Free CAIA Level II Formula Sheet (2026)

Every CAIA Level II formula you need on the test, grouped by topic, rendered with full math notation. 97 formulas across 8 topics, calibrated to the 2026 syllabus. Free forever, no signup required.

97 Formulas
8 Topics
2026 Syllabus
Free Forever

All CAIA Level II Formulas

Institutional Asset Owners 7 items
Expected economic life of a retirement fund
EL=−1ln⁡(1+R)×ln⁡(Payment−R×AssetsPayment)EL = -\dfrac{1}{\ln(1+R)} \times \ln\left(\dfrac{\text{Payment} - R \times \text{Assets}}{\text{Payment}}\right)
Years a fund lasts when inflation-growing withdrawals outpace real return R; a longevity-risk gauge.
Present value of a growing annuity
PV=Payment1r−g[1−(1+g1+r)n]\text{PV} = \dfrac{\text{Payment}_1}{r - g}\left[1 - \left(\dfrac{1+g}{1+r}\right)^{n}\right]
Prices n payments growing at constant rate g; used for inflation-linked retirement income.
Reserve account identity (balance of payments)
ΔReserve=ΔCurrent Account+ΔCapital Account\Delta\text{Reserve} = \Delta\text{Current Account} + \Delta\text{Capital Account}
A country runs a reserve surplus when its current and capital account balances net positive.
After-tax gain
After-tax gain=Pre-tax gain×(1−T)\text{After-tax gain} = \text{Pre-tax gain} \times (1 - T)
Net profit retained after tax rate T; used to compare after-tax outcomes across instruments.
Section 1256 blended tax rate
T1256=0.40 TST+0.60 TLTCGT_{1256} = 0.40\,T_{\text{ST}} + 0.60\,T_{\text{LTCG}}
US tax on listed futures gains: 40% short-term, 60% long-term regardless of holding period.
Pension liability interest-rate sensitivity
% ΔLiabilities=−ModDur×Δy\%\,\Delta\text{Liabilities} = -\text{ModDur} \times \Delta y
Approximates the percent change in a plan's benefit obligation for a small move in the discount yield.
Risk parity portfolio weights
wi∝1/σi∑j1/σjw_i \propto \dfrac{1/\sigma_i}{\sum_j 1/\sigma_j}
Inverse-volatility weights equalize each asset's risk contribution (not capital); often levered 2-3x.
Asset Allocation 22 items
Mean-Variance Utility
E[U(W)]=μ−λ2 σ2E[U(W)] = \mu - \dfrac{\lambda}{2}\,\sigma^2
Core mean-variance objective: expected return penalized by half the variance times the risk-aversion coefficient lambda.
Implied Risk Aversion
λ=E[Rp]−Rfσp2\lambda = \dfrac{E[R_p] - R_f}{\sigma_p^2}
Backs an investor's risk-aversion coefficient out of their chosen optimal portfolio's excess return and variance.
Optimal Weight (One Risky + Riskless Asset)
w=1λ⋅E[R−R0]σ2w = \dfrac{1}{\lambda}\cdot\dfrac{E[R - R_0]}{\sigma^2}
Weight in the risky asset rises with its excess return and falls with its variance and the investor's risk aversion.
MVO Optimal Weights
w=1λ Σ−1 E[R−R0]\mathbf{w} = \dfrac{1}{\lambda}\,\Sigma^{-1}\,E[\mathbf{R} - R_0]
General mean-variance solution: N-asset weights from the inverse covariance matrix times the vector of excess returns.
Optimal Weight with Growing Liabilities
w=1λ⋅E[R−R0]σ2+L δσ2w = \dfrac{1}{\lambda}\cdot\dfrac{E[R - R_0]}{\sigma^2} + L\,\dfrac{\delta}{\sigma^2}
Adds a liability-hedging term so assets whose returns covary with liability growth receive a larger allocation.
Surplus Return (Asset-Liability Framework)
RS=RA A−RL LA−LR_S = \dfrac{R_A\,A - R_L\,L}{A - L}
Return on surplus (assets minus liabilities); pension and insurer programs optimize surplus risk, not pure-asset risk.
Hurdle Rate for Adding an Asset
E[RNew]−Rf>βNew(E[Rp]−Rf)E[R_{New}] - R_f > \beta_{New}\left(E[R_p] - R_f\right)
A new asset improves the optimal portfolio only when its excess return clears this beta-scaled hurdle.
Portfolio Illiquidity Level
Lp=∑i=1NwiLiL_p = \sum_{i=1}^{N} w_i L_i
Weighted-average illiquidity of holdings (L from 0 liquid to 1 illiquid) that feeds the liquidity-penalty objective.
Liquidity Penalty Objective Function
max⁡{R‾p−λ2σp2−ϕ Lp}\max\left\{\overline{R}_p - \dfrac{\lambda}{2}\sigma_p^2 - \phi\,L_p\right\}
Mean-variance objective less an illiquidity penalty (phi), optimized subject to a factor-exposure cap.
Factor Exposure Regression
Rit=ai+biFt+εitR_{it} = a_i + b_i F_t + \varepsilon_{it}
Regresses an asset's return on a risk factor; the slope is its factor exposure, extendable to several factors.
Portfolio Factor Exposure
bP=∑i=1Nwibib_P = \sum_{i=1}^{N} w_i b_i
Portfolio exposure to a factor is the weighted average of asset exposures; optimizers cap it at a chosen target.
Factor Risk Contribution
RCFk=ρFk σFk bkRC_{F_k} = \rho_{F_k}\,\sigma_{F_k}\,b_k
Each factor's share of portfolio volatility; summing all factor contributions plus the residual gives total risk.
Asset Risk Contribution
RCi=ρi σi wiRC_i = \rho_i\,\sigma_i\,w_i
Asset i's share of total portfolio risk; contributions sum to portfolio volatility, the basis of risk budgeting.
Risk-Parity Condition
∂σp∂wi wi=σpN\dfrac{\partial \sigma_p}{\partial w_i}\,w_i = \dfrac{\sigma_p}{N}
Weights chosen so every asset contributes an equal 1/N of risk; inverse-volatility weighting approximates it.
Variance Drain (Geometric Return)
RC≈R‾−12σ2R_C \approx \overline{R} - \dfrac{1}{2}\sigma^2
Compounded growth equals the arithmetic mean minus half the variance, so cutting volatility raises long-run wealth.
CPPI Risky-Asset Exposure
St=m (Vt−Ft)S_t = m\,(V_t - F_t)
Constant-proportion insurance invests a fixed multiple of the cushion (value minus floor) in the risky asset.
CPPI Floor
Ft=AT e−r(T−t)F_t = A_T\,e^{-r(T - t)}
The CPPI floor is the present value of the minimum terminal wealth the investor requires at the horizon.
Constant-Mix Rebalancing
Nt=w VtStN_t = \dfrac{w\,V_t}{S_t}
Constant-mix resets holdings each period to a fixed weight, mechanically selling winners and buying losers.
Put-Call Parity
c+Ke−rT=p+Sc + K e^{-rT} = p + S
Long call plus PV of the strike equals long put plus the share; links protective-put and fiduciary-call insurance.
OBPI Protective-Put Portfolio
VT=N ST+Nmax⁡(K−ST, 0)V_T = N\,S_T + N\max(K - S_T,\,0)
One protective put per share floors value at N times the strike while preserving upside; N = V0/(S0 + put premium).
Single-Factor Model for an Illiquid Asset
rS,t=rf+α+β rq,t+εtr_{S,t} = r_f + \alpha + \beta\,r_{q,t} + \varepsilon_t
Maps an illiquid asset's return onto a correlated liquid futures contract plus alpha and tracking error.
Optimal Futures Overlay (Illiquid Rebalancing)
Ft≈(αrq,t+β)(kt−wt)F_t \approx \left(\dfrac{\alpha}{r_{q,t}} + \beta\right)(k_t - w_t)
Sizes a futures overlay to the gap between target and actual weights, scaled by the asset's alpha and beta.
Risk and Risk Management 16 items
Option delta (call and put)
δcall=N(d1),δput=N(d1)−1\delta_{\text{call}} = N(d_1), \quad \delta_{\text{put}} = N(d_1) - 1
Sensitivity of an option's value to a small move in the underlying; parity links them so put delta equals call delta minus 1.
Delta hedge ratio
h=ΔcΔSh = \dfrac{\Delta c}{\Delta S}
Shares of the underlying to trade per option to stay delta-neutral; equals the option's delta over a binomial step.
Single-factor market model (benchmarking)
Rit−Rf=ai+βi(Rmt−Rf)+eitR_{it} - R_f = a_i + \beta_i\left(R_{mt} - R_f\right) + e_{it}
Regresses excess fund returns on excess market returns; a nonzero intercept signals risk-adjusted over- or under-performance.
Trading level
Trading Level=Funding Level+Notional Level\text{Trading Level} = \text{Funding Level} + \text{Notional Level}
Capital a CTA actively risks: cash and collateral posted plus the notional exposure added through leverage.
Capital at risk (CaR)
CaR%=∑i=1nsi NiAccount Value\text{CaR}\% = \dfrac{\sum_{i=1}^{n} s_i \, N_i}{\text{Account Value}}
Loss as a share of equity if every position hits its stop on the same day; s is each position's stop-loss fraction.
Smoothed price as lags of true prices
Ptrep=αPttrue+α(1−α)Pt−1true+⋯P_{t}^{\text{rep}} = \alpha P_{t}^{\text{true}} + \alpha(1-\alpha)P_{t-1}^{\text{true}} + \cdots
Reported appraisal price is a geometrically decaying weighted average of current and past true prices.
First-order autocorrelation of returns
ρ=corr(Rtrep,Rt−1rep)\rho = \text{corr}\left(R_{t}^{\text{rep}}, R_{t-1}^{\text{rep}}\right)
Correlation of a reported return with its own one-period lag; a high value flags smoothing that understates risk.
Unsmoothed (de-smoothed) return
Rttrue≈Rtrep−ρRt−1rep1−ρR_{t}^{\text{true}} \approx \dfrac{R_{t}^{\text{rep}} - \rho R_{t-1}^{\text{rep}}}{1 - \rho}
Strips serial correlation from reported returns to recover the more volatile underlying true return.
True volatility from smoothed volatility
σtrue=σrep1+ρ1−ρ\sigma_{\text{true}} = \sigma_{\text{rep}} \sqrt{\dfrac{1+\rho}{1-\rho}}
Smoothing biases reported volatility downward; scaling up by this autocorrelation factor recovers the true risk.
Parametric VaR (normal)
VaRα=−(μ+zα σ)\text{VaR}_{\alpha} = -\left(\mu + z_{\alpha}\,\sigma\right)
Normal-distribution loss at confidence alpha (z is -1.645 at 95%, -2.326 at 99%); reported positive and understates fat tails.
Kaplan-Schoar PME (KS-PME)
KS-PME=∑tDt/Mt∑tCt/Mt\text{KS-PME} = \dfrac{\sum_t D_t / M_t}{\sum_t C_t / M_t}
Discounts PE distributions and contributions by a public index; above 1 means the fund beat the public market.
Tracking error
TE=σ(Rp−Rb)\text{TE} = \sigma\left(R_p - R_b\right)
Volatility of active return versus a benchmark; the active-risk budget and the denominator of the information ratio.
Sortino ratio
Sortino=Rp−Rtargetσdown\text{Sortino} = \dfrac{R_p - R_{\text{target}}}{\sigma_{\text{down}}}
Excess return over a target per unit of downside deviation; unlike Sharpe it penalizes only below-target volatility.
Downside deviation
σdown=1n∑min⁡(Ri−Rtarget, 0)2\sigma_{\text{down}} = \sqrt{\dfrac{1}{n}\sum \min\left(R_i - R_{\text{target}},\,0\right)^{2}}
Volatility of returns falling below a target return; serves as the denominator of the Sortino ratio.
Treynor ratio
T=Rp−RfβpT = \dfrac{R_p - R_f}{\beta_p}
Excess return per unit of systematic risk (beta); best for an asset added to a well-diversified portfolio.
Expected loss (credit risk)
EL=PD×LGD×EAD\text{EL} = \text{PD} \times \text{LGD} \times \text{EAD}
Average credit loss: default probability times loss given default (one minus recovery) times exposure at default.
Methods and Models 21 items
Vasicek short-rate model
r~t+1=rt+κ(μ−rt)+σ ε~t+1\tilde{r}_{t+1} = r_t + \kappa(\mu - r_t) + \sigma\,\tilde{\varepsilon}_{t+1}
Short rate mean-reverts to mu at speed kappa; omit the shock for the expected rate, and the CIR variant scales sigma by the square root of r to bar negatives.
Loss given default (LGD)
LGD=EAD×(1−RR)\text{LGD} = \text{EAD} \times (1 - \text{RR})
Exposure at default scaled by the non-recovered fraction; recovery rate RR equals recovered present value divided by EAD.
Expected credit loss
E[Loss]=PD×EAD×(1−RR)E[\text{Loss}] = PD \times \text{EAD} \times (1 - \text{RR})
Multiplies the default probability, the exposure at default, and the loss severity (LGD = 1 minus RR).
Merton d term
d=ln⁡(At/K)+(r+0.5 σA2)τσAτd = \dfrac{\ln(A_t/K) + (r + 0.5\,\sigma_A^2)\tau}{\sigma_A\sqrt{\tau}}
Standardized distance of log asset value above the debt face; this single term feeds every Merton equity, debt, and default formula.
Merton equity value
Et=At N(d)−K e−rτ N ⁣(d−σAτ)E_t = A_t\,N(d) - K\,e^{-r\tau}\,N\!\left(d - \sigma_A\sqrt{\tau}\right)
Prices equity as a Black-Scholes call on firm assets struck at the debt face value K; shareholders own the upside above debt.
Merton put value (bondholders' default exposure)
Pt=K e−rτ N ⁣(−d+σAτ)−At N(−d)P_t = K\,e^{-r\tau}\,N\!\left(-d + \sigma_A\sqrt{\tau}\right) - A_t\,N(-d)
Black-Scholes put on firm assets; bondholders are implicitly short this put, which captures their loss if assets fall below K.
Merton risky debt value
Dt=K e−rτ−PtD_t = K\,e^{-r\tau} - P_t
Risky debt equals a default-free discount bond minus the bondholders' short put; equivalently it is firm assets minus equity.
Merton probability of default
Pr⁡[AT≤K]=1−N ⁣(d−σAτ)\Pr[A_T \leq K] = 1 - N\!\left(d - \sigma_A\sqrt{\tau}\right)
Risk-neutral (q-measure) probability that firm assets finish below the debt face value K at the debt's maturity.
Merton credit spread
st=−1τln⁡ ⁣[N ⁣(d−σAτ)+AtKerτN(−d)]s_t = -\dfrac{1}{\tau}\ln\!\left[N\!\left(d - \sigma_A\sqrt{\tau}\right) + \dfrac{A_t}{K}e^{r\tau}N(-d)\right]
Continuously compounded spread over the riskless rate that prices the firm's default risk into its debt.
Asset volatility from equity volatility
σA≈σE×EA\sigma_A \approx \sigma_E \times \dfrac{E}{A}
Unlevers equity volatility to the firm's asset volatility for Merton; KMV refines it via the equity delta as sigma_E = (A/E) times delta times sigma_A.
Binomial up and down factors
u=eσt,d=1uu = e^{\sigma\sqrt{t}}, \quad d = \dfrac{1}{u}
Up and down jump sizes set from volatility so the price tree recombines; the down move is the reciprocal of the up move.
Risk-neutral binomial probability
p=R−du−dp = \dfrac{R - d}{u - d}
Risk-neutral (q-measure) up probability for pricing options as if investors were risk neutral; R is one plus the periodic risk-free rate.
Binomial backward-induction value
f=p fu+(1−p) fdRf = \dfrac{p\,f_u + (1 - p)\,f_d}{R}
Each node equals its risk-neutral expected next value discounted one period at the riskless rate; roll back to reach today's price.
Convertible bond value at maturity
VT=max⁡ ⁣(F+C,  q ST)V_T = \max\!\left(F + C,\; q\,S_T\right)
At maturity the holder takes the larger of redemption value (face plus coupon) or conversion value, then rolls the tree back by induction.
Bond price node on a binomial rate tree
V=11+i[0.5 (Vu+C)+0.5 (Vd+C)]V = \dfrac{1}{1 + i}\left[0.5\,(V_u + C) + 0.5\,(V_d + C)\right]
Straight-bond node value: discount the equal-weighted average of the next cum-coupon values at the node's one-period rate i.
Callable bond and embedded call value
Vcallable=Vstraight−VcallV_{\text{callable}} = V_{\text{straight}} - V_{\text{call}}
A callable bond is the straight bond minus the issuer's embedded call; cap each node's rolled-back value at the call price.
Fama-French / Carhart factor model
E(Ri)−Rf=βi ⁣[E(Rm)−Rf]+βs SMB+βh HMLE(R_i) - R_f = \beta_i\!\left[E(R_m) - R_f\right] + \beta_s\,\text{SMB} + \beta_h\,\text{HML}
Adds size (SMB) and value (HML) premia to the market factor; Carhart appends a momentum factor (winners minus losers).
Covered interest rate parity
(1+rd)t FtS0=(1+rf)t(1 + r_d)^t\,\dfrac{F_t}{S_0} = (1 + r_f)^t
Pins the forward FX rate to both countries' interest rates, so a fully hedged carry trade earns only the riskless rate.
Merger arbitrage annualized return
Rarb=Poffer−PcurrentPcurrent×365Days to closeR_{\text{arb}} = \dfrac{P_{\text{offer}} - P_{\text{current}}}{P_{\text{current}}} \times \dfrac{365}{\text{Days to close}}
Annualizes the deal spread to closing; multiply by the completion probability for a risk-adjusted expected return.
Convertible arbitrage hedge ratio
h=Δconv×qh = \Delta_{\text{conv}} \times q
Short delta times the conversion ratio (q) shares per bond to neutralize delta; rebalance with gamma; P/L from gamma, carry, and spread tightening.
Fund-of-funds compounded incentive fee
ieff=iHF+iFoF (1−iHF)i_{\text{eff}} = i_{\text{HF}} + i_{\text{FoF}}\,(1 - i_{\text{HF}})
Two fee layers stack: a 20% HF plus 10% FoF incentive takes about 28% of gross profit, atop roughly 3% combined management fees.
Accessing Alternative Investments 13 items
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 (cumulative distributions plus residual NAV) per dollar paid in; above 1.0 signals net value created.
DPI (Distributions to Paid-In)
DPI=∑Dt∑Ct\text{DPI} = \dfrac{\sum D_t}{\sum C_t}
Realized cash returned to investors per dollar of capital paid in; ignores remaining unrealized NAV.
RVPI (Residual Value to Paid-In)
RVPI=NAV∑Ct\text{RVPI} = \dfrac{\text{NAV}}{\sum C_t}
Unrealized residual value still held in the fund per dollar paid in; TVPI equals DPI plus RVPI.
MOIC (Multiple on Invested Capital)
MOIC=Realized+Unrealized ValueInvested Capital\text{MOIC} = \dfrac{\text{Realized} + \text{Unrealized Value}}{\text{Invested Capital}}
Gross deal-level multiple on invested capital; ignores timing of cash flows, unlike IRR.
Interim IRR (IIRR)
∑t=0TDt−Ct(1+IIRR)t+NAVT(1+IIRR)T=0\sum_{t=0}^{T} \dfrac{D_t - C_t}{(1+\text{IIRR})^{t}} + \dfrac{\text{NAV}_T}{(1+\text{IIRR})^{T}} = 0
Rate setting the PV of net cash flows plus the interim NAV to zero for a fund that has not yet fully realized.
Equally-Weighted Portfolio IRR/IIRR
IIRRP=1N∑i=1NIIRRi\text{IIRR}_P = \dfrac{1}{N} \sum_{i=1}^{N} \text{IIRR}_i
Simple arithmetic average of the funds' IRRs; appropriate only when the funds are similarly sized.
Commitment-Weighted Portfolio IRR/IIRR
IIRRP=∑i=1NCCi IIRRi∑i=1NCCi\text{IIRR}_P = \dfrac{\sum_{i=1}^{N} CC_i \, \text{IIRR}_i}{\sum_{i=1}^{N} CC_i}
Averages fund IRRs weighted by committed capital; reflects allocation sizing, not realized asset value.
Pooled Portfolio IRR/IIRR
∑t=0T∑i=1NCFi,t(1+IIRRP)t+∑i=1NNAVi,T(1+IIRRP)T=0\sum_{t=0}^{T} \sum_{i=1}^{N} \dfrac{CF_{i,t}}{(1+\text{IIRR}_P)^{t}} + \sum_{i=1}^{N} \dfrac{\text{NAV}_{i,T}}{(1+\text{IIRR}_P)^{T}} = 0
Merges all funds' cash flows into one stream then solves a single IRR; captures timing and scale (time-zero pooling).
KS-PME (Kaplan-Schoar Public Market Equivalent)
PME=FV(D)+NAVFV(C)\text{PME} = \dfrac{FV(D) + \text{NAV}}{FV(C)}
Distributions and contributions compounded forward at the public index; above 1.0 means the fund beat the market.
Inverse-Volatility (Equal Risk) Weighting
wi=(σi)−1∑k=1N(σk)−1w_i = \dfrac{(\sigma_i)^{-1}}{\sum_{k=1}^{N} (\sigma_k)^{-1}}
Allocates fund-of-funds capital inversely to each fund's volatility; a simplified risk-parity weighting scheme.
Factor-Based Hedge Fund Replication
RHF,t−rf=∑i=1Kβi(Fi,t−rf)+εtR_{HF,t} - r_f = \sum_{i=1}^{K} \beta_i (F_{i,t} - r_f) + \varepsilon_t
Regress excess fund returns on excess factor returns to clone exposures; cash weight equals 1 minus the sum of betas.
Trend-Following Position Sizing (vol targeting)
Position=σtargetσt×Signal\text{Position} = \dfrac{\sigma_{\text{target}}}{\sigma_t} \times \text{Signal}
Scales a long/short trend signal (e.g. moving-average crossover) inversely to recent volatility to hold portfolio vol constant.
Commodity Futures Total Return Decomposition
Rfutures=Rspot+Rroll+RcollateralR_{\text{futures}} = R_{\text{spot}} + R_{\text{roll}} + R_{\text{collateral}}
Futures return splits into spot, roll (positive in backwardation, negative in contango), and collateral yield.
Due Diligence and Selecting Managers 3 items
Information Ratio
IR=αω=Active returnTracking error\text{IR} = \dfrac{\alpha}{\omega} = \dfrac{\text{Active return}}{\text{Tracking error}}
Measures active return earned per unit of active risk (tracking error); a core gauge of manager skill.
Fundamental Law of Active Management
IR≈IC×Breadth\text{IR} \approx \text{IC} \times \sqrt{\text{Breadth}}
Decomposes the information ratio into skill (IC) and breadth, the number of independent active bets per year.
Leverage-Adjusted Return on Equity
ROE=(ROA×L)−[r×(L−1)]\text{ROE} = (\text{ROA} \times L) - [r \times (L - 1)]
Splits equity return into the levered asset return minus interest drag; L is the assets-to-equity ratio.
Volatility and Complex Strategies 14 items
Option Vega
ν=S N′(d)T\nu = S\,N'(d)\sqrt{T}
Option price change per one-point rise in implied volatility; identical for matching calls and puts.
Option Gamma
γ=N′(d)SσT\gamma = \dfrac{N'(d)}{S\sigma\sqrt{T}}
Rate of change of delta; peaks for at-the-money, near-expiry options and is positive for long calls and puts.
Option Theta (at-the-money)
θ=−S N′(d) σ2T\theta = -\dfrac{S\,N'(d)\,\sigma}{2\sqrt{T}}
At-the-money time decay with zero rate; negative and largest in magnitude as expiration nears.
VIX 30-day constant-maturity futures price
P30=Ps(Tl−30)Tl−Ts+Pl(30−Ts)Tl−TsP_{30} = \dfrac{P_s\left(T_l - 30\right)}{T_l - T_s} + \dfrac{P_l\left(30 - T_s\right)}{T_l - T_s}
Interpolates the two nearest VIX futures to a constant 30-day horizon for term-structure and hedging signals.
Portfolio variance
σp2=∑i=1Nwi2σi2+2∑i<jwiwjσiσjρij\sigma_p^{2} = \sum_{i=1}^{N} w_i^{2}\sigma_i^{2} + 2\sum_{i<j} w_i w_j \sigma_i \sigma_j \rho_{ij}
Combines each asset's own variance with the weighted pairwise covariances set by their correlations.
After-tax return, fully taxed
r∗=r(1−T)r^{*} = r(1 - T)
After-tax return when gains are taxed annually at the marginal rate; 10% taxed at 40% nets 6%.
After-tax return, tax-deferred wrapper
r∗={1+[(1+r)N−1](1−T)}1/N−1r^{*} = \left\{ 1 + \left[(1+r)^{N} - 1\right](1 - T) \right\}^{1/N} - 1
Gains compound pre-tax inside the wrapper, then the cumulative gain is taxed once at withdrawal.
After-tax return, deferral plus deduction wrapper
r∗={(1+r)N 1−TN1−T0}1/N−1r^{*} = \left\{ (1+r)^{N}\,\dfrac{1 - T_N}{1 - T_0} \right\}^{1/N} - 1
Deductible, deferred wrapper; matches the pre-tax return when rates are flat and beats it if T_N is below T_0.
Cross-border total return (home currency)
R=(1+r)(1+fx)−1≈r+fxR = (1 + r)(1 + fx) - 1 \approx r + fx
Home-currency return on a foreign asset blends the local return and the currency move; small cross-term dropped.
Variance of home-currency total return
σd2=σfx2+σr2+2 cov(fx,r)\sigma_d^{2} = \sigma_{fx}^{2} + \sigma_r^{2} + 2\,\text{cov}(fx, r)
Currency-return variance plus local-asset variance plus twice their covariance, viewed from the home currency.
Volume-Weighted Average Price (VWAP)
VWAP=∑(P×V)∑V\text{VWAP} = \dfrac{\sum (P \times V)}{\sum V}
Volume-weighted price benchmark; crypto trades above it read bullish, below bearish, for execution timing.
Conditional Value-at-Risk (CVaR / Expected Shortfall)
CVaRα=E[L∣L≥VaRα]\text{CVaR}_{\alpha} = E[L \mid L \geq \text{VaR}_{\alpha}]
Coherent, sub-additive tail risk: the average loss beyond VaR; normal case equals mu plus sigma times phi(z)/(1-alpha).
Omega ratio
Ω(r)=∫r∞(1−F(x))dx∫−∞rF(x) dx\Omega(r) = \dfrac{\int_r^{\infty} \left(1 - F(x)\right)dx}{\int_{-\infty}^{r} F(x)\,dx}
Probability-weighted gains above threshold r over losses below; captures all moments for skewed, fat-tailed returns.
Long-short market-neutral PnL
Rport=wLRL−wSRS−cborrowwS−cfinwLR_{\text{port}} = w_L R_L - w_S R_S - c_{\text{borrow}} w_S - c_{\text{fin}} w_L
Market-neutral return net of borrow and financing costs; dollar-neutral sets the long and short weights equal.
Universal Investment Considerations 1 item
Sustainability Materiality
Materiality=P(issue)×Expected Financial Loss\text{Materiality} = P(\text{issue}) \times \text{Expected Financial Loss}
Sizes an ESG issue's financial materiality as the chance it becomes relevant times the loss expected if it does.

Frequently Asked Questions

Is the CAIA Level II formula sheet free?
Yes. The full CAIA Level II formula sheet is free, with no signup, no email, and no credit card required. 97 formulas across 8 topics, all rendered with the same KaTeX math notation used in the FreeFellow study app.
Will there be a printable PDF version?
A printable PDF is rolling out shortly. In the meantime, the inline page below is print-friendly: most browsers print clean copies via the Print menu (the navigation, footer, and download CTA are hidden in print).
What's covered on the CAIA Level II formula sheet?
Every formula is grouped by official syllabus topic, with the formula in math notation plus a one-line note on when to use it (or a watch-out from CAIA, CFA, or other prep-provider commentary). Coverage is calibrated to the 2026 syllabus and refreshed when the corpus changes.
What is FreeFellow's relationship with CAIA?
FreeFellow LLC is a CAIA Association Preparatory Program Provider. CAIAA does not endorse, promote, review or warrant the accuracy or quality of the products or services offered by FreeFellow LLC.
What else is free at FreeFellow for CAIA Level II candidates?
The full question bank with detailed solutions, mixed practice, readiness tracking, lessons (where available), and the formula sheet are all free forever. Fellow ($59/quarter or $149/year per track) unlocks timed mock exams, spaced-repetition flashcards, performance analytics, AI essay grading, and a personalized study plan.
Practice CAIA Level II questions free →

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About FreeFellow

FreeFellow is a free exam prep library for actuarial (SOA & CAS), CFA, CFP, CPA, CAIA, GARP FRM, IRS Enrolled Agent, IMA CMA, and FINRA / NASAA securities licensing candidates. The entire question bank, written solutions, and lessons are free for every candidate, with no trial period and no credit card. Lessons include narrated audio, and every constructed-response item has a copy-to-AI prompt builder so candidates can paste their answer into their own ChatGPT or Claude for self-graded feedback; Fellow members get instant AI grading on essays against the official rubric (currently CFA Level III, expanding to other essay-bearing sections).

The 70% you need to pass (question bank, written solutions, lessons, formula sheet, mixed practice, readiness tracking) is free forever, with no trial period and no credit card. Become a Fellow ($59/quarter or $149/year per track) to unlock mock exams, flashcards with spaced repetition, performance analytics, AI essay grading, and a personalized study plan.