Free GARP FRM Part I Formula Sheet (2026)

Every FRM Part I formula you need on the test, grouped by topic, rendered with full math notation. 81 formulas across 4 topics, calibrated to the 2026 syllabus. Free forever, no signup required.

81 Formulas
4 Topics
2026 Syllabus
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All FRM Part I Formulas

Foundations of Risk Management 11 items
Annual CDS premium payment
Premium=s×N\text{Premium} = s \times N — s = CDS spread (decimal), N = notional protected. Example: 250 bps on $800M = $20M/year
Expected loss on a credit exposure
EL=PD×LGD×EADEL = PD \times LGD \times EAD — PD = probability of default, LGD = loss given default, EAD = exposure at default
CAPM expected return
E[Ri]=Rf+βi(E[Rm]−Rf)E[R_i] = R_f + \beta_i (E[R_m] - R_f) — R_f = risk-free rate, β_i = asset beta, E[R_m] = expected market return
CDS spread approximation
s≈PD×LGDs \approx PD \times LGD — s = annualized CDS spread, PD = annual probability of default, LGD = loss given default (1 − recovery)
Beta of an asset
βi=Cov(Ri,Rm)σm2\beta_i = \frac{\text{Cov}(R_i, R_m)}{\sigma_m^2} — Cov(R_i,R_m) = covariance of asset and market returns, σ_m² = market return variance
Jensen's alpha
α=Rˉp−[Rf+βp(Rˉm−Rf)]\alpha = \bar R_p - [R_f + \beta_p(\bar R_m - R_f)] — R̄_p = realized portfolio return, R_f = risk-free rate, β_p = portfolio beta, R̄_m = market return
Aggregate firm-wide economic capital variance
σfirm2=∑iσi2+2∑i<jρijσiσj\sigma_{firm}^2 = \sum_i \sigma_i^2 + 2\sum_{i<j} \rho_{ij}\sigma_i\sigma_j — σ_i = standalone EC of silo i, ρ_ij = pairwise correlation between silos i and j
Factor model return decomposition
Ri=E[Ri]+βi,1f1+⋯+βi,kfk+εiR_i = E[R_i] + \beta_{i,1}f_1 + \cdots + \beta_{i,k}f_k + \varepsilon_i — f_k = zero-mean factor surprise, β_{i,k} = factor loading, ε_i = idiosyncratic noise
Risk-adjusted return on capital (RAROC)
RAROC=Risk-adjusted returnEconomic capitalRAROC = \dfrac{\text{Risk-adjusted return}}{\text{Economic capital}} — numerator = return net of EL and funding costs, denominator = economic capital allocated
Fama-French three-factor model
E[Ri]−Rf=βi,M(RM−Rf)+βi,SMB⋅SMB+βi,HML⋅HMLE[R_i] - R_f = \beta_{i,M}(R_M - R_f) + \beta_{i,SMB}\cdot SMB + \beta_{i,HML}\cdot HML — R_M = market return, SMB = small minus big, HML = high minus low book-to-market
APT multifactor expected return
E[Ri]=Rf+βi,1λ1+βi,2λ2+⋯+βi,kλkE[R_i] = R_f + \beta_{i,1}\lambda_1 + \beta_{i,2}\lambda_2 + \cdots + \beta_{i,k}\lambda_k — R_f = risk-free rate, β_{i,k} = asset i loading on factor k, λ_k = factor k risk premium
Quantitative Analysis 23 items
AR(1) mean-reverting level
μ=α1−ϕ\mu = \dfrac{\alpha}{1 - \phi} — α = intercept, φ = AR(1) coefficient with |φ| < 1, μ = unconditional long-run mean
Continuously compounded (log) return
r=ln⁡(Pt/Pt−1)=ln⁡(1+R)r = \ln(P_t / P_{t-1}) = \ln(1 + R) — P_t = price at time t, P_{t-1} = prior price, R = simple return over the period
Sample variance with Bessel's correction
s2=1n−1∑i=1n(Xi−Xˉ)2s^2 = \frac{1}{n-1} \sum_{i=1}^{n} (X_i - \bar{X})^2 — X_i = observation i, Xˉ\bar{X} = sample mean, n = sample size
Bayes' rule
P(A∣B)=P(B∣A) P(A)P(B)P(A \mid B) = \dfrac{P(B \mid A) \, P(A)}{P(B)} — P(A) = prior, P(B|A) = likelihood, P(B) = marginal evidence
Expected value of a lognormal random variable
E[X]=eμ+σ2/2E[X] = e^{\mu + \sigma^2/2} — μ = mean of ln(X), σ² = variance of ln(X); the σ²/2 is the convexity adjustment
F1 score for binary classification
F1=2⋅P⋅RP+RF_1 = \frac{2 \cdot P \cdot R}{P + R} — P = precision = TP/(TP+FP), R = recall = TP/(TP+FN), TP/FP/FN from confusion matrix
Law of total probability (two-event partition)
P(B)=P(B∣A)P(A)+P(B∣Ac)P(Ac)P(B) = P(B \mid A) P(A) + P(B \mid A^c) P(A^c) — A and A^c partition the sample space, P(B|·) = conditional probability of B
Adjusted R-squared
Rˉ2=1−(1−R2)⋅n−1n−k−1\bar{R}^2 = 1 - (1 - R^2) \cdot \frac{n - 1}{n - k - 1} — R² = unadjusted R-squared, n = sample size, k = number of regressors
OLS slope estimator (single regressor)
β^1=Cov(X,Y)Var(X)\hat{\beta}_1 = \frac{\text{Cov}(X, Y)}{\text{Var}(X)} — Cov(X,Y) = sample covariance of X and Y, Var(X) = sample variance of X
AR(1) h-step-ahead forecast
E[Yt+h]=μ+ϕh(Yt−μ)E[Y_{t+h}] = \mu + \phi^h (Y_t - \mu) — μ = mean-reverting level, φ = AR coefficient, h = forecast horizon, Y_t = current value
Pearson correlation coefficient
ρX,Y=Cov(X,Y)σXσY\rho_{X,Y} = \dfrac{\text{Cov}(X,Y)}{\sigma_X \sigma_Y} — Cov = covariance of X and Y; σ_X, σ_Y = standard deviations; ρ ∈ [-1, 1]
Square-root-of-time volatility scaling
σT=σ1T\sigma_T = \sigma_1 \sqrt{T} — σ_T = T-period volatility, σ_1 = one-period volatility, T = number of periods (assumes iid returns)
Variance of a linear transformation
Var(aX+b)=a2 Var(X)\text{Var}(aX + b) = a^2 \, \text{Var}(X) — X = random variable, a = scale factor, b = shift constant
Omitted variable bias in a short regression coefficient
bias(β^1)=β2⋅δ\text{bias}(\hat{\beta}_1) = \beta_2 \cdot \delta — β₂ = true coefficient on omitted X₂, δ = slope from regressing X₂ on included X₁
Variance of a two-asset weighted sum
Var(aX+bY)=a2σX2+b2σY2+2ab Cov(X,Y)\text{Var}(aX + bY) = a^2 \sigma_X^2 + b^2 \sigma_Y^2 + 2ab\,\text{Cov}(X,Y) — a, b = weights; σ = standard deviation; Cov = covariance
Variance inflation factor for regressor j
VIFj=11−Rj2\text{VIF}_j = \frac{1}{1 - R_j^2} — R²_j = R-squared from regressing X_j on the other regressors; VIF > 10 commonly flags collinearity
One-sample t-statistic for testing a mean
t=Xˉ−μ0s/nt = \frac{\bar{X} - \mu_0}{s / \sqrt{n}} — Xˉ\bar{X} = sample mean, μ0\mu_0 = hypothesized mean, s = sample std dev, n = sample size
Box-Pierce Q-statistic for residual autocorrelation
QBP=T∑k=1mρ^k2Q_{BP} = T \sum_{k=1}^{m} \hat{\rho}_k^2 — T = sample size, m = number of lags tested, ρ̂_k = sample autocorrelation of residuals at lag k
LASSO regression penalized objective
min⁡β∑i(yi−y^i)2+λ∑j∣βj∣\min_\beta \sum_i (y_i - \hat{y}_i)^2 + \lambda \sum_j |\beta_j| — y_i = observed, ŷ_i = predicted, β_j = coefficient j, λ = L1 penalty strength
Ridge regression penalized objective
min⁡β∑i(yi−y^i)2+λ∑jβj2\min_\beta \sum_i (y_i - \hat{y}_i)^2 + \lambda \sum_j \beta_j^2 — y_i = observed, ŷ_i = predicted, β_j = coefficient j, λ = L2 penalty strength
Two-sample t-statistic for difference of means
t=Xˉ1−Xˉ2s12/n1+s22/n2t = \frac{\bar{X}_1 - \bar{X}_2}{\sqrt{s_1^2/n_1 + s_2^2/n_2}} — Xˉi\bar{X}_i = sample mean i, sis_i = sample std dev i, nin_i = sample size i
Jarque-Bera test statistic
JB=n6[S2+(K−3)24]JB = \frac{n}{6}\left[S^2 + \frac{(K-3)^2}{4}\right] — n = sample size, S = sample skewness, K = sample kurtosis; chi-square with 2 dof under normality
F-statistic for joint hypothesis test
F=(SSRR−SSRU)/qSSRU/(n−k−1)F = \frac{(\text{SSR}_R - \text{SSR}_U) / q}{\text{SSR}_U / (n - k - 1)} — SSR_R = restricted SSR, SSR_U = unrestricted SSR, q = restrictions, n = sample, k = regressors
Financial Markets and Products 25 items
European call lower bound (no dividend)
c≥max⁡(S0−Ke−rT,  0)c \geq \max(S_0 - K e^{-rT},\; 0) — c = call price, S₀ = spot, K = strike, r = risk-free rate, T = time to maturity
European put lower bound (no dividend)
p≥max⁡(Ke−rT−S0,  0)p \geq \max(K e^{-rT} - S_0,\; 0) — p = put price, K = strike, r = risk-free rate, T = time to maturity, S₀ = spot
Continuous-to-discrete compounding conversion
Rc=mln⁡(1+Rm/m)R_c = m \ln(1 + R_m/m) — R_c = continuous rate, R_m = rate compounded m times per year, m = compounding frequency
Bull call spread maximum profit
Max profit=K2−K1−D\text{Max profit} = K_2 - K_1 - D — K_1 = long (lower) call strike, K_2 = short (higher) call strike, D = net debit paid
Optimal number of futures contracts (untailed)
N∗=h∗⋅QA/QFN^* = h^* \cdot Q_A / Q_F — h* = optimal hedge ratio, Q_A = size of cash exposure, Q_F = futures contract size
Cost-of-carry forward price for a commodity
F0=S0e(r+u−y)TF_0 = S_0 e^{(r+u-y)T} — S₀ = spot, r = risk-free rate, u = storage cost rate, y = convenience yield, T = time to delivery
Forward price on a non-income asset
F0=S0erTF_0 = S_0 e^{rT} — S₀ = spot price, r = continuously compounded risk-free rate, T = time to delivery in years
Relative purchasing power parity
(S1−S0)/S0≈id−if(S_1 - S_0)/S_0 \approx i_d - i_f — S_0 = current spot, S_1 = future spot, i_d = domestic inflation, i_f = foreign inflation
Put-call parity for European options (no dividend)
c+Ke−rT=p+S0c + K e^{-rT} = p + S_0 — c = call price, p = put price, K = strike, r = risk-free rate, T = time to maturity, S₀ = spot price
Long straddle breakeven prices
S∗=K±(c+p)S^* = K \pm (c + p) — K = common strike, c = call premium, p = put premium; profit if |S(T) − K| > c + p
Bond price change with duration and convexity
ΔP/P≈−DmodΔy+12C(Δy)2\Delta P/P \approx -D_{mod} \Delta y + \tfrac{1}{2} C (\Delta y)^2 — D_mod = modified duration, C = convexity, Δy = yield change
Uncovered interest rate parity expected future spot
E[ST]=S⋅(1+rd)/(1+rf)E[S_T] = S \cdot (1 + r_d) / (1 + r_f) — E[S_T] = expected future spot, S = current spot, r_d = domestic rate, r_f = foreign rate
Conditional prepayment rate from single monthly mortality
CPR=1−(1−SMM)12CPR = 1 - (1 - SMM)^{12} — SMM = single monthly mortality, CPR = annualized conditional prepayment rate
Forward price with continuous income yield
F0=S0e(r−q)TF_0 = S_0 e^{(r-q)T} — S₀ = spot, r = risk-free rate, q = continuous income yield (dividend or foreign rate), T = time to delivery
Modified duration from Macaulay duration
Dmod=DMac/(1+y/m)D_{mod} = D_{Mac} / (1 + y/m) — D_Mac = Macaulay duration, y = yield, m = compounding periods per year
Cheapest-to-deliver delivery cost
Cost=Quoted price−(F×CF)\text{Cost} = \text{Quoted price} - (F \times CF) — Quoted price = clean bond price, F = futures settlement, CF = conversion factor
Tailed number of futures contracts
Ntailed∗=h∗⋅VA/VFN^*_{\text{tailed}} = h^* \cdot V_A / V_F — V_A = Q_A × S (dollar value of exposure), V_F = Q_F × F (dollar value of one futures contract)
Number of index futures to adjust portfolio beta
N∗=(β∗−β0)⋅P/FN^* = (\beta^* - \beta_0) \cdot P / F — β* = target beta, β_0 = current portfolio beta, P = portfolio value, F = index futures contract value
Covered interest rate parity forward rate
F=S⋅(1+rd)/(1+rf)F = S \cdot (1 + r_d) / (1 + r_f) — F = forward rate, S = spot, r_d = domestic interest rate, r_f = foreign interest rate (matched to horizon)
Duration-based futures hedge ratio
N∗=P⋅DPVF⋅DFN^* = \frac{P \cdot D_P}{V_F \cdot D_F} — P = portfolio value, D_P = portfolio duration, V_F = futures contract value, D_F = CTD bond duration
Swap value to fixed-receiver (two-bond method)
V=Bfix−BflV = B_{\text{fix}} - B_{\text{fl}} — B_fix = PV of fixed-rate bond (coupons + notional), B_fl = PV of floating-rate bond (next coupon + notional)
Fisher relation linking nominal and real rates
(1+r)=(1+ρ)(1+i)(1 + r) = (1 + \rho)(1 + i) — r = nominal rate, ρ = real rate, i = expected inflation
Long forward payoff at maturity
Payoff=ST−K\text{Payoff} = S_T - K — S_T = spot price at maturity, K = forward strike price
Variance swap payoff at maturity
Payoff=Nvar×(σrealized2−Kvar2)\text{Payoff} = N_{var} \times (\sigma_{realized}^2 - K_{var}^2) — N_var = variance notional, σ²_realized = realized variance, K_var = strike volatility
Long put option payoff at maturity
Payoff=max⁡(K−ST,0)\text{Payoff} = \max(K - S_T, 0) — K = strike price, S_T = spot at maturity
Valuation and Risk Models 22 items
Forward rate between two dates (continuous compounding)
f(t1,t2)=s2t2−s1t1t2−t1f(t_1, t_2) = \frac{s_2 t_2 - s_1 t_1}{t_2 - t_1} — s1, s2 = spot rates to t1 and t2, f = forward rate from t1 to t2
Black-Scholes-Merton call price (no dividends)
C=S0N(d1)−Ke−rTN(d2)C = S_0 N(d_1) - K e^{-rT} N(d_2) — S₀ = spot, K = strike, r = risk-free rate, T = maturity, N = standard normal CDF
Cox-Ross-Rubinstein up and down factors
u=eσΔt, d=1/uu = e^{\sigma \sqrt{\Delta t}},\ d = 1/u — σ = annualized volatility of the underlying, Δt = step length in years
BSM gamma for a European option (no dividends)
Γ=ϕ(d1)/(S0σT)\Gamma = \phi(d_1) / (S_0 \sigma \sqrt{T}) — φ = standard normal density, S₀ = spot, σ = volatility, T = maturity
d₁ in the Black-Scholes-Merton formula (no dividends)
d1=ln⁡(S0/K)+(r+σ2/2)TσTd_1 = \frac{\ln(S_0/K) + (r + \sigma^2/2)T}{\sigma\sqrt{T}} — S₀ = spot, K = strike, r = rate, σ = volatility, T = maturity
Unexpected loss on a single credit exposure (binomial default)
UL=EAD×LGD×PD(1−PD)UL = EAD \times LGD \times \sqrt{PD(1-PD)} — EAD = exposure at default, LGD = loss given default, PD = probability of default
Risk-neutral probability in a binomial tree
p=erΔt−du−dp = \frac{e^{r \Delta t} - d}{u - d} — r = risk-free rate, Δt = step length, u = up-factor, d = down-factor
Unconditional default probability under constant hazard rate
P(τ≤t)=1−e−λtP(\tau \le t) = 1 - e^{-\lambda t} — λ = constant hazard rate (annual default intensity), t = horizon in years, τ = default time
KR01 from key-rate duration
KR01k=KRDk⋅P⋅10−4\text{KR01}_k = \text{KRD}_k \cdot P \cdot 10^{-4} — KRD_k = key-rate duration at rate k, P = bond price; dollar change per 1 bp shift
Survival probability under constant hazard rate
P(τ>t)=e−λtP(\tau > t) = e^{-\lambda t} — λ = constant hazard rate, t = horizon in years, τ = default time
Bond price from discount factors
P=∑i=1nc⋅d(ti)+P⋅d(tn)P = \sum_{i=1}^{n} c \cdot d(t_i) + P \cdot d(t_n) — c = coupon, P = principal, d(t) = discount factor for time t, n = number of payments
Basel total market-risk capital with stressed VaR
K=max⁡(VaR,VaR‾60d)+max⁡(sVaR,sVaR‾60d)K = \max(VaR, \overline{VaR}_{60d}) + \max(sVaR, \overline{sVaR}_{60d}) — VaR = current VaR, sVaR = stressed VaR, 60d-bar = 60-day average
Expected Shortfall under normal returns
ESc=μ+σ⋅ϕ(zc)1−c\text{ES}_c = \mu + \sigma \cdot \frac{\phi(z_c)}{1-c} — μ = mean, σ = std dev, φ(z_c) = standard-normal density at z_c, c = confidence level
Effective duration from shocked prices
Deff=P−−P+2P0ΔyD_{\text{eff}} = \frac{P_- - P_+}{2 P_0 \Delta y} — P₋ = price after yield drop, P₊ = price after yield rise, P₀ = base price, Δy = shock size
Parametric (delta-normal) VaR
VaRc=−μ+zcσ\text{VaR}_c = -\mu + z_c \sigma — μ = expected return, σ = return standard deviation, z_c = one-sided standard-normal quantile at confidence c
Loss given default from recovery rate
LGD=1−RR\text{LGD} = 1 - \text{RR} — LGD = loss given default (fraction of exposure lost), RR = recovery rate (fraction of par recovered after default)
Macaulay duration
Dmac=∑i=1nti⋅PV(Ci)PD_{\text{mac}} = \sum_{i=1}^{n} t_i \cdot \frac{PV(C_i)}{P} — t_i = time to cash flow i, PV(C_i) = present value of cash flow i, P = bond price
Effective annual rate from nominal rate
EAR=(1+r/n)n−1\text{EAR} = (1 + r/n)^n - 1 — r = nominal annual rate, n = compounding periods per year
Dollar value of an 01 (DV01)
DV01≈Dmod×P×10−4\text{DV01} \approx D_{\text{mod}} \times P \times 10^{-4} — D_mod = modified duration, P = bond price; dollar price change for a 1 bp yield change
GARCH(1,1) variance recursion
σt2=ω+αrt−12+βσt−12\sigma_t^2 = \omega + \alpha r_{t-1}^2 + \beta \sigma_{t-1}^2 — ω = constant, α = ARCH weight on lagged squared return, β = GARCH weight on lagged variance
Conditional one-year default probability given survival
P(default in next year∣survived)=1−e−λP(\text{default in next year} \mid \text{survived}) = 1 - e^{-\lambda} — λ = constant annual hazard rate; value is the same each year under constant intensity
Vasicek single-factor credit VaR (Basel IRB)
VaRc=Φ(Φ−1(PD)+ρ Φ−1(c)1−ρ)VaR_c = \Phi\left(\frac{\Phi^{-1}(PD) + \sqrt{\rho}\,\Phi^{-1}(c)}{\sqrt{1-\rho}}\right) — PD = default probability, ρ = asset correlation, c = confidence level, Φ = standard-normal CDF

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