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Free CFA Level III: Portfolio Management Formula Sheet PDF, No Signup (2026)

Index-Based Equity Strategies 5 items

Tracking error decomposition

T E^{2} = T E_{f a c t or}^{2} + T E_{s p ec i f i c}^{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

c_{c a r r y} = r_{f} - d

— r_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

T E = σ (R_{p} - R_{b})

— R_p = portfolio return, R_b = benchmark return, σ = standard deviation of the return differential (annualized)

Cash drag on an index portfolio

Drag = w_{c a s h} \times (R_{e q} - R_{mm})

— w_cash = cash weight, R_eq = equity return, R_mm = money-market rate earned on cash

Index portfolio total return decomposition

R_{p} = R_{i d x} - f - c_{t} - ϵ_{s} + r_{l e n d} + r_{c a s h}

— R_idx = index return, f = fees, c_t = transaction costs, ε_s = sampling residual, r_lend = lending revenue, r_cash = cash spread

Active Equity Investing 5 items

Fundamental law of active management

I R = I C \times B R \times T C

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)

σ_{A} = Var (R_{P} - R_{B})

— R_P = portfolio return, R_B = benchmark return, σ_A = standard deviation of active return

Days-to-trade (capacity)

Days-to-trade = \frac{Position $}{0.10 \times 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

R_{A} = I R \times T E

— IR = information ratio, TE = tracking error (active risk)

Correlation-adjusted effective breadth

B R_{e f f} = \frac{N}{1 + ( N - 1 ) ρ}

— N = raw number of bets, ρ = average pairwise correlation across signals

Liability-Driven & Index-Based Strategies 4 items

DV01 (Dollar Value of 01)

D V 01 = \frac{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

N_{f} = \frac{( D _{t a r g e t} - D _{p or t f o l i o} ) \times V _{p or t f o l i o}}{D _{f u t u r es} \times V _{f u t u r es}}

D = duration, V = dollar value

Positive N_f → buy futures (extend duration)

Negative N_f → sell futures (shorten duration)

Liability-driven surplus duration

Δ S u r pl u s = (D_{A} \times A - D_{L} \times L) \times Δ y

D_A = asset duration, A = asset value

D_L = liability duration, L = liability value

Immunize when:

D_{A} \times A = D_{L} \times L

Macaulay duration of a bond portfolio

D_{M a c} = \frac{\sum _{t = 1}^{N} t \cdot C F _{t} / ( 1 + y ) ^{t}}{\sum _{t = 1}^{N} C F _{t} / ( 1 + y ) ^{t}}

— t = time of cash flow, CF_t = cash flow at t, y = yield, N = number of periods

Yield Curve Strategies 7 items

Key rate duration

%Δ P \approx - \sum_{k} K R D_{k} \times Δ 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):

2 y_{5 Y} < y_{2 Y} + y_{10 Y}

Measures curvature of yield curve

Barbell vs bullet convexity

Barbell (short + long): higher convexity

Bullet (middle): lower convexity

At same duration:

C_{ba r b e l l} > C_{b u l l e t}

Barbell wins in high vol; bullet in stable curves

Expected price change from rate view vs. forwards

E (Δ P) \approx - D (Δ y_{view} - Δ y_{fwd}) + \frac{1}{2} C (Δ y_{view})^{2}

— D = duration, C = convexity, Δy_view = view yield change, Δy_fwd = forward-implied change

Hedged cross-currency bond return under CIP

R_{hedged} \approx R_{local} - (r_{local} - r_{base})

— R_local = local bond return, r_local = local risk-free rate, r_base = base-currency risk-free rate

Expected return of a yield curve strategy

E (R) = Y + Rolldown + E (Δ P_{view}) - E (Credit Loss) \pm E (FX)

— Y = yield income, ΔP_view = price change from rate view, FX = currency return

Expected bond return from a yield change

E (R) \approx Y + Rolldown - D_{mod} Δ y + \frac{1}{2} C (Δ y)^{2}

— Y = yield income, D_mod = modified duration, C = convexity, Δy = yield change

Credit Strategies 6 items

Credit spread duration

%Δ P \approx - Spread duration \times Δ s

Δ 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

X R \approx s \cdot t - Δ s \cdot D_{s} - t \cdot p \cdot L

s=spread, t=horizon,

D_{s}

=spread duration, p=PD, L=loss rate

CDS-cash basis

Basis = CDS spread - Z-spread_{cash}

— CDS spread = single-name or index protection premium (bps), Z-spread_cash = zero-volatility spread on the cash bond (bps)

Option-adjusted spread

OAS = Z-spread - Option cost

— Z-spread = zero-volatility spread (bps), Option cost = embedded option value from lattice/Monte Carlo (bps)

Conditional value at risk

CVaR_{α} = E [L ∣ L > VaR_{α}]

— 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

OAS_{USD-eq} = OAS_{foreign} - (r_{base} - r_{foreign})

— r_base, r_foreign = short rates in base and foreign currency; differential = rolling FX-forward hedge cost

Trade Strategy & Execution 7 items

Implementation shortfall

I S = \frac{Paper portfolio gain - Actual portfolio gain}{Investment decision value}

Components: delay cost + trading cost + opportunity cost

Measures total cost of executing a trade vs decision price

VWAP benchmark

V W A P = \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 = (P_{e x ec} - P_{p r e}) / P_{p r e}

(for buys)

P_exec = average execution price, P_pre = pre-trade benchmark

Temporary impact reverses; permanent impact does not

Higher urgency → more market impact

Explicit fees component of implementation shortfall

Fees = c \times S_{e x ec} + taxes

— c = commission per share, S_exec = shares executed, taxes = transaction taxes

Trading cost component of implementation shortfall

Trading = (P_{e x ec} - P_{a r r i v a l}) \times S_{f i l l e d}

— P_exec = average execution price, P_arrival = arrival price, S_filled = shares filled

Delay cost component of implementation shortfall

Delay = (P_{a r r i v a l} - P_{d ec i s i o n}) \times S_{f i l l e d}

— P_arrival = price when trader gets order, P_decision = PM decision price, S_filled = shares executed

Opportunity cost component of implementation shortfall

Opportunity = (P_{c l ose} - P_{d ec i s i o n}) \times S_{u n f i l l e d}

— P_close = closing price, P_decision = decision price, S_unfilled = shares not filled

Case Study: Endowment 4 items

DV01-based hedge ratio

N = - \frac{D V 0 1 _{p or t f o l i o}}{D V 0 1 _{h e d g e}}

Negative = short the hedge instrument

For cross-hedge:

N = - \frac{D V 0 1 _{p}}{D V 0 1 _{h}} \times β_{s p r e a d}

Effective spending rate

r_{e f f} = \frac{S _{t}}{M V _{t}}

— S_t = current period spending distribution, MV_t = current portfolio market value

Endowment smoothing spending rule

S_{t} = w \cdot S_{t - 1} (1 + π) + (1 - w) \cdot r \cdot M V_{t}

— w = smoothing weight, S = spending, π = inflation, r = target rate, MV = current market value

Illiquidity budget

I B = T P - (S p + C C + R B + S R)

— 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^{*} = \frac{1}{λ} Σ^{- 1} (μ - r_{f} 1)

λ

= risk aversion,

Σ

= covariance matrix,

μ

= expected returns

Corner portfolio blending

w_{A} = \frac{E ( R _{P} ) - E ( R _{B} )}{E ( R _{A} ) - E ( R _{B} )}

,

w_{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 = δ Σ w_{mk t}

Blended:

E (R) = [(τ Σ)^{- 1} + P^{T} Ω^{- 1} P]^{- 1} [(τ Σ)^{- 1} Π + P^{T} Ω^{- 1} Q]

δ

=risk aversion,

w_{mk t}

=mkt cap weights

Portfolio rebalancing trigger (range-based)

Rebalance when:

∣ w_{i} - w_{i}^{*} ∣ > Δ_{i}

w_{i}^{*}

= target weight,

Δ_{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 - Δ S + g + Δ (P / E)

— D/P = dividend yield, ΔS = net share issuance, g = nominal earnings growth, Δ(P/E) = repricing

Taylor rule policy rate

i_{t} = r^{*} + π_{t} + 0.5 (π_{t} - π^{*}) + 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 + i_{d}) / (1 + i_{f})

— F = forward rate, S = spot rate, i_d = domestic interest rate, i_f = foreign interest rate

Fixed-income expected return building blocks

E (R) = r_{f} + π^{e} + T P + C P + L P

— r_f = real risk-free rate, π^e = expected inflation, TP = term premium, CP = credit premium, LP = liquidity premium

Expected fixed-income return decomposition

E (R_{F I}) = Y T M + Roll-down + Δ P_{y i e l d} - L_{cr e d i t} - L_{F X}

— YTM = yield to maturity, ΔP = price change from curve shift, L = expected losses

GARCH(1,1) variance forecast

σ_{t}^{2} = ω + α ε_{t - 1}^{2} + β σ_{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

R P = w (σ ρ \cdot S R_{g}) + (1 - w) (σ \cdot S R_{g})

— w = integration weight, σ = asset vol, ρ = correlation with global portfolio, SR_g = global Sharpe ratio

TAA permitted weight range around SAA

w_{T AA} \in [w_{S AA} - b, w_{S AA} + b]

— w_SAA = policy weight, b = IPS-defined TAA band (e.g., ±5%)

Net after-cost tactical premium

α_{n e t} = α_{g r oss} - T C - T

— α_gross = expected gross tactical alpha, TC = transaction costs, T = realized tax cost

Stress liquidity coverage ratio

L C R_{s t r ess} = \frac{Accessible liquid assets _{s t r ess}}{Cash demands _{s t r ess}}

— minimum prudent target ≥ 2x for illiquid-heavy portfolios

Endowment spending rate

s = \frac{Annual spending}{Portfolio value}

— s = spending rate (e.g., 5% = $40M on $800M)

Number of correlation inputs required for MVO

N_{ρ} = \frac{n ( n - 1 )}{2}

— n = number of asset classes; MVO also needs n expected returns and n standard deviations

Pension funded ratio

F R = \frac{A}{L}

— A = market value of plan assets, L = present value of liabilities (e.g., PBO)

Dollar duration of a portfolio or liability

D D = M V \times D

— MV = market value (or PV of liabilities), D = modified or effective duration; used to size LDI hedges

Pension surplus

S = 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 (R_{p}) - 0.5 \cdot λ \cdot σ_{p}^{2}

— E(R_p) = expected portfolio return, λ = risk aversion coefficient (1-10), σ_p² = portfolio variance

Geometric mean approximation from arithmetic mean and variance

G \approx A - 0.5 σ^{2}

— G = geometric (compound) mean return, A = arithmetic mean return, σ² = variance of returns

Total MVO inputs required for n asset classes

N = 2 n + \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

S F = \frac{E ( R _{p} ) - R _{L}}{σ _{p}}

, where

R_{L}

is the minimum acceptable (threshold) return. The optimal portfolio maximizes SF; under normality this minimizes the probability of a return below

R_{L}

.

Topic 2 26 items

Marginal Contribution to Risk (MCTR)

M C T R_{i} = β_{i} \times σ_{p}

β_{i} = \frac{Cov ( R _{i} , R _{p} )}{σ _{p}^{2}}

Measures risk added by a small increase in asset i's weight

Absolute Contribution to Risk (ACTR)

A C T R_{i} = w_{i} \times M C T R_{i} = w_{i} \times β_{i} \times σ_{p}

\sum_{i} A C T R_{i} = σ_{p}

(contributions sum to total portfolio risk)

Risk budget = set target ACTRs

Tracking error

T E = σ (R_{p} - R_{B}) = \frac{\sum ( r _{p, t} - r _{B, t} - α ˉ ) ^{2}}{T - 1}

Also called active risk or tracking risk

Annualized:

T E_{ann u a l} = T E_{m o n t h l y} \times 12

Implementation shortfall (decomposition)

I S = Explicit + Delay + Impact + Opportunity

— explicit = commissions/fees; delay = decision-to-desk drift; impact = price move from trade; opportunity = unfilled-share return.

Square-root market impact model

Impact \propto shares / ADV

— shares = order size, ADV = average daily volume; doubling order size raises impact by ~41%, not 100%.

VWAP transaction cost (buy and sell)

VWAP cost_{buy} = P_{exec} - VWAP

;

VWAP cost_{sell} = VWAP - P_{exec}

— positive = unfavorable; VWAP = period volume-weighted average price.

Effective equity beta from a PE allocation

β_{e f f} = w_{e q} + w_{P E} \times β_{P E}

— w_eq = public equity weight, w_PE = PE weight, β_PE ≈ 1.3 (PE equity beta)

Effective PE allocation including unfunded commitments

w_{P E}^{e f f} = \frac{N A V + U C}{P + U C}

— NAV = PE net asset value, UC = unfunded commitments, P = total portfolio value

Liquidity coverage ratio for an alternatives program

L C R = \frac{L}{C F _{12 m}}

— L = liquid assets, CF_{12m} = next-12-month committed cash outflows (capital calls + benefits)

Total PE economic exposure

E_{P E} = N A V + U C

— NAV = net asset value of PE holdings, UC = unfunded capital commitments

Endowment real return target

R_{r e a l} = s + c

— s = spending rate, c = management cost ratio

Insurer duration-matched immunization

D_{A} \times A = D_{L} \times L

— D_A = asset duration, A = assets, D_L = liability duration, L = liabilities

Foundation minimum nominal return target

R \geq 0.05 + c + π

— 0.05 = 5% IRS minimum distribution floor, c = costs, π = inflation

DV01 (price value of a basis point)

D V 01 = D_{m o d} \times M V \times 0.0001

— D_mod = modified duration, MV = market value of the bond/portfolio

Active share of an equity portfolio

A S = \frac{1}{2} \sum_{i = 1}^{N} ∣ w_{p, i} - w_{b, i} ∣

— w_{p,i} = portfolio weight in stock i, w_{b,i} = benchmark weight in stock i, N = combined universe

Required pre-tax nominal return for a private client

r = \frac{( S - I ) / V + π}{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

r_{p} = r_{i} + \frac{V _{B}}{V _{E}} (r_{i} - r_{B})

— r_i = asset return, r_B = borrowing cost, V_B = borrowed value, V_E = equity

Human capital as present value of future labor income

H C = \sum_{t = 1}^{N} \frac{E [ w _{t} ]}{( 1 + r ) ^{t}}

— w_t = expected labor income in year t, r = risk-adjusted discount rate, N = remaining working years

Real after-tax return approximation

r_{r e a l, a t} \approx r_{n o m} - π - t \cdot r_{n o m}

— r_nom = nominal return, π = inflation rate, t = tax rate on nominal gain

Modified duration from Macaulay duration

D_{m o d} = \frac{D _{M a c}}{1 + y}

— D_Mac = Macaulay duration, y = periodic yield to maturity

Taxable-equivalent yield on a municipal bond

TEY = \frac{y _{m u ni}}{1 - t}

— y_muni = muni pretax yield, t = investor's marginal tax rate

Economic net worth

E N W = F C + P V (H C) - P V (L) - P V (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

A U M_{s t ab} = 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

T_{ann u a l} = r_{r e a l} \times V_{f u n d}

—

r_{r e a l}

= expected real return (~3%),

V_{f u n d}

= fund market value; principal preserved

Commodity SWF energy-sector exposure cap

w_{e n er g y} \leq w_{b e n c h} - Δ

—

w_{b e n c h}

= benchmark energy weight,

Δ

= transition-risk tilt (e.g., 30%) to offset inflow correlation

SWF maximum single-year withdrawal under charter cap

W_{ma x} = c \times A U M_{3 y}

— c = charter cap (e.g., 5%),

A U M_{3 y}

= 3-year average AUM

Topic 3 12 items

Downside deviation (Sortino denominator)

σ_{D} = \frac{1}{n} \sum_{t} min (R_{t} - M A R, 0)^{2}

Standard deviation of returns below the minimum acceptable return only

Penalizes bad volatility; upside dispersion is ignored

Maximum drawdown

M D D = \frac{Trough value - Peak value}{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

A R = \frac{α}{σ _{ε}}

α

= alpha from a factor regression,

σ_{ε}

= residual (security-specific) risk

Alpha per unit of active bet taken; denominator is residual risk, not tracking error

Brinson allocation effect

A_{i} = (w_{p, i} - w_{b, i}) (R_{b, i} - R_{b})

— w_p = portfolio sector weight, w_b = benchmark sector weight, R_b,i = sector benchmark return, R_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 = \frac{R _{p} - M A R}{σ _{d}}

— R_p = portfolio return, MAR = minimum acceptable return, σ_d = downside deviation of returns below MAR

Brinson selection effect

S_{i} = w_{b, i} (R_{p, i} - R_{b, i})

— w_b = benchmark sector weight, R_p,i = portfolio sector return, R_b,i = benchmark sector return

Downside capture ratio

DC = \frac{R ˉ _{p, down}}{R ˉ _{b, down}}

— averaged over periods when benchmark return is negative; <100% means manager dampens losses

Upside capture ratio

UC = \frac{R ˉ _{p, up}}{R ˉ _{b, up}}

— averaged over periods when benchmark return is positive; >100% means manager amplifies up markets

Up/down capture ratio

Up/Down Capture = \frac{Upside Capture}{Downside Capture}

— ratio above 1.0 indicates favorable asymmetry

Symmetric performance-based fee

Fee = Base + s \times (R_{p} - R_{b})

— Base = base fee, s = sharing rate, R_p = portfolio return, R_b = benchmark return

Topic 4 14 items

Delta of call and put

Call:

Δ_{c} = N (d_{1}) \in (0, 1)

Put:

Δ_{p} = N (d_{1}) - 1 \in (- 1, 0)

Put-call:

Δ_{c} - Δ_{p} = 1

Approx change in option price for $1 change in underlying

Protective put payoff

At expiration:

Payoff = S_{T} + max (X - S_{T}, 0)

=

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:

S_{T} + max (X_{L} - S_{T}, 0) - max (S_{T} - X_{H}, 0)

=

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:

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

Number of bond futures to adjust portfolio duration

N = \frac{D D _{T} - D D _{P}}{D D _{f}}

— DD_T = target dollar duration, DD_P = current dollar duration, DD_f = dollar duration per futures contract (BPV adjusted by conversion factor)

Variance notional converted from vega notional

N_{v a r} = \frac{N _{v e g a}}{2 \times σ _{s t r ik e}}

— N_vega = vega notional ($ per vol point), σ_strike = strike volatility in whole-number percent

Variance swap payoff at maturity

Payoff = N_{v a r} \times (σ_{r e a l i z e d}^{2} - σ_{s t r ik e}^{2})

— N_var = variance notional, σ_realized = realized volatility (%), σ_strike = strike volatility (%)

Number of equity futures to adjust portfolio beta

N = \frac{β _{T} - β _{P}}{β _{F}} \times \frac{V}{P _{f} \times m}

— β_T = target beta, β_P = current beta, β_F = futures beta, V = portfolio value, P_f = futures price, m = multiplier

Roll yield on a currency forward hedge

Roll yield = \frac{F - S}{S}

— F = forward rate, S = spot rate; approximately equals domestic minus foreign interest rate

Minimum-variance hedge ratio (MVHR)

h^{*} = ρ_{A, B} \times \frac{σ _{A}}{σ _{B}}

— A = asset hedged, B = hedging instrument, ρ = correlation, σ = volatility

Domestic-currency return on a foreign asset

R_{D C} = (1 + R_{F C}) (1 + R_{F X}) - 1

; R_FC = foreign asset return, R_FX = % change in exchange rate (domestic per foreign); the approximation R_FC + R_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 - S_{0} + C_{0}

— K = call strike, S_0 = short entry price, C_0 = call premium paid

Long straddle breakeven prices

B E = K \pm (C_{0} + P_{0})

— K = common strike, C_0 = call premium paid, P_0 = put premium paid

Put-call parity

S + P = C + P V (K)

— S = stock price, P = put premium, C = call premium, K = strike, PV(K) = present value of strike

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