Peters (2011a) defined an optimal leverage which maximizes the time-average growth rate of an investment held at constant leverage. It was hypothesized that this optimal leverage is attracted to 1, such that, e.g., leveraging an investment in the market portfolio cannot yield long-term outperformance. This places a strong constraint on the stochastic properties of prices of traded assets, which we call \'leverage efficiency.\' Market conditions that deviate from leverage efficiency are unstable and may create leverage-driven bubbles. Here we expand on the hypothesis and its implications. These include a theory of noise that explains how systemic stability rules out smooth price changes at any pricing frequency; a resolution of the so-called equity premium puzzle; a protocol for central bank interest rate setting to avoid leverage-driven price instabilities; and a method for detecting fraudulent investment schemes by exploiting differences between the stochastic properties of their prices and those of legitimately-traded assets. To submit the hypothesis to a rigorous test we choose price data from different assets: the S&P500 index, Bitcoin, Berkshire Hathaway Inc., and Bernard L. Madoff Investment Securities LLC. Analysis of these data supports the hypothesis.
The large volatility of BTC permits an even more precise prediction of optimal leverage, with a standard error of 0.3, despite the shortness of this time series (10 years). There is no consensus over what asset class BTC is, so it is remarkable that it behaves so similarly to more familiar assets. Berkshire Hathaway, BRK, is a large conglomerate, well known for its sustained rapid growth over the last half century. We include it as an example of a successful and, we assume, legitimate business. As a cherry-picked investment, we anticipate that its optimal leverage will exceed 1, although by how much is an important question for our theory.
id: 62dd7e102eb8ac61c190551d1ffbd7f8 - page: 12
MAD Bernie Mados Ponzi scheme is very interesting. Here is a ctitious asset, whose returns were concocted to defraud investors. We nd its behavior measurably dierent from properly traded assets, supporting our proposal in section 4.4 that fraudulent investments may be detected by their deviations from the predictions of leverage eciency. The FED data are overnight interest rates paid between banks. It is unclear how well an investment in cash or bonds (or whatever one considers a riskless asset) is reected by these rates. For instance, due to falling interest rates, a longer-term government bond would have appreciated considerably in recent decades. To the extent that short-term inter-bank rates are typically lower than real deposit and borrowing rates, we expect the FED data to underestimate the performance of riskless assets and, therefore, to overestimate the optimal allocation to risky assets.
id: e3bdc6606db17bddafe91d4285628ff4 - page: 12
The S&P500 total return includes dividends. Its time series is shorter, but we include it for completeness and to establish bounds on the eects of dividends. The 10 year bond yields, DGS10, help us estimate a range of plausible optimal leverage values that might result from using dierent bond portfolios. Price changes of bonds are not taken into account in any of the analyses we present. 12 104 1980 l=0.0 101 101
id: c1d7bf64c445ef531918a00bd269e4da - page: 12
0 102 102 1950 1960 1990 2010 l=+1.0 1930 1940 100 1970 103 l=+1, S&P500l=0, FEDl=-1l=+3.5 2000 l=+3.5 2020 l (t; ld) for Figure 2: Equity xd investments of initially $1 in the S&P500 at dierent constant leverages, where money is borrowed at overnight federal interest rates. As leverage increases, eventually the uctuations become harmful and the investor loses money. For a given start date, each leverage produces one value for the nal equity. We suspect that, on balance, the competing biases in the data will tend to produce overestimates of real optimal leverage, ld opt. We encourage readers to repeat the analyses for dierent assets and data sets, and to vary parameters and assumptions used in the data analysis. To facilitate this, we provide open-source Python code and data at eciency codes. opt > lr
id: c23473044ca8b6f6b36c4a0866afb5b5 - page: 13