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All FRM Part I Formulas

Foundations of Risk Management 12 items

Annual CDS premium payment

Premium = s \times N

— s = CDS spread (decimal), N = notional protected. Example: 250 bps on $800M = $20M/year

Expected loss on a credit exposure

E L = P D \times L G D \times E A D

— PD = probability of default, LGD = loss given default, EAD = exposure at default

CAPM expected return

E [R_{i}] = R_{f} + β_{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 \approx P D \times L G D

— s = annualized CDS spread, PD = annual probability of default, LGD = loss given default (1 − recovery)

Beta of an asset

β_{i} = \frac{Cov ( R _{i} , R _{m} )}{σ _{m}^{2}}

— Cov(R_i,R_m) = covariance of asset and market returns, σ_m² = market return variance

Jensen's alpha

α = R ˉ_{p} - [R_{f} + β_{p} (R ˉ_{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

σ_{f i r m}^{2} = \sum_{i} σ_{i}^{2} + 2 \sum_{i < j} ρ_{ij} σ_{i} σ_{j}

— σ_i = standalone EC of silo i, ρ_ij = pairwise correlation between silos i and j

Factor model return decomposition

R_{i} = E [R_{i}] + β_{i, 1} f_{1} + \dots + β_{i, k} f_{k} + ε_{i}

— f_k = zero-mean factor surprise, β_{i,k} = factor loading, ε_i = idiosyncratic noise

Risk-adjusted return on capital (RAROC)

R A R O C = \frac{Risk-adjusted return}{Economic capital}

— numerator = return net of EL and funding costs, denominator = economic capital allocated

Fama-French three-factor model

E [R_{i}] - R_{f} = β_{i, M} (R_{M} - R_{f}) + β_{i, S M B} \cdot S M B + β_{i, H M L} \cdot H M L

— R_M = market return, SMB = small minus big, HML = high minus low book-to-market

APT multifactor expected return

E [R_{i}] = R_{f} + β_{i, 1} λ_{1} + β_{i, 2} λ_{2} + \dots + β_{i, k} λ_{k}

— R_f = risk-free rate, β_{i,k} = asset i loading on factor k, λ_k = factor k risk premium

Treynor ratio

T = \frac{R _{p} - R _{f}}{β _{p}}

, excess return per unit of systematic risk (beta). Preferred over the Sharpe ratio for ranking well-diversified sub-portfolios where idiosyncratic risk is already diversified away.

Quantitative Analysis 23 items

AR(1) mean-reverting level

μ = \frac{α}{1 - ϕ}

— α = intercept, φ = AR(1) coefficient with |φ| < 1, μ = unconditional long-run mean

Continuously compounded (log) return

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

s^{2} = \frac{1}{n - 1} \sum_{i = 1}^{n} (X_{i} - X ˉ)^{2}

— X_i = observation i,

X ˉ

= sample mean, n = sample size

Bayes' rule

P (A ∣ B) = \frac{P ( B ∣ A ) P ( A )}{P ( B )}

; P(A) = prior, P(B|A) = likelihood, P(B) = marginal evidence = P(B|A)P(A) + P(B|Aᶜ)P(Aᶜ)

Expected value of a lognormal random variable

E [X] = e^{μ + σ^{2} /2}

— μ = mean of ln(X), σ² = variance of ln(X); the σ²/2 is the convexity adjustment

F1 score for binary classification

F_{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 ∣ A^{c}) P (A^{c})

— A and A^c partition the sample space, P(B|·) = conditional probability of B

Adjusted R-squared

R ˉ^{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} = \frac{Cov ( X , Y )}{Var ( X )}

— Cov(X,Y) = sample covariance of X and Y, Var(X) = sample variance of X

AR(1) h-step-ahead forecast

E [Y_{t + h}] = μ + ϕ^{h} (Y_{t} - μ)

— μ = mean-reverting level, φ = AR coefficient, h = forecast horizon, Y_t = current value

Pearson correlation coefficient

ρ_{X, Y} = \frac{Cov ( X , Y )}{σ _{X} σ _{Y}}

— Cov = covariance of X and Y; σ_X, σ_Y = standard deviations; ρ ∈ [-1, 1]

Square-root-of-time volatility scaling

σ_{T} = σ_{1} T

— σ_T = T-period volatility, σ_1 = one-period volatility, T = number of periods (assumes iid returns)

Variance of a linear transformation

Var (a X + b) = a^{2} Var (X)

— X = random variable, a = scale factor, b = shift constant

Omitted variable bias in a short regression coefficient

bias (β^_{1}) = β_{2} \cdot δ

— β₂ = true coefficient on omitted X₂, δ = slope from regressing X₂ on included X₁

Variance of a two-asset weighted sum

Var (a X + bY) = a^{2} σ_{X}^{2} + b^{2} σ_{Y}^{2} + 2 ab Cov (X, Y)

— a, b = weights; σ = standard deviation; Cov = covariance

Variance inflation factor for regressor j

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 = \frac{X ˉ - μ _{0}}{s / n}

—

X ˉ

= sample mean,

μ_{0}

= hypothesized mean, s = sample std dev, n = sample size

Box-Pierce Q-statistic for residual autocorrelation

Q_{B P} = T \sum_{k = 1}^{m} ρ^_{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} \sum_{i} (y_{i} - y^_{i})^{2} + λ \sum_{j} ∣ β_{j} ∣

— y_i = observed, ŷ_i = predicted, β_j = coefficient j, λ = L1 penalty strength

Ridge regression penalized objective

min_{β} \sum_{i} (y_{i} - y^_{i})^{2} + λ \sum_{j} β_{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 = \frac{X ˉ _{1} - X ˉ _{2}}{s _{1}^{2} / n _{1} + s _{2}^{2} / n _{2}}

—

X ˉ_{i}

= sample mean i,

s_{i}

= sample std dev i,

n_{i}

= sample size i

Jarque-Bera test statistic

J B = \frac{n}{6} [S^{2} + \frac{( K - 3 ) ^{2}}{4}]

— n = sample size, S = sample skewness, K = sample kurtosis; chi-square with 2 dof under normality

F-statistic for joint hypothesis test

F = \frac{( SSR _{R} - SSR _{U} ) / q}{SSR _{U} / ( n - k - 1 )}

— SSR_R = restricted SSR, SSR_U = unrestricted SSR, q = restrictions, n = sample, k = regressors

Financial Markets and Products 26 items

European call lower bound (no dividend)

c \geq max (S_{0} - K e^{- r T}, 0)

— c = call price, S₀ = spot, K = strike, r = risk-free rate, T = time to maturity

European put lower bound (no dividend)

p \geq max (K e^{- r T} - S_{0}, 0)

— p = put price, K = strike, r = risk-free rate, T = time to maturity, S₀ = spot

Continuous-to-discrete compounding conversion

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 = 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^{*} \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

F_{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

F_{0} = S_{0} e^{r T}

— S₀ = spot price, r = continuously compounded risk-free rate, T = time to delivery in years

Relative purchasing power parity

(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 + K e^{- r T} = 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 \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

Δ P / P \approx - D_{m o d} Δ y + \frac{1}{2} C (Δ y)^{2}

— D_mod = modified duration, C = convexity, Δy = yield change

Uncovered interest rate parity expected future spot

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

C P R = 1 - (1 - S M M)^{12}

— SMM = single monthly mortality, CPR = annualized conditional prepayment rate

Forward price with continuous income yield

F_{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

D_{m o d} = D_{M a c} / (1 + y / m)

— D_Mac = Macaulay duration, y = yield, m = compounding periods per year

Cheapest-to-deliver delivery cost

Cost = Quoted price - (F \times C F)

— Quoted price = clean bond price, F = futures settlement, CF = conversion factor

Tailed number of futures contracts

N_{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^{*} - β_{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 \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^{*} = \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 = B_{fix} - B_{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)

— r = nominal rate, ρ = real rate, i = expected inflation

Long forward payoff at maturity

Payoff = S_{T} - K

— S_T = spot price at maturity, K = forward strike price

Variance swap payoff at maturity

Payoff = N_{v a r} \times (σ_{r e a l i z e d}^{2} - K_{v a r}^{2})

— N_var = variance notional, σ²_realized = realized variance, K_var = strike volatility

Long put option payoff at maturity

Payoff = max (K - S_{T}, 0)

— K = strike price, S_T = spot at maturity

FRA settlement payment

Payment = \frac{N ( r _{r e f} - r _{F R A} ) ( d /360 )}{1 + r _{r e f} ( d /360 )}

, settled at the start of the reference period. N = notional, d = day count,

r_{r e f}

= reference rate.

Valuation and Risk Models 23 items

Forward rate between two dates (continuous compounding)

f (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 = S_{0} N (d_{1}) - K e^{- r T} 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 = e^{σ Δ 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 (d_{1}) / (S_{0} σ T)

— φ = standard normal density, S₀ = spot, σ = volatility, T = maturity

d₁ in the Black-Scholes-Merton formula (no dividends)

d_{1} = \frac{l n ( S _{0} / K ) + ( r + σ ^{2} /2 ) T}{σ T}

— S₀ = spot, K = strike, r = rate, σ = volatility, T = maturity

Unexpected loss on a single credit exposure (binomial default)

U L = E A D \times L G D \times P D (1 - P D)

— EAD = exposure at default, LGD = loss given default, PD = probability of default

Risk-neutral probability in a binomial tree

p = \frac{e ^{r Δ 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 \leq t) = 1 - e^{- λ t}

— λ = constant hazard rate (annual default intensity), t = horizon in years, τ = default time

KR01 from key-rate duration

KR01_{k} = KRD_{k} \cdot P \cdot 1 0^{- 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^{- λ t}

— λ = constant hazard rate, t = horizon in years, τ = default time

Bond price from discount factors

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 (V a R, V a R_{60 d}) + max (s V a R, s V a R_{60 d})

— VaR = current VaR, sVaR = stressed VaR, 60d-bar = 60-day average

Expected Shortfall under normal returns

ES_{c} = μ + σ \cdot \frac{ϕ ( 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

D_{eff} = \frac{P _{-} - P _{+}}{2 P _{0} Δ y}

— P₋ = price after yield drop, P₊ = price after yield rise, P₀ = base price, Δy = shock size

Parametric (delta-normal) VaR

VaR_{c} = - μ + z_{c} σ

— 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

— LGD = loss given default (fraction of exposure lost), RR = recovery rate (fraction of par recovered after default)

Macaulay duration

D_{mac} = \sum_{i = 1}^{n} t_{i} \cdot \frac{P V ( 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

— r = nominal annual rate, n = compounding periods per year

Dollar value of an 01 (DV01)

DV01 \approx D_{mod} \times P \times 1 0^{- 4}

— D_mod = modified duration, P = bond price; dollar price change for a 1 bp yield change

GARCH(1,1) variance recursion

σ_{t}^{2} = ω + α r_{t - 1}^{2} + β σ_{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^{- λ}

— λ = constant annual hazard rate; value is the same each year under constant intensity

Vasicek single-factor credit VaR (Basel IRB)

V a R_{c} = Φ (\frac{Φ ^{- 1} ( P D ) + ρ Φ ^{- 1} ( c )}{1 - ρ})

— PD = default probability, ρ = asset correlation, c = confidence level, Φ = standard-normal CDF

Option vega

ν = \frac{\partial V}{\partial σ}

, the change in option value per one-point change in volatility. Positive for long calls and puts; largest for at-the-money options with longer time to maturity.

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