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
cfinancing=rfc_{\text{financing}}=r_f under fair-value pricing. rfr_f = risk-free financing rate. Net carry embedded in futures versus spot is cnet carry=rf−dc_{\text{net carry}}=r_f-d, where dd is the index dividend yield.
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=ICBR TCIR=IC\sqrt{BR}\,TC. IC=forecast accuracy/skill; BR=independent decisions; TC=transfer coefficient (constraint drag). Fundamental approaches typically emphasize research depth; quantitative approaches breadth. IC does not identify strategy type; 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)
Assuming a 10% daily participation limit, Days-to-trade=Position $0.10×ADV $\text{Days-to-trade}=\dfrac{\text{Position \$}}{0.10\times\text{ADV \$}}. Position $ = AUM × target weight; ADV $ = average 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)
Breadth in the fundamental law of active management
BRBR is the number of independent investment decisions made per year. Correlated securities or signals do not each count as independent decisions.
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
S=A−L,DS=(DAA−DLL)/S,ΔS≈−(DAA−DLL)ΔyS=A-L,\quad D_S=(D_AA-D_LL)/S,\quad\Delta S\approx-(D_AA-D_LL)\Delta y. A,LA,L=asset/liability values; DA,DLD_A,D_L=their modified durations; Δy\Delta y=common yield change. Zero first-order surplus sensitivity when DAA=DLLD_AA=D_LL.
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 spread = yield curve humps in the middle
A long-belly/short-wings position gains from falling belly yields and rising wing yields; the butterfly spread need not become negative.
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[(Pview−Pfwd)/P0]≈−Dmod(Δyview−Δyfwd)+12C[(Δyview)2−(Δyfwd)2]E[(P_{view}-P_{fwd})/P_0]\approx-D_{mod}(\Delta y_{view}-\Delta y_{fwd})+\tfrac12 C[(\Delta y_{view})^2-(\Delta y_{fwd})^2]. DmodD_{mod}=modified duration; CC=convexity. Both yield changes and risk measures use the same valuation baseline P0P_0.
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(ΔPbenchmark yields)+E(ΔPyield spreads)+E(FX)E(R)\approx Y+\text{Rolldown}+E(\Delta P_{\text{benchmark yields}})+E(\Delta P_{\text{yield spreads}})+E(\text{FX}).
All terms are return contributions; YY = coupon income including reinvestment income, and the remaining components are signed.
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=credit spread change. For corporates, spread duration ≈\approx modified duration. For floating-rate notes, spread duration measures discount-margin sensitivity over remaining life; rate duration links to next 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 = protection premium for the cash bond's issuer at the same or interpolated maturity (bps); Z-spread of the cash bond = zero-volatility spread (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 foreign-bond return in USD under covered interest parity
RUSD,hedged≈Rforeign+rUSD−rforeignR_{\text{USD,hedged}}\approx R_{\text{foreign}}+r_{\text{USD}}-r_{\text{foreign}}, using risk-free rates and returns for the same horizon. The differential adjusts total return, not OAS.
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 + explicit fees
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 tt, VtV_t = volume at time tt
Trade cost vs VWAP = Pexecution−VWAPP_{\text{execution}}-VWAP for buys
Limitation: manipulable; biased toward zero when the order accounts for much of the benchmark volume, potentially concealing market impact
Market impact cost
Buy cost vs pre-trade benchmark =(Pexec−Ppre)/Ppre=(P_{exec}-P_{pre})/P_{pre}. PexecP_{exec}=average execution price; PpreP_{pre}=pre-trade benchmark. Market impact=adverse price move caused by the order; benchmark cost may include unrelated moves. Temporary impact reverses; permanent does not. Urgency raises impact.
Explicit fees component of implementation shortfall
Fees=c×∣Sexec∣+exchange fees+taxes+other explicit charges\text{Fees}=c\times|S_{\text{exec}}|+\text{exchange fees}+\text{taxes}+\text{other explicit charges}
cc = commission per share; SexecS_{\text{exec}} = shares executed; taxes = transaction taxes.
Trading cost component of implementation shortfall
Trading=(Pexec−Parrival)×Sfilled\text{Trading}=(P_{\text{exec}}-P_{\text{arrival}})\times S_{\text{filled}}
PexecP_{\text{exec}} = average execution price; ParrivalP_{\text{arrival}} = arrival price; SfilledS_{\text{filled}} = signed shares filled: positive for purchases and negative for sales.
Delay cost component of implementation shortfall
Delay=(Parrival−Pdecision)×Sfilled\text{Delay}=(P_{\text{arrival}}-P_{\text{decision}})\times S_{\text{filled}}
ParrivalP_{\text{arrival}} = asset price when the order is released to the market for execution; PdecisionP_{\text{decision}} = PM decision price; SfilledS_{\text{filled}} = signed shares executed: positive for purchases and negative for sales.
Opportunity cost component of implementation shortfall
Opportunity=(Pclose−Pdecision)×Sunfilled\text{Opportunity}=(P_{\text{close}}-P_{\text{decision}})\times S_{\text{unfilled}}
PcloseP_{\text{close}} = closing price; PdecisionP_{\text{decision}} = decision price; SunfilledS_{\text{unfilled}} = signed unfilled shares: positive for purchases and negative for sales.
Case Study: Endowment 4 items
DV01-based hedge ratio
N=−DV01p/DV01hN=-DV01_p/DV01_h; negative means short the hedge. For a first-order rate cross-hedge, N=−(DV01p/DV01h)βyN=-(DV01_p/DV01_h)\beta_y, βy=Cov⁡(Δyp,Δyh)/Var⁡(Δyh)\beta_y=\operatorname{Cov}(\Delta y_p,\Delta y_h)/\operatorname{Var}(\Delta y_h). pp=portfolio, hh=hedge; yield changes match DV01 measures; DV01hDV01_h is per hedge contract.
Effective spending rate
reff=St−GtMVtr_{\text{eff}}=\dfrac{S_t-G_t}{MV_t}
StS_t = current-period spending distribution; GtG_t = gifts and donations received during the period; 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
ww = smoothing weight; SS = spending; π\pi = inflation; rr = target rate; MVMV = moving-average portfolio market value (average AUM) specified by the spending policy.
Illiquidity budget
The illiquidity budget is the maximum portfolio weight permitted in the illiquid time-to-cash category, set alongside minimum allocations to more liquid categories to meet cash-flow needs, including under stress.
Topic 1 23 items
Unconstrained mean-variance optimal risky-asset weights with risk-free borrowing and lending
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 excess returns: Π=δΣwmkt\Pi=\delta\Sigma w_{mkt}. Blended excess: E(R)−rf1=[(τΣ)−1+PTΩ−1P]−1[(τΣ)−1Π+PTΩ−1Q]E(R)-r_f\mathbf{1}=[(\tau\Sigma)^{-1}+P^T\Omega^{-1}P]^{-1}[(\tau\Sigma)^{-1}\Pi+P^T\Omega^{-1}Q]. δ\delta=risk aversion; wmktw_{mkt}=market-cap weights; rfr_f=risk-free rate; QQ=excess-return or return-difference views.
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
i∗=rneutral+πe+0.5(Y^e−Y^trend)+0.5(πe−πtarget)i^*=r_{\mathrm{neutral}}+\pi^e+0.5(\hat Y^e-\hat Y^{\mathrm{trend}})+0.5(\pi^e-\pi^{\mathrm{target}}). i∗i^*: target nominal rate; rneutralr_{\mathrm{neutral}}: neutral real rate; πe,πtarget\pi^e,\pi^{\mathrm{target}}: expected/target inflation; Y^e,Y^trend\hat Y^e,\hat Y^{\mathrm{trend}}: expected/trend real GDP growth
Covered interest rate parity
F/S=(1+id)/(1+if)F/S = (1 + i_d)/(1 + i_f)
FF and SS are forward and spot exchange rates quoted as domestic currency per unit of foreign currency; idi_d and ifi_f are effective domestic and foreign interest rates over the forward contract’s term.
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)≈Coupon income+Rolldown return+E(ΔPbenchmark)+E(ΔPspread)+E(RFX)E(R_{FI}) \approx \text{Coupon income} + \text{Rolldown return} + E(\Delta P_{benchmark}) + E(\Delta P_{spread}) + E(R_{FX})
Coupon income includes coupons and reinvestment income relative to price; rolldown assumes an unchanged yield curve. Price and currency effects are signed return contributions.
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\omega = variance intercept; long-run variance = ω/(1−α−β)\omega/(1 - \alpha - \beta) when α+β<1\alpha + \beta < 1, α\alpha = weight on last squared shock, β\beta = weight on prior variance estimate
Singer-Terhaar blended risk premium
RP=w(σρ⋅SRg)+(1−w)(σ⋅SRs)RP = w(\sigma \rho \cdot SR_g) + (1-w)(\sigma \cdot SR_s)
ww = integration weight, σ\sigma = asset vol, ρ\rho = correlation with global portfolio, SRgSR_g = global Sharpe ratio, SRsSR_s = segmented-market 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}}
Set the coverage target using investor-specific stressed cash demands and available liquidity; there is no universal 2×2\times minimum.
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
AA = market value of plan assets, LL = present value of liabilities; surplus optimization maximizes expected surplus return less a penalty for surplus-return variance, where surplus return = (ΔA−ΔL)/A0(\Delta A - \Delta L)/A_0.
Mean-variance utility function
U=E(Rp)−0.5⋅λ⋅σp2U = E(R_p) - 0.5 \cdot \lambda \cdot \sigma_p^2
E(Rp)E(R_p) = expected portfolio return, λ\lambda = risk aversion coefficient (typically 1–10; 0 for risk neutrality), σp2\sigma_p^2 = 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+Trading+OpportunityIS=\text{Explicit}+\text{Delay}+\text{Trading}+\text{Opportunity}. Explicit=commissions/fees; delay=executed Q × (market-release−decision price); trading=executed Q × (share-weighted execution−market-release price); opportunity=unfilled-share return. Price differences shown for buys; reverse for sells.
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)
VWAPcostbuy=Q(Pexec−VWAP)VWAPcost_{buy}=Q(P_{exec}-VWAP); VWAPcostsell=Q(VWAP−Pexec)VWAPcost_{sell}=Q(VWAP-P_{exec}). QQ=positive executed quantity; PexecP_{exec}=share-weighted average execution price; VWAP=period volume-weighted average price. Positive cost is unfavorable.
Effective equity beta from a PE allocation
βeff=weq+wPE×βPE\beta_{eff} = w_{eq} + w_{PE} \times \beta_{PE}, assuming public-equity beta is 11 and all holdings other than public equity and PE have zero equity beta.
weqw_{eq} = public equity weight, wPEw_{PE} = PE weight, βPE\beta_{PE} = estimated equity beta of the specific PE portfolio.
Effective PE allocation including unfunded commitments
wPEeff=NAV+UCPw_{PE}^{eff} = \dfrac{NAV + UC}{P}
NAVNAV = PE net asset value, UCUC = unfunded commitments, PP = 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
0.050.05 = 5% IRS minimum distribution floor, cc = investment expenses as a fraction of assets, π\pi = inflation
DV01 (price value of a basis point)
DV01=Dmod×MV×0.0001DV01 = D_{mod} \times MV \times 0.0001
DmodD_{mod} = modified duration, MVMV = the bond’s full price per 100 of par
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
Assuming advisory fees reduce taxable investment returns: r=(S−I)/V+π1−t+fr = \dfrac{(S - I)/V + \pi}{1 - t} + f
SS = spending need, II = other income, VV = portfolio value, π\pi = inflation, tt = tax rate, ff = 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
What are the investment priorities of a budget stabilization SWF?
Preserve capital and maintain high liquidity to cover fiscal deficits; avoid assets highly correlated with government revenue.
Norway-style SWF fiscal transfer (spending) rule
Tannual=rreal×VfundT_{annual} = r_{real} \times V_{fund}
rrealr_{real} = expected real return (about 3%), VfundV_{fund} = fund market value; the rule aims to preserve the fund's purchasing power over time.
How should commodity-funded savings SWFs address exposure to their funding source?
Avoid assets whose returns are highly correlated with the non-renewable resources generating government revenue.
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}
Square root of the mean squared shortfall below MARMAR, using all nn observations; returns at or above MARMAR contribute zero.
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, wbw_b = portfolio, benchmark sector weights; Rb,iR_{b,i} = sector benchmark return; RbR_b = total benchmark return
Brinson-Fachler form. Brinson-Hood-Beebower drops RbR_b: Ai=(wp,i−wb,i)Rb,iA_i = (w_{p,i} - w_{b,i})R_{b,i}. Same total, different sector values
Fixed-income attribution components
Total active return = yield + roll + duration + curve + credit (spread) + currency + residual. Duration=parallel moves; curve=steepening/flattening/butterfly; credit=spread moves; currency=FX. Large residual may reflect pricing-approximation error on large yield moves; alone does not prove model failure.
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}}}
Both barred returns are geometric averages over periods when benchmark return is negative; <100%<100\% means the manager dampens losses.
Upside capture ratio
UC=Rˉp,upRˉb,up\text{UC} = \dfrac{\bar{R}_{p,\text{up}}}{\bar{R}_{b,\text{up}}}
Both barred returns are geometric averages over periods when benchmark return is positive; >100%>100\% means the 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 $1 change in underlying
Protective put payoff
At expiration: Payoff=ST+max⁡(X−ST,0)\text{Payoff} = S_T + \max(X - S_T, 0)
= max⁡(ST,X)\max(S_T, X)
Profit = Payoff − (S_0 + p), where p = put premium
Limits downside while preserving upside
Collar payoff at expiration
Long stock + long put (X_L) + short call (X_H)
Payoff: ST+max⁡(XL−ST,0)−max⁡(ST−XH,0)S_T + \max(X_L - S_T, 0) - \max(S_T - X_H, 0)
= min⁡(max⁡(ST,XL),XH)\min(\max(S_T, X_L), X_H)
Limits gains above XHX_H, protects below XLX_L
Covered call payoff at expiration
Long stock + short call (X)
Payoff: ST−max⁡(ST−X,0)=min⁡(ST,X)S_T - \max(S_T - X, 0) = \min(S_T, X)
Profit = Payoff − S0S_0 + c (c = call premium received)
Caps upside; enhances income in flat/down markets
Number of bond futures to adjust portfolio duration
N=DDT−DDPDDfN = \frac{DD_T - DD_P}{DD_f}
DDTDD_T = target dollar duration, DDPDD_P = current dollar duration, DDfDD_f = dollar duration per futures contract (DDCTD/CFDD_{CTD}/CF, where DDCTDDD_{CTD} = modified duration × CTD market value for one contract, and CFCF = conversion factor)
Variance notional converted from vega notional
Nvar=Nvega2×σstrikeN_{var} = \frac{N_{vega}}{2 \times \sigma_{strike}}
NvegaN_{vega} = vega notional ($ 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}
For selling foreign currency forward with spot unchanged: FF = forward rate and SS = spot rate, both quoted as domestic currency per foreign currency; approximately equals the domestic minus foreign interest rate over the contract term.
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 RFC+RFXR_{FC} + R_{FX} drops the cross-product, whose magnitude is ∣RFCRFX∣|R_{FC}R_{FX}|; the approximation is appropriate when both returns are small.
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 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. 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.

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 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.