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Free GARP FRM Part II Formula Sheet PDF, No Signup (2026)

Market Risk 16 items

Sklar's theorem joint distribution decomposition

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 - K e^{- r T}

— 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

t_{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

d r = k (θ - r) d t + σ d W

— k = mean reversion speed, θ = long-run rate, σ = volatility, dW = Brownian increment

Kolmogorov-Smirnov test statistic for PIT uniformity

D = sup_{x} ∣ F_{e m p} (x) - F_{u ni f} (x) ∣

— F_emp = empirical CDF of PITs, F_unif = uniform [0,1] reference CDF

Standard error of empirical VaR quantile

S E (V a R_{c}) \approx \frac{c ( 1 - c ) / n}{f ( V a R _{c} )}

— c = confidence level, n = sample size, f(VaR_c) = density at the VaR quantile

Anderson-Darling test statistic for PIT goodness-of-fit

A^{2} = - n - \sum_{i = 1}^{n} \frac{2 i - 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 [r_{T} ∣ r_{t}] = r_{t} e^{- k (T - t)} + θ (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

σ_{p}^{2} = x^{T} Σ x

— x = vector of dollar exposures to each mapped risk factor, Σ = factor covariance matrix

Mean-reversion regression for equity correlation

Δ ρ_{t} = a (ρ ˉ - ρ_{t - 1}) + ε_{t}

— a = mean-reversion speed, ρ̄ = long-run correlation, ρ_{t-1} = lagged level, ε_t = shock

Regression-based hedge size with beta adjustment

F_{h e d g e} = β \cdot \frac{D V 0 1 _{p os}}{D V 0 1 _{h e d g e}} \cdot F_{p os}

— β = regression slope, DV01 = dollar value of a basis point, F = face amount

Lognormal value at risk

VaR_{c} = S_{0} [1 - exp (μ - z_{c} σ)]

— 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) = Φ_{2} (Φ^{- 1} (p_{1}), Φ^{- 1} (p_{2}); ρ)

— p_i = marginal default prob, ρ = asset correlation, Φ_2 = bivariate normal CDF

Correlation swap payoff

Payoff = N \cdot (ρ_{realized} - K)

— N = notional, ρ_realized = pairwise-average realized correlation over swap life, K = strike correlation

Parametric normal value at risk

VaR_{c} = - μ + z_{c} σ

— μ = mean return, σ = return standard deviation, z_c = one-sided standard normal quantile at confidence c (e.g., 2.326 at 99%)

Age-weighted (BRW) historical simulation weight

w_{i} = \frac{λ ^{i - 1} ( 1 - λ )}{1 - λ ^{N}}

for the observation

i

days ago, decay

0 < λ < 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}

— SMM = monthly prepayment rate, CPR = annualized constant prepayment rate

Tranche impairment fraction from pool loss

impairment = \frac{m a x ( 0 , L _{p oo l} - 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 = 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^{- λ t}

— λ = constant hazard rate, t = time horizon in years

Hazard rate implied by CDS spread

λ \approx \frac{CDS spread}{1 - R}

— λ = annualized hazard rate, R = recovery rate

Expected loss on a credit exposure

EL = PD \times LGD \times EAD

— PD = probability of default, LGD = loss given default, EAD = exposure at default

CDS-bond basis

basis = s_{C D S} - s_{b o n d}

— s_CDS = CDS spread, s_bond = bond credit spread over Treasury; negative basis = buy bond + buy protection

Bilateral CVA

BCVA = UCVA - DVA

— UCVA = unilateral CVA (counterparty default charge), DVA = debt value adjustment (own-default benefit)

Overcollateralization (OC) test ratio

OC ratio = \frac{pool collateral}{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 \times \frac{σ _{e q}}{σ _{b o n d}}

— σ_eq = equity volatility, σ_bond = sovereign bond volatility

Single-loan unexpected loss with deterministic LGD

UL = EAD \times LGD \times PD (1 - PD)

— PD = probability of default, LGD = loss given default, EAD = exposure at default

Single-factor model asset return

R_{i} = ρ M + 1 - ρ ε_{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

ρ_{d} = \frac{P ( D _{i} \cap D _{j} ) - P D _{i} \cdot P D _{j}}{P D _{i} ( 1 - P D _{i} ) P D _{j} ( 1 - P D _{j} )}

— PD_i = marginal default probability, P(D_i ∩ D_j) = joint default probability

Debt service coverage ratio

DSCR = \frac{Net operating income}{Debt service}

— net operating income = property cash flow, debt service = scheduled principal + interest payments

Unilateral CVA on a derivative trade

CVA \approx (1 - R) \sum_{i} EE (t_{i}) \cdot 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

D D = \frac{ln ( V / K ) + ( μ - \frac{1}{2} σ _{V}^{2} ) T}{σ _{V} T}

where V = asset value, K = debt face,

μ

= asset drift,

σ_{V}

= asset volatility, T = horizon

Merton default probability (and KMV EDF)

P D = N (- D D)

where N is the standard normal CDF and DD is the distance to default. Moody’s KMV replaces

N (- D D)

with an empirical EDF.

Merton equity as a call option on firm assets

E = V N (d_{1}) - K e^{- r T} N (d_{2})

,

d_{1} = \frac{ln ( V / K ) + ( r + \frac{1}{2} σ _{V}^{2} ) T}{σ _{V} T}

,

d_{2} = d_{1} - σ_{V} T

Operational Risk and Resilience 10 items

Fault tree AND-gate probability for independent events

P_{A N D} = 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

P_{O R} = 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 \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 = \frac{T _{1} + T _{2}}{RWA} \geq 8%

— T1 = Tier 1 capital, T2 = Tier 2 capital, RWA = risk-weighted assets

Basel III SMA operational risk capital

K_{S M A} = B I \times I L M

— 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

K_{h e l d} = max (K_{r e g}, K_{eco n})

— 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 - β_{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 = \frac{R - E L - E - T + K \cdot r _{f}}{E C}

— R = revenues, EL = expected loss, E = expenses, T = taxes, K·r_f = capital charge income, EC = economic capital

Basel III output floor on RWA

RWA_{eff} = max (RWA_{IRB}, 0.725 \times RWA_{SA})

— IRB = internal ratings-based RWA, SA = standardized-approach RWA, 0.725 = 72.5% floor

Basel 2.5 market risk capital charge

M R C = max (V a R \cdot m_{c}, V a R_{prev}) + max (s V a R \cdot m_{s}, s V a R_{prev}) + I R C + C R M

— m = supervisory multipliers, IRC = incremental risk charge, CRM = comprehensive risk measure

Liquidity and Treasury Risk 13 items

Liquidity-adjusted VaR

LVaR = VaR + \frac{1}{2} P (s ˉ + z σ_{s})

— P = position size,

s ˉ

= average proportional bid-ask spread,

σ_{s}

= spread volatility, z = stress quantile

Required liquid-asset buffer under stress

B = NSO \times k

— NSO = net stressed outflow over survival horizon, k = management cushion factor (typically 1.10-1.25)

Duration gap with leverage adjustment

D_{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 \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

Δ E \approx - D_{gap} \cdot \frac{Δ 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 \cdot d_{stress} \cdot c_{HQLA}

— L = committed line size, d_stress = assumed stress drawdown rate, c_HQLA = HQLA opportunity cost

Historical average cost of funds

r ˉ = \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 = \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 = V_{co l l} \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 = (Δ L + D_{o u t} + S) - (Δ 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

P_{r e p u r c ha se} = P_{s a l e} \times (1 + r_{r e p o} \cdot \frac{n}{360})

— 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 = \frac{HQLA}{Net cash outflows over 30 days} \geq 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 = \frac{Available stable funding}{Required stable funding} \geq 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

I R R \approx T V P I^{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

R_{t r u e, t} = \frac{R _{r e p, t} - ρ R _{r e p, t - 1}}{1 - ρ}

— R_rep = reported return, ρ = first-order autocorrelation

Z-score of a single risk factor

z_{x} = \frac{x - μ _{x}}{σ _{x}}

— x = observation, μ_x = historical mean, σ_x = historical standard deviation

Total value to paid-in multiple (TVPI)

T V P I = \frac{D + N A V}{P I C}

— D = cumulative distributions, NAV = remaining net asset value, PIC = paid-in (called) capital

Smoothed-return autoregression for illiquid assets

R_{r e p, t} = α + ρ R_{r e p, t - 1} + ε_{t}

— R_rep = reported return, ρ = first-order autocorrelation, ε = innovation

Mahalanobis distance for a multi-variable stress scenario

D^{2} = (x - μ)^{T} Σ^{- 1} (x - μ)

— x = observation vector, μ = mean vector, Σ = covariance matrix

Un-smoothed private credit volatility (IMF GFSR April 2024)

σ_{t r u e} \approx σ_{r e p or t e d} /0.40

— σ_reported = quarterly NAV-based vol, 0.40 = IMF-flagged smoothing-to-true ratio

Information ratio

I R = \frac{R ˉ _{p} - R ˉ _{b}}{σ ( 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

I R \approx I C \cdot B R \cdot T C

— IC = information coefficient, BR = breadth (independent bets/year), TC = transfer coefficient

Distributions to paid-in multiple (DPI)

D P I = \frac{D}{P I C}

— D = cumulative cash distributions to LPs, PIC = paid-in (called) capital

Component VaR of a position

C V a R_{i} = w_{i} \cdot M V a R_{i}

, with

\sum_{i} C V a R_{i} = V a R_{d i v}

— w_i = dollar position size, MVaR_i = marginal VaR per dollar of position i

Multifactor model expected return

E [R_{i}] = R_{f} + \sum_{k} β_{i, k} λ_{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 = (R_{p} - R_{f}) - β (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

V a R_{d i v} = z \cdot w^{T} Σ w

— z = confidence multiplier (1.645 at 95%), w = vector of dollar positions, Σ = covariance matrix of returns

Modigliani-squared (M²) risk-adjusted return

M^{2} = R_{f} + \frac{σ _{m}}{σ _{p}} (R ˉ_{p} - R_{f})

— R_f = risk-free rate, σ_m = benchmark volatility, σ_p = portfolio volatility, \bar R_p = mean portfolio return

Fama-French three-factor model

R_{p} - R_{f} = α + β_{M K T} (R_{m} - R_{f}) + β_{S M B} S M B + β_{H M L} H M L

. SMB = small minus big (size factor); HML = high minus low book-to-market (value factor). Carhart adds a momentum factor.

Sharpe ratio

S R = \frac{R _{p} - R _{f}}{σ _{p}}

, excess return per unit of total risk (return volatility

σ_{p}

).

Treynor ratio

T = \frac{R _{p} - R _{f}}{β _{p}}

, excess return per unit of systematic risk (portfolio beta

β_{p}

).

Current Issues in Financial Markets 3 items

BCBS Group 2 risk-weighted assets for unbacked crypto

R W A = Exposure \times 1, 250% = 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 = R W A \times 4.5% = 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 \approx Exposure

— 1,250% weight is calibrated so capital held equals the full position size, effectively expensing it from regulatory capital

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