Free download · 2026 FRM Part 1 curriculum

FRM Part 1 Formula Sheet 2026

Q: Does GARP provide a formula sheet in the FRM Part 1 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.

Q: 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.

Q: How is the FRM Part 1 exam structured?

FRM Part 1 has 100 equally weighted multiple-choice questions to be answered in four hours on computer. Foundations of Risk Management and Quantitative Analysis carry about 20% each; Financial Markets and Products and Valuation and Risk Models carry about 30% each. GARP offers the exam in May, August and November.

Q: Which FRM Part 1 chapters have no formulas to learn?

Of the 62 chapters in FRM Part 1, 13 are tested on concepts alone: FRM 2 How Do Firms Manage Financial Risk?; FRM 3 The Governance of Risk Management; FRM 4 Credit Risk Transfer Mechanisms; FRM 7 Principles for Effective Data Aggregation and Risk Reporting; FRM 8 Enterprise Risk Management and Future Trends; FRM 9 Learning from Financial Disasters; FRM 10 Anatomy of the Great Financial Crisis of 2007-2009; FRM 11 GARP Code of Conduct; FMP 1 Banks; FMP 6 Central Clearing; FMP 7 Futures Markets; VRM 5 Country Risk: Determinants, Measures and Implications; VRM 8 Stress Testing. Each of them is listed on this sheet in its place and marked as concepts only.

Q: How often is this formula sheet updated?

The sheet is revised whenever GARP changes the FRM Part 1 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.

Q: Is the formula sheet updated for the 2026 curriculum?

Yes. The sheet follows the 2026 GARP FRM Part 1 readings chapter by chapter, using the notation of the readings, and was last updated on 1 October 2026.

Q: Is a formula sheet enough to pass FRM Part 1?

No. Part 1 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.

More than 370 formulas for FRM Part 1, set out chapter by chapter across Foundations of Risk Management, Quantitative Analysis, Financial Markets and Products, and Valuation and Risk Models. Every formula is taken from the 2026 GARP readings in their own notation, and the full sheet can be read right here or downloaded as one PDF.

4books
62chapters reviewed
370+formulas
Oct 2026last updated

By Micky Midha Updated 1 October 2026

By Micky Midha, MidhaFin. Updated 1 October 2026. Latest version: midhafin.com/frm/part-1/formula-sheet

Download the formula sheet

Download the PDF 25 pages, A4

Free, no sign-up. The PDF carries the complete sheet in the same order and with the same chapter codes as this page, and it prints cleanly on A4.

Inside the sheet

Foundations of Risk Management3 chapters · 15 formulas
Quantitative Analysis15 chapters · 141 formulas
Financial Markets and Products17 chapters · 120 formulas
Valuation and Risk Models14 chapters · 98 formulas

The research behind this formula sheet

We went through all 62 chapters of the 2026 GARP FRM Part 1 readings before writing this sheet. The aim was a sheet that holds every formula the exam can ask you to use, and says clearly where there is nothing to learn, so your revision time goes only where the marks are.

Every chapter accounted for

49 chapters carry formulas, and all of them are on this sheet. The other 13 are tested on concepts alone. Each of those still appears in its place, marked as concepts only, so you can see that nothing has been missed.

Not required, said plainly

Where a reading shows a result that the exam does not ask you to calculate, the sheet says "not required" instead of leaving a gap, for example the variance of the F and Beta distributions.

In the notation of the readings

Each formula is written the way the 2026 readings write it, so a symbol here means exactly what it means in an exam question, and every line has been checked against its reading.

Related forms kept together

Discrete and continuous compounding, annual and continuous cost of carry, the no-dividend and dividend cases of option bounds sit side by side. Exam tips mark the places where two results are most often confused.

Book 1 of 4 · 20% of the exam

Foundations of Risk Management formulas

FRM 1

The Building Blocks of Risk Management

1 formula

RAROC

\text{RAROC}=\dfrac{\text{Reward}}{\text{Risk}}

in full:

\dfrac{\text{After-tax net risk-adjusted expected return}}{\text{Economic capital}}

FRM 2

How Do Firms Manage Financial Risk?

Concepts only

No formulas to memorise. GARP tests this chapter on concepts, so there is nothing from it to learn for this sheet.

FRM 3

The Governance of Risk Management

Concepts only

No formulas to memorise. GARP tests this chapter on concepts, so there is nothing from it to learn for this sheet.

FRM 4

Credit Risk Transfer Mechanisms

Concepts only

No formulas to memorise. GARP tests this chapter on concepts, so there is nothing from it to learn for this sheet.

FRM 5

Modern Portfolio Theory and the CAPM

10 formulas

Beta

\beta_i=\dfrac{\text{Cov}(R_i,R_M)}{\sigma_M^2}=\rho_{i,M}\,\dfrac{\sigma_i}{\sigma_M}

\ \beta_M=1

Portfolio beta

\beta_P=\sum w_i\beta_i

CAPM (SML)

E(R_i)=r_f+\beta_i\,[E(R_M)-r_f]

slope of the SML $=E(R_M)-r_f$

CML

E(R_P)=r_f+\dfrac{E(R_M)-r_f}{\sigma_M}\,\sigma_P

slope of the CML $=[E(R_M)-r_f]/\sigma_M$

Sharpe performance index

\text{SPI}=\dfrac{E(R_P)-r_f}{\sigma_P}

Treynor performance index

\text{TPI}=\dfrac{E(R_P)-r_f}{\beta_P}

Jensen performance index

R_P-r_f=\alpha_P+\beta_P(R_M-r_f)

\alpha_P=E(R_P)-\{r_f+\beta_P[E(R_M)-r_f]\}

$\alpha_P=0$ in equilibrium; significantly positive is superior, negative is inferior

Tracking error

\text{TE}=\sigma(R_P-R_B)

Information ratio

\text{IR}=\dfrac{E(R_P-R_B)}{\sigma(R_P-R_B)}

active return over tracking error

Sortino ratio

\text{SR}=\dfrac{E(R_P)-\text{MAR}}{\sqrt{\text{MSD}_{\min}}}

\text{MSD}_{\min}=\dfrac1N\sum_{t=1}^{N}\min(0,R_{Pt}-\text{MAR})^2

Exam tip Sharpe divides by total risk $\sigma_P$ ; Treynor divides by systematic risk $\beta_P$ , so it suits a well-diversified portfolio.

FRM 6

The APT and Multifactor Models

4 formulas

APT return model

R_i=E(R_i)+{\beta_{i1}[I_1-E(I_1)]}+\dots+{\beta_{iK}[I_K-E(I_K)]}+e_i

APT expected return

E(R_P)=E(R_Z)+{\beta_{P1}[E(I_1)-E(R_Z)]}+\dots+{\beta_{PK}[E(I_K)-E(R_Z)]}

$E(R_Z)$ = zero-beta (risk-free) return; single factor: $K=1$

Fama-French three-factor

E(R_P)=r_f+\alpha_P+{\beta_{P,M}[E(R_M)-r_f]}+{\beta_{P,SMB}\,E(\text{SMB})}+{\beta_{P,HML}\,E(\text{HML})}

Portfolio factor beta

\beta_{P,k}=\sum w_i\beta_{i,k}

a hedge adds positions until $\beta_{P,k}=0$ for each factor hedged

FRM 7

Principles for Effective Data Aggregation and Risk Reporting

Concepts only

No formulas to memorise. GARP tests this chapter on concepts, so there is nothing from it to learn for this sheet.

FRM 8

Enterprise Risk Management and Future Trends

Concepts only

No formulas to memorise. GARP tests this chapter on concepts, so there is nothing from it to learn for this sheet.

FRM 9

Learning from Financial Disasters

Concepts only

No formulas to memorise. GARP tests this chapter on concepts, so there is nothing from it to learn for this sheet.

FRM 10

Anatomy of the Great Financial Crisis of 2007-2009

Concepts only

No formulas to memorise. GARP tests this chapter on concepts, so there is nothing from it to learn for this sheet.

FRM 11

GARP Code of Conduct

Concepts only

No formulas to memorise. GARP tests this chapter on concepts, so there is nothing from it to learn for this sheet.

Book 2 of 4 · 20% of the exam

Quantitative Analysis formulas

QTA 1

Fundamentals of Probability

8 formulas

Complement

P(A^c)=1-P(A)

Addition rule

P(A\cup B)=P(A)+P(B)-P(A\cap B)

Conditional probability

P(A\mid B)=\dfrac{P(A\cap B)}{P(B)}

Total probability

P(B)=P(B\mid A_1)P(A_1)+\dots+P(B\mid A_n)P(A_n)

$A_i$ mutually exclusive and exhaustive

Total probability, two events

P(A)=P(A\mid B)P(B)+P(A\mid B^c)P(B^c)

Independence

P(A\cap B)=P(A)P(B)

, i.e.

P(A\mid B)=P(A)

n

events:

P(A_1\cap\dots\cap A_n)=P(A_1)\cdots P(A_n)

Conditional independence

P(A\cap B\mid C)=P(A\mid C)\,P(B\mid C)

Bayes' rule

P(B\mid A)=\dfrac{P(A\mid B)\,P(B)}{P(A)}

Exam tip Mutually exclusive is not independent: two mutually exclusive events with positive probabilities are dependent.

QTA 2

Random Variables

10 formulas

Bernoulli PMF

f(x)=p^x(1-p)^{1-x},\ x\in\{0,1\}

Expectation

E[X]=\sum x_ip_i

E[f(X)]=\sum f(x_i)p_i

Properties of E

E[a+bX+cY]=a+bE[X]+cE[Y]

E[g(X)]\ne g(E[X])

in general

Variance

\sigma^2=E[(X-\mu)^2]=E[X^2]-(E[X])^2

Skewness

\text{Skew}(X)=\dfrac{E[(X-\mu)^3]}{\sigma^3}

numerator is the third central moment

Kurtosis

\text{Kurt}(X)=\dfrac{E[(X-\mu)^4]}{\sigma^4}

excess kurtosis

=\text{Kurt}-3

Standardized variable

Z=\dfrac{X-\mu}{\sigma}

mean 0, variance 1

Linear transformation

Y=a+bX

\ E[Y]=a+bE[X]

\ V[Y]=b^2V[X]

\ \sigma_Y=|b|\,\sigma_X

Higher moments of $Y=a+bX$

skewness unchanged if

b>0

(sign flips if

b<0

)kurtosis unchangedmedian and IQR move like the mean and SD

Median and IQR

median

=

50% quantile

\text{IQR}=q_{0.75}-q_{0.25}

QTA 3

Common Univariate Random Variables

23 formulas

Binomial

P(X=x)={}^nC_x\,p^x(1-p)^{n-x}

Poisson

P(X=x)=\dfrac{\lambda^x e^{-\lambda}}{x!}

over a period $t$ , use $\lambda t$

Uniform, interval probability

P(l<X<u)=\dfrac{\min(u,b)-\max(l,a)}{b-a}

Normal

P(X\le x)=\Phi\big((x-\mu)/\sigma\big)

\phi(z)=\dfrac{1}{\sqrt{2\pi}}e^{-z^2/2}

Normal facts

mean

=

median

=

modeskewness 0kurtosis 3

\Phi(-z)=1-\Phi(z)

Normal probability ranges

68%:

\mu\pm\sigma

90%:

\mu\pm1.645\sigma

95%:

\mu\pm1.96\sigma

99%:

\mu\pm2.58\sigma

Lognormal

P(X\le x)=\Phi\big((\ln x-\mu)/\sigma\big)

positively skewed

Exponential

P(X\le x)=1-e^{-x/\beta}

x\ge0

Student's t kurtosis

\text{Kurt}=\dfrac{3(\nu-2)}{\nu-4}

for

\nu>4

, always above 3

Constructions

\chi^2_\nu=\sum_{i=1}^{\nu}Z_i^2

t_\nu=\dfrac{Z}{\sqrt{W/\nu}}

W\sim\chi^2_\nu

F_{\nu_1,\nu_2}=\dfrac{W_1/\nu_1}{W_2/\nu_2}

Mixture

f(x)=\sum w_if_i(x)

E[X]=\sum w_i\mu_i

V[X]=\sum w_i(\sigma_i^2+\mu_i^2)-\left(\sum w_i\mu_i\right)^2

Distribution	Mean	Variance
Bernoulli	$p$	$p(1-p)$
Binomial	$np$	$np(1-p)$
Poisson	$\lambda$	$\lambda$
Uniform $[a,b]$	$(a+b)/2$	$(b-a)^2/12$
Normal	$\mu$	$\sigma^2$
Standard normal	$0$	$1$
Lognormal	$e^{\mu+\sigma^2/2}$	$(e^{\sigma^2}-1)\,e^{2\mu+\sigma^2}$
Student's $t_\nu$	$0$	$\nu/(\nu-2)$
$F_{\nu_1,\nu_2}$	$\nu_2/(\nu_2-2)$	not required
$\chi^2_\nu$	$\nu$	$2\nu$
Exponential	$\beta$	$\beta^2$
Beta $(\alpha,\beta)$	$\alpha/(\alpha+\beta)$	not required

Exam tip In the lognormal, $\mu$ and $\sigma$ are the mean and SD of $\ln X$ , not of $X$ .

QTA 4

Multivariate Random Variables

10 formulas

Marginal and conditional PMF

f_{X_1}(x_1)=\sum_{x_2}f(x_1,x_2)

f(x_1\mid X_2=x_2)=\dfrac{f(x_1,x_2)}{f_{X_2}(x_2)}

Conditional expectation

E[X_1\mid X_2=x_2]=\sum x_1\,f(x_1\mid X_2=x_2)

Conditional variance

{V[X_1\mid X_2=x_2]}={E[X_1^2\mid X_2=x_2]-(E[X_1\mid X_2=x_2])^2}

Covariance

\text{Cov}(X_1,X_2)=E[(X_1-E[X_1])(X_2-E[X_2])]=E[X_1X_2]-E[X_1]E[X_2]

Correlation

\text{Corr}(X_1,X_2)=\dfrac{\text{Cov}(X_1,X_2)}{\sigma_1\sigma_2}

so $\text{Cov}=\text{Corr}\cdot\sigma_1\sigma_2$

Linear transformations

\text{Cov}(a+bX_1,c+dX_2)=bd\,\text{Cov}(X_1,X_2)

\text{Corr}(aX_1,bX_2)=\dfrac{ab}{|a||b|}\,\text{Corr}(X_1,X_2)

unchanged if $a$ , $b$ have the same sign, flips sign otherwise

Covariance with a sum

\text{Cov}(X_1,X_2+X_3)=\text{Cov}(X_1,X_2)+\text{Cov}(X_1,X_3)

Variance of a sum or difference

V[X_1\pm X_2]=V[X_1]+V[X_2]\pm2\,\text{Cov}(X_1,X_2)

Variance of a weighted sum

V[aX_1+bX_2]=a^2V[X_1]+b^2V[X_2]+2ab\,\text{Cov}(X_1,X_2)

Sum of n iid variables

E\!\left[\sum X_i\right]=n\mu

V\!\left[\sum X_i\right]=n\sigma^2

QTA 5

Sample Moments

11 formulas

Sample mean

\hat\mu=\dfrac1n\sum_{i=1}^{n}X_i

Bias

\text{Bias}(\hat\theta)=E[\hat\theta]-\theta

\text{Bias}(\hat\mu)=0

\text{Bias}(\hat\sigma^2)=-\sigma^2/n

Variance estimators

\hat\sigma^2=\dfrac1n\sum(X_i-\hat\mu)^2

(biased)

s^2=\dfrac{1}{n-1}\sum(X_i-\hat\mu)^2

(unbiased)

Standard error of the mean

\sigma/\sqrt n

Central limit theorem

\hat\mu\approx N(\mu,\sigma^2/n)

for large

n

\sqrt n\,(\hat\mu-\mu)/\sigma\to N(0,1)

Scaling the mean

\hat\mu_{\text{annual}}=252\,\hat\mu_{\text{daily}}=52\,\hat\mu_{\text{weekly}}

$n$ periods: $n\mu$

Scaling volatility

\sigma_{\text{annual}}=\sqrt{252}\,\sigma_{\text{daily}}=\sqrt{52}\,\sigma_{\text{weekly}}=\sqrt{12}\,\sigma_{\text{monthly}}

$n$ periods: $\sqrt n\,\sigma$

Exam tip Square-root-of-time scaling assumes iid returns; it fails with autocorrelation.

Sample skewness and kurtosis

\hat S=\dfrac{\frac1n\sum(X_i-\hat\mu)^3}{\hat\sigma^3}

\hat\kappa=\dfrac{\frac1n\sum(X_i-\hat\mu)^4}{\hat\sigma^4}

Median

n

odd:

x_{((n+1)/2)}

n

even:

\dfrac{x_{(n/2)}+x_{(n/2+1)}}{2}

Sample covariance

\hat\sigma_{XY}=\dfrac{1}{n-1}\sum_{i=1}^{n}(X_i-\hat\mu_X)(Y_i-\hat\mu_Y)

Sample correlation

\hat\rho_{XY}=\dfrac{\hat\sigma_{XY}}{\hat\sigma_X\hat\sigma_Y}

QTA 6

Hypothesis Testing

15 formulas

Hypotheses

two-sided

H_0:\mu=\mu_0

H_1:\mu\ne\mu_0

one-sided

H_0:\mu\le\mu_0

H_1:\mu>\mu_0

(or the reverse)

Test statistic, mean

T=\dfrac{\hat\mu-\mu_0}{\sqrt{\hat\sigma^2/n}}\sim N(0,1)

asymptoticallywith

s^2

from a normal sample,

T\sim t_{n-1}

Population	$\sigma^2$	$n\ge30$	$n<30$
Normal	Known	$z$	$z$
Non-normal	Known	$z$	NA
Normal	Unknown	$t$ (or $z$ )	$t$
Non-normal	Unknown	$t$ (or $z$ )	NA

Critical $z$	10%	5%	1%
One-sided	1.28	1.645	2.33
Two-sided	1.645	1.96	2.58

Decision	$H_0$ true	$H_0$ false
Do not reject $H_0$	Correct, $1-\alpha$	Type II error, $\beta$
Reject $H_0$	Type I error, $\alpha$ (size)	Correct, power $1-\beta$

Confidence interval, two-sided

\hat\mu\pm C_\alpha\,\hat\sigma/\sqrt n

Confidence interval, one-sided

H_1:\mu>\mu_0

[\hat\mu-C_\alpha\hat\sigma/\sqrt n,\ \infty)

H_1:\mu<\mu_0

(-\infty,\ \hat\mu+C_\alpha\hat\sigma/\sqrt n]

p-value

two-sided

2[1-\Phi(|T|)]

lower tail

\Phi(T)

upper tail

1-\Phi(T)

Difference in means, paired

T=\dfrac{\hat\mu_X-\hat\mu_Y}{\sqrt{(\hat\sigma_X^2+\hat\sigma_Y^2-2\hat\sigma_{XY})/n}}

Difference in means, independent

T=\dfrac{\hat\mu_X-\hat\mu_Y}{\sqrt{\hat\sigma_X^2/n_X+\hat\sigma_Y^2/n_Y}}

QTA 7

Linear Regression

7 formulas

OLS estimates

\hat\beta=\dfrac{\sum(x_i-\bar X)(y_i-\bar Y)}{\sum(x_i-\bar X)^2}=\dfrac{\hat\sigma_{XY}}{\hat\sigma_X^2}=\hat\rho_{XY}\dfrac{\hat\sigma_Y}{\hat\sigma_X}

\hat\alpha=\bar Y-\hat\beta\bar X

Variance of the shocks

s^2=\dfrac{1}{n-2}\sum\hat\varepsilon_i^2=\dfrac{n}{n-2}\,\hat\sigma_Y^2(1-R^2)

Standard error of the slope

\text{s.e.}(\hat\beta)=\dfrac{s}{\sqrt n\,\hat\sigma_X}

t test, one coefficient

T=\dfrac{\hat\beta-\beta_0}{\text{s.e.}(\hat\beta)}

same form for $\hat\alpha$

Confidence interval, coefficient

\hat\beta\pm C_\alpha\,\text{s.e.}(\hat\beta)

$C_\alpha$ = two-sided critical value (1.96 at 95%)

Sums of squares

\text{ESS}=\sum(\hat Y_i-\bar Y)^2

\text{RSS}=\sum(Y_i-\hat Y_i)^2

\text{TSS}=\sum(Y_i-\bar Y)^2=\text{ESS}+\text{RSS}

R²

R^2=\dfrac{\text{ESS}}{\text{TSS}}=1-\dfrac{\text{RSS}}{\text{TSS}}

one regressor:

R^2=\hat\rho^2_{XY}

QTA 8

Regression with Multiple Explanatory Variables

5 formulas

R², multiple regressors

R^2=\hat\rho^2(Y,\hat Y)

Adjusted R²

\bar R^2=1-\dfrac{\text{RSS}/(n-k-1)}{\text{TSS}/(n-1)}=1-\dfrac{n-1}{n-k-1}(1-R^2)

$k$ excludes the constant

F test, q restrictions

F=\dfrac{(\text{RSS}_R-\text{RSS}_U)/q}{\text{RSS}_U/(n-k_U-1)}=\dfrac{(R^2_U-R^2_R)/q}{(1-R^2_U)/(n-k_U-1)}

$F\sim F_{q,\,n-k_U-1}$

F statistic, all slopes zero

F=\dfrac{(\text{TSS}-\text{RSS})/k}{\text{RSS}/(n-k-1)}\sim F_{k,\,n-k-1}

F from t, uncorrelated regressors

F\approx\dfrac{T_1^2+T_2^2}{2}

QTA 9

Regression Diagnostics

4 formulas

Omitted variable bias

\hat\beta_1\to\beta_1+\beta_2\delta

\ \delta=\dfrac{\text{Cov}(X_1,X_2)}{V(X_1)}

White test

nR^2

from regressing

\hat\varepsilon^2

on the regressors, their squares and cross products

\sim\chi^2_{k(k+3)/2}

Variance inflation factor

\text{VIF}_j=\dfrac{1}{1-R_j^2}

above 10 is excessive

Cook's distance

D_j=\dfrac{\sum_{i=1}^{n}(\hat Y_i^{(-j)}-\hat Y_i)^2}{k\,s^2}

$D_j>1$ flags an outlier

QTA 10

Stationary Time Series

19 formulas

Autocorrelation

\rho_h=\gamma_h/\gamma_0

AR(1)

Y_t=\delta+\phi Y_{t-1}+\varepsilon_t

\ \varepsilon_t\sim WN(0,\sigma^2)

|\phi|<1

stationary,

|\phi|=1

non-stationary

AR(1) moments

\mu=\dfrac{\delta}{1-\phi}

\gamma_0=\dfrac{\sigma^2}{1-\phi^2}

\gamma(h)=\phi^{|h|}\gamma_0

\rho(h)=\phi^{|h|}

$\mu$ is the mean-reverting level

AR(1) PACF

\alpha(h)=\phi^{|h|}

for

|h|\le1

0

for

|h|\ge2

AR(p)

Y_t=\delta+\phi_1Y_{t-1}+\dots+\phi_pY_{t-p}+\varepsilon_t

AR(p) moments

\mu=\dfrac{\delta}{1-\phi_1-\dots-\phi_p}

\gamma_0=\dfrac{\sigma^2}{1-\phi_1\rho_1-\dots-\phi_p\rho_p}

AR(p) ACF and PACF

ACF decays, non-zero at all lagsPACF

\alpha(h)=0

for

h>p

Characteristic equation

roots of

z^p-\phi_1z^{p-1}-\dots-\phi_p=0

stationary if every root has

|z|<1

AR(2): $z^2-\phi_1z-\phi_2=0$

MA(1)

Y_t=\mu+\theta\varepsilon_{t-1}+\varepsilon_t

, always covariance stationary

\theta>0

persistent,

\theta<0

mean-reverting

MA(1) moments

E[Y_t]=\mu

\gamma_0=(1+\theta^2)\sigma^2

\rho(1)=\dfrac{\theta}{1+\theta^2}

\ \rho(h)=0

for

h\ge2

MA(q)

Y_t=\mu+\varepsilon_t+\theta_1\varepsilon_{t-1}+\dots+\theta_q\varepsilon_{t-q}

E[Y_t]=\mu

\gamma_0=\sigma^2(1+\theta_1^2+\dots+\theta_q^2)

MA(q) ACF and PACF

\rho(h)=0

for

h>q

PACF decays, non-zero at all lags

ARMA(1,1)

Y_t=\delta+\phi Y_{t-1}+\theta\varepsilon_{t-1}+\varepsilon_t

\mu=\dfrac{\delta}{1-\phi}

\gamma_0=\dfrac{\sigma^2(1+2\phi\theta+\theta^2)}{1-\phi^2}

\gamma(h)=\phi\,\gamma(h-1)

for

h\ge2

Exam tip Identification: the PACF of an AR(p) cuts off after lag $p$ ; the ACF of an MA(q) cuts off after lag $q$ .

Lag operator

LY_t=Y_{t-1}

1-a_1L

is invertible if

|a_1|<1

Sample autocorrelation

\hat\rho_h=\hat\gamma_h/\hat\gamma_0

Box-Pierce

Q_{BP}=T\sum_{i=1}^{h}\hat\rho_i^2\sim\chi^2_h

Ljung-Box

Q_{LB}=T(T+2)\sum_{i=1}^{h}\dfrac{\hat\rho_i^2}{T-i}\sim\chi^2_h

$Q_{BP}\approx Q_{LB}$ in large samples

Model selection

\hat\sigma^2=\dfrac1T\sum_{t=1}^{T}\hat\varepsilon_t^2

\text{AIC}=T\ln\hat\sigma^2+2k

\text{BIC}=T\ln\hat\sigma^2+k\ln T

Forecasts

AR(1):

E_T[Y_{T+h}]=\mu+\phi^h(Y_T-\mu)\to\mu

ARMA(1,1), one step:

E_T[Y_{T+1}]=\delta+\phi Y_T+\theta\varepsilon_T

QTA 11

Non-Stationary Time Series

7 formulas

Linear trend

Y_t=\delta_0+\delta_1t+\varepsilon_t

E[Y_t]=\delta_0+\delta_1t

Exponential and log-linear trend

Y_t=\beta_0e^{\beta_1t}

\;\Leftrightarrow\;

\ln Y_t=\ln\beta_0+\beta_1t

Seasonal dummies

Y_t=\delta+\gamma_1I_{1t}+\dots+\gamma_{s-1}I_{s-1,t}+\varepsilon_t

Random walk

Y_t=Y_0+\sum_{i=1}^{t}\varepsilon_i

V[Y_t]=t\sigma^2

$\Delta Y_t=Y_t-Y_{t-1}$ removes a unit root

ADF test

\Delta Y_t=\gamma Y_{t-1}+(\delta_0+\delta_1t)+{\lambda_1\Delta Y_{t-1}+\dots+\lambda_p\Delta Y_{t-p}+\varepsilon_t}

statistic: $t$ of $\hat\gamma$ ; $H_0:\gamma=0$ (unit root), $H_1:\gamma<0$ (stationary)

Forecasts

trend:

E_T[Y_{T+h}]=\delta_0+\delta_1(T+h)

seasonal:

E_T[Y_{T+h}]=\delta+\gamma_j

j=(T+h)\bmod s

95% forecast interval

E_T[Y_{T+h}]\pm1.96\,\sigma

QTA 12

Measuring Returns, Volatility and Correlation

6 formulas

Simple return

R_t=\dfrac{P_t-P_{t-1}}{P_{t-1}}

1+R_T=\prod_{t=1}^{T}(1+R_t)

Log return

r_t=\ln P_t-\ln P_{t-1}

r_T=\sum_{t=1}^{T}r_t

1+R_t=e^{r_t}

Jarque-Bera

JB=(T-1)\left[\dfrac{\hat S^2}{6}+\dfrac{(\hat\kappa-3)^2}{24}\right]\sim\chi^2_2

5% critical value 5.99

Power law

P(X>x)=kx^{-\alpha}

\;\Leftrightarrow\;

\ln P(X>x)=\ln k-\alpha\ln x

Spearman rank correlation

\hat\rho_s=1-\dfrac{6\sum d_i^2}{n(n^2-1)}

$d_i=\text{Rank}_{X_i}-\text{Rank}_{Y_i}$ ; distinct ranks

Kendall's τ

\hat\tau=\dfrac{n_c-n_d}{n(n-1)/2}=\dfrac{n_c}{n_c+n_d+n_t}-\dfrac{n_d}{n_c+n_d+n_t}

QTA 13

Simulation and Bootstrapping

3 formulas

Monte Carlo estimate

\hat E[g(X)]=\dfrac1b\sum_{i=1}^{b}g(X_i)

Sampling error

V\big[\hat E[g(X)]\big]=\dfrac{\sigma_g^2}{b}

standard error

\dfrac{\sigma_g}{\sqrt b}

Antithetic variables

standard error

=\dfrac{\sigma_g\sqrt{1+\rho}}{\sqrt b}

smaller when $\rho<0$

QTA 14

Machine-Learning Methods

5 formulas

Rescaling

standardization

\tilde x_{ij}=\dfrac{x_{ij}-\mu_i}{\sigma_i}

normalization

\tilde x_{ij}=\dfrac{x_{ij}-x_{i,\min}}{x_{i,\max}-x_{i,\min}}

Distances

Euclidean

d_E=\sqrt{\sum_{i=1}^{m}(x_{iQ}-x_{iP})^2}

Manhattan

d_M=\sum_{i=1}^{m}|x_{iQ}-x_{iP}|

K-means inertia

I=\sum_{j=1}^{n}d_j^2

Q-value update, Monte Carlo

Q_{\text{new}}(S,A)=Q_{\text{old}}(S,A)+\alpha\,[R-Q_{\text{old}}(S,A)]

$R$ = total subsequent reward

Q-value update, temporal difference

Q_{\text{new}}(S,A)=Q_{\text{old}}(S,A)+\alpha\,[R+V(S')-Q_{\text{old}}(S,A)]

$R$ = reward to the next decision; $V(S')$ = highest current Q-value in the next state

QTA 15

Machine Learning and Prediction

8 formulas

Logistic regression

p=\dfrac{1}{1+\exp[-(\alpha+\beta_1x_1+\dots+\beta_kx_k)]}

\ln\dfrac{p}{1-p}=\alpha+\sum\beta_jx_j

Ridge and LASSO

ridge:

\min\ \text{RSS}+\lambda\sum\beta_j^2

LASSO:

\min\ \text{RSS}+\lambda\sum|\beta_j|

LASSO can set coefficients to exactly 0

Mean squared forecast error

\text{MSFE}=\dfrac{1}{n_{\text{test}}}\sum_{i=1}^{n_{\text{test}}}(y_i-\hat y_i)^2

Confusion matrix	Formula
Accuracy	$\dfrac{TP+TN}{TP+TN+FP+FN}$
Precision	$\dfrac{TP}{TP+FP}$
Recall	$\dfrac{TP}{TP+FN}$
Error rate	$\dfrac{FP+FN}{TP+TN+FP+FN}=1-\text{accuracy}$

Decision tree impurity

entropy

=-\sum_{i=1}^{M}p_i\log_2p_i

Gini

=1-\sum_{i=1}^{M}p_i^2

Book 3 of 4 · 30% of the exam

Financial Markets and Products formulas

FMP 1

Banks

Concepts only

No formulas to memorise. GARP tests this chapter on concepts, so there is nothing from it to learn for this sheet.

FMP 2

Insurance Companies and Pension Plans

8 formulas

Survival probability

P(\text{survive to }N+1)=P(\text{survive to }N)\times(1-q_N)

$q_N$ = probability of death within the year from age $N$ , given alive at $N$

Survival from age X to age Y

P(X\to Y)=\dfrac{P(\text{survive from birth to }Y)}{P(\text{survive from birth to }X)}

Break-even premium

PV of expected premiums

=

PV of expected payouts

each premium weighted by the probability of being alive to pay it; each payout by the probability of death in that year

Ratio	Formula
Loss ratio	Payouts / premiums
Expense ratio	Expenses / premiums
Combined ratio	Loss ratio + expense ratio
Combined ratio after dividends	Combined ratio + dividends as % of premiums
Operating ratio	Combined ratio after dividends − investment income as % of premiums

FMP 3

Fund Management

2 formulas

Net asset value

\text{NAV}=\dfrac{\text{Fund assets}-\text{Fund liabilities}}{\text{Shares outstanding}}

Hedge fund return after fees

investor return

=

gross

-

management fee

-

incentive fee

incentive fee = incentive rate × max(0, return after management fee − hurdle rate), charged only on value above the high-water mark

FMP 4

Introduction to Derivatives

1 formula

Forward payoff

long

S_T-K

short

K-S_T

option payoffs are in FMP 12

FMP 5

Exchanges and OTC Markets

2 formulas

Futures margin

variation margin

=\Delta F\times

contract size

\times

number of contracts

margin call when the balance falls below the maintenance margin; top up to the initial margin

Buying stock on margin

equity

=

market value

-

loanmargin call when

\dfrac{\text{equity}}{\text{market value}}<

maintenance ratiotrigger price

P^*=P_0\,\dfrac{1-\text{IM}}{1-\text{MM}}

IM, MM = initial and maintenance margin ratios

FMP 6

Central Clearing

Concepts only

No formulas to memorise. GARP tests this chapter on concepts, so there is nothing from it to learn for this sheet.

FMP 7

Futures Markets

Concepts only

No formulas to memorise. GARP tests this chapter on concepts, so there is nothing from it to learn for this sheet.

FMP 8

Using Futures for Hedging

6 formulas

Basis

b_t=S_t-F_t

Hedged price

short hedge, price received

=F_0+b_t

long hedge, price paid

=F_0+b_t

Minimum variance hedge ratio

h^*=\rho\,\dfrac{\sigma_S}{\sigma_F}

hedge effectiveness

=\rho^2

Optimal number of contracts

N^*=h^*\dfrac{Q_A}{Q_F}

tailing the hedge:

N^*=h^*\dfrac{V_A}{V_F}=h^*\dfrac{Q_A}{Q_F}\cdot\dfrac{S}{F}

Stock index hedge

N^*=\beta\,\dfrac{V_A}{V_F}

$V_F$ = futures price × multiplier

Changing portfolio beta

N^*=(\beta^*-\beta)\dfrac{V_A}{V_F}

$N^*<0$ : short futures to lower beta; $N^*>0$ : long futures to raise it

FMP 9

Foreign Exchange Markets

7 formulas

Real interest rate

R_{\text{real}}=\dfrac{1+R_{\text{nom}}}{1+R_{\text{infl}}}-1\approx R_{\text{nom}}-R_{\text{infl}}

Bid-ask spread

spread

=

ask

-

bid% spread

=\dfrac{\text{ask}-\text{bid}}{\text{mid}}

Covered interest parity

F=S\left(\dfrac{1+R_{YYY}}{1+R_{XXX}}\right)^T

rate quoted XXXYYY (units of YYY per XXX)

Forward premium and points

\dfrac{F-S}{S}\approx(R_{YYY}-R_{XXX})\,T

forward points

=(F-S)\times10{,}000

Uncovered interest parity

E(S_T)=S_0\left(\dfrac{1+R_{YYY}}{1+R_{XXX}}\right)^T

Purchasing power parity

\dfrac{S_T}{S_0}=\left(\dfrac{1+\pi_{YYY}}{1+\pi_{XXX}}\right)^T

% change in

S\approx\pi_{YYY}-\pi_{XXX}

Appreciation and depreciation

XXXYYY moves from

S_0

S_1

: XXX changes by

\dfrac{S_1-S_0}{S_0}

YYY changes by

\dfrac{S_0-S_1}{S_1}

Exam tip The two percentage changes differ, so work out each currency on its own base.

FMP 10

Pricing Financial Forwards and Futures

9 formulas

Forward price	Annual compounding	Continuous
No income	$S_0(1+R)^T$	$S_0e^{rT}$
Known income, PV $I$	$(S_0-I)(1+R)^T$	$(S_0-I)e^{rT}$
Known yield $Q$ (index: dividend yield)	$S_0\left(\dfrac{1+R}{1+Q}\right)^T$	$S_0e^{(r-q)T}$

Arbitrage

actual

F_0>

theoretical: buy the asset, short the forwardactual

F_0<

theoretical: short the asset, go long the forward

Asset	Value of a long forward
No income	$S_0-\dfrac{K}{(1+R)^T}$
Known income	$(S_0-I)-\dfrac{K}{(1+R)^T}$
Known yield	$\dfrac{S_0}{(1+Q)^T}-\dfrac{K}{(1+R)^T}$
Any asset	$\dfrac{F_0-K}{(1+R)^T}$ ; short $=-$ long

Short sale profit, dividend-paying stock

S_0-S_T-\text{dividends paid while short}

plus interest on the proceeds if earned

FMP 11

Commodity Forwards and Futures

5 formulas

Forward with lease rate

F_0=S_0\left(\dfrac{1+R}{1+l}\right)^T

implied

l=\left(\dfrac{S_0}{F_0}\right)^{1/T}(1+R)-1

Storage cost and convenience yield

F_0=(S_0+U)\left(\dfrac{1+R}{1+Y}\right)^T

implied

Y=\left(\dfrac{S_0+U}{F_0}\right)^{1/T}(1+R)-1

$U$ = PV of storage costs; convenience yield belongs to consumption commodities

No-arbitrage upper bounds

F_0\le S_0(1+R)^T

with storage:

F_0\le(S_0+U)(1+R)^T

Continuous cost of carry

F_0=S_0e^{(c-y)T}

$c$ = interest rate + storage cost rate, $y$ = convenience yield

Expected future spot price

E(S_T)=F_0\,\dfrac{(1+X)^T}{(1+R)^T}

$X$ = expected return on the commodity

FMP 12

Options Markets

4 formulas

Position	Payoff	Profit
Long call	$\max(S_T-K,0)$	$\max(S_T-K,0)-c_0$
Short call	$-\max(S_T-K,0)$	$c_0-\max(S_T-K,0)$
Long put	$\max(K-S_T,0)$	$\max(K-S_T,0)-p_0$
Short put	$-\max(K-S_T,0)$	$p_0-\max(K-S_T,0)$

FMP 13

Properties of Options

19 formulas

European call	Lower bound	Upper
No dividends	$\max(S_0-\text{PV}(K),0)$	$S_0$
Dividends	$\max(S_0-\text{PV}(D)-\text{PV}(K),0)$	$S_0$
Other assets	$\max(\text{PV}(F)-\text{PV}(K),0)$	$\text{PV}(F)$

European put	Lower bound	Upper
No dividends	$\max(\text{PV}(K)-S_0,0)$	$\text{PV}(K)$
Dividends	$\max(\text{PV}(D)+\text{PV}(K)-S_0,0)$	$\text{PV}(K)$
Other assets	$\max(\text{PV}(K)-\text{PV}(F),0)$	$\text{PV}(K)$

American call	Lower bound	Upper
No dividends	$\max(S_0-\text{PV}(K),0)$	$S_0$
Dividends	$\max(S_0-K,\,S_0-\text{PV}(D)-\text{PV}(K),\,0)$	$S_0$

American put	Lower bound	Upper
No dividends	$\max(K-S_0,0)$	$K$
Dividends	$\max(K-S_0,\,\text{PV}(D)+\text{PV}(K)-S_0,\,0)$	$K$

Put-call parity, European

non-dividend

c+\text{PV}(K)=p+S_0

dividend

c+\text{PV}(D)+\text{PV}(K)=p+S_0

forward form

c+\text{PV}(K)=p+\text{PV}(F)

Put-call relation, American

non-dividend

S_0-K\le C-P\le S_0-\text{PV}(K)

dividend

S_0-\text{PV}(D)-K\le C-P\le S_0-\text{PV}(K)

Early exercise of an American call

exercise just before an ex-dividend date can be optimal only if

D>K-K^*

never exercise early without dividends

$K^*$ = $K$ discounted from the next ex-dividend date (or maturity) back to this one

Increase in	European call	European put	American call	American put
$S_0$	+	−	+	−
$K$	−	+	−	+
$T$	?	?	+	+
$\sigma$	+	+	+	+
$r$	+	−	+	−
Dividends	−	+	−	+

FMP 14

Trading Strategies

10 formulas

Option and stock equivalences

protective put

S+P=C+\text{PV}(K)

covered call

S-C=\text{PV}(K)-P

Payoff at $T$	$S_T\le K_1$	$K_1<S_T\le K_2$	$S_T>K_2$
Bull: long call $K_1$ , short call $K_2$	$0$	$S_T-K_1$	$K_2-K_1$
Bear: long put $K_2$ , short put $K_1$	$K_2-K_1$	$K_2-S_T$	$0$
Box: bull call + bear put	$K_2-K_1$	$K_2-K_1$	$K_2-K_1$
Strangle: long put $K_1$ , long call $K_2$	$K_1-S_T$	$0$	$S_T-K_2$

Box spread value

\text{PV}(K_2-K_1)

Butterfly (calls)

0

below

K_1

S_T-K_1

from

K_1

K_2

K_3-S_T

from

K_2

K_3

0

above

K_3

long call $K_1$ , short 2 calls $K_2$ , long call $K_3$ ; $K_2=(K_1+K_3)/2$

Straddle

long call

+

long put, same

K

: payoff

|S_T-K|

Strip and strap

strip

=

1 call

+

2 puts (bearish tilt)strap

=

2 calls

+

1 put (bullish tilt)

same strike and maturity

Principal protected note

zero-coupon bond paying

L

+

call optionfeasible if

L-Le^{-rT}\ge

option cost

FMP 15

Exotic Options

11 formulas

Zero-cost call

payoff

\max(S_T-K-A,\,-A)

A=c(1+R)^T

Gap options

strike

K_1

, trigger

K_2

: call pays

S_T-K_1

S_T>K_2

put pays

K_1-S_T

S_T<K_2

else 0

the payoff can be negative

Payoff	Call	Put
Average price	$\max(S_{\text{avg}}-K,0)$	$\max(K-S_{\text{avg}},0)$
Average strike	$\max(S_T-S_{\text{avg}},0)$	$\max(S_{\text{avg}}-S_T,0)$
Floating lookback	$S_T-S_{\min}$	$S_{\max}-S_T$
Fixed lookback	$\max(S_{\max}-K,0)$	$\max(K-S_{\min},0)$
Cash-or-nothing	$Q$ if $S_T>K$	$Q$ if $S_T<K$
Asset-or-nothing	$S_T$ if $S_T>K$	$S_T$ if $S_T<K$

Barrier parity

knock-in

+

knock-out

=

vanilla option

same barrier, strike and maturity

Exchange option

payoff

\max(U_T-V_T,0)

receive asset $U$ , give up asset $V$

Volatility and variance swaps

receive realized:

L_{\text{vol}}(\sigma-\sigma_K)

L_{\text{var}}(\sigma^2-V_K)

$V_K=\sigma_K^2$ ; $L_{\text{var}}=L_{\text{vol}}/(2\sigma_K)$ ; paying realized reverses the signs

FMP 16

Properties of Interest Rates

10 formulas

Compounding

FV=PV(1+R/m)^{mT}

PV=FV(1+R/m)^{-mT}

continuous:

FV=PVe^{RT}

PV=FVe^{-RT}

Converting rates

R_2=m_2\left[\left(1+\dfrac{R_1}{m_1}\right)^{m_1/m_2}-1\right]

R_m=m\left(e^{R_c/m}-1\right)

R_c=m\ln(1+R_m/m)

Bond price from yield

B=\dfrac c2\sum_{i=1}^{2T}\left(1+\dfrac y2\right)^{-i}+100\left(1+\dfrac y2\right)^{-2T}

Bond price from spot rates

B=\sum c_ie^{-R_it_i}

, or

\sum\dfrac{c_i}{(1+R_i/m)^{mt_i}}

Par yield

c=\dfrac{(100-100\,d_T)\,m}{A}

$d_T$ = final discount factor; $A$ = sum of coupon-date discount factors

Duration

Macaulay

D=\sum t_i\dfrac{PV_i}{B}

modified

D_{\text{mod}}=\dfrac{D_{\text{Mac}}}{1+y/m}

dollar

D_\$=D_{\text{mod}}\,B

Convexity

C=\sum t_i^2\dfrac{PV_i}{B}

Price change

\Delta B=-DB\,\Delta y+\tfrac12CB(\Delta y)^2

with dollar duration:

\Delta B=-D_\$\,\Delta y

Forward rates

discrete

F=\left[\dfrac{(1+R_2)^{T_2}}{(1+R_1)^{T_1}}\right]^{1/(T_2-T_1)}-1

continuous

F=\dfrac{R_2T_2-R_1T_1}{T_2-T_1}

FRA, pay fixed

payoff at the start of the period

\dfrac{(R-R_K)\,\tau L}{1+R\tau}

value today

=(R_F-R_K)\,\tau L

discounted from the period end

receive fixed: signs reversed

FMP 17

Corporate Bonds

2 formulas

Expected loss and return

expected loss rate

=\text{PD}\times(1-\text{RR})

expected return

=r_f+

credit spread

-

expected loss rate

Default rates

issue default rate

=\dfrac{\text{defaulted issues}}{\text{total issues}}

dollar default rate

=\dfrac{\text{par value defaulted}}{\text{total par value}}

FMP 18

Mortgages and Mortgage-Backed Securities

6 formulas

Monthly mortgage payment

X=\dfrac{A\,(R/12)}{1-(1+R/12)^{-12T}}

$A$ = amount borrowed

Interest and principal

\text{Interest}_t=\text{Balance}_{t-1}\times R/12

\text{Principal}_t=X-\text{Interest}_t

\text{Balance}_t=\text{Balance}_{t-1}-\text{Principal}_t

WAC and WAM

\text{WAC}=\sum w_ic_i

\text{WAM}=\sum w_iL_i

w_i=\dfrac{P_i}{\sum P_i}

$P_i$ = remaining principal of mortgage $i$

SMM and CPR

\text{SMM}=\dfrac{\text{prepayment in the month}}{\text{beginning balance}-\text{scheduled principal}}

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

Dollar roll value

A-B+C-D

$A$ = sale price incl. AI; $B$ = repurchase price incl. AI; $C$ = interest on proceeds for the month; $D$ = coupon and principal given up

Refinancing incentive

I=\text{WAC}-R

$R$ = current mortgage rate

FMP 19

Interest Rate Futures

10 formulas

Market	Day count
Treasury bonds (and AI on futures delivery)	Actual/actual
Corporate and municipal bonds	30/360
T-bills and money market	Actual/360

Accrued interest and dirty price

\text{AI}=\dfrac{\text{days since last coupon}}{\text{days in coupon period}}\times\text{coupon}

dirty

=

clean

+

T-bill

Q=\dfrac{360}{n}(100-C)

C=100-\dfrac{Qn}{360}

$Q$ = quote, $C$ = cash price per 100, $n$ = days to maturity

Conversion factor

price per 1 of face of the delivered bond at a 6% yield (semiannual), maturity rounded down to the nearest 3 months

Delivery

cash received

=

settlement price

\times

+

AICTD minimizes quoted price

-

settlement price

\times

Treasury futures price

F_0=(S_0-I)(1+R)^T

, or

(S_0-I)e^{rT}

$I$ = PV of coupons during the futures life

Duration-based hedge

N^*=\dfrac{P\,D_P}{V_F\,D_F}

$P$ , $D_P$ at hedge maturity; $V_F$ = contract price

DV01 hedge

N=-\dfrac{E_V}{E_F}

$E$ = value gain for a 1 bp fall in all rates; $N>0$ long, $N<0$ short

FMP 20

Swaps

8 formulas

Comparative advantage

total gain

=|\Delta_{\text{fixed}}-\Delta_{\text{floating}}|

Swap position	Value
Pay fixed, receive floating	$B_{\text{float}}-B_{\text{fix}}$
Receive fixed, pay floating	$B_{\text{fix}}-B_{\text{float}}$
Currency: receive domestic ( $S_0$ = domestic per foreign)	$B_D-S_0B_F$
Currency: receive foreign	$S_0B_F-B_D$

Bond values inside a swap

B_{\text{fix}}=\sum c\,\exp(-r_it_i)+L\exp(-r_nt_n)

B_{\text{float}}=(L+k^*)\exp(-r^*t^*)

$B_{\text{float}}=L$ immediately after a reset

Swap value from FRAs

pay fixed:

V=\sum\text{PV}\big[(R_{F,i}-R_K)\,\tau L\big]

$R_{F,i}$ = forward rate for period $i$

Currency swap from forwards

receive foreign:

V=\sum\text{PV}(\text{FCF}_i\times F_i-\text{DCF}_i)

FCF, DCF = foreign and domestic cash flows; $F_i$ = forward rate (domestic per foreign); receive domestic: sign reversed

Book 4 of 4 · 30% of the exam

Valuation and Risk Models formulas

VRM 1

Measures of Financial Risk

3 formulas

Two-asset portfolio

\mu_P=w_1\mu_1+w_2\mu_2

\sigma_P=\sqrt{w_1^2\sigma_1^2+w_2^2\sigma_2^2+2\rho w_1w_2\sigma_1\sigma_2}

Normal distribution of	VaR at confidence $X$	ES at confidence $X$
Losses $(\mu_L,\sigma_L)$	$\mu_L+\sigma_Lz$	$\mu_L+\sigma_L\dfrac{e^{-z^2/2}}{(1-X)\sqrt{2\pi}}$
Returns $(\mu_P,\sigma_P)$	$\sigma_Pz-\mu_P$	$\sigma_P\dfrac{e^{-z^2/2}}{(1-X)\sqrt{2\pi}}-\mu_P$

Exam tip VaR from a loss distribution adds the mean; from a return distribution it subtracts the mean.

VRM 2

Calculating and Applying VaR

6 formulas

Linear portfolio

\Delta P=\sum q_i\,\Delta r_i

$q_i=n_iS_i$ (amount invested), $\Delta r_i=\Delta S_i/S_i$

Risk factor sensitivities

linear

\Delta P=\sum\delta_i\,\Delta S_i

delta-gamma

\Delta P=\sum\delta_i\,\Delta S_i+\tfrac12\sum\gamma_i(\Delta S_i)^2

Zero-mean VaR and ES

\text{VaR}=\sigma_Pz

\text{ES}=\sigma_P\dfrac{e^{-z^2/2}}{(1-X)\sqrt{2\pi}}

Square-root-of-time rule

\text{VaR}(T,X)=\sqrt T\,\text{VaR}(1,X)

\text{ES}(T,X)=\sqrt T\,\text{ES}(1,X)

needs iid returns and a zero mean

Delta-normal VaR of an option

\text{VaR}=|\delta|\times S\times\sigma\times z

Historical simulation

VaR

=

the

n(1-X)

-th worst of the

n

scenario lossesES

=

average of the

n(1-X)

worst losses

VRM 3

Measuring and Monitoring Volatility

6 formulas

Variance from daily returns

\sigma_n^2=\dfrac1m\sum_{i=1}^{m}u_{n-i}^2

$u_i=(S_i-S_{i-1})/S_{i-1}$ ; mean return taken as 0

EWMA

\sigma_n^2=\lambda\,\sigma_{n-1}^2+(1-\lambda)\,u_{n-1}^2

RiskMetrics $\lambda=0.94$

GARCH(1,1)

\sigma_n^2=\omega+\alpha u_{n-1}^2+\beta\sigma_{n-1}^2

$\omega=\gamma V_L$ , $\gamma=1-\alpha-\beta$

Long-run variance

V_L=\dfrac{\omega}{1-\alpha-\beta}

GARCH variance forecast

E[\sigma_{n+t}^2]=V_L+(\alpha+\beta)^t(\sigma_n^2-V_L)

EWMA covariance and correlation

\text{cov}_n=\lambda\,\text{cov}_{n-1}+(1-\lambda)\,x_{n-1}y_{n-1}

\rho_n=\dfrac{\text{cov}_n}{\sigma_{x,n}\,\sigma_{y,n}}

$x$ , $y$ = returns on day $n-1$

VRM 4

External and Internal Credit Ratings

6 formulas

Hazard rate

\text{PD}(0,t)=1-e^{-\bar ht}

survival

S(t)=e^{-\bar ht}

Average hazard rate from survival

\bar h=-\dfrac{\ln S(t)}{t}

Default between $t_1$ and $t_2$

\exp(-\bar h_1t_1)-\exp(-\bar h_2t_2)

$\bar h_2$ = average hazard rate to $t_2$

Conditional PD

P(\text{default in year }t\mid\text{survived to }t-1)=\dfrac{S(t-1)-S(t)}{S(t-1)}

Expected loss

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

$\text{LGD}=1-\text{RR}$ ; in % terms $\text{EL}=\text{PD}\times\text{LGD}$

Rating transition matrix

n

-year matrix

=M^n

assumes independence across years; cumulative PD is the default column

VRM 5

Country Risk: Determinants, Measures and Implications

Concepts only

No formulas to memorise. GARP tests this chapter on concepts, so there is nothing from it to learn for this sheet.

VRM 6

Measuring Credit Risk

6 formulas

Loss on one loan

mean

p_iL_i(1-R_i)

\sigma_i=\sqrt{p_i-p_i^2}\;L_i(1-R_i)

Portfolio of identical loans

\sigma_P^2=n\sigma^2+n(n-1)\rho\sigma^2

\dfrac{\sigma_P}{nL}=\dfrac{\sigma\sqrt{1+(n-1)\rho}}{L\sqrt n}

One-factor Gaussian copula

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

\rho(U_i,U_j)=a_ia_j

Vasicek WCDR, 99.9%

\text{WCDR}=N\!\left(\dfrac{N^{-1}(\text{PD})-\sqrt\rho\,N^{-1}(0.001)}{\sqrt{1-\rho}}\right)

Credit risk capital

worst-case loss

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

capital

=\text{UL}=(\text{WCDR}-\text{PD})\times\text{LGD}\times\text{EAD}

loan by loan:

\sum(\text{WCDR}_i-\text{PD}_i)\,\text{LGD}_i\,\text{EAD}_i

Euler contribution

\text{Contribution}_i=x_i\dfrac{\partial F}{\partial x_i}

\sum_i\text{Contribution}_i=F

$F$ = risk measure, $x_i$ = size of loan $i$

VRM 7

Operational Risk

6 formulas

Basic indicator approach

capital

=15\%\times

average annual gross income over 3 years

gross income = interest earned − interest paid + non-interest income

Standardized approach

capital

=

3-year average of

\sum\beta_k\,\text{GI}_k

$\beta$ = 12% retail banking, asset management, retail brokerage; 15% commercial banking, agency services; 18% corporate finance, trading and sales, payment and settlement

SMA loss component

7X+7Y+5Z

$X$ = average annual op loss; $Y$ = from losses above €10m; $Z$ = from losses above €100m (last 10 years)

Loss frequency (Poisson)

P(n)=\dfrac{e^{-\lambda}\lambda^n}{n!}

Scaling external loss data

\text{Loss}_A=\text{Loss}_B\left(\dfrac{\text{Revenue}_A}{\text{Revenue}_B}\right)^{\beta}

$\beta$ = scaling exponent estimated from data

Power law

P(v>x)=Kx^{-\alpha}

VRM 8

Stress Testing

Concepts only

No formulas to memorise. GARP tests this chapter on concepts, so there is nothing from it to learn for this sheet.

VRM 9

Pricing Conventions, Discounting and Arbitrage

3 formulas

Present value

PV=\sum CF_t\,d_t

$d_t$ = discount factor for time $t$

Money-market yield from a discount quote

\dfrac{Qn/360}{100-Qn/360}

360 to 365 day basis

r_{365}=r_{360}\times\dfrac{365}{360}

VRM 10

Interest Rates

4 formulas

Discount factors

semiannual

d_t=(1+z_t/2)^{-2t}

continuous

d_t=e^{-z_tt}

Bond value

V=\dfrac c2A_T+100\,d_T

V=100+\dfrac{c-p}{2}A_T

$A_T$ = sum of discount factors; $p$ = par rate

Six-month forward rate

1+\dfrac f2=\dfrac{(1+z_{T+0.5}/2)^{2T+1}}{(1+z_T/2)^{2T}}

Spot rate from forwards

(1+z_n/2)^n=(1+f_1/2)(1+f_2/2)\cdots(1+f_n/2)

$n$ = number of half-years

VRM 11

Bond Yields and Return Calculations

4 formulas

Annuity and perpetuity

annuity

\dfrac cy\left[1-(1+y/2)^{-2T}\right]

perpetuity

\dfrac cy

$c$ = annual coupon, paid semiannually

Realized return

gross

R=\dfrac{P_{\text{end}}+\text{coupons}-P_{\text{begin}}}{P_{\text{begin}}}

net

=R-\dfrac{\text{financing cost}}{P_{\text{begin}}}

several periods:

(1+R_1)\cdots(1+R_n)-1

Spread

P=\sum_t\dfrac{CF_t}{[1+(z_t+s)/2]^{2t}}

, solve for

s

P&L decomposition

P&L

=

carry roll-down

+

rate changes

+

spread changes

carry roll-down = price change from passing time plus coupon, on the assumed curve (forwards realized, or curve unchanged)

VRM 12

Applying Duration, Convexity and DV01

8 formulas

DV01

\text{DV01}=-\dfrac{\Delta P}{10{,}000\times\Delta r}

$\Delta r$ in decimal

Effective duration

D=\dfrac{P_--P_+}{2P\,\Delta r}

one-sided:

D=-\dfrac{1}{P}\dfrac{\Delta P}{\Delta r}

Effective convexity

C=\dfrac{P_++P_--2P}{P\,(\Delta r)^2}

Price change

\Delta P=-DP\,\Delta r+\tfrac12CP\,(\Delta r)^2

Portfolio measures

\text{DV01}_P=\sum\text{DV01}_i

D_P=\sum\dfrac{V_i}{V}D_i

C_P=\sum\dfrac{V_i}{V}C_i

Hedge face amount

F_H=-F_P\,\dfrac{\text{DV01}_P}{\text{DV01}_H}

DV01s per 100 face

Duration and convexity hedge

one bond:

P_H=-\dfrac{V D_V}{D_H}

two bonds:

P_1D_1+P_2D_2=-VD_V

P_1C_1+P_2C_2=-VC_V

Barbell matching a bullet

V_S+V_L=V_{\text{bullet}}

and

V_SD_S+V_LD_L=V_{\text{bullet}}D_{\text{bullet}}

the barbell has higher convexity

VRM 13

Non-Parallel Term Structure Shifts and Hedging

4 formulas

Key rate and bucket 01s

\text{DV01}=\sum_i\text{KR01}_i

also

\text{DV01}=\sum

forward bucket 01s

Portfolio SD from key rates

\sigma_P=\sqrt{\sum_i\sum_j\rho_{ij}\sigma_i\sigma_j\,\text{KR01}_i\,\text{KR01}_j}

$\sigma_i$ = SD of daily change in key rate $i$ (bp); $\rho_{ij}$ = correlation of those changes

Portfolio SD from principal components

\sigma_P=\sqrt{\sigma_1^2f_1^2+\sigma_2^2f_2^2+\sigma_3^2f_3^2}

$f_i$ = value change per unit of factor $i$

Duration from a 01

\text{Duration}=\dfrac{10{,}000\times\text{01 measure}}{\text{portfolio value}}

VRM 14

Binomial Trees

8 formulas

One step

f=e^{-rT}[pf_u+(1-p)f_d]

p=\dfrac{e^{rT}-d}{u-d}

u=e^{\sigma\sqrt T}

d=1/u

$S_u=S_0u$ , $S_d=S_0d$

Several steps

f=e^{-r\Delta t}[pf_u+(1-p)f_d]

u=e^{\sigma\sqrt{\Delta t}}

d=1/u

American option node

f=\max\big(\text{exercise value},\ e^{-r\Delta t}[pf_u+(1-p)f_d]\big)

Delta

\Delta=\dfrac{f_u-f_d}{S_0u-S_0d}

Underlying	$a$ in $p=(a-d)/(u-d)$
Non-dividend stock	$e^{r\Delta t}$
Dividend yield $q$ or index	$e^{(r-q)\Delta t}$
Currency	$e^{(r-r_f)\Delta t}$
Futures	$1$

VRM 15

The Black-Scholes-Merton Model

11 formulas

Lognormal stock price

\dfrac{\Delta S}{S}\sim N(\mu\Delta t,\ \sigma^2\Delta t)

E(S_T)=S_0e^{\mu T}

Distribution of $\ln S_T$

E(\ln S_T)=\ln S_0+(\mu-\sigma^2/2)T

=\sigma\sqrt T

Realized return

R=\dfrac1T\ln\dfrac{S_T}{S_0}\sim N\!\left(\mu-\dfrac{\sigma^2}{2},\ \dfrac{\sigma^2}{T}\right)

Historical volatility

u_i=\ln(S_i/S_{i-1})

\hat\sigma=\dfrac{s}{\sqrt\tau}

$s$ = sample SD of $u_i$ ; $\tau$ = interval length in years

BSM, no dividends

c=S_0N(d_1)-Ke^{-rT}N(d_2)

p=Ke^{-rT}N(-d_2)-S_0N(-d_1)

d₁ and d₂

d_1=\dfrac{\ln(S_0/K)+(r+\sigma^2/2)T}{\sigma\sqrt T}

d_2=d_1-\sigma\sqrt T

Dividend yield q

c=S_0e^{-qT}N(d_1)-Ke^{-rT}N(d_2)

p=Ke^{-rT}N(-d_2)-S_0e^{-qT}N(-d_1)

d_1=\dfrac{\ln(S_0/K)+(r-q+\sigma^2/2)T}{\sigma\sqrt T}

same for a stock index

Currency options

q=r_f

c=S_0e^{-r_fT}N(d_1)-Ke^{-rT}N(d_2)

p=Ke^{-rT}N(-d_2)-S_0e^{-r_fT}N(-d_1)

Futures options (Black)

c=e^{-rT}[F_0N(d_1)-KN(d_2)]

p=e^{-rT}[KN(-d_2)-F_0N(-d_1)]

d_1=\dfrac{\ln(F_0/K)+\sigma^2T/2}{\sigma\sqrt T}

Discrete dividends

c=[S_0-\text{PV}(D)]N(d_1)-Ke^{-rT}N(d_2)

replace $S_0$ by $S_0-\text{PV}(D)$ in every no-dividend formula, including $d_1$

Warrants

value per warrant

=\dfrac{N}{N+M}\times c

dilution cost to existing shareholders

=M\times

value per warrant

$c$ = BSM call on the stock; $N$ = shares outstanding; $M$ = warrants issued

VRM 16

The Greeks

23 formulas

Greek, no income	Call	Put
Delta	$N(d_1)$	$N(d_1)-1$
Gamma	$\dfrac{N'(d_1)}{S_0\sigma\sqrt T}$	same as call
Vega	$S_0\sqrt T\,N'(d_1)$	same as call
Rho	$KTe^{-rT}N(d_2)$	$-KTe^{-rT}N(-d_2)$

Theta, no income

call

-\dfrac{S_0N'(d_1)\sigma}{2\sqrt T}-rKe^{-rT}N(d_2)

put

-\dfrac{S_0N'(d_1)\sigma}{2\sqrt T}+rKe^{-rT}N(-d_2)

Standard normal density

N'(x)=\dfrac{1}{\sqrt{2\pi}}e^{-x^2/2}

Greek, yield $q$	Call	Put
Delta	$e^{-qT}N(d_1)$	$e^{-qT}[N(d_1)-1]$
Gamma	$\dfrac{N'(d_1)\,e^{-qT}}{S_0\sigma\sqrt T}$	same as call
Vega	$S_0\sqrt T\,N'(d_1)\,e^{-qT}$	same as call
Rho	$KTe^{-rT}N(d_2)$	$-KTe^{-rT}N(-d_2)$

Theta, yield q, call

-\dfrac{S_0N'(d_1)\sigma e^{-qT}}{2\sqrt T}+qS_0N(d_1)e^{-qT}-rKe^{-rT}N(d_2)

Theta, yield q, put

-\dfrac{S_0N'(d_1)\sigma e^{-qT}}{2\sqrt T}-qS_0N(-d_1)e^{-qT}+rKe^{-rT}N(-d_2)

$d_1$ uses $r-q$ , as in VRM 15

Currency and futures options

use

q=r_f

for currencies

q=r

for futures

Long position	Delta	Gamma	Vega
Call	+	+	+
Put	−	+	+
Stock	$1$	$0$	$0$
Forward, non-dividend stock	$1$	$0$	$0$
Futures, non-dividend stock	$e^{rT}$	$0$	$0$
Forward, yield $q$	$e^{-qT}$	$0$	$0$
Futures, yield $q$	$e^{(r-q)T}$	$0$	$0$

Exam tip Short positions reverse every sign in the table above.

Portfolio Greeks

\Delta_P=\sum w_i\Delta_i

\Gamma_P=\sum w_i\Gamma_i

\nu_P=\sum w_i\nu_i

$w_i$ = number of options of type $i$

Gamma-neutral, then delta-neutral

options to trade

n=-\dfrac{\Gamma_P}{\Gamma_T}

then shares

=-(\Delta_P+n\Delta_T)

Theta, delta and gamma

\Theta+rS\Delta+\tfrac12\sigma^2S^2\Gamma=r\Pi

delta-neutral:

\Theta+\tfrac12\sigma^2S^2\Gamma=r\Pi

How to revise with the FRM Part 1 formula sheet

Part 1 questions rarely ask for a formula on its own. They describe a position, a portfolio or a data set and expect you to pick the right relation, 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 Financial Markets and Products and Valuation and Risk Models, which hold most of the calculations. 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

Foundations of Risk Management 20%: Mostly conceptual. The formulas are RAROC, the CAPM with the security and capital market lines, the Sharpe, Treynor and Jensen measures, the information and Sortino ratios, the APT and the Fama-French three-factor model.
Quantitative Analysis 20%: Probability rules and Bayes, distributions and their moments, hypothesis tests, OLS and multiple regression with R squared, adjusted R squared and the F test, AR, MA and ARMA time series, rank correlations, simulation and machine-learning measures.
Financial Markets and Products 30%: Hedging with futures, forward and futures pricing, interest and purchasing power parity, option bounds and put-call parity, trading strategies, exotic payoffs, interest rate conventions, mortgages, Treasury futures and swap valuation.
Valuation and Risk Models 30%: VaR and expected shortfall, EWMA and GARCH volatility, credit ratings and the Vasicek model, operational risk capital, bond pricing, duration, convexity and DV01, key rates, binomial trees, Black-Scholes-Merton and the Greeks.

FRM Part 1 formula sheet questions

Does GARP provide a formula sheet in the FRM Part 1 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 1 exam structured?

FRM Part 1 has 100 equally weighted multiple-choice questions to be answered in four hours on computer. Foundations of Risk Management and Quantitative Analysis carry about 20% each; Financial Markets and Products and Valuation and Risk Models carry about 30% each. GARP offers the exam in May, August and November.

Which FRM Part 1 chapters have no formulas to learn?

Of the 62 chapters in FRM Part 1, 13 are tested on concepts alone: FRM 2 How Do Firms Manage Financial Risk?; FRM 3 The Governance of Risk Management; FRM 4 Credit Risk Transfer Mechanisms; FRM 7 Principles for Effective Data Aggregation and Risk Reporting; FRM 8 Enterprise Risk Management and Future Trends; FRM 9 Learning from Financial Disasters; FRM 10 Anatomy of the Great Financial Crisis of 2007-2009; FRM 11 GARP Code of Conduct; FMP 1 Banks; FMP 6 Central Clearing; FMP 7 Futures Markets; VRM 5 Country Risk: Determinants, Measures and Implications; VRM 8 Stress Testing. 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 1 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 follows the 2026 GARP FRM Part 1 readings chapter by chapter, using the notation of the readings, and was last updated on 1 October 2026.

Is a formula sheet enough to pass FRM Part 1?

No. Part 1 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 1 sample course has 9 hours of lectures by Micky Midha, 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.

1 October 2026 Rebuilt for the 2026 FRM Part 1 curriculum: 49 chapters across all four books.

GARP does not endorse, promote, review or warrant the accuracy of the products or services offered by MidhaFin or any GARP exam related information, nor does it endorse any pass rates that may be claimed by MidhaFin. FRM, GARP and Global Association of Risk Professionals are trademarks owned by the Global Association of Risk Professionals, Inc.

The research behind this formula sheet

Every chapter accounted for

Not required, said plainly

In the notation of the readings

Related forms kept together

RAROC

Beta

Portfolio beta

CAPM (SML)

CML

Sharpe performance index

Treynor performance index

Jensen performance index

Tracking error

Information ratio

Sortino ratio

APT return model

APT expected return

Fama-French three-factor

Portfolio factor beta

Complement

Addition rule

Conditional probability

Total probability

Total probability, two events

Independence

Conditional independence

Bayes' rule

Bernoulli PMF

Expectation

Properties of E

Variance

Skewness

Kurtosis

Standardized variable

Linear transformation

Higher moments of Y=a+bXY=a+bXY=a+bX

Median and IQR

Binomial

Poisson

Uniform, interval probability

Normal

Normal facts

Normal probability ranges

Lognormal

Exponential

Student's t kurtosis

Constructions

Mixture

Marginal and conditional PMF

Conditional expectation

Conditional variance

Covariance

Correlation

Linear transformations

Covariance with a sum

Variance of a sum or difference

Variance of a weighted sum

Sum of n iid variables

Sample mean

Bias

Variance estimators

Standard error of the mean

Central limit theorem

Scaling the mean

Scaling volatility

Sample skewness and kurtosis

Median

Sample covariance

Sample correlation

Hypotheses

Test statistic, mean

Confidence interval, two-sided

Confidence interval, one-sided

p-value

Difference in means, paired

Difference in means, independent

OLS estimates

Variance of the shocks

Standard error of the slope

Higher moments of $Y=a+bX$