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

Every CFA L3 Private Wealth formula you need on the test, grouped by topic, rendered with full math notation. 29 formulas across 7 topics, calibrated to the 2026 syllabus. Free forever, no signup required.

29 Formulas
7 Topics
2026 Syllabus
Free Forever
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All CFA L3 Private Wealth Formulas

Wealth Planning 5 items
Human capital present value
HC0=∑t=1Nwt⋅(1−pdeath,t)(1+r)tHC_0 = \sum_{t=1}^{N} \frac{w_t \cdot (1 - p_{death,t})}{(1 + r)^t}
wtw_t=expected earnings, r=discount rate.
Total wealth = financial + human capital.
Taxable account accumulation
FV=[1+r(1−t)]nFV = [1 + r(1-t)]^n
Annual returns taxed each year; interest/dividend taxed as ordinary income
For deferred capital gains: FV=(1+r)n−[(1+r)n−1]×tCGFV = (1+r)^n - [(1+r)^n - 1] \times t_{CG}
Tax-deferred account accumulation
FV=(1+r)n(1−Tn)FV = (1+r)^n (1 - T_n)
T_n = tax rate at withdrawal
Contributions pre-tax; all withdrawals taxed as ordinary income
Favorable when future tax rate < current rate
Tax-exempt account accumulation
FV=(1+r)nFV = (1+r)^n
After-tax contributions; all growth and withdrawals tax-free
Favorable when future tax rate > current rate
Roth IRA / Roth 401(k) structure
Relative advantage of tax-deferred vs taxable
RA=(1+r)n(1−Tn)[1+r(1−t)]n(1−T0)RA = \frac{(1+r)^n (1-T_n)}{[1+r(1-t)]^n (1-T_0)}
RA > 1 → tax-deferred preferred
T_n = future tax rate, T_0 = current tax rate
Higher returns and longer horizons amplify advantage
Investment Planning 4 items
After-tax return
rAT=rPT×(1−t)r_{AT} = r_{PT} \times (1 - t)
For return with mixed income types:
rAT=rCG(1−tCG)+rInc(1−tInc)+rtax−freer_{AT} = r_{CG}(1-t_{CG}) + r_{Inc}(1-t_{Inc}) + r_{tax-free}
r_PT = pre-tax return, t = tax rate
Core capital estimate
CC=∑t=1TEt(1+rAT)t×psurvive,tCC = \sum_{t=1}^{T} \frac{E_t}{(1+r_{AT})^t} \times p_{survive,t}
E_t = expenses in year t, r_AT = after-tax return
p_survive,t = probability of surviving to year t
Core capital must be preserved; surplus is investable
Tax-loss harvesting benefit
Benefit=tST⋅L−tLT⋅Gf\text{Benefit} = t_{ST} \cdot L - t_{LT} \cdot G_f
L=loss harvested, GfG_f=future gain at lower basis.
Net positive when tST>tLTt_{ST} > t_{LT} or long horizon.
Asset location decision
Tax-inefficient assets (bonds, REITs, high-turnover funds) → tax-deferred accounts
Tax-efficient assets (equities, index funds, muni bonds) → taxable accounts
Rationale: maximize after-tax wealth across entire household balance sheet
Transferring the Wealth 3 items
Estate tax (simple)
Tax=(Estate value−Exemption)×te\text{Tax} = (\text{Estate value} - \text{Exemption}) \times t_e
t_e = estate tax rate
Taxable estate = gross estate − debts − marital deduction − charitable deduction
Relative value of gift vs bequest
RV=(1+rg)n(1−Te)(1+re)n(1−Tg)+Tg−TeRV = \frac{(1+r_g)^n(1-T_e)}{(1+r_e)^n(1-T_g) + T_g - T_e}
r_g = after-tax return if gifted, r_e = after-tax return if bequested
T_e = estate/bequest tax rate, T_g = gift tax rate
RV > 1 → gifting preferred
Generation-Skipping Transfer (GST) tax
Applies to transfers skipping a generation (grandparent → grandchild).
GST tax=Transfer amount×tGST\text{GST tax} = \text{Transfer amount} \times t_{GST}
Separate exemption; dynasty trusts avoid multi-gen estate tax.
Topic 1 4 items
Mean-variance optimal portfolio weight
w∗=1λΣ−1(μ−rf1)\mathbf{w}^* = \frac{1}{\lambda} \Sigma^{-1} (\mu - r_f \mathbf{1})
λ\lambda = risk aversion, Σ\Sigma = covariance matrix, μ\mu = expected returns
Corner portfolio blending
wA=E(RP)−E(RB)E(RA)−E(RB)w_A = \frac{E(R_P) - E(R_B)}{E(R_A) - E(R_B)}, wB=1−wAw_B = 1 - w_A
Blend two adjacent corner portfolios A and B to achieve target return E(R_P)
All blends lie on the efficient frontier
Black-Litterman expected return
Equilibrium: Π=δΣwmkt\Pi = \delta \Sigma w_{mkt}
Blended: E(R)=[(τΣ)−1+PTΩ−1P]−1[(τΣ)−1Π+PTΩ−1Q]E(R) = [(\tau\Sigma)^{-1} + P^T \Omega^{-1} P]^{-1}[(\tau\Sigma)^{-1}\Pi + P^T\Omega^{-1}Q]
δ\delta=risk aversion, wmktw_{mkt}=mkt cap weights
Portfolio rebalancing trigger (range-based)
Rebalance when: ∣wi−wi∗∣>Δi|w_i - w_i^*| > \Delta_i
wi∗w_i^* = target weight, Δi\Delta_i = tolerance band
Wider bands → lower costs, less precision
Correlation-adjusted bands: wider for high-correlation assets
Topic 2 3 items
Marginal Contribution to Risk (MCTR)
MCTRi=βi×σpMCTR_i = \beta_i \times \sigma_p
βi=Cov(Ri,Rp)σp2\beta_i = \frac{\text{Cov}(R_i, R_p)}{\sigma_p^2}
Measures risk added by a small increase in asset i's weight
Absolute Contribution to Risk (ACTR)
ACTRi=wi×MCTRi=wi×βi×σpACTR_i = w_i \times MCTR_i = w_i \times \beta_i \times \sigma_p
∑iACTRi=σp\sum_i ACTR_i = \sigma_p (contributions sum to total portfolio risk)
Risk budget = set target ACTRs
Tracking error
TE=σ(Rp−RB)=∑(rp,t−rB,t−αˉ)2T−1TE = \sigma(R_p - R_B) = \sqrt{\frac{\sum(r_{p,t} - r_{B,t} - \bar{\alpha})^2}{T-1}}
Also called active risk or tracking risk
Annualized: TEannual=TEmonthly×12TE_{annual} = TE_{monthly} \times \sqrt{12}
Topic 3 5 items
Information ratio
IR=Rˉp−RˉBσ(Rp−RB)=αˉTEIR = \frac{\bar{R}_p - \bar{R}_B}{\sigma(R_p - R_B)} = \frac{\bar{\alpha}}{TE}
αˉ\bar{\alpha} = mean active return, TE = tracking error
Measures active return per unit of active risk
Fundamental Law of Active Management
IR=IC×BRIR = IC \times \sqrt{BR}
IC = information coefficient, BR = investment breadth
Expected active return: E(RA)=IC×BR×σAE(R_A) = IC \times \sqrt{BR} \times \sigma_A (TC assumed = 1)
BHB attribution effects
Allocation: (wp,i−wB,i)(RB,i−RB)(w_{p,i} - w_{B,i})(R_{B,i} - R_B)
Selection: wB,i(Rp,i−RB,i)w_{B,i}(R_{p,i} - R_{B,i})
Interaction: (wp,i−wB,i)(Rp,i−RB,i)(w_{p,i} - w_{B,i})(R_{p,i} - R_{B,i})
Sum of all effects = total active return
Sharpe ratio
SRp=Rp−RfσpSR_p = \frac{R_p - R_f}{\sigma_p}
Reward-to-variability ratio using total risk
Sharpe of combined portfolio: SRC2=SRB2+IR2SR_C^2 = SR_B^2 + IR^2
M-squared (M²)
M2=(Rp−Rf)σmσp+RfM^2 = (R_p - R_f) \frac{\sigma_m}{\sigma_p} + R_f
Risk-adjusted return scaled to match market's volatility
M² > R_m → portfolio outperformed on risk-adjusted basis
Topic 4 5 items
Delta of call and put
Call: Δc=N(d1)∈(0,1)\Delta_c = N(d_1) \in (0, 1)
Put: Δp=N(d1)−1∈(−1,0)\Delta_p = N(d_1) - 1 \in (-1, 0)
Put-call: Δc−Δp=1\Delta_c - \Delta_p = 1
Approx change in option price for $1 change in underlying
Protective put payoff
At expiration: Payoff=ST+max⁡(X−ST,0)\text{Payoff} = S_T + \max(X - S_T, 0)
= max⁡(ST,X)\max(S_T, X)
Profit = Payoff − (S_0 + p), where p = put premium
Limits downside while preserving upside
Collar payoff at expiration
Long stock + long put (X_L) + short call (X_H)
Payoff: ST+max⁡(XL−ST,0)−max⁡(ST−XH,0)S_T + \max(X_L - S_T, 0) - \max(S_T - X_H, 0)
= min⁡(max⁡(ST,XL),XH)\min(\max(S_T, X_L), X_H)
Limits gains above X_H, protects below X_L
Covered call payoff at expiration
Long stock + short call (X)
Payoff: ST−max⁡(ST−X,0)=min⁡(ST,X)S_T - \max(S_T - X, 0) = \min(S_T, X)
Profit = Payoff − S_0 + c (c = call premium received)
Caps upside; enhances income in flat/down markets
Variance swap payoff
Payoff=Nvega×(σrealized2−σstrike2)\text{Payoff} = N_{vega} \times (\sigma_{realized}^2 - \sigma_{strike}^2)
Long variance swap profits when realized variance > strike variance
No delta-hedging needed; pure volatility exposure

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