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.

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.