m}}{1 - \left(1 + \frac{k}{m}\right)^{-(n \times m)}} \times \frac{1}{1 + \frac{k}{m}}

Solves for: PMT

Payment from FV (Annual Compounding)

PMT = FV \times \frac{k}{(1 + k)^n - 1} \times \frac{1}{1 + k}

Solves for: PMT

Payment from FV (Periodic Compounding)

PMT = FV \times \frac{\frac{k}{m}}{\left(1 + \frac{k}{m}\right)^{n \times m} - 1} \times \frac{1}{1 + \frac{k}{m}}

Solves for: PMT

FV of Annuity Due (Continuous Compounding)

FVA = PMT \times \frac{e^{k \times n} - 1}{e^{k} - 1} \times e^{k}

Solves for: FVA

PV of Annuity Due (Continuous Compounding)

PVA = PMT \times \frac{1 - e^{-k \times n}}{e^{k} - 1} \times e^{k}

Solves for: PVA

Payment from PV (Continuous Compounding)

PMT = PV \times \frac{e^{k} - 1}{1 - e^{-k \times n}} \times \frac{1}{e^{k}}

Solves for: PMT

Payment from FV (Continuous Compounding)

PMT = FV \times \frac{e^{k} - 1}{e^{k \times n} - 1} \times \frac{1}{e^{k}}

Solves for: PMT

Growing Annuity

PV of Growing Annuity

PVA = \frac{PMT}{k - g} \times \left[1 - \left(\frac{1 + g}{1 + k}\right)^n\right]

Solves for: PVA

PV of Growing Annuity Due

PVA = \frac{PMT}{k - g} \times \left[1 - \left(\frac{1 + g}{1 + k}\right)^n\right] \times (1 + k)

Solves for: PVA

FV of Growing Annuity

FVA = \frac{PMT}{k - g} \times \left[(1 + k)^n - (1 + g)^n\right]

Solves for: FVA

FV of Growing Annuity Due

FVA = \frac{PMT}{k - g} \times \left[(1 + k)^n - (1 + g)^n\right] \times (1 + k)

Solves for: FVA

PV of Growing Annuity (Continuous Compounding)

PVA = PMT \,e^{-k}\sum_{j=0}^{n-1} e^{(g-k)j}

Solves for: PVA

PV of Growing Annuity Due (Continuous Compounding)

PVA = PMT \sum_{j=0}^{n-1} e^{(g-k)j}

Solves for: PVA

FV of Growing Annuity (Continuous Compounding)

FVA = PMT \,e^{kn}\,e^{-k}\sum_{j=0}^{n-1} e^{(g-k)j}

Solves for: FVA

FV of Growing Annuity Due (Continuous Compounding)

FVA = PMT \,e^{kn}\sum_{j=0}^{n-1} e^{(g-k)j}

Solves for: FVA

Perpetuity

PV of Perpetuity (Annual Compounding)

PV = \frac{PMT}{k}

Solves for: PV

PV of Perpetuity (Periodic Compounding)

PV = \frac{PMT}{\frac{k}{m}}

Solves for: PV

PV of Perpetuity Due (Annual Compounding)

PV = PMT + \frac{PMT}{k}

Solves for: PV

PV of Perpetuity Due (Periodic Compounding)

PV = PMT + \frac{PMT}{\frac{k}{m}}

Solves for: PV

PV of Growing Perpetuity

PV = \frac{PMT}{k - g}

Solves for: PV

PV of Growing Perpetuity Due

PV = \frac{PMT \times (1 + k)}{k - g}

Solves for: PV

PV of Perpetuity (Continuous Compounding)

PV = \frac{PMT}{e^{k} - 1}

Solves for: PV

PV of Perpetuity Due (Continuous Compounding)

PV = \frac{PMT \times e^{k}}{e^{k} - 1}

Solves for: PV

Deferred Annuity

PV of Deferred Ordinary Annuity

PV = PMT \times \frac{1 - (1+k)^{-n}}{k} \times (1+k)^{-d}

Solves for: PV

PV of Deferred Annuity Due

PV = PMT \times \frac{1 - (1+k)^{-n}}{k} \times (1+k) \times (1+k)^{-d}

Solves for: PV

Loan Amortization

Remaining Balance (Annual Compounding)

B_p = \max\left(0, PV \times (1+k)^p - PMT \times \frac{(1+k)^p - 1}{k}\right)

Solves for: B

Remaining Balance (Periodic Compounding)

B_p = \max\left(0, PV \times \left(1+\frac{k}{m}\right)^p - PMT \times \frac{\left(1+\frac{k}{m}\right)^p - 1}{\frac{k}{m}}\right)

Solves for: B

Interest Portion of Payment

I_p = B_{p-1}\frac{k}{m},\quad B_{p-1}\geq 0

Solves for: Interest

Principal Portion of Payment

P_p = \min\left(PMT,B_{p-1}(1+k/m)\right)-B_{p-1}k/m

Solves for: Principal

Cumulative Interest Paid

I_{1..p} = \sum_{t=1}^{p} B_{t-1} \times \frac{k}{m}

Solves for: CumInterest

Cumulative Principal Paid

P_{1..p} = PV - B_p

Solves for: CumPrincipal

Total Interest Cost of a Loan

I_{\text{total}} = \sum_{t=1}^{T} B_{t-1} \times \frac{k}{m}, \quad B_T = 0

Solves for: TotalInterest

Interest-Only Payment

PMT_{IO} = PV \times \frac{k}{m}

Solves for: IOPayment

Balloon Payment

B_{nm} = PV(1+k/m)^{nm} - PMT \times \frac{(1+k/m)^{nm} - 1}{k/m}

Solves for: Balloon

Payoff Time with Extra Payments

n = -\frac{\ln\left(1 - \frac{PV \cdot k/m}{PMT + E}\right)}{m \ln(1 + k/m)}

Solves for: PayoffYears

Loan-to-Value Ratio

LTV = \frac{\text{Loan}}{\text{Value}}

Solves for: LTV

Down Payment

\text{Down} = \text{Value} \times d

Solves for: DownPayment

Mortgage Points Break-Even

\text{BE} = \frac{\text{Points} \times \text{Loan}}{PMT(k_{\text{without}}) - PMT(k_{\text{with}})}

Solves for: BreakevenMonths

Refinance Break-Even

\text{BE} = \frac{\text{Closing Costs}}{PMT_{\text{old}} - PMT_{\text{new}}}

Solves for: BreakevenMonths

Effective Borrowing Rate with Fees

PV - \text{Fees} = PMT \times \frac{1 - (1 + r/m)^{-nm}}{r/m}

Solves for: EffRate

Debt Service Coverage Ratio (DSCR)

\text{DSCR} = \frac{\text{NOI}}{\text{Debt Service}}

Solves for: DSCR

Maximum Affordable Loan

\text{Loan} = \left(\frac{\text{Income} \times \text{DTI}}{m} - \text{Other}\right) \times \frac{1 - (1 + k/m)^{-nm}}{k/m}

Solves for: MaxLoan

Rate Conversion

Equity Valuation

Entry P/E Ratio

\text{EntryPE} = \text{ExitPE} \times \frac{(1 + g)^n}{(1 + r)^n} \times (1 - \text{MOS})

Solves for: EntryPE

Entry P/E Ratio with Dividends

\text{EntryPE} \approx \text{ExitPE} \times \left[\frac{(1+g)(1+DY)}{1+r}\right]^n \times (1 - \text{MOS})

Solves for: EntryPE

Maximum Stock Price

P_0 = \frac{E_0 (1+g)^n \times \text{ExitPE}}{(1+r)^n} \times (1 - \text{MOS})

Solves for: P0

Maximum Stock Price with Dividends

P_0 \approx \left[\frac{E_0 (1+g)^n \times \text{ExitPE}}{(1+r)^n}\right] \times (1+DY)^n \times (1 - \text{MOS})

Solves for: P0

Projected EPS from Revenue

\text{EPS}_n = \frac{\text{Rev}_0(1+g_{\text{rev}})^n \times \text{NetMargin}}{\text{Shares}(1+s)^n}

Solves for: EPSn

Fair Buy Price (Revenue Projection)

P_{\text{buy}} = \frac{\text{Rev}_0(1+g_{\text{rev}})^n \times \text{NetMargin} \times \text{ExitPE}}{\text{Shares}(1+s)^n (1+r)^n} \times (1 - \text{MOS})

Solves for: FairPrice

Expected Annual Return (Revenue Projection)

\text{ExpRet} = \left[\frac{\text{Rev}_0(1+g_{\text{rev}})^n \times \text{NetMargin} \times \text{ExitPE}}{\text{Shares}(1+s)^n \times P_{\text{mkt}}}\right]^{1/n} - 1

Solves for: ExpRet

Earnings Yield

\text{Earnings Yield} = \frac{\text{EPS}}{P}

Solves for: EarningsYield

PEG Ratio

\text{PEG} = \frac{\text{P/E}}{\text{Growth Rate (\%)}}

Solves for: PEG

PEGY Ratio

\text{PEGY} = \frac{\text{P/E}}{\text{Growth (\%)} + \text{Dividend Yield (\%)}}

Solves for: PEGY

Implied Growth Rate

\text{ImpliedG} = \left[ \frac{P/E \times (1 + r)^n}{\text{ExitPE}} \right]^{1/n} - 1

Solves for: ImpliedG

Implied Required Return

\text{ImpliedR} = \left[ \frac{(1 + g)^n \times \text{ExitPE}}{P/E} \right]^{1/n} - 1

Solves for: ImpliedR

Implied Required Return with Dividends

\text{ImpliedR} = \left[\left(\frac{\text{ExitPE}}{P/E} \times (1+g)^n\right)^{1/n} \times (1+DY)\right] - 1

Solves for: ImpliedR

Gordon Growth Model

V_0 = \frac{D_1}{r - g}

Solves for: V0

No-Growth DDM (Preferred Stock)

V_0 = \frac{D_0}{r}

Solves for: V0

Two-Stage DDM

V_0 = \sum_{t=1}^{n_1} \frac{D_0(1+g_1)^t}{(1+r)^t} + \frac{D_0(1+g_1)^{n_1}(1+g_2)}{(r-g_2)(1+r)^{n_1}}

Solves for: V0

Sustainable Growth Rate

g = \text{ROE} \times (1 - \text{Payout})

Solves for: SGR

Price-to-Book (P/B)

\text{P/B} = \frac{P}{\text{BVPS}}

Solves for: PB

Book Value per Share (BVPS)

\text{BVPS} = \frac{\text{Equity}}{\text{Shares Outstanding}}

Solves for: BVPS

Graham Number

\text{GrahamNum} = \sqrt{22.5 \times \text{EPS} \times \text{BVPS}}

Solves for: GrahamNum

EV / EBITDA

\text{EV/EBITDA} = \frac{\text{MarketCap} + \text{Debt} - \text{Cash}}{\text{EBITDA}}

Solves for: EVtoEBITDA

Free Cash Flow Yield

\text{FCF Yield} = \frac{\text{FCF}}{\text{MarketCap}}

Solves for: FCFYield

Two-Stage DCF Valuation

V_0 = \sum_{t=1}^{n_1} \frac{\text{FCF}_0(1+g_1)^t}{(1+r)^t} + \frac{\text{FCF}_0(1+g_1)^{n_1}(1+g_2)}{(r-g_2)(1+r)^{n_1}}

Solves for: DCFValue

Total Shareholder Return (TSR)

\text{TSR} = \left( \frac{P_1 + \text{Dividends}}{P_0} \right)^{1/n} - 1

Solves for: TSR

Buyback Yield

\text{Buyback Yield} = \frac{\text{Buyback Spend}}{\text{Market Cap}}

Solves for: BuybackYield

Shareholder Yield

\text{Shareholder Yield} = \text{DivYield} + \text{BuybackYield} + \text{DebtPaydownYield}

Solves for: ShareholderYield

Enterprise Value

EV = \text{MarketCap} + \text{Debt} - \text{Cash}

Solves for: EV

EV/EBIT

\text{EV/EBIT} = \frac{\text{MarketCap} + \text{Debt} - \text{Cash}}{EBIT}

Solves for: EVtoEBIT

EV/Sales

\text{EV/Sales} = \frac{\text{MarketCap} + \text{Debt} - \text{Cash}}{\text{Revenue}}

Solves for: EVtoSales

Price-to-Sales (P/S)

\text{P/S} = \frac{\text{MarketCap}}{\text{Revenue}}

Solves for: PS

Price-to-Free-Cash-Flow (P/FCF)

\text{P/FCF} = \frac{\text{MarketCap}}{\text{FCF}}

Solves for: PFCF

Free Cash Flow per Share

\text{FCF/Share} = \frac{\text{FCF}}{\text{Shares}}

Solves for: FCFPerShare

Free Cash Flow to Equity (FCFE)

\text{FCFE} = \text{CFO} - \text{CapEx} + \text{Net Borrowing}

Solves for: FCFE

Free Cash Flow to the Firm (FCFF)

\text{FCFF} = \text{CFO} + \text{Interest} \times (1 - t) - \text{CapEx}

Solves for: FCFF

Owner Earnings

\text{Owner Earnings} = \text{Net Income} + \text{D\&A} - \text{Maintenance CapEx}

Solves for: OwnerEarnings

Present Value of Growth Opportunities (PVGO)

\text{PVGO} = P - \frac{EPS}{r}

Solves for: PVGO

Portfolio Return & Risk

Sharpe Ratio

\text{Sharpe} = \frac{\overline{R_p} - R_f}{\sigma_p}

Solves for: Sharpe

Sortino Ratio

\text{Sortino} = \frac{\overline R-R_f}{\sqrt{\frac{1}{n}\sum_{i=1}^{n}\min(0,R_i-R_f)^2}}

Solves for: Sortino

Treynor Ratio

\text{Treynor} = \frac{R_p - R_f}{\beta_p}

Solves for: Treynor

Information Ratio

IR = \frac{\overline{R_p - R_b}}{\sigma(R_p - R_b)}

Solves for: IR

Portfolio Variance (2-Asset)

\sigma_p^2 = w_1^2 \sigma_1^2 + w_2^2 \sigma_2^2 + 2 w_1 w_2 \rho_{12} \sigma_1 \sigma_2

Solves for: PortVar

Geometric Mean Return

GMR = \left( \prod_{t=1}^{n} (1 + r_t) \right)^{1/n} - 1

Solves for: GMR

Arithmetic Mean Return

AMR = \frac{1}{n} \sum_{t=1}^{n} r_t

Solves for: AMR

Coefficient of Variation (CV)

CV = \frac{\sigma}{\overline{R}}

Solves for: CV

Jensen's Alpha

\alpha = R_p - \left[R_f + \beta (R_m - R_f)\right]

Solves for: Alpha

Covariance of Returns

\text{Cov}(R_a, R_b) = \frac{\sum (R_{a,t} - \bar{R}_a)(R_{b,t} - \bar{R}_b)}{n - 1}

Solves for: Cov

Correlation of Returns

\rho_{ab} = \frac{\text{Cov}(R_a, R_b)}{\sigma_a \, \sigma_b}

Solves for: Corr

Historical Value at Risk (VaR)

\text{VaR}_c = -\,R_{(\lfloor (1-c)\,n \rfloor)}, \quad R_{(0)} \le R_{(1)} \le \dots

Solves for: VaR

Parametric Value at Risk (VaR)

\text{VaR}_c = z_c \, \sigma - \mu, \quad z_c = \Phi^{-1}(c)

Solves for: VaR

Maximum Drawdown

\text{MDD} = \max_{t} \frac{\max_{s \le t} P_s - P_t}{\max_{s \le t} P_s}

Solves for: MDD

Three-Asset Portfolio Variance

\sigma_p^2 = \sum_{i=1}^{3} w_i^2 \sigma_i^2 + 2 \sum_{i<j} w_i w_j \rho_{ij} \sigma_i \sigma_j

Solves for: PortVar

N-Asset Portfolio Variance

\sigma_p^2 = \mathbf{w}^{\mathsf{T}} \Sigma \mathbf{w} = \sum_i \sum_j w_i w_j \Sigma_{ij}

Solves for: PortVar

Cost of Capital

CAPM (Capital Asset Pricing Model)

r_e = R_f + \beta \times (R_m - R_f)

Solves for: ReCAPM

WACC (Weighted Average Cost of Capital)

\text{WACC} = w_e \times r_e + w_d \times r_d \times (1 - t)

Solves for: WACC

Equity Risk Premium

\text{ERP} = R_m - R_f

Solves for: ERP

Options

Cash-Secured Put Annualized Return

\text{Annualized Return} = \frac{\text{Premium}}{\text{Strike}} \times \frac{365}{\text{Days to Expiration}}

Solves for: IRR

Cash-Secured Put Compounded Annualized Return

\text{Annualized Return} = \left(1 + \frac{\text{Premium}}{\text{Strike}}\right)^{\frac{365}{\text{Days to Expiration}}} - 1

Solves for: IRR

Cash-Secured Put Return on Net Cash

\text{Annualized Return} = \frac{\text{Premium}}{\text{Strike} - \text{Premium}} \times \frac{365}{\text{Days to Expiration}}

Solves for: IRR

Naked Put Return on Margin

\text{Annualized Return} = \frac{\text{Premium}}{\text{MarginReq}} \times \frac{365}{\text{Days to Expiration}}

Solves for: IRR

Covered Call Annualized Return

\text{Annualized Return} = \frac{\text{Premium}}{\text{Strike}} \times \frac{365}{\text{Days to Expiration}}

Solves for: IRR

Covered Call Compounded Annualized Return

\text{Annualized Return} = \left(1 + \frac{\text{Premium}}{\text{Strike}}\right)^{\frac{365}{\text{Days to Expiration}}} - 1

Solves for: IRR

Covered Call Static Return (If-Not-Called)

\text{Static Return} = \frac{\text{Premium}}{\text{StockBasis}} \times \frac{365}{\text{Days to Expiration}}

Solves for: IRR

Covered Call If-Called Total Return

\text{If-Called Return} = \frac{\text{Premium} + (\text{Strike} - \text{StockBasis})}{\text{StockBasis}} \times \frac{365}{\text{Days to Expiration}}

Solves for: IRR

Vertical Credit Spread Return on Risk

\text{Annualized Return} = \frac{\text{NetCredit}}{\text{Width} - \text{NetCredit}} \times \frac{365}{\text{Days to Expiration}}

Solves for: IRR

Iron Condor Return on Risk

\text{Annualized Return} = \frac{\text{NetCredit}}{\text{Width} - \text{NetCredit}} \times \frac{365}{\text{Days to Expiration}}

Solves for: IRR

Short Put Effective Purchase Price

\text{EffectivePrice} = \text{Strike} - \text{Premium}

Solves for: EffectivePrice

Covered Call Effective Sale Price

\text{EffectivePrice} = \text{Strike} + \text{Premium}

Solves for: EffectivePrice

Long Call Break-Even Price

\text{BreakEven} = \text{Strike} + \text{Premium}

Solves for: BreakEven

Long Put Break-Even Price

\text{BreakEven} = \text{Strike} - \text{Premium}

Solves for: BreakEven

Put-Call Parity

C - P = S - K \cdot e^{-rT} \quad \Rightarrow \quad C = P + S - K \cdot e^{-rT}

Solves for: CallPrice

Black-Scholes Call Price

C = S \cdot N(d_1) - K \cdot e^{-rT} \cdot N(d_2), \quad d_1 = \frac{\ln(S/K) + (r + \sigma^2/2)T}{\sigma\sqrt{T}}, \quad d_2 = d_1 - \sigma\sqrt{T}

Solves for: CallPrice

Black-Scholes Put Price

P = K \cdot e^{-rT} \cdot N(-d_2) - S \cdot N(-d_1)

Solves for: PutPrice

Implied Volatility

\text{Find } \sigma \text{ such that } BS(S, K, r, \sigma, T) = \text{Market Price}

Solves for: Sigma

Call Delta

\Delta_{\text{call}} = N(d_1)

Solves for: Delta

Put Delta

\Delta_{\text{put}} = N(d_1) - 1

Solves for: Delta

Gamma

\Gamma = \frac{\varphi(d_1)}{S \sigma \sqrt{T}}

Solves for: Gamma

Call Theta

\Theta_{\text{call}} = -\frac{S \varphi(d_1) \sigma}{2\sqrt{T}} - r K e^{-rT} N(d_2)

Solves for: Theta

Put Theta

\Theta_{\text{put}} = -\frac{S \varphi(d_1) \sigma}{2\sqrt{T}} + r K e^{-rT} N(-d_2)

Solves for: Theta

Vega

\nu = S \varphi(d_1) \sqrt{T}

Solves for: Vega

Call Rho

\rho_{\text{call}} = K T e^{-rT} N(d_2)

Solves for: Rho

Put Rho

\rho_{\text{put}} = -K T e^{-rT} N(-d_2)

Solves for: Rho

Option Intrinsic Value

\text{Intrinsic} = \max(0, \; \text{type} \times (S - K)), \quad \text{type} = +1 \text{ call}, \; -1 \text{ put}

Solves for: Intrinsic

Option Time Value

\text{Time Value} = \text{Option Price} - \text{Intrinsic Value}

Solves for: TimeValue

Forward Price

F_0 = S_0 \, e^{(r - q) T}

Solves for: ForwardPrice

Cost of Carry

c = r + u - y

Solves for: Carry

Futures Price (Cost-of-Carry Model)

F_0 = S_0 \, e^{(r + u - y) T}

Solves for: FuturesPrice

Binomial Option Price (Cox-Ross-Rubinstein)

u = e^{\sigma\sqrt{\Delta t}}, \quad d = \tfrac{1}{u}, \quad p = \frac{e^{r\Delta t} - d}{u - d}, \quad V_{t} = e^{-r\Delta t}\left[p\,V_{t+1}^{u} + (1-p)\,V_{t+1}^{d}\right]

Solves for: BinomialPrice

Bonds

Bond Price

P = \frac{F \cdot CR / m \cdot \left[1 - (1 + y/m)^{-Nm}\right]}{y/m} + \frac{F}{(1 + y/m)^{Nm}}

Solves for: BondPrice

Zero-Coupon Bond Price

P = \frac{F}{(1 + y/m)^{Nm}}

Solves for: ZeroPrice

Accrued Interest

\text{Accrued} = \text{Coupon} \times \frac{\text{AccruedDays}}{\text{PeriodDays}}

Solves for: Accrued

Current Yield

\text{CurrentYield} = \frac{F \cdot CR}{P}

Solves for: CurrentYield

Yield to Maturity (YTM)

P = \sum_{t=1}^{Nm} \frac{F \cdot CR / m}{(1 + y/m)^{t}} + \frac{F}{(1 + y/m)^{Nm}}, \quad \text{solve for } y

Solves for: YTM

Yield to Call (YTC)

P = \sum_{t=1}^{T_c m} \frac{F \cdot CR / m}{(1 + y/m)^{t}} + \frac{\text{Redemption}}{(1 + y/m)^{T_c m}}, \quad \text{solve for } y

Solves for: YTC

Yield to Worst (YTW)

\text{YTW} = \min(\text{YTM}, \text{YTC})

Solves for: YTW

Bond-Equivalent Yield (BEY)

\text{BEY} = 2 \times \text{YSemi}

Solves for: BEY

Holding Period Return (HPR)

\text{HPR} = \frac{P_1 + \text{TotalCoupons} - P_0}{P_0}

Solves for: HPR

Macaulay Duration

D_{Mac} = \frac{1}{P} \sum_{t=1}^{Nm} \frac{t/m \cdot CF_t}{(1 + y/m)^{t}}

Solves for: MacDur

Modified Duration

D_{Mod} = \frac{D_{Mac}}{1 + y/m}

Solves for: ModDur

Convexity

C = \frac{1}{P} \sum_{t=1}^{Nm} \frac{(t/m)(t/m + 1/m) \cdot CF_t}{(1 + y/m)^{t+2}}

Solves for: Conv

DV01 (PVBP)

\text{DV01} \approx D_{Mod} \times P \times 0.0001

Solves for: DV01

Price-Change Approximation

\frac{\Delta P}{P} \approx -D_{Mod} \cdot \Delta y + \frac{1}{2} \cdot C \cdot (\Delta y)^{2}

Solves for: DeltaPpct

Effective Duration

D_{Eff} = \frac{P_{-} - P_{+}}{2 \cdot P_{0} \cdot \Delta y}

Solves for: EffDur

Yield Spread

\text{Spread} = \text{BondYield} - \text{TreasuryYield}

Solves for: Spread

Spot Rate from Zero-Coupon Bond

\text{Spot} = m \cdot \left[ \left(\frac{F}{P}\right)^{1/(Nm)} - 1 \right]

Solves for: Spot

Forward Rate

\left(1 + \frac{\text{Forward}}{m}\right)^{(\text{Years}_2 - \text{Years}_1) m} = \frac{(1 + \text{Spot}_2/m)^{\text{Years}_2 \cdot m}}{(1 + \text{Spot}_1/m)^{\text{Years}_1 \cdot m}}

Solves for: Forward

Implied Default Probability

\text{DefaultProb} \approx \frac{\text{Spread}}{1 - \text{Recovery}}

Solves for: DefaultProb

Fisher Real Yield

\text{Real} = \frac{1 + \text{Nominal}}{1 + \text{Inflation}} - 1

Solves for: Real

Breakeven Inflation

\text{Breakeven} = \frac{1 + \text{TreasuryYield}}{1 + \text{TIPSYield}} - 1 \approx \text{TreasuryYield} - \text{TIPSYield}

Solves for: BreakevenInflation

Present Value of Coupons

PV_{\text{coupons}} = \frac{F \cdot CR}{m} \times \frac{1 - (1 + y/m)^{-Nm}}{y/m}

Solves for: CouponPV

Present Value of Face Value

PV_{\text{face}} = \frac{F}{(1 + y/m)^{Nm}}

Solves for: FacePV

Dollar Duration

\$D = D_{\text{mod}} \times P

Solves for: DollarDur

Effective Annual Yield

EAY = \left(1 + \frac{y}{m}\right)^m - 1

Solves for: EAY

Capital Budgeting

Net Present Value (NPV)

NPV = \sum_{t=0}^{n} \frac{CF_t}{(1 + k)^t}

Solves for: NPV

Internal Rate of Return (IRR)

\text{Find IRR such that: } \sum_{t=0}^{n} \frac{CF_t}{(1 + IRR)^t} = 0

Solves for: IRR

Modified Internal Rate of Return (MIRR)

MIRR = \left( \frac{FV(\text{positive CFs at ReinvestRate})}{|PV(\text{negative CFs at FinanceRate})|} \right)^{1/n} - 1

Solves for: MIRR

Profitability Index (PI)

PI = \frac{PV(\text{future cash flows})}{|CF_0|} = 1 + \frac{NPV}{|CF_0|}

Solves for: PI

Payback Period

\text{Payback} = t^* + \frac{|\text{CumCF}_{t^*}|}{CF_{t^*+1}}

Solves for: Payback

Discounted Payback Period

\text{DiscountedPayback} = t^* + \frac{|\text{CumDiscCF}_{t^*}|}{CF_{t^*+1}/(1+k)^{t^*+1}}

Solves for: DiscountedPayback

Equivalent Annual Annuity (EAA)

EAA = \frac{NPV \times k}{1 - (1 + k)^{-n}}

Solves for: EAA

Economic Value Added (EVA)

\text{EVA} = \text{NOPAT} - \text{WACC} \times \text{Invested Capital}

Solves for: EVA

ROIC − WACC Spread

\text{Spread} = \frac{\text{NOPAT}}{\text{Invested Capital}} - \text{WACC}

Solves for: Spread

Return on Incremental Invested Capital (ROIIC)

\text{ROIIC} = \frac{\text{NOPAT}_1 - \text{NOPAT}_0}{\text{IC}_1 - \text{IC}_0}

Solves for: ROIIC

Financial Ratios

Current Ratio

CR = \frac{\text{Current Assets}}{\text{Current Liabilities}}

Solves for: CR

Quick Ratio (Acid-Test)

\text{Quick Ratio} = \frac{\text{Current Assets} - \text{Inventory} - \text{Other Non-Quick Assets}}{\text{Current Liabilities}}

Solves for: QR

Cash Ratio

CashR = \frac{\text{Cash} + \text{Marketable Securities}}{\text{Current Liabilities}}

Solves for: CashR

Working Capital

WC = \text{Current Assets} - \text{Current Liabilities}

Solves for: WCap

Inventory Turnover

InvTurn = \frac{\text{COGS}}{\text{Average Inventory}}

Solves for: InvTurn

Days Sales Outstanding (DSO)

DSO = \frac{\text{Average Receivables}}{\text{Sales} / 365}

Solves for: DSO

Cash Conversion Cycle

CCC = DIO + DSO - DPO

Solves for: CCC

Debt-to-Equity Ratio

D/E = \frac{\text{Total Debt}}{\text{Shareholders' Equity}}

Solves for: DE

Debt-to-Assets Ratio

D/A = \frac{\text{Total Debt}}{\text{Total Assets}}

Solves for: DA

Interest Coverage Ratio

IC = \frac{EBIT}{\text{Interest Expense}}

Solves for: IC

Degree of Operating Leverage (DOL)

DOL = \frac{\text{Contribution Margin}}{EBIT} = \frac{\text{Sales} - \text{Variable Costs}}{EBIT}

Solves for: DOL

Degree of Financial Leverage (DFL)

DFL = \frac{EBIT}{EBIT - \text{Interest}}

Solves for: DFL

Altman Z-Score

Z = 1.2 \cdot \frac{WC}{TA} + 1.4 \cdot \frac{RE}{TA} + 3.3 \cdot \frac{EBIT}{TA} + 0.6 \cdot \frac{ME}{TL} + 1.0 \cdot \frac{Sales}{TA}

Solves for: Z

Days Inventory Outstanding (DIO)

DIO = \frac{\text{Average Inventory}}{\text{COGS} / 365}

Solves for: DSI

Days Payable Outstanding (DPO)

DPO = \frac{\text{Average Payables}}{\text{COGS} / 365}

Solves for: DPO

Receivables Turnover

\text{Receivables Turnover} = \frac{\text{Sales}}{\text{Average Receivables}}

Solves for: RecTurn

Payables Turnover

\text{Payables Turnover} = \frac{\text{COGS}}{\text{Average Payables}}

Solves for: PayTurn

Gross Profit Margin

\text{Gross Margin} = \frac{\text{Revenue} - \text{COGS}}{\text{Revenue}}

Solves for: GrossMargin

Operating Margin

\text{Operating Margin} = \frac{EBIT}{\text{Revenue}}

Solves for: OperatingMargin

Net Profit Margin

\text{Net Margin} = \frac{\text{Net Income}}{\text{Revenue}}

Solves for: NetMargin

EBITDA Margin

\text{EBITDA Margin} = \frac{\text{EBITDA}}{\text{Revenue}}

Solves for: EBITDAMargin

Effective Tax Rate

\text{Effective Tax Rate} = \frac{\text{Tax Expense}}{\text{Pretax Income}}

Solves for: EffectiveTaxRate

Return on Assets (ROA)

\text{ROA} = \frac{\text{Net Income}}{\text{Total Assets}}

Solves for: ROA

Return on Equity (ROE)

\text{ROE} = \frac{\text{Net Income}}{\text{Equity}}

Solves for: ROE

Return on Invested Capital (ROIC)

\text{ROIC} = \frac{\text{NOPAT}}{\text{Invested Capital}}

Solves for: ROIC

Return on Capital Employed (ROCE)

\text{ROCE} = \frac{EBIT}{\text{Capital Employed}} = \frac{EBIT}{\text{Total Assets} - \text{Current Liabilities}}

Solves for: ROCE

DuPont Decomposition

\text{ROE} = \text{Net Margin} \times \text{Asset Turnover} \times \text{Equity Multiplier}

Solves for: ROE

Free Cash Flow

FCF = CFO - CapEx

Solves for: FCF

Operating Cash Flow Margin

\text{OCF Margin} = \frac{CFO}{\text{Revenue}}

Solves for: OCFMargin

Operating Cash Flow Ratio

\text{OCF Ratio} = \frac{CFO}{\text{Current Liabilities}}

Solves for: OCFRatio

Cash Coverage Ratio

\text{Cash Coverage} = \frac{CFO + \text{Cash Interest Paid} + \text{Cash Taxes Paid}}{\text{Cash Interest Paid}}

Solves for: CashCov

Cash Flow to Debt

\text{CFO/Debt} = \frac{CFO}{\text{Total Debt}}

Solves for: CFOToDebt

Net Debt

\text{Net Debt} = \text{Total Debt} - \text{Cash}

Solves for: NetDebt

Net Debt to EBITDA

\frac{\text{Net Debt}}{\text{EBITDA}} = \frac{\text{Total Debt} - \text{Cash}}{\text{EBITDA}}

Solves for: NetDebtEBITDA

Free Cash Flow Margin

\text{FCF Margin} = \frac{\text{CFO} - \text{CapEx}}{\text{Revenue}}

Solves for: FCFMargin

Dividends per Share (DPS)

DPS = \frac{\text{Total Dividends}}{\text{Shares Outstanding}}

Solves for: DPS

Degree of Combined Leverage

DTL = DOL \times DFL = \frac{\% \Delta EPS}{\% \Delta \text{Sales}}

Solves for: DTL