Multiperiod Excess Returns

Timothy P. Ryan, CIPM
Hartford Investment Management

Excess returns, and their practical use, tend to be glossed over because they are perceived to be straightforward and easy to understand. For example, a 10 percent portfolio return matched against its 8 percent index return means outperformance by 200 bps; the portfolio’s excess return is +2.0 percent. Such utter simplicity can lure one into thinking there is nothing more to say about this topic. It is my hope, however, that this short article will provide insight into where confusion can occur.

Let’s begin by illustrating a typical client inquiry with the following example. A client is convinced there is an error in his fourth quarter performance report, shown in Table 1:

Table 1. Fourth Quarter 2008 Performance Report: Arithmetic Return

	October			November			December
Item	Month	QTD		Month	QTD		Month	QTD
Portfolio return	–9.50%	–9.50%		–15.50%	–23.53%		12.00%	–14.35%
Index return	–15.50	–15.50		–45.00	–53.53		–0.10	–53.57
Excess return (arithmetic)	6.00%	6.00%		29.50%	30.00%		12.10%	39.22%

The client does not grasp how a 6.00 percent excess return in October followed by a 29.50 percent excess return in November can result in a two-month quarter-to-date (QTD) excess return of merely 30.00 percent. Simple arithmetic shows that
[(1 + 6.00%) x (1 + 29.50%)] – 1 ≠ 30.00%
But why doesn’t the standard multiperiod compounding formula work?

The answer to the client’s inquiry is that one cannot directly compound the arithmetic excess returns to arrive at a correct excess return for a longer time period. The key issue, as covered in Ryan (2000/2001), is the order of the mathematical operations. To arrive at the correct arithmetic excess return, the following steps must be taken.

(1) Multiply (separately linking portfolio and index returns):
Portfolio: [(1 + –9.50%) x (1 + –15.50%)] – 1= –23.53%.
Index: [(1 + –15.50%) x (1 + –45.00%)] – 1= –53.53%.

Then (2) subtract the resulting index return from the compound portfolio return:
Arithmetic excess return = Portfolio return – Index return.
30.00%= [(–23.53%) – (–53.53%)].

Subtracting first then multiplying will produce a different and incorrect result. As practitioners, why should we be concerned with this multiperiod linking of excess returns? First, it has a bottom-line focus on the results achieved for the client. Second, it reflects more-frequent reporting, which may mean better communications. Third, it may help detect trends, directions, and contributions from the most recent periods’ performance toward a longer track record. Collectively, these themes represent a frequent source of requests made by clients to investment performance measurement professionals.

Using the same example, let’s look at the resulting excess returns on a geometric basis (see Table 2) where

Geometric excess return =	(1 + Portfolio return)	– 1.
	(1 + Index return)

Table 2. Fourth Quarter 2008 Performance Report: Geometric Return

	October			November			December
Item	Month	QTD		Month	QTD		Month	QTD
Portfolio return	–9.50%	–9.50%		–15.50%	–23.53%		12.00%	–14.35%
Index return	–15.50	–15.50		–45.00	–53.53		–0.10	–53.57
Excess return (geometric)	7.10%	7.10%		53.64%	64.45%		12.11%	84.48%

Now the client can grasp how a 7.10 percent excess return in October followed by a 53.64 percent excess return in November can result in a two-month QTD excess return of 64.45 percent. In other words, now the standard multiperiod compounding formula works:
[(1 + 7.10%) x (1 + 53.64%)] – 1 = 64.55%.

So, multiperiod excess return compounding produces accurate results when one is using geometric excess return calculations but does not work for arithmetic excess return calculations. But which excess return calculation is correct? In reality, both are. But how?

In an article published in the Journal of Performance Measurement, Bacon (2002) states that both geometric and arithmetic excess return calculations produce valid value-added results. But as Bacon further remarks: “The Arithmetic Excess Return explains the added value relative to the initial amount invested and the Geometric Excess Return explains the same added value but relative to the notional fund or the amount expected if the client had invested in the benchmark” (p. 24; italics added).

Both calculations are used throughout the investing community. The arithmetic excess return, calculated as it is by simple subtraction, has the virtue of simplicity. That said, however, Bacon provides several reasons for preferring geometric excess returns. First, they have the arithmetic property of compoundability, as shown earlier. Geometric excess returns also exhibit proportionality. That is, imagine that we invest an equal monetary amount in both a benchmark and a portfolio. If at the end of the time period the portfolio’s asset value is twice as large as the benchmark’s, the geometric excess return will show the same result in percentage terms. Finally, geometric excess returns are convertible. That is, geometric excess returns achieve the same results regardless of which currency is used in their calculation. This last attribute of convertibility explains why geometric returns are more popular in the international community, notably the United Kingdom and Europe, and less so in the United States and Australia. A return should not change simply because it is reported in a different currency; hence, geometric excess returns are preferred.

One other practical insight from Bacon’s article is that “in rising markets the Arithmetic Excess Return is always greater than the Geometric Excess Return and in falling markets the reverse is true” (p. 25). Could this at least partially explain the growing interest in geometric excess returns given the current environment?

Finally, for more information on the nature of compounding itself, one can refer to Ryan (2003).
 
Bibliography
Bacon, Carl. 2002. “Excess Returns – Arithmetic or Geometric?” Journal of Performance Measurement, vol. 6, no. 3 (Spring): 23–31.

Ryan, Timothy P. 2000/2001. “Improving Return Volatility Measurement and Presentation.” Journal of Performance Measurement, vol. 5, no. 2 (Winter): 9–21.

Ryan, Timothy P. 2003. “Return Compounding: Essential Insights and Practical Implications.” Journal of Performance Measurement, vol. 7, no. 3 (Spring): 42–46.