Free CAIA Level I Formula Sheet (2026)

Every CAIA Level I formula you need on the test, grouped by topic and rendered with full math notation. 91 formulas across 5 topics, calibrated to the 2026 syllabus. Free forever, no signup required.

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All CAIA Level I Formulas

Introduction to Alternative Investments 49 items
Annual Hedge Fund Fee
Fee=m+i max⁡[0, RHWM−m−h]\text{Fee} = m + i\,\max\left[0,\ R_{HWM} - m - h\right]
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]\text{Incentive} = i\,\max\left[\text{ENAV} - \text{BNAV},\ 0\right]
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≈i×0.40×NAV×σ\text{Incentive} \approx i \times 0.40 \times \text{NAV} \times \sigma
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×(Profit−Hurdle)\text{Incentive} = i \times (\text{Profit} - \text{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×(Ending NAV−Initial NAV)\text{Carry} = c \times (\text{Ending NAV} - \text{Initial NAV})
GP profit share when no hurdle or other terms apply: the carry rate c times the fund's total profit, its ending NAV minus its initial size.
GP Full Catch-Up (PE Waterfall)
Catch-up=H×c1−c\text{Catch-up} = \dfrac{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=rh×K×tH = r_h \times K \times t
Dollar preferred return the LPs receive before the GP's catch-up: the hurdle rate times the capital the LPs contributed, K, times the years held, t, when the hurdle does not compound.
Internal Rate of Return (IRR)
∑t=0TCFt(1+IRR)t=0\sum_{t=0}^{T} \dfrac{CF_t}{(1+\text{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=(FV(distributions)PV(contributions))1/T−1\text{MIRR} = \left(\dfrac{FV(\text{distributions})}{PV(\text{contributions})}\right)^{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=∑Dt∑Ct\text{DPI} = \dfrac{\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=NAV∑Ct\text{RVPI} = \dfrac{\text{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=∑Dt+NAV∑Ct=DPI+RVPI\text{TVPI} = \dfrac{\sum D_t + \text{NAV}}{\sum C_t} = \text{DPI} + \text{RVPI}
Total value multiple per dollar paid in; the sum of realized (DPI) and unrealized (RVPI) value.
Sample Variance
σ2=1n−1∑i(Ri−R‾)2\sigma^2 = \dfrac{1}{n-1} \sum_i (R_i - \overline{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=σ2\sigma = \sqrt{\sigma^2}
Square root of variance; the volatility measure used interchangeably with total return risk.
Covariance
Cov(Ri,Rj)=E[(Ri−μi)(Rj−μj)]\text{Cov}(R_i, R_j) = \mathrm{E}\left[(R_i - \mu_i)(R_j - \mu_j)\right]
Average co-movement of two return series; the sample estimator divides summed cross-deviations by T-1.
Correlation Coefficient
ρij=σijσi σj\rho_{ij} = \dfrac{\sigma_{ij}}{\sigma_i\,\sigma_j}
Covariance scaled by the two standard deviations; bounded between -1 and +1 for easy interpretation.
Skewness
Skew=E[(R−μ)3]σ3\text{Skew} = \dfrac{\mathrm{E}\left[(R - \mu)^3\right]}{\sigma^3}
Standardized third moment; positive means a longer right tail, negative a longer left tail of returns.
Excess Kurtosis
Excess Kurt=E[(R−μ)4]σ4−3\text{Excess Kurt} = \dfrac{\mathrm{E}\left[(R - \mu)^4\right]}{\sigma^4} - 3
Kurtosis minus 3 (the normal's value); positive signals fat tails and more frequent extreme returns.
Beta
βi=Cov(Ri,Rm)σm2=ρim σiσm\beta_i = \dfrac{\text{Cov}(R_i, R_m)}{\sigma_m^2} = \dfrac{\rho_{im}\,\sigma_i}{\sigma_m}
Systematic risk: covariance with the market over market variance, equivalently correlation times the volatility ratio.
Portfolio Variance
σp2=∑i∑jwiwj Cov(Ri,Rj)\sigma_p^2 = \sum_i \sum_j w_i w_j\,\text{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)
σT=σ1T\sigma_T = \sigma_1 \sqrt{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)=FV(1+rt)tB(t) = \dfrac{FV}{(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
rt=(FVB(t))1/t−1r_t = \left(\dfrac{FV}{B(t)}\right)^{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+π1−τi = r + \dfrac{\pi}{1 - \tau}
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)=[(1+rT)T(1+rt)t]1/(T−t)−1F(t,T) = \left[\dfrac{(1 + r_T)^{T}}{(1 + r_t)^{t}}\right]^{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)
Ft,T=T rT−t rtT−tF_{t,T} = \dfrac{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=∑tt Ct(1+y)tP0D = \dfrac{\sum_t \dfrac{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
Dmod=D1+y/mD_{mod} = \dfrac{D}{1 + y/m}
Scales Macaulay duration for discrete compounding to give the percentage price change per unit change in yield.
Portfolio Duration
Dp=∑iwiDiD_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(Ri)=Rf+βi[E(Rm)−Rf]E(R_i) = R_f + \beta_i\left[E(R_m) - R_f\right]
Single-factor required return: the risk-free rate plus beta times the market risk premium.
Regression t-Statistic
t=∣θ^∣sθ^t = \dfrac{|\hat{\theta}|}{s_{\hat{\theta}}}
Tests whether an estimated alpha or beta differs from zero by comparing the estimate to its standard error.
Jensen's Alpha
αp=E(Rp)−Rf−βp[E(Rm)−Rf]\alpha_p = E(R_p) - R_f - \beta_p\left[E(R_m) - R_f\right]
Expected return above the CAPM benchmark; the ex-ante forecasted alpha, also the intercept of an excess-return regression.
Realized (Ex-Post) Alpha
α=(Rp−Rf)−βp(Rm−Rf)\alpha = (R_p - R_f) - \beta_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
Ft,T=K×P0,TP0,tF_{t,T} = K \times \dfrac{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)
FT=P0 e(r+c−y)TF_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
Vt=Pt e(r+c−y)(T−t)−F0V_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=SC + 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(d1)−K e−rTN(d2)c = S\,N(d_1) - K\,e^{-rT} 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×σ×Days×Value\text{VaR} = Z \times \sigma \times \sqrt{\text{Days}} \times \text{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
SR=E(Rp)−RfσpSR = \dfrac{E(R_p) - R_f}{\sigma_p}
Excess return per unit of total risk; best for evaluating a stand-alone portfolio rather than an added position.
Treynor Ratio
TR=E(Rp)−RfβpTR = \dfrac{E(R_p) - R_f}{\beta_p}
Excess return per unit of systematic risk (beta); suited to an asset added to a well-diversified portfolio.
Sortino Ratio
Sortino=E(Rp)−RTDD\text{Sortino} = \dfrac{E(R_p) - R_{T}}{\text{DD}}
Like Sharpe but measures excess return over target RTR_T against downside deviation DD, penalizing only downside risk.
Information Ratio
IR=E(Rp)−RBTEIR = \dfrac{E(R_p) - R_{B}}{TE}
Active return over benchmark B divided by tracking error TE; excess return per unit of active-return volatility.
Return on VaR (RoVaR)
RoVaR=E(Rp)VaR\text{RoVaR} = \dfrac{E(R_p)}{\text{VaR}}
Expected return scaled by value at risk; useful when VaR captures the relevant downside of the distribution.
M-Squared (M2)
M2=Rf+σmσp[E(Rp)−Rf]M^2 = R_f + \dfrac{\sigma_m}{\sigma_p}\left[E(R_p) - R_f\right]
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+1T(1−α)(r0−rT)\mu^{*} = \mu + \dfrac{1}{T}(1 - \alpha)\left(r_{0} - r_{T}\right)
Gives the mean of stale (appraisal-based) returns, μ∗\mu^{*}, from the true mean μ\mu: the two differ only through the first and last periods, so the gap shrinks as the sample lengthens.
Smoothing Effect on Observed Volatility
σ(r∗)=σ(r)N\sigma(r^{*}) = \dfrac{\sigma(r)}{\sqrt{N}}
Smoothing returns over N periods understates true volatility by the square root of N, biasing reported risk downward.
Portfolio Variance (Markowitz)
σp2=∑i=1n∑j=1nwiwj Cov(Ri,Rj)\sigma_p^2 = \sum_{i=1}^{n} \sum_{j=1}^{n} w_i w_j \, \text{Cov}(R_i, R_j)
Variance of a portfolio of n assets; through the covariance cross-terms, low correlation between holdings lowers portfolio variance.
Portfolio Variance (Uncorrelated Funds)
V(Rp)=∑i=1nwi2 Var(Ri)V(R_p) = \sum_{i=1}^{n} w_i^2 \, \text{Var}(R_i)
Special case for uncorrelated funds: the covariance cross-terms vanish, leaving only the weighted sum of each fund's own variance.
Real Assets 16 items
Land as a call option (binomial)
C0=p max⁡(Vu−K, 0)+(1−p) max⁡(Vd−K, 0)1+rfC_0 = \dfrac{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(Rland)=p Rdev+(1−p) RundevE(R_{\text{land}}) = p\,R_{\text{dev}} + (1-p)\,R_{\text{undev}}
Probability-weighted expected return on land; p is the chance of development, blending the developed and undeveloped outcomes.
Direct capitalization value
V=NOICap RateV = \dfrac{\text{NOI}}{\text{Cap Rate}}
Values stabilized income property or farmland as a perpetuity of net operating income; NOI is income before financing costs and income taxes, with operating costs such as property taxes already deducted.
Capitalization rate
Cap Rate=NOIProperty Value\text{Cap Rate} = \dfrac{\text{NOI}}{\text{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+Rf)(1+E(RLP))(1+E(RRP))−1r = (1+R_f)\left(1+E(R_{LP})\right)\left(1+E(R_{RP})\right) - 1
Build-up discount rate compounding the risk-free rate with liquidity and risk premiums; additive approx r = RfR_f + LP + RP.
DCF income approach (real estate)
V0=∑t=1nNOIt(1+r)t+Vn(1+r)nV_0 = \sum_{t=1}^{n} \dfrac{\text{NOI}_t}{(1+r)^t} + \dfrac{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=Property PriceGross Income\text{GIM} = \dfrac{\text{Property Price}}{\text{Gross Income}}
Quick relative-value gauge; price per dollar of annual gross income, before vacancy and operating expenses.
Cash-on-cash return
CoC=NOI−Debt ServiceEquity Invested\text{CoC} = \dfrac{\text{NOI} - \text{Debt Service}}{\text{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=Stabilized NOITotal Development Cost\text{Development Yield} = \dfrac{\text{Stabilized NOI}}{\text{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=LoanProperty Value\text{LTV} = \dfrac{\text{Loan}}{\text{Property Value}}
Mortgage leverage gauge; first mortgages on stabilized commercial properties typically run 50 to 65 percent loan to value and usually do not exceed 75 percent.
Debt service coverage ratio (DSCR)
DSCR=NOIAnnual Debt Service\text{DSCR} = \dfrac{\text{NOI}}{\text{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)TF \le S\,e^{(r + c - y)T}
Upper bound on a commodity forward: r is financing, c storage, y the convenience yield. Buying and storing the commodity caps the forward here; because a physical commodity is hard to short, the forward can sit below it.
Calendar spread
Calendar Spread=FT+t−FT\text{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 YieldR = \text{Spot Return} + \text{Collateral Yield} + \text{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(PT)=P0 erTE(P_T) = P_0\,e^{rT}
Exhaustible-commodity spot prices should drift up at roughly the risk-adjusted rate; technology gains make this an upper bound.
Basis (futures)
Basis=S−FT\text{Basis} = S - F_T
Spot minus forward; equal in size to the present value of net carrying costs but opposite in sign, and it converges to zero as the contract nears delivery.
Private Equity 9 items
Total Addressable Market (TAM) Valuation
Value=Industry Revenue×Market Share×Revenue Multiple\text{Value} = \text{Industry Revenue} \times \text{Market Share} \times \text{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=EBITDA×EBITDA Multiple(1+IRR)T\text{Value} = \dfrac{\text{EBITDA} \times \text{EBITDA Multiple}}{(1 + \text{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\text{POST} = \text{PRE} + \text{Investment}
Company value right after a financing round: the negotiated pre-money value plus the new capital injected.
VC Ownership Proportion
Ownership=InvestmentPOST\text{Ownership} = \dfrac{\text{Investment}}{\text{POST}}
Equity stake a new round's investor receives, before any dilution from future financing rounds.
Growth Equity Times-Revenue Method
Value=Annual Revenue×Revenue Multiple(1+IRR)T\text{Value} = \dfrac{\text{Annual Revenue} \times \text{Revenue Multiple}}{(1 + \text{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=CFT(1+g)r−g\text{Exit Value} = \dfrac{CF_{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=(Exit Equity ValueInitial Equity)1/T−1\text{IRR} = \left(\dfrac{\text{Exit Equity Value}}{\text{Initial Equity}}\right)^{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=Market PriceNAV−1\text{Premium} = \dfrac{\text{Market Price}}{\text{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)
∑t=0TCFt(1+IRR)t=0\sum_{t=0}^{T}\dfrac{CF_t}{(1 + \text{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 10 items
Credit Loss Rate
Credit Loss Rate=Default Rate×(1−Recovery Rate)\text{Credit Loss Rate} = \text{Default Rate} \times (1 - \text{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≥Credit Loss Rate+Risk Premium\text{Credit Spread} \geq \text{Credit Loss Rate} + \text{Risk Premium}
A distressed-debt spread must at least cover expected credit losses plus a premium for bearing default risk.
Fixed-Rate Mortgage Payment
MP=MB×i1−(1+i)−nMP = MB \times \dfrac{i}{1 - (1+i)^{-n}}
Level monthly payment that fully amortizes a mortgage; underpins MBS and CMO cash flows.
Mortgage Amortization Split
Principalt=MP−i Bt−1\text{Principal}_t = MP - 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=P0 (1+r)n\text{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=∑iwi ri\text{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×Venture Debt\text{Warrant Equity} = \text{Warrant Coverage} \times \text{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
Vpost=Vpre+Equity Raised+Warrant EquityV_{post} = V_{pre} + \text{Equity Raised} + \text{Warrant Equity}
Firm value after a financing round, including the equity created by venture-debt warrant coverage.
Warrant Equity Percentage
Warrant %=Warrant EquityVpost\text{Warrant \%} = \dfrac{\text{Warrant Equity}}{V_{post}}
Share of the firm a venture lender ends up owning once its attached warrants are exercised.
CDS Mark-to-Market Adjustment
MTM≈(St−S0)×D×Notional\text{MTM} \approx (S_{t} - S_{0}) \times D \times \text{Notional}
Value gained by a CDS protection buyer when the market spread St rises above the contracted spread S0, scaled by risky duration.
Hedge Funds 7 items
Simple moving average (SMA)
SMAt(n)=1n∑i=1npt−i\text{SMA}_t(n) = \dfrac{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)
WMAt(n)=∑i=1n(n−i+1) pt−in(n+1)/2WMA_t(n) = \dfrac{\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)
EMAt(λ)=λ Pt−1+(1−λ) EMAt−1(λ)\text{EMA}_t(\lambda) = \lambda\,P_{t-1} + (1-\lambda)\,\text{EMA}_{t-1}(\lambda)
Recursive smoothing with parameter λ (0<λ<1); weights on older prices decay geometrically and higher λ reacts faster.
Relative strength index (RSI)
RSI=100−1001+U/D\text{RSI} = 100 - \dfrac{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×Risk Loading×EquityNotional×RVolTRVolRN = \text{SF} \times \dfrac{\text{Risk Loading}\times\text{Equity}}{\text{Notional}} \times \dfrac{\text{RVol}_T}{\text{RVol}_R}
Scales position size so realized risk tracks vol target RVolTRVol_T; SF sets direction/strength, RVolRRVol_R is forecast vol.
Modified duration price change
ΔPP≈−Dmod×Δy\dfrac{\Delta P}{P} \approx -D_{\text{mod}} \times \Delta 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×max⁡(NAVt−HWMt−1, 0)\text{Fee} = p \times \max(NAV_t - \text{HWM}_{t-1},\, 0)
Performance fee p charged only on NAV above the high-water mark: the highest NAV on which an incentive fee has been paid, read on fee calculation dates, so recouped losses earn no fee.
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