A Note on the Sources of Portfolio Returns: Underlying Stock Returns and the Excess Growth Rate

It is tempting to assume that the performance of a portfolio is equivalent to the performance of its underlying stocks. However, when returns are compounded over time and across firms, the resulting compound return for a portfolio is not necessarily equal to the average compound return of that portfolio's underlying securities. While this mathematical fact is well-known, researchers and investors often ignore its implications when interpreting the sources of returns to portfolios of stocks. We illustrate a method of parsing portfolio returns into their underlying sources drawn from stock-level returns and stock-level variance-covariance properties. We demonstrate the usefulness of this method for analyzing the cross-section of stock returns by showing how some portfolios formed from popular stock characteristics display compound return patterns that are quite different from the average compound returns of the underlying stocks in those portfolios, while other portfolios display return patterns that closely match those of their underlying stocks.

Raw Returns | ||||

Period 1 | Period 2 | Average | Variance | |

A_{1} | +100% | –50% | +25% | 0.5625 |

A_{2} | –50% | +100% | +25% | 0.5625 |

EW_{A} | +25% | +25% | ||

B_{1} | +25% | +25% | +25% | 0.0000 |

B_{2} | +25% | +25% | +25% | 0.0000 |

EW_{B} | +25% | +25% | +25% | 0.0000 |

Log Returns | ||||

Period 1 | Period 2 | Average | Variance | |

A_{1} | +0.6931 | –0.6931 | 0.0000 | 0.4805 |

A_{1} | –0.6931 | +0.6931 | 0.0000 | 0.4805 |

EW_{A} | +0.2231 | +0.2231 | +0.2231 | 0.0000 |

B_{1} | +0.2231 | +0.2231 | +0.2231 | 0.0000 |

B_{2} | +0.2231 | +0.2231 | +0.2231 | 0.0000 |

EW_{B} | +0.2231 | +0.2231 | +0.2231 | 0.0000 |

This illustrates how *compound* portfolio returns are not necessarily related to average *compound* individual constituent returns. This example shows that portfolio returns can be high even when constituent returns are zero.

The following simple example illustrates the underlying economic issue that is the focus of our paper. Suppose there are four stocks with returns over two periods as presented in Table 1. For each stock we present both its raw return and its compound (log) return. First, consider two stocks A_{1} and A_{2}. Each stock doubles in value one period and loses half its value in the other period. Both stocks will have a 25% average raw return over the two periods, but an average log return of zero. Even though the average log return is zero for each stock, an equal-weighted portfolio of the two stocks earns a 22.31% log return per period.

Now consider stocks B_{1} B_{2} and , which both have raw returns of 25% (log returns of 22.31%) each period. An equally-weighted portfolio (EW_{B}) of these two stocks will have a raw return of 25% (log return of 22.31%) per period. Thus, portfolio EW_{B} performs identically to portfolio EW_{A}, but the sources of the returns are strikingly different. In portfolio EW_{A} the compound return is driven by the variance of the underlying stocks. However, the compound return of portfolio EW_{B} is a direct result of the compound returns of the two stocks. We provide a method for parsing portfolio returns into these two distinct sources. Fernholz-Shay (1982), hereafter FS, offer what we believe to be the first rigorous mathematical analysis to identify the sources of the longterm performance of a portfolio. Specifically, FS show that a portfolio that is rebalanced to the same constant weights has a compound return that is a function of the underlying stocks' 3 compound returns and an “excess growth rate” that is due to the difference between the stocks' variances and the (diversified) portfolio's variance.

We illustrate the importance of the FS model by empirically estimating the sources of portfolios' returns for several characteristic-based portfolios that appear frequently in the literature. Figure 1 displays portfolio compound returns, average stock compound returns, and the excess growth rate for portfolios formed from deciles of stocks' market capitalizations. Our results reveal that a portfolio of stocks from the smallest market cap decile has higher average monthly compound returns than a large stock portfolio, but the average compound returns of stocks in the small-cap portfolio are lower than the average compound returns of stocks in the large-cap portfolio. The difference between these two estimates derives from the excess growth rate that is induced by the higher variance of stocks in the small-cap portfolio.

Figure 1 illustrates portfolio compound returns, and the components of portfolio compound returns, for decile portfolios formed from stocks' book-to-market ratios. Portfolio compound returns increase consistently with the book-to-market ratio. In contrast to market-cap portfolios, the difference in portfolio compound returns is driven primarily by the difference in the average underlying stock compound returns for book-to-market portfolios, as the average underlying stock returns also increase with the book-to-market ratio.

This displays average monthly portfolio compound returns, and the components of those returns, for a sample of all valid US-listed common stocks over the 1960-2012 period. Portfolios are formed based on firms' market capitalization. **Interpretation:** Figure 1 shows that average stock returns increase as firm size increases, but portfolio returns decrease.

Our procedure of decomposing portfolio returns can be used to shed light on the sources of portfolio returns for a variety of common portfolio-formation variables. For example, portfolios formed based on long-term past performance (i.e., typical contrarian portfolios) display portfolio-level effects in returns, like size-based portfolios, that are the opposite of their stock-level average returns. On the other hand, portfolios formed based on short-term past performance (such as momentum portfolios) or liquidity characteristics, display portfolio patterns, like B-M portfolios consistent with their underlying stocks.

We extend our analysis by re-examining the impact of variance on the analysis of the cross-section of stock returns. Cross-sectional regressions of monthly raw returns and log returns confirm the results from the decomposition of portfolio returns. Specifically, the apparent cross-sectional relationship between size and stock raw returns is not robust to the inclusion of stock variance in the regression model, while the book-to-market effect is unchanged when variance is included. Furthermore, no cross-sectional relationship is evident between stock log returns and either size or own-return variance, but a firm's book-to-market ratio is positively associated with its average log return.

The implications and applications of decomposing portfolio returns into effects due to stock returns and portfolio excess growth rates are wide-ranging. Whenever the returns of portfolios composed of high-volatility stocks are compared to benchmarks or control portfolios of lower-volatility stocks, we can expect to observe differences due to the excess growth rates of the portfolios. Even in the wider context of other social sciences that study phenomena subject to compound growth rates, any time the growth rate of a high-volatility treatment group is compared to a lower-volatility control group, there could be differences that may be interpreted as an excess growth effect.