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FRM Part I formula sheet

Every core formula for FRM® Part I, grouped by topic, with a note on when each one applies.

Foundations of Risk Management

Asset pricing and performance measures.

CAPM

E(R_i) = R_f + β_i × [E(R_m) − R_f]

β_i = Cov(R_i, R_m) / σ_m².

Multi-factor model

E(R_i) = R_f + Σ β_ik × λ_k

λ_k = risk premium for factor k (APT, Fama–French).

Sharpe ratio

(R_p − R_f) / σ_p

Excess return per unit of total risk.

Treynor ratio

(R_p − R_f) / β_p

Excess return per unit of systematic risk.

Jensen's alpha

α_p = R_p − [R_f + β_p(R_m − R_f)]

Return above the CAPM prediction.

Information ratio and tracking error

IR = (R_p − R_b) / TE TE = σ(R_p − R_b)

Active return per unit of active risk.

Sortino ratio

(R_p − R_min) / Downside deviation

Penalises only returns below the minimum acceptable return.

Quantitative Analysis

Probability, regression and volatility modelling.

Bayes' theorem

P(A | B) = P(B | A) × P(A) / P(B)

The classic FRM application is updating default or manager-skill probabilities.

Binomial and Poisson

P(X = k) = C(n, k) p^k(1 − p)^{n − k} P(X = k) = λ^k e^−λ / k!

Poisson for counts of events such as defaults or losses.

Variance of a sum

Var(aX + bY) = a²Var(X) + b²Var(Y) + 2ab Cov(X, Y)

The basis of portfolio variance.

OLS slope and intercept

β̂ = Cov(X, Y) / Var(X) α̂ = Ȳ − β̂X̄

t = β̂ / SE(β̂).

Adjusted R-squared

R̄² = 1 − (1 − R²) × (n − 1) / (n − k − 1)

k = number of regressors. Penalises adding weak variables.

EWMA volatility

σ_t² = λσ_t−1² + (1 − λ) r_t−1²

RiskMetrics uses λ = 0.94 for daily data.

GARCH(1,1)

σ_t² = ω + α r_t−1² + β σ_t−1²

Long-run variance = ω / (1 − α − β). Persistence = α + β < 1.

Square-root-of-time rule

σ_T = σ₁ × √T

Holds only for independent, identically distributed returns.

Work it through: Square-root-of-time guide.

Financial Markets and Products

Forwards and futures, hedging, swaps and bond futures.

Forward price, continuous compounding

F₀ = S₀ e^{(r − q)T}

q = dividend yield. Currencies: F₀ = S₀ e^{(r_d − r_f)T}.

Commodity forward

F₀ = S₀ e^{(r + u − y)T}

u = storage cost, y = convenience yield.

Minimum-variance hedge ratio

h* = ρ × σ_S / σ_F N* = h* × Q_A / Q_F

Q_A = size of the position, Q_F = size of one contract.

Hedging equity with index futures

N = (β* − β) × P / F

Set β* = 0 to hedge completely.

Duration-based hedge

N = (P × D_P) / (F × D_F)

Values and durations of the portfolio and the futures contract.

Swap valuation

V_{pay fixed} = B_float − B_fixed

Or value as a portfolio of FRAs.

Cheapest-to-deliver bond

Cost = Quoted bond price − (Quoted futures price × Conversion factor)

Deliver the bond with the lowest cost.

Work it through: Swap valuation calculator · Swap valuation guide.

Valuation and Risk Models

VaR and ES, bond risk, options and credit losses.

Parametric VaR

VaR = z_c × σ × V

Assumes normal returns with zero mean. Scale with √T over T days.

Expected shortfall, normal

ES = σ × V × φ(z_c) / (1 − c)

φ = standard normal density. Always above VaR at the same c.

DV01

DV01 = ModDur × P × 0.0001

Price change for a 1 bp move in yield.

Price change with convexity

ΔP / P ≈ −D × Δy + ½ × C × (Δy)²

Duration overstates losses when yields rise for option-free bonds.

Black–Scholes–Merton

c = S₀N(d₁) − X e^−rT N(d₂)

d₁ = [ln(S₀/X) + (r + σ²/2)T] / (σ√T), d₂ = d₁ − σ√T.

Option delta

Δ_call = N(d₁) Δ_put = N(d₁) − 1

With a dividend yield, multiply by e^−qT.

Binomial tree parameters

u = e^σ√Δt d = 1/u p = (e^rΔt − d) / (u − d)

Discount the expected payoff at the risk-free rate.

Expected loss

EL = PD × LGD × EAD

The mean credit loss, covered by pricing and provisions.

Unexpected loss

UL = EAD × √(PD × σ_LGD² + LGD² × σ_PD²)

σ_PD² = PD × (1 − PD). Covered by economic capital.

Work it through: VaR calculator · Duration and convexity calculator · Expected shortfall guide.

Practise with them

FRM® Part I practice questionsExam-style questions with explanations.
FRM® Part I topic weightsWhere each formula sits in the exam.
Finance calculatorsVaR, duration and convexity, swap valuation.
BA II Plus keystrokesThe exam calculator, key by key.

A study aid, not the official formula list: the exam does not provide formulas, and the curriculum is the authority on notation. CFA® and FRM® are registered trademarks of their respective owners; GroundLearn is not affiliated with CFA Institute or GARP.

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