Investment rules and the Ergodic Hypothesis

Abstract

folio rebalancing schemes are special cases of one general investment rule for allocating money between assets. One example of this contention is the strategy of adjusting a bond portfolio’s duration relative to an index, in proportion to the level of yields. This procedure is similar to balancing a portfolio between cash and long zero coupon bonds where the percentage in zeros is related linearly to the interest rate. Another exainple is the strategy of balancing a portfolio between stocks and bonds in proportion to a valuebased measure of the relative attractiveness of these two assets. One such value-based measure is the risk premium, which is commonly defined as the difference between the estimated dividend discount rate for stocks and the yield on long-term bonds. Many stock-bond trading rules can be understood as adjusting the percentage in stocks relative to a constant baseline percentage in proportion to the risk premium. The important benefit of looking at investment rules in this way is that the approach often enables us to derive a formula for the performance of the trading strategy relative to an index consisting of a weighted average of the underlying securities. For example, under certain assumptions about the markets, we can develop a formula for the performance of the bond rule where duration is adjusted relative to an arbitrary index in proportion to the level of yields. These types of formulas illustrate why many standard rebalancing programs are attractive they provide a type of portfolio insurance (purchase or sale depending on how the rebalancing is performed) relative to the index. This is not traditional insurance of an absolute wealth level with fixed expiration, but a type of perpetual insurance of relative performance. Furthermore, the return formulas show why sellers of insurance have more volatile returns than the index over the short run, while over the long run they have higher compound average returns because of the premium income. Similarly, strategies that buy insurance have less volatile returns than the index over the short run, but they will have lower long-run compound average returns. These results are useful in several respects. First, they help illustrate the economic substance of a general investment technique that characterizes some of Wall Street’s most familiar portfolio rebalancing methods. Even some portfolios that are not thought of as being systematically rebalanced will be shown to be the result of trading relative to some other index. For instance, we can closely approximate conventional bond indexes by adjusting duration relative to a constant duration index using one of the methods outlined here. This alone has important implications. Second, the general formulas for return illustrate why certain individuals are buyers and others are sellers of insurance. These formulas show that, over the long run, insurance sale reliably produces