### CFA Practice Question

There are 294 practice questions for this study session.

### CFA Practice Question

Consider the following transactional information for the investment account of an underwriting syndicate:

1st Quarter
Ending portfolio value: \$50,800,000
Total amount invested: \$46,100,000
2nd Quarter
Ending portfolio value: \$51,100,000
Total amount invested: \$50,800,000
3rd Quarter
Ending portfolio value: \$51,000,000
Total amount invested: \$51,100,000
4th Quarter
Ending portfolio value: \$54,500,000
Total amount invested: \$50,000,000

Using this information, what is the annual time-weighted rate of return for this portfolio? Assume no taxes or transaction charges.
A. 20.59% per year
B. 19.59% per year
C. 22.14% per year
Explanation: The time-weighted rate of return is the preferred method of return calculation in the investment management industry, primarily because this method is not sensitive to significant additions and withdrawals of funds from portfolios under examination. The calculation of the time-weighted rate of return involves three steps:

Step 1:
Price the portfolio immediately prior to any significant additions or withdrawals. Separate the portfolio into a series of sub-periods based on the dates of cash inflows and outflows.
Step 2:
Calculate the holding period return for each sub-period.
Step 3:
Determine the annualized holding period return by linking or compounding the holding period return of each sub-period. If the investment is for more than one year, use the geometric mean of the annual returns as the time-weighted rate of return. If the investment is for less than one year, compound the sub-period returns to obtain an annualized measurement.

To begin the process of determining the time-weighted rate of return, we would break the portfolio up into the subsequent series of cash flows. However, in this example, the cash flows are already aggregated for us and we can move on to the next step: determining the holding period return for each sub-period. This process is detailed as follows:
Quarter 1 holding period return = [(\$50,800,000 ending value - \$46,100,000 invested) / \$46,100,000 invested] = 10.19523%
Quarter 2 holding period return = [(\$51,100,000 ending value - \$50,800,000 invested) / \$50,800,000 invested] = 0.59055%
Quarter 3 holding period return = [(\$51,000,000 ending value - \$51,100,000 invested) / \$51,100,000 invested] = (0.1957%)
Quarter 4 holding period return = [(\$54,500,000 ending value - \$50,000,000 invested) / \$50,000,000 invested] = 9.00%
Now that the holding period return for each sub-period has been determined, we must annualize the return measure by taking the product of all four quarterly returns. This process is detailed as follows: [(1 + .10195) * (1 + .00591) * (1 - .00196) * (1 + .09) - 1] = .2059 or 20.59%

When calculating the time-weighted rate of return, remember that the total amount invested is the relevant figure, not the beginning portfolio value. Notice that during the fourth quarter, the total amount invested does not equal the ending amount for the third quarter. This differential could be explained by numerous phenomena. Perhaps the difference is due to a cash withdrawal from the account. Maybe this amount was used to pay expenses or meet an outstanding margin call. What is important to note is the fact that this money was not invested, and thus should not be included in the holding period return for the fourth quarter. With this said, whenever possible you should use the total amount invested rather than the beginning portfolio value in the calculation of the sub-period holding period return. If you chose 19.59%, remember that in the calculation of the time-weighted rate of return, it is the geometric mean that is used rather than the arithmetic mean.

User Comment
eris I undrstand all the way up to the calculation of HPR annualized. I thought we need to calculate the geometric mean? Why 19.59
Shelton Easy:
54.5/50 * 51/51.1 * 51.1/50.8 * 50.8/46.1 - 1 =20.59%
Janey nice work Shelton!!
patsy agree with eris - should we not multiply the 1.2059 by the power of 1/4??
whipp Patsy - since the investment is for only 1 year (broken down into 4 qrt'ly investment periods), compounding the subperiod (qrt'ly) returns produces the annualized time weighted return, no further calculations needed
cpotts17 You are supposed to take the sum of the HPR's to the 1/4th power, then subtract 1. That gives you the time-weighted rate of return.
cpotts17 Oh just 1 year. Disregard my previous comment.
endurance Shelton - of course :-) Nice
Saves us a lot of precious time here..
JoshCA There were no inflows or outflows until the 4th quarter, so quarters 1 through 3 can be calculated as 51m/46.1m
Lambo83 Really the question should be stated as:

Quarter 1
Total invested
Ending Portfolio Value

Quarter 2
Total Invested
Ending Portfolio Value

You know - chronologically.
cfastudypl Nice one Shelton, thanks.
bbodiam The problem is they took out 1m in between Q3 and Q4. meant to test attention to detail not CFA material