Definition of Excess Return

definition of excess return when evaluating investment performance, particularly when comparing a portfolio's return to a benchmark, "excess return" is a key metric however, there are two primary ways to calculate this arithmetically and geometrically understanding the distinction and the benefits of the geometric approach is crucial for accurate performance assessment what is excess return? excess return measures the extent to which an investment has outperformed or underperformed a benchmark or a risk free rate arithmetic excess return the arithmetic excess return is the straightforward difference between the portfolio's return r and the benchmark's return b formula \text{arithmetic excess return} = r b characteristics simplicity it's easy to calculate and understand intuitive for single periods for a single period, it directly shows the added percentage points of return commonly used often seen in client reporting due to its simplicity drawback doesn't account for compounding when looking at multiple periods, simply averaging arithmetic excess returns or linking them directly can be misleading because it ignores the effects of compounding the base upon which returns are earned changes over time, and arithmetic calculations don't fully capture this this can necessitate "smoothing" adjustments for multi period analysis geometric excess return the geometric excess return provides a more accurate measure of performance over time, especially when dealing with multiple periods, because it accounts for the compounding effect it essentially calculates the growth of the portfolio relative to the growth of the benchmark formula \text{geometric excess return} = \frac{(1 + r)}{(1 + b)} 1 characteristics accounts for compounding this is its most significant advantage it reflects how an investment's value has changed relative to the benchmark, considering the compounding of returns over time time consistency geometric excess returns can be compounded across multiple periods without requiring additional adjustments or smoothing factors more accurate for multi period analysis because it incorporates compounding, it generally provides a more realistic picture of an investment's performance over longer horizons main advantages of using the geometric definition using the geometric definition of excess return offers several key advantages, particularly in sophisticated financial analysis and performance attribution accurate reflection of compounded performance investment returns are inherently compounding the geometric approach correctly captures this multiplicative nature of returns over time the arithmetic mean tends to overstate the actual growth, especially with volatile returns the geometric excess return accurately reflects the actual wealth ratio achieved by the portfolio relative to the benchmark compoundability over time geometric excess returns are directly compoundable if you have geometric excess returns for several consecutive periods, you can easily combine them to find the total geometric excess return over the entire timeframe this is not straightforward with arithmetic excess returns, which often require linking or smoothing adjustments to be meaningful over multiple periods currency independence (convertibility) a significant advantage, especially in global investment management, is that geometric active returns (excess returns) are independent of the base currency used for calculation whether returns are measured in dollars, euros, or yen, the geometric excess return will be the same this is not typically true for arithmetic excess returns proportionality and comparability the geometric excess return is a ratio, reflecting the proportional outperformance or underperformance this can make it more suitable when comparing returns, especially if the asset bases have changed significantly over the period theoretically sound for multi period attribution in performance attribution (breaking down excess return into components like allocation and selection effects), the geometric approach is often considered more theoretically sound for multi period analysis it allows for the effects at the segment level to be compounded to equal the total level effects more consistently better for volatile returns for investments with high volatility, the arithmetic average return can be significantly higher than the geometric average return (which reflects the actual change in wealth) geometric excess return, by its nature, better handles this volatility when comparing to a benchmark when might arithmetic still be used? despite the advantages of the geometric approach, arithmetic excess return is still used, primarily because simplicity and intuition for single periods for a single, short period, it's easier to grasp common practice some industry conventions or client reporting preferences may still favor arithmetic calculations for ease of understanding, though this is evolving conclusion while arithmetic excess return offers simplicity for single period snapshots, geometric excess return provides a more accurate, robust, and theoretically sound measure of investment performance over multiple periods and in varying market conditions its ability to correctly account for compounding, its compoundability over time, and its currency independence make it the preferred method for detailed performance analysis and attribution, particularly for investment professionals