282 RISK BUDGETING TIME-WEIGHTED RETURN Ideally, we would want to compute a portfolio's return in such a way as to incorporate

investment funds

the precise time when the cash flows occur. To this end, the time-weighted rate of return (also known as the daily valuation method) for a portfolio is given by RmR = (S1xS2x...xSp)-l (17B.4) where P (p = 1, . . . , P) is the number of subperiods that are defined within the period's return and _MVEp fl7B5) F MVBp where MVE is the market value of the portfolio at the end of the pth subperiod, before any cash flows in period p but including accrued income for the period, and MVB is the market value at the end of the previous subperiod (i.e., beginning of this subperiod), including any cash flows at the end of the previous subperiod and including accrued income up to the end of the previous period.This method is the most exact of the three explained here. Note that the main difference between the do liar-weighted return and the time-weighted return is that the former assumes the same rate of return over the whole period. The time-weighted return, on the other hand, uses the geometric average of returns from each individual period. A good way to understand the methods described is to look at a numerical example. Suppose that on January 1, 2002, we invested $100 in the Nasdaq Composite index. On March 1, 2002, we invest another $100. The total return on the Nasdaq from January 1, 2002, through February 28, 2002, was -11.22 percent. Hence our initial investment of $100 is now worth $88.78. However, since we invested another $100, the total value of our investment is $188.78. By March 28, 2002, the total value of our investment has grown to $201.20 and we sell $100. The Nasdaq then declines until finally, on May 10, 2002, we are left with $87.79. We compute our return on this investment as of May 10, 2002, under the different methods presented above. 11 The ideal time-weighted return is [(88.78/100) x (201.20/188.78) x (87.79/101.20)] - 1 = -17.92% 11 The dollar-weighted annualized return based on the BAI method is 87.79 = 100(1 + IRRATE'f0'252 + 100(1 + IRRATE)50'252 - 100(1 + IRRATE)201252 IRRATE =-25.50%

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