- How to Use This Formula Sheet
- Part I: Quantitative Analysis Formulas
- Part I: Financial Markets & Products Formulas
- Part I: Valuation & Risk Models Formulas
- Part II: Market Risk Formulas
- Part II: Credit Risk Formulas
- Part II: Operational & Liquidity Risk Formulas
- Calculator Tips for Formula Application
- Formula Memorization Strategies
The FRM exam is formula-intensive. While understanding concepts is essential, you'll need to quickly recall and apply dozens of formulas under time pressure. This comprehensive formula sheet covers every critical formula you need to know for both FRM Part I and Part II, organized by topic area with explanations and exam tips.
We've marked each formula with its exam relevance: HIGH YIELD formulas appear frequently on exams and are must-know, while MEDIUM YIELD formulas appear occasionally. Focus your memorization efforts on high-yield formulas first.
How to Use This Formula Sheet
Study Strategy: Don't just memorize formulas in isolation. For each formula, understand: (1) What it measures, (2) When to use it, (3) What each variable represents, (4) How to interpret the result, and (5) Common exam question patterns.
The FRM exam doesn't test formula memorization directly—it tests application. You'll be given scenarios and must identify which formula to use, substitute the correct values, and interpret results. That said, you won't have time to derive formulas from scratch, so committed memorization of the core formulas is essential.
Many FRM formulas require efficient calculator use. We recommend the Texas Instruments BA II Plus Professional for its superior functionality with bond calculations, cash flows, and statistics. Practice each formula type on your calculator until the keystrokes become automatic. See our calculator tips section for specific keystroke sequences.
Part I: Quantitative Analysis Formulas ~35 Formulas
Quantitative Analysis accounts for 20% of FRM Part I. This section covers probability, statistics, regression analysis, and time series—the mathematical foundation for all risk modeling. Strong command of these formulas is essential for both Part I and Part II.
Probability & Statistics Fundamentals
E(X) = Expected value of random variable X
xᵢ = Each possible outcome
P(xᵢ) = Probability of each outcome
On the exam, you'll often calculate expected returns or expected losses using this formula. Make sure probabilities sum to 1.0 before calculating.
Var(X) or σ² = Variance of X
μ = Mean (expected value) of X
E(X²) = Expected value of X squared
The second form (E(X²) - [E(X)]²) is often faster for calculations. Remember: Variance is always non-negative.
σ = Standard deviation (same units as X)
Cov(X,Y) = Covariance between X and Y
E(XY) = Expected value of the product XY
Positive covariance means X and Y move together; negative means they move opposite. Covariance of a variable with itself equals its variance: Cov(X,X) = Var(X).
ρₓᵧ = Correlation coefficient (ranges from -1 to +1)
σₓ, σᵧ = Standard deviations of X and Y
Correlation is the standardized covariance. Always between -1 and +1. ρ = 0 means no linear relationship (but could have nonlinear relationship). ρ = ±1 means perfect linear relationship.
Skewness > 0 = Right (positive) skew - long right tail
Skewness < 0 = Left (negative) skew - long left tail
Skewness = 0 = Symmetric distribution
Excess Kurtosis = Kurtosis - 3
Excess Kurtosis > 0 = Leptokurtic (fat tails) - more extreme values than normal
Excess Kurtosis < 0 = Platykurtic (thin tails) - fewer extreme values
Excess Kurtosis = 0 = Mesokurtic (normal distribution)
Financial return distributions typically exhibit positive excess kurtosis (fat tails). This means VaR models assuming normality underestimate tail risk. Critical concept for risk management!
Portfolio Statistics
E(Rₚ) = Expected portfolio return
w₁, w₂ = Weights of assets 1 and 2 (must sum to 1)
E(R₁), E(R₂) = Expected returns of assets 1 and 2
Or equivalently:
σₚ² = w₁²σ₁² + w₂²σ₂² + 2w₁w₂ρ₁₂σ₁σ₂
σₚ² = Portfolio variance
σ₁², σ₂² = Variances of assets 1 and 2
ρ₁₂ = Correlation between assets 1 and 2
This formula is heavily tested. When ρ = 1, no diversification benefit. When ρ < 1, portfolio risk is less than weighted average of individual risks. When ρ = -1, perfect hedging is possible.
w = Vector of portfolio weights
Σ = Variance-covariance matrix
w' = Transpose of weight vector
Regression Analysis
Y = Dependent variable
β₀ = Intercept (constant term)
β₁ = Slope coefficient
X = Independent variable
ε = Error term (assumed to have mean = 0)
β̂₁ = Estimated slope coefficient
X̄, Ȳ = Sample means of X and Y
Where: SST = SSR + SSE
R² = Proportion of variance explained by the model (0 to 1)
SST = Total Sum of Squares = Σ(Yᵢ - Ȳ)²
SSR = Regression Sum of Squares = Σ(Ŷᵢ - Ȳ)²
SSE = Error Sum of Squares = Σ(Yᵢ - Ŷᵢ)²
For simple linear regression, R² = ρ² (correlation squared). R² of 0.80 means 80% of variation in Y is explained by X. In risk modeling, low R² may be acceptable if the relationship is statistically significant.
t = t-statistic
β̂ = Estimated coefficient
β₀ = Hypothesized value (usually 0)
SE(β̂) = Standard error of the coefficient
Rule of thumb: |t| > 2 generally indicates statistical significance at 5% level. Reject H₀ if |t| > critical value from t-distribution with (n - k - 1) degrees of freedom.
F = F-statistic
k = Number of independent variables
n = Number of observations
Time Series Analysis
Yₜ = Value at time t
φ₁ = Autoregressive coefficient (must be |φ₁| < 1 for stationarity)
Yₜ₋₁ = Value at time t-1
If |φ₁| ≥ 1, the series is non-stationary (has a unit root). Mean reversion speed depends on φ₁: smaller |φ₁| = faster mean reversion.
μ = Mean of the series
θ₁ = Moving average coefficient
εₜ, εₜ₋₁ = Error terms at times t and t-1
σₜ² = Current variance estimate
λ = Decay factor (typically 0.94 for daily data per RiskMetrics)
σₜ₋₁² = Previous variance estimate
rₜ₋₁² = Previous period's squared return
EWMA is a special case of GARCH(1,1) where α + β = 1. Higher λ = more weight on past variance, slower response to shocks. λ = 0.94 is the RiskMetrics standard for daily data, λ = 0.97 for monthly.
ω = Constant term (long-run variance component)
α = Weight on previous squared return (ARCH term)
β = Weight on previous variance (GARCH term)
Stationarity condition: α + β < 1
Long-run variance = ω / (1 - α - β). Persistence = α + β. High persistence (close to 1) means volatility shocks decay slowly. α measures how reactive volatility is to new information.
Part I: Financial Markets & Products Formulas ~40 Formulas
Financial Markets and Products accounts for 30% of FRM Part I—the largest weight. This section covers fixed income, derivatives pricing, and foreign exchange. Mastery of these formulas is essential.
Fixed Income Fundamentals
Or: P = C × [1 - (1+y)⁻ⁿ]/y + FV/(1+y)ⁿ
P = Bond price
C = Coupon payment
y = Yield to maturity (per period)
FV = Face value (par value)
n = Number of periods to maturity
Use your BA II Plus TVM functions: N = periods, I/Y = yield per period, PMT = coupon, FV = face value, CPT PV. Remember to set payment timing (BGN vs END) appropriately.
D_Mac = Macaulay duration (in years)
t = Time to each cash flow (in years)
PV(CFₜ) = Present value of cash flow at time t
P = Bond price (sum of all PVs)
Macaulay duration is the weighted average time to receive cash flows. For a zero-coupon bond, Macaulay duration = time to maturity. For coupon bonds, duration < maturity.
Or approximately: D_Mod ≈ D_Mac / (1 + y)
D_Mod = Modified duration
k = Number of compounding periods per year
y = Yield to maturity (annual)
Or: ΔP ≈ -D_Mod × P × Δy
ΔP = Change in bond price
Δy = Change in yield (in decimal form, e.g., 0.01 for 1%)
This is a linear approximation that underestimates price increases and overestimates price decreases. For large yield changes, add convexity adjustment.
Convexity = Measures curvature of price-yield relationship
First term = Duration effect (linear)
Second term = Convexity adjustment (always positive for option-free bonds)
The convexity term is always positive for vanilla bonds—this explains why duration alone underestimates gains and overestimates losses. Bonds with higher convexity outperform in volatile rate environments.
DV01 = Dollar change in price for 1 basis point (0.01%) yield change
DV01 is extensively used in trading and hedging. To hedge a bond position, match DV01 exposure. For a portfolio, DV01 is additive across positions.
Forward Rate Agreements (FRAs)
Solving for forward rate:
F₁,₂ = [(1 + R₂)^T₂ / (1 + R₁)^T₁]^(1/(T₂-T₁)) - 1
R₁, R₂ = Spot rates for periods T₁ and T₂
F₁,₂ = Forward rate from T₁ to T₂
Derivatives: Futures & Forwards
F₀ = Forward/futures price
S₀ = Current spot price
r = Risk-free rate (continuously compounded)
T = Time to maturity (in years)
I = Present value of known income during contract life
q = Continuous dividend yield or convenience yield
This formula applies to stock indices (q = dividend yield), currencies (q = foreign risk-free rate), and commodities (q = convenience yield - storage cost).
Where: c = r + u - y (cost of carry)
c = Cost of carry
r = Risk-free rate
u = Storage cost (as % of spot price)
y = Convenience yield
Derivatives: Options
Or: c + Ke^(-rT) = p + S₀
c = European call premium
p = European put premium
K = Strike price
S₀ = Current stock price
Put-call parity is frequently tested. If the relationship doesn't hold, there's an arbitrage opportunity. Can be rearranged to solve for any variable.
d₁ = [ln(S₀/K) + (r + σ²/2)T] / (σ√T)
d₂ = d₁ - σ√T
N(d) = Cumulative standard normal distribution function
σ = Volatility of the underlying asset
S₀N(d₁) = Delta-adjusted stock position
Ke^(-rT)N(d₂) = PV of expected strike payment
You won't need to calculate BSM from scratch on the exam, but you MUST understand: (1) how each input affects option price, (2) N(d₁) = Delta for calls, (3) N(d₂) = risk-neutral probability of exercise.
Can also derive put price from call price using put-call parity: p = c - S₀ + Ke^(-rT)
Option Greeks
Δ_put = N(d₁) - 1 = -N(-d₁) [ranges from -1 to 0]
Delta = Change in option price for $1 change in underlying
ATM call delta ≈ 0.5
ATM put delta ≈ -0.5
Delta also represents the hedge ratio (number of shares to short to delta-hedge a long call) and approximates the probability the option finishes in-the-money.
Where N'(d₁) = (1/√2π) × e^(-d₁²/2)
Gamma = Rate of change of delta with respect to underlying price
Gamma is highest for ATM options near expiration
Long options have positive gamma (benefit from large moves). Short options have negative gamma (hurt by large moves). Gamma risk is highest for short-dated ATM options.
Vega = Change in option price for 1% change in volatility
Vega is always positive for both calls and puts
Vega is highest for ATM options with longer time to expiration
Θ_put = -[S₀N'(d₁)σ / (2√T)] + rKe^(-rT)N(-d₂)
Theta = Change in option price for one day passage of time
Theta is usually negative (time decay hurts long option holders)
ρ_put = -KTe^(-rT)N(-d₂)
Rho = Change in option price for 1% change in interest rate
Rho is positive for calls (higher rates increase call value)
Rho is negative for puts (higher rates decrease put value)
Swaps
V_swap = Value of swap (to fixed-rate payer)
B_floating = Value of floating-rate bond (equals notional at reset dates)
B_fixed = Value of fixed-rate bond (PV of fixed payments + notional)
At inception, swap value = 0 (B_floating = B_fixed). Fixed rate is set so that PV of fixed payments equals PV of expected floating payments. After inception, value changes as rates move.
c = Fixed rate (par swap rate)
Z_n = Discount factor for final payment date
Σ(Z_i) = Sum of discount factors for all payment dates
Part I: Valuation & Risk Models Formulas ~30 Formulas
Valuation and Risk Models accounts for 30% of FRM Part I. This section covers Value at Risk (VaR), expected shortfall, and risk model implementation—the core of what makes an FRM.
Value at Risk (VaR)
Or for portfolio: VaR = |z_α| × σ × Portfolio Value
VaR = Value at Risk (maximum loss at confidence level)
μ = Expected return (often assumed to be 0 for short horizons)
z_α = Standard normal quantile for confidence level
σ = Standard deviation (volatility)
90% confidence: z = 1.282 | 95% confidence: z = 1.645 | 99% confidence: z = 2.326
VaR_T = VaR over T periods
VaR_1 = VaR over 1 period
T = Number of periods
This assumes i.i.d. returns and is only valid for short time horizons. 10-day VaR = 1-day VaR × √10 ≈ 1-day VaR × 3.162. Basel requires 10-day VaR for market risk capital.
VaR_p = Portfolio VaR
VaR₁, VaR₂ = Individual position VaRs
ρ₁₂ = Correlation between assets
When ρ = 1: VaR_p = VaR₁ + VaR₂ (no diversification). When ρ = -1: VaR_p = |VaR₁ - VaR₂| (maximum diversification). When ρ = 0: VaR_p = √(VaR₁² + VaR₂²)
Or: MVaRᵢ = VaR_p × βᵢ / P
MVaRᵢ = Marginal VaR of asset i
βᵢ = Beta of asset i with respect to portfolio
Note: Σ CVaRᵢ = VaR_p (component VaRs sum to total VaR)
CVaRᵢ = Component VaR of asset i
wᵢ = Weight of asset i in portfolio
For Normal Distribution:
ES = σ × φ(z_α) / α = σ × φ(z_α) / (1 - confidence)
ES = Expected Shortfall (average loss when VaR is exceeded)
φ(z_α) = Standard normal PDF at the VaR threshold
α = Tail probability (e.g., 0.01 for 99% VaR)
ES is always greater than VaR at the same confidence level. ES is coherent (satisfies subadditivity) while VaR is not. Basel III/IV uses ES at 97.5% confidence for market risk capital.
Binomial Option Pricing
d = e^(-σ√Δt) = 1/u (down move factor)
p = (e^(rΔt) - d) / (u - d) (risk-neutral probability)
u = Multiplicative up factor
d = Multiplicative down factor
p = Risk-neutral probability of up move
Δt = Time step (T/n where n = number of steps)
f = Option value today
f_u = Option value if stock goes up
f_d = Option value if stock goes down
Work backwards through the tree from expiration. For American options, at each node compare continuation value with immediate exercise value, take the maximum.
Part II: Market Risk Formulas ~25 Formulas
Market Risk Measurement and Management accounts for 20% of FRM Part II. This section covers advanced VaR techniques, backtesting, stress testing, and the regulatory framework for market risk capital.
Advanced VaR Methods
Example: 99% VaR with 500 days = 5th worst return
n = Number of historical observations
α = Significance level (e.g., 0.01 for 99% VaR)
HS VaR is non-parametric—no distribution assumption. Advantages: captures fat tails naturally. Disadvantages: dependent on historical period chosen, slow to adapt to changing volatility.
wᵢ = Weight assigned to the i-th most recent observation
λ = Decay factor (typically 0.94-0.99)
n = Total number of observations
Backtesting
Distributed as χ² with 1 degree of freedom
p = Expected failure rate (e.g., 0.01 for 99% VaR)
x = Number of actual exceptions
T = Number of observations in backtest period
Kupiec tests whether the number of exceptions is consistent with the VaR confidence level. Too few exceptions may indicate VaR is too conservative; too many indicates VaR underestimates risk.
Green Zone: 0-4 exceptions (no penalty)
Yellow Zone: 5-9 exceptions (progressively higher capital multiplier)
Red Zone: 10+ exceptions (automatic 4x multiplier)
Expected exceptions = 250 × 0.01 = 2.5. Green zone allows up to 4 exceptions (approximately 2 standard deviations). Memorize these thresholds!
Part II: Credit Risk Formulas ~30 Formulas
Credit Risk Measurement and Management accounts for 20% of FRM Part II. This section covers default probability, loss given default, credit VaR, and credit derivatives.
Credit Risk Fundamentals
EL = Expected Loss
PD = Probability of Default
LGD = Loss Given Default (1 - Recovery Rate)
EAD = Exposure at Default
EL is covered by loan pricing and provisions. Unexpected Loss (UL) requires economic capital. This formula is fundamental to all credit risk analysis.
Simplified (when LGD is constant):
UL = EAD × LGD × √[PD × (1-PD)]
UL = Unexpected Loss (standard deviation of loss)
σ²_LGD = Variance of LGD
With constant hazard rate:
PD_cumulative(T) = 1 - e^(-λT)
λ = Hazard rate (constant intensity of default)
PD₁, PD₂, etc. = Marginal default probabilities for each period
λ = Hazard rate (default intensity)
s = Credit spread over risk-free rate
This approximation assumes risk-neutral pricing. If spread = 2% and recovery = 40%, then λ ≈ 2% / 60% = 3.33%.
Credit VaR & Portfolio Models
PD = N(-DD)
DD = Distance to Default (number of std devs from default)
V = Market value of firm's assets
D = Default point (face value of debt)
μ = Expected return on assets
σ = Volatility of assets
In Merton model, equity = call option on firm's assets. Higher DD = lower PD. The model assumes firm defaults when asset value falls below debt value at maturity.
Credit VaR = (WCDR - PD) × LGD × EAD
WCDR = Worst Case Default Rate at 99.9% confidence
ρ = Asset correlation
N, N⁻¹ = Standard normal CDF and inverse CDF
This is the foundation of Basel IRB capital requirements. Higher correlation = higher WCDR = more capital required. Basel sets ρ based on exposure type (corporate vs. retail).
Credit Derivatives
CDS Spread = Annual premium paid for credit protection
CDS spreads are market-implied credit risk measures. If CDS spread = 200 bps and recovery = 40%, implied PD ≈ 200/(100-40) = 3.33%.
CVA = Credit Valuation Adjustment (price of counterparty credit risk)
EE(tᵢ) = Expected Exposure at time tᵢ
PD(tᵢ₋₁, tᵢ) = Marginal default probability for period
DF(tᵢ) = Discount factor
Part II: Operational & Liquidity Risk Formulas ~20 Formulas
Operational Risk (20%) and Liquidity & Treasury Risk (15%) together account for 35% of FRM Part II. These sections cover Basel capital requirements, liquidity ratios, and operational risk modeling.
Basel Capital Requirements
Basel III Minimum: CET1 ≥ 4.5%, Tier 1 ≥ 6%, Total ≥ 8%
CET1 = Common Equity Tier 1
Tier 1 = CET1 + Additional Tier 1
Total Capital = Tier 1 + Tier 2
With capital conservation buffer (2.5%) and potential countercyclical buffer (0-2.5%), effective CET1 requirement is 7-9.5%. G-SIBs face additional surcharges of 1-2.5%.
Risk weights: Sovereigns (0-150%), Banks (20-150%), Corporates (20-150%), Retail (75%), Residential mortgages (35%)
Liquidity Risk Ratios
HQLA = High Quality Liquid Assets (Level 1, Level 2A, Level 2B)
Net Cash Outflows = Total expected outflows - Min(expected inflows, 75% of outflows)
Level 1 HQLA: cash, central bank reserves, government bonds (no haircut). Level 2A: corporate bonds AA- or better (15% haircut). Level 2B: lower-rated corporate bonds, equities (50% haircut, max 15% of HQLA).
ASF = Liabilities weighted by stability (equity 100%, retail deposits 90-95%, wholesale funding 50%)
RSF = Assets weighted by liquidity (cash 0%, loans 65-100%, illiquid assets 100%)
Operational Risk Capital
BIC = Business Indicator Component
ILM = Internal Loss Multiplier
Business Indicator = Sum of ILDC + SC + FC
ILDC = Interest, Lease, Dividend Component
SC = Services Component
FC = Financial Component
Calculator Tips for Formula Application
Efficient calculator use can save significant time on the FRM exam. Here are essential techniques for the Texas Instruments BA II Plus.
Key Calculator Functions for FRM
| Function | Keystrokes (BA II Plus) | Application |
|---|---|---|
| Natural Log (ln) | [LN] | Black-Scholes d₁, continuous compounding |
| e^x | [2ND][LN] | Continuous compounding, forward prices |
| Square Root | [2ND][x²] | Volatility, standard deviation, VaR scaling |
| y^x (Power) | [yˣ] | Compounding, binomial trees |
| 1/x (Reciprocal) | [1/x] | Yields, rates |
| +/- (Change Sign) | [+/-] | Cash flows, PV calculations |
Store intermediate results: Use STO and RCL functions to save calculation steps. Chain calculations: Leave results on screen and continue operating on them. Practice TVM problems: Bond pricing questions should take under 60 seconds with proper technique. Check your settings: Verify P/Y=1, C/Y=1, and END mode before starting.
Formula Memorization Strategies
With 150+ formulas to learn, a strategic approach to memorization is essential. Here are proven techniques used by successful FRM candidates:
Prioritization Strategy
- Master HIGH YIELD formulas first — These appear on nearly every exam and are tested in multiple ways
- Understand derivations — Knowing where formulas come from helps you reconstruct them under pressure
- Group related formulas — Learn VaR formulas together, Greeks together, credit risk formulas together
- Practice application, not just recall — The exam tests whether you can use formulas in scenarios
Active Learning Techniques
- Flashcards: Create cards with formula on front, variables and application on back
- Practice problems: Work through problems daily to reinforce formula application
- Teach the material: Explaining formulas to others (or yourself) deepens understanding
- Spaced repetition: Review formulas at increasing intervals (1 day, 3 days, 1 week, 2 weeks)
- Write formulas by hand: Physical writing improves retention over typing
Exam Day Tips
- Brain dump: In the first 5 minutes, write down any formulas you're worried about forgetting
- Read carefully: Many errors come from misidentifying which formula to use, not calculation mistakes
- Check units: Ensure time periods, rates, and other units are consistent before calculating
- Estimate first: Before calculating, estimate the answer to catch major errors
Formula memorization is necessary but not sufficient for FRM success. Focus on understanding what each formula measures, when to apply it, and how to interpret results. The exam rewards candidates who can think through problems conceptually, not just plug numbers into equations. Use this formula sheet as a reference during your studies, but aim to internalize the material so deeply that you rarely need to consult it.
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