# Time-weighted returns: unraveling the mystery of investment performance calculations.

 Author: Clarfeld, Robert A.

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Providing clients with fair and meaningful reporting of investment performance is among the primary objectives of investment management services. Often underestimated by CPAs who offer, or are considering offering such services, the process of calculating and presenting investment performance often can be problematic and, in some respects, "counter-intuitive." This article focuses on an important, yet often misunderstood, aspect of the investment reporting process--calculating and presenting time-weighted returns for clients.

CALCULATING INVESTMENT RETURNS

In its simplest form, the total return on an investment is its ending value, plus any distributions, less its beginning value, divided by the beginning value. In the absence of cash flows, the formula is

[R.sub.TR] = (MVE - MVB)/MVB

Where

RTR is total return. MVE is market value--ending. MVB is market value-beginning.

Exhibit 1, page 88, shows a sample calculation.

Exhibit 1: Sample Total Return Calculation

Chuck invests \$100 in Growco, a mutual fund, on April 1. Exactly one year later, he sells his entire position for \$113. Chuck's total return on Growco is 13%, calculated as follows:

(\$113-\$100)/\$100 = 13%.

Unfortunately, investment calculations are rarely as straightforward as this single-period return. Generally, the rate of return on an investment involves an irregular period of time, additional cash contributions, and withdrawals and distributions--all of which the above calculation ignored. A more common method of calculating returns during multiperiod time intervals is internal rate of return or dollar-weighted return. This enables an investment adviser to solve for an investment's return by discounting cash inflows and outflows over a given period of time. The formula for dollar-weighted return is

Initial investment=

[CF.sub.1]/(1+r)+[CF.sub.2]/[(1+r).sup.2]+...+[CF.sub.n]/[(1+r).sup.n]

Where

[CF.sub.(1 to n)] represents the cash flow at each time interval.

r represents the return solved by the equation.

(Investments are shown as negative cash flows, while distributions and the final payment are shown as positive cash flows.) Exhibit 2, page 88, shows a sample calculation.

[Exhibit 2 ILLUSTRATION OMITTED]

Exhibit 2 compares the experiences of two investors--Barry and Samantha--who both invest in the same mutual fund. Dollar-weighted returns reward Samantha's larger investment during the fund's successful first two years. Even though both Barry and Samantha invest in the same...