Free CAIA Level II Formula Sheet (2026)

Every CAIA Level II formula you need on the test, grouped by topic and rendered with full math notation. 89 formulas across 8 topics, calibrated to the 2026 syllabus. Free forever, no signup required.

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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 Account=Current Account+Capital Account\Delta\text{Reserve Account} = \text{Current Account} + \text{Capital Account}
Surpluses count positive and deficits negative. Reserves rise when the current and capital account balances net positive and fall when they net negative.
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 ρi σi=σpNw_i\,\rho_i\,\sigma_i = \dfrac{\sigma_p}{N}
Weights are adjusted until each asset's risk contribution (weight times volatility times correlation with the portfolio) is one Nth of portfolio volatility. They equal inverse-volatility weights only in special cases, such as equal pairwise correlations.
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)
R−L GR - L\,G
Surplus performance is measured per unit of assets as the asset return R minus L times the liability growth rate G, where L is liabilities relative to assets. The plan maximizes expected asset return while penalizing the variance of this quantity, so assets that move with liabilities lower surplus 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
The per-period compounded (geometric) return is roughly the arithmetic mean return minus half the variance, so lowering volatility raises the geometric mean. It does not raise the expected future value, which depends on the arithmetic mean.
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 15 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 (IT/It)+NAVT∑tCt (IT/It)\text{KS-PME} = \dfrac{\sum_t D_t\,(I_T/I_t) + \text{NAV}_T}{\sum_t C_t\,(I_T/I_t)}
Each distribution and contribution is compounded to the terminal date by the public index, and the remaining NAV is added to the distribution side. A ratio above 1.0 means the fund outperformed the index.
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.
Methods and Models 17 items
Expected loss (credit risk)
EL=PD×LGD,LGD=EAD×(1−RR)\text{EL} = \text{PD} \times \text{LGD}, \quad \text{LGD} = \text{EAD} \times (1 - \text{RR})
Expected credit loss is the default probability times loss given default. LGD is a currency amount: exposure at default less the recovered portion, where RR is the PV of recovery divided by EAD.
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.
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 sigmaEsigma_E = (A/E) times delta times sigmaAsigma_A.
Binomial up and down factors
Su=S (1+u),Sd=S (1+d),u>0>dS_u = S\,(1 + u), \quad S_d = S\,(1 + d), \quad u > 0 > d
Each period multiplies the price by one plus the up move u or one plus the down move d, so up-then-down ends where down-then-up does. Moves are given (such as plus or minus 10%), and d need not be the reciprocal of u.
Risk-neutral binomial probability
p=r−du−dp = \dfrac{r - d}{u - d}
With up move 1+u1+u, down move 1+d1+d (d<0<ud<0<u) and periodic riskless rate r, p makes the expected price grow at r, so discounted prices are martingales. It is not the real-world up probability.
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.
Fama-French three-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, small minus big) and value (HML, high minus low book-to-market) premia to the CAPM market factor.
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
Rannual=(1+Total profitInitial investment)365/Days−1R_{\text{annual}} = \left(1 + \dfrac{\text{Total profit}}{\text{Initial investment}}\right)^{365/\text{Days}} - 1
Total profit: long target gain, short acquirer gain or loss, dividends paid on the short and the short rebate. If the deal fails, both legs can lose.
Convertible arbitrage hedge ratio
Nshort=Vconv×ΔSN_{\text{short}} = \dfrac{V_{\text{conv}} \times \Delta}{S}
Delta is the percentage change in the convertible's value for a 1% stock move; this short makes the position delta neutral. Profit comes from gamma when realized volatility beats implied, plus coupon and short rebate, less dividends and financing.
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 11 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).
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
Top-down cloning: regress excess fund returns on excess tradeable factor returns (equity, credit, trend), then hold factors at the betas; cash weight is 1 minus the sum of betas.
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 2 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.
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 TNT_N is below T0T_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.
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What else is free at FreeFellow for CAIA Level II candidates?
The full original question bank is free with an account, subject to usage limits. Worked solutions, written lessons, mixed practice, and your readiness score stay free. The formula sheet is free too. Fellow is $39 per month or $79 per quarter, per exam family (USD). Fellow Plus is $49 per month, $99 per quarter, or $199 per year, per exam family (USD). Every annual plan is Fellow Plus. Fellow adds timed mock exams, spaced-repetition flashcards, performance analytics, and a personalized study plan. AI grading: 5 attempts a day on Fellow; Fellow Plus removes that allowance, subject to grading rate and usage limits.

Formula sheets for other exams

Exam P Exam FM Exam FAM Exam SRM Exam PA Exam ALTAM Exam ASTAM CFA Level I CFA Level II CFA L3 Portfolio Mgmt CFA L3 Private Wealth CFA L3 Private Markets

About FreeFellow

Jeffrey Ting, founder of FreeFellow
Jeffrey Ting
FSA, CFA · Founder

FreeFellow was built by Jeffrey Ting, a credentialed actuary and CFA charterholder who has passed thirteen of the hardest exams in finance, all on his first attempt. He paid four-figure prep fees along the way.

So he started writing his own questions, then lessons, then mock exams, until it grew into a full prep platform covering 40 exams with more than 47,000 original practice questions.

01
Cost shouldn't decide who gets in.

A CFA charter, a CPA license, an actuarial credential. Each opens a real career, but prep fees add to the cost of getting there. FreeFellow keeps its original question bank and written lessons free so you can study even if a paid course is out of reach.

02
Free should mean free.

The original question bank, the worked solutions, the topic lessons, mixed practice, and your readiness score stay free, forever. Full access needs a free account, and practice usage limits apply. Fellow adds AI grading on supported exams, spaced-repetition flashcards, full-length mock exams, analytics, and a study plan that adapts to your progress.

Meet the founder ›Read the values ›

Free forever

Put the formulas to work.

Every formula on this sheet shows up in the free CAIA Level II question bank, and every question carries a step-by-step solution.

Practice CAIA Level II questions free →

No credit card. No trial clock.