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FRM® Exam Part I formula sheet: printable, with checks

A printable formula sheet for Part I of the FRM® Exam: portfolio risk and performance ratios, probability, regression, EWMA and GARCH, forwards and hedge ratios, VaR and expected shortfall, bond duration, binomial trees, Black–Scholes–Merton, Greeks and credit loss – each with a worked check value.

FRM® Exam Part I guideBy High Performer CertsPublished 10 min read

Key facts

  • The sheet lists 38 formulas for the four Part I topic areas, grouped by area.
  • Each formula has a check value – inputs and result – so you can test your recall and your calculator.
  • At 99% confidence, a normal one-day VaR is about 2.33 daily standard deviations of the position's value (zero mean).
  • The print layout keeps only the title and the formula tables.
  • The formulas are our own selection from finance textbooks, lecture notes and papers of central banks and international bodies, not GARP material.

This printable sheet collects 38 core formulas for the four topic areas of Part I of the FRM® Exam, each with a check value you can rework on your calculator and a link to a worked guide or calculator. It is our own selection from finance textbooks, lecture notes and papers of central banks and international bodies, as of 30 October 2026.

The FRM® Exam is run by GARP (GARP, FRM Exam information). This sheet is not GARP material and not a list of what your exam will test: it gathers standard formulas behind the four topic areas – Foundations of Risk Management, Quantitative Analysis, Financial Markets and Products, and Valuation and Risk Models – in our own notation. Our questions and guides complement GARP's own material; they do not replace it.

How do you use this formula sheet?

  1. Print it. Use your browser's print command (Ctrl+P or ⌘P). The print layout keeps the title and the formula tables and leaves out the rest of the page.
  2. Recall before you read. Cover the formula column, write the formula down, then check. Testing yourself this way beats rereading (Roediger and Karpicke, 2006) (Dunlosky et al., 2013).
  3. Rework the check. Enter the inputs on your calculator and compare. The VaR and option checks use exact normal values; with a table value such as 2.33 your answer will differ slightly.
  4. Go deeper where you stumble. Most areas link to a guide or calculator that works the formulas through.

Rates, volatilities and probabilities are decimals inside the formulas: 1.5% is 0.015. VaR and ES are losses, so a positive number is a loss. Which calculators you may use in the exam is explained on calculators allowed in the CFA® Program and the FRM® Exam.

High Performer Certs · https://highperformer-certs.com/blog/risk-part-1-formula-sheet · as of 30 October 2026 · our own selection, not GARP material

Foundations of Risk Management: which formulas do you need?

Go deeper with the CAPM and beta guide and the portfolio risk and Sharpe ratio calculator. Sources: (Naranjo, portfolio risk (secondary source))(Naranjo, CAPM (secondary source))(Sharpe, The Sharpe Ratio (1994))

Foundations of Risk Management: formulas and check values
FormulaWhat it givesCheck: inputs → result
Two-asset portfolio riskσp2=wA2σA2+wB2σB2+2wAwB ρ σAσB\sigma_p^2 = w_A^2\sigma_A^2 + w_B^2\sigma_B^2 + 2w_Aw_B\,\rho\,\sigma_A\sigma_BVariance of a two-asset portfoliow 50% and 50%, σ 18% and 12%, ρ −0.2 → portfolio σ 9.77%
Betaβi=Cov⁡(Ri,Rm)σm2=ρim σiσm\beta_i = \dfrac{\operatorname{Cov}(R_i,R_m)}{\sigma_m^2} = \rho_{im}\,\dfrac{\sigma_i}{\sigma_m}Sensitivity to the marketρ 0.8, σ 30%, market σ 20% → 1.20
CAPME(Ri)=Rf+βi [E(Rm)−Rf]E(R_i) = R_f + \beta_i\,[E(R_m) - R_f]Required return for systematic riskRisk-free 2%, β 1.2, market 7% → 8.00%
Sharpe ratioS=Rp−RfσpS = \dfrac{R_p - R_f}{\sigma_p}Excess return per unit of total riskReturn 9%, risk-free 2%, σ 20% → 0.35
Treynor ratioT=Rp−RfβpT = \dfrac{R_p - R_f}{\beta_p}Excess return per unit of betaReturn 9%, risk-free 2%, β 1.2 → 5.83%
Information ratioIR=Rˉp−Rˉbσ(Rp−Rb)IR = \dfrac{\bar R_p - \bar R_b}{\sigma(R_p - R_b)}Return above a benchmark per unit of tracking errorReturn 9%, benchmark 7.5%, tracking error 3% → 0.50

Quantitative Analysis: which formulas do you need?

Sources: (Chen, probability moments (secondary source))(Naranjo, portfolio risk (secondary source))(NYU Stern, covariance and correlation (secondary source))(NIST, confidence limits for the mean (secondary source))(NIST, skewness and kurtosis (secondary source))(arch documentation, EWMA (secondary source))(NYU V-Lab, GARCH (secondary source))

Quantitative Analysis: formulas and check values
FormulaWhat it givesCheck: inputs → result
Bayes' ruleP(A∣B)=P(B∣A) P(A)P(B∣A) P(A)+P(B∣Ac) P(Ac)P(A\mid B) = \dfrac{P(B\mid A)\,P(A)}{P(B\mid A)\,P(A) + P(B\mid A^c)\,P(A^c)}Updating a probability after new informationP(default) 2%, P(warning | default) 90%, P(warning | no default) 10% → 15.52%
Variance of a sumVar⁡(aX+bY)=a2σX2+b2σY2+2ab ρ σXσY\operatorname{Var}(aX+bY) = a^2\sigma_X^2 + b^2\sigma_Y^2 + 2ab\,\rho\,\sigma_X\sigma_YRisk of a combination of two variablesa 2, b 1, σ of X 3, σ of Y 4, ρ 0.5 → 76
Correlationρ=Cov⁡(X,Y)σXσY\rho = \dfrac{\operatorname{Cov}(X,Y)}{\sigma_X\sigma_Y}Strength of a linear relationship, from −1 to +1Cov 0.0108, σ 0.12 and 0.15 → 0.60
t-statistic for a meant=xˉ−μ0s/nt = \dfrac{\bar x - \mu_0}{s/\sqrt{n}}Tests a sample mean against μ₀x̄ 0.6, μ₀ 0, s 2, n 100 → 3.00
Confidence interval for a meanxˉ±t sn\bar x \pm t\,\dfrac{s}{\sqrt{n}}Range for the population mean; t depends on the level and n − 1x̄ 0.6, s 2, n 100, t 1.984 → 0.203 to 0.997
Regression slope and interceptβ^=Cov⁡(X,Y)Var⁡(X),α^=Yˉ−β^ Xˉ\hat\beta = \dfrac{\operatorname{Cov}(X,Y)}{\operatorname{Var}(X)},\quad \hat\alpha = \bar Y - \hat\beta\,\bar XLeast-squares line Y = α + βXCov 0.00027, Var(X) 0.0003, X̄ 1.0%, Ȳ 1.2% → β 0.90, α 0.30%
R² with one regressorR2=ρXY2R^2 = \rho_{XY}^2Share of the variance of Y the line explainsρ 0.7 → 0.49
Skewness and kurtosisg1=∑i(xi−xˉ)3/ns3,K=∑i(xi−xˉ)4/ns4g_1 = \dfrac{\sum_i (x_i-\bar x)^3/n}{s^3},\quad K = \dfrac{\sum_i (x_i-\bar x)^4/n}{s^4}Asymmetry and tail weight; here s divides by n, not n − 1, and excess kurtosis is K − 3A normal distribution has skewness 0 and kurtosis 3
EWMA varianceσt2=λ σt−12+(1−λ) rt−12\sigma_t^2 = \lambda\,\sigma_{t-1}^2 + (1-\lambda)\,r_{t-1}^2Volatility that weights recent returns moreλ 0.94, yesterday's σ 1.2% and return −2% → σ 1.26%
GARCH(1,1) varianceσt2=ω+α rt−12+β σt−12\sigma_t^2 = \omega + \alpha\,r_{t-1}^2 + \beta\,\sigma_{t-1}^2Volatility that reverts to a long-run levelω 0.000002, α 0.08, β 0.90, same σ and return → σ 1.28%
GARCH long-run varianceVL=ω1−α−βV_L = \dfrac{\omega}{1-\alpha-\beta}The level the GARCH variance reverts to (α + β < 1)Same ω, α, β → long-run σ 1.00% a day

Financial Markets and Products: which formulas do you need?

Go deeper with the forwards and put–call parity guide. Sources: (NYU Stern, time value of money (secondary source))(Back, Liu and Loewenstein, carry (secondary source))(Wolfram, hedge ratio (secondary source))(Wolfram, number of futures contracts (secondary source))(Naranjo, dividend-adjusted parity (secondary source))

Financial Markets and Products: formulas and check values
FormulaWhat it givesCheck: inputs → result
Continuous rate from m-times compoundingRc=m ln⁡ ⁣(1+Rmm)R_c = m\,\ln\!\left(1 + \dfrac{R_m}{m}\right)Converts a rate compounded m times a year: both give the same effective annual rate6% compounded semi-annually → 5.9118%
Forward price with an income yieldF0=S0 e(r−q)TF_0 = S_0\,e^{(r-q)T}Stock index with dividend yield qS₀ 2,000, r 4%, q 2%, T 0.75 → 2,030.23
Currency forwardF0=S0 e(r−rf)TF_0 = S_0\,e^{(r-r_f)T}Spot exchange rate; the second rate is the foreign interest rateS₀ 1.1000, r 4%, foreign rate 2.5%, T 1 → 1.1166
Commodity forwardF0=S0 e(r+u−y)TF_0 = S_0\,e^{(r+u-y)T}u = storage cost, y = convenience yieldS₀ 80, r 4%, u 1%, y 3%, T 0.5 → 80.80
Minimum-variance hedge ratioh∗=ρ σSσFh^* = \rho\,\dfrac{\sigma_S}{\sigma_F}Futures per unit of exposure (price changes)ρ 0.9, σ of spot changes 2.5%, of futures changes 3% → 0.75
Number of futures contractsN∗=h∗ QAQFN^* = \dfrac{h^*\,Q_A}{Q_F}Hedge ratio times the exposure, divided by the size of one contract (same units)h* 0.75, exposure 2,000,000 units, contract 100,000 units → 15.0
Put–call parity with dividendsc0+Ke−rT=p0+S0−PV(D)c_0 + K e^{-rT} = p_0 + S_0 - PV(D)European options; PV(D) = present value of the dividendsc 6, S₀ 50, K 50, r 4%, T 0.5, PV(D) 1 → p 6.01

Valuation and Risk Models: which formulas do you need?

Go deeper with the value at risk guide, the VaR calculator, the binomial guide, the Greeks guide, the expected loss guide and the expected loss calculator. Sources: (RBA, VaR methods, §§2.1–2.7 (secondary source))(IMF, risk measurement, Appendix II (secondary source))(Acerbi and Tasche, expected shortfall)(IMF, duration definitions (secondary source))(CME Group, basis-point value (secondary source))(IMF, convexity and curve assumptions)(Back, Liu and Loewenstein, binomial trees (secondary source))(Back, Liu and Loewenstein, option hedging (secondary source))(BCBS, CRE35)(NIST, binomial moments (secondary source))

Valuation and Risk Models: formulas and check values
FormulaWhat it givesCheck: inputs → result
Normal VaRVaR⁡c=V (zc σ−μ)\operatorname{VaR}_c = V\,(z_c\,\sigma - \mu)Loss not exceeded with confidence cV 1,000,000, σ 1.5% a day, μ 0, 99% (z 2.3263) → 34,895
Square-root-of-time ruleVaR⁡T=VaR⁡1T\operatorname{VaR}_T = \operatorname{VaR}_1\sqrt{T}Scales a one-day VaR (zero mean, independent and identically distributed days)Same position, 10 days → 110,348
Normal expected shortfallES⁡c=V(σ φ(zc)1−c−μ)\operatorname{ES}_c = V\left(\sigma\,\dfrac{\varphi(z_c)}{1-c} - \mu\right)Average loss beyond the VaR levelSame position, 97.5% → 35,067
Bond priceP=∑k=1mNCFk(1+y/m)kP = \sum_{k=1}^{mN}\dfrac{CF_k}{(1+y/m)^k}Price on a coupon date, m coupons a yearCoupon 6% semi-annual, 5 years, y 5%, face 100 → 104.38
Macaulay and modified durationDMac=∑ktk PV(CFk)P,Dmod=DMac1+y/mD_{\text{Mac}} = \dfrac{\sum_k t_k\,PV(CF_k)}{P},\quad D_{\text{mod}} = \dfrac{D_{\text{Mac}}}{1+y/m}Average time to the cash flows; price sensitivity to yieldSame bond → 4.408 and 4.301
DV01DV01≈Dmod×P×0.0001DV01 \approx D_{\text{mod}}\times P\times 0.0001Price change for one basis pointSame bond → 0.0449 per 100 face
Price change with convexityΔPP≈−Dmod Δy+12 C (Δy)2\dfrac{\Delta P}{P} \approx -D_{\text{mod}}\,\Delta y + \tfrac12\,C\,(\Delta y)^2Estimate for a parallel yield changeSame bond, C 22.08, +100 basis points → −4.190% (repriced: −4.193%)
Binomial treeu=eσΔt,d=1u,p=erΔt−du−du = e^{\sigma\sqrt{\Delta t}},\quad d = \dfrac{1}{u},\quad p = \dfrac{e^{r\Delta t} - d}{u - d}Up and down factors and the risk-neutral probability per stepσ 25%, Δt 0.25, r 4% → u 1.1331, d 0.8825, p 0.5089
Black–Scholes–Merton: d₁ and d₂d1=ln⁡(S0/K)+(r−q+σ2/2) TσT,d2=d1−σTd_1 = \dfrac{\ln(S_0/K) + (r - q + \sigma^2/2)\,T}{\sigma\sqrt{T}},\quad d_2 = d_1 - \sigma\sqrt{T}Inputs to the option prices; q is the dividend yieldS₀ 100, K 100, r 5%, σ 20%, T 1, q 0 → d₁ 0.35, d₂ 0.15
Black–Scholes–Merton: call and putc=S0e−qTN(d1)−Ke−rTN(d2),p=Ke−rTN(−d2)−S0e−qTN(−d1)c = S_0e^{-qT}N(d_1) - Ke^{-rT}N(d_2),\quad p = Ke^{-rT}N(-d_2) - S_0e^{-qT}N(-d_1)European option pricesSame inputs → c 10.45, p 5.57
Delta and gammaδcall=e−qTN(d1),δput=e−qT[N(d1)−1],Γ=e−qTφ(d1)S0 σT\delta_{\text{call}} = e^{-qT}N(d_1),\quad \delta_{\text{put}} = e^{-qT}[N(d_1) - 1],\quad \Gamma = \dfrac{e^{-qT}\varphi(d_1)}{S_0\,\sigma\sqrt{T}}Sensitivity to the share price, and how delta changesSame inputs → call δ 0.6368, put δ −0.3632, Γ 0.0188
Delta–gamma approximationΔV≈δ ΔS+12 Γ (ΔS)2\Delta V \approx \delta\,\Delta S + \tfrac12\,\Gamma\,(\Delta S)^2Option value change for a share price moveSame call, ΔS +2 → 1.311 (repriced: 1.310)
Expected lossEL=PD×LGD×EAD,LGD=1−recovery rateEL = PD\times LGD\times EAD,\quad LGD = 1 - \text{recovery rate}Average credit lossPD 2%, recovery 40%, EAD 500,000 → 6,000
Unexpected loss, default onlyUL=LGD×EAD×PD (1−PD)UL = LGD\times EAD\times\sqrt{PD\,(1-PD)}Standard deviation of the loss with fixed LGD and EADSame loan → 42,000

Which formula mistakes are common?

  • Scaling VaR with a mean. The square-root-of-time rule assumes a zero mean; with a mean, the expected profit scales with T, not with √T.
  • Mixing compounding conventions. Forward prices here use continuous rates; convert a periodic rate first.
  • Semi-annual bonds. Divide the yield by m in the discount factor and in modified duration.
  • Volatility or variance. EWMA and GARCH update the variance; take the square root at the end.
  • Recovery for LGD. LGD is one minus the recovery rate.
  • Hedge ratio from price levels. The correlation and volatilities belong to price changes.

What this means for you

A formula sheet shows what you can recall; it does not teach you when to use a formula. Work through one area a week, rework every check value until it comes out right, then move on to mixed questions. Our FRM® Exam Part I course gives you practice questions with worked explanations, and the CFA® Program Level I formula sheet covers the investment side. If you are planning a retake, read what to do after failing Part I.

Frequently asked questions

Which formulas belong on an FRM® Exam Part I formula sheet?

Portfolio risk and performance ratios, probability and regression, EWMA and GARCH volatility, forward prices and hedge ratios, VaR and expected shortfall, bond duration and convexity, binomial trees, Black–Scholes–Merton and the Greeks, and expected and unexpected credit loss. Our sheet lists 38 of them.

Can I print this formula sheet?

Yes. Use your browser's print command (Ctrl+P or ⌘P). The print layout keeps the title and the formula tables and leaves out the rest of the page.

Does this formula sheet come from GARP?

No. It is our own selection of standard formulas from finance textbooks, lecture notes and papers of central banks and international bodies, in our own notation. It is not GARP material and not a list of what your exam will test.

Why does my VaR answer differ slightly from the check value?

The check uses the exact normal quantile, 2.3263 at 99%. With the table value 2.33 the result is a little higher.

Does the square-root-of-time rule always work?

Only with a zero mean and independent, identically distributed daily returns. With a mean, the expected profit scales with T rather than with the square root of T.

Sources

  1. Lorenzo Naranjo, Washington University in St. Louis (lecture notes, secondary source): Portfolios of Risky Assets — expected return, covariance, portfolio variance and the minimum-variance frontier. Accessed 29 October 2026.
  2. Lorenzo Naranjo, Washington University in St. Louis (lecture notes, secondary source): The Capital Asset Pricing Model — assumptions, beta, systematic risk and model limitations. Accessed 29 October 2026.
  3. William F. Sharpe, The Journal of Portfolio Management (author's Stanford reprint): The Sharpe Ratio — The Ratio, Time Dependence, Correlations and Related Measures (1994). Accessed 29 October 2026.
  4. Yen-Chi Chen, University of Washington (lecture notes, secondary source): STAT 535, Lecture 1: Review on Probability and Statistics — expectation, covariance and total expectation. Accessed 29 October 2026.
  5. Aswath Damodaran, NYU Stern School of Business (secondary source): A Primer on Statistics — covariance and correlation. Accessed 29 October 2026.
  6. National Institute of Standards and Technology (statistical handbook, secondary source): NIST/SEMATECH e-Handbook §1.3.5.2 — Confidence limits for the mean and the t test statistic. Accessed 30 October 2026.
  7. National Institute of Standards and Technology (statistical handbook, secondary source): NIST/SEMATECH e-Handbook §1.3.5.11 — Measures of skewness and kurtosis. Accessed 30 October 2026.
  8. Kevin Sheppard, arch (Python package) documentation (secondary source): arch.univariate.EWMAVariance — exponentially weighted moving-average (RiskMetrics) variance, default λ = 0.94. Accessed 30 October 2026.
  9. NYU Stern Volatility and Risk Institute (model documentation, secondary source): V-Lab: GARCH volatility documentation — the GARCH(1,1) recursion and long-run variance ω/(1 − α − β). Accessed 30 October 2026.
  10. Aswath Damodaran, NYU Stern School of Business (secondary source): A Primer on the Time Value of Money. Accessed 29 October 2026.
  11. Kerry Back, Hong Liu and Mark Loewenstein (academic textbook, secondary source): Pricing and Hedging Derivative Securities — Chapter 3, §§3.2–3.3: expectations, cost of carry and convenience yield. Accessed 29 October 2026.
  12. Wolfram Formula Repository (secondary source): Minimum Variance Hedge Ratio — h* = ρσ_S/σ_F for spot and futures price changes. Accessed 30 October 2026.
  13. Wolfram Formula Repository (secondary source): Commodity Contracts Optimal Number — N* = h* × position size ÷ futures contract value. Accessed 30 October 2026.
  14. Lorenzo Naranjo, Washington University in St. Louis (lecture notes, secondary source): The Impact of Dividends — put-call parity adjusted for the present value of cash dividends. Accessed 29 October 2026.
  15. Colleen Cassidy and Marianne Gizycki, Reserve Bank of Australia (methods overview, secondary source): Measuring Traded Market Risk: Value-at-risk and Backtesting Techniques — 2. Value-at-risk (November 1997). Accessed 29 October 2026.
  16. Michael Papaioannou, International Monetary Fund (secondary source): A Primer for Risk Measurement of Bonded Debt from the Perspective of a Sovereign Debt Manager — Appendix II (Working Paper 2006/195) (August 2006). Accessed 29 October 2026.
  17. Carlo Acerbi and Dirk Tasche, arXiv: On the coherence of Expected Shortfall — arXiv v5; definitions 2.1, 2.2 and 2.6; proposition 3.2 (2 May 2002). Accessed 29 October 2026.
  18. International Monetary Fund (methods overview, secondary source): A Primer on Managing Sovereign Debt-Portfolio Risks — Macaulay and modified duration (Working Paper 2018/074). Accessed 29 October 2026.
  19. CME Group (market education, secondary source): How Can You Measure Risk in Treasuries? — Value of a basis point and modified duration. Accessed 29 October 2026.
  20. International Monetary Fund: Financial Soundness Indicators: Compilation Guide — Appendix VI, §§50–55: convexity and the term structure. Accessed 29 October 2026.
  21. Kerry Back, Hong Liu and Mark Loewenstein (academic textbook, secondary source): Pricing and Hedging Derivative Securities — Chapter 4, §§4.1–4.4 and Exercise 4.8: replication, risk-neutral pricing, backward induction, tree calibration and convergence. Accessed 29 October 2026.
  22. Kerry Back, Hong Liu and Mark Loewenstein (academic textbook, secondary source): Pricing and Hedging Derivative Securities — Chapter 9: Greeks, discrete delta hedging and delta-gamma hedging. Accessed 29 October 2026.
  23. Basel Committee on Banking Supervision, Bank for International Settlements: Basel Framework CRE35.1–35.9 — IRB expected losses, eligible provisions and capital treatment. Accessed 29 October 2026.
  24. National Institute of Standards and Technology (statistical handbook, secondary source): NIST/SEMATECH e-Handbook §1.3.6.6.18 — Binomial distribution: mean and standard deviation. Accessed 29 October 2026.
  25. Global Association of Risk Professionals (GARP): FRM Exam information: format and exam topics. Accessed 26 October 2026.
  26. Roediger, H. L. & Karpicke, J. D., Psychological Science 17(3), 249–255: Test-enhanced learning: Taking memory tests improves long-term retention (2006).
  27. Dunlosky, J., Rawson, K. A., Marsh, E. J., Nathan, M. J. & Willingham, D. T., Psychological Science in the Public Interest 14(1), 4–58: Improving students’ learning with effective learning techniques (2013).

High Performer Certs is an independent study platform. It is not affiliated with, endorsed by, or sponsored by CFA Institute or the Global Association of Risk Professionals (GARP). We are not a CFA Institute Prep Provider and not a GARP Exam Preparation Provider. CFA Institute does not endorse, promote, review or warrant the accuracy or quality of our products and services. GARP does not endorse, promote, review or warrant the accuracy of our products or services. All questions and explanations are our own original material and contain no CFA Institute or GARP exam, curriculum or practice questions. CFA®, Chartered Financial Analyst®, CIPM® and GIPS® are registered trademarks owned by CFA Institute. FRM® and GARP® are trademarks owned by the Global Association of Risk Professionals.

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