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A closer approximation

Because daily portfolio valuations are not seen as feasible, the Association of Investment Management and Research (AIMR) accepts approximation methods that utilise time-weighted rate of returns, such as the modified Dietz method. This method has become the industry standard, but its limitations have spurred the creation of a new, more advanced and considerably more accurate method of performance breakdown.
The modified Dietz method assumes a constant rate of return on the portfolio during the evaluated period, weighting each cashflow by the amount of time it is held in the portfolio. Attributing performance using this method requires only the valuation of the portfolio at the beginning and end of the month, along with the dates and amounts of each cashflow during the month. This method was developed to avoid having to revalue the portfolio every time a transaction took place, instead approximating the time-weighted rate of returns using daily sampled cashflows.
Under certain circumstances, however, the results from this method are so approximate that they prove too erroneous for use in performance breakdown. The method becomes less accurate with higher occurrences of cashflows in the period, higher market volatility, and when stocks are added to or removed from the portfolio. Often, under any combination of these conditions, a clear discrepancy between the real performance of a stock and that reported by Dietz can occur.
This example illustrates how the presence of these conditions can create such a discrepancy. Assume that on June 1 the portfolio contains 100 shares of Stock A, valued at e1 per share, and zero shares of Stock B, also valued at e1. Now at the midpoint of the month, June 15, A is priced at e1.5 and B is still priced at e1, when the portfolio experiences its first cashflow: the 100 shares of A, now valued collectively at e150, are sold and 150 shares of B are purchased. At month’s end, the price of B has not changed, with a market value of holdings in Stock B still at e150. Using the modified Dietz method, the results are surprising. The monthly performance for B comes out as expected as 0%, however for A it is somehow calculated as 200%! While the figures used in this example perhaps exaggerate the flaws of the Dietz method, they clearly indicate that under extreme conditions it invites gross miscalculation (the actual figure for A should be 100%).
We have sought to create a new method for performance breakdown that both remedies the highly approximate nature of the modified Dietz method and facilitates performance breakdowns for periods longer than one month. In an attempt to solve these limitations inherent in the Dietz system, several techniques were explored. Eventually the method that proved the most effective and easily implemented was the discretisation of the total period to be measured into elementary periods. These elementary periods are defined in a rather simple way: the beginning of the first corresponds to that of the total period and it ends with the first cashflow, which corresponds to the beginning of the second elementary period, and so on. All the elementary periods are thus defined, with the final period’s completion date corresponding to that of the total period.
Discretisation offers certain obvious advantages, as it is quite simple to break down the performance of funds over a single elementary period. The only difficulty posed is the recombining of the breakdowns of all the elementary periods to obtain the performance breakdown for the total period. Tests have shown that by defining the composition factors of an elementary period by the period directly subsequent, as well by all those that have preceded it, one obtains rather convincing results. Moreover this concept of the composition of an elementary period offers new possibilities for performance breakdown analysis. This method of “composition factors” can be used for the breakdown of longer periods with the same degree of accuracy. Since classically the period of measurement is monthly, it might also prove interesting for those involved in the investment process, and in particular the manager, to know the contribution of each class of asset in a period corresponding to the bets that could be made in a different time horizon (three months, one year, etc.). The new method fulfils this need effectively.
The new method, like Dietz, requires the valuation of the portfolio at the beginning and the end of the month along with the dates and amounts of each cashflow, but also requires the market value of the portfolio at each cashflow. This points to a potential drawback for the new method in that it is more time-consuming than its predecessor, with more data and frequent calculations necessary.
The benefits of this new method, namely the elimination of the highly approximative nature of the Dietz method and the ability to calculate for much longer periods, by far outweigh the negatives and provide performance and reporting teams with an invaluable tool.
Bernard Sancier, Didier Chan-Voc-Chun and Michael Kooris are with the BNP Paribas group in Paris

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