CFA Level III · Private Wealth Pathway

Wealth Planning: formula sheet

Full chapter guide

Key formulas

Overall risk level rule: Overall risk level = lower of (risk tolerance, risk capacity); A working rule, not a law. If tolerance is high and capacity is low, recommend lower risk and explain why. If capacity is high and tolerance is low, you may educate the client, but do not push them past their comfort.
Risk tolerance vs risk capacity: Tolerance = willingness (subjective). Capacity = ability (objective).; Classify each fact in the vignette into one of these two before you decide.
Goal classification: Needs (essential, low flexibility) vs wants (discretionary, flexible); Link needs to lower-risk assets and wants to assets with more risk.
IPS structure: IPS = Objectives (return, risk) + Constraints (liquidity, time horizon, taxes, legal and regulatory, unique circumstances); Use this as your checklist for any IPS question. Do not leave a heading out, even if you write "none stated".
Risk objective rule: Risk tolerance = lower of ability to take risk and willingness to take risk; Applies as the usual approach when the two disagree. State both and say which governs.
Required nominal return: (1 + real return) × (1 + inflation) − 1; Use the multiplicative form when the question asks for precision. The approximation real + inflation is only acceptable if the question allows it.
After-tax return: After-tax return = pre-tax return × (1 − tax rate); Valid when the whole return is taxed at one rate. If income and gains are taxed differently, compute each part separately.
Pre-tax return needed: Pre-tax return = after-tax return required ÷ (1 − tax rate); Use when the client states an after-tax requirement and a single tax rate applies.
Human capital (present value of wages): HC = Σ [ E(t) × P(survive/employed to t) ] ÷ (1 + r)^t; Sum from t = 1 to retirement. Use a discount rate that reflects earnings risk. Safe job: closer to the risk-free rate. Risky job: add a premium.
Human capital for growing wages (constant growth, finite years): HC = [E1 ÷ (r − g)] × [1 − ((1 + g) ÷ (1 + r))^n]; E1 is the first year's earnings, received at the end of year 1. n is the number of working years. Use when r ≠ g.
Total wealth: Total wealth = Human capital + Financial capital; Financial capital is the market value of investable assets.
Economic net worth: Economic net worth = Economic assets − Economic liabilities; Assets include human and financial capital and the PV of pensions. Liabilities include debt and the PV of consumption and bequest needs.
Human capital behaviour rule: Bond-like HC → higher equity in financial portfolio; equity-like HC → lower equity; Judge by earnings volatility and correlation with markets and the client's industry.
Life insurance need (human life value approach): Need = PV(income lost to dependants) − PV(client's own consumption) − existing assets and benefits; Add final expenses, debts and bequest goals where the question asks for them.
Future value of a spending need: FV = PV × (1 + inflation)^n; Use the inflation rate specific to that expense, for example healthcare, if given.
Real return: (1 + nominal) ÷ (1 + inflation) − 1; The approximation nominal − inflation is acceptable only if the question allows it.
Required capital for a level real spending stream: PV of annuity = Spending × [1 − (1 + r)^−n] ÷ r; Use the real return r when spending is in today's money and grows with inflation. The formula assumes end-of-period payments; multiply by (1 + r) for payments at the start of the period.
Spending rate: Spending rate = Annual spending ÷ Portfolio value; Compare with the sustainable rate. A higher rate means a lower probability of success.
Probability of success (Monte Carlo): Successful paths ÷ Total paths; Success means assets last to the end of the horizon. Shortfall probability = 1 − success.
Funding gap: Gap = PV of goals − current assets (and PV of future income); Discount all at the same rate basis. Gap above zero means the plan is underfunded.
Human life value: HLV = Σ [(Earnings_t − Taxes_t − Personal consumption_t) ÷ (1 + r)^t]; Discount at a rate that reflects the risk of the earnings stream. Use after-tax earnings and subtract what the insured would spend on themselves.
Needs approach gap: Insurance needed = PV of family needs − existing financial assets − existing insurance; Family needs include income replacement, debt repayment, education, final expenses and emergency funds. Use a consistent real or nominal basis.
Insurance need from capital: Capital needed = Annual shortfall ÷ sustainable withdrawal rate; A quick perpetuity-style approach. Use a conservative rate if the cover must last many years.
Annuity present value (ordinary): PV = Payment × [1 − (1 + r)^−n] ÷ r; Used to value a fixed-term stream of annuity payments or to size a lump sum.
Cover choice rule: Temporary need → term; permanent need → permanent; longevity risk → annuity; A decision rule, not a law. Always link it to the client's objectives and constraints.
After-tax return on interest: r(after-tax) = r × (1 − t_interest); Use the rate on ordinary income. Applies to taxable bond income.
After-tax return on dividends: r(after-tax) = r × (1 − t_dividend); Dividend rate may differ from the interest rate.
After-tax return on a deferred capital gain (realized at end): FV after tax = V0 × [(1 + r)^n × (1 − t_g) + t_g]; Assumes no interim income and gain taxed on sale, with basis equal to V0. Equivalent to V0(1+r)^n − t_g × [V0(1+r)^n − V0].
Annual after-tax growth, taxed each year: FV = V0 × [1 + r × (1 − t)]^n; Use for fully taxable accrual each year.
Tax-deferred account after-tax value: FV after tax = V0 × (1 + r)^n × (1 − t_withdrawal); Applies when the contribution is pre-tax and withdrawals are taxed. A pre-tax contribution of V0 is compared with an after-tax contribution.
Tax-exempt account after-tax value: FV = V0 × (1 + r)^n; V0 is already after-tax money. Growth and withdrawals are untaxed.
Value of a realized loss: Tax saving = loss × tax rate that the loss offsets; Harvesting defers tax rather than removing it when the lower basis raises later gains.
Future value of a gifted asset: FV = PV × (1 + g)^n; Use to compare keeping an asset in the estate versus gifting it now. Gifting removes the growth from the estate.
After-tax value of a bequest: Net to heir = Asset value × (1 − estate tax rate); Apply the rate to the taxable amount only. Subtract any exemption first.
Gift versus bequest comparison: Gift: heir receives G × (1 + r)^n, with donor cost G × (1 + t_g) (tax-exclusive) or G ÷ (1 − t_g) (tax-inclusive). Bequest: heir receives FV at death × (1 − t_e); Compare on a like-for-like basis: the amounts the heir receives after all taxes, with the donor's gift tax cost kept in view. Ignore taxes on growth unless the question gives them.
Tax-exclusive vs tax-inclusive: Tax-exclusive: cost to donor = gift × (1 + t); Tax-inclusive: cost to donor = gift ÷ (1 − t); Use when the donor pays the gift tax. State the basis first. Basis 1, the same net gift G reaching the recipient. Tax-exclusive: tax is t × G, so the donor's outlay is G × (1 + t). Tax-inclusive: tax is t × the total outlay (gift plus tax), so the outlay is G ÷ (1 − t). For 0 < t < 1, 1 ÷ (1 − t) is greater than 1 + t, so tax-inclusive costs the donor more. Basis 2, the same total outlay X. Tax-exclusive: the recipient gets X ÷ (1 + t), an effective tax rate of t ÷ (1 + t) on the outlay. Tax-inclusive: the recipient gets X × (1 − t), an effective rate of t on the outlay. Tax-inclusive is again the costlier. Example with t = 40% and G = 100: the outlay is 140 tax-exclusive and 166.67 tax-inclusive. Do not mix the two bases in one comparison.
Collar payoff (long stock): Long stock + long put (strike X_L) + short call (strike X_H); Downside floors at X_L and upside caps at X_H. A zero-cost collar sets the call premium equal to the put premium. If the collar is not zero-cost, the floor and cap are adjusted by the net premium paid or received (plus any financing cost).
Protective put floor: Minimum value of the stock-plus-put position = X_put (the strike). Net floor = X_put − premium paid (plus any financing cost of the premium); The client keeps the stock's gains above the strike, but the upside is reduced by the premium paid, so the net gain is the stock gain less the premium. The premium is paid in cash up front. The floor is a value level for the whole position, not a profit figure. Measure profit or loss against the starting value (or the cost basis for tax) plus the premium paid.
Covered call: Maximum value = X_call + premium received; Premium cushions small declines only. Upside is capped and big downside remains.
After-tax proceeds of a sale: Proceeds − [(Sale price − Cost basis) × Capital gains tax rate]; Tax applies to the gain, not the whole sale price. Use per-share or total values consistently.
Prepaid forward (variable): Cash today = loan-to-value % (for example 75-90%) × current market value of the shares; The percentage is set by the dealer's pricing and hedging cost. Client gets cash now and delivers shares or cash value later. Upside is typically partly kept within a range. Tax deferral may be available depending on local tax rules and how the deal is structured, and it can be lost if the arrangement is treated as a constructive sale.
Equity swap (hedge): Client pays equity return, receives a fixed or floating rate; Removes exposure without selling shares, so ownership and voting rights stay. Mind counterparty and tax treatment.

Quick revision

Discovery gathers goals, constraints, risk attitude and facts before any recommendation.
The IPS links objectives (return, risk) and constraints (liquidity, time horizon, tax, legal, unique circumstances) to a plan.
Risk tolerance is willingness; risk capacity is ability. The lower of the two usually limits risk taken.
Economic net worth = human capital + financial capital (including other assets) minus liabilities.
Human capital is the present value of future earnings; stable earnings behave more like a bond.
Separate needs from wants and plan for essential spending with the most certain assets.
Insurance transfers risks that could wreck the plan, such as death, disability and longevity.
Tax drag lowers returns, so place assets in accounts with the tax treatment that suits them.
Estate planning covers who gets assets, when, how and at what tax cost.
Concentrated positions carry single-asset risk; you can sell, hedge, diversify or gift, each with different tax and control effects.
Answer the exact command word and show each calculation step.
Always attempt every question because wrong answers carry no penalty.

Common mistakes

Treating risk tolerance and risk capacity as the same thing. Fix: Tolerance is willingness and feelings. Capacity is ability and numbers. Classify each fact before you conclude.
Using the higher of the two as the overall risk level. Fix: Use the lower one as the governing level unless the question says otherwise, and give the reason.
Treating willingness to take risk as the whole risk objective Fix: Always write ability and willingness as two separate statements. Then state which is lower and let it govern.
Putting a constraint under objectives, or the reverse Fix: Ask whether the fact states what to achieve or what limits the portfolio. A known cash need is a liquidity constraint.
Discounting human capital at the risk-free rate for every client. Fix: Match the rate to income risk. Use a low rate for stable pay and a higher rate for volatile or market-linked pay.
Leaving human capital out of the economic balance sheet, or leaving out future spending needs. Fix: Always list human capital as an asset and the present value of consumption and bequests as liabilities before computing economic net worth.
Mixing nominal returns with real spending Fix: Write 'real' or 'nominal' beside each input before calculating, then convert one to match the other.
Planning only to life expectancy Fix: Life expectancy is the midpoint. Use a longer horizon, and the last survivor for a couple.
Recommending whole life for every client because it builds cash value. Fix: Match the product to the need. For a temporary need, term is usually cheaper, and the saved premium can be invested separately.
Ignoring existing assets and employer cover in the needs analysis. Fix: Always subtract existing financial assets and insurance to get the gap.

Exam tips

Match the command word. 'Determine' wants a conclusion, 'justify' wants a reason tied to a fact, and 'identify' wants only the name.
Always cite a client fact. A bare 'capacity is low' earns less than 'capacity is low because retirement is in two years'.
Know which facts signal tolerance (attitudes, past reactions) and which signal capacity (income, horizon, liabilities, spending needs).
In item sets, watch for answer options that swap tolerance and capacity. Read the definitions before you choose.
Pathway questions mix item sets and essays, so practise both styles and keep essay answers short.
Match the command word. "Determine" wants a conclusion; "justify" wants a conclusion plus a reason from the vignette.
Show every step in a calculation, but type the final number on its own so it earns full credit.
For risk, always write ability and willingness separately before the overall answer.