Leverage efficiency

Leverage efficiency

Ole Peters and Alexander Adamou
London Mathematical Laboratory, 14 Buckingham Street, London WC2N 6DF, UK
Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, NM 87501, USA
Corresponding author. Email: o.peters@lml.org.uk
July 5, 2019

Peters (2011a) defined an optimal leverage which maximizes the time-average growth rate of an investment held at constant leverage. We test the hypothesis that this optimal leverage is attracted to 1, such that leveraging an investment in the market portfolio cannot yield long-run outperformance. Historical data support the hypothesis. This places a strong constraint on the stochastic properties of traded assets, which we call “leverage efficiency.” Market conditions that deviate from leverage efficiency are unstable and may create leverage-driven bubbles. This result resolves the equity premium puzzle, informs interest rate setting, and constitutes a theory of noise in financial markets.

“Neither a borrower nor a lender be”

Hamlet, Act 1, Scene 3, 75

1 Introduction

In section 2 we summarize a few key properties of geometric Brownian motion that were pointed out in (Peters, 2011a). We indicate the main elements of the analogy that is often drawn between this and the dynamics of markets. Section 3 introduces the concept of leverage efficiency, namely the hypothesis that the properties of price fluctuations in real markets are strongly constrained by efficiency arguments so as to make investments of leverage 1 optimal. This hypothesis is motivated by the considerations in section 2 but goes beyond the simple model discussed there. Section 4 constitutes the main body of the study, where the hypothesis is tested empirically using data from American stock markets. The arguments leading to the hypothesis are neither specific to the mathematical model nor to American stock markets. We choose the model because it is a relevant and analytically tractable limiting case, and American stock markets because they have been well observed for a sufficiently long time to test the hypothesis. The arguments are robust enough to yield insights into other assets, such as houses, or indeed national economies or the global economy. Section 5 concludes that optimal leverage is empirically close to 1, as predicted by leverage efficiency. We summarize example applications of this fundamental constraint on the stochastic properties of price dynamics: a resolution of the equity premium puzzle, a protocol for central bank rate setting to avoid leverage bubbles, and an explanation of asset price changes in the absence of new information.

2 Mathematical background

Notation: The present study uses three different levels of realism. To avoid tedious nomenclature and confusion between these, we use three different superscripts:

  1. superscript refers to the mathematical toy model used to motivate and guide our investigations;

  2. superscript refers to data analyses performed to test our main hypothesis empirically; and

  3. superscript refers to corresponding quantities in the context of real people’s behavior.

Toy model: The variable is said to undergo geometric Brownian motion if it obeys the Itô stochastic differential equation,


where denotes time, its infinitesimal increment, and the normally distributed Wiener increment, . We call the drift and the volatility.

We pause here to make an important distinction between averaging over an ensemble and averaging over time. For some special observables, this distinction is unimportant because they have the following ergodic property (Peters and Gell-Mann, 2016):

Equality of averages
The expectation value of the observable is a constant (independent of time) and the finite-time average of the observable converges to this constant with probability one as the averaging time tends to infinity.

The observable defined by equation (1) does not possess this property. Therefore, we cannot assume that the expectation value will be informative of what happens to over time. From now on we will refer to the expectation value as the “ensemble average” because it has little to do with the everyday meaning of the word “expect,” whereas it is by definition the average over an ensemble of systems.

Consider the growth rate estimator,


where indexes realizations of the process described by equation (1) and is a time increment. Taking the limit for finite extracts the behavior of the ensemble average, whereas taking the limit for finite extracts the long-time behavior (Peters and Klein, 2013). This procedure yields clear interpretations of two well-known characteristics of geometric Brownian motion:


shows that the ensemble-average growth rate is equal to the drift; and


shows that the long-time growth rate, i.e. what an individual will experience eventually, is smaller by a correction term . Equation (5), which shows as equal to the ensemble average of the rate of change , is obtained by applying Itô’s formula to equation (1). Indeed, this rate of change is an ergodic observable for the multiplicative dynamics defined by equation (1), as discussed in (Peters and Gell-Mann, 2016).

Referring to the analogy with stock markets, it was pointed out in (Peters, 2011a) that an individual investor should be more concerned about , the long-time growth rate of a single realization of the process, than about , the growth rate of the average over parallel realizations which are inaccessible to him, see also (Fernholz and Shay, 1982). For historical reasons, however, is often mistakenly considered in the literature (Hughson et al., 2006).

We introduce a leverage parameter to extend the market analogy to leveraged investments. We imagine two assets available to an investor: one riskless, whose price obeys equation (1) with drift and volatility zero; and the other risky, whose price obeys equation (1) with drift and volatility . The investor has net resources of to allocate between these assets. A leveraged investment in the risky asset is, in effect, a portfolio in which is held in the risky asset and the remainder is held in the riskless asset. Each holding achieves the same fractional change – which we shall call the “return” – as its respective asset, so that the total resources evolve as


The case , in which the investor holds only the riskless asset, results in deterministic exponential growth at a rate equal to the drift . The case , in which the investor places all his resources in the risky asset, is equivalent to equation (1), where .

We note that is constant in this setup. The fractional holdings in the two assets do not change over time, even though the values of the holdings do change. Unless or , when only one asset is present, this implies that the portfolio is continuously rebalanced, i.e. resources are moved between the risky and riskless assets to maintain the leverage at . In practice, such rebalancing would take place only at finite time intervals and would incur transaction costs. Such effects are included in our empirical work in section 4. Moreover, the leveraged investments imagined in this study should not be confused with “buy-and-hold” portfolios which start with an initial leverage and do not undergo rebalancing. In general, the allocation of such portfolios will change over time.

We will think of the risky asset as resembling an investment in the market portfolio and of the riskless asset as resembling a safe government bond or bank deposit. Leverage reflects short-selling; reflects part of the investor’s equity being invested in the market and part kept safe accruing interest at rate ; and reflects what is commonly referred to as leveraging, i.e. an investment in the market that exceeds the investor’s equity and includes borrowed funds. The volatility in equation (7) is , proportional to the leverage, and the drift is , reflecting a safe interest rate and the excess drift of the market added in proportion to the leverage. Thus leveraging causes both the excess drift and the fluctuation amplitude to increase linearly.

in equation (7) has the leverage-dependent ensemble-average growth rate


and the leverage-dependent time-average growth rate


Crucially equation (9), unlike equation (8), is not monotonic in . Maximizing establishes the existence of an objectively optimal leverage:


Equation (10) implies that, unless , it is possible to choose in equation (7) such that (reflecting a leveraged investment) consistently outperforms in equation (1) (reflecting the market portfolio). For example, would imply that, due to the nonlinear effects of multiplicative fluctuations, a rising market could be beaten by keeping a fixed fraction of one’s resources in a savings account.

In reality, the outcome of an investment held for some finite time is given by the growth of the investment averaged over . The growth rate of the ensemble-average is a priori irrelevant in practice. Maximizing in equation (8) leads to the recommendation of maximizing (or ). But equation (9) shows that this would lead to a negatively diverging time-average growth rate, i.e. to ruin. Thus, if (often called the “expected rate of return”) is falsely believed to reflect the quantity an investor should optimize, and if is interpreted as the leverage used in the investment, then the investor will be led to exceed (positively or negatively) the leverage that would truly be most beneficial. Worse, this excess is likely to ruin the investor.

The history of the struggle to make sense of the misleading recommendations derived from equation (8) is the history of decision theory and of probability theory itself, see (Peters, 2011b; Peters and Gell-Mann, 2016). Under multiplicative growth, such as in equation (1), the difference between the growth rate of the ensemble average and the long-term growth rate of an individual trajectory is the difference between arithmetic and geometric means. This was identified in the context of a repeated gamble as early as (Whitworth, 1870), while the correction term in its present form, , follows directly from (Itô, 1944). Optimal leverage for equation (7) was computed by (Kelly Jr., 1956) and in the form of equation (10) by (Merton, 1969), although neither pointed to the non-ergodicity of as the origin of their findings. The present study is concerned with the dynamic properties of the optimal leverage observed in time series of real markets.

3 Leverage efficiency

The efficient market hypothesis (Bachelier, 1900; Fama, 1965) claims that the price of an asset traded in an efficient market reflects all the information publicly available about the asset. The corollary is that it is impossible for a market participant, without access to privileged (“insider”) information, consistently to achieve growth at a rate exceeding the long-time growth rate of the market (“to beat the market”) by trading assets. We shall refer to this concept as “price efficiency.”

We propose a different, fluctuations-based, market efficiency, which we call

Leverage efficiency:
It is impossible for a market participant without privileged information to beat the market by applying leverage.

Simple strategies such as borrowing money to invest, , or keeping some money in the bank, , should not yield consistent market outperformance, i.e. there should be no leverage-arbitrage. This reasoning was used in (Peters, 2011a) to hypothesize that real markets self-organize so that


is an attractive point for their stochastic properties (represented by , , and in the model).

The hypothesis we are about to test is motivated by the model equation (7) and its properties equation (8), equation (9), and equation (10), in the sense that this model motivates the existence of an optimal leverage. But it is by no means derived from the model, as the hypothesis requires the dynamic adjustment, or self-organization, of the stochastic properties of the system, which, in the model, are represented by fixed parameters. One would have to think of , and as slowly varying (compared to the fluctuations) functions of time, related to one another as well as to through a dynamic which has as an attractor.

Although inspired by a mathematical toy model, the hypothesis in equation (11) does not rest on model-specific properties. Crucial for it are the identification of the time-average growth rate and the consequent establishment of an optimal leverage, about which economic arguments may be framed.

Leverage efficiency is a tantalizing concept. It posits that the market has a different quality of knowledge than implied by price efficiency. Price efficiency is essentially a static concept, as it states that prices coincide with some form of value. Leverage efficiency, on the other hand, constrains price dynamics and predicts properties of fluctuations.

To argue convincingly for leverage efficiency we must elucidate those aspects of the dynamics which give rise to it. We propose a dynamical feedback in which prices and their fluctuations respond to changes in optimal leverage, in a manner reminiscent of the basic feedback between prices and supply-demand imbalances familiar in economics. We augment this with criteria for global stability derived from the “no leverage-arbitrage” argument. This mechanism, detailed below, suggests that both and are particularly attractive and that the interval constitutes a stable regime, whereas values outside it are unstable.

  1. Leverage feedbacks:

    1. If , investors will eventually borrow money to invest. High demand for risky assets will lead to price increases and low demand for riskless deposits to an increase in yields on safe bonds . Both effects reduce . In addition, highly leveraged investments are liable to margin calls and tend to increase volatility. The fall in and the rise in act to decrease .

    2. If , there is no feedback. If we imagine leverage decreasing from scenario (a), asset prices fall and bond yields drop as investors withdraw from the market and move some of their resources to safe deposits. Both effects increase . Optimally leveraged investments require no borrowing, so volatility-increasing margin calls do not occur. Thus the pressures in (a) bearing down on are relaxed. This regime is marginally stable.

    3. If , investors will eventually borrow stock to short-sell. Low demand for risky assets will lead to price decreases and high demand for riskless deposits to a decrease in yields on safe bonds . Both effects make less negative. Highly negatively leveraged investments are liable to margin calls and tend to increase volatility. The increase in and the rise in imply that becomes less negative.

  2. Global stability: It is difficult to envisage globally stable economies existing with optimal leverage outside the interval because:

    1. If , everyone should invest in the market more than he owns. This is not possible because the funds to be invested must be provided by someone.

    2. If , everyone should sell more market shares than he owns. This is not possible because the assets to be sold must be provided by someone.

    Thus the range is special in not being globally unstable.

We believe the above to be the main drivers behind stochastic efficiency. There are additional effects, however, which reinforce it.

  1. Economic paralysis: In an economy with there is no incentive to invest in risky assets, which may limit productive economic activity. Policy makers will tend to steer away from such conditions, perceiving as more desirable.

  2. Covered short-selling: An investment with is punished by the costs of borrowing stock to short-sell, i.e. covered as opposed to naked short-selling.

  3. Asymmetric interest rates: The interest received by a depositor is typically less than the interest paid by a borrower. Therefore, an investment with is punished by low deposit interest rates and an investment with is punished by high borrowing costs. This reinforces as an attractive point.

  4. Transaction costs: The costs of buying and selling assets (fees, market spreads, and so on) punish any strategy that requires trading. Holding an investment of constant leverage generally requires trading to rebalance the ratio of assets to equity. The two exceptions are investments with and .

Following these considerations we arrive at a refined hypothesis: on sufficiently long time scales is a strong attractor (which we refer to as “strong” leverage efficiency). Deviations from this attractor are likely to be confined to the interval (“weak” leverage efficiency), whose end points are sticky. In the following we submit this hypothesis to an empirical test using market data.

4 Tests of leverage efficiency in historical data

We test the leverage efficiency hypothesis by computing the growth rates of constant-leverage investments in the Standard & Poor’s index of 500 leading U.S. companies (S&P500) using its historical daily closing prices over the last 62 years.

4.1 Data sets

The data used in this study are publicly available from the Federal Reserve Economic Data (FRED) website: http://research.stlouisfed.org/fred2. We use the daily closing prices, adjusted for dividends and stock splits, of the S&P500 (FRED time series: SP500) from August 1955 to March 2017. Additionally we use the daily effective federal funds rate (FRED time series: DFF) and the daily bank prime loan rate (FRED time series: DPRIME) over the same period. The estimate of optimal leverage using these data is generous, as the S&P500 represents a well-diversified portfolio of large and successful companies, and – since bankrupt companies are replaced – is positively affected by survivorship bias. All studies were repeated for Dow Jones Industrial Average and NASDAQ data with essentially identical results (not shown).

4.2 Data analysis

The performance of an investment of constant leverage over a given time period, or “window,” is computed as follows. At the start of the first day we assume unit equity, comprising holdings of in the risky asset (S&P500) and cash deposits of . At the end of the day the values of these holdings and deposits are updated according to the historical market returns and interest rates. The portfolio is then rebalanced, i.e. the holdings in the risky asset are adjusted so that their ratio to the total equity remains . On non-trading days the return of the market is zero, whereas deposits continue to earn interest payments, which leads to an unrealistic but negligible rebalancing on those days. The investment proceeds in this fashion until the final day of the window, when the final equity is recorded. If at any time the total equity falls below zero, the investment is declared bankrupt and the computation stops, i.e. we do not allow recovery from negative equity. The optimal leverage, , is the leverage for which the final equity is maximized. This is found using a golden section search algorithm (Press et al., 2002, Chap. 9).

Figure 1 shows the total return (i.e. the effective fractional change from start to finish) as a function of leverage for a hypothetical investment in the S&P500 over the largest possible window, namely the entire time series. The four curves in the figure correspond to four sets of assumptions about interest rates and transaction costs, mentioned as additional effects in section 3. We list these in order of increasing complexity and resemblance to actual conditions and practices in financial markets:

  • Data analysis 1 (red line in figure 1) is the simple case, where the effective federal funds rate is applied to all cash, whether deposited or borrowed. No costs are incurred for short-selling (), akin to naked short-selling, wherefore market returns apply to negative stock holdings exactly as they apply to positive holdings. Transaction costs are neglected. This results in a smooth curve.

  • Data analysis 2 (yellow line) is like the first case, but federal interest rates are paid on short positions, corresponding to fees for borrowed stock. This penalization of negative holdings in the market introduces a discontinuity in the derivative, or “kink,” at .

  • Data analysis 3 (green line) is like the second case, but now federal interest rates are received on cash deposits, whereas prime interest rates are paid on borrowed funds or stock. This resembles the effect of asymmetric interest rates and introduces a kink at .

  • Data analysis 4 (blue line), the complex case, is like the third case, but whenever the portfolio is rebalanced a loss in equity of 0.2% of the value of the assets traded is incurred. This resembles transaction costs.

Figure 1: Return-leverage curves for the S&P500.
Total return from a constant-leverage investment in the S&P500, starting August 1955 and ending March 2017, as a function of leverage. Data analyses 1 (red), 2 (yellow), 3 (green), and 4 (blue). For descriptions of the computations, see text, section 4.2.

As discussed in section 3, covered short-selling, asymmetric interest rates, and transaction costs tend to penalize investments with leverages other than 0 or 1. This is reflected in the empirical results by the kinks described above and visible in figure 1. For many time windows the discontinuity in the derivative of the return-leverage curve at or is accompanied by a change in sign of the derivative, making the point a global maximum and fixing there. This likely corresponds to a real effect observed in real markets. However, even without it, the red line shows that . This, being the simplest case with the fewest assumptions and approximations, provides the strongest support for our hypothesis.

4.3 The entire time series

The return-leverage curve for an investment window spanning the entire time series over the last 62 years shows an optimal leverage of for the simple case (data analysis 1) and for the complex case (data analysis 4). We discuss in section 4.6 the extent to which this confirms the hypothesis.

The time-average growth rate in equation (9), specific to the model in equation (7), is parabolic in . We show in figure 2 the time-average growth rate for the simple case, which is the logarithm of the total return, shown in figure 1, divided by the window length. Given the known deficiencies of the geometric Brownian motion model, the parabolic fit (black dashed line) is remarkably good within the range of the model’s validity. It is simultaneously remarkably bad outside this range: daily rebalanced investments in the S&P500 with leverage or would have been lost entirely (producing a negative divergence in the logarithmic return) due to extreme events. For highly leveraged investors, the non-Gaussian tails of the return distribution, which determine their ruin probability, are far more important than any other property.

The parameters of the fitted parabola are a fit of the model equation (9) and can be taken as meaningful definitions of the empirical riskless drift , excess drift , and volatility for the S&P500 over the last 62 years. A least-squares fit estimates these parameters as p.a., p.a., and per square root of one year. We performed a one-parameter fit, fixing the co-ordinates of the maximum and then fitting on the range . The meanings of these numbers warrant some remarks. The riskless drift is practically identical to the time-average growth rate of a cash deposit over the 62-year window. However, the excess drift, , does not correspond to the excess growth rate of stock over cash. Instead, due to the wealth-depleting effect of the volatility – manifested in the model as in equation (9) – a real investment in the S&P500 outgrows federal deposits at only 1.7% p.a. over the window, which is less than the excess drift. Investing almost entirely in stock, , proves growth-optimal over this period: investments with higher leverage suffer the aforementioned volatility penalty; while investments with lower leverage fail to exploit well the outperformance of stocks relative to cash.

4.4 Equity premium puzzle

The term “equity premium” has been used to describe a form of compensation investors demand for holding a risky asset. One way of quantifying the idea is this: imagine you hold a riskless asset with a given time-average growth rate, which you may swap for a risky asset with a given volatility; how much larger must the risky asset’s time-average growth rate be to justify the swap?

The literature on this question is large and often takes a psychological and individual-specific perspective. For instance, a more risk-averse individual will demand a higher equity premium. Models of human behaviour enter both into the definition of the equity premium – which lacks consensus (Fernández, 2009) – and into its analysis. Much of the literature comes to the conclusion that dominant behavioural models are inconsistent with the observed equity premium, and this is known as the “equity premium puzzle” (Mehra and Prescott, 1985).

The framework we have developed here takes a psychologically naïve perspective. We define the equity premium, , without reference to human behaviour as the difference between the time-average growth rates of the risky () and riskless () assets:


We ask what value we expect the equity premium to take in a real market. Leverage efficiency dictates how large volatility must be for the market to avoid a leverage instability (and, therefore, to survive so that we can ask the question). Substituting equation (10) into equation (11) yields as an attractor. It follows from equation (13) that the equity premium is attracted to


Our analysis reveals this to be a very accurate prediction. The empirical equity premium – the rate at which the S&P500 outgrew federal deposits – was p.a., close to our prediction of p.a.. We regard the consistency of the observed equity premium with the leverage efficiency hypothesis a resolution of the equity premium puzzle.

Figure 2: Parabolic fit of leveraged time-average growth rate.
Computed time-average growth rates closely follow a parabola as a function of leverage. The deviation from parabolic form for extreme leverages is due to crashes and sudden recoveries. The 20.47% drop of 19 October 1987 leads to bankruptcy (an infinitely negative logarithmic return) for leverage , and the 11.58% rise of 13 October 2008 leads to bankruptcy for , showing the well-known asymmetry between negative and positive extreme events.

4.5 Central bank rate setting

Our observations are also relevant to the issue of setting a central bank’s lending rate. The rate setter would view the total drift of an appropriate asset or index as given, and the risk-free drift as the central bank’s rate. If the aim is to achieve full investment in productive activity without fuelling an asset bubble, then this rate should be set so that . Since and , this is achieved by setting


Using and of section 4.3, the optimal interest rate comes out as 5.3% p.a., a historically typical value. The task of the central banker can then be seen as the task of estimating and in the relevant way. This will involve choices about data and timescales which are far from trivial. For instance, in our data analyses at any given time there is an estimate for and one for for each possible length of lookback window. Operational matters aside, stability with respect to leverage is an important consideration for any central bank. Leverage efficiency provides a simple quantitative basis for a rate setting protocol and may frame qualitative discussions about interest rates in a useful way.

4.6 Shorter time scales

Using the full time series as our window we find an optimal leverage between zero and one. How significant a corroboration of our hypothesis is this? Even assuming that is attracted to a particular value, we expect random deviations from it to increase as the investment window gets shorter, since returns over shorter windows are more heavily influenced by noise. To take an extreme example, in the simple data analysis of a daily rebalanced portfolio, the observed optimal leverage over a window of one day does not exist. Either if the market beats the federal funds rate on that day; or if federal funds beat the market. Indeed, the magnitude of the observed optimal leverage will diverge for any window over which the daily market returns are either all greater than, or all less than, the daily returns on federal funds. This is unlikely for windows of months or years but occurs commonly over windows of days or weeks. The longest run of consecutive up-moves relative to the federal funds rate in the S&P500 was 14 trading days from 26 March 1971 to 15 April 1971, and the longest draw-down relative to the federal funds rate was 12 trading days from 22 April 1966 to 9 May 1966. Even without this divergence, shorter windows are more likely to result in larger positive and negative optimal leverages because relative fluctuations are larger over shorter time scales.

We quantify this idea in the model. Solving equation (7) yields the following estimate for the time-average growth rate after a finite time :


Maximizing this generates an estimate for the optimal leverage over a window of length :


Thus, in the model, optimal leverage for finite-time windows is normally distributed with mean and standard deviation


To assess the significance of finding (or for the complex data analysis), we estimate the time scale at which the standard deviation of is 1, i.e. . For shorter time windows a single measurement does not corroborate significantly the assertion that is confined to a range of size 1. Substituting in the computed volatility per square-root of one year fixes this time scale at around 40 years. However, to avoid over-reliance on the specific form of the model, we test empirically the relation suggested by equation (18).

Likening to the observed optimal leverage over a window of size , we investigate how well equation (18) predicts the fluctuations in . We compile histograms of by moving windows of size across the record and compare the standard deviation of found in these histograms to the standard deviation of . For windows considerably shorter than the entire record (months or a few years), the standard deviations of the corresponding histograms are considered meaningful, and the relation in equation (18) can be tested.

Figure 3 shows, on double-logarithmic scales, the standard deviation of against the window length for the simple case. Good agreement is found with the model-specific prediction in equation (18). We note that, for shorter time scales, the standard deviation is slightly higher than predicted, which may have to do with data discreteness and the divergence of optimal leverage for short windows.

Figure 3: Standard deviation of observed optimal leverage.
The standard deviation of in the simple case (symbols) as a function of window length can be predicted based on the specific model equation (1) (straight line), using the parameters found in section 4.3. Only non-overlapping windows were used.

In figure 4 the diminishing fluctuations in are illustrated as follows: for every day the optimal leverage is computed for the longest available window (the window starting on 4 August 1955) for the simple and complex cases. As time passes, the optimal leverage is seen to be consistent with an approach to . About one third of the measurements lie outside the one-standard deviation band, as would be the case in the model. No measurements lie outside the two-standard deviations band, whereas this would occur about 5% of the time in the model. The period investigated could be atypical, but we attribute the lack of large deviations to the inadequacy of the model equation (1): extreme fluctuations in daily closing prices, whose likelihoods are underestimated by equation (1), prohibit very large values of by causing bankruptcy. Our prediction gets better, the fatter the tails of the empirical return distribution.

Figure 4: Observed optimal leverage for an expanding window.
Daily optimal leverages for an expanding window, starting on 4 August 1955. Also shown are the one- and two-standard deviation envelopes about , based on the estimate per square-root of one year in section 4.3.

Figure 4 illustrates the convergence of over time, but provides no information regarding the typicality of the time series. Further insight into the dynamics of can be gained by examining time series for fixed window lengths. Figure 5 (a) shows for the simple case for windows ranging from 5 years to 40 years as a function of the end date of the window. Figure 5 (b) shows the same for the complex case, which we claim is a more realistic representation of market conditions and practices. From the strong fluctuations over short time scales emerges attractive behavior consistent with both the strong and weak forms of the leverage efficiency hypothesis. The effects of the stickiness of the points and in the complex model are clearly visible and lend additional support to both forms of the hypothesis as it applies to real markets.

Figure 5: Observed optimal leverage for fixed-length windows.
(a) In data analysis 1, see text, observed optimal leverage fluctuates strongly on short time scales but appears to converge to on long time scales, which constitutes the central result of the study. (b) In data analysis 4, see text, the kinks in figure 1 ensure that and are often found exactly. The 40-year window supports the strong leverage efficiency hypothesis, , with a dip to only after the financial crisis of 2008.

4.7 A theory of noise

As well as providing estimates of the uncertainties in measurements of , the analysis of section 4.6 sheds light on the existence and nature of “noise” in financial markets (Black, 1986). According to leverage efficiency, prices of risky assets must fluctuate if an excess drift exists, , simply because the market would otherwise become unstable. Furthermore, leverage efficiency tells us the fluctuation amplitude required to avoid instability: . But, if price fluctuations are necessary for stability, then the intellectual basis for price efficiency – that changes in price are driven by the arrival of new economic information – cannot be the whole truth. At least some component of observed fluctuations must be driven by the leverage feedbacks described in section 3, which enforce leverage efficiency and which have little to do with information arrival.

Black (1986) differentiated between information-based and other types of price fluctuation, referring to the latter as “noise” and regarding it as a symptom of inaccurate information and market in-efficiency. However, the strong empirical confirmation in figure 3 of the relation in equation (18) over a very wide range of time scales – from weeks to decades – suggests that real prices have fluctuation amplitudes close to the levels required for leverage stability. Their fluctuations are, therefore, consistent with and predicted by leverage efficiency. Prices “discovered” at ever higher trading frequencies will always show more ups and downs, as seen in figure 3, but this noise is self-generated, imposed by the requirement of leverage stability. That stability is the genesis of volatility constitutes a theory of noise requiring no appeal to the arrival of unspecified information, accurate or not.

5 Discussion

Nothing in nature, not even Brown’s pollen (Mazo, 2002), truly follows Brownian motion, whether geometric or not. Nor is anything in nature knowably faithfully described by any mathematical expression (Rényi, 1967). However, just as the movements of Brown’s pollen, in the appropriate regime, have some properties in common with a Wiener noise, so the movements of share prices have some properties in common with geometric Brownian motion. Specifically, the daily excess returns for the markets investigated – like the fractional changes in geometric Brownian motion – are sometimes positive and sometimes negative. For any time-window that includes both positive and negative daily excess returns, regardless of their distribution, a well-defined optimal constant leverage exists in our computations, section 4.2. We have investigated empirically the properties of such optimal leverages.

Stability arguments, which do not depend on the specific distribution of returns and go beyond the model of geometric Brownian motion, led us to the quantitative prediction that on sufficiently long time scales real optimal leverage is attracted to (or, in the strong form of our hypothesis, to ).

We used specific properties of geometric Brownian motion to estimate the time necessary to obtain a meaningful empirical test of this prediction. Over short time scales, the fluctuations in are too large for a single measurement of to falsify our hypothesis that it lies in a range of size 1. The model predicts a required observation time scale of  years, which estimate we confirmed in the scaling of the standard deviation of in figure 3. We therefore consider our main finding (or in the complex case) for the longest possible window of 62 years a significant corroboration of both strong () and weak () hypotheses. Both end points 0 and 1 are special due to the kinks in figure 1. The economic paralysis argument suggests that is a stronger attractor than , and our observations support this argument, especially the expanding window in figure 4 and the 40-year window in figure 5.

Leverage efficiency also suggests a fundamental explanation for the existence of volatility in markets and, specifically, for its observed levels. Price fluctuations are necessary to avoid leverage instability and their observed amplitude is consistent with predictions that assume strong leverage efficiency, , as in figure 3. The corollary is that mainstream theories in which price fluctuations are caused by the public disclosure of information or by market dysfunction are, at best, incomplete. Trading at arbitrarily high frequencies will reveal structure, but this structure does not necessarily have economic meaning beyond imposing market stability.

The existence of optimal leverage is important conceptually, and its observed value and associated stability arguments are of practical significance. While these arguments do not preclude special conditions under which it is optimal to invest more than one’s equity or to short-sell an asset, they give a fundamental scale to leverage in general. In other words, if it appears that optimal leverage is outside the band , then a special reason – such as insider knowledge or a tax incentive – for this violation of leverage efficiency must exist. Artificially maintaining such conditions will lead to instabilities. Consider housing: many societies consider it desirable for an individual to be able to purchase a home whose price exceeds his equity without having to take reckless risks. Without carefully designed restrictions on speculative home purchases, policies which aim to achieve the corresponding market conditions, i.e. , will defeat their purpose and create investment bubbles followed by crashes.

Leverage efficiency is “accountable” in the sense of Popper (1982, Chap. I.2), who demanded that a “theory will have to account for the imprecision of the prediction”.111Popper does not refer to stochastic theories in this discussion. To apply his arguments to our case, we replace “precision in the initial conditions” in his Chap. I.3 by “window length”. Both concepts quantify the information available about the system. Leverage efficiency predicts its own imprecision, equation (18), and the degree of its validity can be meaningfully and objectively tested. This is particularly important given the complexity of the systems involved.

We emphasize that our work is in no way meant to advocate or evaluate constant-leverage or any other investment strategies. Leverage efficiency is a fundamental organizing principle for the stochastic properties of markets. The data analysis in this study is an empirical test of this fundamental principle.

It has been argued that central banks, focusing their attention on interest rates, pay insufficient attention to leverage (Geanakoplos, 2010). Arguing in the context of the model, a strong link between the two is equation (10): reducing the risk-free interest rate (something we liken to the rate at which governments lend) increases optimal leverage because, assuming that overall drift does not change, it implicitly increases by creating an incentive to invest rather than save. This tends to lead to an eventual increase in real leverage. We agree with the criticism in (Geanakoplos, 2010): effecting an increase (or decrease) in real leverage through a decrease (or increase) in is rather indirect. This appears problematic given how sensitive is to – an increase of by (estimated at 2.9% p.a. over the last 62 years) sets to zero, making it optimal to invest nothing. An increase by about , i.e. 1.5% p.a., is enough to remove any incentive to invest. Conversely, a decrease of by doubles , generating instability.

Our results are relevant to the equity premium puzzle (Mehra and Prescott, 1985). The equity premium, defined as the long-term outgrowth of the S&P500 compared to federal deposits, is 1.7% p.a.. This value is in line with our prediction: given the fluctuations in stock returns and short-term government bond returns the equity premium is such that optimal leverage converges to 1 in the long run. From this perspective, only an equity premium violating leverage efficiency would constitute a “puzzle” requiring further explanation.

We do not consider our arguments specific to financial markets. They are relevant also to other regularly traded assets and commodities, related even to such basic needs as food and shelter, such as the price of wheat or apartments in Manhattan. They are relevant to macroeconomic decisions. Indeed, the same type of dynamics – multiplicative growth with fluctuations – is at work in many other systems. Equation (1) has been used to describe the growth of populations in ecology (Lewontin and Cohen, 1969), the early spread of a disease in epidemiology (Daley and Gani, 1999), and as the basis for the evolution of cooperation (Peters and Adamou, 2015).

We have argued that is a natural attractor for an economic or market system, with the end points of the interval being sticky. We note that a sticky may correspond to a depression: in this case there is no incentive to invest and to take risks. The aim of economic policy may be viewed as creating conditions where for the entire economy is within and close to the upper bound. Recent observations in figure 5(a) suggest that the policy response to the 2008 financial crisis has now somewhat overshot this aim.


The authors thank Zonlab ltd. for support and G. D. Nystrom for bringing the equity premium puzzle to their attention.


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