A portfolio had a value of $10 million at the beginning of a quarter and a value of $102 million half way through the quarter. At this point, there was a deposit of $2 million and the portfolio value at the end of the quarter was $105 million. The time-weighted return for the quarter was
A) 2.98%.
B) -.96%.
C) 1.99%.
D) 4.21%.

Question

Quizplus · Accepted Answer

The time-weighted return (TWR) is calculated by taking the geometric mean of the returns for each sub-period, factoring in any external cash flows (deposits or withdrawals). In this case, there are two sub-periods: from the beginning to the midpoint of the quarter, and from after the deposit to the end of the quarter.1. For the first sub-period, the return is calculated as: $\frac{102}{10} - 1 = 9.2$ or 920%.2. After the deposit of $2 million, the portfolio's value is adjusted to $104 million just before the end of the quarter. The return for the second sub-period is: $\frac{105}{104} - 1 = 0.009615$ or approximately 0.96%.To find the TWR, we multiply the growth factors of each period and subtract 1: $(1 + 9.2) * (1 + 0.009615) - 1 = (10.2) * (1.009615) - 1 = 10.29763 - 1 = 9.29763$ or 929.763%.However, the calculation above seems to have been misunderstood in the context of the question's choices, suggesting a mistake in the calculation process. The correct approach to calculate the TWR should involve correctly applying the formula for each sub-period and then combining these correctly. Given the options and the context, let's correct the approach:1. First period return: $\frac{102}{10} = 10.2$ or 1020% increase (this step was initially miscalculated as a 920% return).2. Adjust for the deposit: The portfolio increases to $104 million after the deposit.3. Second period return: $\frac{105}{104} = 1.009615$ or approximately 0.9615% increase.The correct TWR calculation should involve finding the product of (1 + each period's return) for all periods, then subtracting 1. The mistake in the initial explanation was in the interpretation of the returns and their calculation. Given the options and the typical way TWR is presented, the correct answer should reflect a reasonable calculation based on the given data, which involves correctly applying the TWR formula:$$TWR = (1 + r_1) \times (1 + r_2) - 1$$where $r_1$ and $r_2$ are the returns for each period. The correct calculation should reflect the geometric linking of returns across the two periods, considering the impact of the cash flow. The provided options suggest a misunderstanding in the calculation process, as none directly match the recalculated approach. The correct TWR should be calculated by accurately applying the returns for each period, considering the deposit as an adjustment to the portfolio's value before calculating the second period's return. Given the mistake in the initial explanation, the focus should be on correctly applying the TWR formula to the given scenario.

A Portfolio Had a Value of $10 Million at the Beginning