Formula Cheat Sheet

\[ R = \frac{P_1 - P_0 + I_1}{P_0} = \frac{P_1 + I_1}{P_0} - 1 \]

\[ (1 + r_{\text{nominal}}) = (1 + r_{\text{real}})\,(1 + \pi)\,(1 + RP) \] \[ \text{Approximation: } r_{\text{nominal}} \approx r_{\text{real}} + \pi + RP \]

\[ \bar{R}_a = \frac{1}{N}\sum_{i=1}^{N} R_i = \frac{R_1 + R_2 + \dots + R_N}{N} \]

\[ \bar{R}_g = \left[\prod_{i=1}^{N}(1 + R_i)\right]^{1/N} - 1 \]

\[ \bar{X}_H = \frac{N}{\displaystyle\sum_{i=1}^{N}\frac{1}{X_i}} \]

\[ \sum_{t=0}^{N} \frac{CF_t}{(1 + MWR)^t} = 0 \]

\[ TWR = \left[(1+R_1)(1+R_2)\cdots(1+R_N)\right]^{1/N} - 1 \]

\[ R_{annual} = (1 + R_{period})^{c} - 1 \]

\[ r_{cc} = \ln(1 + R) = \ln\!\left(\frac{P_1}{P_0}\right) \]

\[ R_{net} = R_{gross} - \text{fees} - \text{expenses} \] \[ R_{after\text{-}tax} = R_{pre\text{-}tax}\,(1 - t) \] \[ R_{leveraged} = \frac{V_E\,R_{portfolio} - V_B\,r_D}{V_E} \]

\[ FV_N = PV\,(1 + r)^{N} \qquad PV = \frac{FV_N}{(1 + r)^{N}} \]

\[ PV_{\text{ord}} = A \cdot \frac{1 - (1+r)^{-N}}{r} \] \[ FV_{\text{ord}} = A \cdot \frac{(1+r)^{N} - 1}{r} \]

\[ PV_{\text{due}} = PV_{\text{ord}} \cdot (1 + r) \qquad FV_{\text{due}} = FV_{\text{ord}} \cdot (1 + r) \]

\[ PV_{\text{perp}} = \frac{A}{r} \quad \text{(level)} \qquad PV = \frac{D_1}{r - g} \quad \text{(growing, } r > g\text{)} \]

\[ EAR = \left(1 + \frac{r_s}{m}\right)^{m} - 1 \]

\[ EAR_{cc} = e^{r_s} - 1 \qquad FV = PV \cdot e^{r_s \cdot N} \]

\[ P_0 = \sum_{t=1}^{N} \frac{C}{(1 + r)^{t}} + \frac{F}{(1 + r)^{N}} \]

\[ L_y = (n + 1)\,\frac{y}{100} \]

\[ \text{Range} = X_{\max} - X_{\min} \] \[ MAD = \frac{1}{n}\sum_{i=1}^{n}\,|X_i - \bar{X}| \]

\[ \sigma^{2} = \frac{1}{N}\sum_{i=1}^{N}(X_i - \mu)^{2} \qquad s^{2} = \frac{1}{n - 1}\sum_{i=1}^{n}(X_i - \bar{X})^{2} \] \[ \sigma = \sqrt{\sigma^{2}} \qquad s = \sqrt{s^{2}} \]

\[ CV = \frac{s}{\bar{X}} \]

\[ \text{Skew} = \frac{1}{n}\sum_{i=1}^{n}\frac{(X_i - \bar{X})^{3}}{s^{3}} \]

\[ K = \frac{1}{n}\sum_{i=1}^{n}\frac{(X_i - \bar{X})^{4}}{s^{4}} \] \[ K_E = K - 3 \]

\[ s_{\text{target}} = \sqrt{\frac{\displaystyle\sum_{X_i < B}(X_i - B)^{2}}{n - 1}} \]

\[ \text{Sharpe} = \frac{\bar{R}_p - R_f}{s_p} \]

\[ \text{Cov}(X,Y) = \frac{1}{n - 1}\sum_{i=1}^{n}(X_i - \bar{X})(Y_i - \bar{Y}) \] \[ \rho_{X,Y} = \frac{\text{Cov}(X,Y)}{s_X \cdot s_Y}, \qquad -1 \le \rho \le 1 \]

\[ P(A \mid B) = \frac{P(AB)}{P(B)},\quad P(B) > 0 \]

\[ P(AB) = P(A \mid B) \cdot P(B) = P(B \mid A) \cdot P(A) \] \[ \text{If A, B independent: } P(AB) = P(A) \cdot P(B) \]

\[ P(A \cup B) = P(A) + P(B) - P(AB) \]

\[ P(A) = \sum_{i=1}^{n} P(A \mid S_i) \cdot P(S_i) \]

\[ P(S \mid I) = \frac{P(I \mid S) \cdot P(S)}{P(I)} \]

\[ E(X) = \sum_{i=1}^{n} P(X_i) \cdot X_i \]

\[ \sigma^{2}(X) = E[(X - E(X))^{2}] = \sum_{i=1}^{n} P(X_i)\,[X_i - E(X)]^{2} \]

\[ E(X) = \sum_{i=1}^{n} P(S_i) \cdot E(X \mid S_i) \]

\[ E(R_p) = \sum_{i=1}^{n} w_i \cdot E(R_i),\qquad \sum w_i = 1 \]

\[ \sigma_p^{2} = w_1^{2}\sigma_1^{2} + w_2^{2}\sigma_2^{2} + 2\,w_1\,w_2\,\rho_{1,2}\,\sigma_1\,\sigma_2 \]

\[ \sigma_p^{2} = \sum_{i=1}^{n}\sum_{j=1}^{n} w_i\,w_j\,\text{Cov}(R_i, R_j) \]

\[ \text{Cov}(X, Y) = \rho_{X,Y} \cdot \sigma_X \cdot \sigma_Y \]

\[ SFR = \frac{E(R_p) - R_L}{\sigma_p} \]

\[ P(R_p < R_L) = N(-SFR) \]

\[ P_T = P_0 \cdot e^{r_{cc} \cdot T},\qquad r_{cc} \sim N(\mu, \sigma^{2}) \implies P_T \sim \text{Lognormal} \]

\[ r_{cc,\,0 \to T} \sim N(\mu \cdot T,\ \sigma^{2} \cdot T) \] \[ \sigma_{0 \to T} = \sigma \cdot \sqrt{T} \quad \text{(volatility scales with } \sqrt{T}\text{)} \]

\[ P_{t+1} = P_t \cdot e^{\,(\mu - \tfrac{1}{2}\sigma^{2})\,\Delta t \,+\, \sigma\sqrt{\Delta t}\, Z} \]

\[ E(\bar{X}) = \mu, \qquad \text{Var}(\bar{X}) = \frac{\sigma^{2}}{n} \]

\[ \bar{X} \overset{\text{approx}}{\sim} N\!\left(\mu,\ \frac{\sigma^{2}}{n}\right) \]

\[ \sigma_{\bar{X}} = \frac{\sigma}{\sqrt{n}} \quad \text{(known } \sigma\text{)} \qquad s_{\bar{X}} = \frac{s}{\sqrt{n}} \quad \text{(unknown } \sigma\text{)} \]

\[ \bar{X} \pm z_{\alpha/2} \cdot \frac{\sigma}{\sqrt{n}} \]

\[ \bar{X} \pm t_{\alpha/2,\,n - 1} \cdot \frac{s}{\sqrt{n}} \]

\[ z = \frac{\bar{X} - \mu_0}{\sigma/\sqrt{n}} \]

\[ t = \frac{\bar{X} - \mu_0}{s/\sqrt{n}}, \quad df = n - 1 \]

\[ t = \frac{(\bar{X}_1 - \bar{X}_2) - (\mu_1 - \mu_2)_0}{\sqrt{\dfrac{s_p^{2}}{n_1} + \dfrac{s_p^{2}}{n_2}}} \] \[ s_p^{2} = \frac{(n_1 - 1)\,s_1^{2} + (n_2 - 1)\,s_2^{2}}{n_1 + n_2 - 2}, \quad df = n_1 + n_2 - 2 \]

\[ \chi^{2} = \frac{(n - 1)\,s^{2}}{\sigma_0^{2}}, \quad df = n - 1 \]

\[ F = \frac{s_1^{2}}{s_2^{2}}, \quad \text{put the larger variance on top} \]

\[ t = \frac{r\,\sqrt{n - 2}}{\sqrt{1 - r^{2}}}, \quad df = n - 2 \]

\[ r_s = 1 - \frac{6\sum_{i=1}^{n} d_i^{2}}{n\,(n^{2} - 1)} \]

\[ \chi^{2} = \sum_{i=1}^{r}\sum_{j=1}^{c}\frac{(O_{ij} - E_{ij})^{2}}{E_{ij}}, \quad df = (r - 1)(c - 1) \] \[ E_{ij} = \frac{(\text{row}_i\,\text{total})(\text{col}_j\,\text{total})}{n} \]

\[ Y_i = b_0 + b_1\,X_i + \varepsilon_i \] \[ \hat{Y}_i = \hat{b}_0 + \hat{b}_1\,X_i \]

\[ \hat{b}_1 = \frac{\text{Cov}(X, Y)}{\text{Var}(X)} = \frac{\sum (X_i - \bar{X})(Y_i - \bar{Y})}{\sum (X_i - \bar{X})^{2}} \] \[ \hat{b}_0 = \bar{Y} - \hat{b}_1\,\bar{X} \]

\[ \underbrace{\sum (Y_i - \bar{Y})^{2}}_{SST} = \underbrace{\sum (\hat{Y}_i - \bar{Y})^{2}}_{SSR} + \underbrace{\sum (Y_i - \hat{Y}_i)^{2}}_{SSE} \]

\[ R^{2} = \frac{SSR}{SST} = 1 - \frac{SSE}{SST} \]

\[ SEE = \sqrt{\frac{SSE}{n - 2}} = \sqrt{MSE} \]

\[ F = \frac{MSR}{MSE} = \frac{SSR/k}{SSE/(n - k - 1)} \]

\[ t = \frac{\hat{b}_1 - b_{1,\,\text{H}_0}}{s_{\hat{b}_1}}, \quad df = n - 2 \]

\[ \hat{b}_1 \pm t_{\alpha/2,\,n - 2} \cdot s_{\hat{b}_1} \]

\[ \hat{Y}_f \pm t_{\alpha/2,\,n - 2} \cdot s_f \]

\[ MR = MC \]

\[ CR_N = \sum_{i=1}^{N} s_i \]

\[ HHI = \sum_{i=1}^{n} s_i^{2} \]

\[ CPI = \frac{\text{Cost of basket at current prices}}{\text{Cost of basket at base-year prices}} \times 100 \]

\[ \text{Unemployment rate} = \frac{\text{Unemployed}}{\text{Labor force}} \times 100\% \]

\[ \text{Multiplier} = \frac{1}{1 - MPC \cdot (1 - t)} \]

\[ \text{Tax multiplier} = -\,\frac{MPC}{1 - MPC \cdot (1 - t)} \]

\[ \text{Money multiplier} = \frac{1}{\text{Reserve requirement}} \]

\[ M \cdot V = P \cdot Y \]

\[ R_{nom} \approx R_{real} + \pi^{e} \]

\[ R_{target} = R_{neutral} + 0.5\,(\pi - \pi^{*}) + 0.5\,(Y - Y^{*}) \]

\[ \text{Current} + \text{Capital} + \text{Financial} = 0 \]

\[ CA = (S - I) + (T - G) \]

\[ \text{Quote: } \frac{\text{Price currency}}{\text{Base currency}} \quad e.g. \frac{USD}{EUR} = 1.10 \]

\[ \text{Spread} = \text{Ask} - \text{Bid} \]

\[ \%\Delta = \frac{S_t - S_0}{S_0} \times 100\% \]

\[ \frac{A}{C} = \frac{A}{B} \times \frac{B}{C} \]

\[ F = S \cdot \frac{1 + r_{P} \cdot (T/360)}{1 + r_{B} \cdot (T/360)} \]

\[ V_{f} = \frac{(F_{t} - F_{0}) \cdot \text{Notional}}{1 + r_{P} \cdot (T_{\text{remain}}/360)} \]

\[ \text{Basic EPS} = \frac{\text{Net income} - \text{Preferred dividends}}{\text{Weighted avg. common shares}} \]

\[ \text{Diluted EPS} = \frac{NI - \text{Pref. div.} + \text{Conv. pref. div.} + \text{Conv. debt int.}(1 - t)}{\text{WASO} + \text{Conv. pref. shares} + \text{Conv. debt shares} + \text{Options/warrants}} \]

\[ \text{Net new shares} = N - N \cdot \frac{K}{P} = N \cdot \frac{P - K}{P} \]

\[ \text{Comprehensive income} = \text{Net income} + \text{OCI} \]

\[ \text{Assets} = \text{Liabilities} + \text{Equity} \]

\[ \text{Common-size \%} = \frac{\text{Line item}}{\text{Total assets}} \times 100\% \]

\[ \text{Goodwill} = \text{Purchase price} - \text{FV of identifiable net assets acquired} \]

\[ CFO = NI + \text{Non-cash items} + \Delta\text{Working capital} \]

\[ \text{Cash from customers} = \text{Revenue} - \Delta AR \] \[ \text{Cash to suppliers} = COGS + \Delta\text{Inv} - \Delta AP \]

\[ FCFF = NI + NCC + \text{Int}\,(1 - t) - FCInv - WCInv \]

\[ FCFE = FCFF - \text{Int}\,(1 - t) + \text{Net borrowing} \]

\[ \text{CF/Rev} = \frac{CFO}{\text{Revenue}} \quad \text{(cash generation per dollar of sales)} \] \[ \text{CF coverage} = \frac{CFO + \text{Int} + \text{Tax}}{\text{Int}} \] \[ \text{Reinvestment} = \frac{CFO}{\text{CapEx}} \] \[ \text{Debt coverage} = \frac{CFO}{\text{Total debt}} \]

\[ \text{Beg. Inv} + \text{Purchases} = COGS + \text{End. Inv} \]

\[ \text{LIFO reserve} = \text{Inv}_{FIFO} - \text{Inv}_{LIFO} \]

\[ \text{Inventory turnover} = \frac{COGS}{\text{Avg. inventory}} \] \[ DOH = \frac{365}{\text{Inventory turnover}} \]

\[ \text{Straight-line} = \frac{\text{Cost} - \text{Salvage}}{\text{Useful life}} \] \[ \text{DDB}_t = \text{NBV}_{t-1} \times \frac{2}{\text{Useful life}} \] \[ \text{Units-of-prod.} = \frac{\text{Cost} - \text{Salvage}}{\text{Total est. units}} \times \text{Units this period} \]

\[ \text{Int. expense}_t = \text{Carrying value}_{t-1} \times r_{market} \] \[ \text{Coupon paid} = \text{Face} \times r_{coupon} \] \[ \Delta\text{Carrying} = \text{Int. expense} - \text{Coupon paid} \]

\[ \text{Funded status} = \text{FV of plan assets} - \text{PBO} \]

\[ \text{Tax expense} = \text{Current tax} + \Delta\,DTL - \Delta\,DTA \]

\[ \text{Effective tax rate} = \frac{\text{Tax expense}}{\text{Pretax income}} \]

\[ Z = 1.2 X_1 + 1.4 X_2 + 3.3 X_3 + 0.6 X_4 + 1.0 X_5 \]

\[ \text{Receivables turnover} = \frac{\text{Revenue}}{\text{Avg. AR}}, \quad DSO = \frac{365}{\text{turnover}} \] \[ \text{Inventory turnover} = \frac{COGS}{\text{Avg. Inv}}, \quad DOH = \frac{365}{\text{turnover}} \] \[ \text{Payables turnover} = \frac{\text{Purchases}}{\text{Avg. AP}}, \quad DPO = \frac{365}{\text{turnover}} \] \[ \text{Total asset turnover} = \frac{\text{Revenue}}{\text{Avg. Total Assets}} \] \[ \text{Cash conversion cycle} = DSO + DOH - DPO \]

\[ \text{Current ratio} = \frac{\text{Current assets}}{\text{Current liabilities}} \] \[ \text{Quick (acid-test)} = \frac{\text{Cash} + \text{ST inv} + \text{AR}}{\text{Current liab}} \] \[ \text{Cash ratio} = \frac{\text{Cash} + \text{ST inv}}{\text{Current liab}} \] \[ \text{Defensive interval} = \frac{\text{Cash} + \text{ST inv} + \text{AR}}{\text{Daily expenditures}} \]

\[ \text{Debt-to-equity} = \frac{\text{Total debt}}{\text{Total equity}} \] \[ \text{Debt-to-assets} = \frac{\text{Total debt}}{\text{Total assets}} \] \[ \text{Financial leverage} = \frac{\text{Avg. TA}}{\text{Avg. Equity}} \] \[ \text{Interest coverage (TIE)} = \frac{EBIT}{\text{Interest expense}} \] \[ \text{Fixed charge coverage} = \frac{EBIT + \text{Lease}}{\text{Interest} + \text{Lease}} \]

\[ \text{Gross margin} = \frac{\text{Gross profit}}{\text{Revenue}} \] \[ \text{Operating margin} = \frac{\text{Operating income}}{\text{Revenue}} \] \[ \text{Net margin} = \frac{NI}{\text{Revenue}} \] \[ \text{ROA} = \frac{NI}{\text{Avg. TA}}, \qquad \text{ROE} = \frac{NI}{\text{Avg. Equity}} \] \[ \text{Operating ROA} = \frac{EBIT}{\text{Avg. TA}} \]

\[ \text{ROE} = \underbrace{\frac{NI}{\text{Rev}}}_{\text{Net margin}} \times \underbrace{\frac{\text{Rev}}{\text{TA}}}_{\text{Asset turnover}} \times \underbrace{\frac{\text{TA}}{\text{Equity}}}_{\text{Leverage}} \] \[ \text{5-step ROE} = \frac{NI}{EBT} \times \frac{EBT}{EBIT} \times \frac{EBIT}{\text{Rev}} \times \frac{\text{Rev}}{\text{TA}} \times \frac{\text{TA}}{\text{Eq}} \]

\[ \text{P/E} = \frac{\text{Price}}{EPS}, \qquad \text{P/B} = \frac{\text{Price}}{\text{Book value per share}} \] \[ \text{P/S} = \frac{\text{Price}}{\text{Sales per share}}, \qquad \text{P/CF} = \frac{\text{Price}}{\text{CFO per share}} \] \[ \text{Dividend yield} = \frac{D_0}{P_0}, \qquad \text{Payout ratio} = \frac{\text{Dividends}}{NI} \] \[ \text{Retention} = 1 - \text{Payout}, \qquad g = \text{ROE} \times \text{Retention} \]

\[ WC = \text{Current assets} - \text{Current liabilities} \]

\[ CCC = DSO + DOH - DPO \]

\[ \text{Cost} = \left(1 + \frac{\text{Discount}}{1 - \text{Discount}}\right)^{365 / (\text{Net} - \text{Discount period})} - 1 \]

\[ NPV = -CF_0 + \sum_{t=1}^{N} \frac{CF_t}{(1 + r)^{t}} \]

\[ \sum_{t=0}^{N} \frac{CF_t}{(1 + IRR)^{t}} = 0 \]

\[ PI = \frac{PV(\text{future CF})}{|CF_0|} = 1 + \frac{NPV}{|CF_0|} \]

\[ WACC = \frac{E}{V}\,r_E + \frac{D}{V}\,r_D\,(1 - t) + \frac{P}{V}\,r_P \]

\[ V_L = V_U + t \cdot D \]

\[ \text{Leverage} = \frac{1}{\text{Initial margin}\,\%} \]

\[ P_{\text{call}} = P_0 \cdot \frac{1 - \text{IM}}{1 - \text{MM}} \]

\[ \text{Price-weighted}: w_i = \frac{P_i}{\sum P_i} \] \[ \text{Equal-weighted}: w_i = \frac{1}{N} \] \[ \text{Mkt-cap weighted}: w_i = \frac{P_i \cdot Q_i}{\sum P_i Q_i} \] \[ \text{Float-adjusted MCAP}: w_i = \frac{P_i \cdot \text{Float}_i}{\sum P_i \cdot \text{Float}_i} \] \[ \text{Fundamental-weighted}: w_i \propto \text{Fundamental}_i \text{ (sales, BV, CF, dividends)} \]

\[ \text{Index} = \frac{\sum P_i}{\text{Divisor}} \]

\[ r_E = R_f + \beta \cdot (R_m - R_f) \quad \text{(CAPM)} \]

\[ E(\text{value}) = \sum_{i} P_i \cdot V_i \]

\[ V_0 = \sum_{t=1}^{\infty} \frac{D_t}{(1 + r)^{t}} \]

\[ V_0 = \frac{D_1}{r - g} = \frac{D_0(1 + g)}{r - g}, \quad r > g \]

\[ V_0 = \sum_{t=1}^{n}\frac{D_t}{(1 + r)^{t}} + \frac{V_n}{(1 + r)^{n}}, \quad V_n = \frac{D_{n+1}}{r - g_L} \]

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

\[ \frac{P_0}{E_1} = \frac{1 - \text{Payout}}{\,r - g\,}\,\,? \quad \text{(no — see below)} \] \[ \frac{P_0}{E_1} = \frac{\text{Payout ratio}}{r - g} \] \[ \frac{P_0}{E_0} = \frac{\text{Payout}\,(1 + g)}{r - g} \quad \text{(trailing)} \]

\[ P/B = \frac{\text{Price}}{\text{BV per share}} \] \[ P/S = \frac{\text{Price}}{\text{Sales per share}} \] \[ P/CF = \frac{\text{Price}}{\text{CF per share}} \] \[ EV/EBITDA = \frac{\text{Mkt cap} + \text{Debt} - \text{Cash}}{EBITDA} \]

\[ P_0 = \sum_{t=1}^{N} \frac{C}{(1 + r)^{t}} + \frac{F}{(1 + r)^{N}} \]

\[ \text{Full (dirty) price} = PV_{\text{flat}} \cdot (1 + r)^{t/T} \] \[ \text{Accrued interest} = C \cdot \frac{t}{T} \] \[ \text{Flat (clean) price} = \text{Full price} - \text{Accrued interest} \]

\[ P_0 = \sum_{t=1}^{N} \frac{C}{(1 + YTM)^{t}} + \frac{F}{(1 + YTM)^{N}} \]

\[ \text{Current yield} = \frac{\text{Annual coupon}}{P_0} \]

\[ BEY = 2 \times \text{Semi-annual YTM} \]

\[ EAY = (1 + YTM_{\text{periodic}})^{m} - 1 \]

\[ \text{Coupon}_t = (\text{Reference rate}_{t-1} + \text{Quoted margin}) \cdot \frac{\text{Days in period}}{\text{Day count}} \]

\[ \text{Discount basis: } r_{BD} = \frac{F - P_0}{F} \cdot \frac{360}{T} \] \[ \text{Add-on rate (AOR): } r_{AOR} = \frac{F - P_0}{P_0} \cdot \frac{360}{T} \] \[ \text{Bond Equivalent Yield: BEY} = \frac{F - P_0}{P_0} \cdot \frac{365}{T} \]

\[ \text{Discount factor}: DF_T = \frac{1}{(1 + S_T)^{T}} \]

\[ (1 + S_B)^{B} = (1 + S_A)^{A} \cdot (1 + f_{A,\,B-A})^{B-A} \]

\[ HPR_{annual} = \left(\frac{\text{Total ending value}}{\text{Initial price}}\right)^{1/N} - 1 \]

\[ \text{MacD} = \sum_{t=1}^{N} \frac{t \cdot PV(CF_t)}{P_0} \]

\[ \text{ModD} = \frac{\text{MacD}}{1 + \tfrac{YTM}{m}} \]

\[ \text{Money duration} = \text{ModD} \times P_0 \] \[ PVBP = \frac{\text{Money duration}}{10{,}000} = \text{ModD} \times P_0 \times 0.0001 \]

\[ \text{Portfolio ModD} = \sum_{i=1}^{n} w_i \cdot \text{ModD}_i \]

\[ \%\Delta P \approx -\text{ModD} \cdot \Delta y + \tfrac{1}{2} \cdot \text{Convexity} \cdot (\Delta y)^{2} \]

\[ \text{EffD} = \frac{P_{-} - P_{+}}{2 \cdot P_0 \cdot \Delta\text{Curve}} \]

\[ \text{EffC} = \frac{P_{-} + P_{+} - 2P_0}{P_0 \cdot (\Delta\text{Curve})^{2}} \]

\[ EL = POD \times LGD \times EAD \]

\[ \text{Yield} = \text{Risk-free} + \text{Liquidity premium} + \text{Credit spread} \]

\[ \text{Debt/EBITDA} = \frac{\text{Total debt}}{EBITDA} \] \[ \text{Debt/Capital} = \frac{\text{Debt}}{\text{Debt} + \text{Equity}} \] \[ \text{FFO/Debt} = \frac{\text{Funds from operations}}{\text{Debt}} \]

\[ \text{EBITDA/Interest} = \frac{EBITDA}{\text{Interest}} \] \[ \text{EBIT/Interest} = \frac{EBIT}{\text{Interest}} \] \[ \text{FCF/Debt} = \frac{FCF}{\text{Debt}} \]

\[ SMM = 1 - (1 - CPR)^{1/12} \] \[ \text{PSA 100\%}: CPR \text{ ramps from 0.2\% in month 1 to 6\% by month 30, then constant} \]

\[ F_0 = S_0 \cdot (1 + r)^{T} + FV(\text{costs}) - FV(\text{benefits}) \]

\[ F_0(T) = S_0 \cdot (1 + r)^{T} \]

\[ F_0(T) = (S_0 - PV_0(I)) \cdot (1 + r)^{T} \]

\[ F_0(T) = (S_0 + PV_0(\text{costs})) \cdot (1 + r)^{T} \]

\[ V_t(\text{long}) = \frac{F_t(T) - F_0(T)}{(1 + r)^{T - t}} \]

\[ F_0(T) = S_0 \cdot (1 + r)^{T} - FV(\text{benefits}) + FV(\text{costs}) \]

\[ \text{Basis} = S_t - F_t \]

\[ V_{\text{pay-fixed swap}} = B_{\text{floating}} - B_{\text{fixed}} \]

\[ \text{Swap rate} = \frac{1 - PV(\text{final})}{\sum_{t=1}^{N} PV_t} \]

\[ \text{Call payoff: } \max(S_T - K,\,0) \] \[ \text{Put payoff: } \max(K - S_T,\,0) \]

\[ \text{Intrinsic (call)} = \max(S - K,\,0), \quad \text{Intrinsic (put)} = \max(K - S,\,0) \] \[ \text{Option price} = \text{Intrinsic} + \text{Time value} \]

\[ \max(0,\,S_0 - K(1+r)^{-T}) \le c_0 \le S_0 \] \[ \max(0,\,K(1+r)^{-T} - S_0) \le p_0 \le K(1+r)^{-T} \]

\[ c_0 + \frac{K}{(1 + r)^{T}} = p_0 + S_0 \]

\[ \text{Synthetic call} = p_0 + S_0 - K(1+r)^{-T} \] \[ \text{Synthetic put} = c_0 + K(1+r)^{-T} - S_0 \] \[ \text{Synthetic stock} = c_0 - p_0 + K(1+r)^{-T} \] \[ \text{Synthetic bond} = p_0 - c_0 + S_0 \]

\[ c_0 + \frac{K}{(1 + r)^{T}} = p_0 + S_0 - PV(\text{div}) \]

\[ \pi = \frac{(1 + r) - d}{u - d} \]

\[ c_0 = \frac{\pi \cdot c^{+} + (1 - \pi) \cdot c^{-}}{1 + r} \]

\[ h = \frac{c^{+} - c^{-}}{S_0(u - d)} \]

\[ \text{IRR}: \sum_{t=0}^{N}\frac{CF_t}{(1+r)^{t}} = 0 \] \[ MOIC = \frac{\text{Total distributions} + \text{Residual NAV}}{\text{Total contributions}} \] \[ DPI = \frac{\text{Distributions}}{\text{Contributions}}, \quad RVPI = \frac{\text{Residual NAV}}{\text{Contributions}}, \quad TVPI = DPI + RVPI \]

\[ NOI = \text{Rental income} - \text{Op. expenses} \] \[ \text{Cap rate} = \frac{NOI}{\text{Property value}} \] \[ \text{Property value} = \frac{NOI}{\text{Cap rate}} \]

\[ R_{\text{commodity}} = R_{\text{spot}} + R_{\text{roll}} + R_{\text{collateral}} \]

\[ E(R_p) = w_1 E(R_1) + w_2 E(R_2) \] \[ \sigma_p^{2} = w_1^{2}\sigma_1^{2} + w_2^{2}\sigma_2^{2} + 2 w_1 w_2 \rho_{12} \sigma_1 \sigma_2 \]

\[ E(R_p) = R_f + \frac{E(R_T) - R_f}{\sigma_T} \cdot \sigma_p \]

\[ E(R_i) = R_f + \beta_i \cdot [E(R_m) - R_f] \]

\[ \beta_i = \frac{\text{Cov}(R_i, R_m)}{\sigma_m^{2}} = \rho_{i,m} \cdot \frac{\sigma_i}{\sigma_m} \]

\[ \text{Sharpe} = \frac{R_p - R_f}{\sigma_p} \quad \text{(total risk)} \] \[ \text{Treynor} = \frac{R_p - R_f}{\beta_p} \quad \text{(systematic risk)} \] \[ \alpha_p = R_p - [R_f + \beta_p (R_m - R_f)] \quad \text{(Jensen's alpha)} \] \[ M^{2} = (R_p - R_f) \cdot \frac{\sigma_m}{\sigma_p} - (R_m - R_f) \quad \text{(M-squared)} \] \[ \text{Information ratio} = \frac{R_p - R_b}{\sigma(R_p - R_b)} \]