Free download · 2026 FRM Part 2 curriculum

FRM Part 2 Formula Sheet 2026

270 formulas for FRM Part 2, set out chapter by chapter across Market Risk, Credit Risk, Operational Risk and Resilience, Liquidity and Treasury Risk, and Risk Management and Investment Management. The sheet was built from the 2026 GARP readings for the learning objectives that need a formula, and the full sheet can be read right here or downloaded as one PDF.

6books
107chapters reviewed
270formulas
Oct 2026last updated

By Micky Midha Updated 7 October 2026

By Micky Midha, MidhaFin. Updated 7 October 2026. Latest version: midhafin.com/frm/part-2/formula-sheet

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Inside the sheet

Market Risk Measurement and Management17 chapters · 65 formulas
Credit Risk Measurement and Management18 chapters · 74 formulas
Operational Risk and Resilience6 chapters · 39 formulas
Liquidity and Treasury Risk Measurement and Management11 chapters · 39 formulas
Risk Management and Investment Management12 chapters · 53 formulas
Current Issues in Financial Markets8 chapters · concepts only

The research behind this formula sheet

This sheet was built from the 2026 GARP FRM Part 2 readings, for the learning objectives that need a formula. The aim was a sheet that holds those formulas chapter by chapter and says clearly where there is nothing to learn, so your revision time goes only where the marks are.

Every chapter accounted for

64 chapters carry formulas, and all of them are on this sheet. The other 43, including all eight Current Issues readings, have no formula to learn. Each of those still appears in its place, marked as concepts only.

Built from the learning objectives

A formula is on this sheet because a 2026 GARP learning objective needs it. That keeps the whole of Part 2 to 270 formulas, which fit on a 15-page PDF.

Read from the books, checked twice

Every formula was read from the 2026 GARP readings and then checked a second time before it went on the sheet.

Tips and tables where they help

18 short exam tips sit next to the formulas they concern, and 21 tables gather families of results in one place, such as the backtesting zones and the extreme value tail index.

Book 1 of 6 · 20% of the exam

Market Risk Measurement and Management formulas

MR 1

Estimating Market Risk Measures

7 formulas

Historical simulation VaR and ES

VaR

=

the

[(1-\alpha)n+1]

th highest loss (the

n(1-\alpha)

th is also accepted)ES

=

average of the

(1-\alpha)n

highest losses

$n$ observations, confidence $\alpha$ ; 1,000 obs at 95%: 51st highest loss

Normal VaR, P/L and L/P data

P/L data:

\alpha\text{VaR}=-\mu_{P/L}+\sigma_{P/L}\,z_\alpha

L/P data:

\alpha\text{VaR}=\mu_{L/P}+\sigma_{L/P}\,z_\alpha

Normal VaR, arithmetic returns

\alpha\text{VaR}=-(\mu_r-\sigma_rz_\alpha)\,P_{t-1}

r_t=\dfrac{P_t+D_t-P_{t-1}}{P_{t-1}}

Lognormal VaR

\alpha\text{VaR}=P_{t-1}\,(1-\exp[\mu_R-\sigma_Rz_\alpha])

R_t=\ln\dfrac{P_t+D_t}{P_{t-1}}

, normally distributed

ES as average of tail VaRs

\approx

average of the

n-1

VaRs at confidence

\alpha+k(1-\alpha)/n,\ k=1,\dots,n-1

tail beyond $\alpha$ cut into $n$ equal-probability slices; $n=10$ at 95%: VaRs at 95.5%, 96%, ..., 99.5%

Coherent risk measure from quantiles

M_\phi=\displaystyle\int_0^1\phi(p)\,q_p\,dp

ES:

\phi(p)=0

for

p<\alpha,\ \phi(p)=\dfrac{1}{1-\alpha}

for

p\ge\alpha

Standard error of a quantile

\text{var}(q)=\dfrac{p(1-p)}{n\,[f(q)]^2}

\text{se}(q)=\dfrac{\sqrt{p(1-p)/n}}{f(q)}

90% CI:

[q-1.645\,\text{se}(q),\ q+1.645\,\text{se}(q)]

$p$ = prob. of a loss above $q+h/2$ ; $f(q)$ = prob. mass in $q\pm h/2$ ; $h$ = bin width

Exam tip The CI multiplier is two-tailed (90% CI uses 1.645, 95% CI uses 1.96) and is separate from the $z$ that gives $q$ itself (95% VaR: $q=1.645$ ).

QQ plot	Reading
Linear	data match the reference distribution
Intercept, slope	$\approx$ location, scale ( $\approx0$ and $1$ : standard normal)
Steeper in the tails than the middle	heavier tails than the reference
Isolated points off the line	outliers

MR 2

Non-Parametric Approaches

4 formulas

Bootstrapped HS

draw

B

resamples of size

n

with replacementVaR (ES)

=

mean of the

B

resample VaRs (ESs)

Age-weighted HS (BRW)

w(i)=\dfrac{\lambda^{i-1}(1-\lambda)}{1-\lambda^n}

w(i)=\lambda\,w(i-1)

$\lambda\to1$ gives equal weights $1/n$

Volatility-weighted HS (Hull and White)

r^*_{t,i}=\dfrac{\sigma_{T,i}}{\sigma_{t,i}}\,r_{t,i}

$\sigma_{T,i}$ = current forecast; $\sigma_{t,i}$ = forecast made for day $t$

Correlation-weighted HS

\bar R=\bar A\,A^{-1}R

2\times2

a_{11}=1,\ a_{12}=0,\ a_{21}=\rho,\ a_{22}=\sqrt{1-\rho^2}

$A$ , $\bar A$ = Choleski factors of the historical and current correlation matrices

MR 3

Parametric Approaches: Extreme Value

3 formulas

GEV VaR

Fréchet (

\xi>0

\text{VaR}=\mu_n-\dfrac{\sigma_n}{\xi_n}\left[1-(-n\ln\alpha)^{-\xi_n}\right]

Gumbel (

\xi=0

\text{VaR}=\mu_n-\sigma_n\ln[-n\ln\alpha]

$\alpha$ = VaR confidence level; $n$ = block size

POT VaR

\text{VaR}=u+\dfrac{\beta}{\xi}\left\{\left[\dfrac{n}{N_u}(1-\alpha)\right]^{-\xi}-1\right\}

$u$ = threshold; $N_u$ = number of excesses over $u$ ; $\beta$ = scale

POT ES

\text{ES}=\dfrac{\text{VaR}}{1-\xi}+\dfrac{\beta-\xi u}{1-\xi},\ \xi<1

Tail index	Distribution	Tails
$\xi>0$	Fréchet	heavy ( $t$ , Pareto, Lévy); returns usually $0<\xi<0.35$
$\xi=0$	Gumbel	exponential (normal, lognormal)
$\xi<0$	Weibull	lighter than normal

MR 4

Backtesting VaR

4 formulas

Exceptions as binomial

failure rate

=N/T

f(x)={}^TC_x\,p^x(1-p)^{T-x}

E(x)=pT

V(x)=p(1-p)T

$p=1-c$

Normal approximation test

z=\dfrac{x-pT}{\sqrt{p(1-p)T}}\approx N(0,1)

two-tailed 95% test: reject if

|z|>1.96

Kupiec unconditional coverage

\text{LR}_{uc}=-2\ln[(1-p)^{T-N}p^N]+2\ln\{[1-(N/T)]^{T-N}(N/T)^N\}\sim\chi^2(1)

reject if $\text{LR}_{uc}>3.841$ (95%)

Conditional coverage

\text{LR}_{cc}=\text{LR}_{uc}+\text{LR}_{ind}\sim\chi^2(2)

reject if

\text{LR}_{cc}>5.991

reject independence alone if

\text{LR}_{ind}>3.841

Exceptions ( $T=250$ , 99%)	Zone	Increase in $k$ (from 3)
0 to 4	Green	+0.00
5	Yellow	+0.40
6	Yellow	+0.50
7	Yellow	+0.65
8	Yellow	+0.75
9	Yellow	+0.85
10 or more	Red	+1.00

Exam tip The test confidence level (cutoffs 1.96, 3.841) is chosen separately from the VaR level $c$ , and a Type I error rejects a correct model (5 or more exceptions: 10.8% at 99%, $T=250$ ).

MR 5

VaR Mapping

9 formulas

General and specific risk

R_i=\alpha_i+\beta_iR_m+\varepsilon_i

\beta_p=\sum w_i\beta_i

V(R_p)=\beta_p^2V(R_m)+\sum w_i^2\sigma_{\varepsilon_i}^2

first term general (market) risk, second term specific risk

Principal and duration mapping

principal: VaR

=

\times

VaR% of the zero at the average maturityduration: VaR

=

\times

VaR% of the zero at the portfolio duration

VaR% interpolated linearly between vertices

Cash-flow mapping VaR

undiversified VaR

=\sum|x_i|V_i

diversified VaR

=\alpha\sqrt{x'\Sigma x}=\sqrt{(x\times V)'R\,(x\times V)}

two vertices:

\sqrt{\text{VaR}_1^2+\text{VaR}_2^2+2\rho\,\text{VaR}_1\text{VaR}_2}

\text{VaR}_i=|x_i|\times V_i

$x$ = PV of cash flows at each vertex; $V$ = VaR% of each vertex; $R$ = correlation matrix

Tracking error VaR

\text{TE-VaR}=\alpha\sqrt{(x-x_0)'\,\Sigma\,(x-x_0)}

variance improvement

=1-\left(\dfrac{\text{TE-VaR}}{\text{VaR of index}}\right)^2

$x$ , $x_0$ = portfolio and benchmark exposures

Mapping a currency forward

f_t=S_te^{-r^*\tau}-Ke^{-r\tau}=(F_t-K)e^{-r\tau}

long forward

=

long foreign currency spot

(Se^{-r^*\tau})

+

long foreign currency bill

(Se^{-r^*\tau})

+

short domestic bill

(Ke^{-r\tau})

$r^*$ = foreign rate

Mapping a commodity forward

exposure

=Q\times F\times e^{-r\tau}

(PV of the forward position)VaR

=

exposure

\times

VaR% of the contract

Mapping an FRA

long

\tau_1\times\tau_2

FRA

=

long

\tau_1

bill

+

short

\tau_2

billeach leg

x=\dfrac{N}{1+R_1\tau_1}

(1+R_2\tau_2)=(1+R_1\tau_1)[1+F_{1,2}(\tau_2-\tau_1)]

Mapping an interest rate swap

pay fixed, receive floating

=

long FRN

+

short fixed-coupon bondequivalently a portfolio of forward contracts

FRN: cash at reset; after reset, a bill to the next reset

Mapping an option (delta-normal)

\Delta=e^{-r^*\tau}N(d_1)

dc=x_1\dfrac{dS}{S}+x_2\dfrac{dP^*}{P^*}+x_3\dfrac{dP}{P}

x_1=x_2=Se^{-r^*\tau}N(d_1),\ x_3=-Ke^{-r\tau}N(d_2)

$P^*$ , $P$ = foreign and domestic bills

MR 6

Validating Bank Holding Companies’ Value-at-Risk Models for Market Risk

Concepts only

No formulas to memorise. None of the 2026 learning objectives for this chapter needs a formula, so there is nothing from it to learn for this sheet.

MR 7

Beyond Exceedance-Based Backtesting

1 formula

Probability integral transform

p_{t+1}=F_t(\text{PL}_{t+1})

accurate model: PITs i.i.d.

U(0,1)

$F_t$ = forecast CDF of P&L, evaluated at the realized P&L

PIT histogram	Model
Flat	accurate
Hump in the middle	too wide, conservative
Spikes at both ends (U shape)	too narrow, aggressive, tails too thin
Excess mass near 0	tail losses understated

MR 8

Correlation Basics

5 formulas

Realized correlation

\rho_{\text{realized}}=\dfrac{2}{n^2-n}\sum_{i>j}\rho_{i,j}

number of pairs

=n(n-1)/2

Correlation swap payoff

fixed-rate payer (buys correlation):

N(\rho_{\text{realized}}-\rho_{\text{fixed}})

fixed-rate receiver: the negative

VaR with correlated assets

\text{VaR}_P=\sigma_P\,\alpha\sqrt{x}

\sigma_P=\sqrt{\beta_hC\beta_v}

two assets:

\sigma_P=\sqrt{\beta_1^2\sigma_1^2+\beta_2^2\sigma_2^2+2\rho\beta_1\beta_2\sigma_1\sigma_2}

$\beta$ = amounts invested; $\sigma$ daily; $x$ = days; $C$ = covariance matrix; $\alpha$ = normal quantile

Joint default probability

P(X\cap Y)=\rho_{XY}\sqrt{P_X(1-P_X)}\sqrt{P_Y(1-P_Y)}+P_XP_Y

Exchange option volatility

\sigma_E=\sqrt{\sigma_A^2+\sigma_B^2-2\,\text{Cov}_{AB}}

Option	Higher price with
Better of two, call on max, exchange, spread, better of two or cash, dual-strike	lower correlation
Worse of two, basket	higher correlation
Quanto call	lower $\rho(\text{asset},\text{FX})$

MR 9

Empirical Properties of Correlation

4 formulas

Mean reversion model

S_t-S_{t-1}=a(\mu_S-S_{t-1})\Delta t+\sigma_S\varepsilon\sqrt{\Delta t}

\Delta t=1

, no noise:

E[S_t]=S_{t-1}+a(\mu_S-S_{t-1})

Mean reversion rate from regression

regress

S_t-S_{t-1}

S_{t-1}

S_t-S_{t-1}=a\mu_S-aS_{t-1}

slope

\beta=-a

, so

a=-\beta

intercept

=a\mu_S

Autocorrelation

one-period autocorrelation

=1-a

mean reversion $+$ autocorrelation $=1$

Best-fit correlation distributions

equity: Johnson SBbond: generalized extreme value (normal also fits quite well)default probability: Johnson SB

MR 10

Financial Correlation Modeling: Bottom-Up

2 formulas

Joint default probability, two assets

Q(t_B\le t_1\cap t_{Caa}\le t_2)=M_2\big(x_B\le N^{-1}(Q_B(t_1))\cap x_{Caa}\le N^{-1}(Q_{Caa}(t_2));\rho\big)

n

assets:

M_n

with correlation matrix

\rho_M

$M_n$ = $n$ -variate standard normal CDF

Correlated default time of asset i

M_n(\cdot)=Q_i(\tau_i)\Rightarrow\tau_i=Q_i^{-1}(M_n(\cdot))

repeat over many draws from

M_n

and average

\tau_i

MR 11

Regression Hedging and PCA

5 formulas

DV01-neutral hedge

F^{30}=-F^{\text{JNJ}}\times\dfrac{\text{DV01}^{\text{JNJ}}}{\text{DV01}^{30}}

(risk weight 100%)

Regression hedge

\Delta y_t^{\text{JNJ}}=\alpha+\beta\,\Delta y_t^{30}+\varepsilon_t

F^{30}=-F^{\text{JNJ}}\times\dfrac{\text{DV01}^{\text{JNJ}}}{\text{DV01}^{30}}\times\hat\beta

risk weight:

\dfrac{-F^{30}\,\text{DV01}^{30}}{F^{\text{JNJ}}\,\text{DV01}^{\text{JNJ}}}=\hat\beta

SD of regression-hedged P&L

\sigma_{\text{P\&L}}=\left|\dfrac{F^{\text{JNJ}}\times\text{DV01}^{\text{JNJ}}}{100}\right|\times\sigma_\varepsilon

$\sigma_\varepsilon$ = standard error of the regression

Two-variable regression hedge

\Delta y_t^{20}=\alpha+\beta^{10}\Delta y_t^{10}+\beta^{30}\Delta y_t^{30}+\varepsilon_t

F^{10}=-F^{20}\times\dfrac{\text{DV01}^{20}}{\text{DV01}^{10}}\times\hat\beta^{10}

F^{30}=-F^{20}\times\dfrac{\text{DV01}^{20}}{\text{DV01}^{30}}\times\hat\beta^{30}

risk weights:

\dfrac{-F^{10}\,\text{DV01}^{10}}{F^{20}\,\text{DV01}^{20}}=\hat\beta^{10},\ \dfrac{-F^{30}\,\text{DV01}^{30}}{F^{20}\,\text{DV01}^{20}}=\hat\beta^{30}

Reverse regression hedge

regress Treasury on JNJ (slope

\hat\beta_{\text{rev}}

): risk weight of the hedge

=1/\hat\beta_{\text{rev}}

(not

\hat\beta

)

Exam tip Level regression $y_t=\alpha+\beta x_t+\varepsilon_t$ vs change regression $\Delta y_t=\beta\Delta x_t+\Delta\varepsilon_t$ : level errors are usually serially correlated ( $\varepsilon_t=\rho\varepsilon_{t-1}+v_t$ ), so OLS stays unbiased and consistent but may not be efficient.

MR 12

Arbitrage Pricing with Term Structure Models

5 formulas

Binomial tree pricing

V_0=\dfrac{p\,V_u+(1-p)\,V_d}{1+r/2}

(six-month steps, semiannual rates)general step

\Delta t

: divide by

(1+r\Delta t)

Replicating portfolio

F^{.5}+P_u(1)\,F^1=V_u

F^{.5}+P_d(1)\,F^1=V_d

option price

=P_0(.5)\,F^{.5}+P_0(1)\,F^1

$F$ = face amounts of the six-month and one-year zeros; $P_u(1)$ , $P_d(1)$ = date 1 prices of the one-year zero

Risk-neutral probability and drift

solve

P_0=\dfrac{p\,P_u+(1-p)\,P_d}{1+r/2}

for

p

risk-neutral drift

=p(r_u-r_0)+(1-p)(r_d-r_0)

CMT swap

payoff each six months

=

Notional

\times\dfrac{y_{\text{CMT}}-K}{2}

node value

=\dfrac{p\,V_u+(1-p)\,V_d}{1+r/2}+

payoff at that node

Option-adjusted spread

market price

=

model value with every discount rate

r

replaced by

r+\text{OAS}

(cash flows still from unshifted rates)

\text{OAS}>0

: cheap,

\ \text{OAS}<0

: rich

E[dP/P]=(r_0+\text{OAS})\,dt

MR 13

Expectations, Risk Premium, and Convexity

3 formulas

Jensen's inequality and convexity effect

E\left[\dfrac{1}{1+r}\right]>\dfrac{1}{E[1+r]}=\dfrac{1}{1+E[r]}\Rightarrow f<E[r]

P(2)=\dfrac{\frac12\left[\frac{1}{1+r_u}+\frac{1}{1+r_d}\right]}{1+r_0}=\dfrac{1}{(1+r_0)(1+f)}

convexity effect

=E[r]-f

, rises with maturity and volatility

$f$ = forward rate

Bond return decomposition

\dfrac{dP}{P}=f(T)\,dt-D\,dr+\tfrac12C\sigma^2dt

E\left[\dfrac{dP}{P}\right]=f(T)\,dt-D\,E[dr]+\tfrac12C\sigma^2dt

$D=-\frac1P\,\frac{\partial P}{\partial r},\ C=\frac1P\,\frac{\partial^2P}{\partial r^2}$

Expected return, risk-averse investor

risk-neutral:

E[dP/P]=r_0\,dt

risk-averse:

E[dP/P]=r_0\,dt+\lambda D\,dt

\text{SR}=\lambda/\sigma

Exam tip The forward rate splits into expectations, risk premium and convexity: $f(T)=\{r_0+E[dr/dt]\,D\}+\lambda D-\tfrac12C\sigma^2$ .

MR 14

Term Structure Models: Drift

5 formulas

Model 1 (no drift)

dr=\sigma\,dw

E[dr]=0

\text{SD}(dr)=\sigma\sqrt{dt}

tree:

r_0\pm\sigma\sqrt{dt}

, then

r_0\pm2\sigma\sqrt{dt}

and

r_0

, probabilities

\tfrac12

$\sigma$ = annual basis-point volatility

Model 2 (constant drift)

dr=\lambda\,dt+\sigma\,dw

tree:

r_0+\lambda dt\pm\sigma\sqrt{dt}

, then

r_0+2\lambda dt\pm2\sigma\sqrt{dt}

and

r_0+2\lambda dt

after

T

years: mean

r_0+\lambda T

, SD

\sigma\sqrt{T}

Ho-Lee (time-dependent drift)

dr=\lambda_t\,dt+\sigma\,dw

tree:

r_0+\lambda_1dt\pm\sigma\sqrt{dt}

, then

r_0+(\lambda_1+\lambda_2)dt\pm2\sigma\sqrt{dt}

and

r_0+(\lambda_1+\lambda_2)dt

Vasicek rate change

dr=k(\theta-r)\,dt+\sigma\,dw

\theta=r_\infty+\lambda/k

E[dr]=k(\theta-r)\,dt,\ \text{SD}(dr)=\sigma\sqrt{dt}

Vasicek in T years, half-life

E[r_T]=r_0e^{-kT}+\theta(1-e^{-kT})

\text{SD}=\sqrt{\dfrac{\sigma^2}{2k}(1-e^{-2kT})}

half-life

\tau=\dfrac{\ln2}{k}

Exam tip $dw$ has mean 0 and SD $\sqrt{dt}$ : plug a given realization of $dw$ straight into $dr$ , and from a standard normal $Z$ use $dw=Z\sqrt{dt}$ .

MR 15

Term Structure Models: Volatility

3 formulas

Model 3 (time-dependent volatility)

dr=\lambda(t)\,dt+\sigma e^{-\alpha t}dw

SD of

dr

at time

t

\sigma e^{-\alpha t}\sqrt{dt}

terminal SD equals the Vasicek SD with $k=\alpha$

CIR and lognormal rate change

CIR:

dr=k(\theta-r)\,dt+\sigma\sqrt{r}\,dw

Model 4 (lognormal):

dr=ar\,dt+\sigma r\,dw

Courtadon:

dr=k(\theta-r)\,dt+\sigma r\,dw

Basis-point volatility

normal:

\sigma

CIR:

\sigma\sqrt{r}

lognormal:

\sigma r

e.g.

\sigma_{\text{CIR}}\sqrt{8\%}=1\%\Rightarrow\sigma_{\text{CIR}}=3.54\%

\sigma_y\times8\%=1\%\Rightarrow\sigma_y=12.5\%

lognormal $\sigma$ = yield volatility, % of the rate

MR 16

The Vasicek and Gauss+ Models

1 formula

Gauss+ factor changes

dr_t=-\alpha_r(r_t-m_t)\,dt

dm_t=-\alpha_m(m_t-l_t)\,dt+\sigma_m\big(\rho\,dW_t^1+\sqrt{1-\rho^2}\,dW_t^2\big)

dl_t=-\alpha_l(l_t-\mu)\,dt+\sigma_l\,dW_t^1

E[dW_t^1\,dW_t^2]=0,\ dW=Z\sqrt{dt}

MR 17

Volatility Smiles and Surfaces

1 formula

Put-call parity and implied volatility

p+S_0e^{-qT}=c+Ke^{-rT}

p_{\text{BS}}-p_{\text{mkt}}=c_{\text{BS}}-c_{\text{mkt}}\Rightarrow

implied vol of a European call

=

implied vol of a European put (same

K

T

)

FX: $q$ = foreign risk-free rate

Options	Implied vol	Implied distribution
FX	smile: higher away from ATM	heavier both tails, more peaked
Equity	skew: falls as $K$ rises	heavier left tail, thinner right tail
Single large jump expected	frown: ATM highest	bimodal

MR 18

Fundamental Review of the Trading Book

3 formulas

97.5% ES vs 99% VaR (normal losses)

\text{VaR}_{99\%}=\mu+2.326\sigma

\text{ES}_{97.5\%}=\mu+2.338\sigma

stressed 97.5% ES replaces 99% VaR; liquidity horizons replace the 10-day horizon

Liquidity horizon (days)	Level	Volatility
Interest rate	10 to 60	60
Equity, large cap	10	20
Equity, small cap	20	60
FX rate	10 to 40	40
Energy price	20	60
Precious metal price	20	60
Other commodities price	60	120

Liquidity horizon, other	Days
Credit spread: sovereign IG	20
Credit spread: sovereign non-IG	40
Credit spread: corporate IG	40
Credit spread: corporate non-IG	60
Credit spread: other	120
Credit spread volatility	120
Equity other	60
Commodity other	120

Liquidity-adjusted ES

\text{ES}=\sqrt{\text{ES}_1^2+\displaystyle\sum_{j=2}^{5}\left(\text{ES}_j\sqrt{\dfrac{\text{LH}_j-\text{LH}_{j-1}}{10}}\right)^2}=\sqrt{\text{ES}_1^2+\text{ES}_2^2+2\text{ES}_3^2+2\text{ES}_4^2+6\text{ES}_5^2}

$\text{LH}=10, 20, 40, 60, 120$ ; $\text{ES}_j$ = ES with 10-day shocks to categories $j$ and above only

Backtesting and P&L attribution

backtest of 1-day VaR over 12 months: more than 12 exceptions at 99% or more than 30 at 97.5%

\Rightarrow

standardized approachPLA ratios:

\text{mean}(U)/\text{SD}(V)

within

-10\%

+10\%

and

\text{var}(U)/\text{var}(V)<20\%

four or more PLA ratio breaches in 12 months

\Rightarrow

standardized approach

$U$ = actual minus model P&L; $V$ = actual P&L

Exam tip Under the IMA, jump-to-default risk uses a 1-year, 99.9% VaR, while credit spread risk sits in ES with liquidity horizons of 20 to 120 days.

Book 2 of 6 · 20% of the exam

Credit Risk Measurement and Management formulas

CR 1

Fundamentals of Credit Risk

Concepts only

No formulas to memorise. None of the 2026 learning objectives for this chapter needs a formula, so there is nothing from it to learn for this sheet.

CR 2

Governance

Concepts only

No formulas to memorise. None of the 2026 learning objectives for this chapter needs a formula, so there is nothing from it to learn for this sheet.

CR 3

Credit Risk Management

1 formula

Expected loss

\text{EL}=\text{PD}\times\text{EAD}\times\text{LGD}

EL in currency; PD, LGD in %

IFRS 9 stage	Impairment	Interest on
1: performing	12-month ECL	Gross amount
2: arrears or significant credit deterioration	Lifetime ECL	Gross amount
3: nonperforming	Lifetime ECL	Net (carrying) amount

CR 4

Capital Structure in Banks

7 formulas

Expected loss, horizon H

\text{EL}_H=\text{EA}_H-E(\text{EA}_H)=\text{PD}_H\cdot\text{EA}_H\cdot\text{LR}_H

\text{EL}_P=\sum\text{EL}_i=\sum\text{EA}_i\cdot\text{PD}_i\cdot\text{LR}_i

EA = exposure amount, LR = loss rate (LGD)

Standalone UL

\text{UL}=\text{EA}\cdot\sqrt{\text{PD}\cdot\sigma_{\text{LR}}^2+\text{LR}^2\cdot\sigma_{\text{PD}}^2}

EA, PD, LR independent

Variance of default

\sigma_{\text{PD}}^2=\text{PD}\,(1-\text{PD})

Portfolio UL

\text{UL}_P=\sqrt{\sum_i\sum_j\rho_{ij}\,\text{UL}_i\,\text{UL}_j}

two loans:

\text{UL}_P=\sqrt{\text{UL}_1^2+\text{UL}_2^2+2\rho_{12}\,\text{UL}_1\text{UL}_2}

$\rho_{ij}$ = default correlation over the horizon

UL contribution

\text{ULC}_i=\dfrac{\text{UL}_i\sum_j\text{UL}_j\,\rho_{ij}}{\text{UL}_P}

\sum_i\text{ULC}_i=\text{UL}_P

Homogeneous portfolio

\text{UL}_P=\text{UL}_i\sqrt{n+\rho\,(n^2-n)}

large

n

\text{ULC}_i\approx\text{UL}_i\sqrt{\rho}

$n$ similar loans, common $\rho$

Economic capital

\text{EC}_P=\text{UL}_P\times\text{CM}

\text{EC}_i=\text{ULC}_i\times\text{CM}

CM = capital multiplier

CR 5

Credit Risk Modeling and Assessment

6 formulas

Capital adequacy ratio

\text{CAR}=\dfrac{\text{Capital}}{\text{Risk-weighted assets}}>\alpha

Capital

=

Tier 1

+

Tier 2

\alpha=8\%

(Basel II)Basel III:

\text{CAR}\ge10.5\%

Risk-weighted assets

IRB:

\text{RWA}=K\times12.5\times\text{EAD}

standardized:

\text{RWA}=\sum\text{risk weight}\times\text{exposure}

$K$ = capital requirement (function of PD, LGD, $M$ , $R$ ); standardized uses prescribed supervisory risk weights

Merton PD and DD, real world

\text{PD}=N\left[-\dfrac{\ln(A/L)+(\mu-\sigma_A^2/2)T}{\sigma_A\sqrt{T}}\right]

\text{DD}=\dfrac{\ln(A/L)+(\mu-\sigma_A^2/2)T}{\sigma_A\sqrt{T}}

$A$ = asset value, $L$ = face value of debt, $\mu$ = expected asset return

KMV default point

Default point

=

short-term debt

+

\tfrac12\times

long-term debt

RAROC

\text{RAROC}=\dfrac{\text{Loan revenues}}{\text{Capital at risk}}

Loan revenues

=(s+f-l-c)(1-x)\times

loan valueaccept if RAROC

>

cost of capital

$s$ = spread, $f$ = fees, $l$ = expected loan loss, $c$ = operating cost, $x$ = tax rate

RAROC capital at risk

Duration approach:

\Delta L=-L\cdot D\cdot\dfrac{\Delta i}{1+i}

default-data approach: unexpected loan loss

=\alpha\times\text{LGD}\times\text{EAD}

$\alpha=2.6\sigma$ at 99.5% if normal; 5 to 6 for skewed losses

CR 6

Credit Scoring and Rating

Concepts only

No formulas to memorise. None of the 2026 learning objectives for this chapter needs a formula, so there is nothing from it to learn for this sheet.

CR 7

Credit Scoring and Retail Credit Risk

2 formulas

Accuracy ratio

\text{AR}=\dfrac{A_R}{A_P}

closer to 1 is better

$A_R$ = area between actual CAP and random 45° line; $A_P$ = same for the perfect model

Implied PD of a score band

Implied PD

=\dfrac{\text{Loss rate}}{\text{LGD}}

CR 8

Sovereign Default Risk

1 formula

Sovereign default spread

Default spread

=

yield on sovereign bond in a currency

-

yield on riskless (e.g., US Treasury) bond in the same currency

CR 9

Estimating Default Probabilities

7 formulas

Altman Z-score

Z=1.2X_1+1.4X_2+3.3X_3+0.6X_4+0.999X_5

Z>3.0

: unlikely to default2.7 to 3.0: on alert1.8 to 2.7: good chance of default

Z<1.8

: very high

$X_1$ = working capital/TA, $X_2$ = retained earnings/TA, $X_3$ = EBIT/TA, $X_4$ = market value of equity/book value of total liabilities, $X_5$ = sales/TA

Marginal and conditional PD

Unconditional PD in year

n=Q(n)-Q(n-1)

conditional PD in year

n=\dfrac{Q(n)-Q(n-1)}{1-Q(n-1)}

$Q$ = cumulative PD

Hazard rate, survival, cumulative PD

V(t)=e^{-\int_0^t\lambda(\tau)\,d\tau}

Q(t)=1-e^{-\bar\lambda(t)\,t}

\bar\lambda(t)=-\dfrac1t\ln[1-Q(t)]

$V$ = survival probability, $\bar\lambda$ = average hazard rate

Average hazard rate from a spread

\bar\lambda=\dfrac{s(T)}{1-R}

hazard between

T_1

and

T_2

\dfrac{\bar\lambda_2T_2-\bar\lambda_1T_1}{T_2-T_1}

$s$ = CDS, bond yield or asset swap spread

CDS-bond basis

=

CDS spread

-

bond yield spread

Merton model

E_0=V_0N(d_1)-De^{-rT}N(d_2)

d_1=\dfrac{\ln(V_0/D)+(r+\sigma_V^2/2)T}{\sigma_V\sqrt{T}},\ d_2=d_1-\sigma_V\sqrt{T}

\sigma_EE_0=N(d_1)\,\sigma_VV_0

debt

=V_0-E_0

expected loss on debt

=\dfrac{De^{-rT}-(V_0-E_0)}{De^{-rT}}

$V_0$ = firm value, $D$ = face value of debt

Merton PD and DD, risk neutral

\text{PD}=N(-d_2)

\text{DD}=d_2=\dfrac{\ln V_0-\ln D+(r-\sigma_V^2/2)T}{\sigma_V\sqrt{T}}

Exam tip Risk-neutral PD (from spreads, $N(-d_2)$ ) exceeds real-world PD (historical); use risk-neutral PD for valuation and real-world PD for scenario analysis and credit VaR.

CR 10

Credit Value at Risk

4 formulas

Transition matrix, other horizons

n

-year matrix

=

(one-year matrix)

^n

matrix for

1/n

of a year

=

n

th root of the one-year matrix

assumes independent periods

Vasicek WCDR and credit loss

\text{WCDR}(T,X)=N\left[\dfrac{N^{-1}(\text{PD})+\sqrt{\rho}\,N^{-1}(X)}{\sqrt{1-\rho}}\right]

loss at

X\approx\sum_i\text{WCDR}_i(T,X)\times\text{EAD}_i\times\text{LGD}_i

Basel IRB: $T$ = 1 year, $X$ = 99.9% (times maturity adjustment)

CreditRisk+ number of defaults

\text{Prob}(m\text{ defaults})=\dfrac{e^{-qn}(qn)^m}{m!}

$n$ loans, each with PD $q$

CreditMetrics rating thresholds

Sample

x\sim N(0,1)

, correlated across obligors by equity correlationnew rating boundaries at

N^{-1}

of cumulative transition probabilities from the best ratingdefault if

x>N^{-1}(1-\text{PD})

Exam tip Here credit VaR is the credit loss over $T$ not exceeded with confidence $X$ (the loss quantile itself, EL not deducted), while CR 11 deducts EL.

CR 11

Portfolio Credit Risk

6 formulas

Default correlation of two credits

\rho_{12}=\dfrac{\pi_{12}-\pi_1\pi_2}{\sqrt{\pi_1(1-\pi_1)}\sqrt{\pi_2(1-\pi_2)}}

\pi_{12}=\rho_{12}\sqrt{\pi_1(1-\pi_1)}\sqrt{\pi_2(1-\pi_2)}+\pi_1\pi_2

P[1\text{ or }2\text{ or both default}]=\pi_1+\pi_2-\pi_{12}

$\pi$ = PD, $\pi_{12}$ = joint PD

Single-factor model

a_i=\beta_im+\sqrt{1-\beta_i^2}\,\varepsilon_i

\text{corr}(a_i,a_j)=\beta_i\beta_j

default if

a_i\le k_i,\ \pi_i=\Phi(k_i)

$m$ = market factor, $k_i$ = default threshold

Conditional default probability

Given

m

a_i

normal, mean

\beta_im

, variance

1-\beta_i^2

p(m)=\Phi\left[\dfrac{k_i-\beta_im}{\sqrt{1-\beta_i^2}}\right]

Joint default, single factor

\pi_{12}=\Phi_2(k,k;\beta^2)

\rho=\dfrac{\Phi_2(k,k;\beta^2)-\pi^2}{\pi(1-\pi)}

$\Phi_2$ = bivariate standard normal, correlation $\beta^2$

Granular portfolio loss distribution

P[X\le x]=\Phi\left[\dfrac{k-\sqrt{1-\beta^2}\,\Phi^{-1}(x)}{\beta}\right]

market factor for loss level

x

m=\dfrac{k-\sqrt{1-\beta^2}\,\Phi^{-1}(x)}{\beta}

Credit VaR and granularity

Credit VaR

=

loss quantile

-

\rho=1

: portfolio behaves as one credit

\rho=0

: number of defaults

\sim\text{Binomial}(n,\pi)

, credit VaR falls as

n

rises,

\to0

for a very granular portfolio

CR 12

Credit Risk

3 formulas

Value with CVA and DVA

f=f_{\text{nd}}-\text{CVA}+\text{DVA}

\text{CVA}=\sum_iq_iv_i

\text{DVA}=\sum_iq_i^*v_i^*

$f_{\text{nd}}$ = no-default value; $q_i$ ( $q_i^*$ ) = risk-neutral PD of counterparty (bank) in interval $i$ ; $v_i$ ( $v_i^*$ ) = PV of expected loss to bank (counterparty)

Interval PD from spreads

q_i=\exp\left[-\dfrac{s(t_{i-1})\,t_{i-1}}{1-R}\right]-\exp\left[-\dfrac{s(t_i)\,t_i}{1-R}\right]

survival to

t_i=\exp\left[-\dfrac{s(t_i)\,t_i}{1-R}\right]

One-factor Gaussian copula PD

x_i=a_iF+\sqrt{1-a_i^2}\,Z_i

Q(T\mid F)=N\left[\dfrac{N^{-1}(Q(T))-\sqrt{\rho}\,F}{\sqrt{1-\rho}}\right]

$a_i=\sqrt{\rho}$ , $F$ = common factor

CR 13

Credit Derivatives

5 formulas

CDS spread valuation

\text{PV}(\text{expected payments})+\text{PV}(\text{expected accrual})=\text{PV}(\text{expected payoff})

payments:

\sum s\cdot S(t_j)\cdot e^{-rt_j}

payoff:

\sum(1-R)\cdot\text{PD}(\text{year }j)\cdot e^{-rt_{j,\text{mid}}}

accrual:

\sum\tfrac12s\cdot\text{PD}(\text{year }j)\cdot e^{-rt_{j,\text{mid}}}

defaults mid-year, annual payments in arrears; $S$ = survival probability

Marking a CDS to market

Value to protection seller

=s_{\text{contract}}\times(\text{PV payments}+\text{accrual per unit spread})-\text{PV}(\text{expected payoff})

buyer value

=

negative of this

Exam tip Use risk-neutral PDs (implied from bond prices or CDS quotes); CDS value is insensitive to $R$ if the same $R$ is used to imply PD and to value.

Binary CDS spread

Payoff

=1

(not

1-R

)binary spread

\approx\dfrac{\text{vanilla spread}}{1-R}

Index CDS, fixed coupon price

P=100-100\times D\times(s-c)

per 100buyer pays

100-P

upfront (negative: buyer receives)then pays coupon

c

on remaining notional (names not yet defaulted)

$D$ = PV factor of spread payments, $s$ = index spread, $c$ = fixed coupon

Synthetic CDO breakeven spread

s=\dfrac{C}{A+B}

$A$ = PV of spread payments per unit spread, $B$ = PV of accrual payments, $C$ = PV of expected payoffs

CR 14

Derivatives

Concepts only

No formulas to memorise. None of the 2026 learning objectives for this chapter needs a formula, so there is nothing from it to learn for this sheet.

CR 15

Counterparty Risk and Beyond

Concepts only

No formulas to memorise. None of the 2026 learning objectives for this chapter needs a formula, so there is nothing from it to learn for this sheet.

CR 16

Netting and Close-out

2 formulas

Net notional, trade compression

Net notional

=\sum

long notionals

-\sum

short notionals (same reference, maturity, coupon)

net contract typically held with the counterparty of the largest original position on the same side as the net position, as a reduction of that trade

Ring (multilateral) compression

In a closed ring of fungible obligations, every leg is reduced by the smallest notional in the ring (net positions unchanged)

CR 17

Margin and Settlement

3 formulas

Margin, one party

\text{Margin}=\max(\text{value}-K_C,0)+\text{IM}_C

$K_C$ = counterparty threshold, $\text{IM}_C$ = counterparty initial margin

Credit support amount, two-way VM

\text{Amount}=\max(\text{value}-K_C,0)-\max(-\text{value}-K_P,0)-C

positive: call margin, negative: return or posttransfer only if the amount exceeds the MTA (rounding may apply)IM is not netted against VM

$K_C$ , $K_P$ = counterparty and own thresholds, $C$ = margin already held

Haircut

Value credited

=(1-x)\times

market valueto deliver margin worth

M

post

M/(1-x)

haircut

=\Phi^{-1}(\alpha)\times\sigma_C\times\sqrt{\tau}

$\tau$ = liquidation time; for a bond, $\sigma_C$ is a yield volatility and the result is multiplied by duration

Bilateral (non-cleared)	Rule
VM	Regular (daily), zero threshold, MTA $\le$ €500,000 (same in USD), may be rehypothecated
IM	Both parties, gross (no netting), 99% confidence, 10-day horizon, segregated, no rehypothecation

CR 18

Central Clearing

0 formulas

Order	CCP loss waterfall
1	Defaulter IM
2	Defaulter default fund
3	CCP skin in the game
4	Non-defaulters' default fund
5	Rights of assessment and other loss allocation (VM gains haircutting, tear-up, forced allocation)
6	Remaining CCP capital
7	Liquidity support or CCP fails

CCP item	Rule
IM confidence	$\ge$ 99%
MPoR, OTC CCPs	About 5 business days
MPoR, exchange-traded	1 to 2 days
MPoR, bilateral	Minimum 10 days
Default fund	Cover 1: largest member and affiliates
Default fund, more complex risk profile or systemically important in multiple jurisdictions	Cover 2: two largest members

CR 19

Future Value and Exposure

5 formulas

Positive and negative exposure

Positive exposure

=\max(\text{value},0)

negative exposure

=\min(\text{value},0)

with margin:

\max(\text{value}-\text{margin},0)

\min(\text{value}-\text{margin},0)

Metric (scenario set)	Definition
EFV	Average of all values
EPE (EE)	Average with negative values set to 0
ENE (NEE)	Average with positive values set to 0
PFE	Exposure at a high confidence level
Maximum PFE	Highest PFE over the profile
Average EPE	Time-weighted average of EPE (loan equivalent)

Square-root-of-time profiles

Single final payment (FX forward): exposure

\propto\sqrt{t}

swap (roll-off): exposure

\propto(T-t)\sqrt{t}

, maximum at

t=T/3

Margin, exposure and funding

Positive exposure

=\max(\text{value}-\text{VM}-\text{IM}^R,0)

negative exposure

=\min(\text{value}-\text{VM}+\text{IM}^P,0)

funding

=\text{value}-\text{VM}+\text{IM}^P

$\text{IM}^R$ = IM received, $\text{IM}^P$ = IM posted

MPoR	Days
Basel II minimum (OTC, daily calls)	10 business days; add contractual days between calls
Basel III, certain cases	20 days
CCPs	About 5 days

Effective EE and effective EPE

\text{Effective EE}_k=\max(\text{Effective EE}_{k-1},\text{EE}_k)

Effective EPE

=

time-weighted average of effective EE over the first year

Netting factor

\text{Netting factor}=\dfrac{\sqrt{n+n(n-1)\bar{\rho}}}{n}

\bar{\rho}=0

1/\sqrt{n}

(two exposures 71%, four 50%)

\bar{\rho}=1

: 100%

\bar{\rho}\ge-1/(n-1)

$\bar{\rho}$ = average correlation

CR 20

CVA

6 formulas

Unilateral CVA

\text{UCVA}(t)=-\text{LGD}\sum_{i=1}^{m}\text{EPE}(t,t_i)\times\text{PD}(t_{i-1},t_i)

EPE discounted; PD = marginal default probability in each interval; CVA negative (a cost) in this chapter

CVA as a running spread

\text{UCVA}\approx-\text{average EPE}\times\text{spread}

running spread

=\dfrac{\text{upfront CVA}}{\text{risky annuity}}

per annum, same units as the credit spread; risky annuity = risky duration $\times$ notional

CVA, actual vs market LGD

\text{UCVA}(t)=-\text{LGD}_{\text{actual}}\sum\text{EPE}(t,t_i)\times\left[e^{-s_{i-1}t_{i-1}/\text{LGD}_{\text{mkt}}}-e^{-s_it_i/\text{LGD}_{\text{mkt}}}\right]

Exam tip If $\text{LGD}_{\text{actual}}=\text{LGD}_{\text{mkt}}$ the LGD terms cancel to first order (higher recovery raises implied PD but cuts LGD, so CVA barely moves), and CVA generally rises in magnitude as spreads widen.

CVA and DVA with survival

\text{CVA}=-\text{LGD}_C\sum\text{EPE}(t,t_i)\times\text{PD}_C(t_{i-1},t_i)\times[1-\text{PD}_P(0,t_{i-1})]

\text{DVA}=-\text{LGD}_P\sum\text{ENE}(t,t_i)\times\text{PD}_P(t_{i-1},t_i)\times[1-\text{PD}_C(0,t_{i-1})]

\text{BCVA}=\text{CVA}+\text{DVA}

$P$ = party calculating, $C$ = counterparty; DVA positive since ENE negative

BCVA as a spread

\text{BCVA}=-\,\text{average EPE}\times\text{Spread}_C-\text{average ENE}\times\text{Spread}_P

if average EPE

=-

average ENE:

\text{BCVA}=-\text{EPE}\times(\text{Spread}_C-\text{Spread}_P)

Incremental and marginal CVA

\text{CVA}^{NS}\ge\sum_i\text{CVA}_i

(netted CVA never more negative than the sum of standalone)

\text{CVA}^{NS\to NS^*}=\text{CVA}^{NS^*}-\text{CVA}^{NS}=-\text{LGD}\sum\text{EPE}^{NS\to NS^*}(t,t_i)\times\text{PD}(t_{i-1},t_i)

marginal CVA: same with marginal EPE, contributions sum to total CVA

$NS$ = netting set, $NS^*$ = netting set with the new trade

CR 21

Stress Testing Counterparty Exposures

4 formulas

Loan portfolio stress loss

\text{EL}=\sum_i p_i\cdot\text{ead}_i\cdot\text{lgd}_i

\text{EL}_s=\sum_i p_i^s\cdot\text{ead}_i\cdot\text{lgd}_i

stress loss

=\text{EL}_s-\text{EL}

Derivative portfolio stress loss

\text{EL}=\sum_i p_i\cdot\alpha\cdot\text{epe}_i\cdot\text{lgd}_i

\text{EL}_s=\sum_i p_i^s\cdot\alpha\cdot\text{epe}_i^s\cdot\text{lgd}_i

stress loss

=\text{EL}_s-\text{EL}

stress PD, EPE or both

Stressed CVA and stress loss

\text{CVA}_n=\text{LGD}_n^*\sum_j\text{EE}_n^*(t_j)\cdot q_n^*(t_{j-1},t_j)

\text{CVA}=\sum_n\text{CVA}_n

\text{CVA}_s

uses stressed LGD, EE,

q

stress loss

=\text{CVA}_s-\text{CVA}

$\text{EE}^*$ discounted; $q^*$ = risk-neutral marginal PD

Bilateral CVA with DVA

\text{BCVA}=\sum_n\text{LGD}_n^*\sum_j\text{EE}_n^*(t_j)\cdot q_n^*(t_{j-1},t_j)\cdot S_I^*(t_{j-1})-\sum_n\text{LGD}_I^*\sum_j\text{NEE}_n^*(t_j)\cdot q_I^*(t_{j-1},t_j)\cdot S_n^*(t_{j-1})

DVA

=

the second sumstress loss

=

stressed BCVA

-

BCVA

$I$ = the institution, $S$ = survival probability

Exam tip CR 21 writes CVA as a positive cost and $\text{BCVA}=\text{CVA}-\text{DVA}$ ; CR 20 writes CVA as negative and $\text{BCVA}=\text{CVA}+\text{DVA}$ .

CR 22

Structured Credit Risk

6 formulas

Tranche loss

Tranche writedown

=\dfrac{\min[\max(\text{pool loss}-A,0),\ D-A]}{\text{tranche size}}

$A$ = attachment, $D$ = detachment

Collateral interest

L_t=(r+\text{spread})\times\left(N-\sum_{\tau\le t}d_\tau\right)\times

loan parexcess spread

=L_t-B

$N$ = number of loans, $d$ = defaults, $B$ = bond coupons due

OC diversion and equity flow

\text{OC}_t=\min(L_t-B,K)

L_t\ge B

\text{OC}_t=\max\{L_t-B,\ -[(1+r)\,\text{OC account}_{t-1}+R_t]\}

L_t<B

equity flow

=\max(L_t-B-\text{OC}_t,0)

$K$ = cap on diversion

Recovery and OC account

R_t=0.4\times d_t\times

loan par

\text{OC account}_t=\text{OC}_t+R_t+(1+r)\times\text{OC account}_{t-1}

Terminal funds and tranche losses

F=(1+r)\,\text{OC account}_{T-1}+\left[\left(N-\sum d\right)(1+c_{\text{loan}})+0.4\,d_T\right]\times

loan parsenior loss

=\max[P_S(1+c_S)-F,0]

mezzanine loss

=\max[P_M(1+c_M)-(F-P_S(1+c_S)),0]

equity

=\max[F-P_S(1+c_S)-P_M(1+c_M),0]

$P$ , $c$ = tranche par and coupon ( $S$ senior, $M$ mezzanine); $r$ = OC account rate; $c_{\text{loan}}$ = loan coupon

default01

\text{default01}=\dfrac{1}{20}\times[V(\pi+0.0010)-V(\pi-0.0010)]

$V$ = mean value or loss; quoted positive, per 1 bp of default probability

CR 23

Introduction to Securitization

6 formulas

Delinquency and default ratios

Delinquency ratio

=\dfrac{\text{receivables overdue}>90\text{ days}}{\text{total receivables}}

default ratio

=\dfrac{\text{receivables written off in period}}{\text{total receivables at period end}}

credit cards

Monthly payment rate

\text{MPR}=\dfrac{\text{principal and interest repaid in period}}{\text{pool balance}}

collections / outstanding pool balance; not a prepayment measure

DSCR

\text{DSCR}=\dfrac{\text{net operating income}}{\text{debt payments}}

below 1.0: cash flow does not cover debt payments

WAC and WAM

\text{WAC}=\dfrac{\sum(\text{loan rate}\times\text{loan balance})}{\sum\text{loan balance}}

\text{WAM}=\dfrac{\sum(\text{remaining term in months}\times\text{loan balance})}{\sum\text{loan balance}}

WAL

\text{WAL}=\sum\dfrac{a}{365}\times\text{PF}(t)

$a$ = actual days in each payment period (period length, not elapsed time), PF = pool factor

SMM, CPR, PSA

\text{SMM}=\dfrac{\text{prepayment in month}}{\text{start-of-month balance}-\text{scheduled principal}}

\text{CPR}=1-(1-\text{SMM})^{12}

\text{PSA}=\dfrac{\text{CPR}}{0.2\times m}\times100

100% PSA: CPR rises 0.2% a month to 6% at month 30, then flat

$m$ = months since origination

Book 3 of 6 · 20% of the exam

Operational Risk and Resilience formulas

OR 1

Introduction to Operational Risk and Resilience

Concepts only

No formulas to memorise. None of the 2026 learning objectives for this chapter needs a formula, so there is nothing from it to learn for this sheet.

OR 2

Risk Governance

Concepts only

No formulas to memorise. None of the 2026 learning objectives for this chapter needs a formula, so there is nothing from it to learn for this sheet.

OR 3

Risk Identification

Concepts only

No formulas to memorise. None of the 2026 learning objectives for this chapter needs a formula, so there is nothing from it to learn for this sheet.

OR 4

Risk Measurement and Assessment

3 formulas

Fault tree, AND conditions

P(\text{scenario})=P_1\times P_2\times\cdots\times P_n

e.g. three controls each failing 10%:

10\%^3=1/1{,}000

independent conditions; the theoretical minimum likelihood

Fault tree, OR conditions

P(\text{at least one})=1-(1-P_1)(1-P_2)\cdots(1-P_n)

independent conditions

Scaling by unit of exposure

Expected occurrences

=

probability per unit

\times

number of units of exposuree.g.

1/1{,}000{,}000

per employee

\times100{,}000

employees

=10\%

Exam tip Heatmap ratings are ordinal, so never multiply likelihood by impact: $1\times4$ (frequent, low impact) is not the same risk as $4\times1$ (remote, extreme impact).

OR 5

Risk Mitigation

Concepts only

No formulas to memorise. None of the 2026 learning objectives for this chapter needs a formula, so there is nothing from it to learn for this sheet.

OR 6

Risk Reporting

Concepts only

No formulas to memorise. None of the 2026 learning objectives for this chapter needs a formula, so there is nothing from it to learn for this sheet.

OR 7

Integrated Risk Management

Concepts only

No formulas to memorise. None of the 2026 learning objectives for this chapter needs a formula, so there is nothing from it to learn for this sheet.

OR 8

Cyber-resilience: Range of practices

Concepts only

No formulas to memorise. None of the 2026 learning objectives for this chapter needs a formula, so there is nothing from it to learn for this sheet.

OR 9

Cyberthreats and Information Security Risks

Concepts only

No formulas to memorise. None of the 2026 learning objectives for this chapter needs a formula, so there is nothing from it to learn for this sheet.

OR 11

Financial Crime and Fraud

Concepts only

No formulas to memorise. None of the 2026 learning objectives for this chapter needs a formula, so there is nothing from it to learn for this sheet.

OR 12

Guidance on Managing Outsourcing Risk

Concepts only

No formulas to memorise. None of the 2026 learning objectives for this chapter needs a formula, so there is nothing from it to learn for this sheet.

OR 13

Third-Party Risk Management

Concepts only

No formulas to memorise. None of the 2026 learning objectives for this chapter needs a formula, so there is nothing from it to learn for this sheet.

OR 14

Investor Protection and Compliance Risks in Investment Activities

Concepts only

No formulas to memorise. None of the 2026 learning objectives for this chapter needs a formula, so there is nothing from it to learn for this sheet.

OR 15

Supervisory Guidance on Model Risk Management

Concepts only

No formulas to memorise. None of the 2026 learning objectives for this chapter needs a formula, so there is nothing from it to learn for this sheet.

OR 16

Model Risk and Model Validation

Concepts only

No formulas to memorise. None of the 2026 learning objectives for this chapter needs a formula, so there is nothing from it to learn for this sheet.

OR 17

Stress Testing Banks

Concepts only

No formulas to memorise. None of the 2026 learning objectives for this chapter needs a formula, so there is nothing from it to learn for this sheet.

OR 18

Risk Capital Attribution and RAROC

9 formulas

RAROC

\text{RAROC}=\dfrac{\text{After-tax expected risk-adjusted net income}}{\text{Economic capital}}

capital budgeting form:

\text{RAROC}=\dfrac{\begin{gathered}\text{Expected revenues}-\text{Costs}-\text{EL}-\text{Taxes}\\+\,\text{Return on risk capital}\pm\text{Transfers}\end{gathered}}{\text{Economic capital}}

RAROC inputs

Return on risk capital

=

risk capital

\times r_f

after-tax income

=

pre-tax

\times(1-t)

ex post performance: realized revenues and losses in place of expected

$t$ = effective tax rate

Economic capital

\text{EC}=

risk capital

+

strategic risk capitalstrategic risk capital

=

goodwill

+

burned-out capitalrisk capital

=

worst-case loss at the confidence level

-\,\text{EL}

(unexpected loss)

Confidence level from target rating

Confidence level

=1-

target-rating one-year PDAAA

0.01\%\to99.99\%

0.03\%\to99.97\%

(AA 3 to 5 bp

\to

99.95 to 99.97%)

Hurdle rate

h_{AT}=\dfrac{CE\times r_{CE}+PE\times r_{PE}}{CE+PE}

r_{CE}=r_f+\beta_{CE}(R_M-r_f)

accept if

\text{RAROC}>h_{AT}

$CE$ , $PE$ = market values of common and preferred equity; $r_{PE}$ = preferred yield

Adjusted RAROC

\text{Adjusted RAROC}=\text{RAROC}-\beta_E(R_M-r_f)

accept if adjusted RAROC

>r_f

$\beta_E$ = beta of the firm's equity

Market risk capital over one year

Naive: annualized VaR

=

daily VaR

\times\sqrt{252}

\text{RC}=\sqrt{\sum\text{VaR}_t^2+\text{VaR}_{\text{core}}^2\times\text{remaining days}}

with a core risk level; sum over the time-to-reduce days; days total 252

Aggregate risk capital bounds

Perfect correlation:

\sum\text{RC}_i

zero correlation:

\sqrt{\sum\text{RC}_i^2}

firm capital lies between the two

Allocating capital in a business unit

Diversification effect

=\sum\text{stand-alone RC}-\text{RC}_{BU}

fully diversified

\text{RC}_i=

stand-alone

\text{RC}_i-\text{effect}\times\dfrac{\text{RC}_i}{\sum\text{RC}}

(pro rata)marginal

\text{RC}_i=\text{RC}_{BU}-\text{RC}_{BU\text{ without }i}

\sum\text{marginal}<\text{RC}_{BU}

OR 19

Range of practices and issues in economic capital frameworks

Concepts only

No formulas to memorise. None of the 2026 learning objectives for this chapter needs a formula, so there is nothing from it to learn for this sheet.

OR 20

Capital Planning at Large Bank Holding Companies

Concepts only

No formulas to memorise. None of the 2026 learning objectives for this chapter needs a formula, so there is nothing from it to learn for this sheet.

OR 21

Capital Regulation Before the GFC

11 formulas

Basel I capital ratios

\text{Tier 1}/\text{RWA}>4\%

\text{total capital}/\text{RWA}>8\%

(Cooke ratio)total capital

=

Tier 1

+

Tier 2Tier 2

\le

half of total capitalloan loss reserves in capital

\le1.25\%

of RWA (originally 2%)

Basel I RWA

\text{RWA}=\sum w_iA_i

capital

=8\%\times\text{RWA}

off-balance sheet: credit equivalent

=\text{CCF}\times

principal, then

\times

counterparty risk weightderivative risk weight capped at 50%

Weight	Basel I on-balance-sheet exposures
0%	Cash, claims on OECD governments, full OECD government guarantee
20%	OECD banks and OECD public sector entities (municipalities, Fannie Mae, Freddie Mac)
50%	Uninsured residential mortgages
100%	All other (corporate, consumer loans)

CCF	Basel I off-balance-sheet items
100%	Guarantees on loans and bonds, bankers acceptances
50%	Warranties, transaction-related standby letters of credit
20%	Loan commitments, original maturity $\ge1$ year
0%	Loan commitments $<1$ year

Derivatives, current exposure method

Credit equivalent

=\max(V,0)+\text{add-on factor}\times L

$L$ = notional; add-on factors in % of notional below

Maturity	Interest rate add-on	FX add-on
$<1$ yr	0%	1%
1 to 5 yr	0.5%	5%
$>5$ yr	1.5%	7.5%

Netting (1995 amendment)

\text{CEA}=\sum_c\max(\sum_{i\in c}V_i,0)+\sum_j(0.4\,D_j+0.6\,\text{NRR}\,D_j)

\text{NRR}=\dfrac{\sum_c\max(\sum_{i\in c}V_i,0)}{\sum_i\max(V_i,0)}=\dfrac{\text{net current exposure}}{\text{gross positive exposure}}

netting only within each counterparty $c$ ; $D_j$ = add-on factor $\times$ total notional of derivative type $j$ (a currency amount)

1996 market risk charge (internal models)

\text{MR}=\max(\text{VaR}_{t-1},\ m\times\text{VaR}_{\text{avg}})

10-day 99% VaR (

\sqrt{10}\times

one-day allowed),

\text{VaR}_{\text{avg}}

over 60 days,

m\ge3

specific risk multiplier 4market RWA

=12.5\times\text{MR}\ (12.5=1/8\%)

Backtesting multiplier

1-day 99% VaR vs actual loss over the last 250 days

m=3

for fewer than 5 exceptions5, 6, 7, 8, 9 exceptions: 3.40, 3.50, 3.65, 3.75, 3.8510 or more: 4

same zones as the MR 4 backtesting table

IRB capital

\text{Capital}=\sum[\text{EAD}_i\times\text{LGD}_i\times\text{DR}_{99.9,i}]-\text{EL}

\text{EL}=\sum\text{EAD}_i\times\text{LGD}_i\times\text{PD}_i

\text{DR}_{99.9,i}=N\left[\dfrac{N^{-1}(\text{PD}_i)+\sqrt{\rho}\,N^{-1}(0.999)}{\sqrt{1-\rho}}\right]

IRB RWA, bank, corporate, sovereign

\text{RWA}=12.5\times\text{EAD}\times\text{LGD}\times(\text{DR}-\text{PD})\times\text{MA}

\text{MA}=\dfrac{1+(M-2.5)\,b}{1-1.5\,b}

b=[0.11852-0.05478\ln(\text{PD})]^2

F-IRB:

\text{PD}\ge0.03\%

(bank, corporate), LGD 45% senior, 75% subordinated,

M=2.5

in most cases

$M$ = effective maturity; MA = maturity adjustment

IRB correlation

Bank, corporate, sovereign:

\rho=0.12\,\dfrac{1-e^{-50\text{PD}}}{1-e^{-50}}+0.24\left[1-\dfrac{1-e^{-50\text{PD}}}{1-e^{-50}}\right]

retail: residential mortgages

\rho=0.15

, qualifying revolving

\rho=0.04

, other retail same form with 0.03, 0.16 and

-35\text{PD}

retail: no MA

Basic indicator approach

Capital

=15\%\times

average annual gross income over the past three years, ignoring years of negative gross income

Standardized approach (operational)

Each year: capital

=\sum\beta_i\times\text{GI}_i

over 8 business lines (negatives offset within the year)requirement

=

average over three years, ignoring years with negative total

$\beta$	Business lines
18%	Corporate finance, trading and sales, payment and settlement
15%	Commercial banking, agency services
12%	Retail banking, asset management, retail brokerage

Exam tip A year with a negative total is excluded from the BIA or TSA average (divide by the number of positive years), not floored at zero and kept.

Standard	Confidence	Horizon
Market risk (1996)	99%	10-day
IRB credit	99.9%	1 year
AMA operational (insurance offset $\le20\%$ )	99.9%	1 year
Solvency II SCR	99.5%	1 year

OR 22

Post-Crisis Solvency and Liquidity Regulation

9 formulas

Basel 2.5 market risk charge

\text{MR}_{2.5}=\max(\text{VaR}_{t-1},m_c\text{VaR}_{\text{avg}})+\max(\text{SVaR}_{t-1},m_s\text{SVaR}_{\text{avg}})

10-day 99%, averages over 60 days,

m_c,m_s\ge3

SVaR from the most stressful one-year (250-day) period in the past 7 years

Exam tip With equal multipliers, $\text{MR}_{2.5}\ge2\times\text{MR}_{1996}$ : stressed VaR is added to VaR, it does not replace it.

Incremental risk charge

99.9th percentile, one-year horizon, constant level of risk (downgraded or defaulted positions replaced)liquidity horizon never less than 3 months

Basel III minimum ratios

\text{CET1}/\text{RWA}\ge4.5\%

\text{Tier 1 }(\text{CET1+AT1})/\text{RWA}\ge6\%

\text{total capital }(\text{Tier 1+Tier 2})/\text{RWA}\ge8\%

general reserves in Tier 2

\le1.25\%

of standardized RWA or

0.6\%

of IRB RWA

Basel III capital components

Core Tier 1

=

common equity

+

retained earnings

+

limited minority interest and unrealized gains and losses

-

goodwill and other intangibles

-

deferred tax assets

-

shortfall of reserves vs IRB expected lossesAdditional Tier 1

=

non-cumulative perpetual preferred (callable only after

\ge5

years)

+

debt with conversion or write-down triggers

+

minority interest not in Core Tier 1Tier 2

=

subordinated debt (

\ge5

years original maturity, callable only after

\ge5

years)

+

general loan loss reserves (cap above)other deductions: defined-benefit pension deficits, certain cross-holdings, mortgage servicing rights

>10\%

of common equity

Basel III buffers

\text{CCB}=2.5\%

of RWA

\text{CCyB}=0

2.5\%

of RWA (international bank: weighted average of each jurisdiction's CCyB)G-SIB surcharge

=1

, 1.5, 2, 2.5 or 3.5%CET1 requirement

=4.5\%+2.5\%+\text{G-SIB surcharge}\ (+\,\text{CCyB})

Leverage ratio

\text{Leverage ratio}=\dfrac{\text{Tier 1 capital}}{\text{Exposure measure}}\ge3\%

exposure measure

=

on-balance-sheet assets

+

fractions of off-balance-sheet exposures

the OR 22 text uses core Tier 1

Liquidity coverage ratio

\text{LCR}=\dfrac{\text{HQLA}}{\text{Net cash outflows over 30 days}}\ge100\%

\text{HQLA}=\sum\text{assets}\times(1-\text{haircut})

Level 2 assets

\le40\%

of HQLA after haircut

Net stable funding ratio

\text{NSFR}=\dfrac{\text{Available stable funding}}{\text{Required stable funding}}\ge100\%

\text{ASF}=\sum\text{funding}\times\text{ASF factor}

\text{RSF}=\sum\text{assets}\times\text{RSF factor}

one-year horizon

CoCo capital classification

Trigger

\text{CET1}/\text{RWA}\ge5.125\%

: Additional Tier 1lower trigger: Tier 2

OR 23

Basel III Reforms Summary

2 formulas

Output floor

\text{RWA}=\max(\text{RWA}_{\text{IM}},\ 72.5\%\times\text{RWA}_{\text{SA}})

$\text{RWA}_{\text{IM}}$ = total RWA using approved approaches, including internal models; $\text{RWA}_{\text{SA}}$ = total RWA using only the standardized approaches

G-SIB leverage ratio buffer

Buffer

=50\%\times

G-SIB risk-weighted higher-loss-absorbency requirement, met with Tier 1Tier 1 leverage requirement

=3\%+

buffer

OR 24

Basel III Operational Risk Capital

5 formulas

Business indicator

\text{BI}=\text{ILDC}+\text{SC}+\text{FC}

\text{ILDC}=\min[|\text{II}-\text{IE}|,\ 2.25\%\times\text{IEA}]+\text{DI}

\text{SC}=\max(\text{OOI},\text{OOE})+\max(\text{FI},\text{FE})

\text{FC}=|\text{Net P\&L trading book}|+|\text{Net P\&L banking book}|

II, IE = interest income, expense; IEA = interest earning assets; DI = dividend income; OOI, OOE = other operating income, expense; FI, FE = fee income, expense; each term a three-year average ( $t$ , $t-1$ , $t-2$ ), absolute values taken year by year before averaging

Business indicator component

\text{BIC}=\sum\alpha_i\,\text{BI}_i

(marginal)bucket 1,

\text{BI}\le

€1bn: 12%bucket 2, €1bn

<\text{BI}\le

€30bn: 15%bucket 3,

\text{BI}>

€30bn: 18%

Exam tip Each coefficient applies only to the slice of BI inside its bucket (like tax slabs), never to the whole BI.

Loss component and ILM

\text{LC}=15\times

average annual operational risk losses over the previous 10 years

\text{ILM}=\ln\!\left(e^1-1+(\text{LC}/\text{BIC})^{0.8}\right)

\text{LC}=\text{BIC}\Rightarrow\text{ILM}=1

\text{LC}>\text{BIC}\Rightarrow\text{ILM}>1

Operational risk capital

\text{ORC}=\text{BIC}\times\text{ILM}

\text{RWA}_{\text{op}}=12.5\times\text{ORC}

bucket 1:

\text{ILM}=1

\text{ORC}=12\%\times\text{BI}

supervisor may set

\text{ILM}=1

for all banksloss data standards not met: capital

\ge100\%

of BICfewer than 5 years of data: BIC only

Loss data rules

10-year observation period (5 years acceptable on first moving to the SA)loss inclusion threshold €20,000, raisable to €100,000 for buckets 2 and 3losses net of recoveries, including insurance (receivables and tax effects are not recoveries)

Book 4 of 6 · 15% of the exam

Liquidity and Treasury Risk Measurement and Management formulas

LR 1

Liquidity Risk

5 formulas

Bid-offer spread

p=\text{Offer price}-\text{Bid price}

s=\dfrac{\text{Offer price}-\text{Bid price}}{\text{Mid-market price}}

Cost of liquidation, normal market

\displaystyle\sum_i\dfrac{s_i\alpha_i}{2}

$s_i$ = proportional spread, $\alpha_i$ = dollar mid-market value of position $i$

Cost of liquidation, stressed market

\displaystyle\sum_i\dfrac{(\mu_i+\lambda\sigma_i)\,\alpha_i}{2}

$\mu_i$ , $\sigma_i$ = mean and SD of the proportional spread; $\lambda=2.326$ at 99% if spreads are normal

Liquidity-adjusted VaR

\text{LVaR}=\text{VaR}+\displaystyle\sum_i\dfrac{s_i\alpha_i}{2}

stressed:

\text{LVaR}=\text{VaR}+\displaystyle\sum_i\dfrac{(\mu_i+\lambda\sigma_i)\,\alpha_i}{2}

Basel III LCR and NSFR

\text{LCR}=\dfrac{\text{High-quality liquid assets}}{\text{Net cash outflows in 30 days}}\ge100\%

\text{NSFR}=\dfrac{\text{Amount of stable funding}}{\text{Required amount of stable funding}}\ge100\%

NSFR: funding $\times$ ASF factor over assets and off-balance-sheet items $\times$ RSF factor

LR 2

Liquidity and Leverage

6 formulas

Leverage ratio

L=\dfrac{A}{E}=\dfrac{E+D}{E}=1+\dfrac{D}{E}

collateralized position with haircut

h

L=\dfrac{1}{h}

Leverage effect

r^e=Lr^a-(L-1)\,r^d

\dfrac{\partial r^e}{\partial L}=r^a-r^d

$r^a$ = return on assets, $r^d$ = cost of debt, $r^e$ = return on equity

Levered return on repo-financed bond

\dfrac{c-(1-h)\,r}{h}=c+\dfrac{1-h}{h}\,(c-r)

$c$ = coupon, $h$ = haircut, $r$ = repo rate

Expected transactions cost

E[P_{t+1}]\times\dfrac{\bar s}{2}

s=\dfrac{2(\text{ask}-\text{bid})}{\text{ask}+\text{bid}}=\dfrac{\text{ask}-\text{bid}}{\text{midprice}}

Spread risk factor (99%)

\tfrac12(\bar s+2.33\,\sigma_s)

99% CI on transactions cost per unit:

\pm\bar P\times\tfrac12(\bar s+2.33\,\sigma_s)

LVaR for liquidation over T days

\text{VaR}\times\sqrt{\dfrac{(1+T)(1+2T)}{6T}}

T=5

\times1.4832

T=10

: about

\times2

equal daily sales; VaR = one-day position VaR

Exam tip Malz scales VaR by $\sqrt{(1+T)(1+2T)/(6T)}$ , not by $\sqrt{T}$ , while Hull adds the spread cost to VaR.

LR 3

Early Warning Indicators

Concepts only

No formulas to memorise. None of the 2026 learning objectives for this chapter needs a formula, so there is nothing from it to learn for this sheet.

LR 4

The Investment Function

2 formulas

After-tax yield and TEY

\text{After-tax gross yield}=\text{Before-tax gross yield}\times(1-t)

\text{TEY}=\dfrac{\text{After-tax return on tax-exempt investment}}{1-t}

$t$ = marginal income tax rate

Duration price change and immunization

\%\Delta P\approx-D\,\dfrac{\Delta i}{1+i}

m

payments a year:

\%\Delta P\approx-D\,\dfrac{\Delta i}{1+i/m}

immunize:

D=

planned holding period

LR 5

Liquidity and Reserves Management

4 formulas

Net liquidity position

L=

(Incoming deposits + Revenues from nondeposit services + Customer loan repayments + Sales of assets + Borrowings from money market)

-

(Deposit withdrawals + Volume of acceptable loan requests + Repayments of borrowings + Other operating expenses + Dividend payments)

$L<0$ : deficit; $L>0$ : surplus

Sources and uses of funds

Estimated liquidity deficit

(-)

or surplus

(+)

=

Estimated change in deposits

-

Estimated change in loans

Total liquidity requirement (structure of funds)

\text{Total}=0.95\,(\text{Hot money funds}-\text{RR})+0.30\,(\text{Vulnerable funds}-\text{RR})+0.15\,(\text{Stable funds}-\text{RR})+1.00\,(\text{Potential loans}-\text{Actual loans})

RR = required legal reserves held against that category

Expected liquidity requirement

\sum P(\text{outcome})\times

estimated liquidity surplus or deficit in that outcome

Liquidity indicator	Ratio
Cash position	Cash and due from depository institutions / Total assets
Liquid securities	U.S. government securities / Total assets
Net fed funds and repo position	(Fed funds sold + reverse repos $-$ Fed funds purchased $-$ repos) / Total assets
Capacity (negative)	Net loans and leases / Total assets
Pledged securities (negative)	Pledged securities / Total securities
Hot money	Money market assets / Volatile liabilities
Deposit brokerage	Brokered deposits / Total deposits
Core deposit	Core deposits / Total assets
Deposit composition	Demand deposits / Time deposits
Loan commitments	Unused loan commitments / Total assets

LR 6

Intraday Liquidity Risk Management

Concepts only

No formulas to memorise. None of the 2026 learning objectives for this chapter needs a formula, so there is nothing from it to learn for this sheet.

LR 7

Monitoring Liquidity

4 formulas

Cumulated expected cash flow

CF(t_0,t_a,t_b)=\displaystyle\sum_{i=a}^{b}\left[cf_e^+(t_0,t_i)+cf_e^-(t_0,t_i)\right]

\text{TSECCF}=\{CF(t_0,t_0,t_1),\dots,CF(t_0,t_0,t_b)\}

Expected liquidity

\text{TSL}_e(t_0,t_i)=\text{TSECCF}(t_0,t_i)+\text{TSCLGC}(t_0,t_i)

\text{TSECCF}(t_0,t_0)=\text{Cash}(t_0)

solvency requires

\text{TSL}_e\ge0

at every date

Cash flow at risk

\text{cfaR}_\alpha(t_0,t_i)=cf_\alpha(t_0,t_i;x)-cf_e(t_0,t_i;x)

\text{cfaR}_{1-\alpha}(t_0,t_i)=cf_{1-\alpha}(t_0,t_i;x)-cf_e(t_0,t_i;x)

$cf$ = total net cash flow on $t_i$ ; $cf_e=E[cf]$ ; $x$ = vector of risk factors

Repo cash and repurchase amount

\text{Cash received}=N\times(\text{Clean price}+\text{Accrued})\times(1-h)

Repayment

=

Cash received

\times\,(1+r_{\text{repo}}\,\tau)

$N$ = notional, $h$ = haircut, $\tau$ = repo term in years

LR 8

The Failure Mechanics of Dealer Banks

Concepts only

No formulas to memorise. None of the 2026 learning objectives for this chapter needs a formula, so there is nothing from it to learn for this sheet.

LR 9

Liquidity Stress Testing

1 formula

Stressed liquid asset buffer

Normal liquid asset buffer

-

Stressed cash outflows

+

Stressed cash inflows

LR 10

Liquidity Risk Reporting and Stress Testing

Concepts only

No formulas to memorise. None of the 2026 learning objectives for this chapter needs a formula, so there is nothing from it to learn for this sheet.

LR 11

Contingency Funding Planning

Concepts only

No formulas to memorise. None of the 2026 learning objectives for this chapter needs a formula, so there is nothing from it to learn for this sheet.

LR 12

Managing Non-Deposit Liabilities

4 formulas

Available funds gap

AFG

=

Current and projected loans and investments the institution desires to make

-

Current and expected deposit inflows and other available funds

Effective cost of a funding source

\text{Effective cost rate}=\dfrac{\text{Current interest cost}+\text{Noninterest costs}}{\text{Net investable funds}}

Interest cost

=

Money market rate

\times

Amount borrowedNoninterest cost

=

Cost rate

\times

Amount borrowedNet investable funds

=

Amount borrowed

-

Legal reserves

-

Deposit insurance

-

Funds placed in nonearning assets

Historical average cost of funds

Weighted average interest expense

=\dfrac{\text{All interest paid}}{\text{Total funds raised}}

Break-even cost rate

=\dfrac{\text{Interest}+\text{Other operating costs}}{\text{All earning assets}}

Weighted average overall cost of capital

=

Break-even cost

+\left[\dfrac{\text{After-tax cost of equity}}{1-t}\right]\times\dfrac{\text{Stockholders’ investment}}{\text{Earning assets}}

Pooled-funds cost and hurdle rate

Pooled deposit and nondeposit funds expense

=\dfrac{\text{All expected operating expenses}}{\text{All new funds expected}}

Hurdle rate over all earning assets

=\dfrac{\text{All expected operating costs}}{\text{Dollars available to place in earning assets}}

LR 13

Repurchase Agreements and Financing

3 formulas

Repo repurchase price

Repurchase price

=

Invoice price (incl. accrued)

\times\left(1+\text{repo rate}\times\dfrac{\text{days}}{360}\right)

actual/360

Special spread and fails penalty

Special spread

=

GC rate

-

Special rateFails penalty rate

=\max(3\%-\text{fed funds target},\ 0)

upper limit of the special spread: GC rate before May 2009, penalty rate after

Financing advantage of a special

Financing value

=

Market value

\times

Special spread

\times\dfrac{\text{days}}{360}

e.g. $100\times 125\times 0.22\%/360=0.076$ per 100 market value

LR 14

Liquidity Transfer Pricing

2 formulas

Cost of contingent liquidity risk

Charge rate

=\dfrac{\text{Limit}-\text{Drawn amount}}{\text{Limit}}\times

Likelihood of drawdown

\times

Cost of funding liquidity cushionDollar charge

=

Charge rate

\times

Limit

e.g. $(10-4)/10\times 0.6\times 0.0018=6.48\text{ bps}$ ; $\quad\times\,\$10\text{m}=\$6{,}480$

Funds transfer price

\text{FTP}=\text{Base rate}+\text{Term liquidity premium}+\text{Liquidity premium}

liquidity premium = cost of carrying the liquidity cushion / total assets, e.g. $\$30\text{m}/\$300\text{bn}=1\text{ bp}$

LR 15

The US Dollar Shortage in Global Banking and the International Policy Response

Concepts only

No formulas to memorise. None of the 2026 learning objectives for this chapter needs a formula, so there is nothing from it to learn for this sheet.

LR 16

Covered Interest Parity Lost

2 formulas

Covered interest parity

\dfrac{F}{S}=\dfrac{1+r}{1+r^*}

USD at a premium in FX swaps:

F-S>S\left[\dfrac{1+r}{1+r^*}-1\right]

$S$ , $F$ in USD per unit of foreign currency; $r$ = USD rate, $r^*$ = foreign rate

Cross-currency basis

F-S=S\,\dfrac{1+r+b}{1+r^*}-S

with

r^*

small:

b=\dfrac{F-S}{S}-(r-r^*)

drop in market liquidity:

b=\dfrac{F}{S}-\dfrac{F_b}{S_a}

credit or other risk premia:

b=rp

hedging demand with balance sheet costs:

b\propto rp\times\text{FX hedging demand}

$S_a$ = spot ask, $F_b$ = forward bid

LR 17

Asset-Liability Management

6 formulas

Net interest margin

NIM

=

(Interest income from loans and investments

-

Interest expense on deposits and other borrowed funds) / Total earning assets

Interest-sensitive gap measures

\text{ISGAP}=\text{ISA}-\text{ISL}

\text{Relative ISGAP}=\dfrac{\text{ISGAP}}{\text{Total assets}}

\text{Interest sensitivity ratio}=\dfrac{\text{ISA}}{\text{ISL}}

Change in net interest income

\Delta\text{NII}=\Delta i\times\text{Cumulative gap}

$\Delta i$ in percentage points, gap in dollars

Duration

D=\dfrac{\sum_t CF_t\,t/(1+\text{YTM})^t}{\sum_t CF_t/(1+\text{YTM})^t}=\dfrac{\sum_t CF_t\,t/(1+\text{YTM})^t}{\text{Price}}

D_{\text{portfolio}}=\sum(\text{Market value weight}\times D_i)

Leverage-adjusted duration gap

\text{Duration gap}=D_A-D_L\times\dfrac{\text{Total liabilities}}{\text{Total assets}}

Change in net worth

\Delta\text{NW}=\left[-D_A\dfrac{\Delta i}{1+i}A\right]-\left[-D_L\dfrac{\Delta i}{1+i}L\right]

\approx-\left[D_A-\dfrac{L}{A}D_L\right]\times A\times\dfrac{\Delta i}{1+i}

$i$ = original rate

Gap	Rates up	Rates down
IS gap $>0$ (asset sensitive; relative ISGAP $>0$ , ISR $>1$ )	NIM up	NIM down
IS gap $<0$ (liability sensitive)	NIM down	NIM up
Duration gap $>0$	NW down	NW up
Duration gap $<0$	NW up	NW down
Zero gap	Immunized	Immunized

Exam tip The duration gap scales $D_L$ by $L/A$ , so a zero gap needs $D_L=D_A\times A/L$ (slightly above $D_A$ ).

Book 5 of 6 · 15% of the exam

Risk Management and Investment Management formulas

IM 1

Factor Theory

3 formulas

CAPM: market risk premium and SML

E(r_m)-r_f=\bar\gamma\,\sigma_m^2

E(r_i)-r_f=\beta_i\,[E(r_m)-r_f]

\ \beta_i=\dfrac{\text{cov}(r_i,r_m)}{\text{var}(r_m)}

$\bar\gamma$ = risk aversion of the average investor

SDF pricing

P_i=E[m\times\text{payoff}_i]

1=E[m(1+r_i)]

risk-free:

\dfrac{1}{1+r_f}=E[m]

$m$ = stochastic discount factor

SDF risk premium (beta form)

E(r_i)-r_f=\left[\dfrac{\text{cov}(r_i,m)}{\text{var}(m)}\right]\times\left[\dfrac{-\text{var}(m)}{E(m)}\right]=\beta_{i,m}\times\lambda_m

payoff high in bad times (high cov with $m$ ) earns a lower premium

IM 2

Factors

1 formula

Fama-French model (plus momentum)

E(r_i)=r_f+\beta_{i,\text{MKT}}\,E(r_m-r_f)+\beta_{i,\text{SMB}}\,E(\text{SMB})+\beta_{i,\text{HML}}\,E(\text{HML})

momentum version adds

+\,\beta_{i,\text{WML}}\,E(\text{WML})

IM 3

Alpha and the Low-Risk Anomaly

5 formulas

Alpha, tracking error, IR

r_t^{ex}=r_t-r_t^{bmk}

\alpha=\dfrac{1}{T}\sum r_t^{ex}

\text{TE}=\bar\sigma=\text{stdev}(r_t^{ex})

\text{IR}=\dfrac{\alpha}{\bar\sigma}

Sharpe ratio (benchmark = $r_f$ )

\alpha=\text{mean}(r_t-r_{ft})

\text{SR}=\dfrac{\text{mean}(r_t-r_{ft})}{\sigma}

IR with a risk-free benchmark = SR

Fundamental law of active management

\text{IR}\approx\text{IC}\times\sqrt{\text{BR}}

IC = information coefficient; BR = number of independent bets per year

CAPM benchmark (mimicking portfolio)

E(r_i)=(1-\beta)\,r_f+\beta\,E(r_m)

r_t^{bmk}=(1-\beta)\,r_t^f+\beta\,r_t^m

\alpha=\text{mean}(r_t-r_t^{bmk})

Factor regression (CAPM, Fama-French, momentum)

r_{it}-r_{ft}=\alpha+\beta(r_{mt}-r_{ft})+s\,\text{SMB}_t+h\,\text{HML}_t\,(+\,u\,\text{UMD}_t)+\varepsilon_{it}

benchmark

=(1-\beta)

in T-bills

+\,\beta

in market

+\,s(\text{small}-\text{large})+h(\text{value}-\text{growth})

IM 4

Portfolio Construction

8 formulas

Alpha structure and scale

\alpha=\text{volatility}\times\text{IC}\times\text{score}

\text{Std}\{\alpha\}\approx\text{volatility}\times\text{IC}

score has mean 0 and SD 1

Exam tip Trim outliers: examine alphas with magnitude above 3 times the scale, and pull genuine ones back to plus or minus 3 times the scale.

Benchmark neutralization

\alpha_n(\text{neutral})=\alpha_n-\beta_n\times\alpha_B

Optimal active risk aversion

\lambda_A=\dfrac{\text{IR}}{2\psi_P}

$\psi_P$ = active risk, in percent (not decimals)

Annualized transaction cost

annualized TC

=\dfrac{\text{round-trip cost}}{\text{holding period (years)}}

Marginal contribution to value added

\text{MCVA}_n=\alpha_n-2\lambda_A\times\psi\times\text{MCAR}_n

MCAR = marginal contribution to active risk

No-trade region

-\text{SC}_n\le\text{MCVA}_n\le\text{PC}_n

2\lambda_A\psi\,\text{MCAR}_n-\text{SC}_n\le\alpha_n\le\text{PC}_n+2\lambda_A\psi\,\text{MCAR}_n

band width

=\text{PC}_n+\text{SC}_n

PC, SC = purchase and sale costs

Value added objective

\alpha_P-\lambda_A\psi_P^2-\text{TC}

split aversions:

U=\alpha_P-(\lambda_{A,CF}\Psi_{P,CF}^2+\lambda_{A,SP}\Psi_{P,SP}^2)

CF = common factor, SP = specific

Dispersion bound

\psi^2\le\dfrac{\text{TC}}{2\lambda_A}

E\{r_{PA,\text{max}}-r_{PA,\text{min}}\}=2\times\Phi^{-1}\{(1/2)^{1/N}\}\times\psi

$\psi$ and TC in percent

IM 5

Portfolio Risk: Analytical Methods

10 formulas

Diversified (portfolio) VaR

\text{VaR}_p=\alpha\sigma_pW=\alpha\sqrt{x'\Sigma x}

two assets:

\text{VaR}_p=\alpha\sqrt{w_1^2\sigma_1^2+w_2^2\sigma_2^2+2w_1w_2\rho_{12}\sigma_1\sigma_2}\times W

$x$ = dollar exposures

Individual and undiversified VaR

\text{VaR}_i=\alpha\sigma_i|W_i|=\alpha\sigma_i|w_i|W

undiversified VaR

=\sum\text{VaR}_i

\rho=0

\text{VaR}_p=\sqrt{\text{VaR}_1^2+\text{VaR}_2^2}

\rho=1

\text{VaR}_p=\text{VaR}_1+\text{VaR}_2

Equal-weight portfolio risk

\sigma_p=\sigma\sqrt{\dfrac{1}{N}+\left(1-\dfrac{1}{N}\right)\rho}\ \to\ \sigma\sqrt{\rho}

N\to\infty

Beta of position

\beta_i=\dfrac{\text{cov}(R_i,R_p)}{\sigma_p^2}=\rho_{ip}\dfrac{\sigma_i}{\sigma_p}

vector:

\beta=\dfrac{\Sigma w}{w'\Sigma w}

Marginal VaR

\Delta\text{VaR}_i=\dfrac{\partial\text{VaR}}{\partial x_i}=\alpha\dfrac{\text{cov}(R_i,R_p)}{\sigma_p}=\alpha\,\beta_i\,\sigma_p=\dfrac{\text{VaR}}{W}\,\beta_i

Incremental VaR

\text{Incremental VaR}=\text{VaR}_{p+a}-\text{VaR}_p\approx(\Delta\text{VaR})'\times a

$a$ = vector of new positions

Component VaR

\text{CVaR}_i=(\Delta\text{VaR}_i)\,w_iW=\text{VaR}\,\beta_i\,w_i=\text{VaR}_i\,\rho_i

\sum\text{CVaR}_i=\text{VaR}

Percent contribution to VaR

\dfrac{\text{CVaR}_i}{\text{VaR}}=w_i\beta_i

Best hedge (risk-minimizing trade)

a^*=-W\dfrac{\sigma_{ip}}{\sigma_i^2}=-W\beta_i\dfrac{\sigma_p^2}{\sigma_i^2}

Minimum-risk and optimal portfolio

min risk:

\Delta\text{VaR}_i=\dfrac{\text{VaR}}{W}\times\beta_i=

constant (all

\beta_i

equal)max Sharpe:

\dfrac{E_i}{\Delta\text{VaR}_i}=\dfrac{E_i}{\beta_i}=

constant,

E_i=E_m\beta_i

$E_i$ = excess return

IM 6

VaR and Risk Budgeting

5 formulas

Absolute vs relative VaR

absolute:

\text{VaR}=\alpha W\sigma(R_{\text{asset}})

relative:

E=R_{\text{asset}}-R^b,\ \text{VaR}=\alpha W_0\sigma_E

Policy mix vs active management return

R_{\text{asset}}=\sum w_i^bR_i^b+\sum\big(w_iR_i-w_i^bR_i^b\big)

first term policy mix, second active management

Surplus return (funding risk, SAR)

R_S=\dfrac{\Delta S}{A}=R_{\text{asset}}-R_{\text{liabilities}}\times\dfrac{L}{A}

SAR from

\alpha W\sigma_{\text{surplus}}

Risk budget per asset class

\text{risk budget}_i=\alpha\times\sigma_i\times(w_iW)

\sum\text{risk budgets}=

undiversified VaR

>

fund VaR

Risk budgeting across active managers

\text{IR}=\dfrac{\mu}{\omega}

\mu_p=\sum x_i\mu_i=\sum x_i(\text{IR}_i\times\omega_i)

\omega_p=\sqrt{\sum x_i^2\omega_i^2}

optimal

x_i\omega_i=\text{IR}_i\times\dfrac{\omega_p}{\text{IR}_p}

\text{IR}_p=\sqrt{\sum\text{IR}_i^2}

$\omega$ = active risk; independent active returns

IM 7

Portfolio Performance Evaluation

9 formulas

Time-weighted vs dollar-weighted return

\text{TWR}=[(1+r_1)(1+r_2)\cdots(1+r_n)]^{1/n}-1

DWR = IRR:

\sum\dfrac{\text{CF}_t}{(1+r)^t}=0

TWR is the geometric average of period returns

Risk-adjusted measures

\text{Sharpe}=\dfrac{\bar r_P-\bar r_f}{\sigma_P}

\text{Treynor}=\dfrac{\bar r_P-\bar r_f}{\beta_P}

Jensen

\alpha_P=\bar r_P-[\bar r_f+\beta_P(\bar r_M-\bar r_f)]

\text{IR}=\dfrac{\alpha_P}{\sigma(e_P)}

M²

M_P^2=r_{P^*}-r_M

$P^*$ = mix of P and T-bills with $\sigma_{P^*}=\sigma_M$ (weight in P $=\sigma_M/\sigma_P$ )

T² (Treynor square)

T^2=r_{P^*}-r_M

$P^*$ = mix of P and T-bills with $\beta_{P^*}=1$ (weight in P $=1/\beta_P$ )

Sharpe gain from active portfolio

S_P^2=S_M^2+\left[\dfrac{\alpha_H}{\sigma(e_H)}\right]^2

Standard error and t-statistic of alpha

\hat\sigma(\alpha)=\dfrac{\hat\sigma(e)}{\sqrt{N}}

t(\hat\alpha)=\dfrac{\hat\alpha}{\hat\sigma(\alpha)}=\dfrac{\hat\alpha\sqrt{N}}{\hat\sigma(e)}

Market timing regressions

Treynor-Mazuy:

r_P-r_f=a+b(r_M-r_f)+c(r_M-r_f)^2+e_P

Henriksson-Merton:

r_P-r_f=a+b(r_M-r_f)+c(r_M-r_f)D+e_P

D=1

r_M>r_f

$c>0$ means timing ability

Value of market timing (call option)

perfect timer:

C=2N\left(\tfrac{1}{2}\sigma_M\sqrt{T}\right)-1

per dollar of assetsimperfect:

\text{MV}=(P_1+P_2-1)\times C

timing ability $=P_1+P_2-1$

Performance attribution

bogey

=\sum w_{Bi}r_{Bi}

excess

=r_P-r_B

asset allocation

=\sum(w_{Pi}-w_{Bi})\,r_{Bi}

selection

=\sum w_{Pi}(r_{Pi}-r_{Bi})

sector allocation within equity

=\sum

(active sector weight

\times

sector return)

IM 8

Hedge Fund Investment Strategies

2 formulas

Merger arbitrage annualized return

\text{Exp. annualized return}=\dfrac{C\times G-L\times(1-C)}{Y\times P}

$C$ = probability of completion; $Y$ = years to close; $P$ = current price; $G$ = offer $-$ current price; $L$ = current price $-$ fallback price

Convertible arbitrage income

income = coupon + interest on short proceeds

-

stock dividend

-

stock borrow costshort proceeds = share price

\times

conversion ratio

\times

% shorted

shares shorted depend on delta $\times$ conversion ratio

IM 9

Risk, Regulation and Organizational Structure

Concepts only

No formulas to memorise. None of the 2026 learning objectives for this chapter needs a formula, so there is nothing from it to learn for this sheet.

IM 10

The Last Mile: Financial Vulnerabilities and Risks

Concepts only

No formulas to memorise. None of the 2026 learning objectives for this chapter needs a formula, so there is nothing from it to learn for this sheet.

IM 11

Private Markets Investing

2 formulas

Multiples

\text{DPI}=\dfrac{\sum\text{distributions}}{\text{paid-in}}

\text{RVPI}=\dfrac{\text{residual value}}{\text{paid-in}}

\text{TVPI}=\text{DPI}+\text{RVPI}=\dfrac{\sum\text{distributions}+\text{residual value}}{\text{paid-in}}

maturity $=$ DPI / TVPI

Public market equivalent

\text{PME}=\dfrac{\sum\text{discounted distributions}}{\sum\text{discounted capital calls}}

discounted with the index total return; PME > 1 means outperformance

IM 12

Performing Due Diligence on Specific Managers and Funds

Concepts only

No formulas to memorise. None of the 2026 learning objectives for this chapter needs a formula, so there is nothing from it to learn for this sheet.

IM 13

Distress Symptoms and Remedies

Concepts only

No formulas to memorise. None of the 2026 learning objectives for this chapter needs a formula, so there is nothing from it to learn for this sheet.

IM 14

A Riot of Red Flags

Concepts only

No formulas to memorise. None of the 2026 learning objectives for this chapter needs a formula, so there is nothing from it to learn for this sheet.

IM 15

Market-Driven Scenarios

2 formulas

Mahalanobis distance and scenario z-score

\text{MD}(r,\Sigma)=\sqrt{(r-\bar r)'\,\Sigma^{-1}(r-\bar r)}

Z(r,\Sigma)=\dfrac{\text{MD}(r,\Sigma)}{\sqrt{n}}

$n$ = number of policy variables; lower $Z$ = more plausible

Volatility and correlation z-scores

V(z)=\sqrt{\dfrac{z'z}{n}}

C(r,\Sigma)=\dfrac{Z(r,\Sigma)}{V(z)}=\sqrt{\dfrac{z'}{\sqrt{z'z}}\,\Lambda^{-1}\,\dfrac{z}{\sqrt{z'z}}}

$z$ = vector of factor z-scores (shock $/\sigma$ ); $\Lambda$ = correlation matrix; $C>1$ : shocks inconsistent with the correlations

IM 16

Liquidity Risk Management

4 formulas

Days to liquidate

\text{Days to liquidate}=\dfrac{\text{size of trade}}{\text{participation rate}\times\text{ADV}}

Expected volume, infrequently traded bond

E(V_{t+1}^b)=p_{t+1}^b\times E(V_{t+1}^b\mid\text{TRADE})

$p^b$ = probability the bond trades

T-cost model

\text{t-cost}=[k_1\times\text{BAS}]+\left[k_2\times D\times S\times\left(\dfrac{\text{trade notional volume}}{\text{ADV}}\right)^{\gamma}\right]

fixed cost plus market impact; BAS = % bid-ask spread; $D$ = spread duration; $S$ = OAS

Implementation shortfall

\text{IS (bp)}=\text{sign}\times\dfrac{\text{traded price}-\text{benchmark price}}{\text{benchmark price}}\times 10{,}000

sign $=+1$ buy, $-1$ sell

IM 17

Illiquid Assets

2 formulas

Unsmoothing (Geltner-Ross-Zisler)

r_t=\dfrac{1}{1-\varphi}\,r_t^*-\dfrac{\varphi}{1-\varphi}\,r_{t-1}^*

equivalently

r_t^*=(1-\varphi)\,r_t+\varphi\,r_{t-1}^*

$r^*$ = reported, $r$ = true, $\varphi$ = autocorrelation of reported returns

Effect on volatility

\text{var}(r_t)\ge\text{var}(r_t^*)

unsmoothed volatility is higher; means unchanged

Book 6 of 6 · 10% of the exam

Current Issues in Financial Markets formulas

CI 1

Advances in Artificial Intelligence: Implications for Capital Markets Activities

Concepts only

No formulas to memorise. None of the 2026 learning objectives for this chapter needs a formula, so there is nothing from it to learn for this sheet.

CI 2

The Financial Stability Implications of Artificial Intelligence

Concepts only

No formulas to memorise. None of the 2026 learning objectives for this chapter needs a formula, so there is nothing from it to learn for this sheet.

CI 3

The Global Drivers of Private Credit

Concepts only

No formulas to memorise. None of the 2026 learning objectives for this chapter needs a formula, so there is nothing from it to learn for this sheet.

CI 4

Global Financial Stability Report, IMF, April 2025

Concepts only

No formulas to memorise. None of the 2026 learning objectives for this chapter needs a formula, so there is nothing from it to learn for this sheet.

CI 5

Monetary and Fiscal Policy: Safeguarding Stability and Trust

Concepts only

No formulas to memorise. None of the 2026 learning objectives for this chapter needs a formula, so there is nothing from it to learn for this sheet.

CI 6

Regulating the Crypto Ecosystem: The Case of Unbacked Crypto Assets

Concepts only

No formulas to memorise. None of the 2026 learning objectives for this chapter needs a formula, so there is nothing from it to learn for this sheet.

CI 7

Tokenization and Financial Market Inefficiencies

Concepts only

No formulas to memorise. None of the 2026 learning objectives for this chapter needs a formula, so there is nothing from it to learn for this sheet.

CI 8

Digital Resilience and Financial Stability: The Quest for Policy Tools in the Financial Sector

Concepts only

No formulas to memorise. None of the 2026 learning objectives for this chapter needs a formula, so there is nothing from it to learn for this sheet.

How to revise with the FRM Part 2 formula sheet

Part 2 questions usually describe a bank, a portfolio or a position before asking for a number, so the real test is whether you can name the row of this sheet a question needs before you reach for the calculator. Work through a chapter here only after you have studied it, then cover the formula column and rewrite each line from its label.

In the last six weeks before the exam, go through the sheet book by book and give most of the time to Market Risk and Credit Risk, which hold more than half of the formulas. The formulas you cannot write from memory on a second pass are the ones to drill with timed practice questions.

Where the formulas sit in each book

Market Risk Measurement and Management 20%: Historical, normal and lognormal VaR and expected shortfall, weighted historical simulation, extreme value theory, VaR backtesting, VaR mapping, correlation, regression hedging, term structure models and the FRTB.
Credit Risk Measurement and Management 20%: Expected and unexpected loss, economic capital, Merton and KMV default probabilities, hazard rates, credit VaR, portfolio credit risk, CDS valuation, margin, exposure, CVA and DVA, structured credit and securitization.
Operational Risk and Resilience 20%: Fault trees, RAROC and risk capital attribution, the Basel I, Basel 2.5 and Basel III capital rules, the leverage, LCR and NSFR ratios, the output floor and the Basel III operational risk capital.
Liquidity and Treasury Risk Measurement and Management 15%: Liquidity-adjusted VaR and the cost of liquidation, leverage, liquidity requirements and cash flow at risk, the cost of funds, repo pricing, funds transfer pricing, covered interest parity and asset-liability management with the duration gap.
Risk Management and Investment Management 15%: Factor models and the SDF, alpha and the information ratio, portfolio construction, marginal, incremental and component VaR, risk budgeting, performance evaluation and attribution, private market multiples and liquidity measures.
Current Issues in Financial Markets 10%: No formulas. All eight readings are listed on the sheet as concepts only.

FRM Part 2 formula sheet questions

Does GARP provide a formula sheet in the FRM Part 2 exam?

No. GARP does not give candidates a formula sheet, and notes cannot be taken into the exam, so every formula on this page has to be recalled from memory. Only an approved business calculator is allowed.

Which calculators are allowed in the FRM exam?

GARP permits the Texas Instruments BA II Plus (including the Professional), the HP 12C (including the Platinum, Anniversary and Prestige editions), the HP 10B II, the HP 10BII+ and the HP 20B. No other calculator is allowed.

How is the FRM Part 2 exam structured?

FRM Part 2 has 80 equally weighted multiple-choice questions to be answered in four hours on computer. Market Risk, Credit Risk, and Operational Risk and Resilience carry about 20% each; Liquidity and Treasury Risk and Risk Management and Investment Management carry about 15% each; Current Issues in Financial Markets carries about 10%. GARP offers the exam in May, August and November.

Which FRM Part 2 chapters have no formulas to learn?

Of the 107 chapters in FRM Part 2, 43 have no formula to learn: MR 6 Validating Bank Holding Companies’ Value-at-Risk Models for Market Risk; CR 1 Fundamentals of Credit Risk; CR 2 Governance; CR 6 Credit Scoring and Rating; CR 14 Derivatives; CR 15 Counterparty Risk and Beyond; OR 1 Introduction to Operational Risk and Resilience; OR 2 Risk Governance; OR 3 Risk Identification; OR 5 Risk Mitigation; OR 6 Risk Reporting; OR 7 Integrated Risk Management; OR 8 Cyber-resilience: Range of practices; OR 9 Cyberthreats and Information Security Risks; OR 10 Sound Management of Risks related to Money Laundering and Financing of Terrorism; OR 11 Financial Crime and Fraud; OR 12 Guidance on Managing Outsourcing Risk; OR 13 Third-Party Risk Management; OR 14 Investor Protection and Compliance Risks in Investment Activities; OR 15 Supervisory Guidance on Model Risk Management; OR 16 Model Risk and Model Validation; OR 17 Stress Testing Banks; OR 19 Range of practices and issues in economic capital frameworks; OR 20 Capital Planning at Large Bank Holding Companies; LR 3 Early Warning Indicators; LR 6 Intraday Liquidity Risk Management; LR 8 The Failure Mechanics of Dealer Banks; LR 10 Liquidity Risk Reporting and Stress Testing; LR 11 Contingency Funding Planning; LR 15 The US Dollar Shortage in Global Banking and the International Policy Response; IM 9 Risk, Regulation and Organizational Structure; IM 10 The Last Mile: Financial Vulnerabilities and Risks; IM 12 Performing Due Diligence on Specific Managers and Funds; IM 13 Distress Symptoms and Remedies; IM 14 A Riot of Red Flags; CI 1 Advances in Artificial Intelligence: Implications for Capital Markets Activities; CI 2 The Financial Stability Implications of Artificial Intelligence; CI 3 The Global Drivers of Private Credit; CI 4 Global Financial Stability Report, IMF, April 2025; CI 5 Monetary and Fiscal Policy: Safeguarding Stability and Trust; CI 6 Regulating the Crypto Ecosystem: The Case of Unbacked Crypto Assets; CI 7 Tokenization and Financial Market Inefficiencies; CI 8 Digital Resilience and Financial Stability: The Quest for Policy Tools in the Financial Sector. Each of them is listed on this sheet in its place and marked as concepts only.

How often is this formula sheet updated?

The sheet is revised whenever GARP changes the FRM Part 2 readings, and whenever a correction is made. Every revision is dated in the update history at the end of this page, and the PDF is replaced at the same time, so the download always matches what you see here.

Is the formula sheet updated for the 2026 curriculum?

Yes. The sheet was built from the 2026 GARP FRM Part 2 readings for the learning objectives that need a formula, and was last updated on 7 October 2026.

Is a formula sheet enough to pass FRM Part 2?

No. Part 2 questions test whether you can choose the right formula for a situation and apply it under time pressure, so the sheet works best for revision alongside the full readings and timed practice questions.

Plan the rest of your preparation

See these formulas taught

The free FRM Part 2 sample course has 12 hours of lectures, with the lecture PDFs and class notes.

Start the free sample course

Update history

The sheet and the PDF are revised together, and every change is dated here.

7 October 2026 Built for the 2026 FRM Part 2 curriculum: 270 formulas from 64 chapters, with all 107 chapters listed.

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