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Free CAIA Level II Formula Sheet PDF, No Signup (2026)

Institutional Asset Owners 7 items

Expected economic life of a retirement fund

E L = - \frac{1}{ln ( 1 + R )} \times ln (\frac{Payment - R \times Assets}{Payment})

Years a fund lasts when inflation-growing withdrawals outpace real return R; a longevity-risk gauge.

Present value of a growing annuity

PV = \frac{Payment _{1}}{r - g} [1 - (\frac{1 + g}{1 + r})^{n}]

Prices n payments growing at constant rate g; used for inflation-linked retirement income.

Reserve account identity (balance of payments)

Δ Reserve = Δ Current Account + Δ 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 \times (1 - T)

Net profit retained after tax rate T; used to compare after-tax outcomes across instruments.

Section 1256 blended tax rate

T_{1256} = 0.40 T_{ST} + 0.60 T_{LTCG}

US tax on listed futures gains: 40% short-term, 60% long-term regardless of holding period.

Pension liability interest-rate sensitivity

% Δ Liabilities = - ModDur \times Δ y

Approximates the percent change in a plan's benefit obligation for a small move in the discount yield.

Risk parity portfolio weights

w_{i} \propto \frac{1/ σ _{i}}{\sum _{j} 1/ σ _{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 - \frac{λ}{2} σ^{2}

Core mean-variance objective: expected return penalized by half the variance times the risk-aversion coefficient lambda.

Implied Risk Aversion

λ = \frac{E [ R _{p} ] - R _{f}}{σ _{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 = \frac{1}{λ} \cdot \frac{E [ R - R _{0} ]}{σ ^{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 = \frac{1}{λ} Σ^{- 1} E [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 = \frac{1}{λ} \cdot \frac{E [ R - R _{0} ]}{σ ^{2}} + L \frac{δ}{σ ^{2}}

Adds a liability-hedging term so assets whose returns covary with liability growth receive a larger allocation.

Surplus Return (Asset-Liability Framework)

R_{S} = \frac{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 [R_{N e w}] - R_{f} > β_{N e w} (E [R_{p}] - R_{f})

A new asset improves the optimal portfolio only when its excess return clears this beta-scaled hurdle.

Portfolio Illiquidity Level

L_{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} - \frac{λ}{2} σ_{p}^{2} - ϕ L_{p}}

Mean-variance objective less an illiquidity penalty (phi), optimized subject to a factor-exposure cap.

Factor Exposure Regression

R_{i t} = a_{i} + b_{i} F_{t} + ε_{i t}

Regresses an asset's return on a risk factor; the slope is its factor exposure, extendable to several factors.

Portfolio Factor Exposure

b_{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

R C_{F_{k}} = ρ_{F_{k}} σ_{F_{k}} b_{k}

Each factor's share of portfolio volatility; summing all factor contributions plus the residual gives total risk.

Asset Risk Contribution

R C_{i} = ρ_{i} σ_{i} w_{i}

Asset i's share of total portfolio risk; contributions sum to portfolio volatility, the basis of risk budgeting.

Risk-Parity Condition

\frac{\partial σ _{p}}{\partial w _{i}} w_{i} = \frac{σ _{p}}{N}

Weights chosen so every asset contributes an equal 1/N of risk; inverse-volatility weighting approximates it.

Variance Drain (Geometric Return)

R_{C} \approx R - \frac{1}{2} σ^{2}

Compounded growth equals the arithmetic mean minus half the variance, so cutting volatility raises long-run wealth.

CPPI Risky-Asset Exposure

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

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

N_{t} = \frac{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 + K e^{- r T} = 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

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

r_{S, t} = r_{f} + α + β r_{q, t} + ε_{t}

Maps an illiquid asset's return onto a correlated liquid futures contract plus alpha and tracking error.

Optimal Futures Overlay (Illiquid Rebalancing)

F_{t} \approx (\frac{α}{r _{q, t}} + β) (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)

δ_{call} = N (d_{1}), δ_{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 = \frac{Δ c}{Δ 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)

R_{i t} - R_{f} = a_{i} + β_{i} (R_{m t} - R_{f}) + e_{i t}

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

Capital a CTA actively risks: cash and collateral posted plus the notional exposure added through leverage.

Capital at risk (CaR)

CaR % = \frac{\sum _{i = 1}^{n} s _{i} N _{i}}{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

P_{t}^{rep} = α P_{t}^{true} + α (1 - α) P_{t - 1}^{true} + \dots

Reported appraisal price is a geometrically decaying weighted average of current and past true prices.

First-order autocorrelation of returns

ρ = corr (R_{t}^{rep}, R_{t - 1}^{rep})

Correlation of a reported return with its own one-period lag; a high value flags smoothing that understates risk.

Unsmoothed (de-smoothed) return

R_{t}^{true} \approx \frac{R _{t}^{rep} - ρ R _{t - 1}^{rep}}{1 - ρ}

Strips serial correlation from reported returns to recover the more volatile underlying true return.

True volatility from smoothed volatility

σ_{true} = σ_{rep} \frac{1 + ρ}{1 - ρ}

Smoothing biases reported volatility downward; scaling up by this autocorrelation factor recovers the true risk.

Parametric VaR (normal)

VaR_{α} = - (μ + z_{α} σ)

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 = \frac{\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 (R_{p} - R_{b})

Volatility of active return versus a benchmark; the active-risk budget and the denominator of the information ratio.

Sortino ratio

Sortino = \frac{R _{p} - R _{target}}{σ _{down}}

Excess return over a target per unit of downside deviation; unlike Sharpe it penalizes only below-target volatility.

Downside deviation

σ_{down} = \frac{1}{n} \sum min (R_{i} - R_{target}, 0)^{2}

Volatility of returns falling below a target return; serves as the denominator of the Sortino ratio.

Treynor ratio

T = \frac{R _{p} - R _{f}}{β _{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 \times LGD \times 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} = r_{t} + κ (μ - r_{t}) + σ ε ~_{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 \times (1 - 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] = P D \times EAD \times (1 - RR)

Multiplies the default probability, the exposure at default, and the loss severity (LGD = 1 minus RR).

Merton d term

d = \frac{ln ( A _{t} / K ) + ( r + 0.5 σ _{A}^{2} ) τ}{σ _{A} τ}

Standardized distance of log asset value above the debt face; this single term feeds every Merton equity, debt, and default formula.

Merton equity value

E_{t} = A_{t} N (d) - K e^{- r τ} N (d - σ_{A} τ)

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)

P_{t} = K e^{- r τ} N (- d + σ_{A} τ) - 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

D_{t} = K e^{- r τ} - 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 [A_{T} \leq K] = 1 - N (d - σ_{A} τ)

Risk-neutral (q-measure) probability that firm assets finish below the debt face value K at the debt's maturity.

Merton credit spread

s_{t} = - \frac{1}{τ} ln [N (d - σ_{A} τ) + \frac{A _{t}}{K} e^{r τ} N (- d)]

Continuously compounded spread over the riskless rate that prices the firm's default risk into its debt.

Asset volatility from equity volatility

σ_{A} \approx σ_{E} \times \frac{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 = e^{σ t}, d = \frac{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 = \frac{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 = \frac{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

V_{T} = max (F + C, q S_{T})

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 = \frac{1}{1 + i} [0.5 (V_{u} + C) + 0.5 (V_{d} + C)]

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

V_{callable} = V_{straight} - V_{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 (R_{i}) - R_{f} = β_{i} [E (R_{m}) - R_{f}] + β_{s} SMB + β_{h} 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 + r_{d})^{t} \frac{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

R_{arb} = \frac{P _{offer} - P _{current}}{P _{current}} \times \frac{365}{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 = Δ_{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

i_{eff} = i_{HF} + i_{FoF} (1 - i_{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 = \frac{\sum D _{t} + NAV}{\sum C _{t}} = DPI + 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 = \frac{\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 = \frac{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 = \frac{Realized + Unrealized Value}{Invested Capital}

Gross deal-level multiple on invested capital; ignores timing of cash flows, unlike IRR.

Interim IRR (IIRR)

\sum_{t = 0}^{T} \frac{D _{t} - C _{t}}{( 1 + IIRR ) ^{t}} + \frac{NAV _{T}}{( 1 + 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

IIRR_{P} = \frac{1}{N} \sum_{i = 1}^{N} IIRR_{i}

Simple arithmetic average of the funds' IRRs; appropriate only when the funds are similarly sized.

Commitment-Weighted Portfolio IRR/IIRR

IIRR_{P} = \frac{\sum _{i = 1}^{N} C C _{i} IIRR _{i}}{\sum _{i = 1}^{N} C C _{i}}

Averages fund IRRs weighted by committed capital; reflects allocation sizing, not realized asset value.

Pooled Portfolio IRR/IIRR

\sum_{t = 0}^{T} \sum_{i = 1}^{N} \frac{C F _{i, t}}{( 1 + IIRR _{P} ) ^{t}} + \sum_{i = 1}^{N} \frac{NAV _{i, T}}{( 1 + 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 = \frac{F V ( D ) + NAV}{F V ( C )}

Distributions and contributions compounded forward at the public index; above 1.0 means the fund beat the market.

Inverse-Volatility (Equal Risk) Weighting

w_{i} = \frac{( σ _{i} ) ^{- 1}}{\sum _{k = 1}^{N} ( σ _{k} ) ^{- 1}}

Allocates fund-of-funds capital inversely to each fund's volatility; a simplified risk-parity weighting scheme.

Factor-Based Hedge Fund Replication

R_{H F, t} - r_{f} = \sum_{i = 1}^{K} β_{i} (F_{i, t} - r_{f}) + ε_{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 = \frac{σ _{target}}{σ _{t}} \times 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

R_{futures} = R_{spot} + R_{roll} + R_{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 = \frac{α}{ω} = \frac{Active return}{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 \approx IC \times 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 \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

Option price change per one-point rise in implied volatility; identical for matching calls and puts.

Option Gamma

γ = \frac{N ^{'} ( d )}{S σ 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 = - \frac{S N ^{'} ( d ) σ}{2 T}

At-the-money time decay with zero rate; negative and largest in magnitude as expiration nears.

VIX 30-day constant-maturity futures price

P_{30} = \frac{P _{s} ( T _{l} - 30 )}{T _{l} - T _{s}} + \frac{P _{l} ( 30 - T _{s} )}{T _{l} - T _{s}}

Interpolates the two nearest VIX futures to a constant 30-day horizon for term-structure and hedging signals.

Portfolio variance

σ_{p}^{2} = \sum_{i = 1}^{N} w_{i}^{2} σ_{i}^{2} + 2 \sum_{i < j} w_{i} w_{j} σ_{i} σ_{j} ρ_{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)

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} - 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} \frac{1 - T _{N}}{1 - T _{0}}}^{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 + f x) - 1 \approx r + f x

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

σ_{d}^{2} = σ_{f x}^{2} + σ_{r}^{2} + 2 cov (f x, r)

Currency-return variance plus local-asset variance plus twice their covariance, viewed from the home currency.

Volume-Weighted Average Price (VWAP)

VWAP = \frac{\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)

CVaR_{α} = E [L ∣ L \geq VaR_{α}]

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) = \frac{\int _{r}^{\infty} ( 1 - F ( x ) ) d x}{\int _{- \infty}^{r} F ( x ) d x}

Probability-weighted gains above threshold r over losses below; captures all moments for skewed, fat-tailed returns.

Long-short market-neutral PnL

R_{port} = w_{L} R_{L} - w_{S} R_{S} - c_{borrow} w_{S} - c_{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) \times 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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