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Free CAIA Level I Formula Sheet (2026)

Every CAIA Level I formula you need on the test, grouped by topic, rendered with full math notation. 94 formulas across 6 topics, calibrated to the 2026 syllabus. Free forever, no signup required.

94 Formulas

6 Topics

2026 Syllabus

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All CAIA Level I Formulas

Introduction to Alternative Investments 48 items

Annual Hedge Fund Fee

Fee = m + i max [0, R_{H W M} - m - h]

Management fee m plus incentive rate i applied to the gross return above the high-water mark, net of the fee and hurdle h.

Incentive Fee as a Call Option

Incentive = i max [ENAV - BNAV, 0]

The incentive fee behaves like a call on NAV struck at the beginning NAV or high-water mark, whichever is greater.

At-the-Money Incentive Fee Value

Incentive \approx i \times 0.40 \times NAV \times σ

Approximate value of an at-the-money incentive fee; it rises directly with the fund's annual asset volatility.

Hedge Fund Fee with a Hard Hurdle

Incentive = i \times (Profit - Hurdle)

Under a hard hurdle the incentive fee is charged only on profit exceeding the hurdle, not on the whole gain.

Carried Interest (No Hurdle)

Carry = c \times (Ending NAV - Initial NAV)

GP profit share equal to the carry rate c times the fund's gain over committed capital when no hurdle applies.

GP Full Catch-Up (PE Waterfall)

Catch-up = \frac{H \times c}{1 - c}

After LPs receive capital plus hurdle profit H, the GP catches up so its carry equals rate c on total profit.

Hurdle Amount (Dollar)

H = r_{h} \times K \times t

Dollar profit a deal must clear before carry: hurdle rate on committed capital K over holding period t.

Internal Rate of Return (IRR)

\sum_{t = 0}^{T} \frac{C F _{t}}{( 1 + IRR ) ^{t}} = 0

Discount rate that sets the net present value of a deal's cash-flow stream (outflows negative) to zero.

Modified IRR (MIRR)

MIRR = (\frac{F V ( distributions )}{P V ( contributions )})^{1/ T} - 1

Compounds distributions at a realistic reinvestment rate and discounts contributions, fixing the IRR's reinvestment flaw.

DPI (Distributions to Paid-In)

DPI = \frac{\sum D _{t}}{\sum C _{t}}

Realized multiple: cumulative distributions per dollar of capital called; above 1 means more cash returned than invested.

RVPI (Residual Value to Paid-In)

RVPI = \frac{NAV}{\sum C _{t}}

Unrealized multiple: residual fund NAV per dollar of paid-in capital; mark-based and subject to valuation risk.

TVPI (Total Value to Paid-In)

TVPI = \frac{\sum D _{t} + NAV}{\sum C _{t}} = DPI + RVPI

Total value multiple per dollar paid in; the sum of realized (DPI) and unrealized (RVPI) value.

MOIC (Multiple on Invested Capital)

MOIC = \frac{Realized + Unrealized Value}{Invested Capital}

Total value over invested (deployed) capital; cleaner than TVPI for deal-level returns since it excludes uncalled fees.

Sample Variance

σ^{2} = \frac{1}{n - 1} \sum_{i} (R_{i} - R)^{2}

Average squared deviation from the mean; the n-1 denominator corrects bias when the mean is estimated from the sample.

Standard Deviation

σ = σ^{2}

Square root of variance; the volatility measure used interchangeably with total return risk.

Covariance

Cov (R_{i}, R_{j}) = E [(R_{i} - μ_{i}) (R_{j} - μ_{j})]

Average co-movement of two return series; the sample estimator divides summed cross-deviations by T-1.

Correlation Coefficient

ρ_{ij} = \frac{σ _{ij}}{σ _{i} σ _{j}}

Covariance scaled by the two standard deviations; bounded between -1 and +1 for easy interpretation.

Skewness

Skew = \frac{E [ ( R - μ ) ^{3} ]}{σ ^{3}}

Standardized third moment; positive means a longer right tail, negative a longer left tail of returns.

Excess Kurtosis

Excess Kurt = \frac{E [ ( R - μ ) ^{4} ]}{σ ^{4}} - 3

Kurtosis minus 3 (the normal's value); positive signals fat tails and more frequent extreme returns.

Beta

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

Systematic risk: covariance with the market over market variance, equivalently correlation times the volatility ratio.

Portfolio Variance

σ_{p}^{2} = \sum_{i} \sum_{j} w_{i} w_{j} Cov (R_{i}, R_{j})

Total portfolio variance as the weighted sum of every pairwise covariance among the holdings.

Volatility Time-Scaling (square root of time)

σ_{T} = σ_{1} T

Annualizes volatility for i.i.d. returns; variance scales linearly with T, so std dev scales with its square root.

Zero-Coupon Bond Price

B (t) = \frac{F V}{( 1 + r _{t} ) ^{t}}

Prices a zero by discounting face value at the annual spot rate; use (1+r/m) to the mt power for m-times compounding.

Spot Rate from a Zero-Coupon Bond Price

r_{t} = (\frac{F V}{B ( t )})^{1/ t} - 1

Backs out the annual zero-coupon (spot) rate implied by an observed bond price; the term-structure building block.

Fisher Equation (After-Tax)

i = r + \frac{π}{1 - τ}

Nominal rate built from the real rate and inflation, grossed up by the income tax rate; with no tax the inflation term is simply added.

Implied Forward Rate (annual compounding)

F (t, T) = [\frac{( 1 + r _{T} ) ^{T}}{( 1 + r _{t} ) ^{t}}]^{1/ (T - t)} - 1

No-arbitrage forward rate between two future dates t and T derived from the annual-compounded spot curve.

Implied Forward Rate (maturity-weighted)

F_{t, T} = \frac{T r _{T} - t r _{t}}{T - t}

Forward (FRA) rate as the maturity-weighted spread between the long and short spot rates; continuous-compounding form.

Macaulay Duration

D = \frac{\sum _{t} \frac{t C _{t}}{( 1 + y ) ^{t}}}{P _{0}}

Present-value-weighted average time to receive a bond's cash flows; its effective-maturity measure.

Modified Duration

D_{m o d} = \frac{D}{1 + y / m}

Scales Macaulay duration for discrete compounding to give the percentage price change per unit change in yield.

Portfolio Duration

D_{p} = \sum_{i} w_{i} D_{i}

Market-value-weighted average of the component bonds' durations; the bond portfolio's interest-rate sensitivity.

Capital Asset Pricing Model (CAPM)

E (R_{i}) = R_{f} + β_{i} [E (R_{m}) - R_{f}]

Single-factor required return: the risk-free rate plus beta times the market risk premium.

Regression t-Statistic

t = \frac{∣ θ ^ ∣}{s _{θ^}}

Tests whether an estimated alpha or beta differs from zero by comparing the estimate to its standard error.

Jensen's Alpha

α_{p} = E (R_{p}) - R_{f} - β_{p} [E (R_{m}) - R_{f}]

Expected return above the CAPM benchmark; the ex-ante forecasted alpha, also the intercept of an excess-return regression.

Realized (Ex-Post) Alpha

α = (R_{p} - R_{f}) - β_{p} (R_{m} - R_{f})

Alpha from realized returns; its gap from forecasted alpha helps separate manager skill from luck.

Forward Price of a Default-Free Zero-Coupon Bond

F_{t, T} = K \times \frac{P _{0, T}}{P _{0, t}}

No-arbitrage forward price to deliver a default-free zero, from the ratio of far- to near-dated spot zero prices.

Commodity Forward Price (Cost of Carry)

F_{T} = P_{0} e^{(r + c - y) T}

Spot grown by net carry: financing rate plus storage cost minus convenience yield (an upper bound under short-sale limits).

Mark-to-Market Value of a Forward

V_{t} = P_{t} e^{(r + c - y) (T - t)} - F_{0}

Marks a long forward to market as the carried spot value of the deliverable minus the locked-in original forward price.

Put-Call Parity

C + B - P = S

Long call plus a bond (PV of strike) minus a put equals the underlying; the basis for collars and synthetic positions.

Black-Scholes Call Option

c = S N (d_{1}) - K e^{- r T} N (d_{2})

European call without dividends; N(d1) and N(d2) are cumulative-normal terms and the put follows from put-call parity.

Parametric VaR

VaR = Z \times σ \times Days \times Value

Scales one-day volatility to the horizon by the square root of time at z-score Z, assuming a zero mean return.

Sharpe Ratio

S R = \frac{E ( R _{p} ) - R _{f}}{σ _{p}}

Excess return per unit of total risk; best for evaluating a stand-alone portfolio rather than an added position.

Treynor Ratio

T R = \frac{E ( R _{p} ) - R _{f}}{β _{p}}

Excess return per unit of systematic risk (beta); suited to an asset added to a well-diversified portfolio.

Sortino Ratio

Sortino = \frac{E ( R _{p} ) - R _{T}}{DD}

Like Sharpe but measures excess return over target R_T against downside deviation DD, penalizing only downside risk.

Information Ratio

I R = \frac{E ( R _{p} ) - R _{B}}{T E}

Active return over benchmark B divided by tracking error TE; excess return per unit of active-return volatility.

Return on VaR (RoVaR)

RoVaR = \frac{E ( R _{p} )}{VaR}

Expected return scaled by value at risk; useful when VaR captures the relevant downside of the distribution.

M-Squared (M2)

M^{2} = R_{f} + \frac{σ _{m}}{σ _{p}} [E (R_{p}) - R_{f}]

Risk-adjusted return a portfolio would earn if levered to the market's volatility; lets you compare at equal total risk.

Stale-Pricing Mean-Return Correction

μ^{*} = μ + \frac{1}{T} (1 - α) (r_{0} - r_{T})

Corrects an observed (stale) mean for lagged pricing; the endpoint adjustment shrinks as the sample lengthens.

Smoothing Effect on Observed Volatility

σ (r^{*}) = \frac{σ ( r )}{N}

Smoothing returns over N periods understates true volatility by the square root of N, biasing reported risk downward.

Real Assets 16 items

Land as a call option (binomial)

C_{0} = \frac{p max ( V _{u} - K , 0 ) + ( 1 - p ) max ( V _{d} - K , 0 )}{1 + r _{f}}

One-period binomial value of undeveloped land as a call; K is the development cost, p the risk-neutral up-probability.

Expected return of land

E (R_{land}) = p R_{dev} + (1 - p) R_{undev}

Probability-weighted expected return on land; p is the chance of development, blending the developed and undeveloped outcomes.

Direct capitalization value

V = \frac{NOI}{Cap Rate}

Values stabilized income property or farmland as a perpetuity of net operating income; NOI excludes debt service and taxes.

Capitalization rate

Cap Rate = \frac{NOI}{Property Value}

Property income yield; implicitly Cap Rate = r - g, so a lower cap rate signals higher expected rent growth or lower risk.

Required return on real estate (build-up)

r = (1 + R_{f}) (1 + E (R_{L P})) (1 + E (R_{R P})) - 1

Build-up discount rate compounding the risk-free rate with liquidity and risk premiums; additive approx r = R_f + LP + RP.

DCF income approach (real estate)

V_{0} = \sum_{t = 1}^{n} \frac{NOI _{t}}{( 1 + r ) ^{t}} + \frac{V _{n}}{( 1 + r ) ^{n}}

Income-approach value for non-stabilized property: discount projected NOI plus the terminal (reversion) value at the required return.

Gross income multiplier

GIM = \frac{Property Price}{Gross Income}

Quick relative-value gauge; price per dollar of annual gross income, before vacancy and operating expenses.

Cash-on-cash return

CoC = \frac{NOI - Debt Service}{Equity Invested}

Annual pre-tax cash flow per dollar of equity; leverage lifts it when the property yield exceeds the borrowing cost.

Development yield

Development Yield = \frac{Stabilized NOI}{Total Development Cost}

Return on cost for a development; its spread over market cap rates measures the developer's profit margin.

Loan-to-value (LTV)

LTV = \frac{Loan}{Property Value}

Mortgage leverage gauge; commercial deals typically run 60 to 75 percent loan to value.

Debt service coverage ratio (DSCR)

DSCR = \frac{NOI}{Annual Debt Service}

Cash-flow cushion for lenders; loan covenants commonly require a DSCR of at least 1.25.

Cost of carry (commodity forward price)

F = S e^{(r + c - y) T}

Carries spot to delivery: r is financing, c storage, y the convenience yield; arbitrage caps the forward at this level.

Calendar spread

Calendar Spread = F_{T + t} - F_{T}

Price gap between two delivery dates on the same commodity; widens or narrows as net carry shifts across maturities.

Fully collateralized commodity futures return

R = Spot Return + Collateral Yield + Roll Yield

CAIA decomposition of a fully collateralized futures position into spot change, cash collateral interest, and roll (basis) yield.

Hotelling's theory (commodity prices)

E (P_{T}) = P_{0} e^{r T}

Exhaustible-commodity spot prices should drift up at roughly the risk-adjusted rate; technology gains make this an upper bound.

Basis (futures)

Basis = S - F_{T}

Spot minus forward; equals the present value of net carrying costs and converges to zero as the contract nears delivery.

Private Equity 9 items

Total Addressable Market (TAM) Valuation

Value = Industry Revenue \times Market Share \times Revenue Multiple

Top-down VC valuation: size the full market, apply the firm's expected capture share, then a price-to-sales multiple.

Venture Capital Method (operating income)

Value = \frac{EBITDA \times EBITDA Multiple}{( 1 + IRR ) ^{T}}

Values a later-stage venture by discounting a projected EBITDA-multiple exit value at the investor's high target return.

Post-Money Valuation

POST = PRE + Investment

Company value right after a financing round: the negotiated pre-money value plus the new capital injected.

VC Ownership Proportion

Ownership = \frac{Investment}{POST}

Equity stake a new round's investor receives, before any dilution from future financing rounds.

Growth Equity Times-Revenue Method

Value = \frac{Annual Revenue \times Revenue Multiple}{( 1 + IRR ) ^{T}}

Values a not-yet-profitable growth firm off a projected revenue-multiple exit, discounted back to today.

LBO Projected Exit Value

Exit Value = \frac{C F _{T} ( 1 + g )}{r - g}

Terminal value at LBO exit via the Gordon growth model on the cash flow one year past the exit date.

LBO Equity Rate of Return

IRR = (\frac{Exit Equity Value}{Initial Equity})^{1/ T} - 1

Annualized return to the levered equity; more leverage widens the gap from the unlevered return.

Closed-End Fund Premium (or Discount)

Premium = \frac{Market Price}{NAV} - 1

Positive means the fund trades above NAV (premium), negative below (discount); usually quoted as a percentage.

Money-weighted return (IRR for PE funds)

\sum_{t = 0}^{T} \frac{C F _{t}}{( 1 + IRR ) ^{t}} = 0

The discount rate that zeros a PE fund's NPV; sensitive to LP cash-flow timing (the J-curve effect).

Private Debt 11 items

Credit Loss Rate

Credit Loss Rate = Default Rate \times (1 - Recovery Rate)

Expected annual default loss on a debt portfolio: default rate times loss given default (1 minus recovery).

Minimum Credit Spread Criterion

Credit Spread \geq Credit Loss Rate + Risk Premium

A distressed-debt spread must at least cover expected credit losses plus a premium for bearing default risk.

Fixed-Rate Mortgage Payment

M P = M B \times \frac{i}{1 - ( 1 + i ) ^{- n}}

Level monthly payment that fully amortizes a mortgage; underpins MBS and CMO cash flows.

Mortgage Amortization Split

Principal_{t} = M P - i B_{t - 1}

Each payment first pays interest (i x prior balance); the remainder amortizes principal and feeds sequential-pay CMO tranches.

PIK Accrued Balloon Balance

Balloon = P_{0} (1 + r)^{n}

Payment-in-kind interest compounds onto principal each year and is repaid as a single balloon at maturity.

Weighted Average Cost of Capital

WACC = \sum_{i} w_{i} r_{i}

Blended cost of capital across funding layers; inserting cheaper mezzanine between senior debt and equity can lower it.

Warrant Coverage Equity Value

Warrant Equity = Warrant Coverage \times Venture Debt

Dollar value of equity a venture lender gains via warrants attached to the loan, set as a percent of principal.

Post-Money Valuation with Warrants

V_{p os t} = V_{p r e} + Equity Raised + Warrant Equity

Firm value after a financing round, including the equity created by venture-debt warrant coverage.

Warrant Equity Percentage

Warrant % = \frac{Warrant Equity}{V _{p os t}}

Share of the firm a venture lender ends up owning once its attached warrants are exercised.

CDS Mark-to-Market Adjustment

MTM \approx (S_{t} - S_{0}) \times D \times Notional

Value gained by a CDS protection buyer when the market spread St rises above the contracted spread S0, scaled by risky duration.

PV of a Life Insurance Policy (Life Settlement)

PV = \frac{D B}{( 1 + r ) ^{L E}} - \sum_{t = 1}^{L E} \frac{Prem _{t}}{( 1 + r ) ^{t}}

Value of a settled policy: death benefit discounted over the insured's life expectancy minus the PV of premiums paid until then.

Hedge Funds 7 items

Simple moving average (SMA)

SMA_{t} (n) = \frac{1}{n} \sum_{i = 1}^{n} p_{t - i}

Equal-weighted mean of the last n prices; a rising SMA signals an uptrend in trend-following systems.

Weighted moving average (WMA)

W M A_{t} (n) = \frac{\sum _{i = 1}^{n} ( n - i + 1 ) p _{t - i}}{n ( n + 1 ) /2}

Linearly declining weights stress the newest price; the denominator normalizes the integer weights to sum to one.

Exponential moving average (EMA)

EMA_{t} (λ) = λ P_{t - 1} + (1 - λ) EMA_{t - 1} (λ)

Recursive smoothing with parameter λ (0<λ<1); weights on older prices decay geometrically and higher λ reacts faster.

Relative strength index (RSI)

RSI = 100 - \frac{100}{1 + U / D}

Momentum oscillator from 0 to 100; U and D are average up/down moves. Below 30 oversold, above 70 overbought.

Futures contracts for volatility targeting

N = SF \times \frac{Risk Loading \times Equity}{Notional} \times \frac{RVol _{T}}{RVol _{R}}

Scales position size so realized risk tracks vol target RVol_T; SF sets direction/strength, RVol_R is forecast vol.

Modified duration price change

\frac{Δ P}{P} \approx - D_{mod} \times Δ y

Approximates a bond's percentage price move for a small yield change Δy; longer duration means greater rate sensitivity.

High-water mark performance fee

Fee = p \times max (N A V_{t} - HWM_{t - 1}, 0)

Performance fee p charged only on gains above the prior peak, so a manager can't double-charge after a drawdown.

Funds of Funds 3 items

Portfolio Variance (Markowitz)

σ_{p}^{2} = \sum_{i = 1}^{n} \sum_{j = 1}^{n} w_{i} w_{j} Cov (R_{i}, R_{j})

Variance of a multi-fund portfolio; the covariance cross-terms mean low inter-fund correlation is what delivers FoF diversification.

Portfolio Variance (Uncorrelated Funds)

V (R_{p}) = \sum_{i = 1}^{n} w_{i}^{2} Var (R_{i})

Special case for uncorrelated funds: the covariance cross-terms vanish, leaving only the weighted sum of each fund's own variance.

Fund-of-Funds Risk Reduction (Square-Root-of-n)

σ_{p} = \frac{σ _{f}}{n}

Equally weighting n uncorrelated funds of equal volatility cuts portfolio volatility by the square root of the fund count.

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