CHAPTER 4
45
46 RETURN MEASUREMENT
sures the performance of a dollar invested in the fund for the complete mea- surement period. The TWR eliminates the timing effect that external portfolio cash flows have on performance, leaving only the effects of the mar- ket and manager decisions. TWRs are used to make legitimate comparisons of manager performance, both to other managers with a similar style and to the market. To calculate a time-weighted return, we break the measurement period into shorter subperiods, calculate the returns for the subperiods, and then compound them together to derive the TWR for the period. The subpe- riod boundaries are determined by the dates of each cash flow.
Taking these steps we can calculate a TWR:
1. Start with the market value at the beginning of the period.
2. Move forward through time toward the end of the period.
3. Note the value of the portfolio the instant before a cash flow into or out of the portfolio.
4. At each valuation date, calculate a subperiod return, which is the return for the subperiod since the last cash flow.
5. Use the market value at the end of the period to calculate the subperiod return for the last period.
6. Compute the TWR, equal to the product of (1 + the subperiod returns).
The last step is called geometric linking, or chain linking of the returns. Chain linking has the same function as compounding in the future value calculation: each subperiod ending value including income earned is invested into the next period’s return. Chain linking is used instead of the future value formula to employ the compounding because the periodic returns are usually different for each subperiod:
(4.1) whereRN are the subperiod returns.
We often use the word geometricwhen we are referring to the com- pounding process. Geometric mathematical operations relate to propor- tions, whereas arithmetic operations refer to differences. We use multiplication when we work with proportions and addition when we work with differences. To illustrate, in an arithmetic sequence 3, 6, 9, 12 we add the fixed number 3 to derive the next result. In a geometric sequence we multiply to get 3, 9, 27, 81. Fortunately for us, investing is a geometric process!
The TWR assumes compounding and reinvestment of gains and income earned in prior subperiods. The expression (1 + the subperiod decimal return) is called a wealth relative, return relative, orgrowth rate.
Time-weighted return 1+R1
( )×(1+R2) …× (1+RN)–1
[ ]×100
=
Time-Weighted Return 47
The growth rate represents the increase in capital over the subperiod, which is the ratio of the ending market value to the beginning market value. For example, if a portfolio is worth $100 at the beginning of the subperiod, and $105 at the end of the subperiod before the next cash flow, the subperiod return is 5% and the growth rate for the subperiod equals 1.05. The TWR requires as inputs the date and value of each cash flow into and out of the portfolio and the values of the fund at the beginning and end of each subperiod bracketed by the cash flow dates.
The next three sections illustrate the steps to calculate a TWR where we want to evaluate the performance of an investment manager over a month where the fund the fund values were:
And there were two cash flows during the month:
1. Divide into Subperiods Based on Cash Flow Dates
The first step in the TWR calculation is to divide the period we are interested in into subperiods, where the subperiods are separated by the cash flow dates. The next step is to note the value of the portfolio the instant before the cash flow. If we are working with a beginning of day assumption for the timing of the cash flows, we use the valuation for the night prior to the date of the cash flow into or out of the portfolio:
Date End of Day Valuation
5/31 1000
6/9 1100
6/19 1200
6/30 1200
Date Cash Flow
6/10 200
6/20 −100
Date Begin of Day Valuation Cash Flow End of Day Valuation
5/31 1000
6/9 1100
6/10 1100 200
6/19 1200
6/20 1200 −100
6/30 1200 1200
48 RETURN MEASUREMENT
In this case, there are two cash flows and three subperiods, from:
1. 5/31 to the end of day 6/9 2. 6/10 to the end of day 6/19 3. 6/20 to the end of day 6/30
Note that there are (1 + the number of cash flow dates) subperiods.
2. Calculate Subperiod Returns
Single period returns are calculated for each subperiod. The assumption regarding the time of day that cash flows are made available to the man- ager controls the treatment of the cash flow adjustments in the formula.
Here we assume that cash flows occur at the beginning of the day, so we adjust the market value for the beginning of the day by the cash flow in order to form the denominator of the return calculation. Cash flows into the portfolio are added to the denominator, cash flows out of the portfolio are subtracted. If there is more than one cash flow during the day we net the flows together in order to calculate the cash flow adjust- ment:
(4.2)
The numerator in the calculation is the ending market value for the subperiod. The denominator is the beginning market value adjusted for the cash flow in or out of the portfolio. This cash flow adjustment is required to adjust the valuation just prior to the cash flow for the amount of the flow. The purpose of the cash flow adjustment is to negate the effect of the contributions/withdrawals from the return cal- culation.
The subperiod returns calculated in terms of our example are 10%,
−7.69%, and 9.09%. Exhibit 4.1 summarizes the calculation of the sub- period returns.
EXHIBIT 4.1 Subperiod Returns
Subperiod return (start of day flow assumed) MVE
MVB+Net cash inflows ---
=
Time-Weighted Return 49
EXHIBIT 4.2 Time-Weighted Return
3. Link the Subperiod Returns
The percentage return for the month is calculated by chain linking the subperiod returns:
By calculating returns in this manner, we have completely eliminated the impact of cash flows into or out of the portfolio. TWR is used in per- formance measurement applications where we are interested in isolating just the decisions of the portfolio manager. These applications include the comparison of manager performance to benchmarks. Exhibit 4.2 shows how the TWR eliminates the effect of cash flows from the example return calculation.
There are some exceptions to the general rule that TWR is the appro- priate measure of investment manager performance. In some situations, the portfolio manager does have discretion over the timing of cash flows.
The development of the TWR was an important milestone in invest- ment analysis. A study commissioned in 1968 by the Bank Administra- tion Institute (BAI), recommended the TWR as the appropriate method of calculating a return for the purpose of manager evaluation and com- parison.1 The BAI study, authored by leading academics, is the most influential document developed in the field of investment performance measurement. Although some aspects are dated, such as certain techniques for dealing with portfolios measured infrequently, the recommendations of
1Bank Administration Institute, Measuring the Investment Performance of Pension Funds for the Purpose of Inter-Fund Comparison, Bank Administration Institute, Park Ridge, Illinois 1968.
1+0.10
( )×(1+(–0.0769))×(1+0.0909)–1
[ ]×100 = 10.77%
50 RETURN MEASUREMENT
the BAI study remain the template for the measurement of portfolio per- formance.