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. 76 formulas across 5 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
Half-life of a Vasicek rate shock
t1/2=ln⁡2/kt_{1/2}=\ln 2/k, for k>0k>0. k is Vasicek mean-reversion speed. This is the time for the conditional expected deviation from the long-run level to halve, not a guarantee about a realized rate path.
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
Standard error of empirical VaR quantile
SE(VaR^c)≈c(1−c)/n/fL(VaRc)SE(\widehat{VaR}_c)\approx\sqrt{c(1-c)/n}/f_L(VaR_c). c: confidence; n: sample size; f_L: loss density at the quantile. Large-sample iid continuous-loss result with positive density there. Dependence changes the variance.
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
Var⁡(ΔV)=xTΣRx\operatorname{Var}(\Delta V)=\mathbf{x}^{T}\Sigma_R\mathbf{x}, for ΔV=xTR\Delta V=\mathbf{x}^{T}R. x: mapped dollar exposures; R: factor returns over the horizon; ΣR\Sigma_R: their covariance matrix. The result is in currency squared.
Mean-reversion regression for equity correlation
Δρt=a(ρˉ−ρt−1)+εt\Delta\rho_t=a(\bar\rho-\rho_{t-1})+\varepsilon_t. a: discrete mean-reversion fraction per observation; ρˉ\bar\rho: long-run level. The level AR(1) coefficient is 1−a1-a. Continuous speed is −ln⁡(1−a)/Δt-\ln(1-a)/\Delta t when 0<a<10<a<1.
Regression-based hedge size with beta adjustment
Fh=−β(dp/dh)FpF_h=-\beta(d_p/d_h)F_p, signed hedge face. Regress position yield changes on hedge yield changes; each d is DV01 per equal face unit. If position DV01 is total, use Fh=−βDV01p/dhF_h=-\beta DV01_p/d_h, with dhd_h per dollar face. Positive beta for a long bond means a short hedge.
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(\rho_{\rm realized}-K) to the receiver of realized correlation. The contract defines the basket and averaging weights. With correlations in decimals, N is currency per unit correlation; per-point notional requires conversion.
Parametric normal value at risk
VaRc=V0(zcσR−μR)VaR_c=V_0(z_c\sigma_R-\mu_R) for return R∼N(μR,σR2)R\sim N(\mu_R,\sigma_R^2) over the chosen horizon and loss L=−V0RL=-V_0R. zc=Φ−1(c)z_c=\Phi^{-1}(c); at 99%, 2.326. Divide by V0V_0 for return units.
Age-weighted (BRW) historical simulation weight
wi=λi−1(1−λ)/(1−λN)w_i=\lambda^{i-1}(1-\lambda)/(1-\lambda^N), for N observations, i=1i=1 the newest, and 0<λ<10<\lambda<1. The weights sum to 1 and decline with age; as λ→1\lambda\to1, they approach equal weights.
POT/GPD tail loss quantile
qc=u+βξ[(n(1−c)/Nu)−ξ−1]q_c=u+\frac{\beta}{\xi}[(n(1-c)/N_u)^{-\xi}-1]. For ξ=0\xi=0: qc=u−βln⁡[n(1−c)/Nu]q_c=u-\beta\ln[n(1-c)/N_u]. n observations, NuN_u losses above u, β>0\beta>0 scale, ξ\xi shape; c>1−Nu/nc>1-N_u/n, within GPD support.
Risk-neutral bond-tree backward induction
Bt=[q(Bu+Cu)+(1−q)(Bd+Cd)]/(1+rt)ΔtB_t=[q(B_u+C_u)+(1-q)(B_d+C_d)]/(1+r_t)^{\Delta t}. q: up probability under Q; B: next-node continuation value; C: next-node coupon; r: annual effective node rate. For continuous rates, discount by e−rtΔte^{-r_t\Delta t}.
CIR local rate change and volatility scaling
Δr≈k(θ−r)Δt+σrΔt Z\Delta r\approx k(\theta-r)\Delta t+\sigma\sqrt{r\Delta t}\,Z; local SD =σrΔt=\sigma\sqrt{r\Delta t}, Z∼N(0,1)Z\sim N(0,1), r≥0r\geq0. k: reversion speed; θ\theta: target; σ\sigma: diffusion coefficient. Small-step approximation; naive Euler can go negative.
Credit Risk 22 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
ℓA,D=min⁡(max⁡(L−A,0),D−A)/(D−A)\ell_{A,D}=\min(\max(L-A,0),D-A)/(D-A). L=pool loss fraction; A/D=attachment/detachment on the same pool basis. Multiply by the tranche-position notional for dollars. Loss begins above A and is capped at 100%.
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
b=sCDS−sbondb=s_{CDS}-s_{bond}, using comparable maturity and spread conventions. Negative basis motivates buying the bond and protection, but funding, liquidity, deliverability and counterparty risk prevent guaranteed arbitrage.
Bilateral CVA
BCVA=CVAFTD−DVAFTDBCVA=CVA_{FTD}-DVA_{FTD}; adjusted value =V0−BCVA=V_0-BCVA. FTD terms stop at the first default. The shortcut UCVA−UDVAUCVA-UDVA ignores first-to-default effects; DVA uses expected negative exposure and own-default inputs.
Overcollateralization (OC) test ratio
OC=adjusted collateral/covered debtOC=\text{adjusted collateral}/\text{covered debt}. Covered debt includes the tested class and senior classes as the indenture specifies. If the threshold fails, the waterfall redirects cash to restore coverage or repay senior debt.
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 LGDPD(1−PD)UL=EAD\,LGD\sqrt{PD(1-PD)}. This is the standard deviation of a single Bernoulli default loss with fixed EAD and LGD. It is not tail capital at a chosen confidence. PD=default probability; LGD=loss fraction; EAD=exposure.
Single-factor model asset return
Ri=ρM+1−ρεiR_i=\sqrt{\rho}M+\sqrt{1-\rho}\varepsilon_i. M and all epsilon terms are independent standard normals; rho is common asset correlation. More generally Ri=βiM+1−βi2εiR_i=\beta_iM+\sqrt{1-\beta_i^2}\varepsilon_i, so Corr(Ri,Rj)=βiβjCorr(R_i,R_j)=\beta_i\beta_j.
Default correlation between two obligors
ρd=[P(Di∩Dj)−pipj]/pi(1−pi)pj(1−pj)\rho_d=[P(D_i\cap D_j)-p_ip_j]/\sqrt{p_i(1-p_i)p_j(1-p_j)}. Here p denotes horizon PD and D denotes a default event. In a Gaussian factor model, joint PD is a bivariate-normal CDF at the inverse-PD thresholds.
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
UCVA≈(1−RC)∑iDiEEiΔQC,iUCVA\approx(1-R_C)\sum_i D_iEE_i\Delta Q_{C,i}. Q gives risk-neutral default probabilities; delta Q is unconditional period default mass, D is discounting, EE is mean positive exposure. Assumes exposure-default independence and fixed recovery.
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)
PDP=Φ(−DDP)PD_P=\Phi(-DD_P); using risk-free drift instead gives risk-neutral PDQ=Φ(−d2)PD_Q=\Phi(-d_2). Phi is the standard normal CDF. KMV maps distance to default to empirical EDF instead of using the normal tail.
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}
Accuracy ratio from cumulative accuracy profile areas
AR=(Amodel−0.5)/(Aperfect−0.5)AR=(A_{\rm model}-0.5)/(A_{\rm perfect}-0.5). Areas lie under CAP curves with borrowers ranked riskiest first; random area is 0.5. If default fraction is pp, Aperfect=1−p/2A_{\rm perfect}=1-p/2.
Vasicek stressed loss quantile and unexpected-loss capital
qc=Φ[(Φ−1(p)+ρΦ−1(c))/1−ρ]q_c=\Phi[(\Phi^{-1}(p)+\sqrt{\rho}\Phi^{-1}(c))/\sqrt{1-\rho}]. Lc=EAD LGD qcL_c=EAD\,LGD\,q_c; capital =Lc−EAD LGD p=L_c-EAD\,LGD\,p. Homogeneous infinite portfolio; p is PD, ρ\rho asset correlation, c confidence. Excludes maturity adjustments.
Positive exposure within an enforceable netting set
E=max⁡(∑iVi,0)E=\max(\sum_i V_i,0); without netting, E=∑imax⁡(Vi,0)E=\sum_i\max(V_i,0). Values are from one party's perspective. Compute each legally separate set independently; collateral is excluded here.
Time-weighted EPE and effective EPE
EPE=T−1∑iEEiΔtiEPE=T^{-1}\sum_i EE_i\Delta t_i; EEieff=max⁡(EEi−1eff,EEi)EE_i^{eff}=\max(EE_{i-1}^{eff},EE_i). Effective EPE uses EEieffEE_i^{eff} in that weighted average. EE=mean positive exposure; initialize with current exposure and state the averaging horizon.
Stressed expected loss on a loan portfolio
ELs=∑iPDi,sLGDi,sEADi,sEL_s=\sum_i PD_{i,s}LGD_{i,s}EAD_{i,s}. Use consistent stressed horizons and inputs. Incremental expected-loss stress is ELs−EL0EL_s-EL_0. A named-default scenario instead sums losses on the assumed defaults without PD weighting.
Operational Risk and Resilience 8 items
Fault tree AND-gate probability for independent events
P(A∩B)=pApBP(A\cap B)=p_Ap_B
For independent child events that must both occur. With dependence, use P(A)P(B∣A)P(A)P(B\mid A).
Fault tree OR-gate probability for two independent events
P(A∪B)=pA+pB−pApBP(A\cup B)=p_A+p_B-p_Ap_B
For independent events. The small-probability approximation is pA+pBp_A+p_B; generally subtract P(A∩B)P(A\cap B).
Basel total capital ratio
Total capital ratio=(T1+T2)/RWA≥8%\text{Total capital ratio}=(T_1+T_2)/RWA\geq8\%
Eligible Tier 1 plus Tier 2 over risk-weighted assets. This baseline minimum excludes buffers and additional requirements.
Basel III SMA operational risk capital
K=BIC×ILMK=BIC\times ILM
BIC marginal coefficients: 12% of first EUR1bn BI, 15% of next EUR29bn, 18% of excess. ILM=ln⁡(e−1+(LC/BIC)0.8)ILM=\ln(e-1+(LC/BIC)^{0.8}); LC: 15 times ten-year average loss. Default bucket 1 ILM is 1; national discretion may set all ILMs to 1.
Adjusted RAROC with systematic-risk correction
RAROCadj=RAROC−βE(RM−RF)RAROC_{adj}=RAROC-\beta_E(R_M-R_F)
Equity beta βE\beta_E; market return RMR_M; risk-free rate RFR_F. Compare with RFR_F, equivalently RAROC with RF+βE(RM−RF)R_F+\beta_E(R_M-R_F).
Risk-adjusted return on capital (RAROC)
RAROC=(R−EL−E−T+CI)/ECRAROC=(R-EL-E-T+CI)/EC
R: net revenue; EL: expected loss; E: expenses; T: taxes; CI: capital income excluded from R; EC: economic capital. Use consistent tax treatment and count income once.
Basel III output floor on RWA
RWAfinal=max⁡(RWApre,0.725RWASA)RWA_{final}=\max(RWA_{pre},0.725RWA_{SA})
At full phase-in: aggregate pre-floor RWA versus 72.5% of corresponding total under prescribed standardized approaches. Use a stated transitional rate if applicable.
Basel 2.5 market risk capital charge
K=max⁡(Vt−1,mcVˉ)+max⁡(St−1,msSˉ)K=\max(V_{t-1},m_c\bar V)+\max(S_{t-1},m_s\bar S)
V: VaR; S: stressed VaR; bars: 60-day means; m: multipliers. Add IRC and comprehensive risk charges only for their applicable, non-overlapping portfolios.
Liquidity and Treasury Risk 15 items
Liquidity-adjusted VaR
LVaR≈VaRschedule+12P(sˉ+zσs)\mathrm{LVaR}\approx\mathrm{VaR}_{schedule}+\tfrac12P(\bar s+z\sigma_s)
P = value sold; s = proportional full spread. Price VaR must match the liquidation schedule. The sum is an approximation, not an exact joint quantile.
Required liquid-asset buffer under stress
B=kmax⁡(NSO,0)B=k\max(\mathrm{NSO},0)
NSO = net stressed outflow; k = supplied management cushion. This simplified target also needs timing, usable-asset and peak-deficit checks; k is not universal.
Duration gap with leverage adjustment
Dgap=DA−(L/A)DLD_{gap}=D_A-(L/A)D_L
Durations must share a convention; A and L are market values of assets and liabilities. A positive gap means rising yields reduce economic equity under the parallel-shift approximation.
Covered interest parity forward exchange rate
F=S(1+rd)/(1+rf)F=S(1+r_d)/(1+r_f)
S and F = domestic currency per foreign unit; rates cover the same tenor. For annual simple rates, use rT. Frictionless comparable funding and hedging assumed.
Change in equity from a duration-gap rate shock
ΔE≈−DgapAΔr/(1+r)\Delta E\approx-D_{gap}A\Delta r/(1+r)
Macaulay durations, common yield r, small parallel shock and market values assumed. With modified durations, omit the extra division by 1+r.
Contingent liquidity risk charge for committed lines
Annual charge=L dstress cbuffer\text{Annual charge}=L\,d_{stress}\,c_{buffer}
L = undrawn commitment; d = assumed stress draw fraction; c = annual net buffer holding cost. Credit, capital and other fee components are separate.
Historical average cost of funds
rˉ=∑iBiri/∑iBi\bar r=\sum_i B_i r_i/\sum_i B_i
B = existing funding balance; r = annual source rate. This is interest-only historical cost; add specified noninterest expenses for an all-in measure.
Net stressed outflow over a defined horizon
NSO=∑iriDi+dL−I\mathrm{NSO}=\sum_i r_iD_i+dL-I
D = funding balances; r = runoff fractions; L = undrawn lines; d = draw fraction; I = reliable inflows. Add other cash demands once; preserve timing.
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
G=(ΔL+J+Dout+S)−(ΔD+Lin)G=(\Delta L+J+D_{out}+S)-(\Delta D+L_{in})
L = new loans; J = investment purchases; Dout = withdrawals; S = debt service; D = new deposits; Lin = repayments. Positive gap needs funding. Avoid double-counting flows already netted.
Repo repurchase price
P1=P0(1+rrepon/b)P_1=P_0(1+r_{repo}n/b)
P0 = post-haircut cash advance; r = annual simple rate; n = days; b = specified day-count basis (often 360). Repurchase cash includes principal and interest.
Basel III Liquidity Coverage Ratio
LCR=HQLA/[O−min⁡(I,0.75O)]\mathrm{LCR}=\mathrm{HQLA}/[O-\min(I,0.75O)]
O and I = stressed 30-day outflows and eligible inflows; HQLA is adjusted for eligibility, haircuts and caps. Normal-time minimum 100%; buffers can be used during stress.
Basel III Net Stable Funding Ratio
NSFR=ASF/RSF≥100%\mathrm{NSFR}=\mathrm{ASF}/\mathrm{RSF}\ge100\%
ASF = capital/liabilities weighted by stability and tenor; RSF = assets and off-balance-sheet exposures weighted by funding need. One-year structural horizon.
Signed downside cash-flow-at-risk from a net cash-flow quantile
CFaR1−α−=q1−α(CF)−E[CF]\mathrm{CFaR}^{-}_{1-\alpha}=q_{1-\alpha}(CF)-E[CF]
Receipts positive; same horizon. Lower quantile minus mean is signed downside CFaR. Reverse the sign for positive shortfall magnitude; actual funding need also depends on opening resources.
Change in net interest income from a repricing gap
ΔNII≈(RSA−RSL)Δr T\Delta NII\approx(RSA-RSL)\Delta r\,T
Immediate common repricing, fixed balances; T in years. Weight later resets by time remaining. NII is currency; divide by average earning assets for the NIM change.
Risk and Investment Management 15 items
Unsmoothed true return from reported series
Rt∗=Rrep,t−ρRrep,t−11−ρR_t^*=\frac{R_{\mathrm{rep},t}-\rho R_{\mathrm{rep},t-1}}{1-\rho}.
Assumes reported returns follow this smoothing filter, 0≤ρ<10\le\rho<1. Autocorrelation identifies ρ\rho only with independent economic returns.
Total value to paid-in multiple (TVPI)
TVPI=(D+NAV)/PICTVPI=(D+NAV)/PIC.
DD: cumulative distributions; NAVNAV: remaining net asset value; PICPIC: paid-in capital. Includes unrealized marks and ignores cash-flow timing.
Smoothed-return autoregression for illiquid assets
Rrep,t=a+ρRrep,t−1+εtR_{\mathrm{rep},t}=a+\rho R_{\mathrm{rep},t-1}+\varepsilon_t.
Under the simple filter, a=(1−ρ)μa=(1-\rho)\mu. Independent economic returns make ρ\rho reported autocorrelation; other serial dependence breaks that identification.
Mahalanobis distance for a multi-variable stress scenario
D=(x−μ)TΣ−1(x−μ)D=\sqrt{(x-\mu)^T\Sigma^{-1}(x-\mu)}.
xx: shock vector; μ\mu: reference mean; Σ\Sigma: covariance. Scenario z-score is D/nD/\sqrt n, with nn variables. Distance is not an event probability.
Information ratio
IR=Rp−Rb‾σ(Rp−Rb)IR=\frac{\overline{R_p-R_b}}{\sigma(R_p-R_b)}.
Numerator: mean active return; denominator: tracking error. Match frequencies. A factor-residual ratio uses residual volatility instead.
Grinold's fundamental law of active management
IR≈ICBR TCIR\approx IC\sqrt{BR}\,TC.
ICIC: forecast skill; BRBR: independent bets per year; TCTC: implementation transfer. Ideal maximum has TC=1TC=1. Correlated bets reduce effective breadth.
Distributions to paid-in multiple (DPI)
DPI=D/PICDPI=D/PIC.
DD: cumulative cash distributions; PICPIC: paid-in capital. DPI excludes remaining NAV and does not measure the timing of distributions.
Component VaR of a position
CVaRi=xi ∂VaR/∂xiCVaR_i=x_i\,\partial VaR/\partial x_i, and ∑iCVaRi=VaR\sum_i CVaR_i=VaR for differentiable, degree-one homogeneous VaR.
xix_i is a dollar position; the derivative is VaR per dollar. Components can be negative.
Multifactor model expected return
E[Ri]=Rf+∑kβikλkE[R_i]=R_f+\sum_k\beta_{ik}\lambda_k.
RfR_f: risk-free rate; βik\beta_{ik}: factor loading; λk\lambda_k: priced factor premium. Statistical factors need not all be priced.
Jensen's alpha (single-factor)
α=Rp−Rf‾−β Rm−Rf‾\alpha=\overline{R_p-R_f}-\beta\,\overline{R_m-R_f}.
β\beta is market exposure. Use the same periods and market benchmark. A single-period residual does not by itself establish skill.
Diversified portfolio VaR
VaR=zxTΣxVaR=z\sqrt{x^T\Sigma x}.
xx: dollar positions; Σ\Sigma: return covariance for the horizon; zz: one-sided normal quantile (about 1.645 at 95%). Assumes linear exposures and zero or ignored mean.
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).
σm,σp\sigma_m,\sigma_p: benchmark and portfolio volatility. This is the benchmark-risk return level; subtract Rˉm\bar R_m for the excess-return convention. Match periods.
Fama-French three-factor model
Rp−Rf=α+βm(Rm−Rf)+βsSMB+βvHML+εR_p-R_f=\alpha+\beta_m(R_m-R_f)+\beta_s SMB+\beta_v HML+\varepsilon.
SMB: small minus big; HML: high minus low book-to-market. ε\varepsilon is residual return. Adding momentum gives the Carhart model.
Sharpe ratio
SR=(Rˉp−Rf)/σpSR=(\bar R_p-R_f)/\sigma_p.
Mean excess return per unit of total return volatility. Use matching periods and a consistent risk-free return.
Treynor ratio
T=(Rˉp−Rf)/βpT=(\bar R_p-R_f)/\beta_p.
Mean excess return per unit of market beta. A zero or negative beta makes conventional ranking problematic; match benchmark and periods.
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