# Time-Weighted Returns

##### Unraveling the mystery of investment performance calculations.
BY ROBERT A. CLARFELD

ROBERT A. CLARFELD, CPA/PFS, CFP, is president of Clarfeld & Co., PC, a New York City CPA financial planning firm. He is a member of the American Institute of CPAs personal financial planning executive committee and chairman of its investment services task force.

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 processcalculating 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 TR =(MVE-MVB)/MVB

Where
RTR is total return.
MVE is market value—ending.
MVB is market value—beginning.

Exhibit 1 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. Chucks 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 distributionsall 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 investments return by discounting cash inflows and outflows over a given period of time. The formula for dollar-weighted return is

Initial investment=
CF 1 /(1+r)+CF 2 /(1+r) 2 +. . . +CF n /(1+r) n

Where
CF (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 shows a sample calculation.

Exhibit 2 compares the experiences of two investorsBarry and Samanthawho both invest in the same mutual fund. Dollar-weighted returns reward Samanthas larger investment during the funds successful first two years. Even though both Barry and Samantha invest in the same mutual fund over the same time period, their opinions of the fund managers abilitybased on their very different ending account balancesmay differ. Accordingly, the principal limitation of this investor-centered means of measuring performance is that dollar-weighted returns do not isolate a funds performance from an investors timing, luck or lack thereof. Although the ending account balance may be of ultimate importance to an investor, dollar-weighted returns do not adequately reflect a fund managers performance.

Exhibit 2: Sample Dollar-Weighted Return Calculation

Barry and Samantha both invest in Riskco, a mutual fund. All distributions are reinvested.

 Barry Samantha Investment Unrealized Gain or Loss Ending Value Fund Return Investment Unrealized Gain or Loss Ending Value Year 1 \$100 \$ 50 \$150 50% \$300 \$150 \$450 Year 2 100 100 350 40 0 180 630 Year 3 100 (180) 270 (40) 0 (252) 378 Dollar-weighted return. (5.2%)* annualized 8.0%** annualized

* \$100 = -\$100/(1+r) + -\$100/(1+r) 2 +\$270/(1+r) 3 . Additional investments are shown as negative cash flows and the ending value is shown as an inflow. All unrealized gains and losses are assumed to be reinvested, resulting in a net cash flow of \$0.

** \$300 = 0/(1+r) + 0/(1+r) 2 + \$378/(1+r) 3 . All unrealized gains and losses are assumed to be reinvested, resulting in a net cash flow of \$0.

Time-Weighted Returns
Acknowledging the need for consistency in reporting investment returns separate from investors actions, the investment community uses a standard performance measurethe time-weighted returnwhich essentially is a calculation of the investment return generated by a manager over specific time periods that are geometrically linked or compounded. This isolates a managers specific performance from investor timing and the direction of cash flows. Time-weighted returns allow for a consistent measure of a managers performance relative to other managers and to alternative investments and indices.

Ideally, the daily linking of time intervalsthe practice of the mutual fund industryprovides the most accurate performance calculations. One commonly used formula for calculating time-weighted returns when daily valuations are not available is the modified Dietz method.

Return=(MVE-MVB-F) / (MVB+FW).

Where
MVE is market value—ending.
MVB is market value—beginning.
F is net cash inflows (outflows are negative).
FW is the net cash inflows weighted by the days invested.

Exhibit 3, shows how this calculation works.

 Exhibit 3: Sample Modified Dietz Method Calculation Phyllis invests \$100 in Safeco, a mutual fund, on August 1. On August 10, she adds \$25 to her investment. On August 31, the market value of her shares is \$150. Her time-weighted return is 21.4%, calculated as follows: (\$150 - \$100 - \$25) / (\$100 + \$25 * 21/31) = 21.4%.

The modified Dietz method assumes an investment earns a constant rate of return over a selected period, eliminating the need to know the exact valuation on the date of each cash flow. Intuitively, this return might be thought of as a weighted average of achieving a 25% return (earning \$25 on \$100 invested) for one-third of the period under analysis and a 20% return (earning \$25 on \$125 invested) for two-thirds of the period. Calculating Barrys and Samanthas investments in Riskco using time-weighted returns provides an interesting insight, as shown in exhibit 4.

The time-weighted return for Riskco is the same for both Barry and Samantha. (After all, both investors were part of the same pool of commingled funds). In addition, Samanthas returns are identical to her dollar-weighted return, since there were no additional cash flows after her initial investment.

Barry did experience a very real economic loss over the period the fund reported a positive rate of return. However, the difference between these two calculations rests squarely on Barrys timing of his investmentscrushing his belief in the infallibility of dollar-cost averaging, at least over the short term. In the process, Barry learned the meaning of still another expressiongetting whipsawed by the markets.

Choice of Time Intervals
When providing investment advisory services, CPAs should consider following the guidelines of the Association for Investment Management and Research (AIMR), a self-governing organization recognized as the investment communitys resource for the fair and accurate reporting of investment performance. The AIMR performance presentation standards established the voluntary standards investment professionals and organizations follow to promote uniformity and comparability among investment performance presentations. The AIMR Performance Presentation Standards Handbook1997 is an invaluable resource for investment advisers.

Exhibit 4: Time-Weighted Return for Barry and Samantha
 Barry Samantha Year 1 (\$150 - \$100) / \$100 = 50% (\$450 - \$300) / \$300 = 50% Year 2 (\$350 - \$150 - \$100) / (\$150 + \$100) = 40% (\$630 - \$450) / \$450 = 40% Year 3 (\$270 - \$350 - \$100) / (\$350 + \$100) = (40%) (\$378 - \$630) / \$630 = (40%) Time-weighted return: (1+.5) x (1+.4) x (1-.4) = (1+.26) = 8.0% annualized (1+.5) x (1+.4) x (1-.4) = (1+.26) = 8.0% annualized

The shorter the time between valuations, the more accurate the measurement. AIMR encourages geometrically linked daily portfolio valuations. Generally, monthly performance calculations are adequate for most purposes, with quarterly intervals being the absolute minimum. The AIMR standard is as follows: Performance must be valued at least quarterly, and periodic returns must be geometrically linked. AIMR also recommends that if portfolios are not valued daily, they should be valued whenever cash flows and market action combine to materially distort performance. CPAs should consider several factors when selecting time intervals, including the nature and magnitude of investor cash flows, the marginal accuracy gained by using shorter time intervals, the availability of investment data and the methodology used to perform the calculations.

 Investment Outlook Advisory Board Robert A. Clarfeld, CPA/PFS, CFP, Editor Steven I. Levey, CPA/PFS Eric A. Norberg, CPA/PFS, CFP Phyllis J. Bernstein, CPA Susan A. Frohlich, CPA

The Need for Time-Weighted Performance
A modest investment that has performed quite well over the past several years may experience a disastrous run immediately after a large additional investment, leaving a client with a very real aggregate economic loss and a biased perception of the managers performance. Although the clients own timing created the loss, many are slow to acknowledge their contribution. For investors to rely on historical performance as a factor in future investment decisions, they need the objectivity and consistency of time-weighted performance reporting.

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