Free GARP FRM Part II Formula Sheet (2026)

Every FRM Part II formula you need on the test, grouped by topic and rendered with full math notation. 78 formulas across 6 topics, calibrated to the 2026 syllabus. Free forever, no signup required.

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All FRM Part II Formulas

Market Risk 16 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, FiF_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)|
FempF_{emp} = empirical CDF of PITs, FunifF_{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)})
rtr_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, ρt−1ρ_{t-1} = lagged level, εtε_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)]
S0S_0 = position value, μ = mean log return, σ = log-return volatility, zcz_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)
pip_i = marginal default prob, ρ = asset correlation, Φ2Φ_2 = bivariate normal CDF
Correlation swap payoff
Payoff=N⋅(ρrealized−K)\text{Payoff} = N \cdot (\rho_{\text{realized}} - K)
N = notional, ρrealizedρ_{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, zcz_c = one-sided standard normal quantile at confidence c (e.g., 2.326 at 99%)
Age-weighted (BRW) historical simulation weight
wi=λi−1(1−λ)1−λNw_i = \dfrac{\lambda^{i-1}(1 - \lambda)}{1 - \lambda^N} for the observation ii days ago, decay 0<λ<10 < \lambda < 1. Recent observations get more weight, so VaR reacts faster to volatility changes than equal-weighted historical simulation.
Credit Risk 18 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}
LpoolL_{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}
sCDSs_{CDS} = CDS spread, sbonds_{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}}
σeqσ_{eq} = equity volatility, σbondσ_{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), εiε_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)}}
PDiPD_i = marginal default probability, P(D_i ∩ DjD_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
Merton distance to default
DD=ln⁡(V/K)+(μ−12σV2)TσVTDD = \dfrac{\ln(V/K) + (\mu - \tfrac{1}{2}\sigma_V^2)T}{\sigma_V\sqrt{T}} where V = asset value, K = debt face, μ\mu = asset drift, σV\sigma_V = asset volatility, T = horizon
Merton default probability (and KMV EDF)
PD=N(−DD)PD = N(-DD) where N is the standard normal CDF and DD is the distance to default. Moody’s KMV replaces N(−DD)N(-DD) with an empirical EDF.
Merton equity as a call option on firm assets
E=V N(d1)−Ke−rTN(d2)E = V\,N(d_1) - K e^{-rT} N(d_2), d1=ln⁡(V/K)+(r+12σV2)TσVTd_1 = \dfrac{\ln(V/K) + (r + \tfrac{1}{2}\sigma_V^2)T}{\sigma_V\sqrt{T}}, d2=d1−σVTd_2 = d_1 - \sigma_V\sqrt{T}
Operational Risk and Resilience 10 items
Fault tree AND-gate probability for independent events
PAND=p1×p2P_{AND} = p_1 \times p_2
p1p_1, p2p_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
p1p_1, p2p_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})
KregK_{reg} = Basel regulatory capital floor, KeconK_{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)
βEβ_E = activity equity beta, RMR_M = market return, RFR_F = risk-free rate; compare to RFR_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·rfr_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}
DAD_A = asset duration, DLD_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, rdr_d = domestic interest rate, rfr_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
DgapD_{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, dstressd_{stress} = assumed stress drawdown rate, cHQLAc_{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}
BiB_i = balance of funding source i, rir_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
rir_i = category run-off rate, DiD_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)
VcollV_{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, DoutD_{out} = deposit run-off, S = contractual debt service, ΔD = new deposit growth, LinL_{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)
PsaleP_{sale} = post-haircut cash advance, rrepor_{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 18 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}
RrepR_{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, μxμ_x = historical mean, σxσ_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
RrepR_{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
σreportedσ_{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)}
RpR_p = portfolio return, RbR_b = benchmark return, σ(R_p − RbR_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}
wiw_i = dollar position size, MVaRiMVaR_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
RfR_f = risk-free rate, βi,kβ_{i,k} = asset i's loading on factor k, λkλ_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)
RpR_p = portfolio return, RbR_b = benchmark return, RfR_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)
RfR_f = risk-free rate, σmσ_m = benchmark volatility, σpσ_p = portfolio volatility, \bar RpR_p = mean portfolio return
Fama-French three-factor model
Rp−Rf=α+βMKT(Rm−Rf)+βSMB SMB+βHML HMLR_p - R_f = \alpha + \beta_{MKT}(R_m - R_f) + \beta_{SMB}\,SMB + \beta_{HML}\,HML. SMB = small minus big (size factor); HML = high minus low book-to-market (value factor). Carhart adds a momentum factor.
Sharpe ratio
SR=Rp−RfσpSR = \dfrac{R_p - R_f}{\sigma_p}, excess return per unit of total risk (return volatility σp\sigma_p).
Treynor ratio
T=Rp−RfβpT = \dfrac{R_p - R_f}{\beta_p}, excess return per unit of systematic risk (portfolio beta βp\beta_p).
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
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