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Free CFA Level II Formula Sheet (2026)

Every CFA Level II formula you need on the test, grouped by topic and rendered with full math notation. 116 formulas across 9 topics, calibrated to the 2026 syllabus. Free forever, no signup required.

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All CFA Level II Formulas

Quantitative Methods 15 items
R-squared (coefficient of determination)
R2=RSSSST=1−SSESSTR^2 = \frac{RSS}{SST} = 1 - \frac{SSE}{SST}
RSS = regression sum of squares, SSE = error sum of squares
SST = total sum of squares. Fraction of variation explained.
Adjusted R-squared
Rˉ2=1−n−1n−k−1(1−R2)\bar{R}^2 = 1 - \frac{n-1}{n-k-1}(1 - R^2)
n = observations, k = number of independent variables
Penalizes for adding irrelevant predictors
F-statistic (overall regression)
F=RSS/kSSE/(n−k−1)=MSRMSEF = \frac{RSS/k}{SSE/(n-k-1)} = \frac{MSR}{MSE}
k = predictors, n = observations
Tests H0:allslopecoefficients=0H_0: all slope coefficients = 0
Standard error of regression (SEE)
SEE=SSEn−k−1=MSESEE = \sqrt{\frac{SSE}{n - k - 1}} = \sqrt{MSE}
SSE = sum of squared errors, n = observations, k = independent variables
Measures typical prediction error of the model
AR(1) model one-period forecast
x^t+1∣t=b^0+b^1xt\hat{x}_{t+1\mid t}=\hat{b}_0+\hat{b}_1x_t, where xtx_t is the observed current value and b^0,b^1\hat{b}_0,\hat{b}_1 are estimated coefficients. The future error has conditional mean zero; stationarity requires ∣b1∣<1|b_1|<1.
Multiple regression model equation
Y=b0+b1X1+b2X2+⋯+bkXk+eY = b_0 + b_1 X_1 + b_2 X_2 + \dots + b_k X_k + e, Y = dependent variable, b0 = intercept, bj = partial slope on Xj, k = number of independent variables, e = error term
t-statistic for an individual regression coefficient
t=bj−Bjsbjt=\frac{b_j-B_j}{s_{b_j}}, where bjb_j is the estimated slope coefficient, BjB_j is the hypothesized coefficient, and sbjs_{b_j} is its standard error. For Bj=0B_j=0, this reduces to bj/sbjb_j/s_{b_j}. Degrees of freedom: n−k−1n-k-1, where nn is observations and kk is independent variables.
Partial F-test for a group of regression variables
F=(SSER−SSEU)/qSSEU/(n−kU−1)F = \frac{(SSE_R - SSE_U)/q}{SSE_U/(n - k_U - 1)}, SSERSSE_R = restricted SSE, SSEUSSE_U = unrestricted SSE, q = variables tested, n = observations, kUk_U = unrestricted predictors
Bias-variance decomposition of total prediction error
Total error=bias2+variance+irreducible noise\text{Total error} = \text{bias}^2 + \text{variance} + \text{irreducible noise}, bias = error from oversimplification, variance = error from sensitivity to training data, irreducible noise = inherent randomness
Classification precision and recall
precision=TPTP+FP,recall=TPTP+FN\text{precision} = \frac{TP}{TP + FP}, \quad \text{recall} = \frac{TP}{TP + FN}, TP = true positives, FP = false positives, FN = false negatives
Breusch-Pagan test statistic
BP=n×R2BP = n \times R^2, n = number of observations, R2R^2 = R-squared from regressing squared residuals on the original X variables; compared to chi-squared with k df
Variance inflation factor
VIFj=11−Rj2VIF_j = \dfrac{1}{1 - R_j^2}, Rj2R_j^2 = R-squared from regressing XjX_j on all other independent variables; VIF > 10 = severe multicollinearity
Odds ratio from a logistic coefficient
OR=ebjOR = e^{b_j}, OR = odds ratio (multiplier on odds per unit increase in XjX_j), bjb_j = logistic coefficient on XjX_j; OR>1 increases odds, OR<1 decreases
Logistic regression probability from log-odds
p=11+e−(b0+b1X1+b2X2+… )p = \dfrac{1}{1 + e^{-(b_0 + b_1 X_1 + b_2 X_2 + \dots)}}, p = probability of outcome, b = coefficients, X = predictors, log-odds = b0+b1X1+…b_0 + b_1 X_1 + \dots
F1 score and accuracy (confusion matrix)
F1=2PRP+RF1=\frac{2PR}{P+R}, Accuracy=TP+TNTP+FP+TN+FN\text{Accuracy}=\frac{TP+TN}{TP+FP+TN+FN}
P = precision, R = recall, TP/TN = true positives/negatives, FP/FN = false positives/negatives. F1 is the harmonic mean of precision and recall; prefer it to accuracy when the class distribution is unequal.
Economics 7 items
Covered Interest Rate Parity (CIP)
FS=1+rd1+rf\frac{F}{S} = \frac{1 + r_d}{1 + r_f}
F = forward rate (d/f), S = spot rate (d/f)
rdr_d = domestic rate, rfr_f = foreign rate
No-arbitrage; holds in practice
International Fisher effect
rnom,d−rnom,f≈E(πd)−E(πf)r_{nom,d}-r_{nom,f}\approx E(\pi_d)-E(\pi_f)
Nominal interest rate differentials reflect expected inflation differentials.
Combines the Fisher effect with real interest rate parity.
Relative PPP
%ΔSd/f≈πd−πf\%\Delta S_{d/f}\approx\pi_d-\pi_f
Actual percentage change in the spot rate is approximately the actual inflation differential.
Holds better over long horizons.
Unhedged foreign asset return (in domestic currency)
(1+RDC)=(1+RFC)(1+RFX)(1 + R_{\text{DC}}) = (1 + R_{\text{FC}})(1 + R_{FX}); approx RDC≈RFC+RFXR_{\text{DC}} \approx R_{\text{FC}} + R_{FX}. RFXR_{FX} = foreign-currency appreciation vs. domestic.
Labor productivity growth
gLP=gY−gLg_{LP} = g_Y - g_L, gLPg_{LP} = growth in output per worker, gYg_Y = GDP growth, gLg_L = labor force growth
Uncovered interest rate parity expected spot change
E[%ΔSprice/base]≈rprice−rbaseE[\%\Delta S_{price/base}] \approx r_{price} - r_{base}, E[%ΔS] = expected change in spot (price per base), rpricer_{price} = price-currency rate, rbaser_{base} = base-currency rate
Taylor rule policy rate
i=rn+π+0.5(π−π∗)+0.5(y−y∗)i = r_n + \pi + 0.5(\pi - \pi^*) + 0.5(y - y^*), i = policy rate, rnr_n = neutral real rate, π = inflation, π* = target inflation, y - y* = output gap
Financial Statement Analysis 12 items
Net pension expense components (ASC 715)
ASC 715: NPPC=SC+IC−ER+APSC±AGL\mathrm{NPPC}=SC+IC-ER+A_{PSC}\pm A_{GL} (current service, interest, expected return, amortized past service and gains/losses). IAS 19: P&L=SC+PSC+NI\mathrm{P\&L}=SC+PSC+NI; NI=opening net pension liability×discount rateNI=\text{opening net pension liability}\times\text{discount rate} (asset: income). Remeasurements: OCI, no P&L amortization.
US GAAP net periodic pension cost
PE=SC+IC−ER+APSC±APE=SC+IC-ER+A_{PSC}\pm A. SCSC: current service cost; ICIC: interest on beginning PBO; ERER: expected asset return; APSCA_{PSC}: amortized past service cost. AA: corridor amortization of actuarial obligation changes and actual-minus-expected asset returns; add losses, subtract gains.
Total periodic pension cost
TPPC=−ΔFS+CTPPC = -\Delta FS + C, ΔFS\Delta FS = change in funded status (plan assets − PBO), C = employer contributions; equals SC + IC − actual return ± actuarial gains/losses
Bank efficiency ratio
Efficiency=Non-interest expenseNII+Non-interest income\text{Efficiency} = \dfrac{\text{Non-interest expense}}{\text{NII} + \text{Non-interest income}}, NII = net interest income; provisions deliberately excluded; lower is better
Full goodwill under the acquisition method
GW=(C+NCIFV)−FVNAGW = (C + NCI_{FV}) - FV_{NA}, GW = goodwill, C = consideration transferred, NCIFVNCI_{FV} = non-controlling interest at fair value, FVNAFV_{NA} = fair value of identifiable net assets
Equity method investment carrying value (CAID)
CV=BegCV+s×NI−Amort−s×Div+s×OCICV = BegCV + s \times NI - Amort - s \times Div + s \times OCI, CV = carrying value, s = ownership share, NI = investee net income, Amort = amortization of excess purchase price, Div = dividends
Pre-CTA translated equity build-up (current rate method)
Epre=Ebeg+NIavg−DivtxnE_{pre}=E_{beg}+NI_{avg}-Div_{txn}, where EbegE_{beg} is beginning translated equity excluding the opening cumulative translation adjustment, NIavgNI_{avg} is net income at the average rate, and DivtxnDiv_{txn} is dividends at the transaction rate.
Cumulative translation adjustment plug (current rate method)
CTA=Ereq−EpreCTA = E_{req} - E_{pre}, EreqE_{req} = translated net assets (assets − liabilities at current rate), EpreE_{pre} = pre-CTA translated equity
Balance sheet accruals ratio
Accruals ratio=ΔNOAAverage NOA\text{Accruals ratio}=\dfrac{\Delta NOA}{\text{Average }NOA}, where NOANOA is net operating assets =(total assets−cash and short-term investments)−(total liabilities−financial debt)=(\text{total assets}-\text{cash and short-term investments})-(\text{total liabilities}-\text{financial debt}), and ΔNOA\Delta NOA is the change in NOANOA over the period.
Cash flow accruals ratio
Accruals ratio=NI−CFO−CFIAverage NOA\text{Accruals ratio} = \dfrac{NI - CFO - CFI}{\text{Average } NOA}, NI = net income, CFO = cash flow from operations, CFI = cash flow from investing, NOA = net operating assets
LIFO-to-FIFO equity adjustment
EquityFIFO=EquityLIFO+LR×(1−t)Equity_{FIFO} = Equity_{LIFO} + LR \times (1 - t), LR = LIFO reserve, t = tax rate; inventory rises by full LR and COGS falls by the change in LR
Five-component DuPont decomposition of ROE
ROE=NIEBT×EBTEBIT×EBITRev×RevAssets×AssetsEquityROE = \frac{NI}{EBT} \times \frac{EBT}{EBIT} \times \frac{EBIT}{Rev} \times \frac{Rev}{Assets} \times \frac{Assets}{Equity}, NI = net income, EBT = pretax income, EBIT = operating income, Rev = revenue, Assets = avg total assets, Equity = avg equity
Corporate Issuers 7 items
MM Propositions I & II (with taxes)
Prop I: VL=VU+tDV_L = V_U + tD
debt tax shield adds value.
Prop II: re=r0+(r0−rd)(1−t)DEr_e = r_0 + (r_0 - r_d)(1-t)\frac{D}{E}; WACC declines with leverage. Theoretical optimum: 100% debt.
Pure-play unlevered (asset) beta via Hamada equation
βasset=βequity1+(1−t) (D/E)\beta_{asset} = \dfrac{\beta_{equity}}{1 + (1 - t)\,(D/E)}, β = beta, t = comparable's tax rate, D/E = comparable's debt-to-equity ratio
Weighted average cost of capital (WACC)
WACC=wd rd(1−t)+wp rp+we reWACC = w_d \, r_d(1 - t) + w_p \, r_p + w_e \, r_e, w = target weight, r = component cost, t = tax rate; subscripts d=debt, p=preferred, e=equity
Lintner target payout adjustment model expected dividend
D1=D0+a×(TPR×EPS1−D0)D_1 = D_0 + a \times (\text{TPR} \times \text{EPS}_1 - D_0), D1D_1 = expected dividend, D0D_0 = current dividend, a = adjustment factor, TPR = target payout ratio, EPS1EPS_1 = projected EPS
Post-repurchase earnings per share
EPSnew=EN−C/P\text{EPS}_{new} = \dfrac{E}{N - C/P}, E = total earnings, N = original shares, C = cash spent on repurchase, P = repurchase price per share
LBO multiple of invested capital
MOIC=Exit EquityInitial EquityMOIC = \frac{Exit\ Equity}{Initial\ Equity}, Exit Equity = Exit EV − remaining debt, Initial Equity = sponsor's equity contribution at entry
Acquirer's gain in an acquisition
Gain=PV(Synergies)−PremiumGain = PV(Synergies) - Premium, PV(Synergies) = present value of net after-tax synergies, Premium = offer value above target's pre-announcement market value
Equity Valuation 21 items
FCFF from EBIT
FCFF=EBIT(1−t)+D&A−ΔWC−CapExFCFF = EBIT(1-t) + D\&A - \Delta WC - CapEx
t = tax rate, D&A = depreciation & amortization
ΔWC\Delta WC = change in working capital
FCFE from FCFF
FCFE=FCFF−Int(1−t)+ΔDebtFCFE = FCFF - Int(1-t) + \Delta\text{Debt}
Int = interest expense, t = tax rate, ΔDebt\Delta\text{Debt} = net new borrowing.
Residual income model
V0=B0+∑t=1∞(ROEt−re)Bt−1(1+re)tV_0 = B_0 + \sum_{t=1}^{\infty} \frac{(ROE_t - r_e) B_{t-1}}{(1+r_e)^t}
B0B_0 = book value, ROE = return on equity, rer_e = cost of equity.
Sustainable growth rate
g=ROE×bg = ROE \times b
b = retention ratio = 1 − payout ratio
ROE = net income / equity
Growth rate achievable without changing capital structure or issuing equity
Two-stage DDM
V0=∑t=1nD0(1+gS)t(1+r)t+Vn(1+r)nV_0 = \sum_{t=1}^{n} \frac{D_0(1+g_S)^t}{(1+r)^t} + \frac{V_n}{(1+r)^n}
Vn=Dn+1r−gLV_n = \frac{D_{n+1}}{r - g_L}
gSg_S = high growth (stage 1), gLg_L = long-run growth (stage 2)
FCFE from net income
FCFE=NI+NCC−ΔWC−CapEx+ΔDebtFCFE = NI + NCC - \Delta WC - \text{CapEx} + \Delta\text{Debt}
NI = net income, NCC = non-cash charges (D&A, deferred tax), ΔDebt\Delta\text{Debt} = net new borrowing.
FCFE constant-growth valuation
V0=FCFE1re−gV_0 = \frac{FCFE_1}{r_e - g}. Requires re>gr_e > g; g = sustainable FCFE growth. Gordon-growth analog using FCFE instead of dividends.
Build-up method for cost of equity (private company)
re=rf+ERP+Size premium+Specific-company premium±Industry premiumr_e = r_f + \text{ERP} + \text{Size premium} + \text{Specific-company premium} \pm \text{Industry premium}. Used when CAPM fails for illiquid/private firms.
Sum-of-the-parts enterprise value
VSOTP=∑iVi−(m×OH)V_{SOTP} = \sum_i V_i - (m \times OH), ViV_i = standalone value of segment i, m = overhead capitalization multiple, OH = annual corporate overhead
Expected return as dividend yield plus capital gains yield
E(r)=D1P0+E(P1)−P0P0E(r)=\frac{D_1}{P_0}+\frac{E(P_1)-P_0}{P_0}, where D1D_1 is the next dividend, P0P_0 is the current price, and E(P1)E(P_1) is the expected end-of-period price.
H-model value
V0=D0(1+gL)r−gL+D0⋅H(gS−gL)r−gLV_0 = \dfrac{D_0(1+g_L)}{r-g_L} + \dfrac{D_0 \cdot H (g_S-g_L)}{r-g_L}, D0D_0 = current dividend, gSg_S = initial high growth, gLg_L = long-run growth, H = half the high-growth period, r = required return
Residual income continuing value with persistence factor
CV=ω×RIT1+r−ωCV = \dfrac{\omega \times RI_T}{1 + r - \omega}, ω = persistence factor, RITRI_T = final-year residual income, r = cost of equity
Residual income (spread form)
RI=B×(ROE−r)RI = B \times (ROE - r), B = beginning book value, ROE = return on equity, r = cost of equity
Capitalized cash flow method value
V=CFr−gV = \frac{CF}{r - g}, VV = firm value, CFCF = expected normalized next-period FCFF, rr = WACC, gg = stable growth rate
FCFF from net income
FCFF=NI+NCC+Int(1−t)−CapEx−ΔWCFCFF=NI+NCC+Int(1-t)-CapEx-\Delta WC, where NINI = net income, NCCNCC = net noncash charges, IntInt = interest expense, tt = tax rate, CapExCapEx = capital expenditures, and ΔWC\Delta WC = change in working capital.
Single-stage FCFF enterprise value
EV=FCFF1WACC−gEV = \frac{FCFF_1}{WACC - g}, EV = enterprise value, FCFF₁ = next-period free cash flow to firm, WACC = weighted average cost of capital, g = stable growth rate
Gordon Growth Model value
V0=D1r−gV_0 = \dfrac{D_1}{r - g}, V0V_0 = intrinsic value, D1=D0(1+g)D_1 = D_0(1+g) = next-period dividend, r = required return on equity, g = constant growth (g < r)
Discount for lack of control from a control premium
DLOC=1−11+CPDLOC = 1 - \frac{1}{1 + CP}, DLOC = discount for lack of control, CP = control premium
Justified price-to-book ratio
PB=ROE−gr−g\frac{P}{B} = \frac{ROE-g}{r-g}, ROE = return on equity, g = growth rate, r = required return on equity
Justified leading price-to-earnings ratio
PE1=1−br−g\frac{P}{E_1} = \frac{1-b}{r-g}, b = retention ratio (1-b = payout), r = required return on equity, g = growth rate
P/E-to-growth (PEG) ratio
PEG=P/Eg\text{PEG}=\frac{P/E}{g}
P/E = stock's price-to-earnings ratio, g = expected earnings growth rate in percentage points (12% enters as 12). Lower PEG is more attractive, all else equal; it assumes a linear P/E-growth relation and ignores risk and growth duration.
Fixed Income 11 items
Option-Adjusted Spread (OAS)
Callable: OAS=Z-spread−call-option spread equivalent\mathrm{OAS}=\text{Z-spread}-\text{call-option spread equivalent}. Putable: OAS=Z-spread+put-option spread equivalent\mathrm{OAS}=\text{Z-spread}+\text{put-option spread equivalent}. Both option spread equivalents are positive amounts in bps.
Callable: OAS<Z-spread\mathrm{OAS}<\text{Z-spread}. Putable: OAS>Z-spread\mathrm{OAS}>\text{Z-spread}.
Effective duration
Deff=P−Δy−P+Δy2P0ΔyD_{\text{eff}} = \frac{P_{-\Delta y} - P_{+\Delta y}}{2 P_0 \Delta y}. Used for bonds with embedded options (callable, putable, MBS).
Portfolio effective duration (market-value weighted)
Deff, port=∑iwiDeff,iD_{\text{eff, port}} = \sum_i w_i D_{\text{eff},i}; wiw_i = market-value weight. Assumes parallel yield-curve shift; use KRDs for non-parallel.
Credit spread decomposition (term structure)
Observed Spread=Expected Loss+Credit Risk Premium+Liquidity Premium+Tax Component\text{Observed Spread}=\text{Expected Loss}+\text{Credit Risk Premium}+\text{Liquidity Premium}+\text{Tax Component}. Expected Loss=PD×LGD\text{Expected Loss}=PD\times LGD.
Hazard rate from credit spread (reduced-form model)
λ≈spread1−R\lambda\approx\dfrac{\text{spread}}{1-R}, where λ\lambda is the hazard rate (default intensity), spread is the credit spread, RR is the recovery rate, and 1−R1-R is loss given default.
Value of a callable bond
Vcallable=Voption−free−VcallV_{callable} = V_{option-free} - V_{call}, VoptionV_{option}-free = value of identical option-free bond, VcallV_{call} = value of the embedded call option held by the issuer
Value of a putable bond
Vputable=Voption−free+VputV_{putable} = V_{option-free} + V_{put}, VoptionV_{option}-free = value of identical option-free bond, VputV_{put} = value of the embedded put option held by the bondholder
Rolldown return from riding the yield curve
Rroll≈−D×ΔyR_{roll}\approx-D\times\Delta y, where Δy\Delta y is the shorter-maturity yield minus the initial yield and DD is modified duration. This is the price-return component and assumes a stable yield curve over the holding period.
CDS mark-to-market gain to protection buyer
MTM≈(scur−scontract)×D×N\text{MTM}\approx(s_{cur}-s_{contract})\times D\times N, where scurs_{cur} is the current spread, scontracts_{contract} is the contracted entry spread, not the fixed coupon, DD is effective spread duration, and NN is notional.
Backward induction node value in a binomial interest rate tree
Vnode=0.5×(Vup+Vdown)+C1+rnodeV_{node} = \dfrac{0.5 \times (V_{up} + V_{down}) + C}{1 + r_{node}}, V = bond value, C = coupon, r = node rate, 0.5 = risk-neutral probabilities
Lognormal up and down rate relationship in a binomial tree
rup=rdown×e2σr_{up} = r_{down} \times e^{2\sigma}, rupr_{up} = higher node rate, rdownr_{down} = lower node rate, σ = annual std dev of ln of short rate
Derivatives 13 items
Binomial option pricing (one period)
c=πucu+πdcd1+rc = \frac{\pi_u c_u + \pi_d c_d}{1+r}; πu=(1+r)−du−d\pi_u = \frac{(1+r) - d}{u - d}, πd=1−πu\pi_d = 1 - \pi_u (risk-neutral probs). u, d = up/down factors.
Put-call parity
c+X(1+r)T=p+S0c+\frac{X}{(1+r)^T}=p+S_0. Continuous: c+Xe−rT=p+S0c+Xe^{-rT}=p+S_0. Same non-dividend-paying stock, strike and expiry; European options.
Option gamma
Γ=∂2V∂S2=∂Δ∂S\Gamma = \frac{\partial^2 V}{\partial S^2} = \frac{\partial \Delta}{\partial S}. Highest ATM near expiry; long options are positive gamma, short are negative.
Option vega
ν=∂V∂σ\nu=\frac{\partial V}{\partial\sigma}. For σ\sigma in decimal units, a +1-percentage-point volatility move changes the option price by approximately 0.01ν0.01\nu. Long call or put = positive vega. Highest ATM with longer expiry.
Delta-gamma-vega option price approximation
ΔC≈δ dS+12γ(dS)2+ν dσ\Delta C \approx \delta \, dS + \tfrac{1}{2} \gamma (dS)^2 + \nu \, d\sigma, δ = delta, γ = gamma, ν = vega, dS = stock move, dσ = vol change
Black-Scholes-Merton call price
C=S N(d1)−Ke−rTN(d2)C = S \, N(d_1) - K e^{-rT} N(d_2), C = call value, S = spot, K = strike, r = risk-free rate, T = time, N = standard normal CDF, d1,d2d_1,d_2 = moneyness terms
Forward rate agreement settlement to the long (pay-fixed, receive-floating) position
P=NA×(rf−rFRA)×DCF1+rd×DCFP=\dfrac{NA\times(r_f-r_{FRA})\times DCF}{1+r_d\times DCF}, where NANA is notional, rfr_f is the floating reference rate, rdr_d is the settlement discount rate, rFRAr_{FRA} is the contract rate, and DCFDCF is the day-count fraction of the reference period. Set rd=rfr_d=r_f when that equality is assumed.
Fixed swap rate from discount factors
s=1−Z(n)∑t=1nZ(t)s = \dfrac{1 - Z(n)}{\sum_{t=1}^{n} Z(t)}, s = periodic fixed swap rate, Z(t) = discount factor for period t, n = number of settlement periods
Option theta (time decay)
Θ=∂V∂t\Theta = \dfrac{\partial V}{\partial t}, the change in an option value per unit of calendar time. Usually negative for long options (value erodes toward expiry); time decay is largest for at-the-money options near expiration.
Payer swaption payoff
Payer=N⋅max⁡(Rmkt−RK,0)⋅A\text{Payer}=N\cdot\max(R_{\text{mkt}}-R_K,0)\cdot A, where NN is notional, RmktR_{\text{mkt}} is the market swap rate at option expiration TT, RKR_K is the strike rate, and A=AP∑iPVT,tiA=AP\sum_i PV_{T,t_i}. Here APAP is the fixed-leg accrual fraction and PVT,tiPV_{T,t_i} discounts the payment at tit_i to TT.
Caplet payoff
Caplet=N⋅max⁡(Rref−Rcap,0)⋅days360\text{Caplet} = N \cdot \max(R_{\text{ref}} - R_{\text{cap}}, 0) \cdot \frac{\text{days}}{360}, N = notional, RrefR_{ref} = reference rate, RcapR_{cap} = cap strike rate, days/360 = day-count fraction
Risk-neutral probability in a binomial tree
p=1+r−du−d p = \frac{1 + r - d}{u - d} , p = risk-neutral (up) probability, r = periodic risk-free rate, u = up factor, d = down factor
Binomial hedge ratio (option delta)
Δ=cu−cdSu−Sd\Delta=\frac{c_u-c_d}{S_u-S_d}, where Δ\Delta is shares per option, cuc_u and cdc_d are option values at the next up and down nodes, and SuS_u and SdS_d are stock prices at those nodes. The option values are terminal payoffs when the next step is expiration.
Alternative Investments 14 items
TVPI (Total Value to Paid-In)
TVPI=Distributions+Residual NAVPaid-inTVPI = \frac{\text{Distributions} + \text{Residual NAV}}{\text{Paid-in}} = DPI + RVPI (realized + unrealized multiples).
Direct capitalization (real estate)
V=NOI1Cap RateV=\frac{NOI_1}{\text{Cap Rate}}; NOI1NOI_1 = stabilized first-year NOI (after operating expenses, before debt service and income taxes). Implicit: cap rate=r−g\text{cap rate}=r-g.
Equity REIT NAV per share
NAV/share=Property value+Other assets−Total liabilitiesShares outstanding\text{NAV/share} = \frac{\text{Property value} + \text{Other assets} - \text{Total liabilities}}{\text{Shares outstanding}}
Property value typically from cap-rate or DCF on stabilized NOI. Price/NAV reveals premium or discount.
Pre-money and post-money valuation
Post-money=Pre-money+New investment\text{Post-money} = \text{Pre-money} + \text{New investment}
Investor share=InvestmentPost-money\text{Investor share} = \frac{\text{Investment}}{\text{Post-money}}
Price per share = pre-money / pre-money shares. PE / VC funding rounds.
Total return on a commodity futures position
Rtotal=Rspot+Rroll+RcollateralR_{total} = R_{spot} + R_{roll} + R_{collateral}, RspotR_{spot} = spot price return, RrollR_{roll} = roll yield, R_collateral = interest earned on posted collateral
Commodity forward price under cost-of-carry
F=S×e(r+c−y)×TF = S \times e^{(r + c - y) \times T}, F = forward price, S = spot price, r = risk-free rate, c = storage cost rate, y = convenience yield, T = time in years
Hard-hurdle incentive fee
Fee=p(R−h)×AUM\text{Fee}=p(R-h)\times AUM when R>hR>h, and Fee=0\text{Fee}=0 when R≤hR\le h. Here pp is the incentive rate, RR is return after management fees and expenses but before incentive fees, hh is the hard-hurdle rate, and AUMAUM is beginning-of-period assets under management.
Variance correction under first-order return smoothing
For Rtreported=(1−ρ)Rttrue+ρRt−1reportedR_t^{\text{reported}}=(1-\rho)R_t^{\text{true}}+\rho R_{t-1}^{\text{reported}}, with stationary, serially uncorrelated true returns and 0≤ρ<10\le\rho<1, σtrue2=σreported21+ρ1−ρ\sigma_{\text{true}}^2=\sigma_{\text{reported}}^2\dfrac{1+\rho}{1-\rho}. Here ρ\rho is the first-order autocorrelation of reported returns.
Funds from operations for a REIT
FFO=NI+DepRE−GsaleFFO = NI + Dep_{RE} - G_{sale}, NI = GAAP net income, DepREDep_{RE} = real estate depreciation/amortization, GsaleG_{sale} = gains on sale of properties
Adjusted funds from operations for a REIT
AFFO=FFO−Capexmaint−SLRAFFO=FFO-Capex_{maint}-SLR, where FFOFFO is funds from operations, CapexmaintCapex_{maint} is recurring maintenance-type capital expenditures and leasing costs, including leasing commissions and tenant improvement allowances, and SLRSLR is the straight-line rent adjustment.
Net operating income for a property
NOI=PGI−V−OpEx−MANOI=PGI-V-OpEx-MA. PGIPGI: full-occupancy gross income; VV: vacancy/collection losses; OpExOpEx: operating expenses, excluding debt service, capex, and income tax; MAMA: maintenance allowance, including capex needed to sustain current economic use.
Cash-on-cash return on a leveraged property
CoC=NOI−DSECoC = \frac{NOI - DS}{E}, NOI = net operating income, DS = annual debt service, E = equity invested
Alpha of an equity long/short fund
α=Rgross−(Enet×Rm)\alpha = R_{gross} - (E_{net} \times R_m), RgrossR_{gross} = fund gross return, EnetE_{net} = net market exposure, RmR_m = market return
Net and gross exposure of a long/short fund
Enet=L−SE_{net} = L - S, Egross=L+SE_{gross} = L + S, LL = long exposure (% of capital), SS = short exposure (% of capital); net = direction, gross = leverage
Portfolio Management 16 items
Treynor ratio
Tp=Rp−RfβpT_p = \frac{R_p - R_f}{\beta_p}
RpR_p = portfolio return, RfR_f = risk-free rate, βp\beta_p = portfolio beta
Excess return per unit of systematic risk
Jensen's alpha
αp=Rp−[Rf+βp(Rm−Rf)]\alpha_p = R_p - [R_f + \beta_p(R_m - R_f)]
Actual return minus CAPM expected return
Positive α\alpha = outperformance after adjusting for systematic risk
Fundamental Law of Active Management
IR=TC×IC×BRIR=TC\times IC\times\sqrt{BR}, where TCTC is the transfer coefficient; TC=1TC=1 for an unconstrained optimal portfolio.
ICIC = information coefficient (skill), BRBR = breadth (independent bets), IRIR = information ratio.
M-squared (Modigliani-Modigliani)
M2=(Rp−Rf)×σmσp+RfM^2 = (R_p - R_f) \times \frac{\sigma_m}{\sigma_p} + R_f
Levered/delevered portfolio return at market's risk level
Expressed in %, directly comparable across portfolios
Sharpe-information ratio relationship
SRP=SRB2+IR2SR_P=\sqrt{SR_B^2+IR^2}, where SRPSR_P is the maximum expected portfolio Sharpe ratio after optimally scaling active risk by combining the active portfolio and benchmark, SRBSR_B is the benchmark Sharpe ratio, and IRIR is the information ratio. The relationship is Pythagorean, never additive.
Optimal level of active risk for an unconstrained portfolio with benchmark beta equal to \(1\)
σA∗=IRSRB×σB\sigma_A^* = \frac{IR}{SR_B} \times \sigma_B, σAσ_A* = optimal active risk, IR = information ratio, SRBSR_B = benchmark Sharpe ratio, σBσ_B = benchmark standard deviation
Break-even inflation rate
BEI=ynom−yrealBEI=y_{nom}-y_{real}, where BEIBEI is expected average inflation over the bonds’ maturity plus an inflation risk premium, ynomy_{nom} is the nominal government bond yield, and yrealy_{real} is the yield on an inflation-linked bond of the same maturity.
Forward-looking equity risk premium (DDM)
ERP=(DY+g)−rfERP=(DY+g)-r_f, where DYDY is the forward dividend yield D1/P0D_1/P_0, gg is the constant expected nominal dividend growth rate, equal to earnings growth when the payout ratio is constant, and rfr_f is the risk-free government bond rate.
Authorized participant creation arbitrage profit
π=(Pmkt−NAV−c)×Q\pi = (P_{mkt} - NAV - c) \times Q, PmktP_{mkt} = ETF market price per share, NAV = net asset value per share, c = transaction cost per share, Q = creation unit size (shares)
Total cost of ETF ownership
TC=ER×H+Crt+Srt+PD+OHTC=ER\times H+C_{rt}+S_{rt}+PD+O_H. ERER=annual expense ratio; HH=years held; Crt,SrtC_{rt},S_{rt}=round-trip commissions and bid-ask spread; PDPD=net entry/exit premium/discount; OHO_H=other costs (turnover, taxes, tracking effects) less securities lending income. No double count; one return basis.
Arbitrage Pricing Theory expected return
E(Ri)=Rf+∑k=1Kβi,kλkE(R_i) = R_f + \sum_{k=1}^{K} \beta_{i,k} \lambda_k, RfR_f = risk-free rate, β = factor sensitivity, λ = factor risk premium, K = number of factors
Active specific risk from active risk decomposition
Active specific risk=(Total active risk)2−Active factor risk\text{Active specific risk}=(\text{Total active risk})^2-\text{Active factor risk}. Total active risk is tracking error, a standard deviation; active factor risk and active specific risk are variance contributions.
Square root of time VaR scaling
Assuming zero expected return, VaRT=VaRdaily×TVaR_T=VaR_{daily}\times\sqrt{T}, where VaRTVaR_T is VaR over TT days, VaRdailyVaR_{daily} is one-day VaR, and TT is the number of trading days. Variance scales linearly, so standard deviation scales with T\sqrt{T}.
Parametric (variance-covariance) Value at Risk
VaR=V(zσ−μ)VaR=V(z\sigma-\mu), where VV is portfolio value, zz is the z-score (1.651.65 at 95%95\%, 2.332.33 at 99%99\%), σ\sigma is periodic return standard deviation, and μ\mu is expected portfolio return over the same period as σ\sigma.
Annual transaction cost drag from turnover
Drag=T×cDrag = T \times c, T = annual portfolio turnover, c = cost per trade (including market impact, in bps or %)
Bonferroni correction adjusted significance threshold
αadj=αm\alpha_{adj} = \dfrac{\alpha}{m}, αadj\alpha_{adj} = per-test threshold, α\alpha = desired overall significance level, m = number of tests run
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