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[EQUATION] where [MATH] is the spectral density function of [MATH] . Thus, [MATH] can be estimated in the same manner as in the previous section, by first estimating the spectral density through a smoothed periodogram |
, and then using quadrature techniques to estimate the integral. Maximum-Entropy Estimation : The maximum-entropy estimation of the measure of auto-predictability is easily found from Section 3.4.3 to read |
[EQUATION] where [MATH] [MATH] and [MATH] are as per Section 3.4.3 4.3 Exogenous Predictability The measure of auto-predictability can easily be extended to vector-valued time series as |
[EQUATION] for which Proposition 4.1 still holds. We further extend this measure of predictability to capture the reduction in uncertainty about future returns resulting from knowing not just current and past returns but, more generally, current and past values of any given set of factors or signals. |
Definition 4.1 We denote measure of predictability of [MATH] using [MATH] [EQUATION] [MATH] represents the maximum amount of uncertainty reduction about a future return value one can achieve by observing factors or signals [MATH] and past values of [MATH] . Intuitively, we would expect [MATH] to be the smallest when th... |
Proposition 4.2 Let [MATH] and [MATH] be two stationary discrete-time stochastic processes such that [MATH] . Then, [EQUATION] Moreover, |
[EQUATION] if and only if [MATH] and [MATH] are independent, and [EQUATION] if and only if [MATH] and [MATH] are independent and [MATH] is jointly independent for every [MATH] |
Proof. Similar to the proof of Proposition 4.1 As in the one-dimensional case, the measure of predictability of [MATH] using [MATH] can also be interpreted as the rate of KL-divergence between the joint process [MATH] and the process whose distribution is identical to that of [MATH] , except that the first coordinate p... |
4.4 Illustration Let us consider the AR(1) time series [EQUATION] Intuitively, the lower [MATH] , the closer this time series is to its innovation white noise, and therefore the less we would expect the time series to be auto-predictable. Moreover, when the time series is equal to its innovation white noise, that is wh... |
Next, we consider assessing how predictable the most actively traded global currencies, U.S. blue chip stocks and U.S. futures are. We use as proxy for the most active U.S. blue chip stocks constituents of the Dow Jones Industrial Average. For futures, we select the 20 most active futures by average daily volume over t... |
Overall, it can be seen that auto-predictabilities of financial assets are fairly low as expected, much lower than the values obtained in our synthetic experiment (Figures ( 11a ) and ( 11b )). However, returns time series are not white noises, they do exhibit memory, some more than others. As it turns out, stocks are ... |
QUANTIFYING IMPACT ON DISTRIBUTION TAILS Our measure of the impact an asset’s returns [MATH] could have on the tails of those in a reference pool [MATH] should implicitly or explicitly address two concerns: how big are tail events which the new asset can undergo, and how do tail events of the new asset compare to those... |
Traditionally, in the one-dimensional case, whether a distribution has heavy tails is often associated with whether its even moments higher than [MATH] are greater than those of the Gaussian distribution with the same mean and variance. Examples such distributions are the so-called leptokurtic distributions, defined as... |
[EQUATION] is higher than that of a Gaussian, which we recall is [MATH] . This measure was extended to the multidimensional case in |
as [EQUATION] where [EQUATION] which we recall is equal to [MATH] for multivariate normal vectors of length [MATH] . The kurtosis ratio |
[EQUATION] can therefore be regarded as a measure of the tails of returns of assets in the reference pool. Noting that, when [MATH] is a multivariate Gaussian, [MATH] follows a [MATH] distribution with [MATH] degrees of freedom, and recalling that the [MATH] -th moment of a [MATH] distribution with [MATH] degrees of fr... |
[EQUATION] The kurtosis ratio corresponds to the special case [MATH] . In general however, the larger [MATH] , the more sensitive the tail ratio is to extreme events. |
A natural approach for quantifying the impact of a new asset on the tails of a reference pool is to compare the tail ratio of the reference pool with and without the new asset, for instance through the difference |
[EQUATION] The main limitation with this idea is that, when the number of assets [MATH] in the reference pool is very large, inverting the covariance matrix [MATH] would be numerically intractable or unstable (i.e. prone to ill-conditioning). To circumvent this limitation, we swap [MATH] for best replicating portfolio ... |
[EQUATION] To confirm that [MATH] indeed measures the impact of the new asset on the existing reference pool, we review a few scenarios. When [MATH] are jointly Gaussian, so are [MATH] and [MATH] for all [MATH] . When [MATH] is Gaussian, so is [MATH] , and [MATH] so that [MATH] only reflects the non-Gaussianity of tail... |
Our working assumption is that, when [MATH] , the extent to which the tails of [MATH] impact those of [MATH] is fully reflected in the best replicating portfolio and the tracking error, and we use [MATH] to quantify the impact on portfolio tails. When [MATH] , the new asset can be perfectly replicated using existing as... |
[EQUATION] for every [MATH] and [MATH] As for estimation, [MATH] can be estimated in a consistent, fast, and robust manner, by noting that, thanks to our stationary ergodic assumption, the expectation in Equation ( 57 ) can be replaced by sample average, and that [MATH] and [MATH] are decorrelated, so that only varianc... |
5.1 Illustration We compare the tails of stocks and currencies. As proxy for stocks we use constituents of the Dow Jones Industrial Average (DJIA), and as proxy for currencies, we use [MATH] of the most liquid electronically traded foreign currencies against the U.S. dollar. For each asset we compute daily close-to-clo... |
The excessive kurtosis of CHFUSD can be partly attributed to the [MATH] daily move in September 2011 due to the Swiss Franc starting to peg the Euro, and the [MATH] daily move in January 2015 due the Swiss Franc unpegging the Euro. Considering how unusual such moves and the corresponding sample kurtosis (250) are, we w... |
QUANTIFYING SUITABILITY FOR PASSIVE INVESTMENT Whether a new asset is suitable for passive investment or not boils down to whether one can achieve a decent level of risk-adjusted returns without changing the investment decision too often, an extreme example of which being buying and holding the asset or short-selling t... |
In the interest of providing a unified treatment of both long and short passive strategies, we introduce the bidirectional Sharpe ratio of an asset A with stationary returns series [MATH] , which we define as |
[EQUATION] where [MATH] is the expected gross return of the asset (i.e. prior to any cost such as transaction cost, exchange and brokerage fees, short-selling borrowing cost etc.), [MATH] is the total operating cost incurred per time period and per unit of wealth that can be attributed to holding the asset long when [M... |
[EQUATION] and the expected net excess return over the risk-free rate would read [EQUATION] When [MATH] , buying and holding the asset is a winning strategy gross of operating costs, and the expected net excess return over the risk-free rate reads |
[EQUATION] Overall, the bidirectional Sharpe ratio therefore represents a measure of the best expected net excess return per unit of risk that can be obtained by passively investing in an asset, long or short. |
Our working assumption is that the extent to which an asset is suitable for passive investment is the extent to which its bidirectional Sharpe ratio is typical of those of a reference pool of assets known to be sought after by large investment managers. Quantifying suitability for passive investment therefore requires ... |
A critical issue needs addressing as part of our approach to quantifying suitability for passive investment. In the interest of clarity, let us consider a statistical test of suitability for passive investment that has false positive probability [MATH] , that is, the probability that the test concludes that an asset is... |
6.1 Estimating Bidirectional Sharpe Ratios We estimate a bidirectional Sharpe ratio by replacing [MATH] and [MATH] in Equation ( 59 ) by their sample estimates. It follows from our ergodic assumption that the resulting estimator is consistent. In order to make the bidirectional Sharpe ratio independent of the sample fr... |
would imply occurs often, the annualized bidirectional Sharpe ratio estimator using daily returns will overshoot. We find that monthly returns of stocks, currencies and futures exhibit little to no auto-predictability. |
6.2 Characterizing the Bidirectional Sharpe Ratio of Passive Investments Once the bidirectional Sharpe ratio of the new asset has been estimated, we need to determine how similar it is to those of assets we know to be suitable for passive investment. To do so, we assume that all assets suitable for passive investment a... |
[EQUATION] which we complement with the conjugate Normal-Gamma prior on [MATH] , which we recall means that [EQUATION] and [EQUATION] |
Upon estimating the bidirectional Sharpe ratios [MATH] of [MATH] assets that are known to be suitable for passive investment (e.g. passive funds, stocks and bonds ETFs, blue chip stocks, etc.), the predictive distribution |
[EQUATION] forms our best guess, in light of observed data, about the characteristic distribution of bidirectional Sharpe ratios of assets that are suitable for passive investment. |
We recall that the predictive distribution is available in closed-form and reads [EQUATION] where [EQUATION] with [EQUATION] and where [MATH] is the Student-t distribution with [MATH] degrees of freedom, location parameter [MATH] , and scale parameter [MATH] . The associated probability density function therefore reads |
[EQUATION] where [MATH] is the gamma function. Prior parameters [MATH] can be set to express uninformativeness. Moreover, we recommend setting [MATH] so as to avoid expressing (a priori) that assets suitable for passive investment should be expected to return more or less than the risk-free rate, net of operating costs... |
6.3 Testing for Suitability for Passive Investment Once the distribution of bidirectional Sharpe ratios of assets known to be suitable for passive investment has been estimated as the posterior predictive distribution |
[EQUATION] of Equation ( 63 ), we are ready to test whether a new asset A is suitable for passive investment. One Trial Allowed : We begin by assuming that A is the only asset that we will put to our suitability for passive investment test. |
In general, the log-predictive posterior [EQUATION] reflects the ‘likelihood’ that a bidirectional Sharpe ratio [MATH] is consistent with observations [MATH] , and therefore the extent to which the asset whose bidirectional Sharpe ratio is [MATH] is suitable for passive investment. Denoting [MATH] our point estimate of... |
[EQUATION] As for a statistical hypothesis test of suitability for passive investment, we note that [EQUATION] reflects the probability that an asset suitable to passive investment, as per observations [MATH] , presents less passive investment opportunities than A, or equivalently the probability that A presents more p... |
[EQUATION] The [MATH] -value [MATH] is the probability that our test makes a false positive (or type I) error, and should therefore be set to a small value, for instance [MATH] |
Multiple Trials Allowed : When running multiple tests of suitability for passive investment, care should be taken while assessing the false positive or type I error rate. |
If [MATH] tests of suitability for passive investment are independently run on [MATH] assets, then the overall false positive rate, defined as the expected number of false positive errors divided by the number of tests, remains the test’s [MATH] -value. In practice however, this might not be the best metric to rely on ... |
If a test to act on is chosen uniformly at random among all [MATH] tests, then the expected false positive rate is also [MATH] . In general, nevertheless, when the test result to act on is selected among all available [MATH] tests using a different strategy, one cannot conclude. To see why, let us denote [MATH] the ind... |
[EQUATION] for some [MATH] . Clearly, the probability that [MATH] , and consequently the false positive rate, depends on [MATH] , despite the fact that [MATH] are i.i.d. Whence, our statistical test needs to be adapted to have a known and configurable false positive rate for any number of trials [MATH] |
In order to adapt our statistical test to multiple trials, let us consider a selection strategy typical of an investment manager looking for a new investment opportunity. We assume the investment manager keeps testing assets until he/she finds one that is suitable for passive investment. Let us assume [MATH] assets hav... |
[EQUATION] Thus, [EQUATION] is ill-suited to measure the likelihood that the [MATH] -th asset is suitable for passive investment as it compares the best of [MATH] independent attempts at finding an asset suitable for passive investment to a single attempt [MATH] at generating a bidirectional Sharpe ratio similar to the... |
[EQUATION] Hence, the investment manager willing to test whether the best of [MATH] assets chosen at random is suitable for passive investment should use the following statistical test: |
[EQUATION] Remark 6.1 We stress that the false positive or type I error rate of the test above is always [MATH] , irrespective of [MATH] . When the asset with the highest bidirectional Sharpe ratio is the [MATH] -th asset [MATH] , accounting for previously failed attempts (Test ( 74 )) is essentially the same as assumi... |
[EQUATION] or [EQUATION] both tests agree. However, when [EQUATION] both tests disagree, and the one not accounting for previously failed attempts is wrong—this is the manifestation of so-called backtest overfitting . To see how pervasive this issue is, we note that when [MATH] and [MATH] [MATH] ! When [MATH] , the ove... |
The measure of suitability for passive investment [MATH] previously introduced can also be extended to the multiple-trials case as the log-predictive posterior |
[EQUATION] where [MATH] and [MATH] are i.i.d. drawn from the posterior distribution [MATH] . Specifically, denoting [MATH] the cumulative density function associated to predictive density [MATH] (Equation ( 68 )), we define measure of suitability for passive investment of the best of [MATH] assets [MATH] the quantity |
[EQUATION] The previous analysis was based on a selection strategy that keeps testing assets for suitability for passive investment until one such asset is found. When the investment manager does not stop at the first asset suitable for passive investment, but instead continuously tests assets/strategies, selecting the... |
Remark 6.2 We consciously make the conservative/overly penalizing assumption that bidirectional Sharpe ratios of tested assets are independent. In practice, we understand that assets tested by the investment manager might be positively correlated, for instance when optimizing a parametric family of trading strategies. ... |
6.4 Illustration To illustrate our approach, we use as reference set of assets suitable for passive investment U.S. blue chip stocks, specifically, constituents of the Dow Jones Industrial Average at the time of writing of this paper. Figure ( 13 ) illustrates the posterior distributions [MATH] for various cost and ris... |
Using the foregoing reference set, we consider testing whether the most actively traded global currencies and U.S. futures are suitable for passive investment at a [MATH] -value of [MATH] , and for an aggregate risk-free rate and operating cost of [MATH] basis points per month, and we compute their suitability for pass... |
Interestingly, no currency or currency future is found to be suitable for passive investment. This makes intuitive sense. Indeed, had a foreign currency been found to be suitable for passive investment, this would have suggested that it would have had tendency to either appreciate relative to the U.S. dollar in the lon... |
Out of the [MATH] futures considered, only [MATH] were found to be suitable for passive investment, namely the CBOT [MATH] -year U.S. Treasury Note (TY), the NYMEX Natural Gas (NG), the CBOT Soybeans (S), the CBOT [MATH] -year U.S. Treasury Note (FV), the CME E-mini Dow Jones, and the CBOE VIX. |
CONCLUSION In this paper, we provide a quantitative framework for answering a basic, yet fundamental question: what makes an asset useful to an investment manager? The notion of asset in the aforementioned question includes all investments resulting in a periodic stream of returns, realized or marked-to-market. This al... |
Summary and Contributions: We argue that the usefulness of a new asset to an investment manager is relative to the pool of assets he/she already has access to and factors he/she would like to avoid exposure to. Indeed, if the new asset can easily be replicated using existing assets and factors, intuitively it is of lit... |
We identify four key criteria a new asset should exhibit to be considered useful to an investment manager, two primary and two secondary, each corresponding to a motivation an investment manager might have for broadening the universe of assets he/she trades, and all four are independent from the investment manager’s as... |
As primary criteria, we propose that, to be useful, a new asset should sufficiently diversify the pool of assets and factors the investment manager already has access to, and the new asset’s returns time series should be sufficiently predictable. These two criteria are primary criteria in that an investment manager, ac... |
Additionally, we propose as secondary criteria that, to be useful, a new asset should not have an excessive adverse impact on the tails of assets the investment manager currently trades, and it should be suitable for passive investment. The first secondary criteria caters to investment managers interested in broadening... |
We introduce the mutual information timescale as measure of how much incremental diversification a new asset adds to a reference pool of assets and factors. Simply put, the mutual information timescale quantifies the amount of time required to see a bit of mutual/shared information between the new asset and the referen... |
We use as measure of predictability of returns the maximum reduction in uncertainty about future returns that can be achieved by knowing past returns and possibly other set of signals. Crucially, our approach does not make any assumption on how one would go about predicting future values of returns of the new asset. It... |
We measure impact on tails by comparing the tails of returns of the portfolio of assets in the reference pool that best replicates the new asset, to the tails of returns of the replication error/innovation. In simpler terms, we measure whether the ‘beta’ component of the new asset’s returns time series with respect to ... |
As for quantifying suitability for passive investment, we propose proceeding in two steps. First, we quantify how much risk adjusted net return above the risk-free rate one can get by investing passively in the new asset, long or short. We call the corresponding metric bidirectional Sharpe ratio. We build a statistical... |
The pertinence of proposed approaches is demonstrated on a wide ranges of experiments on synthetic and real data. Empirical Findings: By applying the proposed techniques to real data we are able to recover well known stylized facts, as well as new findings. |
By comparing our measure of incremental diversification to correlation (in the two-assets case), we find evidence that daily returns of constituents of the S&P 100 exhibit a nonlinear relationship and/or a relationship that is both cross-sectional and temporal in nature (i.e. exhibits lead-lag or mutual excitation acro... |
We find that cross asset class diversification works better than within asset class diversification, as expected. However, how much more incremental diversification one can get through cross asset class diversification, as opposed to within asset class diversification, varies drastically as a function of the asset clas... |
We also find that, as well-known to practitioners, time series of asset returns are not white noises; they have memory and, consequently, are predictable, some more than others. We find that currencies are less predictable than stocks and futures overall, stocks are more predictable than futures overall, and the predic... |
In relation to impact on tails, we find that currencies have the heaviest tails in isolation. However, adding currencies to a basket of U.S. blue chip stocks, more often than not, has a positive impact on tails, while adding blue chips to currencies on average has a negative effect on tails (surprisingly). |
As for suitability for passive investment, we find that foreign currencies are (unsurprisingly) not suitable for passive investment, which suggests that one can only make money trading currencies through active management. |
Coming Up: Exchange-traded assets, typically represented through ticker symbols, are by far the most commonly used representation of financial markets through which investment managers seek to find investment opportunities. As much as it is the only representation that is consistent with the institutional segmentation ... |
Indeed, any set of time series of investment decisions [MATH] , where [MATH] denotes a portfolio weight vector, forms a valid tradeable representation of financial markets. Each time series of investment decisions [MATH] , when executed, will result in a time series of returns, and consequently can be regarded as an as... |
Clearly, there are infinitely many such tradeable representations of financial markets, and it is unlikely that the one defined by ticker symbols is the most useful to investment managers, or the most conducive to finding investment opportunities; companies don’t IPO so that hedge funds can generate alpha, government a... |
Pit.AI Technologies will use the framework developed in this paper to lauch a marketplace for incrementally constructing an alternative tradeable representation of financial markets, engineered from the ground up to be useful to investment managers, that quantitative investment managers can use as building blocks in th... |
Appendix A Algorithms Algorithm 1 Direct Model-Free Estimation of Differential Entropy Rate. [MATH] : discretization precision. [MATH] : sample path of [MATH] -valued process [MATH] |
An estimate of [MATH] A1: [MATH] is stationary and ergodic. Step 1. Discretize [MATH] using the scheme of Theorem ( 3.2 ) with precision [MATH] , to obtain the sequence of [MATH] discrete tuples (or characters) [MATH] |
Step 2. Compute the Lempel-Ziv complexity [MATH] of [MATH] using Listing 1. Step 3. Draw [MATH] sequences [MATH] where each character is sampled independently and uniformly at random with replacement from [MATH] |
Step 4. Compute the Lempel-Ziv complexity [MATH] of [MATH] using Listing 1. Step 6. Compute the frequency of occurrence of each character in [MATH] and compute the corresponding estimate of discrete entropy [MATH] |
Step 5. [EQUATION] Algorithm 2 Nonparametric Estimation of Differential Entropy Rate. [MATH] : sample path of [MATH] -valued process [MATH] |
An estimate of [MATH] A1: [MATH] is stationary and ergodic. A2: [MATH] is a Gaussian process. Step 1. Compute an estimate [MATH] for the matrix-valued spectral densities function of [MATH] as a smoothed periodogram, for instance using Welch’s method |
Step 2. Approximate Equation ( 39 ) using previously estimated spectral density functions and Bayesian Quadrature. Algorithm 3 Maximum-Entropy Estimation of Differential Entropy Rate. |
[MATH] : sample path of [MATH] -valued process [MATH] An estimate of [MATH] A3: All maximum-entropy constraints are of the autocovariance type. |
Step 1. Define [MATH] , and for [MATH] compute sample cross-covariance terms [MATH] Step 2. Denote [MATH] the corresponding sample autocovariance matrix (Equation ( 41 )). |
Step 3. [EQUATION] Algorithm 4 Estimation of Order- [MATH] Incremental Entropy. [MATH] : sample path of returns of new asset A. [MATH] : sample path of returns of the [MATH] assets and factors in the reference pool P. |
[MATH] : Sparsity parameter. An estimate of [MATH] A1: [MATH] is stationary and ergodic. Step 0. Normalize [MATH] and [MATH] so that each column has sample variance [MATH] |
Step 1. Sample [MATH] random partitions of [MATH] into subsets of size [MATH] Step 2. For each subset [MATH] in random partition [MATH] [MATH] , define [MATH] by selecting the column of [MATH] whose indexes are in subset [MATH] |
Step 3. For each [MATH] [MATH] , use either one of Algorithms , or to estimate entropy rates, first with [MATH] then with [MATH] , and finally with [MATH] , and denote [MATH] the difference between the sum of the first two estimated entropy rates and the last. |
Step 4. [EQUATION] Appendix B Experimental Setup Throughout this paper, unless stated otherwise, asset returns are daily close-to-close returns. For foreign exchange rates, we use 5PM Eastern Time as daily cutoff for the close. For futures, we use exchange settlement prices instead of the close. All futures contracts a... |
# Source: arxiv 1806.08476 # Title: A General Approach to State Complexity of Operations: Formalization and Limitations # Sections: all # Downloaded: 2026-03-03T01:49:22.051830+00:00 |
A General Approach to State Complexity of Operations: Formalization and Limitations Abstract The state complexity of the result of a regular operation is often positively correlated with the number of distinct transformations induced by letters in the minimal deterministic finite automaton of the input languages. That ... |
Introduction Given a regular operation, how do we determine its (deterministic) state complexity? There is probably no universal method for solving these problems. Nonetheless, for many operations there is a common approach we can take. While this approach is not new, it does not seem to be universally known to researc... |
, but it has seldom been used in the context of state complexity. In cases where it was used, authors typically did not acknowledge the full power and generality of the approach. This paper attempts to give a formal, general account of the approach and its uses in state complexity. |
We will refer to the approach in question as the “one letter per action” (OLPA) approach. We give an informal description of the OLPA approach below. The root of the approach is to think about regular operations in terms of how they affect deterministic finite automata (DFAs). Typically, a regular operation takes some ... |
If a new letter is added to the input DFAs, then the state complexity of the output DFA will not decrease. If a new letter is added to the input DFAs, and in each DFA this letter acts the same as an existing letter, then the state complexity of the output DFA will stay the same. |
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