CAIA Level II — Formula Sheet

Grouped by exam topic. Every entry carries a plain-English line saying what the formula actually means — cover that column and reproduce the formula, then cover the formula and reproduce the meaning.

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Performance Risk & VaR Models & credit Allocation Private markets Real assets Options & vol Asset owners

Performance & risk-adjusted return

Which denominator you divide by is the assumption you are making about what risk means.

Sharpe ratio
SR = (Rp − Rf) / σp
Excess return per unit of total volatility. Treats upside and downside deviation identically, so it flatters negatively skewed strategies.
Treynor ratio
T = (Rp − Rf) / βp
Excess return per unit of systematic risk. The right measure when the position is one sleeve inside an already diversified portfolio.
Jensen's alpha
α = Rp − [Rf + β(Rm − Rf)]
What is left after paying for the risk you took — but only for the risks the model actually contains. Every omitted factor lands here.
Information ratio
IR = (Rp − Rb) / TE,  TE = σ(Rp − Rb)
Active return per unit of active risk. The measure of whether deviating from the benchmark was worth it.
Appraisal ratio
AR = α / σ(ε)
Alpha per unit of idiosyncratic (unhedgeable) risk taken to get it. The cleanest measure of security-selection skill.
Sortino ratio & downside deviation
Sortino = (Rp − MAR) / σd,  σd = √[Σ min(Rt − MAR, 0)² / n]
Sharpe with only the bad half of the distribution in the denominator. Use whenever returns are skewed — i.e. nearly always in alternatives.
Calmar ratio & maximum drawdown
Calmar = annualised return / |max drawdown|
Return per unit of worst peak-to-trough pain. Speaks to the real constraint: the point at which a board or an investor gives up.
M² (Modigliani risk-adjusted performance)
M² = Rf + SRp × σm
Restates the Sharpe ratio as a return the client can understand: what the portfolio would have earned levered to market volatility.
Skewness & excess kurtosis
S = E[(R−μ)³]/σ³  ·  Kex = E[(R−μ)⁴]/σ⁴ − 3
Asymmetry and tail fatness. Negative skew plus high kurtosis is the signature of a strategy that is short insurance.
Volatility scaling (i.i.d.)
σannual = σmonthly × √12
Only valid without serial correlation. With positive autocorrelation (smoothed marks) it understates true multi-period risk.

VaR, CVaR & risk decomposition

VaR tells you the threshold; CVaR tells you what is past it.

Parametric VaR
VaRα = −(μ − zασ) × V  (z95=1.65, z99=2.33)
The loss threshold under a normal assumption. Fast and additive, and wrong in exactly the direction that matters for alternatives.
Cornish–Fisher adjusted z
z* = z + (z²−1)S/6 + (z³−3z)Kex/24 − (2z³−5z)S²/36
Bends the normal quantile to account for skew and fat tails. The standard patch when you must keep a parametric VaR.
Conditional VaR (expected shortfall)
CVaRα = E[ L | L > VaRα ]
The average loss given that you are in the tail. Coherent and convex, so it can be optimised directly — the reason mean-CVaR beats mean-VaR.
Marginal VaR
MVaRi = ∂VaRp/∂wi = (VaRp/σp) × cov(Ri,Rp)/σp
What the next dollar into this position costs in portfolio risk. Use for trade sizing; equal across positions at the risk minimum.
Component VaR
CVaRi = MVaRi × wi,  Σ CVaRi = VaRp
How existing risk is split across sleeves. Additive, so it is the basis of every risk budget.
Marginal risk contribution (volatility)
MRCi = ∂σp/∂wi = (Σw)i/σp;  RCi = wi × MRCi
The volatility version of the same decomposition. Risk parity is the portfolio where every RCi is equal.
Two-asset portfolio variance
σ²p = w²1σ²1 + w²2σ²2 + 2w1w2ρσ1σ2
All diversification lives in the third term. When ρ rises toward 1 in a crisis, that term stops helping.
Beta
βi = cov(Ri, Rm)/σ²m = ρi,m σi/σm
Sensitivity to the market. Note the second form: smoothing lowers measured σi and ρ, so private-asset betas are biased downward twice over.

Factor & credit models

CAPM
E(Ri) = Rf + βi[E(Rm) − Rf]
Only non-diversifiable risk is paid for. Every multi-factor model is a claim that CAPM left something priced out.
Multi-factor / Fama–French–Carhart
Ri − Rf = α + β1MKT + β2SMB + β3HML + β4MOM + ε
Expand the factor set until the intercept stops being an excuse. What remains in α is skill or an omitted factor you have not named yet.
Merton structural model
E0 = V0N(d1) − De−rTN(d2);  risky debt = risk-free debt − put on assets
Equity is a call on the firm's assets struck at the debt's face value. Which is why raising asset volatility transfers value from creditors to shareholders.
Distance to default (KMV)
DD = (VA − DP) / (VA σA);  DP ≈ STD + ½ LTD
How many standard deviations of asset value sit between the firm and its default point. Mapped to an empirical EDF, not to a normal tail.
Reduced-form survival & spread
P(survive to T) = e−λT;  spread ≈ λ(1 − R);  λ ≈ spread/(1 − R)
The spread pays you for expected loss. λ and recovery cannot be separated from the spread alone — you must assume one.
Expected loss
EL = PD × LGD × EAD,  LGD = 1 − recovery rate
The average credit cost you should already be pricing. Unexpected loss — the volatility around it — is what capital is held against.
Altman Z-score (public manufacturers)
Z = 1.2X₁ + 1.4X₂ + 3.3X₃ + 0.6X₄ + 1.0X₅  (Z < 1.81 distress)
A purely statistical distress screen from accounting ratios — no economic model of default, and it is calibrated to a specific population.
Vasicek and CIR short-rate processes
Vasicek: dr = κ(θ − r)dt + σ dW  ·  CIR: dr = κ(θ − r)dt + σ√r dW
Both pull the rate back to θ at speed κ. Only CIR's √r term keeps the rate non-negative by shutting off volatility at zero.
Risk-neutral binomial probability
p = (erΔt − d)/(u − d)
Not a forecast — the weight that makes today's price arbitrage-free. Never use it to estimate real-world outcomes.

Allocation, unsmoothing & risk budgeting

Mean-variance utility
U = E(Rp) − ½ λ σ²p
The objective MVO actually maximises. λ is the investor's risk aversion — and it is the only place their preferences enter.
Black–Litterman equilibrium returns
Π = λ Σ wmkt
Reverse-optimisation: the returns that would make the market portfolio optimal. Your neutral prior, so no view means no bet.
Black–Litterman posterior
E(R) = [(τΣ)−1 + PᵀΩ−1P]−1 [(τΣ)−1Π + PᵀΩ−1Q]
A precision-weighted blend of equilibrium and your views. Confident views (small Ω) move the answer; vague ones barely register.
Unsmoothing (Geltner)
r*t = (rt − α rt−1) / (1 − α),  α = first-order autocorrelation
Strips the appraiser's averaging out of the series. Same mean, higher volatility, higher correlation — and a much lower Sharpe.
Inverse-volatility (naive risk parity) weights
wi = (1/σi) / Σj(1/σj)
Equalises risk contributions when correlations are equal. Ignores expected returns entirely, and needs leverage to reach an equity-like target.
Leverage and levered return
Rlevered = L × Rasset − (L − 1) × Rborrow;  σlevered = L × σasset
Leverage scales both return and risk, but the Sharpe ratio only improves if the borrowing rate is below the risk-free assumption — it never is.
Fund-of-funds double fee (net to investor)
Rnet = [(Rgross − mf1) − c1·max(0, ·)] − mf2 − c2·max(0, ·)
Fees compound in layers, and the second layer's carry is charged on returns already net of the first. Always compute sequentially.

Private markets & fee waterfalls

IRR
Σ CFt / (1 + IRR)t = 0
A money-weighted return, so it is sensitive to when cash moves. Not additive across funds, and inflatable by delaying capital calls.
DPI, RVPI, TVPI (MOIC)
DPI = distributions/paid-in · RVPI = NAV/paid-in · TVPI = DPI + RVPI
DPI is money actually returned; RVPI is the manager's own mark. A high TVPI carried by RVPI is a valuation claim, not a result.
PME (Kaplan–Schoar)
KS-PME = PV(distributions, discounted at index) / PV(calls, discounted at index)
Did the fund beat putting identical cash flows into the public index? Above 1.0 means yes. Timing-manipulation-proof, unlike IRR.
Direct alpha
IRR of index-discounted cash flows = annualised out/under-performance
Converts the PME comparison into an annualised excess-return number, which is what committees actually want to hear.
European (whole-fund) waterfall order
1 return of capital → 2 preferred return → 3 GP catch-up → 4 80/20 split
Nothing to the GP until every LP dollar and the hurdle are back. The marks are in the ordering, not the arithmetic.
GP catch-up
catch-up amount = [c/(1−c)] × preferred paid  (c = carry %, 100% catch-up)
With 20% carry, the GP receives 0.25 × the preferred return before the 80/20 split resumes — restoring 20% of total profit.
Hedge fund fee with hurdle & high-water mark
fee = mf × NAV + c × max(0, NAVend − max(HWM, NAVstart(1+h)))
No performance fee until past both the prior peak and the hurdle. More frequent crystallisation raises the effective fee on a volatile path.
Takahashi–Alexander pacing (structure)
Ct = RC × uncalled; Dt = NAVt(1+g) × (t/L)B; NAVt = NAVt−1(1+g) + Ct − Dt
Projects the exposure path so you can size commitments. Its three sensitivities — call rate, distribution timing, growth — are where to stress it.

Real assets & commodities

Cap rate & property value
cap rate = NOI / value;  value = NOI / cap rate;  cap ≈ r − g
A perpetuity yield. Rising rates raise the cap rate and cut value unless NOI growth rises to match — the whole 2022 real-estate story.
NOI, DSCR, LTV
NOI = EGI − opex; DSCR = NOI / debt service; LTV = loan / value
The three covenant numbers. DSCR tests whether the cash flow services the debt; LTV tests whether the collateral covers it.
Equity multiple & levered equity return
multiple = total distributions / equity invested
The multiple ignores time; IRR ignores scale. Neither is sufficient alone, which is why real-estate underwriting quotes both.
Futures cost of carry & convenience yield
F = S e(r + u − y)T  (u = storage, y = convenience yield)
Contango when carry costs dominate; backwardation when the convenience of holding the physical does. Scarcity shows up as y.
Commodity futures total return
total = spot return + roll yield + collateral yield
You never earn the spot price move alone. In persistent contango, roll yield is negative and can dominate everything else.

Options, Greeks & volatility

Put–call parity
C + Ke−rT = P + S
The identity behind every structured payoff. Rearrange it to decompose any note or collar into parts you can price.
Black–Scholes
C = S N(d1) − Ke−rTN(d2); d1 = [ln(S/K)+(r+σ²/2)T]/(σ√T); d2 = d1 − σ√T
One volatility for all strikes and maturities — which is exactly the assumption the observed surface refutes. N(d2) is the risk-neutral probability of exercise.
The Greeks
Δ = ∂V/∂S · Γ = ∂²V/∂S² · ν = ∂V/∂σ · Θ = ∂V/∂t · ρ = ∂V/∂r
Gamma peaks at-the-money near expiry; vega peaks at-the-money for long-dated options. Different exposures to different kinds of "movement".
Delta-hedged option P&L
P&L ≈ ½ Γ S² (σ²realized − σ²implied) Δt
You are paid the difference between what actually moved and what you paid for. This is the volatility risk premium, expressed as a trade.
Variance swap payoff
payoff = Nvar × (σ²realized − K²var)
Linear in variance means convex in volatility: double the volatility, quadruple the payoff. Why short variance breaks so violently.
CPPI exposure rule
cushion = V − floor;  risky exposure = m × cushion
Synthetically replicates a protected payoff — until a gap larger than 1/m of the cushion breaks the floor before you can trade.

Asset owners & spending rules

Required return for a perpetual endowment
Rrequired ≈ spending rate + inflation + costs  (multiplicative: (1+s)(1+i)(1+c) − 1)
What you must earn to spend forever without shrinking in real terms. The additive version is an approximation; know both.
Geometric smoothing spending rule
St = w × St−1(1+i) + (1−w) × Rtarget × MVt−1
The weight w is the trade-off: high w gives a stable operating budget and more corpus drift; low w tracks the endowment's actual size.
Funded ratio & surplus
FR = assets / PV(liabilities);  surplus = A − L
The pension's real scoreboard. A falling discount rate raises L, so the ratio can deteriorate on a day when assets rose.
Surplus return & surplus volatility
Rsurplus = RA − (L/A) RL;  σ²s = σ²A + (L/A)²σ²L − 2(L/A)ρσAσL
Risk is measured against the liability, not against zero. Correlation with the liability is the entire point of an LDI hedging sleeve.
Duration hedge ratio
hedge ratio = (DA × A) / (DL × L)
Matches dollar duration, not duration. A fully hedged plan is immunised against a parallel rate move — and only that.
After-tax return (family office)
Rafter-tax = Rpre-tax × (1 − teffective);  teff weights income vs gains by turnover
Turnover converts deferred gains into current income, so a high-turnover strategy must clear a much higher pre-tax hurdle to win.

Real assets & private market mechanics

Textbook Chapters 13–16 — the most computable relationships in the private-asset parts.

Smoothing process (reported return)
Rt,rep = ρRt−1,rep + (1 − ρ)Rt,true
An appraisal index reports only part of this period's true move; the rest arrives later. That lag is what creates the autocorrelation.
Unsmoothed (true) return
Rt,true = (Rt,rep − ρRt−1,rep) / (1 − ρ)
Reverses the lag. Small reported moves can become extreme unsmoothed ones, because the correction amplifies noise as well as signal.
Variance and beta of the true series
σ²true = σ²rep × (1 + ρ)/(1 − ρ);  βtrue scales the same way
Always raises risk. Reported appraisal statistics understate volatility, beta and correlation — so they overstate Sharpe and diversification.
Cap rate and expected total return
cap rate = NOI / value;  E(R) ≈ cap rate + g  ⇔  cap rate ≈ E(R) − g
Cap-rate compression is a discount-rate event, not an income event. Decompose realised return into income, NOI growth and cap-rate movement.
Overcommitment ratio
OC = total commitments / resources available for commitment
Numerator is total commitments, not the excess. Below 100% wastes capacity; documented practice runs roughly 125–140% given that not all commitments are called.

Commodities, credit & insurance-linked

Textbook Chapters 22–24, 29 and 36.

Cost of carry / futures price
F = S(1 + r + storage − convenience yield)
Convenience yield rises as inventories fall, which is what flips the curve from contango into backwardation.
Commodity index return decomposition
total return = collateral return + spot return + roll return
Roll is negative in contango, positive in backwardation; spot mean-reverts, so over long horizons collateral and roll dominate.
Basis and roll yield
basis = S − F;  roll yield ≈ (Fnear − Ffar) / Fnear
Basis converges to zero at expiry; the roll captures that convergence each time the position is moved along the curve.
Hazard rate survival (reduced-form)
P(survive to t) = e−λt;  spread ≈ λ × (1 − recovery)
λ and recovery are not separately identified from spreads alone — one must be assumed. Spreads give risk-neutral, not physical, probabilities.
Distance to default (KMV)
DD = [E(VA) − DP] / (σA × E(VA))
Standard deviations of cushion above the default point, then mapped empirically to an expected default frequency rather than a normal distribution.
Cat bond expected loss & spread multiple
EL ≈ P(attachment) × expected severity;  multiple = spread / EL
Pricing convention is the multiple of modelled expected loss — so the whole valuation rests on the catastrophe model, not on a discount rate.
Convertible arbitrage hedge
shares shorted = Δ × conversion ratio × bonds held
Delta-hedging the equity leg leaves long volatility plus credit exposure — and a position whose survival depends on borrow and repo financing.

Variable key

Rp, Rf, Rm, Rb — portfolio, risk-free, market, benchmark return
σ, σd, TE — volatility, downside deviation, tracking error
β, ρ, ε — beta, correlation, residual
V, wi, λ — portfolio value, weight, risk aversion
Σ, Π, P, Q, Ω, τ — covariance, equilibrium returns, view matrix, view returns, view uncertainty, scalar
PD, LGD, EAD, R — default probability, loss given default, exposure, recovery
λ (credit) — default intensity / hazard rate
VA, σA, DP, DD — asset value, asset volatility, default point, distance to default
κ, θ, r — mean-reversion speed, long-run rate, short rate
S, K, T, C, P — spot, strike, time to maturity, call, put
Δ Γ ν Θ ρ — delta, gamma, vega, theta, rho
NOI, EGI, DSCR, LTV — net operating income, effective gross income, debt service coverage, loan-to-value
DA, DL, FR — asset duration, liability duration, funded ratio
mf, c, h, HWM — management fee, carry rate, hurdle, high-water mark
ρ (smoothing) — first-order autocorrelation of a reported series
OC, DP — overcommitment ratio, default point
F, S, EL — futures price, spot price, expected loss
RC, g, L, B — call rate, growth rate, fund life, bow (pacing model)
m — CPPI multiplier
Conventions vary between the curriculum, providers and practice. Where an exam question specifies a convention (e.g. setting μ = 0 in VaR, or a soft versus hard hurdle), follow the question. Back to the study desk.