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Why is gifting treated as an annuity due rather than an ordinary annuity?
Annuity due = payment at start of period (extra $(1+r)$ compounding factor). Ordinary annuity = payment at end. Gifting strategies are typically annuity due because grandparents write checks on Jan 5. For PMT=\$19K, $r=5\%$, $N=30$: ordinary gives \$1.26M, annuity due gives \$1.33M ($5\%$ higher). Always check the timing convention before computing...
Should I use the IRS Single Life Expectancy table or a personalized mortality estimate when planning gifting horizons?
For CFA exam purposes: use IRS Single Life Expectancy table (unisex). For HNW client practice: personalized estimate adjusting for sex ($+3$ to $+5$ for F), smoking ($-10$ to $-15$), family history ($\pm 3$ to 7), concierge healthcare ($+3$ to $+5$). IRS tables are required for gift-tax valuation but personalized can drive planning. Joint life expectancy for married couples is meaningfully longer...
Why is "pulling the constant out of the limit" a valid algebra step in deriving the infinite geometric sum?
It's legal because $\lim_{n \to \infty}[g \cdot f(n)] = g \cdot \lim f(n)$ when $g$ is a constant. The geometric sum rewrites as $S_n = [a/(1-r)] \cdot (1-r^n)$; pulling out the constant $a/(1-r)$ is just rescaling the limit. Same pattern as $E[cX]=cE[X]$ or $\int c \cdot f\,dx = c \cdot \int f\,dx$ — linearity...
In private wealth planning, when do you use the finite-$N$ annuity formula vs the infinite-perpetuity formula?
Finite-$N$ for annual gifting plans, GRATs, term-certain CRTs, defined planning horizons. Infinite for dynasty trusts, university endowments, Gordon DDM, very-long-horizon family wealth. After 40+ years the two formulas converge — the finite captures $86\%+$ of the infinite value...
When does the geometric-series sum $a/(1-r)$ converge as $n$ approaches infinity?
The convergence condition is strictly $|r| < 1$. For wealth planning, $r$ is typically a discount factor ($< 1$) so convergence is automatic. Gordon DDM $PV = D_1/(r - g)$ is the same formula — breaks when $g \ge r$. Standard convention: PV of perpetuity is finite, FV of perpetuity is infinite...
What happens to the unamortised excess if the investee sells the underlying asset before amortisation is complete?
When the investee sells the underlying asset, the remaining unamortised excess is written off immediately through equity income, alongside the investor's share of the investee's gain/loss. This typically creates a net loss (because the investor paid more for the asset than book value). The investment carrying value is reduced accordingly...
How does the amortisation of excess under the equity method affect the investor's reported net income?
Yes — amortisation of equity-method excess reduces the investor's reported NI. It flows through "Equity in earnings of affiliates" net of the share of NI. Affects ROE, ROA, and net margin. Embedded in equity income, not separately disclosed unless material. Cash flow adjustment: subtract equity income (non-cash) and add dividends (cash)...
When do I need to amortise the excess purchase price under the equity method?
Amortisation is required for the excess attributed to depreciable PP&E, finite-life intangibles, and inventory. Land, indefinite-life intangibles, and goodwill are NOT amortised. Annual amortisation = (excess attributed to finite-life asset) / (asset's remaining useful life). Common mistake is forgetting this step in roll-forward...
How does goodwill impairment work for an equity-method investment?
Under the equity method, goodwill is NOT separately impaired. Instead, the whole investment carrying value is tested for "other-than-temporary impairment" when triggered. The test compares carrying value to fair value, and judgment determines whether decline is temporary or permanent...
Why do we attribute the excess purchase price to specific assets before calculating goodwill?
Attribution matters because goodwill and identifiable-asset step-ups have different downstream treatments: depreciable-asset step-ups must be amortised against future equity income, while goodwill stays on the books indefinitely (subject to impairment). The two scenarios produce different earnings trajectories even for the same total excess paid...
Where exactly does goodwill appear on the balance sheet under the equity method?
Goodwill is embedded inside the Investment in Investee line — it does NOT appear separately on the balance sheet. The equity method is a one-line consolidation. Compare to full consolidation where goodwill is a separate Intangibles line. Implications for ratio analysis: D/E and ROA look better under equity method...
When exactly does "significant influence" trigger the equity method? Is it always at 20%?
It is a presumption, not a hard rule. Significant influence can apply below 20% (with board seats, intercompany transactions, or technology dependence) and may NOT apply above 20% (with voting restrictions or shareholder agreements). The CFA exam usually relies on the 20% shortcut but tests the nuanced framework in 1-2 vignettes per cycle...
What is the BASE rule the instructor uses to verify the equity-method roll-forward?
BASE = Beginning + Additions − Subtractions = Ending. It's a universal sanity check for any balance that rolls forward. For equity method: B + Σ(share of NI) − Σ(share of dividends) = E. Catches arithmetic errors and works on PP&E, debt, retained earnings, inventory...
Why are dividends NOT counted as income under the equity method?
Dividends are not income under the equity method because that would double-count earnings. When the investee earns $300K and the investor (30% owner) picks up $90K of equity income, a $60K dividend is just paying out cash that the investor already recognised...
How does delta change as time passes or volatility changes? Are there separate Greeks for those effects?
Three effects on delta: gamma ($\partial \Delta / \partial S$), charm ($\partial \Delta / \partial T$ as time passes), vanna ($\partial \Delta / \partial \sigma$ when vol shifts). Gamma is most watched. Charm matters for weekend drift on ATM options. Vanna matters for skew traders. Knowing names and signs covers exam conceptual questions...
After I understand the delta derivation from BSM, what other Greeks can I derive using the same technique?
All five Greeks derive from BSM using the same machinery. Difficulty: Delta easiest, then Gamma (differentiate delta again), Vega (chain rule on $\sigma$), Rho (one term in $r$), Theta hardest (multiple time dependencies). Every derivation uses the same $N'$ identity...
Why does the $N'(d_1) \cdot S \cdot e^{-qT} = N'(d_2) \cdot K \cdot e^{-rT}$ identity hold in the BSM Greeks derivation?
The identity $S e^{-qT} N'(d_1) = K e^{-rT} N'(d_2)$ follows from $d_2^2 - d_1^2 = -2 \sigma \sqrt{T} d_1 + \sigma^2 T$ and the $d_1$ definition. The proof simplifies to ratio $= 1$. Intuitively it reflects the change of numeraire between the risk-neutral and stock-numeraire measures...
Do I need to know the full delta derivation for the CFA Level II exam?
You only need the result $\Delta_{\text{call}}=N(d_1)$ and $\Delta_{\text{put}}=N(d_1)-1$ (times $e^{-qT}$ for dividend stocks). The exam does not test derivation. But walking through it once helps you avoid confusing $N(d_1)$ with $N(d_2)$ and handles dividend stocks correctly...
How do I approximate delta in an Excel spreadsheet without using the closed-form formula?
Use a finite-difference approximation: $\Delta \approx (V(S+\varepsilon) - V(S)) / \varepsilon$ (forward) or $(V(S+\varepsilon) - V(S-\varepsilon)) / (2\varepsilon)$ (central, more accurate). Sweet spot for $\varepsilon$ is $0.001S$ to $0.01S$. The Excel approach matches the closed-form $\Delta = N(d_1)$ to 4-5 decimals...
Why is delta a *partial* derivative and not just an ordinary derivative?
Delta is a partial derivative because $V$ depends on multiple variables ($S$, $K$, $r$, $q$, $\sigma$, $T$). The $\partial V/\partial S$ notation explicitly means "rate of change with respect to $S$, holding everything else constant." Each Greek isolates one variable...
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