Free CFA Level III: Portfolio Management Formula Sheet (2026)

Every CFA L3 Portfolio Mgmt formula you need on the test, grouped by topic, rendered with full math notation. 32 formulas across 11 topics, calibrated to the 2026 syllabus. Free forever, no signup required.

32 Formulas
11 Topics
2026 Syllabus
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All CFA L3 Portfolio Mgmt Formulas

Index-Based Equity Strategies 1 item
Tracking error decomposition
TE2=TEfactor2+TEspecific2TE^2 = TE_{factor}^2 + TE_{specific}^2
Factor TE from active factor exposures × factor covariance matrix
Specific TE from active stock-specific bets
Index funds minimize both; active strategies accept higher TE
Active Equity Investing 2 items
Jensen's alpha
α=Rp−[Rf+βp(Rm−Rf)]\alpha = R_p - [R_f + \beta_p(R_m - R_f)]
CAPM-adjusted outperformance
Fundamental law link: E(α)=IC×BR×σA×TCE(\alpha) = IC \times \sqrt{BR} \times \sigma_A \times TC
Treynor ratio
Tp=Rp−RfβpT_p = \frac{R_p - R_f}{\beta_p}
Excess return per unit of systematic (beta) risk
Use when portfolio is one component of a larger portfolio
Liability-Driven & Index-Based Strategies 3 items
DV01 (Dollar Value of 01)
DV01=Modified duration×V10000DV01 = \frac{\text{Modified duration} \times V}{10{}000}
Dollar price change for a 1 bp decline in yield
V = portfolio market value
Futures contracts for duration adjustment
Nf=(Dtarget−Dportfolio)×VportfolioDfutures×VfuturesN_f = \frac{(D_{target} - D_{portfolio}) \times V_{portfolio}}{D_{futures} \times V_{futures}}
D = duration, V = dollar value
Positive N_f → buy futures (extend duration)
Negative N_f → sell futures (shorten duration)
Liability-driven surplus duration
ΔSurplus=(DA×A−DL×L)×Δy\Delta Surplus = (D_A \times A - D_L \times L) \times \Delta y
D_A = asset duration, A = asset value
D_L = liability duration, L = liability value
Immunize when: DA×A=DL×LD_A \times A = D_L \times L
Yield Curve Strategies 3 items
Key rate duration
%ΔP≈−∑kKRDk×Δyk\%\Delta P \approx -\sum_k KRD_k \times \Delta y_k
KRD_k = key rate duration at maturity k
Used for non-parallel yield curve shifts; sum of KRDs ≈ effective duration
Butterfly spread conditions
Net position: long belly, short wings (or reverse)
Positive butterfly = yield curve humps in the middle
Profit condition (long butterfly): 2y5Y<y2Y+y10Y2 y_{5Y} < y_{2Y} + y_{10Y}
Measures curvature of yield curve
Barbell vs bullet convexity
Barbell (short + long): higher convexity
Bullet (middle): lower convexity
At same duration: Cbarbell>CbulletC_{barbell} > C_{bullet}
Barbell wins in high vol; bullet in stable curves
Credit Strategies 2 items
Credit spread duration
%ΔP≈−Spread duration×Δs\%\Delta P \approx -\text{Spread duration} \times \Delta s
Δs\Delta s = change in credit spread
For corporate bonds: spread duration ≈ modified duration
For floating-rate notes: spread duration ≈ time to reset
Excess return over Treasuries
XR≈s⋅t−Δs⋅Ds−t⋅p⋅LXR \approx s \cdot t - \Delta s \cdot D_s - t \cdot p \cdot L
s=spread, t=horizon, DsD_s=spread duration, p=PD, L=loss rate
Trade Strategy & Execution 3 items
Implementation shortfall
IS=Paper portfolio gain−Actual portfolio gainInvestment decision valueIS = \frac{\text{Paper portfolio gain} - \text{Actual portfolio gain}}{\text{Investment decision value}}
Components: delay cost + trading cost + opportunity cost
Measures total cost of executing a trade vs decision price
VWAP benchmark
VWAP=∑t(Pt×Vt)∑tVtVWAP = \frac{\sum_t (P_t \times V_t)}{\sum_t V_t}
P_t = price at time t, V_t = volume at time t
Trade cost vs VWAP = (VWAP − execution price) for buys
Limitation: manipulable; meaningless for large orders
Market impact cost
Market impact=(Pexec−Ppre)/Ppre\text{Market impact} = (P_{exec} - P_{pre}) / P_{pre} (for buys)
P_exec = average execution price, P_pre = pre-trade benchmark
Temporary impact reverses; permanent impact does not
Higher urgency → more market impact
Case Study: Endowment 1 item
DV01-based hedge ratio
N=−DV01portfolioDV01hedgeN = -\frac{DV01_{portfolio}}{DV01_{hedge}}
Negative = short the hedge instrument
For cross-hedge: N=−DV01pDV01h×βspreadN = -\frac{DV01_p}{DV01_h} \times \beta_{spread}
Topic 1 4 items
Mean-variance optimal portfolio weight
w∗=1λΣ−1(μ−rf1)\mathbf{w}^* = \frac{1}{\lambda} \Sigma^{-1} (\mu - r_f \mathbf{1})
λ\lambda = risk aversion, Σ\Sigma = covariance matrix, μ\mu = expected returns
Corner portfolio blending
wA=E(RP)−E(RB)E(RA)−E(RB)w_A = \frac{E(R_P) - E(R_B)}{E(R_A) - E(R_B)}, wB=1−wAw_B = 1 - w_A
Blend two adjacent corner portfolios A and B to achieve target return E(R_P)
All blends lie on the efficient frontier
Black-Litterman expected return
Equilibrium: Π=δΣwmkt\Pi = \delta \Sigma w_{mkt}
Blended: E(R)=[(τΣ)−1+PTΩ−1P]−1[(τΣ)−1Π+PTΩ−1Q]E(R) = [(\tau\Sigma)^{-1} + P^T \Omega^{-1} P]^{-1}[(\tau\Sigma)^{-1}\Pi + P^T\Omega^{-1}Q]
δ\delta=risk aversion, wmktw_{mkt}=mkt cap weights
Portfolio rebalancing trigger (range-based)
Rebalance when: ∣wi−wi∗∣>Δi|w_i - w_i^*| > \Delta_i
wi∗w_i^* = target weight, Δi\Delta_i = tolerance band
Wider bands → lower costs, less precision
Correlation-adjusted bands: wider for high-correlation assets
Topic 2 3 items
Marginal Contribution to Risk (MCTR)
MCTRi=βi×σpMCTR_i = \beta_i \times \sigma_p
βi=Cov(Ri,Rp)σp2\beta_i = \frac{\text{Cov}(R_i, R_p)}{\sigma_p^2}
Measures risk added by a small increase in asset i's weight
Absolute Contribution to Risk (ACTR)
ACTRi=wi×MCTRi=wi×βi×σpACTR_i = w_i \times MCTR_i = w_i \times \beta_i \times \sigma_p
∑iACTRi=σp\sum_i ACTR_i = \sigma_p (contributions sum to total portfolio risk)
Risk budget = set target ACTRs
Tracking error
TE=σ(Rp−RB)=∑(rp,t−rB,t−αˉ)2T−1TE = \sigma(R_p - R_B) = \sqrt{\frac{\sum(r_{p,t} - r_{B,t} - \bar{\alpha})^2}{T-1}}
Also called active risk or tracking risk
Annualized: TEannual=TEmonthly×12TE_{annual} = TE_{monthly} \times \sqrt{12}
Topic 3 5 items
Information ratio
IR=Rˉp−RˉBσ(Rp−RB)=αˉTEIR = \frac{\bar{R}_p - \bar{R}_B}{\sigma(R_p - R_B)} = \frac{\bar{\alpha}}{TE}
αˉ\bar{\alpha} = mean active return, TE = tracking error
Measures active return per unit of active risk
Fundamental Law of Active Management
IR=IC×BRIR = IC \times \sqrt{BR}
IC = information coefficient, BR = investment breadth
Expected active return: E(RA)=IC×BR×σAE(R_A) = IC \times \sqrt{BR} \times \sigma_A (TC assumed = 1)
BHB attribution effects
Allocation: (wp,i−wB,i)(RB,i−RB)(w_{p,i} - w_{B,i})(R_{B,i} - R_B)
Selection: wB,i(Rp,i−RB,i)w_{B,i}(R_{p,i} - R_{B,i})
Interaction: (wp,i−wB,i)(Rp,i−RB,i)(w_{p,i} - w_{B,i})(R_{p,i} - R_{B,i})
Sum of all effects = total active return
Sharpe ratio
SRp=Rp−RfσpSR_p = \frac{R_p - R_f}{\sigma_p}
Reward-to-variability ratio using total risk
Sharpe of combined portfolio: SRC2=SRB2+IR2SR_C^2 = SR_B^2 + IR^2
M-squared (M²)
M2=(Rp−Rf)σmσp+RfM^2 = (R_p - R_f) \frac{\sigma_m}{\sigma_p} + R_f
Risk-adjusted return scaled to match market's volatility
M² > R_m → portfolio outperformed on risk-adjusted basis
Topic 4 5 items
Delta of call and put
Call: Δc=N(d1)∈(0,1)\Delta_c = N(d_1) \in (0, 1)
Put: Δp=N(d1)−1∈(−1,0)\Delta_p = N(d_1) - 1 \in (-1, 0)
Put-call: Δc−Δp=1\Delta_c - \Delta_p = 1
Approx change in option price for $1 change in underlying
Protective put payoff
At expiration: Payoff=ST+max⁡(X−ST,0)\text{Payoff} = S_T + \max(X - S_T, 0)
= max⁡(ST,X)\max(S_T, X)
Profit = Payoff − (S_0 + p), where p = put premium
Limits downside while preserving upside
Collar payoff at expiration
Long stock + long put (X_L) + short call (X_H)
Payoff: ST+max⁡(XL−ST,0)−max⁡(ST−XH,0)S_T + \max(X_L - S_T, 0) - \max(S_T - X_H, 0)
= min⁡(max⁡(ST,XL),XH)\min(\max(S_T, X_L), X_H)
Limits gains above X_H, protects below X_L
Covered call payoff at expiration
Long stock + short call (X)
Payoff: ST−max⁡(ST−X,0)=min⁡(ST,X)S_T - \max(S_T - X, 0) = \min(S_T, X)
Profit = Payoff − S_0 + c (c = call premium received)
Caps upside; enhances income in flat/down markets
Variance swap payoff
Payoff=Nvega×(σrealized2−σstrike2)\text{Payoff} = N_{vega} \times (\sigma_{realized}^2 - \sigma_{strike}^2)
Long variance swap profits when realized variance > strike variance
No delta-hedging needed; pure volatility exposure

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