77 core formulas for CFA Level I, CFA Level II and FRM Part I, grouped by the official CFA Institute and GARP topic areas, with variable definitions and when to use each one. Official topic weights are linked to their sources. Last checked 2026-10-12.
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CFA Level I (44 formulas)
Official topic weights (2026)
- Ethical and Professional Standards: 15–20%
- Quantitative Methods: 6–9%
- Economics: 6–9%
- Financial Statement Analysis: 11–14%
- Corporate Issuers: 6–9%
- Equity Investments: 11–14%
- Fixed Income: 11–14%
- Derivatives: 5–8%
- Alternative Investments: 7–10%
- Portfolio Management: 8–12%
Source: https://www.cfainstitute.org/programs/cfa-program/candidate-resources/level-i-examQuantitative Methods
Future value (discrete compounding)\[FV = PV\,(1+r)^{N}\]
PV present value; r periodic rate; N number of periods — Growing a single cash flow forward Present value of an ordinary annuity\[PV = A\,\frac{1-(1+r)^{-N}}{r}\]
A level payment at end of each period; r periodic rate; N periods — Loan payments, bond coupons, retirement income Present value of a perpetuity\[PV = \frac{A}{r}\]
A constant payment per period; r discount rate — Preferred stock, consols Effective annual rate\[EAR = \left(1+\frac{r_s}{m}\right)^{m}-1\]
r_s stated annual rate; m compounding periods per year — Comparing rates with different compounding frequencies Continuous compounding\[FV = PV\,e^{r_s N}\]
r_s stated annual rate; N years — Continuously compounded returns and growth Holding period return\[HPR = \frac{P_1 - P_0 + D_1}{P_0}\]
P_0 beginning price; P_1 ending price; D_1 income received — Single-period return including income Geometric mean return\[\bar R_G = \left[\prod_{t=1}^{T}(1+R_t)\right]^{1/T}-1\]
R_t return in period t; T number of periods — Compound average growth rate over multiple periods Sample variance\[s^2 = \frac{\sum_{i=1}^{n}(X_i-\bar X)^2}{n-1}\]
X_i observations; \bar X sample mean; n sample size — Dispersion of a sample Coefficient of variation\[CV = \frac{s}{\bar X}\]
s sample standard deviation; \bar X sample mean — Risk per unit of return Sharpe ratio\[S_p = \frac{\bar R_p - R_f}{\sigma_p}\]
\bar R_p portfolio mean return; R_f risk-free rate; \sigma_p portfolio standard deviation — Risk-adjusted performance Sample correlation\[r_{XY} = \frac{s_{XY}}{s_X\, s_Y}\]
s_{XY} sample covariance; s_X, s_Y sample standard deviations — Strength of linear relationship Bayes' formula\[P(A\mid B) = \frac{P(B\mid A)\,P(A)}{P(B)}\]
P(A) prior; P(B|A) likelihood; P(B) unconditional probability — Updating probabilities with new information z-statistic / standardization\[z = \frac{X-\mu}{\sigma}\]
X value; \mu mean; \sigma standard deviation — Standard normal probabilities t-statistic for a mean\[t = \frac{\bar X-\mu_0}{s/\sqrt{n}}\]
\bar X sample mean; \mu_0 hypothesized mean; s sample std dev; n sample size (df = n-1) — Hypothesis test of a population mean Simple linear regression slope\[\hat b_1 = \frac{\operatorname{Cov}(X,Y)}{\operatorname{Var}(X)}\]
Cov(X,Y) covariance; Var(X) variance of the independent variable — Estimating the slope coefficient Coefficient of determination (simple regression)\[R^2 = \frac{SSR}{SST} = 1-\frac{SSE}{SST}\]
SSR regression sum of squares; SSE sum of squared errors; SST total sum of squares — Share of variation explained Economics
Price elasticity of demand\[E_P = \frac{\%\Delta Q_d}{\%\Delta P}\]
Q_d quantity demanded; P price — Responsiveness of demand to price Covered interest rate parity (forward rate)\[F_{f/d} = S_{f/d}\,\frac{1+i_f\left(\frac{\text{days}}{360}\right)}{1+i_d\left(\frac{\text{days}}{360}\right)}\]
S spot rate (price currency f per base currency d); i_f, i_d money-market rates — Pricing currency forwards Financial Statement Analysis
Current ratio\[\text{Current ratio} = \frac{\text{Current assets}}{\text{Current liabilities}}\]
— — Short-term liquidity Cash conversion cycle\[CCC = DOH + DSO - \text{Days of payables}\]
DOH days of inventory on hand; DSO days of sales outstanding — Working-capital efficiency DuPont (three-step) ROE\[ROE = \frac{NI}{\text{Sales}}\times\frac{\text{Sales}}{\text{Assets}}\times\frac{\text{Assets}}{\text{Equity}}\]
NI net income; average assets and equity — Decomposing return on equity Basic EPS\[EPS = \frac{NI - \text{Preferred dividends}}{\text{Weighted avg. common shares}}\]
NI net income — Earnings per common share Interest coverage\[\text{Interest coverage} = \frac{EBIT}{\text{Interest payments}}\]
EBIT earnings before interest and taxes — Solvency Corporate Issuers
Weighted average cost of capital\[WACC = w_d\,r_d\,(1-t) + w_p\,r_p + w_e\,r_e\]
w weights (market values); r_d before-tax cost of debt; t marginal tax rate; r_p cost of preferred; r_e cost of equity — Hurdle rate for average-risk projects Net present value\[NPV = \sum_{t=0}^{N}\frac{CF_t}{(1+r)^t}\]
CF_t cash flow at t (CF_0 usually negative); r required return — Capital budgeting accept/reject Degree of operating leverage\[DOL = \frac{\%\Delta EBIT}{\%\Delta \text{Sales}} = \frac{Q(P-V)}{Q(P-V)-F}\]
Q units; P price; V variable cost per unit; F fixed operating costs — Operating risk Degree of financial leverage\[DFL = \frac{\%\Delta NI}{\%\Delta EBIT} = \frac{Q(P-V)-F}{Q(P-V)-F-C}\]
C fixed financial costs (interest) — Financial risk CAPM cost of equity\[r_e = R_f + \beta\,[E(R_M)-R_f]\]
R_f risk-free rate; \beta equity beta; E(R_M) expected market return — Required return on equity Equity Investments
Gordon (constant) growth model\[V_0 = \frac{D_1}{r-g}\]
D_1 next dividend; r required return; g constant growth rate (g < r) — Valuing mature dividend payers Justified forward P/E\[\frac{P_0}{E_1} = \frac{D_1/E_1}{r-g}\]
D_1/E_1 payout ratio; r required return; g growth — Relative valuation from fundamentals Preferred stock value\[V_0 = \frac{D_p}{r_p}\]
D_p fixed preferred dividend; r_p required return — Non-callable perpetual preferred shares Margin call price (long)\[P_{call} = P_0\,\frac{1-IM}{1-MM}\]
P_0 purchase price; IM initial margin; MM maintenance margin — Leveraged long position Fixed Income
Bond price from yield\[P = \sum_{t=1}^{N}\frac{PMT}{(1+r)^t} + \frac{FV}{(1+r)^N}\]
PMT coupon per period; FV par value; r market discount rate per period; N periods — Pricing option-free bonds Modified duration\[\text{ModDur} = \frac{\text{MacDur}}{1+r}\]
MacDur Macaulay duration; r yield per period — Price sensitivity to yield Percentage price change (duration + convexity)\[\%\Delta P \approx -\text{AnnModDur}\,\Delta y + \tfrac12\,\text{AnnConvexity}\,(\Delta y)^2\]
\Delta y change in annual yield (decimal) — Estimating price change for a yield shift Money duration and PVBP\[\text{MoneyDur} = \text{AnnModDur}\times P^{\text{full}}, \quad PVBP = \frac{PV_- - PV_+}{2}\]
P^full full price; PV_- / PV_+ prices for 1 bp down / up yield — Dollar interest-rate exposure Forward rate from spot rates\[(1+S_{A+B})^{A+B} = (1+S_A)^A\,(1+{}_AF_B)^B\]
S spot rates; {}_AF_B forward rate starting at A for B periods — Implied forward rates / bootstrapping Derivatives
Forward price (no income)\[F_0(T) = S_0\,(1+r)^T\]
S_0 spot price; r risk-free rate; T years — No-arbitrage forward pricing Put–call parity\[S_0 + p_0 = c_0 + \frac{X}{(1+r)^T}\]
S_0 spot; p_0, c_0 European put/call; X strike; r risk-free; T years — Relating European put and call prices One-period binomial risk-neutral probability\[\pi = \frac{1+r-d}{u-d}\]
u, d up/down factors; r risk-free rate per period — Binomial option pricing Portfolio Management
Two-asset portfolio variance\[\sigma_p^2 = w_1^2\sigma_1^2 + w_2^2\sigma_2^2 + 2w_1w_2\rho_{12}\sigma_1\sigma_2\]
w weights; \sigma std devs; \rho_{12} correlation — Diversification effect Beta\[\beta_i = \frac{\operatorname{Cov}(R_i,R_m)}{\sigma_m^2} = \rho_{i,m}\frac{\sigma_i}{\sigma_m}\]
R_i asset return; R_m market return — Systematic risk Treynor ratio and Jensen's alpha\[T_p = \frac{R_p-R_f}{\beta_p},\quad \alpha_p = R_p - [R_f + \beta_p(R_m-R_f)]\]
R_p portfolio return; \beta_p portfolio beta — Performance vs systematic risk Alternative Investments
Hedge fund 2-and-20 fee (simplified)\[\text{Fee} = 0.02\,AUM + 0.20\,\max(0,\ \text{Profit}-\text{Hurdle})\]
AUM assets under management; hurdle/high-water mark terms vary by fund — Net-of-fee return calculations Practise these with the CFA Level I formula reference deck from UniPrep2Go.
CFA Level II (16 formulas)
Official topic weights
- Quantitative Methods: 5–10%
- Economics: 5–10%
- Financial Statement Analysis: 10–15%
- Corporate Finance: 5–10%
- Equities: 10–15%
- Fixed Income: 10–15%
- Derivatives and Risk Management: 5–10%
- Alternative Investments: 5–10%
- Portfolio Construction: 10–15%
- Ethical and Professional Standards: 10–15%
Source: https://www.cfainstitute.org/programs/cfa-program/candidate-resources/level-ii-examQuantitative Methods
Adjusted R-squared\[\bar R^2 = 1-\left[\frac{n-1}{n-k-1}\right](1-R^2)\]
n observations; k independent variables — Comparing multiple regression models F-statistic (regression)\[F = \frac{SSR/k}{SSE/(n-k-1)}\]
SSR regression SS; SSE error SS; k slopes; n observations — Joint significance of slope coefficients AR(1) mean-reverting level\[\hat x_t = \frac{b_0}{1-b_1}\]
b_0, b_1 AR(1) coefficients (|b_1| < 1) — Time-series forecasting Equities
Free cash flow to the firm\[FCFF = NI + NCC + Int(1-t) - FCInv - WCInv\]
NCC non-cash charges; Int interest expense; FCInv fixed capital investment; WCInv working capital investment — FCFF valuation Free cash flow to equity\[FCFE = FCFF - Int(1-t) + \text{Net borrowing}\]
— — FCFE valuation Sustainable growth rate\[g = b \times ROE\]
b earnings retention ratio — Long-run growth input for DDM Residual income\[RI_t = E_t - r\,B_{t-1}\]
E_t earnings; r cost of equity; B_{t-1} beginning book value — Residual income valuation Residual income model\[V_0 = B_0 + \sum_{t=1}^{\infty}\frac{E_t - rB_{t-1}}{(1+r)^t}\]
B_0 current book value per share — Valuing firms with no/irregular dividends Fixed Income
Binomial interest-rate tree node value\[V = \frac{0.5\,(V_H + C) + 0.5\,(V_L + C)}{1+i}\]
V_H, V_L values at next nodes; C coupon; i one-period forward rate at the node — Valuing bonds with embedded options Value of callable / putable bond\[V_{call} = V_{straight} - V_{call\ option},\quad V_{put} = V_{straight} + V_{put\ option}\]
— — Embedded option valuation Credit valuation adjustment\[CVA = \sum_{t}\text{LGD}_t \times \text{POD}_t \times DF_t\]
LGD loss given default; POD probability of default; DF discount factor — Valuing risky bonds Derivatives and Risk Management
Forward with carry benefits/costs\[F_0(T) = (S_0 - PV_0(I) + PV_0(C))(1+r)^T\]
PV_0(I) PV of income; PV_0(C) PV of carry costs — Forwards on income-paying assets Black–Scholes–Merton call price\[c_0 = S_0 N(d_1) - e^{-rT}X\,N(d_2),\ d_1 = \frac{\ln(S_0/X)+(r+\sigma^2/2)T}{\sigma\sqrt T},\ d_2 = d_1 - \sigma\sqrt T\]
N(·) standard normal CDF; \sigma volatility; r continuously compounded rate — European option pricing Swap fixed rate\[s_N = \frac{1 - PV_N}{\sum_{i=1}^{N} PV_i}\]
PV_i discount factor for period i — Pricing plain-vanilla interest rate swaps Portfolio Construction
Arbitrage pricing theory\[E(R_p) = R_F + \lambda_1\beta_{p,1} + \dots + \lambda_K\beta_{p,K}\]
\lambda_j factor risk premium; \beta_{p,j} factor sensitivity — Multifactor expected return Parametric VaR (single asset)\[VaR = (z_\alpha\,\sigma - \mu)\,V\]
z_\alpha standard normal quantile; \mu, \sigma period mean/std of return; V position value — Downside risk measurement Practise these with the CFA Level II formula reference deck from UniPrep2Go.
FRM Part I (17 formulas)
Official topic weights (2023 curriculum)
- Foundations of Risk Management: 20%
- Quantitative Analysis: 20%
- Financial Markets and Products: 30%
- Valuation and Risk Models: 30%
Source: https://www.garp.org/hubfs/Website/FRM/PDF/Study%20Materials/2023/FRM2023-Learning-Objectives.pdfFoundations of Risk Management
CAPM expected return\[E(R_i) = R_F + \beta_i\,[E(R_M)-R_F]\]
\beta_i asset beta — Required return / performance evaluation Information ratio\[IR = \frac{R_P - R_B}{\sigma(R_P-R_B)}\]
R_B benchmark return; denominator tracking error — Active-management skill Sortino ratio\[\text{Sortino} = \frac{R_P - R_{min}}{\sigma_{downside}}\]
R_min minimum acceptable return; downside deviation — Downside-risk-adjusted performance Quantitative Analysis
Variance of a sum\[\operatorname{Var}(aX+bY) = a^2\operatorname{Var}(X) + b^2\operatorname{Var}(Y) + 2ab\operatorname{Cov}(X,Y)\]
a, b constants — Portfolio and aggregation math OLS slope and intercept\[\hat\beta = \frac{\sum (X_i-\bar X)(Y_i-\bar Y)}{\sum (X_i-\bar X)^2},\quad \hat\alpha = \bar Y - \hat\beta\bar X\]
— — Linear regression estimation EWMA volatility\[\sigma_t^2 = \lambda\,\sigma_{t-1}^2 + (1-\lambda)\,r_{t-1}^2\]
\lambda decay factor (e.g. 0.94 in RiskMetrics) — Volatility forecasting GARCH(1,1)\[\sigma_t^2 = \omega + \alpha\,r_{t-1}^2 + \beta\,\sigma_{t-1}^2,\quad \sigma_L^2 = \frac{\omega}{1-\alpha-\beta}\]
\omega constant; \alpha, \beta weights (\alpha+\beta < 1); \sigma_L long-run variance — Volatility forecasting with mean reversion Financial Markets and Products
Minimum-variance hedge ratio\[h^* = \rho\,\frac{\sigma_S}{\sigma_F}\]
\rho correlation of spot and futures changes; \sigma_S, \sigma_F std devs — Futures hedging Optimal number of futures contracts\[N^* = h^*\,\frac{Q_A}{Q_F}\]
Q_A size of position hedged; Q_F size of one contract — Futures hedging Equity beta hedge\[N^* = (\beta^* - \beta)\,\frac{P}{F}\]
\beta target beta; P portfolio value; F value of one futures contract — Changing portfolio beta with index futures Futures price with storage cost and yield\[F_0 = S_0\,e^{(r+u-q)T}\]
u storage cost rate; q income/convenience yield rate (continuous) — Commodity and asset futures pricing Interest rate parity (continuous)\[F_0 = S_0\,e^{(r-r_f)T}\]
r domestic rate; r_f foreign rate (continuous) — FX forward pricing Valuation and Risk Models
Parametric VaR\[VaR_\alpha = -\mu + z_\alpha\,\sigma\]
z_\alpha standard normal quantile (e.g. 1.645 at 95%); \mu, \sigma of P&L/returns — Market risk measurement Expected shortfall (normal)\[ES_\alpha = -\mu + \sigma\,\frac{\phi(z_\alpha)}{1-\alpha}\]
\phi standard normal pdf; \alpha confidence level — Tail risk beyond VaR Expected loss (credit)\[EL = PD \times LGD \times EAD\]
PD probability of default; LGD loss given default; EAD exposure at default — Credit risk Option delta (BSM, non-dividend call/put)\[\Delta_c = N(d_1),\quad \Delta_p = N(d_1) - 1\]
N(d_1) from BSM — Delta hedging Effective duration and convexity\[D_{eff} = \frac{V_- - V_+}{2V_0\,\Delta y},\quad C_{eff} = \frac{V_- + V_+ - 2V_0}{V_0\,(\Delta y)^2}\]
V_-/V_+ values for yield down/up by \Delta y — Interest rate risk (incl. embedded options) Practise these with the FRM Part I flashcard deck from UniPrep2Go.