## Sunday, July 25, 2010

### Why Do We Call it "Time-Weighted?" Recently, my boss David Spaulding commented regarding the origin of the term "time-weighted return."

The CIPM curriculum at the Principles Level gives a description as to the origin of the name. Unfortunately, an original source is not cited. But, I remind candidates of this description given in the curriculum, as they are responsible for knowing this information.

I quote from the candidate reading, Chapter 12, Evaluating Portfolio Performance:

"The TWR derives its name from the fact that each subperiod return within the full evaluation period receives a weight proportional to the length of the subperiod relative to the length of the full evaluation period. That relationship becomes apparent if each subperiod return is expressed as the cumulative return over smaller time units."

The curriculum goes on to describe an example of a time-weighted return calculated for a month as follows:

• beginning value - 1,000,000
• contribution of 30,000 on the 5th of the month; market value at that time is 1,045,000
• contribution of 20,000 on the 16th of the month; market value at that time is 1,060,000
• ending value is 1,080,000
The two external cash flows break the month into 3 sub-periods, with returns as follows:

• rt,1 = [(\$1,045,000 − \$30,000) − \$1,000,000]/\$1,000,000 = 1.50%
• rt,2 = [(\$1,060,000 − \$20,000) − \$1,045,000]/\$1,045,000 = -0.48%
• rt,3 = (\$1,080,000 − \$1,060,000)/\$1,060,000 = 1.89%

Thus, the time-weighted return for the month is the geometric linking of the subperiod returns: rtwr = (1 + 0.0150) × (1 + −0.0048) × (1 + 0.0189) − 1 = 2.92%

The reading goes on to point out that one could calculate a daily compounded return for each subperiod:

• period 1 corresponds to a daily compounded return of 0.30% for 5 days
• period 2 corresponds to a daily compounded return of -0.04% for 11 days
• period 3 corresponds to a daily compounded return of 0.13% for 14 days

Thus, the time-weighting, the curriculum indicates, corresponds to the exponent used to compound the daily returns:

rtwr = (1 + 0.0030)**5 × (1 + −0.0004)**11 × (1 + 0.0013)**14 – 1 = 2.92%