Free GARP FRM Part II Formula Sheet (2026)

Every FRM Part II formula you need on the test, grouped by topic, rendered with full math notation. 71 formulas across 6 topics, calibrated to the 2026 syllabus. Free forever, no signup required.

71 Formulas
6 Topics
2026 Syllabus
Free Forever
Print-ready PDF: 1080x1350 portrait, math pre-rendered, fonts embedded. Download once, study anywhere.
Download PDF →

All FRM Part II Formulas

Market Risk 15 items
Sklar's theorem joint distribution decomposition
F(x1,…,xn)=C(F1(x1),…,Fn(xn))F(x_1,\dots,x_n) = C(F_1(x_1),\dots,F_n(x_n)) — F = joint CDF, F_i = marginal CDFs, C = copula on the unit hypercube
Put-call parity
C−P=S−Ke−rTC - P = S - K e^{-rT} — C = call price, P = put price, S = spot, K = strike, r = risk-free rate, T = time to expiry
Half-life of a Vasicek rate shock
t1/2=ln⁡2/kt_{1/2} = \ln 2 / k — k = mean reversion speed; time for a rate shock to decay to 50% of its initial magnitude
Vasicek short-rate dynamics
dr=k(θ−r) dt+σ dWdr = k(\theta - r)\,dt + \sigma\,dW — k = mean reversion speed, θ = long-run rate, σ = volatility, dW = Brownian increment
Kolmogorov-Smirnov test statistic for PIT uniformity
D=sup⁡x∣Femp(x)−Funif(x)∣D = \sup_x |F_{emp}(x) - F_{unif}(x)| — F_emp = empirical CDF of PITs, F_unif = uniform [0,1] reference CDF
Standard error of empirical VaR quantile
SE(VaRc)≈c(1−c)/nf(VaRc)SE(VaR_c) \approx \frac{\sqrt{c(1-c)/n}}{f(VaR_c)} — c = confidence level, n = sample size, f(VaR_c) = density at the VaR quantile
Anderson-Darling test statistic for PIT goodness-of-fit
A2=−n−∑i=1n2i−1n[ln⁡u(i)+ln⁡(1−u(n+1−i))]A^2 = -n - \sum_{i=1}^{n}\frac{2i-1}{n}[\ln u_{(i)} + \ln(1 - u_{(n+1-i)})] — n = sample size, u_(i) = i-th order statistic of PITs
Vasicek conditional expected short rate
E[rT∣rt]=rte−k(T−t)+θ(1−e−k(T−t))E[r_T \mid r_t] = r_t e^{-k(T-t)} + \theta(1 - e^{-k(T-t)}) — r_t = current rate, k = mean reversion speed, θ = long-run rate, T-t = horizon
Portfolio variance via VaR factor mapping
σp2=xTΣx\sigma_p^2 = \mathbf{x}^T \boldsymbol{\Sigma} \mathbf{x} — x = vector of dollar exposures to each mapped risk factor, Σ = factor covariance matrix
Mean-reversion regression for equity correlation
Δρt=a(ρˉ−ρt−1)+εt\Delta \rho_t = a(\bar\rho - \rho_{t-1}) + \varepsilon_t — a = mean-reversion speed, ρ̄ = long-run correlation, ρ_{t-1} = lagged level, ε_t = shock
Regression-based hedge size with beta adjustment
Fhedge=β⋅DV01posDV01hedge⋅FposF_{hedge} = \beta \cdot \frac{DV01_{pos}}{DV01_{hedge}} \cdot F_{pos} — β = regression slope, DV01 = dollar value of a basis point, F = face amount
Lognormal value at risk
VaRc=S0 [1−exp⁡(μ−zcσ)]\text{VaR}_c = S_0 \,[1 - \exp(\mu - z_c \sigma)] — S_0 = position value, μ = mean log return, σ = log-return volatility, z_c = normal quantile at c
Bivariate Gaussian copula joint default probability
P(both default)=Φ2(Φ−1(p1),Φ−1(p2);ρ)P(\text{both default}) = \Phi_2(\Phi^{-1}(p_1), \Phi^{-1}(p_2); \rho) — p_i = marginal default prob, ρ = asset correlation, Φ_2 = bivariate normal CDF
Correlation swap payoff
Payoff=N⋅(ρrealized−K)\text{Payoff} = N \cdot (\rho_{\text{realized}} - K) — N = notional, ρ_realized = pairwise-average realized correlation over swap life, K = strike correlation
Parametric normal value at risk
VaRc=−μ+zcσ\text{VaR}_c = -\mu + z_c \sigma — μ = mean return, σ = return standard deviation, z_c = one-sided standard normal quantile at confidence c (e.g., 2.326 at 99%)
Credit Risk 15 items
Single-month mortality from constant prepayment rate
SMM=1−(1−CPR)1/12\text{SMM} = 1 - (1 - \text{CPR})^{1/12} — SMM = monthly prepayment rate, CPR = annualized constant prepayment rate
Tranche impairment fraction from pool loss
impairment=max⁡(0,Lpool−A)D−A\text{impairment} = \frac{\max(0, L_{pool} - A)}{D - A} — L_pool = pool loss, A = attachment, D = detachment; clip to [0, 1]
Mezzanine tranche price via base correlation
V[A,D]=V[0,D]−V[0,A]V_{[A,D]} = V_{[0,D]} - V_{[0,A]} — V = tranche PV priced under each slice's own base correlation; A = attachment, D = detachment
Cumulative default probability with constant hazard rate
P(default by t)=1−e−λtP(\text{default by } t) = 1 - e^{-\lambda t} — λ = constant hazard rate, t = time horizon in years
Hazard rate implied by CDS spread
λ≈CDS spread1−R\lambda \approx \frac{\text{CDS spread}}{1 - R} — λ = annualized hazard rate, R = recovery rate
Expected loss on a credit exposure
EL=PD×LGD×EAD\text{EL} = \text{PD} \times \text{LGD} \times \text{EAD} — PD = probability of default, LGD = loss given default, EAD = exposure at default
CDS-bond basis
basis=sCDS−sbond\text{basis} = s_{CDS} - s_{bond} — s_CDS = CDS spread, s_bond = bond credit spread over Treasury; negative basis = buy bond + buy protection
Bilateral CVA
BCVA=UCVA−DVA\text{BCVA} = \text{UCVA} - \text{DVA} — UCVA = unilateral CVA (counterparty default charge), DVA = debt value adjustment (own-default benefit)
Overcollateralization (OC) test ratio
OC ratio=pool collateraltranche balance\text{OC ratio} = \frac{\text{pool collateral}}{\text{tranche balance}} — failure diverts pool cash from mezzanine and equity to pay down senior
Damodaran country equity risk premium adjustment
ERP adj=Sovereign default spread×σeqσbond\text{ERP adj} = \text{Sovereign default spread} \times \frac{\sigma_{eq}}{\sigma_{bond}} — σ_eq = equity volatility, σ_bond = sovereign bond volatility
Single-loan unexpected loss with deterministic LGD
UL=EAD×LGD×PD(1−PD)\text{UL} = \text{EAD} \times \text{LGD} \times \sqrt{\text{PD}(1-\text{PD})} — PD = probability of default, LGD = loss given default, EAD = exposure at default
Single-factor model asset return
Ri=ρ M+1−ρ εiR_i = \sqrt{\rho}\,M + \sqrt{1-\rho}\,\varepsilon_i — M = common factor ~N(0,1), ε_i = idiosyncratic shock ~N(0,1) independent across firms, ρ = asset correlation
Default correlation between two obligors
ρd=P(Di∩Dj)−PDi⋅PDjPDi(1−PDi) PDj(1−PDj)\rho_d = \frac{P(D_i \cap D_j) - PD_i \cdot PD_j}{\sqrt{PD_i(1-PD_i)\,PD_j(1-PD_j)}} — PD_i = marginal default probability, P(D_i ∩ D_j) = joint default probability
Debt service coverage ratio
DSCR=Net operating incomeDebt service\text{DSCR} = \dfrac{\text{Net operating income}}{\text{Debt service}} — net operating income = property cash flow, debt service = scheduled principal + interest payments
Unilateral CVA on a derivative trade
CVA≈(1−R)∑iEE(ti)⋅PD(ti−1,ti)⋅D(ti)\text{CVA} \approx (1 - R) \sum_{i} \text{EE}(t_i) \cdot \text{PD}(t_{i-1}, t_i) \cdot D(t_i) — R = recovery, EE = expected exposure, PD = marginal default prob, D = discount factor
Operational Risk and Resilience 10 items
Fault tree AND-gate probability for independent events
PAND=p1×p2P_{AND} = p_1 \times p_2 — p_1, p_2 = probabilities of independent child events that must both occur for parent event
Fault tree OR-gate probability for two independent events
POR=p1+p2−p1p2≈p1+p2P_{OR} = p_1 + p_2 - p_1 p_2 \approx p_1 + p_2 — p_1, p_2 = probabilities of child events; approximation valid for small probabilities
RCSA multiplicative risk score
Score=L×I\text{Score} = L \times I — L = likelihood rating (1-5 scale), I = impact rating (1-5 scale); applied to both inherent and residual risk
Basel total capital ratio
Total capital ratio=T1+T2RWA≥8%\text{Total capital ratio} = \frac{T_1 + T_2}{\text{RWA}} \geq 8\% — T1 = Tier 1 capital, T2 = Tier 2 capital, RWA = risk-weighted assets
Basel III SMA operational risk capital
KSMA=BI×ILMK_{SMA} = BI \times ILM — BI = Business Indicator (size proxy from financials), ILM = Internal Loss Multiplier (scales from 1 by 10-yr loss / BI)
Capital held by a bank under integrated risk management
Kheld=max⁡(Kreg, Kecon)K_{held} = \max(K_{reg},\, K_{econ}) — K_reg = Basel regulatory capital floor, K_econ = internal economic capital at target confidence (typically 99.95-99.97%)
Adjusted RAROC with systematic-risk correction
Adjusted RAROC=RAROC−βE(RM−RF)\text{Adjusted RAROC} = \text{RAROC} - \beta_E (R_M - R_F) — β_E = activity equity beta, R_M = market return, R_F = risk-free rate; compare to R_F (not cost of equity)
Risk-adjusted return on capital (RAROC)
RAROC=R−EL−E−T+K⋅rfEC\text{RAROC} = \frac{R - EL - E - T + K \cdot r_f}{EC} — R = revenues, EL = expected loss, E = expenses, T = taxes, K·r_f = capital charge income, EC = economic capital
Basel III output floor on RWA
RWAeff=max⁡(RWAIRB,0.725×RWASA)\text{RWA}_{\text{eff}} = \max(\text{RWA}_{\text{IRB}}, 0.725 \times \text{RWA}_{\text{SA}}) — IRB = internal ratings-based RWA, SA = standardized-approach RWA, 0.725 = 72.5% floor
Basel 2.5 market risk capital charge
MRC=max⁡(VaR‾⋅mc,VaRprev)+max⁡(sVaR‾⋅ms,sVaRprev)+IRC+CRMMRC = \max(\overline{VaR} \cdot m_c, VaR_{\text{prev}}) + \max(\overline{sVaR} \cdot m_s, sVaR_{\text{prev}}) + IRC + CRM — m = supervisory multipliers, IRC = incremental risk charge, CRM = comprehensive risk measure
Liquidity and Treasury Risk 13 items
Liquidity-adjusted VaR
LVaR=VaR+12P(sˉ+zσs)\text{LVaR} = \text{VaR} + \tfrac{1}{2} P (\bar{s} + z \sigma_s) — P = position size, sˉ\bar{s} = average proportional bid-ask spread, σs\sigma_s = spread volatility, z = stress quantile
Required liquid-asset buffer under stress
B=NSO×kB = \text{NSO} \times k — NSO = net stressed outflow over survival horizon, k = management cushion factor (typically 1.10-1.25)
Duration gap with leverage adjustment
Dgap=DA−DL⋅LAD_{\text{gap}} = D_A - D_L \cdot \frac{L}{A} — D_A = asset duration, D_L = liability duration, L = total liabilities, A = total assets
Covered interest parity forward exchange rate
F=S⋅1+rd1+rfF = S \cdot \frac{1 + r_d}{1 + r_f} — F = forward rate (domestic per foreign), S = spot rate, r_d = domestic interest rate, r_f = foreign interest rate
Change in equity from a duration-gap rate shock
ΔE≈−Dgap⋅Δr1+r⋅A\Delta E \approx -D_{\text{gap}} \cdot \frac{\Delta r}{1 + r} \cdot A — D_gap = duration gap, Δr = parallel rate shock, r = current rate, A = asset value
Contingent liquidity risk charge for committed lines
Charge=L⋅dstress⋅cHQLA\text{Charge} = L \cdot d_{\text{stress}} \cdot c_{\text{HQLA}} — L = committed line size, d_stress = assumed stress drawdown rate, c_HQLA = HQLA opportunity cost
Historical average cost of funds
rˉ=∑iBiri∑iBi\bar{r} = \frac{\sum_i B_i r_i}{\sum_i B_i} — B_i = balance of funding source i, r_i = rate paid on source i; blended carrying cost across the existing funding stack
Net stressed outflow over a defined horizon
NSO=∑iriDi+d⋅L−I\text{NSO} = \sum_i r_i D_i + d \cdot L - I — r_i = category run-off rate, D_i = deposit balance, d = line drawdown rate, L = undrawn commitments, I = reliable inflows
Repo cash advance after haircut
Cash=Vcoll×(1−h)\text{Cash} = V_{coll} \times (1 - h) — V_coll = market value of collateral pledged, h = haircut percentage; interest accrues on this cash amount, not on collateral value
Available funds gap
Gap=(ΔL+Dout+S)−(ΔD+Lin)\text{Gap} = (\Delta L + D_{out} + S) - (\Delta D + L_{in}) — ΔL = new loan demand, D_out = deposit run-off, S = contractual debt service, ΔD = new deposit growth, L_in = loan repayments
Repo repurchase price
Prepurchase=Psale×(1+rrepo⋅n360)P_{repurchase} = P_{sale} \times \left(1 + r_{repo} \cdot \frac{n}{360}\right) — P_sale = post-haircut cash advance, r_repo = repo rate, n = days to maturity, 360 = money-market day-count
Basel III Liquidity Coverage Ratio
LCR=HQLANet cash outflows over 30 days≥100%\text{LCR} = \dfrac{\text{HQLA}}{\text{Net cash outflows over 30 days}} \ge 100\% — HQLA = high-quality liquid assets after haircuts and caps; denominator = stressed 30-day outflows minus capped inflows
Basel III Net Stable Funding Ratio
NSFR=Available stable fundingRequired stable funding≥100%\text{NSFR} = \dfrac{\text{Available stable funding}}{\text{Required stable funding}} \ge 100\% — ASF = liabilities/capital weighted by tenor and stickiness; RSF = assets weighted by liquidity profile over 1-year horizon
Risk and Investment Management 15 items
IRR approximation from TVPI and average cash-flow duration
IRR≈TVPI1/T−1IRR \approx TVPI^{1/T} - 1 — TVPI = total value to paid-in multiple, T = average duration of net cash flows in years
Unsmoothed true return from reported series
Rtrue,t=Rrep,t−ρRrep,t−11−ρR_{true,t} = \frac{R_{rep,t} - \rho R_{rep,t-1}}{1-\rho} — R_rep = reported return, ρ = first-order autocorrelation
Z-score of a single risk factor
zx=x−μxσxz_x = \frac{x - \mu_x}{\sigma_x} — x = observation, μ_x = historical mean, σ_x = historical standard deviation
Total value to paid-in multiple (TVPI)
TVPI=D+NAVPICTVPI = \dfrac{D + NAV}{PIC} — D = cumulative distributions, NAV = remaining net asset value, PIC = paid-in (called) capital
Smoothed-return autoregression for illiquid assets
Rrep,t=α+ρRrep,t−1+εtR_{rep,t} = \alpha + \rho R_{rep,t-1} + \varepsilon_t — R_rep = reported return, ρ = first-order autocorrelation, ε = innovation
Mahalanobis distance for a multi-variable stress scenario
D2=(x−μ)TΣ−1(x−μ)D^2 = (x - \mu)^T \Sigma^{-1} (x - \mu) — x = observation vector, μ = mean vector, Σ = covariance matrix
Un-smoothed private credit volatility (IMF GFSR April 2024)
σtrue≈σreported/0.40\sigma_{true} \approx \sigma_{reported} / 0.40 — σ_reported = quarterly NAV-based vol, 0.40 = IMF-flagged smoothing-to-true ratio
Information ratio
IR=Rˉp−Rˉbσ(Rp−Rb)IR = \frac{\bar R_p - \bar R_b}{\sigma(R_p - R_b)} — R_p = portfolio return, R_b = benchmark return, σ(R_p − R_b) = tracking error
Grinold's fundamental law of active management
IR≈IC⋅BR⋅TCIR \approx IC \cdot \sqrt{BR} \cdot TC — IC = information coefficient, BR = breadth (independent bets/year), TC = transfer coefficient
Distributions to paid-in multiple (DPI)
DPI=DPICDPI = \dfrac{D}{PIC} — D = cumulative cash distributions to LPs, PIC = paid-in (called) capital
Component VaR of a position
CVaRi=wi⋅MVaRiCVaR_i = w_i \cdot MVaR_i, with ∑iCVaRi=VaRdiv\sum_i CVaR_i = VaR_{div} — w_i = dollar position size, MVaR_i = marginal VaR per dollar of position i
Multifactor model expected return
E[Ri]=Rf+∑kβi,kλkE[R_i] = R_f + \sum_k \beta_{i,k} \lambda_k — R_f = risk-free rate, β_{i,k} = asset i's loading on factor k, λ_k = risk premium on factor k
Jensen's alpha (single-factor)
α=(Rp−Rf)−β(Rb−Rf)\alpha = (R_p - R_f) - \beta(R_b - R_f) — R_p = portfolio return, R_b = benchmark return, R_f = risk-free rate, β = portfolio beta to benchmark
Diversified portfolio VaR
VaRdiv=z⋅wTΣwVaR_{div} = z \cdot \sqrt{w^T \Sigma w} — z = confidence multiplier (1.645 at 95%), w = vector of dollar positions, Σ = covariance matrix of returns
Modigliani-squared (M²) risk-adjusted return
M2=Rf+σmσp(Rˉp−Rf)M^2 = R_f + \frac{\sigma_m}{\sigma_p}(\bar R_p - R_f) — R_f = risk-free rate, σ_m = benchmark volatility, σ_p = portfolio volatility, \bar R_p = mean portfolio return
Current Issues in Financial Markets 3 items
BCBS Group 2 risk-weighted assets for unbacked crypto
RWA=Exposure×1,250%=Exposure×12.5RWA = \text{Exposure} \times 1{,}250\% = \text{Exposure} \times 12.5 — Exposure = bank position in unbacked crypto or non-qualifying stablecoin
Minimum CET1 capital required against Group 2 crypto exposure
Required CET1=RWA×4.5%=Exposure×12.5×4.5%\text{Required CET1} = RWA \times 4.5\% = \text{Exposure} \times 12.5 \times 4.5\% — RWA = risk-weighted assets, 4.5% = Basel III CET1 minimum ratio
Effective dollar-for-dollar capital charge on Group 2 crypto
Required CET1≈Exposure\text{Required CET1} \approx \text{Exposure} — 1,250% weight is calibrated so capital held equals the full position size, effectively expensing it from regulatory capital

Frequently Asked Questions

Is the FRM Part II formula sheet free?
Yes. The full FRM Part II formula sheet is free, with no signup, no email, and no credit card required. 71 formulas across 6 topics, all rendered with the same KaTeX math notation used in the FreeFellow study app.
Can I download the FRM Part II formula sheet as a printable PDF?
Yes. A 1080x1350 portrait PDF (Instagram and LinkedIn carousel native size, also great for tablet study) is linked at the top of this page. The PDF is fully self-contained: math is pre-rendered, fonts are embedded, no internet connection needed once downloaded.
What's covered on the FRM Part II formula sheet?
Every formula is grouped by official syllabus topic, with the formula in math notation plus a one-line note on when to use it (or a watch-out from CAIA, CFA, or other prep-provider commentary). Coverage is calibrated to the 2026 syllabus and refreshed when the corpus changes.
What is FreeFellow's relationship with GARP?
No. FreeFellow is not affiliated with GARP. GARP does not endorse, promote, review, or warrant the accuracy of the products or services offered by FreeFellow LLC.
What else is free at FreeFellow for FRM Part II candidates?
The full question bank with detailed solutions, mixed practice, readiness tracking, lessons (where available), and the formula sheet are all free forever. Fellow ($59/quarter or $149/year per track) unlocks timed mock exams, spaced-repetition flashcards, performance analytics, AI essay grading, and a personalized study plan.
Practice FRM Part II questions free →

Formula sheets for other exams

Exam P Exam FM Exam FAM Exam SRM Exam ALTAM Exam ASTAM CFA Level I CFA Level II CFA L3 Portfolio Mgmt CFA L3 Private Wealth CFA L3 Private Markets CFP

About FreeFellow

FreeFellow is an AI-native exam prep platform for actuarial (SOA & CAS), CFA, CFP, CPA, CAIA, and securities licensing candidates — built around modern AI as a core capability rather than as a bolt-on. Every lesson ships with AI-narrated audio. Every constructed-response item has a copy-to-AI prompt builder so candidates can paste their answer into their own ChatGPT or Claude for self-graded feedback. Fellow members get instant AI grading on essays against the official rubric (currently CFA Level III, expanding to other essay-bearing sections).

The 70% you need to pass — question bank, written solutions, lessons, formula sheet, mixed practice, readiness tracking — is free forever, with no trial period and no credit card. Become a Fellow ($59/quarter or $149/year per track) to unlock mock exams, flashcards with spaced repetition, performance analytics, AI essay grading, and a personalized study plan.