# CFA Level I (Session 2) — Equity Valuation, Fixed Income (Duration & Convexity), Derivatives (Put-Call Parity), Alternatives & Portfolio Management

> Comprehensive CFA Level I Session 2 flashcards covering Market Microstructure & Indexes, Gordon Growth Model & Enterprise Value, Spot/Forward Rates, Z-Spread & OAS, Macaulay/Modified/Effective Duration & Convexity, MBS Prepayment Risk, Forward Pricing, Put-Call Parity & Binomial Option Valuation, Private Equity/Hedge Fund Waterfalls, Real Estate NOI & Commodity Backwardation/Contango, CAPM (CML vs. SML), Sharpe/Treynor/M2/Information Ratios, IPS & Behavioral Finance.

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- **Level:** intermediate
- **Languages:** en → en
- **Total Cards:** 32

## Flashcards (32 Cards)

| # | Front (Term / Prompt) | Back (Definition / Answer) | Example | Multiple-Choice Options |
| :--- | :--- | :--- | :--- | :--- |
| 1 | A $200 million multi-asset portfolio has a 1-day 95% Value at Risk (5% daily VaR) of $3.2 million and a Conditional VaR (CVaR / Expected Shortfall) of $4.8 million. Which statement accurately interprets these risk metrics? | Over a 1-day horizon, there is a 5% probability that the portfolio will lose AT LEAST $3.2 million (or a 95% probability that the daily loss will NOT exceed $3.2 million), and on those worst 5% of days when the $3.2 million VaR threshold is breached, the average expected loss is $4.8 million — Value at Risk (VaR) is the MINIMUM loss expected over a given time horizon at a stated tail probability (here 5% tail = 95% confidence), or equivalently the maximum loss expected to be exceeded only 5% of the time. Because VaR ignores the magnitude of tail losses beyond the cutoff, Conditional VaR (CVaR / Expected Shortfall) measures the probability-weighted average loss given that the VaR threshold is exceeded ($4.8M). | Value at Risk (VaR) is the MINIMUM loss expected over a given time horizon at a stated tail probability (here 5% tail = 95% confidence), or equivalently the maximum loss expected to be exceeded only 5% of the time. Because VaR ignores the magnitude of tail losses beyond the cutoff, Conditional VaR (CVaR / Expected Shortfall) measures the probability-weighted average loss given that the VaR threshold is exceeded ($4.8M). | A) Over a 1-day horizon, there is a 5% probability that the portfolio will lose AT LEAST $3.2 million (or a 95% probability that the daily loss will NOT exceed $3.2 million), and on those worst 5% of days when the $3.2 million VaR threshold is breached, the average expected loss is $4.8 million — Value at Risk (VaR) is the MINIMUM loss expected over a given time horizon at a stated tail probability (here 5% tail = 95% confidence), or equivalently the maximum loss expected to be exceeded only 5% of the time. Because VaR ignores the magnitude of tail losses beyond the cutoff, Conditional VaR (CVaR / Expected Shortfall) measures the probability-weighted average loss given that the VaR threshold is exceeded ($4.8M). ✓; B) The maximum possible loss the portfolio can suffer in a single day is $3.2 million with 95% certainty and $4.8 million with 100% certainty; C) Over 95% of trading days, the portfolio will lose an average of $3.2 million per day; D) VaR measures only systematic beta risk, whereas CVaR measures liquidity risk |
| 2 | A non-dividend-paying stock currently trades at S_0 = $100.00. A 1-year European call option with strike price X = $105.00 trades at c_0 = $8.00, and the annual risk-free interest rate is r = 5.00%. Using Put-Call Parity, what is the no-arbitrage price of a 1-year European put option with strike X = $105.00 (p_0), and what synthetic position equals a Protective Put? | p_0 = $8.00 (since c_0 + X/(1 + r)^T = $8.00 + $105/1.05 = $8.00 + $100.00 = $108.00 = p_0 + S_0, so p_0 = $108.00 - $100.00 = $8.00); a Fiduciary Call (Long Call + Long Risk-Free Zero-Coupon Bond paying X at maturity) equals a Protective Put (Long Put + Long Stock) — By European Put-Call Parity, Fiduciary Call = Protective Put: c_0 + X / (1 + r)^T = p_0 + S_0. Substituting the values: $8.00 + $105.00 / 1.05 = p_0 + $100.00 -> $8.00 + $100.00 = p_0 + $100.00 -> p_0 = $8.00. Notice that because the strike X = $105 equals the 1-year forward price F_0(T) = $100(1.05) = $105 (At-The-Money Forward), the call and put have identical values ($8.00) by Put-Call-Forward Parity! | By European Put-Call Parity, Fiduciary Call = Protective Put: c_0 + X / (1 + r)^T = p_0 + S_0. Substituting the values: $8.00 + $105.00 / 1.05 = p_0 + $100.00 -> $8.00 + $100.00 = p_0 + $100.00 -> p_0 = $8.00. Notice that because the strike X = $105 equals the 1-year forward price F_0(T) = $100(1.05) = $105 (At-The-Money Forward), the call and put have identical values ($8.00) by Put-Call-Forward Parity! | A) p_0 = $8.00 (since c_0 + X/(1 + r)^T = $8.00 + $105/1.05 = $8.00 + $100.00 = $108.00 = p_0 + S_0, so p_0 = $108.00 - $100.00 = $8.00); a Fiduciary Call (Long Call + Long Risk-Free Zero-Coupon Bond paying X at maturity) equals a Protective Put (Long Put + Long Stock) — By European Put-Call Parity, Fiduciary Call = Protective Put: c_0 + X / (1 + r)^T = p_0 + S_0. Substituting the values: $8.00 + $105.00 / 1.05 = p_0 + $100.00 -> $8.00 + $100.00 = p_0 + $100.00 -> p_0 = $8.00. Notice that because the strike X = $105 equals the 1-year forward price F_0(T) = $100(1.05) = $105 (At-The-Money Forward), the call and put have identical values ($8.00) by Put-Call-Forward Parity! ✓; B) p_0 = $7.62 ($8.00 / 1.05); Put-Call Parity applies only to American options; C) p_0 = $13.00 ($8.00 + $105.00 - $100.00); a Long Call + Short Stock equals a Protective Put; D) p_0 = $3.00 ($8.00 - $5.00); a Short Put + Long Bond equals a Protective Put |
| 3 | During the past year, Portfolio P earned R_p = 14.0% with standard deviation sigma_p = 20.0%, Beta = 1.25, and Tracking Error (sigma_(p-B)) = 4.0%. The Market Benchmark earned R_M = 11.0% with standard deviation sigma_M = 15.0%, and the risk-free rate was R_f = 3.0%. Calculate Portfolio P's Sharpe Ratio, Treynor Ratio, M-Squared (M^2), and Information Ratio. | Sharpe Ratio = 0.55 ((14% - 3%)/20%); Treynor Ratio = 8.80% ((14% - 3%)/1.25); M^2 = 11.25% (3% + 0.55 × 15%, giving M^2 Alpha = +0.25%); Information Ratio = 0.75 ((14% - 11%)/4%) — (1) Sharpe Ratio = (R_p - R_f) / sigma_p = (14% - 3%) / 20% = 0.55 (vs. Market Sharpe = (11% - 3%)/15% = 0.5333). (2) Treynor Ratio = (R_p - R_f) / Beta_p = 11% / 1.25 = 8.80% (vs. Market Treynor = 8.00%). (3) M^2 = R_f + Sharpe_p × sigma_M = 3.0% + 0.55 × 15.0% = 11.25% (M^2 Alpha = 11.25% - 11.00% = +0.25%). (4) Information Ratio = Active Return / Tracking Error = (14.0% - 11.0%) / 4.0% = 0.75. | (1) Sharpe Ratio = (R_p - R_f) / sigma_p = (14% - 3%) / 20% = 0.55 (vs. Market Sharpe = (11% - 3%)/15% = 0.5333). (2) Treynor Ratio = (R_p - R_f) / Beta_p = 11% / 1.25 = 8.80% (vs. Market Treynor = 8.00%). (3) M^2 = R_f + Sharpe_p × sigma_M = 3.0% + 0.55 × 15.0% = 11.25% (M^2 Alpha = 11.25% - 11.00% = +0.25%). (4) Information Ratio = Active Return / Tracking Error = (14.0% - 11.0%) / 4.0% = 0.75. | A) Sharpe Ratio = 0.70; Treynor Ratio = 11.20%; M^2 = 14.00%; Information Ratio = 2.75; B) Sharpe Ratio = 0.55; Treynor Ratio = 8.80%; M^2 = 8.25%; Information Ratio = 0.15; C) Sharpe Ratio = 0.55 ((14% - 3%)/20%); Treynor Ratio = 8.80% ((14% - 3%)/1.25); M^2 = 11.25% (3% + 0.55 × 15%, giving M^2 Alpha = +0.25%); Information Ratio = 0.75 ((14% - 11%)/4%) — (1) Sharpe Ratio = (R_p - R_f) / sigma_p = (14% - 3%) / 20% = 0.55 (vs. Market Sharpe = (11% - 3%)/15% = 0.5333). (2) Treynor Ratio = (R_p - R_f) / Beta_p = 11% / 1.25 = 8.80% (vs. Market Treynor = 8.00%). (3) M^2 = R_f + Sharpe_p × sigma_M = 3.0% + 0.55 × 15.0% = 11.25% (M^2 Alpha = 11.25% - 11.00% = +0.25%). (4) Information Ratio = Active Return / Tracking Error = (14.0% - 11.0%) / 4.0% = 0.75. ✓; D) Sharpe Ratio = 0.533; Treynor Ratio = 8.00%; M^2 = 11.00%; Information Ratio = 0.75 |
| 4 | Why must Effective Duration (rather than Macaulay or Modified Duration) be used to measure the interest rate sensitivity of a callable bond or a residential Mortgage-Backed Security (MBS), and how does a callable bond behave as yields fall sharply? | Because embedded call/prepayment options cause future cash flows to change when benchmark interest rates change (violating Modified Duration's fixed-cash-flow assumption); as yields fall below the coupon rate, a callable bond exhibits NEGATIVE convexity and its Effective Duration shrinks as price compresses toward the call price — Macaulay and Modified Duration assume bond cash flows do not change when yields shift. For bonds with embedded options (callable bonds, putable bonds, MBS), cash flows are path- and rate-dependent, requiring Effective Duration = (PV_- - PV_+) / (2 × Delta_Curve × PV_0). When yields fall below a callable bond's coupon, call probability rises, capping price appreciation near the call price (negative convexity) and reducing Effective Duration. | Macaulay and Modified Duration assume bond cash flows do not change when yields shift. For bonds with embedded options (callable bonds, putable bonds, MBS), cash flows are path- and rate-dependent, requiring Effective Duration = (PV_- - PV_+) / (2 × Delta_Curve × PV_0). When yields fall below a callable bond's coupon, call probability rises, capping price appreciation near the call price (negative convexity) and reducing Effective Duration. | A) Because embedded call/prepayment options cause future cash flows to change when benchmark interest rates change (violating Modified Duration's fixed-cash-flow assumption); as yields fall below the coupon rate, a callable bond exhibits NEGATIVE convexity and its Effective Duration shrinks as price compresses toward the call price — Macaulay and Modified Duration assume bond cash flows do not change when yields shift. For bonds with embedded options (callable bonds, putable bonds, MBS), cash flows are path- and rate-dependent, requiring Effective Duration = (PV_- - PV_+) / (2 × Delta_Curve × PV_0). When yields fall below a callable bond's coupon, call probability rises, capping price appreciation near the call price (negative convexity) and reducing Effective Duration. ✓; B) Modified Duration is still accurate for callable bonds provided Yield to Call (YTC) is substituted into the denominator; callable bonds exhibit positive convexity at all yield levels; C) Effective Duration measures sensitivity to credit spread changes while holding the benchmark yield curve constant; D) As yields fall, a putable bond exhibits negative convexity and its Effective Duration decreases toward zero |
| 5 | In a plain-vanilla Fixed-for-Floating Interest Rate Swap, how are cash flows exchanged, what is the swap's initial market value at inception, and which party benefits when market interest rates rise unexpectedly after inception? | The swap has zero initial market value at inception (priced as a series of off-market Forward Rate Agreements); notional principal is NOT exchanged—only net interest payments are exchanged each settlement period; the Fixed-Rate Payer (Floating-Rate Receiver) gains positive mark-to-market value when interest rates rise — A plain-vanilla interest rate swap exchanges fixed interest payments for floating-rate payments (e.g., SOFR) on a stated notional principal (which is never exchanged). At initiation, the fixed swap rate is set so PV(Fixed Leg) = PV(Floating Leg), giving zero initial value. When interest rates rise after inception, the Pay-Fixed / Receive-Floating party receives higher floating cash flows while paying the locked-in low fixed rate, gaining positive mark-to-market value. | A plain-vanilla interest rate swap exchanges fixed interest payments for floating-rate payments (e.g., SOFR) on a stated notional principal (which is never exchanged). At initiation, the fixed swap rate is set so PV(Fixed Leg) = PV(Floating Leg), giving zero initial value. When interest rates rise after inception, the Pay-Fixed / Receive-Floating party receives higher floating cash flows while paying the locked-in low fixed rate, gaining positive mark-to-market value. | A) Both counterparties exchange the full notional principal at inception and maturity; the Fixed-Rate Receiver (Floating-Rate Payer) gains when interest rates rise; B) The swap has zero initial market value at inception (priced as a series of off-market Forward Rate Agreements); notional principal is NOT exchanged—only net interest payments are exchanged each settlement period; the Fixed-Rate Payer (Floating-Rate Receiver) gains positive mark-to-market value when interest rates rise — A plain-vanilla interest rate swap exchanges fixed interest payments for floating-rate payments (e.g., SOFR) on a stated notional principal (which is never exchanged). At initiation, the fixed swap rate is set so PV(Fixed Leg) = PV(Floating Leg), giving zero initial value. When interest rates rise after inception, the Pay-Fixed / Receive-Floating party receives higher floating cash flows while paying the locked-in low fixed rate, gaining positive mark-to-market value. ✓; C) The Fixed-Rate Payer pays a upfront premium equal to the swap rate multiplied by notional principal at t = 0; D) An interest rate swap is traded exclusively on futures exchanges with zero counterparty credit risk and no ISDA Master Agreement |
| 6 | Under the expectations, liquidity preference, segmented markets, and preferred habitat theories of the term structure of interest rates, how is an upward-sloping yield curve explained? | Pure Expectations Theory attributes an upward-sloping yield curve solely to market expectations that future short-term rates will rise (forward rates = expected future spot rates); Liquidity Preference Theory states that investors demand a positive, maturity-increasing liquidity risk premium to hold longer-term bonds (so the yield curve can slope upward even if short-term rates are expected to stay flat); Segmented Markets and Preferred Habitat emphasize institutional maturity supply/demand and risk premiums required to leave a preferred maturity habitat — Under Pure Expectations Theory, investors are risk-neutral across maturities, so implied forward rates equal expected future spot rates (an upward-sloping spot curve implies short rates are expected to rise). Under Liquidity Preference Theory, forward rates = expected future spot rates + a positive liquidity premium that increases with maturity. | Under Pure Expectations Theory, investors are risk-neutral across maturities, so implied forward rates equal expected future spot rates (an upward-sloping spot curve implies short rates are expected to rise). Under Liquidity Preference Theory, forward rates = expected future spot rates + a positive liquidity premium that increases with maturity. | A) Segmented Markets Theory assumes borrowers and lenders freely substitute across all maturities whenever forward rates exceed expected spot rates; B) Pure Expectations Theory attributes an upward-sloping yield curve solely to market expectations that future short-term rates will rise (forward rates = expected future spot rates); Liquidity Preference Theory states that investors demand a positive, maturity-increasing liquidity risk premium to hold longer-term bonds (so the yield curve can slope upward even if short-term rates are expected to stay flat); Segmented Markets and Preferred Habitat emphasize institutional maturity supply/demand and risk premiums required to leave a preferred maturity habitat — Under Pure Expectations Theory, investors are risk-neutral across maturities, so implied forward rates equal expected future spot rates (an upward-sloping spot curve implies short rates are expected to rise). Under Liquidity Preference Theory, forward rates = expected future spot rates + a positive liquidity premium that increases with maturity. ✓; C) Pure Expectations Theory states that long-term bonds always carry a negative term premium, whereas Liquidity Preference Theory assumes investors are risk-neutral; D) Under Pure Expectations Theory, an upward-sloping yield curve implies that long-term bond yields are expected to fall immediately |
| 7 | A commercial office property has Gross Potential Rent of $1,200,000, Other Income of $50,000, a Vacancy & Collection Loss rate of 6% of total gross income, Operating Expenses (property taxes, insurance, maintenance, utilities, management) of $425,000, Annual Debt Service of $250,000, and Depreciation of $180,000. If comparable properties trade at a Going-In Capitalization Rate (Cap Rate) of 7.50%, what are the property's Net Operating Income (NOI) and Direct Capitalization Value? | Effective Gross Income = $1,175,000 ($1,250,000 × 0.94); NOI = $750,000 ($1,175,000 - $425,000, excluding debt service and depreciation); Property Value = $10,000,000 ($750,000 / 0.075) — Net Operating Income (NOI) is an unlevered pre-tax operating cash flow measure before financing costs (debt service) and non-cash accounting depreciation: Total Potential Gross Income = $1,200,000 + $50,000 = $1,250,000; less 6% Vacancy Loss ($75,000) = Effective Gross Income of $1,175,000; less Operating Expenses ($425,000) = NOI of $750,000. By Direct Capitalization, Value = NOI_1 / Cap Rate = $750,000 / 0.075 = $10,000,000. | Net Operating Income (NOI) is an unlevered pre-tax operating cash flow measure before financing costs (debt service) and non-cash accounting depreciation: Total Potential Gross Income = $1,200,000 + $50,000 = $1,250,000; less 6% Vacancy Loss ($75,000) = Effective Gross Income of $1,175,000; less Operating Expenses ($425,000) = NOI of $750,000. By Direct Capitalization, Value = NOI_1 / Cap Rate = $750,000 / 0.075 = $10,000,000. | A) NOI = $320,000 (after deducting debt service and depreciation); Property Value = $4,266,667; B) Effective Gross Income = $1,175,000 ($1,250,000 × 0.94); NOI = $750,000 ($1,175,000 - $425,000, excluding debt service and depreciation); Property Value = $10,000,000 ($750,000 / 0.075) — Net Operating Income (NOI) is an unlevered pre-tax operating cash flow measure before financing costs (debt service) and non-cash accounting depreciation: Total Potential Gross Income = $1,200,000 + $50,000 = $1,250,000; less 6% Vacancy Loss ($75,000) = Effective Gross Income of $1,175,000; less Operating Expenses ($425,000) = NOI of $750,000. By Direct Capitalization, Value = NOI_1 / Cap Rate = $750,000 / 0.075 = $10,000,000. ✓; C) NOI = $500,000 (after deducting debt service); Property Value = $6,666,667; D) NOI = $825,000; Property Value = $11,000,000 |
| 8 | Given a 1-year zero-coupon spot rate z_1 = 3.00% and a 2-year zero-coupon spot rate z_2 = 4.00% (annual compounding), what is the implied 1-year forward rate one year from today (1f1)? | 5.01% (calculated as (1.04)^2 / (1.03)^1 - 1 = 1.0816 / 1.03 - 1 = 0.050097) — By no-arbitrage forward rate bootstrapping: (1 + z_2)^2 = (1 + z_1)(1 + 1f1). Solving for 1f1 gives (1.04)^2 / 1.03 - 1 = 1.0816 / 1.03 - 1 = 5.0097% (or approximately 2×z_2 - z_1 = 2(4%) - 3% = 5.00%). When the spot curve is upward-sloping, implied forward rates lie ABOVE the spot curve. | By no-arbitrage forward rate bootstrapping: (1 + z_2)^2 = (1 + z_1)(1 + 1f1). Solving for 1f1 gives (1.04)^2 / 1.03 - 1 = 1.0816 / 1.03 - 1 = 5.0097% (or approximately 2×z_2 - z_1 = 2(4%) - 3% = 5.00%). When the spot curve is upward-sloping, implied forward rates lie ABOVE the spot curve. | A) 3.50% (the arithmetic average of 3.00% and 4.00%); B) 4.50% (calculated as 4.00% + 0.50%); C) 2.01% (calculated as (1.03)^2 / 1.04 - 1); D) 5.01% (calculated as (1.04)^2 / (1.03)^1 - 1 = 1.0816 / 1.03 - 1 = 0.050097) — By no-arbitrage forward rate bootstrapping: (1 + z_2)^2 = (1 + z_1)(1 + 1f1). Solving for 1f1 gives (1.04)^2 / 1.03 - 1 = 1.0816 / 1.03 - 1 = 5.0097% (or approximately 2×z_2 - z_1 = 2(4%) - 3% = 5.00%). When the spot curve is upward-sloping, implied forward rates lie ABOVE the spot curve. ✓ |
| 9 | How do Exchange-Traded Funds (ETFs) create and redeem shares via Authorized Participants (APs), and why are ETFs generally more tax-efficient for taxable US investors than traditional open-end mutual funds? | Authorized Participants (APs) create and redeem ETF shares in large blocks ('creation units') with the ETF issuer in the primary market through IN-KIND transfers of the underlying securities basket; in-kind redemptions allow the ETF manager to hand off low-cost-basis shares without triggering realized capital gains distributions to remaining ETF shareholders — In the ETF primary market, Authorized Participants (APs) exchange a specified basket of underlying securities in-kind for creation units of ETF shares (and vice versa on redemption), keeping the ETF's secondary market price tightly arbitraged to its intraday NAV (iNAV). Because redemptions are settled in-kind rather than by selling securities for cash inside the fund, the ETF avoids realizing embedded capital gains (and can purge its lowest-tax-basis lots). | In the ETF primary market, Authorized Participants (APs) exchange a specified basket of underlying securities in-kind for creation units of ETF shares (and vice versa on redemption), keeping the ETF's secondary market price tightly arbitraged to its intraday NAV (iNAV). Because redemptions are settled in-kind rather than by selling securities for cash inside the fund, the ETF avoids realizing embedded capital gains (and can purge its lowest-tax-basis lots). | A) ETFs are exempt from federal capital gains taxation under the Investment Company Act of 1940; B) Authorized Participants (APs) create and redeem ETF shares in large blocks ('creation units') with the ETF issuer in the primary market through IN-KIND transfers of the underlying securities basket; in-kind redemptions allow the ETF manager to hand off low-cost-basis shares without triggering realized capital gains distributions to remaining ETF shareholders — In the ETF primary market, Authorized Participants (APs) exchange a specified basket of underlying securities in-kind for creation units of ETF shares (and vice versa on redemption), keeping the ETF's secondary market price tightly arbitraged to its intraday NAV (iNAV). Because redemptions are settled in-kind rather than by selling securities for cash inside the fund, the ETF avoids realizing embedded capital gains (and can purge its lowest-tax-basis lots). ✓; C) Retail investors redeem ETF shares directly with the fund sponsor at 4:00 PM ET for cash, triggering capital gains taxes only for the redeeming investor; D) Authorized Participants widen the spread between ETF market price and intraday NAV (iNAV) by prohibiting arbitrage |
| 10 | In commodity futures markets, what causes the futures term structure to be in Backwardation versus Contango, and what is the sign of the Roll Yield in each regime? | Futures Price ≈ Spot Price(1 + r) + Storage Costs - Convenience Yield; when non-monetary Convenience Yield (benefit of holding physical inventory during shortages) exceeds risk-free financing + storage costs, Futures Price < Spot Price (Backwardation), generating a POSITIVE roll yield for a long futures investor rolling expiring contracts into cheaper longer-dated contracts; when Storage + Financing > Convenience Yield, Futures Price > Spot Price (Contango), generating a NEGATIVE roll yield — By the Theory of Storage, Futures Price = Spot Price + Interest + Storage Cost - Convenience Yield. When physical inventories are tight, the Convenience Yield of holding physical commodity is high, pushing Futures < Spot (Backwardation) and creating a positive roll yield as long investors roll out of higher-priced expiring contracts into lower-priced deferred contracts. When inventories are plentiful, Convenience Yield is near zero, so Futures > Spot (Contango) and roll yield is negative. | By the Theory of Storage, Futures Price = Spot Price + Interest + Storage Cost - Convenience Yield. When physical inventories are tight, the Convenience Yield of holding physical commodity is high, pushing Futures < Spot (Backwardation) and creating a positive roll yield as long investors roll out of higher-priced expiring contracts into lower-priced deferred contracts. When inventories are plentiful, Convenience Yield is near zero, so Futures > Spot (Contango) and roll yield is negative. | A) Backwardation (Futures > Spot) occurs when storage costs are high and produces a negative roll yield; Contango (Futures < Spot) produces a positive roll yield; B) Convenience Yield is highest when physical commodity inventories are abundant, driving the market into steep Contango; C) Roll yield is always zero unless the spot price of the physical commodity changes during the holding period; D) Futures Price ≈ Spot Price(1 + r) + Storage Costs - Convenience Yield; when non-monetary Convenience Yield (benefit of holding physical inventory during shortages) exceeds risk-free financing + storage costs, Futures Price < Spot Price (Backwardation), generating a POSITIVE roll yield for a long futures investor rolling expiring contracts into cheaper longer-dated contracts; when Storage + Financing > Convenience Yield, Futures Price > Spot Price (Contango), generating a NEGATIVE roll yield — By the Theory of Storage, Futures Price = Spot Price + Interest + Storage Cost - Convenience Yield. When physical inventories are tight, the Convenience Yield of holding physical commodity is high, pushing Futures < Spot (Backwardation) and creating a positive roll yield as long investors roll out of higher-priced expiring contracts into lower-priced deferred contracts. When inventories are plentiful, Convenience Yield is near zero, so Futures > Spot (Contango) and roll yield is negative. ✓ |
| 11 | How does the Macaulay Duration of a zero-coupon bond compare to a coupon-paying bond of the same maturity, and what is the Macaulay Duration of a level perpetuity paying annual coupons at yield y? | A zero-coupon bond has a Macaulay Duration exactly equal to its time to maturity T (whereas any coupon-paying bond with the same maturity T has Macaulay Duration < T); a level perpetuity has Macaulay Duration = (1 + y) / y and Modified Duration = 1 / y — Because Macaulay Duration is the present-value-weighted average time until cash flows are received, a zero-coupon bond (single cash flow at maturity T) has MacDur = T. Any positive coupon pulls the weighted-average time earlier (MacDur < T). For a level perpetuity at annual yield y, MacDur = (1 + y) / y and ModDur = MacDur / (1 + y) = 1 / y (e.g., at y = 5%, MacDur = 1.05 / 0.05 = 21.0 years and ModDur = 1 / 0.05 = 20.0 years). | Because Macaulay Duration is the present-value-weighted average time until cash flows are received, a zero-coupon bond (single cash flow at maturity T) has MacDur = T. Any positive coupon pulls the weighted-average time earlier (MacDur < T). For a level perpetuity at annual yield y, MacDur = (1 + y) / y and ModDur = MacDur / (1 + y) = 1 / y (e.g., at y = 5%, MacDur = 1.05 / 0.05 = 21.0 years and ModDur = 1 / 0.05 = 20.0 years). | A) A zero-coupon bond has a Macaulay Duration equal to T / 2, and a perpetuity has an infinite Macaulay Duration; B) A coupon-paying bond has a higher Macaulay Duration than a zero-coupon bond of the same maturity because more cash is received; C) A zero-coupon bond has a Macaulay Duration exactly equal to its time to maturity T (whereas any coupon-paying bond with the same maturity T has Macaulay Duration < T); a level perpetuity has Macaulay Duration = (1 + y) / y and Modified Duration = 1 / y — Because Macaulay Duration is the present-value-weighted average time until cash flows are received, a zero-coupon bond (single cash flow at maturity T) has MacDur = T. Any positive coupon pulls the weighted-average time earlier (MacDur < T). For a level perpetuity at annual yield y, MacDur = (1 + y) / y and ModDur = MacDur / (1 + y) = 1 / y (e.g., at y = 5%, MacDur = 1.05 / 0.05 = 21.0 years and ModDur = 1 / 0.05 = 20.0 years). ✓; D) A perpetuity has Macaulay Duration = 1 / y and Modified Duration = (1 + y) / y |
| 12 | In credit analysis of corporate debt, distinguish between the Structural Model (Merton Model) and Reduced-Form Models, and calculate the Expected Loss on a bond with a 2.5% Probability of Default (POD) and a 40% Recovery Rate. | Expected Loss = 1.50% (POD × LGD = 2.5% × (1 - 0.40) = 2.5% × 60%); in the Merton Structural Model, company equity is modeled as a European Call option on firm assets (with strike equal to face value of debt D) and risky debt equals risk-free debt minus a European Put option on firm assets — Loss Given Default (LGD) = 1 - Recovery Rate = 1 - 0.40 = 60%, so Expected Loss (and approximate fair credit spread) = POD × LGD = 2.5% × 60% = 1.50% (150 bps). In Merton's Structural Model, limited liability gives shareholders a payoff of max(0, A_T - D)—identical to a European Call option on firm assets A_T with strike D—while bondholders receive min(A_T, D) = D - max(0, D - A_T), which is risk-free debt D minus a European Put option written to shareholders. | Loss Given Default (LGD) = 1 - Recovery Rate = 1 - 0.40 = 60%, so Expected Loss (and approximate fair credit spread) = POD × LGD = 2.5% × 60% = 1.50% (150 bps). In Merton's Structural Model, limited liability gives shareholders a payoff of max(0, A_T - D)—identical to a European Call option on firm assets A_T with strike D—while bondholders receive min(A_T, D) = D - max(0, D - A_T), which is risk-free debt D minus a European Put option written to shareholders. | A) Expected Loss = 1.50%; in the Merton Model, bondholders have unlimited upside and shareholders have capped upside equal to debt face value; B) Expected Loss = 1.00% (2.5% × 40%); in the Merton Model, risky debt is modeled as a long call option on firm equity; C) Expected Loss = 1.50% (POD × LGD = 2.5% × (1 - 0.40) = 2.5% × 60%); in the Merton Structural Model, company equity is modeled as a European Call option on firm assets (with strike equal to face value of debt D) and risky debt equals risk-free debt minus a European Put option on firm assets — Loss Given Default (LGD) = 1 - Recovery Rate = 1 - 0.40 = 60%, so Expected Loss (and approximate fair credit spread) = POD × LGD = 2.5% × 60% = 1.50% (150 bps). In Merton's Structural Model, limited liability gives shareholders a payoff of max(0, A_T - D)—identical to a European Call option on firm assets A_T with strike D—while bondholders receive min(A_T, D) = D - max(0, D - A_T), which is risk-free debt D minus a European Put option written to shareholders. ✓; D) Expected Loss = 6.25% (2.5% / 0.40); Reduced-Form Models cannot incorporate macroeconomic variables |
| 13 | An investor purchases 1,000 shares of a stock at $80 per share using 50% initial margin (borrowing $40,000). If the broker's maintenance margin requirement is 25%, at what stock price will the investor receive a margin call, and what is the leveraged return if the stock rises to $96 (ignoring interest and commissions)? | Margin Call Price = $53.33 ($80 × (1 - 0.50) / (1 - 0.25)); Leveraged Return at $96 = +40.0% (+20.0% unleveraged × 2.0 leverage factor) — Margin Call Trigger Price = P_0 × (1 - Initial Margin %) / (1 - Maintenance Margin %) = $80 × (0.50 / 0.75) = $53.33. With 50% initial margin, the leverage factor is 1 / 0.50 = 2.0, so a +20% stock price increase ($80 to $96) produces a +40.0% return on the investor's $40,000 equity. | Margin Call Trigger Price = P_0 × (1 - Initial Margin %) / (1 - Maintenance Margin %) = $80 × (0.50 / 0.75) = $53.33. With 50% initial margin, the leverage factor is 1 / 0.50 = 2.0, so a +20% stock price increase ($80 to $96) produces a +40.0% return on the investor's $40,000 equity. | A) Margin Call Price = $60.00 ($80 × 0.75); Leveraged Return at $96 = +20.0%; B) Margin Call Price = $40.00 ($80 × 0.50); Leveraged Return at $96 = +40.0%; C) Margin Call Price = $53.33 ($80 × (1 - 0.50) / (1 - 0.25)); Leveraged Return at $96 = +40.0% (+20.0% unleveraged × 2.0 leverage factor) — Margin Call Trigger Price = P_0 × (1 - Initial Margin %) / (1 - Maintenance Margin %) = $80 × (0.50 / 0.75) = $53.33. With 50% initial margin, the leverage factor is 1 / 0.50 = 2.0, so a +20% stock price increase ($80 to $96) produces a +40.0% return on the investor's $40,000 equity. ✓; D) Margin Call Price = $53.33; Leveraged Return at $96 = +30.0% |
| 14 | How does an increase in (1) Volatility of the underlying (sigma), (2) the Risk-Free Interest Rate (r), and (3) Cash Dividends paid by the underlying stock affect the values of European Call and European Put options? | Higher Volatility (sigma) INCREASES both Call and Put values; Higher Risk-Free Rate (r) INCREASES Call values and DECREASES Put values; Higher Dividends on the underlying DECREASE Call values and INCREASE Put values — Because both calls and puts have asymmetric bounded downside (zero intrinsic value floor) and unbounded/large upside, higher underlying volatility (sigma) increases the value of BOTH European calls and puts. A higher risk-free rate (r) lowers the PV of the exercise price PV(X), which increases call values and decreases put values. Dividends reduce the ex-dividend stock price, which decreases call values and increases put values. | Because both calls and puts have asymmetric bounded downside (zero intrinsic value floor) and unbounded/large upside, higher underlying volatility (sigma) increases the value of BOTH European calls and puts. A higher risk-free rate (r) lowers the PV of the exercise price PV(X), which increases call values and decreases put values. Dividends reduce the ex-dividend stock price, which decreases call values and increases put values. | A) Higher Volatility INCREASES Call values and DECREASES Put values; Higher Risk-Free Rate INCREASES both Call and Put values; B) Higher Volatility (sigma) INCREASES both Call and Put values; Higher Risk-Free Rate (r) INCREASES Call values and DECREASES Put values; Higher Dividends on the underlying DECREASE Call values and INCREASE Put values — Because both calls and puts have asymmetric bounded downside (zero intrinsic value floor) and unbounded/large upside, higher underlying volatility (sigma) increases the value of BOTH European calls and puts. A higher risk-free rate (r) lowers the PV of the exercise price PV(X), which increases call values and decreases put values. Dividends reduce the ex-dividend stock price, which decreases call values and increases put values. ✓; C) Higher Risk-Free Rate DECREASES Call values and INCREASES Put values; Higher Dividends INCREASE Call values; D) Higher Volatility has no effect on Out-of-the-Money options, and Higher Dividends increase both Call and Put values |
| 15 | A portfolio invests 60% in Asset A (standard deviation sigma_A = 20%) and 40% in Asset B (standard deviation sigma_B = 15%). If the correlation between Asset A and Asset B is rho = 0.25, what are the covariance Cov(R_A, R_B) and the portfolio standard deviation (sigma_p)? | Cov(R_A, R_B) = 0.0075 (75 bps^2); Portfolio Variance = 0.0216; Portfolio Standard Deviation sigma_p = 14.70% (sqrt(0.0216)) — First, Cov(R_A, R_B) = rho × sigma_A × sigma_B = 0.25 × 0.20 × 0.15 = 0.0075. Next, Portfolio Variance sigma_p^2 = w_A^2(sigma_A^2) + w_B^2(sigma_B^2) + 2(w_A)(w_B)Cov(R_A, R_B) = (0.60)^2(0.04) + (0.40)^2(0.0225) + 2(0.60)(0.40)(0.0075) = 0.0144 + 0.0036 + 0.0036 = 0.0216. Taking the square root gives sigma_p = sqrt(0.0216) = 0.14697 (14.70%), which is 3.30% lower than the 18.00% weighted average due to diversification. | First, Cov(R_A, R_B) = rho × sigma_A × sigma_B = 0.25 × 0.20 × 0.15 = 0.0075. Next, Portfolio Variance sigma_p^2 = w_A^2(sigma_A^2) + w_B^2(sigma_B^2) + 2(w_A)(w_B)Cov(R_A, R_B) = (0.60)^2(0.04) + (0.40)^2(0.0225) + 2(0.60)(0.40)(0.0075) = 0.0144 + 0.0036 + 0.0036 = 0.0216. Taking the square root gives sigma_p = sqrt(0.0216) = 0.14697 (14.70%), which is 3.30% lower than the 18.00% weighted average due to diversification. | A) Cov(R_A, R_B) = 0.0075; Portfolio Standard Deviation sigma_p = 13.42% (ignoring the factor of 2 in 2×w_A×w_B×Cov); B) Cov(R_A, R_B) = 0.0300; Portfolio Standard Deviation sigma_p = 18.00% (weighted average 0.60(20%) + 0.40(15%)); C) Cov(R_A, R_B) = 0.0075 (75 bps^2); Portfolio Variance = 0.0216; Portfolio Standard Deviation sigma_p = 14.70% (sqrt(0.0216)) — First, Cov(R_A, R_B) = rho × sigma_A × sigma_B = 0.25 × 0.20 × 0.15 = 0.0075. Next, Portfolio Variance sigma_p^2 = w_A^2(sigma_A^2) + w_B^2(sigma_B^2) + 2(w_A)(w_B)Cov(R_A, R_B) = (0.60)^2(0.04) + (0.40)^2(0.0225) + 2(0.60)(0.40)(0.0075) = 0.0144 + 0.0036 + 0.0036 = 0.0216. Taking the square root gives sigma_p = sqrt(0.0216) = 0.14697 (14.70%), which is 3.30% lower than the 18.00% weighted average due to diversification. ✓; D) Cov(R_A, R_B) = 0.0150; Portfolio Standard Deviation sigma_p = 16.00% |
| 16 | An option-free bond has an Annual Modified Duration of 7.00 and an Annual Convexity of 60.0. If the bond's Yield to Maturity increases by 150 basis points (+1.50%, or +0.0150), what is the estimated percentage change in the bond's full price including both the duration and convexity adjustments? | -9.825% (Duration effect = -7.00 × 0.015 = -10.50%; Convexity adjustment = +0.5 × 60.0 × (0.015)^2 = +0.675%; Total = -10.50% + 0.675% = -9.825%) — Full Percentage Price Change ≈ (-AnnModDur × Delta_YTM) + (0.5 × AnnConvexity × (Delta_YTM)^2) = (-7.00 × 0.015) + (0.5 × 60.0 × 0.000225) = -0.1050 + 0.00675 = -0.09825 (-9.825%). Positive convexity always cushions the price drop when yields rise and magnifies the price gain when yields fall. | Full Percentage Price Change ≈ (-AnnModDur × Delta_YTM) + (0.5 × AnnConvexity × (Delta_YTM)^2) = (-7.00 × 0.015) + (0.5 × 60.0 × 0.000225) = -0.1050 + 0.00675 = -0.09825 (-9.825%). Positive convexity always cushions the price drop when yields rise and magnifies the price gain when yields fall. | A) -9.825% (Duration effect = -7.00 × 0.015 = -10.50%; Convexity adjustment = +0.5 × 60.0 × (0.015)^2 = +0.675%; Total = -10.50% + 0.675% = -9.825%) — Full Percentage Price Change ≈ (-AnnModDur × Delta_YTM) + (0.5 × AnnConvexity × (Delta_YTM)^2) = (-7.00 × 0.015) + (0.5 × 60.0 × 0.000225) = -0.1050 + 0.00675 = -0.09825 (-9.825%). Positive convexity always cushions the price drop when yields rise and magnifies the price gain when yields fall. ✓; B) -10.500% (Duration effect only) or -11.175% (subtracting the convexity adjustment); C) -9.150% (calculated as -7.00 × 0.015 + 60.0 × (0.015)^2 without the 1/2 factor); D) +9.825% |
| 17 | A 1-year forward contract on a dividend-paying stock is initiated when the spot price is S_0 = $120, the present value of dividends over the year is PV_0(I) = $3.00, and the annual risk-free rate is r = 5.00%. What is the no-arbitrage Forward Price F_0(T), and what is the value of the long forward position 6 months later (t = 0.5) if the stock price is S_0.5 = $125 and remaining dividends have present value PV_0.5(I) = $1.50? | Initial Forward Price F_0(1) = $122.85 (($120.00 - $3.00) × 1.05); Value at t = 0.5 is V_0.5(1) = +$3.61 ($125.00 - $1.50 - $122.85 / (1.05)^0.5 = $123.50 - $119.89) — At initiation (t = 0), Forward Price F_0(T) = [S_0 - PV_0(I)](1 + r)^T = ($120 - $3.00)(1.05)^1 = $117.00 × 1.05 = $122.85 (and initial contract value V_0 = $0). At t = 0.5 years (0.5 years remaining to maturity), the value to the long party is V_0.5 = S_0.5 - PV_0.5(I) - F_0(T) / (1 + r)^(1 - 0.5) = $125.00 - $1.50 - ($122.85 / 1.024695) = $123.50 - $119.89 = +$3.61. | At initiation (t = 0), Forward Price F_0(T) = [S_0 - PV_0(I)](1 + r)^T = ($120 - $3.00)(1.05)^1 = $117.00 × 1.05 = $122.85 (and initial contract value V_0 = $0). At t = 0.5 years (0.5 years remaining to maturity), the value to the long party is V_0.5 = S_0.5 - PV_0.5(I) - F_0(T) / (1 + r)^(1 - 0.5) = $125.00 - $1.50 - ($122.85 / 1.024695) = $123.50 - $119.89 = +$3.61. | A) Initial Forward Price F_0(1) = $129.15 (($120.00 + $3.00) × 1.05); Value at t = 0.5 is -$2.54; B) Initial Forward Price F_0(1) = $122.85; Value at t = 0.5 is $0.00 because forward contracts are marked to market daily; C) Initial Forward Price F_0(1) = $126.00; Value at t = 0.5 is +$2.15 ($125.00 - $122.85); D) Initial Forward Price F_0(1) = $122.85 (($120.00 - $3.00) × 1.05); Value at t = 0.5 is V_0.5(1) = +$3.61 ($125.00 - $1.50 - $122.85 / (1.05)^0.5 = $123.50 - $119.89) — At initiation (t = 0), Forward Price F_0(T) = [S_0 - PV_0(I)](1 + r)^T = ($120 - $3.00)(1.05)^1 = $117.00 × 1.05 = $122.85 (and initial contract value V_0 = $0). At t = 0.5 years (0.5 years remaining to maturity), the value to the long party is V_0.5 = S_0.5 - PV_0.5(I) - F_0(T) / (1 + r)^(1 - 0.5) = $125.00 - $1.50 - ($122.85 / 1.024695) = $123.50 - $119.89 = +$3.61. ✓ |
| 18 | In agency Residential Mortgage-Backed Securities (RMBS) and Collateralized Mortgage Obligations (CMOs), distinguish between Contraction Risk and Extension Risk, and explain how Planned Amortization Class (PAC) tranches and Support (Companion) tranches allocate these risks. | Contraction Risk occurs when falling interest rates accelerate mortgage prepayments (shortening average life and forcing reinvestment at lower yields); Extension Risk occurs when rising interest rates slow prepayments (lengthening average life); in a CMO, Support/Companion tranches absorb both contraction and extension prepayment risk first so PAC tranches maintain predictable scheduled cash flows within a PSA collar — Homeowners hold an embedded prepayment (call) option: when mortgage rates fall, refinancing surges, returning principal faster than expected (Contraction Risk). When mortgage rates rise, refinancing dries up and homeowners stay in low-rate mortgages longer (Extension Risk). In a PAC/Support CMO structure, Support (Companion) tranches act as sponges—absorbing excess prepayments when rates fall and forgoing principal when rates rise—to keep PAC tranche cash flows on schedule within the initial PSA prepayment collar. | Homeowners hold an embedded prepayment (call) option: when mortgage rates fall, refinancing surges, returning principal faster than expected (Contraction Risk). When mortgage rates rise, refinancing dries up and homeowners stay in low-rate mortgages longer (Extension Risk). In a PAC/Support CMO structure, Support (Companion) tranches act as sponges—absorbing excess prepayments when rates fall and forgoing principal when rates rise—to keep PAC tranche cash flows on schedule within the initial PSA prepayment collar. | A) Both Contraction Risk and Extension Risk are forms of credit default risk eliminated by Fannie Mae and Ginnie Mae guarantees; B) Contraction Risk occurs when interest rates rise; Extension Risk occurs when interest rates fall; PAC tranches absorb all prepayment volatility to protect Support tranches; C) Contraction Risk occurs when falling interest rates accelerate mortgage prepayments (shortening average life and forcing reinvestment at lower yields); Extension Risk occurs when rising interest rates slow prepayments (lengthening average life); in a CMO, Support/Companion tranches absorb both contraction and extension prepayment risk first so PAC tranches maintain predictable scheduled cash flows within a PSA collar — Homeowners hold an embedded prepayment (call) option: when mortgage rates fall, refinancing surges, returning principal faster than expected (Contraction Risk). When mortgage rates rise, refinancing dries up and homeowners stay in low-rate mortgages longer (Extension Risk). In a PAC/Support CMO structure, Support (Companion) tranches act as sponges—absorbing excess prepayments when rates fall and forgoing principal when rates rise—to keep PAC tranche cash flows on schedule within the initial PSA prepayment collar. ✓; D) Sequential-pay CMO tranches eliminate all Contraction and Extension risk for the senior A tranche without needing Support tranches |
| 19 | A stock trades at $60.00 per share, has expected next-year earnings E_1 = $4.50 per share, and a required return on equity r = 10.0%. What is the Present Value of Growth Opportunities (PVGO), and what percentage of the stock's current price is attributable to PVGO? | No-Growth Value (E_1 / r) = $45.00; PVGO = $15.00 ($60.00 - $45.00); PVGO accounts for 25.0% ($15 / $60) of the current share price — A stock's value decomposes into its no-growth perpetuity value plus the Present Value of Growth Opportunities: P_0 = (E_1 / r) + PVGO. Here, E_1 / r = $4.50 / 0.10 = $45.00, so PVGO = $60.00 - $45.00 = $15.00 per share (25.0% of the $60.00 market price). | A stock's value decomposes into its no-growth perpetuity value plus the Present Value of Growth Opportunities: P_0 = (E_1 / r) + PVGO. Here, E_1 / r = $4.50 / 0.10 = $45.00, so PVGO = $60.00 - $45.00 = $15.00 per share (25.0% of the $60.00 market price). | A) No-Growth Value = $60.00; PVGO = $0.00 because E_1 / P_0 = 7.5% < 10%; B) No-Growth Value = $40.91; PVGO = $19.09; PVGO accounts for 31.8% of the current share price; C) No-Growth Value (E_1 / r) = $45.00; PVGO = $15.00 ($60.00 - $45.00); PVGO accounts for 25.0% ($15 / $60) of the current share price — A stock's value decomposes into its no-growth perpetuity value plus the Present Value of Growth Opportunities: P_0 = (E_1 / r) + PVGO. Here, E_1 / r = $4.50 / 0.10 = $45.00, so PVGO = $60.00 - $45.00 = $15.00 per share (25.0% of the $60.00 market price). ✓; D) No-Growth Value = $45.00; PVGO = $105.00 ($60.00 + $45.00) |
| 20 | In Private Equity fund waterfalls, compare a Deal-by-Deal (American) Waterfall with a Whole-of-Fund (European) Waterfall, and define a Clawback Provision and a High-Water Mark (HWM). | An American (Deal-by-Deal) Waterfall pays carried interest to the GP after each profitable exit (more favorable to the GP, requiring a Clawback clause so LPs can recover excess carry if later deals lose money); a European (Whole-of-Fund) Waterfall returns 100% of LP contributed capital and the preferred hurdle return before the GP receives any carried interest; a High-Water Mark in hedge funds prevents charging incentive fees twice on recovering past losses — In a European (Whole-of-Fund) waterfall, LPs receive all capital contributions plus the hurdle return across the entire fund before the GP collects carried interest. In an American (Deal-by-Deal) waterfall, the GP receives carry on early winning exits before LPs are made whole on the entire fund—making a Clawback Provision essential so LPs can reclaim overpaid carry if subsequent investments incur losses. | In a European (Whole-of-Fund) waterfall, LPs receive all capital contributions plus the hurdle return across the entire fund before the GP collects carried interest. In an American (Deal-by-Deal) waterfall, the GP receives carry on early winning exits before LPs are made whole on the entire fund—making a Clawback Provision essential so LPs can reclaim overpaid carry if subsequent investments incur losses. | A) A High-Water Mark guarantees that LPs cannot lose more than 20% of their initial investment in a hedge fund; B) A European Waterfall pays carried interest after each individual deal, whereas an American Waterfall waits until the entire fund is liquidated; C) A Clawback Provision allows the GP to demand additional capital above an LP's committed capital; D) An American (Deal-by-Deal) Waterfall pays carried interest to the GP after each profitable exit (more favorable to the GP, requiring a Clawback clause so LPs can recover excess carry if later deals lose money); a European (Whole-of-Fund) Waterfall returns 100% of LP contributed capital and the preferred hurdle return before the GP receives any carried interest; a High-Water Mark in hedge funds prevents charging incentive fees twice on recovering past losses — In a European (Whole-of-Fund) waterfall, LPs receive all capital contributions plus the hurdle return across the entire fund before the GP collects carried interest. In an American (Deal-by-Deal) waterfall, the GP receives carry on early winning exits before LPs are made whole on the entire fund—making a Clawback Provision essential so LPs can reclaim overpaid carry if subsequent investments incur losses. ✓ |
| 21 | A stock currently priced at S_0 = $50 can either rise by 25% to S_1+ = $62.50 (u = 1.25) or fall by 20% to S_1- = $40.00 (d = 0.80) over the next year. The 1-year risk-free rate is r = 4.00%. Using the One-Period Binomial Option Pricing Model, calculate the risk-neutral probability of an up move (pi) and the current value of a 1-year European Call option with strike X = $50.00. | Risk-neutral probability pi = 0.5333 ((1.04 - 0.80) / (1.25 - 0.80) = 0.24 / 0.45); Call payoffs are c_1+ = $12.50 and c_1- = $0; Call value c_0 = $6.41 ((0.5333 × $12.50 + 0.4667 × $0) / 1.04) — In the one-period binomial model, the risk-neutral probability pi = (1 + r - d) / (u - d) = (1.04 - 0.80) / (1.25 - 0.80) = 0.24 / 0.45 = 8/15 = 0.5333. At expiration, c_1+ = max(0, $62.50 - $50) = $12.50 and c_1- = max(0, $40 - $50) = $0. Discounting the risk-neutral expected payoff at the risk-free rate gives c_0 = [(8/15)($12.50) + (7/15)($0)] / 1.04 = $6.6667 / 1.04 = $6.41. Actual real-world probabilities do not affect no-arbitrage option prices. | In the one-period binomial model, the risk-neutral probability pi = (1 + r - d) / (u - d) = (1.04 - 0.80) / (1.25 - 0.80) = 0.24 / 0.45 = 8/15 = 0.5333. At expiration, c_1+ = max(0, $62.50 - $50) = $12.50 and c_1- = max(0, $40 - $50) = $0. Discounting the risk-neutral expected payoff at the risk-free rate gives c_0 = [(8/15)($12.50) + (7/15)($0)] / 1.04 = $6.6667 / 1.04 = $6.41. Actual real-world probabilities do not affect no-arbitrage option prices. | A) Risk-neutral probability pi = 0.5000; Call value c_0 = $6.01 ($6.25 / 1.04); B) Risk-neutral probability pi = 0.5556 (0.25 / 0.45); Call value c_0 = $6.68; C) Risk-neutral probability pi = 0.5333 ((1.04 - 0.80) / (1.25 - 0.80) = 0.24 / 0.45); Call payoffs are c_1+ = $12.50 and c_1- = $0; Call value c_0 = $6.41 ((0.5333 × $12.50 + 0.4667 × $0) / 1.04) — In the one-period binomial model, the risk-neutral probability pi = (1 + r - d) / (u - d) = (1.04 - 0.80) / (1.25 - 0.80) = 0.24 / 0.45 = 8/15 = 0.5333. At expiration, c_1+ = max(0, $62.50 - $50) = $12.50 and c_1- = max(0, $40 - $50) = $0. Discounting the risk-neutral expected payoff at the risk-free rate gives c_0 = [(8/15)($12.50) + (7/15)($0)] / 1.04 = $6.6667 / 1.04 = $6.41. Actual real-world probabilities do not affect no-arbitrage option prices. ✓; D) Cannot be calculated without knowing the real-world econometric probability of the stock rising |
| 22 | Compare the Zero-Volatility Spread (Z-Spread) and the Option-Adjusted Spread (OAS) for a callable corporate bond versus a putable corporate bond. | For a callable bond, OAS < Z-Spread (because OAS = Z-Spread - Option Cost, as the investor is short the call option); for a putable bond, OAS > Z-Spread (because OAS = Z-Spread + Option Benefit, as the investor is long the put option); if interest rate volatility increases, the callable bond's OAS decreases — The Z-Spread is the constant spread added to every spot rate on the benchmark spot curve to equate PV of cash flows to the bond's market price. It includes both credit/liquidity spread (OAS) and embedded option value. For a callable bond, Z-Spread = OAS + Call Option Cost, so OAS = Z-Spread - Call Option Cost < Z-Spread (and higher volatility raises Call Option Cost, lowering OAS when holding price fixed). For a putable bond, OAS > Z-Spread. | The Z-Spread is the constant spread added to every spot rate on the benchmark spot curve to equate PV of cash flows to the bond's market price. It includes both credit/liquidity spread (OAS) and embedded option value. For a callable bond, Z-Spread = OAS + Call Option Cost, so OAS = Z-Spread - Call Option Cost < Z-Spread (and higher volatility raises Call Option Cost, lowering OAS when holding price fixed). For a putable bond, OAS > Z-Spread. | A) Z-Spread and OAS are always equal unless the benchmark Treasury yield curve is inverted; B) For a callable bond, OAS > Z-Spread; for a putable bond, OAS < Z-Spread; higher interest rate volatility increases the OAS of a callable bond; C) For a callable bond, OAS < Z-Spread (because OAS = Z-Spread - Option Cost, as the investor is short the call option); for a putable bond, OAS > Z-Spread (because OAS = Z-Spread + Option Benefit, as the investor is long the put option); if interest rate volatility increases, the callable bond's OAS decreases — The Z-Spread is the constant spread added to every spot rate on the benchmark spot curve to equate PV of cash flows to the bond's market price. It includes both credit/liquidity spread (OAS) and embedded option value. For a callable bond, Z-Spread = OAS + Call Option Cost, so OAS = Z-Spread - Call Option Cost < Z-Spread (and higher volatility raises Call Option Cost, lowering OAS when holding price fixed). For a putable bond, OAS > Z-Spread. ✓; D) Z-Spread is the yield spread over the single on-the-run Treasury bond of closest maturity (G-Spread) |
| 23 | Compare the Capital Market Line (CML) and the Security Market Line (SML) with respect to (1) the risk measure on the horizontal axis, (2) the slope of the line, and (3) which assets/portfolios plot on the line in equilibrium. | The CML plots Expected Return against Total Risk (standard deviation sigma_p) with slope equal to the Market Sharpe Ratio (E(R_M) - R_f) / sigma_M, and ONLY efficient portfolios (combinations of the risk-free asset and the Market Portfolio M) lie on the CML; the SML plots Expected Return against Systematic Risk (beta_i) with slope equal to the Market Risk Premium (E(R_M) - R_f), and ALL properly priced individual securities and portfolios (efficient or inefficient) lie on the SML — The CML is the special Capital Allocation Line where the risky tangency portfolio is the Market Portfolio M: E(R_p) = R_f + [(E(R_M) - R_f) / sigma_M] × sigma_p. Because unsystematic risk is diversified away in efficient portfolios, only efficient portfolios plot on the CML (x-axis = sigma_p). The SML graphs the CAPM: E(R_i) = R_f + beta_i × [E(R_M) - R_f] (x-axis = beta_i), pricing every individual asset and portfolio in equilibrium. | The CML is the special Capital Allocation Line where the risky tangency portfolio is the Market Portfolio M: E(R_p) = R_f + [(E(R_M) - R_f) / sigma_M] × sigma_p. Because unsystematic risk is diversified away in efficient portfolios, only efficient portfolios plot on the CML (x-axis = sigma_p). The SML graphs the CAPM: E(R_i) = R_f + beta_i × [E(R_M) - R_f] (x-axis = beta_i), pricing every individual asset and portfolio in equilibrium. | A) A stock that plots ABOVE the SML has a negative Jensen's Alpha and is overvalued; B) The CML uses Beta on the x-axis and prices all individual stocks; the SML uses Standard Deviation on the x-axis and applies only to efficient portfolios; C) The CML plots Expected Return against Total Risk (standard deviation sigma_p) with slope equal to the Market Sharpe Ratio (E(R_M) - R_f) / sigma_M, and ONLY efficient portfolios (combinations of the risk-free asset and the Market Portfolio M) lie on the CML; the SML plots Expected Return against Systematic Risk (beta_i) with slope equal to the Market Risk Premium (E(R_M) - R_f), and ALL properly priced individual securities and portfolios (efficient or inefficient) lie on the SML — The CML is the special Capital Allocation Line where the risky tangency portfolio is the Market Portfolio M: E(R_p) = R_f + [(E(R_M) - R_f) / sigma_M] × sigma_p. Because unsystematic risk is diversified away in efficient portfolios, only efficient portfolios plot on the CML (x-axis = sigma_p). The SML graphs the CAPM: E(R_i) = R_f + beta_i × [E(R_M) - R_f] (x-axis = beta_i), pricing every individual asset and portfolio in equilibrium. ✓; D) Both the CML and SML use Standard Deviation on the x-axis, but the CML includes borrowing at the risk-free rate while the SML does not |
| 24 | In Behavioral Finance, distinguish between Cognitive Errors (Belief Perseverance and Information-Processing Biases) and Emotional Biases ('LOSSES'), and explain when an adviser should Moderate vs. Adapt to a client's bias. | Cognitive Errors (e.g., Conservatism, Confirmation, Representativeness, Anchoring, Mental Accounting, Framing, Availability) arise from faulty statistical/information processing and can be MODERATED through education; Emotional Biases (LOSSES: Loss Aversion, Overconfidence, Self-Control, Status Quo, Endowment, Regret Aversion) arise from feelings/impulses and are hard to correct, so advisers typically ADAPT to them (especially when the client has HIGH ability to bear risk) and MODERATE when ability is low — Cognitive errors stem from faulty reasoning or memory/statistical shortcuts and respond well to structured education (Moderating). Emotional biases ('LOSSES': Loss Aversion, Overconfidence, Self-Control, Status Quo, Endowment, Regret Aversion) stem from visceral feelings. Matrix rule: High Wealth (High Ability) + Emotional Bias -> Adapt; Low Wealth (Low Ability) + Cognitive Error -> Moderate. | Cognitive errors stem from faulty reasoning or memory/statistical shortcuts and respond well to structured education (Moderating). Emotional biases ('LOSSES': Loss Aversion, Overconfidence, Self-Control, Status Quo, Endowment, Regret Aversion) stem from visceral feelings. Matrix rule: High Wealth (High Ability) + Emotional Bias -> Adapt; Low Wealth (Low Ability) + Cognitive Error -> Moderate. | A) When a wealthy client has high ability to take risk and exhibits Emotional Biases, the adviser must strictly force the client onto the mathematically optimal mean-variance portfolio; B) Emotional Biases are easily eliminated through quantitative spreadsheets, whereas Cognitive Errors can never be moderated; C) Cognitive Errors (e.g., Conservatism, Confirmation, Representativeness, Anchoring, Mental Accounting, Framing, Availability) arise from faulty statistical/information processing and can be MODERATED through education; Emotional Biases (LOSSES: Loss Aversion, Overconfidence, Self-Control, Status Quo, Endowment, Regret Aversion) arise from feelings/impulses and are hard to correct, so advisers typically ADAPT to them (especially when the client has HIGH ability to bear risk) and MODERATE when ability is low — Cognitive errors stem from faulty reasoning or memory/statistical shortcuts and respond well to structured education (Moderating). Emotional biases ('LOSSES': Loss Aversion, Overconfidence, Self-Control, Status Quo, Endowment, Regret Aversion) stem from visceral feelings. Matrix rule: High Wealth (High Ability) + Emotional Bias -> Adapt; Low Wealth (Low Ability) + Cognitive Error -> Moderate. ✓; D) Loss Aversion and Endowment Bias are Information-Processing Cognitive Errors, whereas Anchoring and Framing are Emotional Biases |
| 25 | In constructing an Investment Policy Statement (IPS), what are the five standard portfolio constraints ('LLTTU'), and how should a wealth manager resolve a conflict between a client's Ability to take risk and Willingness to take risk? | The 5 IPS constraints (LLTTU) are Liquidity, Legal and Regulatory, Time Horizon, Tax Concerns, and Unique Circumstances; when a client's objective Ability to bear risk (financial capacity, horizon, wealth relative to liabilities) conflicts with their subjective Willingness to bear risk (psychological risk tolerance), the adviser must follow the LOWER of the two and educate the client — Every IPS defines Return and Risk objectives alongside the five LLTTU constraints: Liquidity, Legal/Regulatory, Time Horizon, Tax, and Unique Needs. Risk tolerance depends on both Ability (objective financial metrics: time horizon, net worth, income stability, liquidity needs) and Willingness (subjective psychological comfort). Prudent fiduciary practice dictates resolving any mismatch in favor of the LOWER risk level. | Every IPS defines Return and Risk objectives alongside the five LLTTU constraints: Liquidity, Legal/Regulatory, Time Horizon, Tax, and Unique Needs. Risk tolerance depends on both Ability (objective financial metrics: time horizon, net worth, income stability, liquidity needs) and Willingness (subjective psychological comfort). Prudent fiduciary practice dictates resolving any mismatch in favor of the LOWER risk level. | A) The 5 IPS constraints (LLTTU) are Liquidity, Legal and Regulatory, Time Horizon, Tax Concerns, and Unique Circumstances; when a client's objective Ability to bear risk (financial capacity, horizon, wealth relative to liabilities) conflicts with their subjective Willingness to bear risk (psychological risk tolerance), the adviser must follow the LOWER of the two and educate the client — Every IPS defines Return and Risk objectives alongside the five LLTTU constraints: Liquidity, Legal/Regulatory, Time Horizon, Tax, and Unique Needs. Risk tolerance depends on both Ability (objective financial metrics: time horizon, net worth, income stability, liquidity needs) and Willingness (subjective psychological comfort). Prudent fiduciary practice dictates resolving any mismatch in favor of the LOWER risk level. ✓; B) When ability is high and willingness is low, the adviser should invest aggressively without informing the client to maximize terminal wealth; C) The 5 IPS constraints are Leverage, Liquidity, Turnover, Taxes, and Underwriting; willingness to take risk always overrides ability to take risk; D) When ability is low and willingness is high, the adviser should take the arithmetic average of the two risk levels |
| 26 | A company has 10 million common shares trading at $35 per share, $40 million in preferred stock (market value), $150 million in total debt (market value), $10 million in noncontrolling (minority) interest, and $50 million in cash and short-term marketable securities. If EBITDA is $80 million, what are Enterprise Value (EV) and the EV/EBITDA multiple? | Enterprise Value = $500 million ($350M + $40M + $150M + $10M - $50M); EV/EBITDA = 6.25x ($500M / $80M) — Enterprise Value (EV) = Market Value of Common Equity ($350M) + Market Value of Preferred Stock ($40M) + Market Value of Total Debt ($150M) + Noncontrolling Interest ($10M) - Cash & Short-Term Investments ($50M) = $500 million (cash is subtracted because an acquirer takes over the cash to pay down takeover cost). EV/EBITDA = $500M / $80M = 6.25x. | Enterprise Value (EV) = Market Value of Common Equity ($350M) + Market Value of Preferred Stock ($40M) + Market Value of Total Debt ($150M) + Noncontrolling Interest ($10M) - Cash & Short-Term Investments ($50M) = $500 million (cash is subtracted because an acquirer takes over the cash to pay down takeover cost). EV/EBITDA = $500M / $80M = 6.25x. | A) Enterprise Value = $450 million (excluding preferred stock and minority interest); EV/EBITDA = 5.63x; B) Enterprise Value = $550 million ($350M + $40M + $150M + $10M); EV/EBITDA = 6.88x; C) Enterprise Value = $600 million (adding cash to debt and equity); EV/EBITDA = 7.50x; D) Enterprise Value = $500 million ($350M + $40M + $150M + $10M - $50M); EV/EBITDA = 6.25x ($500M / $80M) — Enterprise Value (EV) = Market Value of Common Equity ($350M) + Market Value of Preferred Stock ($40M) + Market Value of Total Debt ($150M) + Noncontrolling Interest ($10M) - Cash & Short-Term Investments ($50M) = $500 million (cash is subtracted because an acquirer takes over the cash to pay down takeover cost). EV/EBITDA = $500M / $80M = 6.25x. ✓ |
| 27 | Under the Multifactor Arbitrage Pricing Theory (APT) model, what are the key assumptions compared to the CAPM, and what is the expected return of a well-diversified portfolio with GDP Factor Sensitivity beta_GDP = 1.4 (Factor Risk Premium = 3.0%) and Inflation Surprise Sensitivity beta_INF = -0.5 (Factor Risk Premium = -2.0%) when the risk-free rate is 4.0%? | Expected Return = 9.20% (4.0% + 1.4(3.0%) + (-0.5)(-2.0%) = 4.0% + 4.2% + 1.0%); unlike CAPM, APT does NOT require the unobservable 'Market Portfolio,' assumes asset returns are generated by multiple systematic factor risks, and relies on no-arbitrage pricing across well-diversified portfolios — Under Ross's Arbitrage Pricing Theory (APT), E(R_p) = R_f + beta_1(lambda_1) + beta_2(lambda_2) + ... = 4.0% + (1.4)(3.0%) + (-0.5)(-2.0%) = 4.0% + 4.2% + 1.0% = 9.20%. APT makes fewer restrictive assumptions than CAPM: it requires only a factor model of returns, a large number of assets to diversify away idiosyncratic risk, and the absence of arbitrage opportunities (without requiring the true Market Portfolio). | Under Ross's Arbitrage Pricing Theory (APT), E(R_p) = R_f + beta_1(lambda_1) + beta_2(lambda_2) + ... = 4.0% + (1.4)(3.0%) + (-0.5)(-2.0%) = 4.0% + 4.2% + 1.0% = 9.20%. APT makes fewer restrictive assumptions than CAPM: it requires only a factor model of returns, a large number of assets to diversify away idiosyncratic risk, and the absence of arbitrage opportunities (without requiring the true Market Portfolio). | A) Expected Return = 9.20% (4.0% + 1.4(3.0%) + (-0.5)(-2.0%) = 4.0% + 4.2% + 1.0%); unlike CAPM, APT does NOT require the unobservable 'Market Portfolio,' assumes asset returns are generated by multiple systematic factor risks, and relies on no-arbitrage pricing across well-diversified portfolios — Under Ross's Arbitrage Pricing Theory (APT), E(R_p) = R_f + beta_1(lambda_1) + beta_2(lambda_2) + ... = 4.0% + (1.4)(3.0%) + (-0.5)(-2.0%) = 4.0% + 4.2% + 1.0% = 9.20%. APT makes fewer restrictive assumptions than CAPM: it requires only a factor model of returns, a large number of assets to diversify away idiosyncratic risk, and the absence of arbitrage opportunities (without requiring the true Market Portfolio). ✓; B) Expected Return = 9.20%; APT is a single-factor equilibrium model identical to the Security Market Line; C) Expected Return = 7.20% (4.0% + 4.2% - 1.0%); APT requires that all investors hold mean-variance efficient portfolios and that returns are normally distributed; D) Expected Return = 5.20%; APT uses only firm-specific unsystematic risk factors |
| 28 | A bond trading at a full price of $102.00 has a Macaulay Duration of 6.30 years and a Yield to Maturity of 5.00% (paid annually). If its price rises to $105.05 when YTM falls by 50 bps (-0.0050) and falls to $99.05 when YTM rises by 50 bps (+0.0050), what are its exact Modified Duration, Approximate Modified Duration, and Duration Gap for an investor with a 4.00-year investment horizon? | Exact Modified Duration = 6.00 years (6.30 / 1.05); Approximate Modified Duration = 5.88 years (($105.05 - $99.05) / (2 × 0.005 × $102.00)); Duration Gap = +2.30 years (6.30 - 4.00, meaning price risk dominates reinvestment risk and the investor is hurt by rising yields) — Exact Modified Duration = Macaulay Duration / (1 + YTM/m) = 6.30 / 1.05 = 6.00 years. Approximate Modified Duration = (PV_- - PV_+) / (2 × Delta_Yield × PV_0) = ($105.05 - $99.05) / (2 × 0.005 × $102.00) = $6.00 / $1.02 = 5.88 years. Duration Gap = Macaulay Duration - Investment Horizon = 6.30 - 4.00 = +2.30 years (positive gap means market price risk dominates coupon reinvestment risk, exposing the investor to rising interest rates). | Exact Modified Duration = Macaulay Duration / (1 + YTM/m) = 6.30 / 1.05 = 6.00 years. Approximate Modified Duration = (PV_- - PV_+) / (2 × Delta_Yield × PV_0) = ($105.05 - $99.05) / (2 × 0.005 × $102.00) = $6.00 / $1.02 = 5.88 years. Duration Gap = Macaulay Duration - Investment Horizon = 6.30 - 4.00 = +2.30 years (positive gap means market price risk dominates coupon reinvestment risk, exposing the investor to rising interest rates). | A) Exact Modified Duration = 6.30 years; Duration Gap = 0.00 years; B) Exact Modified Duration = 6.00 years (6.30 / 1.05); Approximate Modified Duration = 5.88 years (($105.05 - $99.05) / (2 × 0.005 × $102.00)); Duration Gap = +2.30 years (6.30 - 4.00, meaning price risk dominates reinvestment risk and the investor is hurt by rising yields) — Exact Modified Duration = Macaulay Duration / (1 + YTM/m) = 6.30 / 1.05 = 6.00 years. Approximate Modified Duration = (PV_- - PV_+) / (2 × Delta_Yield × PV_0) = ($105.05 - $99.05) / (2 × 0.005 × $102.00) = $6.00 / $1.02 = 5.88 years. Duration Gap = Macaulay Duration - Investment Horizon = 6.30 - 4.00 = +2.30 years (positive gap means market price risk dominates coupon reinvestment risk, exposing the investor to rising interest rates). ✓; C) Exact Modified Duration = 6.615 years (6.30 × 1.05); Approximate Modified Duration = 6.00 years; Duration Gap = -2.30 years (exposed to falling yields); D) Exact Modified Duration = 6.00 years; Approximate Modified Duration = 2.94 years; Duration Gap = +2.00 years |
| 29 | Why do bond dealers quote the Flat (Clean) Price rather than the Full (Dirty/Invoice) Price, and how are the two related between coupon dates? | Full (Dirty) Price = Flat (Clean) Price + Accrued Interest (where AI = (t / T) × PMT); dealers quote the Clean Price so daily price quotes reflect true yield/credit changes rather than the predictable saw-tooth upward drift of accrued interest between coupon payments, while the buyer pays the Full (Dirty) Price at settlement — At settlement, the buyer must pay the seller the Full (Dirty/Invoice) Price = Clean Price + Accrued Interest (where Accrued Interest = (days since last coupon t / days in coupon period T) × Coupon Payment). Quoting the Clean Price avoids misleading daily price increases caused merely by the passage of time between coupon dates. | At settlement, the buyer must pay the seller the Full (Dirty/Invoice) Price = Clean Price + Accrued Interest (where Accrued Interest = (days since last coupon t / days in coupon period T) × Coupon Payment). Quoting the Clean Price avoids misleading daily price increases caused merely by the passage of time between coupon dates. | A) Full (Dirty) Price = Flat (Clean) Price + Accrued Interest (where AI = (t / T) × PMT); dealers quote the Clean Price so daily price quotes reflect true yield/credit changes rather than the predictable saw-tooth upward drift of accrued interest between coupon payments, while the buyer pays the Full (Dirty) Price at settlement — At settlement, the buyer must pay the seller the Full (Dirty/Invoice) Price = Clean Price + Accrued Interest (where Accrued Interest = (days since last coupon t / days in coupon period T) × Coupon Payment). Quoting the Clean Price avoids misleading daily price increases caused merely by the passage of time between coupon dates. ✓; B) Flat (Clean) Price = Full (Dirty) Price + Accrued Interest; the buyer pays the Clean Price at settlement and the seller pays Accrued Interest; C) Dealers quote the Dirty Price for corporate bonds and the Clean Price only for zero-coupon Treasury bills; D) Accrued Interest is calculated by compounding the coupon rate at the bond's current YTM over the remaining days to maturity |
| 30 | A hedge fund with $100 million beginning-of-year NAV charges a '2 and 20' fee structure (2% management fee based on end-of-year NAV, and a 20% incentive fee calculated NET of the management fee, subject to a 5% HARD hurdle rate). If the fund generates a 25% gross return before fees during the year (ending gross value = $125 million), what are the total fees and the investor's net percentage return? | Management Fee = $2.50M (2% of $125M); Return net of management fee = $22.50M (22.5%); Hard Hurdle = $5.00M (5% of $100M); Incentive Fee = $3.50M (20% of ($22.50M - $5.00M)); Total Fees = $6.00M; Investor Net Return = +19.00% — Step 1: Management Fee = 2% × $125M (ending gross value) = $2.50M, leaving $122.50M ($22.50M gain net of management fee). Step 2: Because the 5% hurdle ($5.00M) is a HARD hurdle and incentive fees are calculated net of management fees, Incentive Fee = 20% × ($22.50M - $5.00M) = 20% × $17.50M = $3.50M. Step 3: Total Fees = $2.50M + $3.50M = $6.00M, leaving Ending Net NAV = $119.00M (+19.00% net return). | Step 1: Management Fee = 2% × $125M (ending gross value) = $2.50M, leaving $122.50M ($22.50M gain net of management fee). Step 2: Because the 5% hurdle ($5.00M) is a HARD hurdle and incentive fees are calculated net of management fees, Incentive Fee = 20% × ($22.50M - $5.00M) = 20% × $17.50M = $3.50M. Step 3: Total Fees = $2.50M + $3.50M = $6.00M, leaving Ending Net NAV = $119.00M (+19.00% net return). | A) Management Fee = $2.50M (2% of $125M); Return net of management fee = $22.50M (22.5%); Hard Hurdle = $5.00M (5% of $100M); Incentive Fee = $3.50M (20% of ($22.50M - $5.00M)); Total Fees = $6.00M; Investor Net Return = +19.00% — Step 1: Management Fee = 2% × $125M (ending gross value) = $2.50M, leaving $122.50M ($22.50M gain net of management fee). Step 2: Because the 5% hurdle ($5.00M) is a HARD hurdle and incentive fees are calculated net of management fees, Incentive Fee = 20% × ($22.50M - $5.00M) = 20% × $17.50M = $3.50M. Step 3: Total Fees = $2.50M + $3.50M = $6.00M, leaving Ending Net NAV = $119.00M (+19.00% net return). ✓; B) Management Fee = $2.50M; Incentive Fee = $4.50M (20% of $22.50M because 22.5% > 5% soft hurdle); Investor Net Return = +18.00%; C) Management Fee = $2.00M; Incentive Fee = $4.00M (20% of ($25M - $5M)); Total Fees = $6.00M; Investor Net Return = +19.00%; D) Management Fee = $2.50M; Incentive Fee = $5.00M (20% of $25M); Investor Net Return = +17.50% |
| 31 | A company has a Return on Equity (ROE) of 15.0%, a dividend payout ratio of 40%, a just-paid annual dividend D_0 of $2.00, and a required return on equity r of 12.0%. Using the Gordon Growth Model (GGM), calculate the company's sustainable growth rate g, intrinsic value V_0, and Justified Leading P/E multiple (P_0 / E_1). | Sustainable growth rate g = 9.0% ((1 - 0.40) × 15%); Intrinsic Value V_0 = $72.67 ($2.00 × 1.09 / (0.12 - 0.09)); Justified Leading P/E = 13.33x (0.40 / (0.12 - 0.09)) — Retention ratio b = 1 - 0.40 = 0.60, so sustainable growth g = b × ROE = 0.60 × 15% = 9.0%. Next year's expected dividend D_1 = D_0(1 + g) = $2.00 × 1.09 = $2.18. By the Gordon Growth Model, V_0 = D_1 / (r - g) = $2.18 / (0.12 - 0.09) = $2.18 / 0.03 = $72.67. Justified Leading P/E = (1 - b) / (r - g) = 0.40 / 0.03 = 13.33x. | Retention ratio b = 1 - 0.40 = 0.60, so sustainable growth g = b × ROE = 0.60 × 15% = 9.0%. Next year's expected dividend D_1 = D_0(1 + g) = $2.00 × 1.09 = $2.18. By the Gordon Growth Model, V_0 = D_1 / (r - g) = $2.18 / (0.12 - 0.09) = $2.18 / 0.03 = $72.67. Justified Leading P/E = (1 - b) / (r - g) = 0.40 / 0.03 = 13.33x. | A) Sustainable growth rate g = 6.0% (0.40 × 15%); Intrinsic Value V_0 = $35.33; Justified Leading P/E = 6.67x; B) Sustainable growth rate g = 9.0% ((1 - 0.40) × 15%); Intrinsic Value V_0 = $72.67 ($2.00 × 1.09 / (0.12 - 0.09)); Justified Leading P/E = 13.33x (0.40 / (0.12 - 0.09)) — Retention ratio b = 1 - 0.40 = 0.60, so sustainable growth g = b × ROE = 0.60 × 15% = 9.0%. Next year's expected dividend D_1 = D_0(1 + g) = $2.00 × 1.09 = $2.18. By the Gordon Growth Model, V_0 = D_1 / (r - g) = $2.18 / (0.12 - 0.09) = $2.18 / 0.03 = $72.67. Justified Leading P/E = (1 - b) / (r - g) = 0.40 / 0.03 = 13.33x. ✓; C) Sustainable growth rate g = 9.0%; Intrinsic Value V_0 = $66.67 ($2.00 / 0.03); Justified Leading P/E = 20.00x (0.60 / 0.03); D) Sustainable growth rate g = 15.0%; GGM cannot be applied because g > r |
| 32 | Stock Z has a return standard deviation of sigma_Z = 24%, the Market Portfolio has a standard deviation of sigma_M = 16%, and the correlation between Stock Z and the Market is rho_(Z,M) = 0.75. The risk-free rate is R_f = 4.0% and the expected return on the Market Portfolio is E(R_M) = 10.0%. If an analyst forecasts that Stock Z will return 12.5% over the next year, what are Stock Z's Beta, CAPM Required Return, Jensen's Alpha, and valuation status? | Beta = 1.125 (0.75 × (24% / 16%)); CAPM Required Return = 10.75% (4.0% + 1.125 × (10.0% - 4.0%)); Expected Jensen's Alpha = +1.75% (12.50% - 10.75%); Stock Z plots ABOVE the SML and is UNDERVALUED (Buy) — Step 1: Beta_Z = rho_(Z,M) × (sigma_Z / sigma_M) = 0.75 × (0.24 / 0.16) = 0.75 × 1.50 = 1.125. Step 2: CAPM Required Return = R_f + Beta_Z × (E(R_M) - R_f) = 4.0% + 1.125 × (10.0% - 4.0%) = 4.0% + 6.75% = 10.75%. Step 3: Expected Alpha = Forecast Return - Required Return = 12.50% - 10.75% = +1.75%. Because Forecast Return > Required Return (plots ABOVE the SML), the stock's price is too low relative to its future cash flows—it is UNDERVALUED (Buy). | Step 1: Beta_Z = rho_(Z,M) × (sigma_Z / sigma_M) = 0.75 × (0.24 / 0.16) = 0.75 × 1.50 = 1.125. Step 2: CAPM Required Return = R_f + Beta_Z × (E(R_M) - R_f) = 4.0% + 1.125 × (10.0% - 4.0%) = 4.0% + 6.75% = 10.75%. Step 3: Expected Alpha = Forecast Return - Required Return = 12.50% - 10.75% = +1.75%. Because Forecast Return > Required Return (plots ABOVE the SML), the stock's price is too low relative to its future cash flows—it is UNDERVALUED (Buy). | A) Beta = 0.500 (0.75 × (16% / 24%)); CAPM Required Return = 7.00%; Expected Jensen's Alpha = +5.50%; OVERVALUED; B) Beta = 1.125; CAPM Required Return = 10.75%; Expected Jensen's Alpha = +1.75%; Stock Z plots ABOVE the SML and is OVERVALUED (Sell); C) Beta = 1.125 (0.75 × (24% / 16%)); CAPM Required Return = 10.75% (4.0% + 1.125 × (10.0% - 4.0%)); Expected Jensen's Alpha = +1.75% (12.50% - 10.75%); Stock Z plots ABOVE the SML and is UNDERVALUED (Buy) — Step 1: Beta_Z = rho_(Z,M) × (sigma_Z / sigma_M) = 0.75 × (0.24 / 0.16) = 0.75 × 1.50 = 1.125. Step 2: CAPM Required Return = R_f + Beta_Z × (E(R_M) - R_f) = 4.0% + 1.125 × (10.0% - 4.0%) = 4.0% + 6.75% = 10.75%. Step 3: Expected Alpha = Forecast Return - Required Return = 12.50% - 10.75% = +1.75%. Because Forecast Return > Required Return (plots ABOVE the SML), the stock's price is too low relative to its future cash flows—it is UNDERVALUED (Buy). ✓; D) Beta = 1.500 (24% / 16%); CAPM Required Return = 13.00%; Expected Jensen's Alpha = -0.50%; OVERVALUED |

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