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Free CFA Level III: Private Markets Formula Sheet (2026)

Every CFA L3 Private Markets formula you need on the test, grouped by topic and rendered with full math notation. 101 formulas across 11 topics, calibrated to the 2026 syllabus. Free forever, no signup required.

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All CFA L3 Private Markets Formulas

Private Investments & Structures 2 items
Distributed to Paid-In (DPI)
DPI=Cumulative DistributionsPaid-In Capital\text{DPI} = \dfrac{\text{Cumulative Distributions}}{\text{Paid-In Capital}}
Distributions = realized cash returned to LPs, Paid-In Capital = capital actually called and funded
Total Value to Paid-In (TVPI)
TVPI=DPI+RVPI=Distributions+NAVPaid-In Capital\text{TVPI} = \text{DPI} + \text{RVPI} = \dfrac{\text{Distributions} + \text{NAV}}{\text{Paid-In Capital}}
Distributions = realized cash, NAV = unrealized value, Paid-In Capital = called and funded
GP & Investor Perspectives 4 items
RVPI (Residual Value to Paid-In)
RVPI=Residual NAVPaid-in capital\mathrm{RVPI} = \dfrac{\text{Residual NAV}}{\text{Paid-in capital}}
Unrealized portion of TVPI
Declining RVPI may reflect realizations, write-downs, or additional paid-in capital; assess realization progress alongside distributions.
TVPI=DPI+RVPI\mathrm{TVPI} = \mathrm{DPI} + \mathrm{RVPI}
J-curve
Early: negative cash flows (calls and fees, no distributions) → negative or low net IRR
Positive net cash flow: distributions exceed contributions; this need not coincide with the net IRR turning point.
Curve shape depends on deployment timing, fees, valuations, and realizations.
Management fee base
Investment period: fees on committed capital.
Once fully deployed: fees on committed capital less the cumulative cost basis of exited and written-off investments.
Fee=Rate×Base\text{Fee} = \text{Rate} \times \text{Base} (typically 1.5–2%).
GP catch-up amount (100% catch-up provision)
X=c1−c×PrefX = \dfrac{c}{1-c} \times \text{Pref}
X = GP catch-up, c = carry rate (e.g. 0.20), Pref = LP preferred return distributed
Private Equity 7 items
MOIC (Multiple on Invested Capital)
MOIC=Distributions+Residual NAVInvested capitalMOIC = \frac{\text{Distributions} + \text{Residual NAV}}{\text{Invested capital}}
Also called TVPI. TVPI = DPI + RVPI.
Levered equity return
re=ru+DE(ru−rd)r_e = r_u + \frac{D}{E}(r_u - r_d)
rur_u = unlevered asset return, rdr_d = cost of debt
D/E = debt-to-equity ratio
Leverage amplifies equity returns (and risk)
Waterfall distribution (PE)
Order: Return of capital → Preferred return (hurdle) → GP catch-up → Carry split
American: deal-by-deal. European: whole-fund before carry.
Typical: 20% carry above 8% hurdle.
LBO value creation decomposition
ΔEquity=ΔEBITDA×Mentry+EBITDAexit×ΔM+ΔDebt\Delta \text{Equity} = \Delta \text{EBITDA} \times M_{\text{entry}} + \text{EBITDA}_{\text{exit}} \times \Delta M + \Delta \text{Debt}
M = EV/EBITDA multiple, ΔDebt = debt paid down
Post-money valuation (venture capital method)
Post-money=Exit Value(1+r)n\text{Post-money} = \dfrac{\text{Exit Value}}{(1 + r)^n}
Exit Value = projected exit equity value, r = required IRR, n = years to exit
VC ownership percentage and pre-money valuation
Ownership%=InvestmentPost-money\text{Ownership\%} = \dfrac{\text{Investment}}{\text{Post-money}}; Pre-money=Post-money−Investment\text{Pre-money} = \text{Post-money} - \text{Investment}
Investment = new capital injected
Sponsor IRR in a leveraged buyout
With one initial sponsor investment and no interim sponsor cash flows, IRR=(Exit Equity/Entry Equity)1/n−1\text{IRR}=(\text{Exit Equity}/\text{Entry Equity})^{1/n}-1. Exit Equity=sponsor proceeds after debt repayment, adjusted for ownership, preferred-share terms and dilution; Entry Equity=sponsor equity check; nn=hold years.
Private Debt 4 items
Loan-to-cost ratio (LTC)
LTC=Loan AmountTotal Project Cost\text{LTC} = \dfrac{\text{Loan Amount}}{\text{Total Project Cost}}
Loan Amount = principal of debt facility; Total Project Cost = acquisition price plus development and soft costs
Interest coverage ratio
Interest Coverage=EBITDAInterest Expense\text{Interest Coverage} = \dfrac{\text{EBITDA}}{\text{Interest Expense}}
EBITDA = earnings before interest, tax, depreciation, and amortization; Interest Expense = total interest expense, including payment-in-kind interest.
Total debt to EBITDA leverage ratio
Leverage=Total DebtEBITDA\text{Leverage} = \dfrac{\text{Total Debt}}{\text{EBITDA}}
Total Debt = senior + subordinated principal; typical buyout range 5-7x on quality credits
Fixed charge coverage ratio (FCCR)
FCCR=EBITDA−CapExInterest+Mandatory Principal Amortization\text{FCCR} = \dfrac{\text{EBITDA} - \text{CapEx}}{\text{Interest} + \text{Mandatory Principal Amortization}}
CapEx = capital expenditures; denominator = total debt service
Private Special Situations 1 item
IRR implied by expected terminal recovery
r=(E[Recovery]P)1/T−1r = \left(\dfrac{E[\text{Recovery}]}{P}\right)^{1/T} - 1
E[Recovery]E[\text{Recovery}] = probability-weighted recovery per unit face, PP = purchase price, TT = fixed recovery horizon in years. Assumes one terminal recovery and no interim cash flows; rr is the IRR of the expected cash flow.
Private Real Estate 7 items
Loan-to-Value (LTV)
LTV=Loan amountProperty valueLTV = \frac{\text{Loan amount}}{\text{Property value}}
Higher LTV → higher leverage, more risk
Typical real estate: 60–75% LTV
Lenders covenant on LTV and DSCR
Net Operating Income (NOI)
NOI=EGI−Operating expenses−Property maintenance allowance\mathrm{NOI} = \mathrm{EGI} - \text{Operating expenses} - \text{Property maintenance allowance}
EGI=Potential gross income−Vacancy and credit losses\mathrm{EGI} = \text{Potential gross income} - \text{Vacancy and credit losses}
Operating expenses exclude debt service, depreciation, and income taxes.
Direct capitalization value
V=NOIrcapV = \frac{NOI}{r_{cap}}
Stabilized NOI used (normalized for occupancy, expenses)
Cap rate sourced from comparable sales
Used for income-producing properties
Discounted cash flow value of real estate
V0=∑t=1nCFt(1+r)t+NOIn+1/rexit(1+r)nV_0 = \sum_{t=1}^{n} \dfrac{CF_t}{(1+r)^t} + \dfrac{NOI_{n+1}/r_{exit}}{(1+r)^n}
CF = annual cash flow, r = discount rate, rexitr_{exit} = exit cap rate, n = hold period
Gordon link between cap rate and discount rate
rcap≈r−gr_{cap} \approx r - g
rcapr_{cap} = going-in cap rate, r = property discount rate (IRR target), g = expected NOI growth rate
Levered equity cash flow for real estate
Before-tax cash flow: CFequity,t=NOIt−Interestt−Principal repaymentst−CapExtCF_{\text{equity},t} = \mathrm{NOI}_t - \text{Interest}_t - \text{Principal repayments}_t - \text{CapEx}_t
NOI = net operating income, Interest = debt interest expense, CapEx = additional capital expenditures in period tt not already deducted in NOI.
Cap-rate sensitivity of property value
%ΔV≈−Δrcaprcap\%\Delta V \approx -\dfrac{\Delta r_{cap}}{r_{cap}}
%ΔV = percent change in value, ΔrcapΔr_{cap} = change in cap rate (decimal), rcapr_{cap} = current cap rate
Infrastructure 1 item
Allowed revenue under the regulated asset base approach
Allowed Revenue=RAB×rallowed+Depreciation+Opex \text{Allowed Revenue} = \text{RAB} \times r_{\text{allowed}} + \text{Depreciation} + \text{Opex}
RAB = regulated asset base, rallowedr_{allowed} = allowed return, Opex = operating costs
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
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