CORRECT: C, The holding period return captures both components of what an investor earned: the change in price (P₁ − P₀) and any income received during the period (I₁). Both go in the numerator; the beginning price (P₀) is the denominator because it represents the starting investment. An investor who pays USD 1,000 for a bond that returns USD 50 in interest and sells for USD 1,030 earns (1,030 − 1,000 + 50) / 1,000 = 8%. Omitting the income understates the return.

Why not A? Option A divides by the ending price P₁ rather than the beginning price P₀. Returns are always expressed relative to what you paid at the start, not what the asset is worth at the end. Using P₁ as the denominator produces a different and non-standard number that does not represent the investor's percentage gain or loss on their original outlay.

Why not B? Option B uses the correct denominator (P₀) and captures the price change, but omits income. For assets that pay dividends, interest, or rent, this systematically understates the total return. A bond paying 5% annual interest alongside a 2% price gain has a 7% holding period return, not a 2% one.

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CORRECT: B, The correct calculation is (1 + (−0.20))(1 + 0.20) − 1 = (0.80)(1.20) − 1 = 0.96 − 1 = −4%. The loss in Year 1 reduces the portfolio to 80 cents on the dollar. The 20% gain in Year 2 then applies to this reduced base: 0.80 × 1.20 = 0.96. The two changes do not cancel because they operate on different bases. This asymmetry explains why the geometric mean is always less than or equal to the arithmetic mean when returns vary.

Why not A? Adding −20% and +20% and dividing by 2 gives an arithmetic mean of 0%. That is the average of the two single-year returns, not the compound result. It answers a different question: what might this portfolio return in a single future year? For actual compound growth over two years, the returns must be chained by multiplication.

Why not C? −20% would be the result only if both years produced −20%. The second year's gain of +20% does partially recover the loss, just not fully. The terminal value of 96 cents implies a two-year return of −4%, not −20%. Selecting −20% ignores the Year 2 return entirely.

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CORRECT: B, Chain the return factors: (1.14)(0.90)(0.98) = 1.00548. Subtract one: 0.00548 = 0.55%. This is the correct multi-period holding period return, reflecting the compound effect of all three years.

Why not A? 0.67% appears to result from a small arithmetic error in the multiplication. The correct product (1.14)(0.90)(0.98) is 1.00548, not 1.00670. Rounding each factor before multiplying introduces error. Always multiply the full decimal expressions before rounding.

Why not C? 2.0% is the arithmetic sum of the three annual returns: 14% − 10% − 2% = 2%. This adds the returns rather than multiplying the return factors. The arithmetic sum ignores compounding and produces the wrong result for a multi-period holding period return. Note that 2.0% and 0.55% are not rounding variants of each other: they come from fundamentally different calculations.

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CORRECT: C, Both the arithmetic mean (2.67%) and geometric mean (0.52%) are calculated correctly. The error is in the application. Projecting EUR 1 at 2.67% per year for three years gives (1.0267)³ − 1 = 8.1%, implying a terminal value of EUR 1.081. The actual terminal value is (1.22)(0.75)(1.11) = EUR 1.016. The geometric mean of 0.52% replicates the actual terminal value: (1.0052)³ = 1.016. The arithmetic mean is a forward-looking estimator for one period; it cannot represent cumulative compound growth.

Why not A? The arithmetic mean of (22% − 25% + 11%) / 3 = 8% / 3 = 2.67% is correct. Option A introduces a calculation error that does not exist in the question. The issue is not whether the arithmetic mean is correctly computed, but whether it is the right tool for this purpose.

Why not B? The choice between arithmetic and geometric means has nothing to do with the type of asset. The geometric mean is the appropriate measure of actual compound growth for any investment across multiple periods: equities, hedge funds, bonds, real estate, or any other asset class. Restricting the geometric mean to fixed income would be incorrect.

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CORRECT: A, When a fixed monetary amount is invested at varying prices, the harmonic mean correctly captures the average cost per unit: X̄_H = 4 / [(1/62) + (1/76) + (1/84) + (1/90)] = 4 / 0.0523 = EUR 76.48. At EUR 62, EUR 5,000 buys 80.6 shares. At EUR 90, it buys only 55.6 shares. More units were acquired at lower prices. The harmonic mean accounts for this by giving lower weight to higher prices, which is exactly what the buying pattern produces.

Why not B? The arithmetic mean of EUR 78.00 assumes equal quantities were purchased at each price. That is not what happened. Equal monetary amounts produce unequal share quantities. Using the arithmetic mean overstates the true average cost because it gives the high prices (EUR 84, EUR 90) more weight than the investor's actual purchases warrant.

Why not C? Prices are not compounding returns. The geometric mean is designed for rates of the form (1 + R). Applying it to raw price levels does not correspond to any meaningful financial concept for this type of cost-averaging problem. EUR 77.26 would emerge mechanically from the geometric formula but does not describe the average cost per share acquired.

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CORRECT: C, The actual terminal value is EUR 1,000 × (0.50)(1.35)(1.27) = EUR 1,000 × 0.857 = EUR 857. The geometric mean of this sequence is (0.857)^(1/3) − 1 = −5.0% per year. EUR 1,000 growing at −5% per year for three years: (0.95)³ × 1,000 = EUR 857. The large Year 1 loss permanently reduces the base. The subsequent gains, while positive, apply to that reduced base and cannot recover it.

Why not A? EUR 1,124 comes from compounding at the arithmetic mean of 4.0%: EUR 1,000 × (1.04)³ = EUR 1,124. This is the answer the trap is designed to produce. The arithmetic mean of 4.0% is calculated correctly, but it is the wrong tool. It systematically overstates compound growth when returns vary because it does not account for the asymmetric impact of losses on the base amount.

Why not B? EUR 1,040 represents EUR 1,000 growing at 4.0% for just one year. This is a misapplication in the opposite direction: using the arithmetic mean as if it applies to a single year rather than three. The question asks for a three-year terminal value, so a one-year projection answers the wrong question entirely. Neither EUR 1,124 nor EUR 1,040 reflects what the fund actually delivered.

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CORRECT: B, The money-weighted return is the IRR applied to all net cash flows. Money flowing into the investment is a negative outflow from the investor's perspective; money received back is positive. The discount rate that sets the net present value of all these flows to zero is the MWR. Critically, this rate is sensitive to the amounts invested at each date, giving more influence to periods with more capital at risk.

Why not A? Option A describes the time-weighted return. The TWR chains sub-period HPRs by multiplication, explicitly removing the influence of cash flow timing. The MWR does the opposite: it is the one measure that is explicitly sensitive to when and how much is invested. They are conceptual opposites on the question of cash-flow timing.

Why not C? Option C is also the time-weighted return, specifically its interpretation: what one unit of currency invested at inception would have grown to. The MWR does not track a hypothetical unit, it tracks the actual dollars the investor deployed at actual dates. Two investors in the same fund with different deposit histories will have different MWRs and the same TWR.

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CORRECT: B, TWR = (1.25)(1.05) − 1 = 31.25%. The large deposit arrived at the start of Year 2, just before the lower 5% return. More capital endured the weak year. The MWR is pulled toward 5% and falls well below 31.25%. TWR remains 31.25% regardless of when the investor added money, because it weights each year equally.

Why not A? The two measures are equal only when no mid-period cash flows exist, or when the flows are negligible. A large deposit creates a meaningful divergence. The sign of the returns (both positive here) tells you MWR will also be positive, but it says nothing about whether MWR will equal, exceed, or fall short of TWR. The timing of the deposit determines the direction of the gap.

Why not C? Both annual returns being positive means both MWR and TWR are positive. The direction of the MWR, TWR gap is determined by whether large cash flows arrived before the stronger or the weaker year. The large deposit arrived before Year 2 (the weaker year, at 5%, not the 25% year). This pulls MWR down, not up. MWR falls below TWR in this scenario.

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CORRECT: B, Enter CF0 = −200, C01 = −220, C02 = +480 in the BA II Plus CF worksheet and press IRR CPT. The result is 9.39%. This is the single rate that satisfies: −200 + (−220/1.0939) + (480/1.0939²) = 0. Alternatively, at 9.39%: −200 − 201.12 + 401.12 = 0. ✓

Why not A? At r = 8%: −200 + (−220/1.08) + (480/1.08²) = −200 − 203.7 + 411.5 = +7.8. The NPV is still positive at 8%, meaning the actual IRR is higher than 8%. The true rate must be raised until the NPV equals zero. 8% is too low.

Why not C? At r = 10%: −200 + (−220/1.10) + (480/1.10²) = −200 − 200 + 396.7 = −3.3. The NPV is slightly negative at 10%, meaning the true IRR is just below 10%. The exact answer of 9.39% lies between these two bounds and is the only rate that produces NPV = 0.

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CORRECT: B, Break at the 1 July cash flow. Sub-period 1 (Jan, Jun): HPR₁ = (130 − 100)/100 = 30%. Sub-period 2 (Jul, Dec): HPR₂ = (270 − 300)/300 = −10%. Chain: (1.30)(0.90) − 1 = 1.17 − 1 = 17.0%. The sub-period boundary must be set at the cash flow date, using the portfolio value before the deposit arrives.

Why not A? 10.0% is the arithmetic average of the two sub-period returns: (30% + (−10%))/2 = 10%. The TWR requires multiplicative chaining, not addition. The difference between 10% and 17% arises from the compounding effect of the strong first half applied to the second half. Averaging single-period returns always understates the TWR when the first period is strongly positive.

Why not C? 0.0% would emerge from calculating (ending − beginning − deposit)/beginning = (270 − 100 − 170)/100 = 0. This treats the year as a single unsplit period and uses the full beginning value as the denominator, ignoring that a large new investment arrived mid-year. The TWR requires splitting at each cash flow date precisely to avoid this distortion. The deposit belongs to the second sub-period, not the first.

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CORRECT: B, Both managers earned an identical TWR of 18.0%. The TWR is the appropriate measure for evaluating manager skill because it strips out the effect of cash flows that managers do not control. Nakamura's MWR exceeds 18% (clients withdrew before a strong quarter, meaning less money earned the high return). Bellini's MWR falls below 18% (clients added before a weak quarter). These differences reflect investor timing, not manager decisions. Comparing managers on MWR would produce misleading conclusions.

Why not A? The statement is factually correct about the direction of the MWR differences, but it draws the wrong conclusion. A higher MWR for Nakamura does not indicate superior manager performance, it indicates that Nakamura's clients happened to withdraw at a fortunate time. Manager evaluation requires TWR, which shows both managers performed identically at 18.0%.

Why not C? The volume of new client investment reflects marketing success, client confidence, or market timing by investors, none of which are indicators of manager investment skill. Bellini attracted capital, but that capital arrived before a poor period, which is precisely why Bellini's MWR fell below their TWR. Using capital flows as a performance signal would reward managers for attracting poorly timed deposits.

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CORRECT: A, The correct entry nets the t = 1 flows: −225 + 5 = −220. This gives CF0 = −100, CF1 = −220, CF2 = +480, yielding MWR = 9.39%. The analyst's entry introduces a fourth row: the +5 is entered as what the calculator reads as CF3 (a third period beyond t = 0). This pushes the USD 480 inflow to what becomes a fourth period, discounting it more heavily. The higher discount applied to the terminal inflow reduces the IRR. In practice, the calculator produces a materially lower result, not the 9.39% that correctly reflects Aiko's performance.

Why not B? The total values are the same: −100 − 225 + 5 + 480 = 160 in both cases. But present value is not just about totals, it depends on timing. The calculator worksheet assigns each new entry to the next sequential time period. Entering the t = 1 outflow and inflow as two separate entries shifts the terminal cash flow to t = 3 instead of t = 2. The timing error changes the IRR even though the undiscounted sum is identical.

Why not C? Entering the flows separately does not amplify the cost. The outflow of −225 is entered at its actual t = 1 position. The error is that the +5 inflow is then assigned to t = 2, and the +480 to t = 3, one period later than intended. This delays the receipt of the large positive cash flow, which reduces the IRR. The cost side is unchanged; only the timing of the receipt is distorted.

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CORRECT: B, With monthly compounding, there are 12 compounding periods in one year. The TVM worksheet requires both inputs in matching units. N = 12 total monthly periods. I/Y = 12%/12 = 1% per month. Entering N = 12 and I/Y = 1 and pressing CPT FV correctly applies monthly compounding for one year. The effective annual rate from these inputs is (1.01)¹² − 1 = 12.68%, higher than the 12% stated rate.

Why not A? N = 1 and I/Y = 12 calculates annual compounding for one year, producing (1.12)¹ − 1 = 12%. This ignores the monthly compounding frequency. Monthly compounding on the same stated rate produces a higher effective return (12.68%) because each month's interest earns interest in subsequent months. Using annual inputs when compounding is monthly understates the future value.

Why not C? I/Y must reflect the periodic rate, not the annual rate. Entering I/Y = 12 tells the calculator that each of the 12 monthly periods earns 12%, implying an annual effective rate of (1.12)¹² − 1 = 241%. This is obviously wrong. The stated rate of 12% per year becomes 1% per month, which is the only correct I/Y input when N counts monthly periods.

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CORRECT: C, Multiplying 0.25% by 52 assumes each week earns 0.25% on the original base with no reinvestment. Compounding assumes each week's return earns additional returns in subsequent weeks. After week one, the base is 1.0025 rather than 1. Week two's 0.25% is applied to that larger base, and so on. The difference (13.86% − 13.00% = 0.86%) is the accumulated interest-on-interest over 52 weeks. On the exam, the compounded answer is always required.

Why not A? The approximation excuse does not apply here. When exam answer choices are separated by less than 1%, using the linear multiplication result will select the wrong option. The curriculum explicitly teaches the compounded formula for a reason. "Small return" approximations are appropriate only for the real rate / nominal rate formula (LO 1), not for annualising returns.

Why not B? The compounded formula is fully valid for any holding period, including weekly returns. The constraint is the assumption of reinvestment at the same rate each period, the exam acknowledges this limitation but still requires the compounded formula. The reliability concern exists in practice but does not affect the calculation methodology tested in this LO.

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CORRECT: B, Apply the non-annual compounding formula: FV = 20,000 × (1 + 0.10/4)^(4×2) = 20,000 × (1.025)^8 = 20,000 × 1.2184 = USD 24,368. On the BA II Plus: N = 8, I/Y = 2.5, PV = −20,000, PMT = 0, CPT FV gives 24,367.91.

Why not A? USD 24,200 results from annual compounding at 10% for two years: 20,000 × (1.10)² = 24,200. This ignores the quarterly compounding frequency, it fails to account for interest earned on previously accumulated quarterly interest. The quarterly rate of 2.5% applied eight times always produces more than the annual rate of 10% applied twice.

Why not C? USD 24,491 corresponds to monthly compounding: 20,000 × (1 + 0.10/12)^(24) = 24,491. Monthly compounding (12 periods per year) produces a higher result than quarterly compounding (4 periods per year) because the compounding frequency is higher. The question specifies quarterly, not monthly. Using m = 12 instead of m = 4 produces this inflated answer.

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CORRECT: B, ETF 1: (1.0425)^(365/125) − 1 = 12.92%. ETF 2: (1.0195)^(52/8) − 1 = 13.37%. ETF 3: (1.1718)^(12/16) − 1 = 12.63%. ETF 2 has the highest annualised return despite having the lowest raw return (1.95%). Its eight-week holding period implies an aggressive annualisation exponent of 52/8 = 6.5. Short-term returns with high compounding factors annualise powerfully because each period's gain builds on an already-compounded base.

Why not A? ETF 1 is close but falls short at 12.92%. The 125-day holding period with 4.25% generates a strong annualised result, but ETF 2's 1.95% over only 8 weeks compounds more efficiently when projected to an annual basis. The shorter the period, the larger the exponent, and the larger the exponent, the more aggressively a positive return grows when expressed annually.

Why not C? ETF 3 has both the longest holding period (16 months, more than one year) and the highest raw return (17.18%). Annualising a return measured over more than one year requires an exponent below one (12/16 = 0.75), which shrinks the return to 12.63%. A high multi-month return does not automatically translate into the highest annual rate, duration matters as much as magnitude.

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CORRECT: A, Apply the formula: r_cc = ln(P₁/P₀) = ln(208.25/186.75) = ln(1.1151) = 0.1090 = 10.90%. On the BA II Plus: enter 1.1151, then press the LN key (left side of the calculator, same row as the 7, 8, and 9 digits). The LN key computes the natural logarithm directly. The continuously compounded return is always less than the holding period return for positive returns.

Why not B? 11.51% is the holding period return, not the continuously compounded return. The two are related by the formula r_cc = ln(1 + HPR) = ln(1.1151) = 10.90%. Treating them as equal ignores the mathematical distinction between discrete compounding (the HPR framework) and continuous compounding (the ln framework). They are equivalent representations of the same price change but are numerically different.

Why not C? 12.17% would result from applying a formula in the wrong direction or from an arithmetic error. One possible source: computing 1/ln(1.1151) or confusing the exponential and logarithm functions. The continuously compounded return is always below the holding period return for positive values; any answer above 11.51% should be immediately rejected as inconsistent with the direction of the relationship.

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CORRECT: C, The correct annualised return is (1.062)^(365/100) − 1 = (1.062)^3.65 − 1 = 24.55%. The exponent 3.65 captures the compounding of the 6.2% return across 3.65 equivalent 100-day periods per year. Each period's gain builds on the accumulated gains of previous periods. The analyst's linear multiplication of 6.2% × 3.65 = 22.63% ignores this interest-on-interest effect.

Why not B? 22.63% is the analyst's linear extrapolation result, which assumes each 100-day period earns 6.2% on the same original base with no reinvestment. This matches the concept of simple interest, not compound interest. On an exam where answer choices are 20%, 22.6%, and 24.6%, linear multiplication selects option B, which is incorrect. The compounded figure (24.55%) is the only answer consistent with the reinvestment assumption embedded in the standard annualisation formula.

Why not A? 20.00% has no direct connection to the correct calculation or the linear trap. It might appear as a round-number distractor. The three meaningful numbers in this question are: the raw HPR (6.2%), the linear annualisation (22.63%), and the correct compounded annualisation (24.55%). Any answer below 22.63% would imply a smaller compounding effect than even linear multiplication, which has no mathematical justification.

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CORRECT: B, Trading commissions are directly tied to generating investment returns. They reduce the prices at which assets are bought or increase the prices at which they are sold, and are therefore already embedded in the gross return figure. Gross return is computed after trading costs, not before them. This is why gross return is appropriate for evaluating manager skill: it includes the costs the manager directly incurs in pursuing returns.

Why not A? Management fees are not deducted from gross return. They are deducted to convert gross return into net return. Management fees compensate the firm for running the portfolio, but their size often depends on the amount of assets under management, not directly on the trading activity that generated returns. Subtracting them from gross return to get net return shows what the investor actually receives.

Why not C? Custodial fees are administrative expenses unrelated to the trading or portfolio decisions that generated returns. They are excluded from gross return and deducted when computing net return, alongside management fees. The distinction to remember: any cost that directly affects the prices at which trades are executed belongs in gross return; any cost that is a separate fee for services belongs in the gross-to-net deduction.

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CORRECT: C, Real return = (1.080 / 1.021) − 1 = 1.0578 − 1 = 5.78%. The exact division formula accounts for the fact that inflation reduces the value of the nominal return in every period, not just once at the end. The division removes the purchasing-power erosion embedded in the 8.0% nominal return.

Why not A? 5.90% results from the simple subtraction approximation: 8.0% − 2.1% = 5.9%. This approximation works adequately when both rates are very small, but here the difference versus the correct answer is 0.12 percentage points, sufficient for exam answer choices to distinguish them. When choices include both 5.78% and 5.90%, the exact formula selects the correct one.

Why not B? 6.20% does not correspond to any standard real return calculation. It is higher than even the approximation (5.90%). If this number appears on an exam, it signals an arithmetic error in the direction of overstating the real return, perhaps from incorrect formula manipulation or using (1.08 × 1.021) rather than dividing.

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CORRECT: A, Identify the inputs: VB = 20% × USD 25M = USD 5M. VE = 80% × USD 25M = USD 20M. VB/VE = 5/20 = 0.25. Rp = 10%, rD = 6%. Apply the formula: RL = 10% + 0.25 × (10% − 6%) = 10% + 0.25 × 4% = 10% + 1.0% = 11.0%. The spread of 4% between portfolio return and borrowing cost, scaled by the 0.25 leverage ratio, adds 1 percentage point to the unlevered return.

Why not B? 10.5% results from using the wrong leverage ratio. If a candidate uses VB/(VB + VE) = 0.20 instead of VB/VE = 0.25, they compute 10% + 0.20 × 4% = 10.8%, which rounds to neither 10.5% nor 11.0%. The formula specifies VB divided by VE (equity only), not VB divided by total assets. The distinction between these two ratios is the most common source of error in leveraged return calculations.

Why not C? 10.0% is the portfolio return on total assets before any leverage effect. This would be the correct answer only if no debt were used. Adding leverage when Rp > rD always increases the return on equity above the unlevered portfolio return. Reporting 10.0% as the leveraged return ignores the amplification effect entirely, it answers a different question (what did the portfolio earn?) rather than the question asked (what did the equity investor earn, given the borrowing?).

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