FRM Exam Formula Sheet: Key Formulas You Must Know

Q: What are the most important formulas for the FRM exam?

The highest-frequency formulas are: parametric VaR and Expected Shortfall, EWMA and GARCH volatility, portfolio variance, modified duration and DV01, Black-Scholes and the Greeks, Expected Loss (PD × LGD × EAD), Basel III capital and liquidity ratios (CET1, LCR, NSFR), and risk-adjusted return measures (Sharpe, Information Ratio, RAROC).

Q: Should I memorize all FRM formulas?

Focus on understanding rather than rote memorization. Know the most frequently tested formulas cold (VaR, duration, Greeks, EL, capital ratios), understand the derivations and assumptions behind them, and practice applying them under timed conditions. Understanding when a formula applies — and when it breaks down — is more valuable than perfect recall.

Q: What is the difference between VaR and Expected Shortfall formulas?

Parametric VaR = μ + z_α × σ gives you the loss threshold at confidence level α. Expected Shortfall = μ + σ × φ(z_α)/(1-α) gives the average loss beyond VaR, where φ is the standard normal PDF. ES is always larger than VaR and better captures tail risk severity.

Q: How do I apply the square root of time rule?

VaR_T = VaR_1 × √T scales one-period VaR to a T-period horizon. This assumes returns are independent and identically distributed (i.i.d.). The rule breaks down when returns are autocorrelated, volatility clusters (as in GARCH), or there is mean reversion. Use with caution for horizons beyond a few weeks.

Q: What Basel III ratios should I know for the FRM exam?

Key ratios: CET1 ≥ 4.5% of RWA, Tier 1 ≥ 6%, Total Capital ≥ 8%, plus Capital Conservation Buffer (2.5%), Countercyclical Buffer (0-2.5%), G-SIB surcharge (1-3.5%). Leverage Ratio ≥ 3% (Tier 1 / Total Exposure). LCR ≥ 100% (HQLA / 30-day net outflows). NSFR ≥ 100% (ASF / RSF).

FRM Exam Formula Sheet: Key Formulas You Must Know

The FRM exam is quantitative and formula-heavy. While understanding concepts is paramount, being able to quickly recall and apply key formulas under time pressure can make the difference between passing and failing. Here are the essential formulas organized by topic area.

Quantitative Analysis

Portfolio variance (two assets):

σ²_p = w₁²σ₁² + w₂²σ₂² + 2w₁w₂ρ₁₂σ₁σ₂

Linear regression:

β = Cov(X,Y) / Var(X) = ρ(X,Y) · σ_Y / σ_X

EWMA volatility (RiskMetrics):

σ²_t = λσ²_{t-1} + (1-λ)r²_{t-1}, where λ typically = 0.94

GARCH(1,1):

σ²_t = ω + αr²_{t-1} + βσ²_{t-1}, where α + β < 1 for stationarity

Long-run variance = ω / (1 - α - β)

Value at Risk

Parametric VaR (normal):

VaR_α = μ + z_α · σ

For 99% confidence: z = 2.326; for 95%: z = 1.645

Portfolio VaR:

VaR_p = z_α · σ_p · √t · Portfolio Value

Expected Shortfall:

ES_α = μ + σ · φ(z_α) / (1 - α)

where φ is the standard normal PDF

VaR scaling (square root of time):

VaR_T = VaR_1 · √T (assumes i.i.d. returns)

Fixed Income & Interest Rate Risk

Duration and convexity:

Modified Duration = Macaulay Duration / (1 + y/m)

ΔP/P ≈ -D_mod · Δy + ½ · Convexity · (Δy)²

DV01 (Dollar Value of a Basis Point):

DV01 = (D_mod · P) / 10,000

Options & Derivatives

Black-Scholes call price:

C = S₀N(d₁) - Ke^{-rT}N(d₂)

d₁ = [ln(S₀/K) + (r + σ²/2)T] / (σ√T)

d₂ = d₁ - σ√T

Key Greeks:

Delta (Δ) = ∂C/∂S = N(d₁) for calls
Gamma (Γ) = ∂²C/∂S² = φ(d₁) / (S₀σ√T)
Vega (ν) = ∂C/∂σ = S₀φ(d₁)√T
Theta (Θ) = ∂C/∂T (time decay; negative for long options)
Rho (ρ) = ∂C/∂r = KTe^{-rT}N(d₂) for calls

Put-Call Parity:

C - P = S₀ - Ke^{-rT}

Credit Risk

Expected Loss:

EL = PD × LGD × EAD

Unexpected Loss (single exposure):

UL = EAD × √[PD × σ²_LGD + LGD² × PD × (1-PD)]

Merton Model (Distance to Default):

DD = [ln(V/D) + (μ - σ²_V/2)T] / (σ_V√T)

CDS pricing relationship:

CDS spread ≈ (1 - Recovery Rate) × Hazard Rate

Basel III & Regulatory Capital

Capital Ratios:

CET1 Ratio = CET1 Capital / Risk-Weighted Assets ≥ 4.5%

Tier 1 Ratio = Tier 1 Capital / RWA ≥ 6%

Total Capital Ratio = Total Capital / RWA ≥ 8%

Leverage Ratio:

Leverage Ratio = Tier 1 Capital / Total Exposure ≥ 3%

Liquidity Ratios:

LCR = HQLA / Net Cash Outflows (30 days) ≥ 100%

NSFR = Available Stable Funding / Required Stable Funding ≥ 100%

Risk-Adjusted Performance

Sharpe Ratio:

Sharpe = (R_p - R_f) / σ_p

Information Ratio:

IR = (R_p - R_b) / σ(R_p - R_b)

Treynor Ratio:

Treynor = (R_p - R_f) / β_p

RAROC:

RAROC = Risk-Adjusted Return / Economic Capital

Study Tips for Formulas

Understand derivations — Don't just memorize; know why each formula works
Practice calculations — Use your calculator extensively before exam day
Know the assumptions — When does square-root-of-time scaling fail? When does Black-Scholes break down?
Link formulas to concepts — Duration measures interest rate sensitivity; VaR measures potential loss at a confidence level
Focus on high-frequency formulas — VaR, ES, duration, Greeks, and Basel ratios appear most frequently

Review your FRM study materials regularly, practice with timed questions, and apply the study strategies that work best for your learning style. Good luck on exam day!

Frequently Asked Questions

What are the most important formulas for the FRM exam?▼

The highest-frequency formulas are: parametric VaR and Expected Shortfall, EWMA and GARCH volatility, portfolio variance, modified duration and DV01, Black-Scholes and the Greeks, Expected Loss (PD × LGD × EAD), Basel III capital and liquidity ratios (CET1, LCR, NSFR), and risk-adjusted return measures (Sharpe, Information Ratio, RAROC).

Should I memorize all FRM formulas?▼

Focus on understanding rather than rote memorization. Know the most frequently tested formulas cold (VaR, duration, Greeks, EL, capital ratios), understand the derivations and assumptions behind them, and practice applying them under timed conditions. Understanding when a formula applies — and when it breaks down — is more valuable than perfect recall.

What is the difference between VaR and Expected Shortfall formulas?▼

Parametric VaR = μ + z_α × σ gives you the loss threshold at confidence level α. Expected Shortfall = μ + σ × φ(z_α)/(1-α) gives the average loss beyond VaR, where φ is the standard normal PDF. ES is always larger than VaR and better captures tail risk severity.

How do I apply the square root of time rule?▼

VaR_T = VaR_1 × √T scales one-period VaR to a T-period horizon. This assumes returns are independent and identically distributed (i.i.d.). The rule breaks down when returns are autocorrelated, volatility clusters (as in GARCH), or there is mean reversion. Use with caution for horizons beyond a few weeks.

What Basel III ratios should I know for the FRM exam?▼

Key ratios: CET1 ≥ 4.5% of RWA, Tier 1 ≥ 6%, Total Capital ≥ 8%, plus Capital Conservation Buffer (2.5%), Countercyclical Buffer (0-2.5%), G-SIB surcharge (1-3.5%). Leverage Ratio ≥ 3% (Tier 1 / Total Exposure). LCR ≥ 100% (HQLA / 30-day net outflows). NSFR ≥ 100% (ASF / RSF).

FRM Exam Formula Sheet: Key Formulas You Must Know

FRM Exam Formula Sheet: Key Formulas You Must Know

Quantitative Analysis

Value at Risk

Fixed Income & Interest Rate Risk

Options & Derivatives

Credit Risk

Basel III & Regulatory Capital

Risk-Adjusted Performance

Study Tips for Formulas

Frequently Asked Questions

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