Wednesday, January 5, 2011

Attribution question...answered!

A colleague sent me an interesting question, which I will summarize here:

An account has a single cash flow during the month, so there are two sub periods for performance to generate a TWRR. Although we can link the returns to get the full month return, it isn't as easy to derive a single weight that would maintain the integrity of a weighted average calculation. So, the preference is to calculate attribution for each sub period and link. However, we still face a scenario where many indices aren't available daily, so how do we get index return periodicity to match the account returns? This poses a number of questions:
  • Can we assume linear performance for the benchmark?
  • Do the cons of making assumptions around indexes outweigh the benefits? Should we be driven by the least common denominator? In other words if index data is only available monthly then attribution should only be produced monthly. This then means you need to explain the difference between the published return and attribution analysis.
  • Can we manipulate portfolio weights so that we can use monthly analysis and linked returns?
In my view this essentially speaks to the holdings- versus transaction-based issue with attribution. There is a misperception many hold that daily transaction-based attribution is a superior methodology; the reality is that monthly is sufficient for transaction-based attribution; holdings-based, on the other hand, warrants a consideration for daily in order to improve accuracy. And there's the rub: once we move to daily we need to be able to capture the index details for the corresponding periods. And as noted in the question, even breaking a month into two pieces would require this. And, as correctly noted, not all indices provide daily information.

However, if we employ a transaction-based approach, then we don't have to segment the month nor do we have to worry about index details for the sub periods. By using the beginning sector weights, plus weighted flows (just as we do with Modified Dietz) we can accomplish our objective. The returns, too, have to take into consideration the intra-month activity to ensure accuracy.

    1 comment:

    1. David

      You say that “monthly is sufficient for transaction-based attribution… By using the beginning sector weights, plus weighted flows (just as we do with Modified Dietz) we an accomplish our objective.”
      But, consider a fund that starts the month with $100 of cash and only one share of issue z in sector Sz priced at $100. On the first business day of the 20-day month the share price increases to $110 and the fund sells the share into (i.e. purchasing) cash. No more trades are done this month and cash receives no interest.

      In general, Modified Dietz weights and returns are
      Wj = (Voj + Wpj*Pj – Wsj*Sj)/Vo
      Rj = (Vfj – Voj + Sj - Pj)/ (Voj + Wpj*Pj – Wsj*Sj)

      Using Modified Dietz for the month we have
      For cash:
      Wc = (100 + 0.95*110)/200 = 102.25%
      Rc = (210 – 100 + 0 – 110)/(100 + 0.95*110) = 0

      For the issue z and, thus, for sector Sz:
      Wz = (100 – 0.95*110)/200 = -2.25%
      Rz = (0 – 100 + 110)/ (100 – 0.95*110) = -222%

      Rolling up to the fund level we find:
      Wc + Wz = 100% and
      R = (102.25%)*0 + (-2.25%)*(-222%) = 5%,
      Which is the correct fund level return: (210 – 200)/200 = 5%.

      Thus, in this long-only fund, the cash weight is greater than 100% and the issue and sector weight is less than zero. Also, this issue and its sector that did so well are assigned a return of less than negative 100%.
      Despite the fact that these weights and returns roll up to the correct fund-level values for the month, it is easy to see how such strange values for the weights and returns of the cash and the issue/sector can cause havoc in the subsequent attribution calculation that motivated this question. Also, matters get even worse if we tried to link sector Sz’s return of -222% for this month to its return for the next month where, say, we buy some other issue in the Sz sector and keep it till the end of the month where it achieves a positive return. The next month’s positive return makes the two-month period to date return even more negative.
      Quite reasonable examples like this lead me to think that it is not the case that the Modified Dietz approach to performance applied “monthly is sufficient for transaction-based attribution.”



    Note: Only a member of this blog may post a comment.