CFA Institute Prep Provider

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

Every CFA L3 Portfolio Mgmt formula you need on the test, grouped by topic and rendered with full math notation. 113 formulas across 11 topics, calibrated to the 2026 syllabus. Free forever, no signup required.

113 formulas 11 topics 2026 syllabus Free forever
Print-ready PDF: 1080x1350 portrait, math pre-rendered, fonts embedded. Download once, study anywhere.
Download PDF →

All CFA L3 Portfolio Mgmt Formulas

Index-Based Equity Strategies 5 items
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
Implicit financing cost of an equity index future
ccarry=rf−dc_{carry} = r_f - d
rfr_f = risk-free rate (cost of carry), d = dividend yield of the index; embedded in the futures price versus spot
Tracking error of an index portfolio
TE=σ(Rp−Rb)TE = \sigma(R_p - R_b)
RpR_p = portfolio return, RbR_b = benchmark return, σ = standard deviation of the return differential (annualized)
Cash drag on an index portfolio
Drag=wcash×(Req−Rmm)\text{Drag} = w_{cash} \times (R_{eq} - R_{mm})
wcashw_{cash} = cash weight, ReqR_{eq} = equity return, RmmR_{mm} = money-market rate earned on cash
Index portfolio total return decomposition
Rp=Ridx−f−ct−ϵs+rlend+rcashR_p = R_{idx} - f - c_t - \epsilon_s + r_{lend} + r_{cash}
RidxR_{idx} = index return, f = fees, ctc_t = transaction costs, εsε_s = sampling residual, rlendr_{lend} = lending revenue, rcashr_{cash} = cash spread
Active Equity Investing 5 items
Fundamental law of active management
IR=IC×BR×TCIR = IC \times \sqrt{BR} \times TC
IC = information coefficient (forecast skill), BR = breadth (independent decisions), TC = transfer coefficient (constraint drag)
High IC + low BR = fundamental; low IC + high BR = quant; constraints push TC below 1
Active risk (tracking error)
σA=Var(RP−RB)\sigma_A = \sqrt{\text{Var}(R_P - R_B)}
RPR_P = portfolio return, RBR_B = benchmark return, σAσ_A = standard deviation of active return
Days-to-trade (capacity)
Days-to-trade=Position $0.10×ADV $\text{Days-to-trade} = \dfrac{\text{Position \$}}{0.10 \times \text{ADV \$}}. Position $ = AUM × target weight; ADV $ = avg daily dollar volume; if days-to-trade exceeds the strategy horizon, the fund is capacity-constrained.
Expected active return from information ratio
RA=IR×TER_A = IR \times TE
IR = information ratio, TE = tracking error (active risk)
Correlation-adjusted effective breadth
BReff=N1+(N−1)ρBR_{eff} = \frac{N}{1 + (N-1)\rho}
N = raw number of bets, ρ = average pairwise correlation across signals
Liability-Driven & Index-Based Strategies 4 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 NfN_f → buy futures (extend duration)
Negative NfN_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
DAD_A = asset duration, A = asset value
DLD_L = liability duration, L = liability value
Immunize when: DA×A=DL×LD_A \times A = D_L \times L
Macaulay duration of a bond portfolio
DMac=∑t=1Nt⋅CFt/(1+y)t∑t=1NCFt/(1+y)tD_{Mac} = \frac{\sum_{t=1}^{N} t \cdot CF_t/(1+y)^t}{\sum_{t=1}^{N} CF_t/(1+y)^t}
t = time of cash flow, CFtCF_t = cash flow at t, y = yield, N = number of periods
Yield Curve Strategies 7 items
Key rate duration
%ΔP≈−∑kKRDk×Δyk\%\Delta P \approx -\sum_k KRD_k \times \Delta y_k
KRDkKRD_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
Expected price change from rate view vs. forwards
E(ΔP)≈−D(Δyview−Δyfwd)+12C(Δyview)2E(\Delta P) \approx -D(\Delta y_{\text{view}} - \Delta y_{\text{fwd}}) + \tfrac{1}{2} C (\Delta y_{\text{view}})^2
D = duration, C = convexity, ΔyviewΔy_{view} = view yield change, ΔyfwdΔy_{fwd} = forward-implied change
Hedged cross-currency bond return under CIP
Rhedged≈Rlocal−(rlocal−rbase)R_{\text{hedged}} \approx R_{\text{local}} - (r_{\text{local}} - r_{\text{base}})
RlocalR_{local} = local bond return, rlocalr_{local} = local risk-free rate, rbaser_{base} = base-currency risk-free rate
Expected return of a yield curve strategy
E(R)=Y+Rolldown+E(ΔPview)−E(Credit Loss)±E(FX)E(R) = Y + \text{Rolldown} + E(\Delta P_{\text{view}}) - E(\text{Credit Loss}) \pm E(\text{FX})
Y = yield income, ΔPviewΔP_{view} = price change from rate view, FX = currency return
Expected bond return from a yield change
E(R)≈Y+Rolldown−DmodΔy+12C(Δy)2E(R) \approx Y + \text{Rolldown} - D_{\text{mod}} \Delta y + \tfrac{1}{2} C (\Delta y)^2
Y = yield income, DmodD_{mod} = modified duration, C = convexity, Δy = yield change
Credit Strategies 6 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
CDS-cash basis
Basis=CDS spread−Z-spreadcash\text{Basis} = \text{CDS spread} - \text{Z-spread}_{\text{cash}}
CDS spread = single-name or index protection premium (bps), Z-spreadcashspread_{cash} = zero-volatility spread on the cash bond (bps)
Option-adjusted spread
OAS=Z-spread−Option cost\text{OAS} = \text{Z-spread} - \text{Option cost}
Z-spread = zero-volatility spread (bps), Option cost = embedded option value from lattice/Monte Carlo (bps)
Conditional value at risk
CVaRα=E[L∣L>VaRα]\text{CVaR}_{\alpha} = E[L \mid L > \text{VaR}_{\alpha}]
L = portfolio loss, VaR_α = value at risk at confidence level α, E[·] = expected value conditional on tail breach
FX-hedged USD-equivalent OAS on foreign credit
OASUSD-eq=OASforeign−(rbase−rforeign)\text{OAS}_{\text{USD-eq}} = \text{OAS}_{\text{foreign}} - (r_{\text{base}} - r_{\text{foreign}})
rbaser_{base}, rforeignr_{foreign} = short rates in base and foreign currency; differential = rolling FX-forward hedge cost
Trade Strategy & Execution 7 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}
PtP_t = price at time t, VtV_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)
PexecP_{exec} = average execution price, PpreP_{pre} = pre-trade benchmark
Temporary impact reverses; permanent impact does not
Higher urgency → more market impact
Explicit fees component of implementation shortfall
Fees=c×Sexec+taxes\text{Fees} = c \times S_{exec} + \text{taxes}
c = commission per share, SexecS_{exec} = shares executed, taxes = transaction taxes
Trading cost component of implementation shortfall
Trading=(Pexec−Parrival)×Sfilled\text{Trading} = (P_{exec} - P_{arrival}) \times S_{filled}
PexecP_{exec} = average execution price, ParrivalP_{arrival} = arrival price, SfilledS_{filled} = shares filled
Delay cost component of implementation shortfall
Delay=(Parrival−Pdecision)×Sfilled\text{Delay} = (P_{arrival} - P_{decision}) \times S_{filled}
ParrivalP_{arrival} = price when trader gets order, PdecisionP_{decision} = PM decision price, SfilledS_{filled} = shares executed
Opportunity cost component of implementation shortfall
Opportunity=(Pclose−Pdecision)×Sunfilled\text{Opportunity} = (P_{close} - P_{decision}) \times S_{unfilled}
PcloseP_{close} = closing price, PdecisionP_{decision} = decision price, SunfilledS_{unfilled} = shares not filled
Case Study: Endowment 4 items
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}
Effective spending rate
reff=StMVtr_{eff} = \dfrac{S_t}{MV_t}
StS_t = current period spending distribution, MVtMV_t = current portfolio market value
Endowment smoothing spending rule
St=w⋅St−1(1+π)+(1−w)⋅r⋅MVtS_t = w \cdot S_{t-1}(1+\pi) + (1-w) \cdot r \cdot MV_t
w = smoothing weight, S = spending, π = inflation, r = target rate, MV = current market value
Illiquidity budget
IB=TP−(Sp+CC+RB+SR)IB = TP - (Sp + CC + RB + SR)
TP = total portfolio, Sp = spending reserve, CC = capital calls, RB = rebalancing buffer, SR = stress reserve
Topic 1 23 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
Grinold-Kroner expected equity return
E(R)=D/P−ΔS+g+Δ(P/E)E(R) = D/P - \Delta S + g + \Delta(P/E)
D/P = dividend yield, ΔS = net share issuance, g = nominal earnings growth, Δ(P/E) = repricing
Taylor rule policy rate
it=r∗+πt+0.5(πt−π∗)+0.5(yt−y∗)i_t = r^* + \pi_t + 0.5(\pi_t - \pi^*) + 0.5(y_t - y^*)
r* = neutral real rate, π = inflation, π* = inflation target, y - y* = output gap
Covered interest rate parity
F/S=(1+id)/(1+if)F/S = (1 + i_d)/(1 + i_f)
F = forward rate, S = spot rate, idi_d = domestic interest rate, ifi_f = foreign interest rate
Fixed-income expected return building blocks
E(R)=rf+πe+TP+CP+LPE(R) = r_f + \pi^e + TP + CP + LP
rfr_f = real risk-free rate, π^e = expected inflation, TP = term premium, CP = credit premium, LP = liquidity premium
Expected fixed-income return decomposition
E(RFI)=YTM+Roll-down+ΔPyield−Lcredit−LFXE(R_{FI}) = YTM + \text{Roll-down} + \Delta P_{yield} - L_{credit} - L_{FX}
YTM = yield to maturity, ΔP = price change from curve shift, L = expected losses
GARCH(1,1) variance forecast
σt2=ω+αεt−12+βσt−12\sigma_t^2 = \omega + \alpha \varepsilon_{t-1}^2 + \beta \sigma_{t-1}^2
ω = long-run vol anchor, α = weight on last squared shock, β = weight on prior variance estimate
Singer-Terhaar blended risk premium
RP=w(σρ⋅SRg)+(1−w)(σ⋅SRg)RP = w(\sigma \rho \cdot SR_g) + (1-w)(\sigma \cdot SR_g)
w = integration weight, σ = asset vol, ρ = correlation with global portfolio, SRgSR_g = global Sharpe ratio
TAA permitted weight range around SAA
wTAA∈[wSAA−b,  wSAA+b]w_{TAA} \in [w_{SAA} - b,\; w_{SAA} + b]
wSAAw_{SAA} = policy weight, b = IPS-defined TAA band (e.g., ±5%)
Net after-cost tactical premium
αnet=αgross−TC−T\alpha_{net} = \alpha_{gross} - TC - T
αgrossα_{gross} = expected gross tactical alpha, TC = transaction costs, T = realized tax cost
Stress liquidity coverage ratio
LCRstress=Accessible liquid assetsstressCash demandsstressLCR_{stress} = \frac{\text{Accessible liquid assets}_{stress}}{\text{Cash demands}_{stress}}
minimum prudent target ≥ 2x for illiquid-heavy portfolios
Endowment spending rate
s=Annual spendingPortfolio values = \frac{\text{Annual spending}}{\text{Portfolio value}}
s = spending rate (e.g., 5% = $40M on $800M)
Number of correlation inputs required for MVO
Nρ=n(n−1)2N_\rho = \frac{n(n-1)}{2}
n = number of asset classes; MVO also needs n expected returns and n standard deviations
Pension funded ratio
FR=ALFR = \frac{A}{L}
A = market value of plan assets, L = present value of liabilities (e.g., PBO)
Dollar duration of a portfolio or liability
DD=MV×DDD = MV \times D
MV = market value (or PV of liabilities), D = modified or effective duration; used to size LDI hedges
Pension surplus
S=A−LS = A - L
A = market value of plan assets, L = present value of liabilities; surplus optimization maximizes return on S
Mean-variance utility function
U=E(Rp)−0.5⋅λ⋅σp2U = E(R_p) - 0.5 \cdot \lambda \cdot \sigma_p^2
E(R_p) = expected portfolio return, λ = risk aversion coefficient (1-10), σpσ_p² = portfolio variance
Geometric mean approximation from arithmetic mean and variance
G≈A−0.5σ2G \approx A - 0.5 \sigma^2
G = geometric (compound) mean return, A = arithmetic mean return, σ² = variance of returns
Total MVO inputs required for n asset classes
N=2n+n(n−1)2N = 2n + \frac{n(n-1)}{2}
N = total inputs, n = asset classes; counts n expected returns, n standard deviations, n(n-1)/2 correlations
Roy's safety-first ratio
SF=E(Rp)−RLσpSF = \dfrac{E(R_p) - R_L}{\sigma_p}, where RLR_L is the minimum acceptable (threshold) return. The optimal portfolio maximizes SF; under normality this minimizes the probability of a return below RLR_L.
Topic 2 26 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}
Implementation shortfall (decomposition)
IS=Explicit+Delay+Impact+OpportunityIS = \text{Explicit} + \text{Delay} + \text{Impact} + \text{Opportunity}
explicit = commissions/fees; delay = decision-to-desk drift; impact = price move from trade; opportunity = unfilled-share return.
Square-root market impact model
Impact∝shares/ADV\text{Impact} \propto \sqrt{\text{shares}/\text{ADV}}
shares = order size, ADV = average daily volume; doubling order size raises impact by ~41%, not 100%.
VWAP transaction cost (buy and sell)
VWAP costbuy=Pexec−VWAP\text{VWAP cost}_{\text{buy}} = P_{\text{exec}} - \text{VWAP}; VWAP costsell=VWAP−Pexec\text{VWAP cost}_{\text{sell}} = \text{VWAP} - P_{\text{exec}}
positive = unfavorable; VWAP = period volume-weighted average price.
Effective equity beta from a PE allocation
βeff=weq+wPE×βPE\beta_{eff} = w_{eq} + w_{PE} \times \beta_{PE}
weqw_{eq} = public equity weight, wPEw_{PE} = PE weight, βPEβ_{PE} ≈ 1.3 (PE equity beta)
Effective PE allocation including unfunded commitments
wPEeff=NAV+UCP+UCw_{PE}^{eff} = \dfrac{NAV + UC}{P + UC}
NAV = PE net asset value, UC = unfunded commitments, P = total portfolio value
Liquidity coverage ratio for an alternatives program
LCR=LCF12mLCR = \dfrac{L}{CF_{12m}}
L = liquid assets, CF12mCF_{12m} = next-12-month committed cash outflows (capital calls + benefits)
Total PE economic exposure
EPE=NAV+UCE_{PE} = NAV + UC
NAV = net asset value of PE holdings, UC = unfunded capital commitments
Endowment real return target
Rreal=s+cR_{real} = s + c
s = spending rate, c = management cost ratio
Insurer duration-matched immunization
DA×A=DL×LD_A \times A = D_L \times L
DAD_A = asset duration, A = assets, DLD_L = liability duration, L = liabilities
Foundation minimum nominal return target
R≥0.05+c+πR \geq 0.05 + c + \pi
0.05 = 5% IRS minimum distribution floor, c = costs, π = inflation
DV01 (price value of a basis point)
DV01=Dmod×MV×0.0001DV01 = D_{mod} \times MV \times 0.0001
DmodD_{mod} = modified duration, MV = market value of the bond/portfolio
Active share of an equity portfolio
AS=12∑i=1N∣wp,i−wb,i∣AS = \tfrac{1}{2} \sum_{i=1}^{N} |w_{p,i} - w_{b,i}|
wp,iw_{p,i} = portfolio weight in stock i, wb,iw_{b,i} = benchmark weight in stock i, N = combined universe
Required pre-tax nominal return for a private client
r=(S−I)/V+π1−t+fr = \dfrac{(S - I)/V + \pi}{1 - t} + f
S = spending need, I = other income, V = portfolio value, π = inflation, t = tax rate, f = advisory fees
Leveraged portfolio return on equity
rp=ri+VBVE(ri−rB)r_p = r_i + \dfrac{V_B}{V_E}(r_i - r_B)
rir_i = asset return, rBr_B = borrowing cost, VBV_B = borrowed value, VEV_E = equity
Human capital as present value of future labor income
HC=∑t=1NE[wt](1+r)tHC = \sum_{t=1}^{N} \dfrac{E[w_t]}{(1+r)^t}
wtw_t = expected labor income in year t, r = risk-adjusted discount rate, N = remaining working years
Real after-tax return approximation
rreal,at≈rnom−π−t⋅rnomr_{real,at} \approx r_{nom} - \pi - t \cdot r_{nom}
rnomr_{nom} = nominal return, π = inflation rate, t = tax rate on nominal gain
Modified duration from Macaulay duration
Dmod=DMac1+yD_{mod} = \dfrac{D_{Mac}}{1+y}
DMacD_{Mac} = Macaulay duration, y = periodic yield to maturity
Taxable-equivalent yield on a municipal bond
TEY=ymuni1−t\text{TEY} = \dfrac{y_{muni}}{1-t}
ymuniy_{muni} = muni pretax yield, t = investor's marginal tax rate
Economic net worth
ENW=FC+PV(HC)−PV(L)−PV(C)ENW = FC + PV(HC) - PV(L) - PV(C); FC = financial capital, PV(HC) = PV of human capital, PV(L) = PV of liabilities, PV(C) = PV of future consumption needs
SWF stabilization sub-fund sizing rule
AUMstab=f×G×nAUM_{stab} = f \times G \times n
f = fiscal dependence on commodity, G = annual government spending, n = years of shortfall coverage
Norway-style SWF fiscal transfer (spending) rule
Tannual=rreal×VfundT_{annual} = r_{real} \times V_{fund}
rrealr_{real} = expected real return (~3%), VfundV_{fund} = fund market value; principal preserved
Commodity SWF energy-sector exposure cap
wenergy≤wbench−Δw_{energy} \leq w_{bench} - \Delta
wbenchw_{bench} = benchmark energy weight, Δ\Delta = transition-risk tilt (e.g., 30%) to offset inflow correlation
SWF maximum single-year withdrawal under charter cap
Wmax=c×AUM‾3yW_{max} = c \times \overline{AUM}_{3y}
c = charter cap (e.g., 5%), AUM‾3y\overline{AUM}_{3y} = 3-year average AUM
Topic 3 12 items
Downside deviation (Sortino denominator)
σD=1n∑tmin⁡(Rt−MAR, 0)2\sigma_D = \sqrt{\frac{1}{n}\sum_{t} \min(R_t - MAR,\, 0)^2}
Standard deviation of returns below the minimum acceptable return only
Penalizes bad volatility; upside dispersion is ignored
Maximum drawdown
MDD=Trough value−Peak valuePeak valueMDD = \frac{\text{Trough value} - \text{Peak value}}{\text{Peak value}}
Largest peak-to-trough decline over the evaluation period
Captures the worst realized experience that volatility metrics smooth over
Drawdown duration
Time from the portfolio's peak until it recovers that prior peak
Measures how long the portfolio stayed underwater
Long durations test investor patience and can force liquidation at the bottom
Appraisal ratio
AR=ασεAR = \frac{\alpha}{\sigma_{\varepsilon}}
α\alpha = alpha from a factor regression, σε\sigma_{\varepsilon} = residual (security-specific) risk
Alpha per unit of active bet taken; denominator is residual risk, not tracking error
Brinson allocation effect
Ai=(wp,i−wb,i)(Rb,i−Rb)A_i = (w_{p,i} - w_{b,i})(R_{b,i} - R_b)
wpw_p = portfolio sector weight, wbw_b = benchmark sector weight, RbR_b,i = sector benchmark return, RbR_b = total benchmark return
Fixed-income attribution components
Total active return = duration effect + curve effect + credit (spread) effect + currency effect + residual
Duration = parallel moves, curve = steepening/flattening/butterfly, credit = spread changes, currency = FX
A large residual signals a broken attribution model
Sortino ratio
Sortino=Rp−MARσd\text{Sortino} = \frac{R_p - MAR}{\sigma_d}
RpR_p = portfolio return, MAR = minimum acceptable return, σdσ_d = downside deviation of returns below MAR
Brinson selection effect
Si=wb,i(Rp,i−Rb,i)S_i = w_{b,i}(R_{p,i} - R_{b,i})
wbw_b = benchmark sector weight, RpR_p,i = portfolio sector return, RbR_b,i = benchmark sector return
Downside capture ratio
DC=Rˉp,downRˉb,down\text{DC} = \dfrac{\bar{R}_{p,\text{down}}}{\bar{R}_{b,\text{down}}}
averaged over periods when benchmark return is negative; <100% means manager dampens losses
Upside capture ratio
UC=Rˉp,upRˉb,up\text{UC} = \dfrac{\bar{R}_{p,\text{up}}}{\bar{R}_{b,\text{up}}}
averaged over periods when benchmark return is positive; >100% means manager amplifies up markets
Up/down capture ratio
Up/Down Capture=Upside CaptureDownside Capture\text{Up/Down Capture} = \dfrac{\text{Upside Capture}}{\text{Downside Capture}}
ratio above 1.0 indicates favorable asymmetry
Symmetric performance-based fee
Fee=Base+s×(Rp−Rb)\text{Fee} = \text{Base} + s \times (R_p - R_b)
Base = base fee, s = sharing rate, RpR_p = portfolio return, RbR_b = benchmark return
Topic 4 14 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 $1 change in underlying
Protective put payoff
At expiration: Payoff=ST+max⁡(X−ST,0)\text{Payoff} = S_T + \max(X - S_T, 0)
= max⁡(ST,X)\max(S_T, X)
Profit = Payoff − (S_0 + p), where p = put premium
Limits downside while preserving upside
Collar payoff at expiration
Long stock + long put (X_L) + short call (X_H)
Payoff: ST+max⁡(XL−ST,0)−max⁡(ST−XH,0)S_T + \max(X_L - S_T, 0) - \max(S_T - X_H, 0)
= min⁡(max⁡(ST,XL),XH)\min(\max(S_T, X_L), X_H)
Limits gains above XHX_H, protects below XLX_L
Covered call payoff at expiration
Long stock + short call (X)
Payoff: ST−max⁡(ST−X,0)=min⁡(ST,X)S_T - \max(S_T - X, 0) = \min(S_T, X)
Profit = Payoff − S0S_0 + c (c = call premium received)
Caps upside; enhances income in flat/down markets
Number of bond futures to adjust portfolio duration
N=DDT−DDPDDfN = \frac{DD_T - DD_P}{DD_f}
DDTDD_T = target dollar duration, DDPDD_P = current dollar duration, DDfDD_f = dollar duration per futures contract (BPV adjusted by conversion factor)
Variance notional converted from vega notional
Nvar=Nvega2×σstrikeN_{var} = \frac{N_{vega}}{2 \times \sigma_{strike}}
NvegaN_{vega} = vega notional ($ per vol point), σstrikeσ_{strike} = strike volatility in whole-number percent
Variance swap payoff at maturity
Payoff=Nvar×(σrealized2−σstrike2)\text{Payoff} = N_{var} \times (\sigma^2_{realized} - \sigma^2_{strike})
NvarN_{var} = variance notional, σrealizedσ_{realized} = realized volatility (%), σstrikeσ_{strike} = strike volatility (%)
Number of equity futures to adjust portfolio beta
N=βT−βPβF×VPf×mN = \frac{\beta_T - \beta_P}{\beta_F} \times \frac{V}{P_f \times m}
βTβ_T = target beta, βPβ_P = current beta, βFβ_F = futures beta, V = portfolio value, PfP_f = futures price, m = multiplier
Roll yield on a currency forward hedge
Roll yield=F−SS\text{Roll yield} = \dfrac{F - S}{S}
F = forward rate, S = spot rate; approximately equals domestic minus foreign interest rate
Minimum-variance hedge ratio (MVHR)
h∗=ρA,B×σAσBh^* = \rho_{A,B} \times \dfrac{\sigma_A}{\sigma_B}
A = asset hedged, B = hedging instrument, ρ = correlation, σ = volatility
Domestic-currency return on a foreign asset
RDC=(1+RFC)(1+RFX)−1R_{DC} = (1 + R_{FC})(1 + R_{FX}) - 1; RFCR_{FC} = foreign asset return, RFXR_{FX} = % change in exchange rate (domestic per foreign); the approximation RFCR_{FC} + RFXR_{FX} drops the cross-product, material when either exceeds 5-10%
Maximum loss on short stock plus long call (synthetic long put)
Max Loss=K−S0+C0\text{Max Loss} = K - S_0 + C_0
K = call strike, S0S_0 = short entry price, C0C_0 = call premium paid
Long straddle breakeven prices
BE=K±(C0+P0)BE = K \pm (C_0 + P_0)
K = common strike, C0C_0 = call premium paid, P0P_0 = put premium paid
Put-call parity
S+P=C+PV(K)S + P = C + PV(K)
S = stock price, P = put premium, C = call premium, K = strike, PV(K) = present value of strike
Take the free CFA L3 Portfolio Mgmt diagnostic quiz →
Instant readiness score in about 6 minutes. No signup to start.

Frequently Asked Questions

Is the CFA L3 Portfolio Mgmt formula sheet free?
Yes. The full CFA L3 Portfolio Mgmt formula sheet is free, with no signup, no email, and no credit card required. 113 formulas across 11 topics, all rendered with the same KaTeX math notation used in the FreeFellow study app.
Can I download the CFA L3 Portfolio Mgmt formula sheet as a printable PDF?
Yes. A 1080x1350 portrait PDF (Instagram and LinkedIn carousel native size, also great for tablet study) is linked at the top of this page. The PDF is fully self-contained: math is pre-rendered, fonts are embedded, no internet connection needed once downloaded.
What's covered on the CFA L3 Portfolio Mgmt 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 CFA?
FreeFellow LLC is a CFA Institute Prep Provider. CFA Institute 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 CFA L3 Portfolio Mgmt 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 Private Wealth CFA L3 Private Markets CFP

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 passed thirteen of the hardest exams in finance on the first attempt, and paid four-figure prep fees for every one. The learning itself was always free. The price was a moat.

So he started writing his own questions, then lessons, then mock exams, until it grew into a full prep platform covering 40 finance credentials with more than 45,000 original practice questions. The name says exactly what it is: the question bank is free, and Fellow is what you become once you pass. FreeFellow LLC is a CFA Institute Prep Provider. Its CFA® exam materials are validated by CFA Institute for substantial curriculum coverage and updated annually.

01
Cost shouldn't decide who gets in.

The exam is a fair gate. A four-figure prep course is not. FreeFellow takes the second gate down, so the exam is the only one left.

02
Free should mean free.

No trial clock, no email gate, no credit card. The question bank, worked solutions, lessons, and readiness score stay free, and they are enough to pass.

03
Built by someone who sat where you sit.

He paid for the big-name courses, found nothing he respected, and built the prep he wished had existed. Not a marketing team that has never sat an exam.

Meet the founder ›Read the values ›

Free forever

Put the formulas to work.

Every formula on this sheet shows up in the free CFA L3 Portfolio Mgmt question bank, and every question carries a step-by-step solution.

Practice CFA L3 Portfolio Mgmt questions free →

No credit card. No trial clock.