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This paper defines theoretical lower bounds of uncertainty of observations of macroeconomic variables that depend on statistical moments and correlations of random values and volumes of market trades. Any econometric assessments of macroeconomic variables have greater uncertainty. We consider macroeconomic variables as random that depend on random values and volumes of trades. To predict random macroeconomic variables, one should forecast their probabilities. Upper limits on the accuracy of the forecasts of probabilities of macroeconomic variables, prices, and returns depend on the number of predicted statistical moments. We consider economic obstacles that limit by the first two the number of predicted statistical moments. The accuracy of any forecasts of probabilities of random macroeconomic variables, prices, returns, and market trades doesn't exceed the accuracy of Gaussian approximations. Any forecasts of macroeconomic variables have uncertainty higher than one determined by predictions of coefficients of variation of random values and volumes of trades., Comment: 18 pages

This paper studies the links between the descriptions of macroeconomic variables and statistical moments of market trade, price, and return. The randomness of market trade values and volumes during the averaging interval {\Delta} results in the random properties of price and return. We describe how averages and volatilities of price and return depend on the averages, volatilities, and correlations of market trade values and volumes. The averages, volatilities, and correlations of market trade, price, and return can behave randomly during the long interval {\Delta}2>>{\Delta}. To describe their statistical properties during the long interval {\Delta}2, we introduce the secondary averaging procedure of trade, price, and return. We explain why, in the coming years, predictions of market-based probabilities of price and return will be limited by Gaussian distributions. We discuss the roots of the internal weakness of the commonly used hedging tool, Value-at-Risk, that cannot be solved and remains the source of additional risks and losses. One should consider theoretical economics as a set of successive approximations, each of which describes the next array of the n-th statistical moments of market trades, price, return, and macroeconomic variables, which are repeatedly averaged during the sequence of increasing time intervals., Comment: 17 pages

We discuss the economic reasons why the predictions of price and return statistical moments in the coming decades, in the best case, will be limited by their averages and volatilities. That limits the accuracy of the forecasts of price and return probabilities by Gaussian distributions. The economic origin of these restrictions lies in the fact that the predictions of the market-based n-th statistical moments of price and return for n=1,2,.., require the description of the economic variables of the n-th order that are determined by sums of the n-th degrees of values or volumes of market trades. The lack of existing models that describe the evolution of the economic variables determined by the sums of the 2nd degrees of market trades results in the fact that even predictions of the volatilities of price and return are very uncertain. One can ignore existing economic barriers that we highlight but cannot overcome or resolve them. The accuracy of predictions of price and return probabilities substantially determines the reliability of asset pricing models and portfolio theories. The restrictions on the accuracy of predictions of price and return statistical moments reduce the reliability and veracity of modern asset pricing and portfolio theories., Comment: 23 pages

We describe how the market-based average and volatility of the "actual" return, which the investors gain within their market sales, depend on the statistical moments, volatilities, and correlations of the current and past market trade values. We describe three successive approximations. First, we derive the dependence of the market-based average and volatility of a single sale return on market trade statistical moments determined by multiple purchases in the past. Then, we describe the dependence of average and volatility of return that a single investor gains during the "trading day." Finally, we derive the market-based average and volatility of return of different investors during the "trading day" as a function of volatilities and correlations of market trade values. That highlights the distribution of the "actual" return of market trade and can serve as a benchmark for "purchasing" investors., Comment: 16 pages

Markets possess all available information on stock returns. The randomness of market trade determines the statistics of stock returns. This paper describes the dependence of the first four market-based statistical moments of stock returns on statistical moments and correlations of current and past trade values. The mean return of trades during the averaging period coincides with Markowitz's definition of portfolio value weighted return. We derive the market-based volatility of return and return-value correlations. We present approximations of the characteristic functions and probability measures of stock return by a finite number of market-based statistical moments. To forecast market-based average return or volatility of return, one should predict the statistical moments and correlations of current and past market trade values at the same time horizon., Comment: 17 pages

Accuracy of economic theories and efficiency of economic policy strictly depend on the choice of the economic variables and processes mostly liable for description of economic reality. That states the general problem of assessment of any possible economic variables and processes chargeable for economic evolution. We show that economic variables and processes described by current economic theories constitute only a negligible fraction of factors responsible for economic dynamics. We consider numerous unnoted economic variables and overlooked economic processes those determine the states and predictions of the real economics. We regard collective economic variables, collective transactions and expectations, mean risks of economic variables and transactions, collective velocities and flows of economic variables, transactions and expectations as overlooked factors of economic evolution. We introduce market-based probability of the asset price and consider unnoticed influence of market stochasticity on randomness of macroeconomic variables. We introduce economic domain composed by continuous numeric risk grades and outline that the bounds of the economic domain result in unnoticed inherent cyclical motion of collective variables, transactions and expectations those are responsible for observed business cycles. Our treatment of unnoticed and overlooked factors of theoretical economics and policy decisions preserves a wide field of studies for many decades for academic researchers, economic authorities and high-level politicians., Comment: 26 pages

We consider the randomness of market trade values and volumes as the origin of asset price stochasticity. We define the first four market-based price statistical moments that depend on statistical moments and correlations of market trade values and volumes. Market-based price statistical moments coincide with conventional frequency-based ones if all trade volumes are constant during the time averaging interval. We present approximations of market-based price probability by a finite number of price statistical moments. We consider the consequences of the use of market-based price statistical moments for asset-pricing models and Value-at-Risk. We show that the use of volume weighted average price results in zero price-volume correlations. We derive market-based correlations between price and squares of volume and between squares of price and volume. To forecast market-based price volatility at horizon T one should predict the first two statistical moments of market trade values and volumes and their correlations at the same horizon T., Comment: 18 pages

This paper highlights the hidden dependence of the basic pricing equation of a multi-period consumption-based asset pricing model on price and payoff autocorrelations. We obtain the approximations of the basic pricing equation that describe the mean price "to-day," mean payoff "next-day," price and payoff volatilities, and price and payoff autocorrelations. The deep conjunction of the consumption-based model with other versions of asset pricing, such as ICAPM, APM, etc. (Cochrane, 2001), emphasizes that our results are valid for other pricing models., Comment: 10 pages

This paper assumes that the randomness of market trade values and volumes determines the properties of stochastic market prices. We derive the direct dependence of the first two price statistical moments and price volatility on statistical moments, volatilities, and correlations of market trade values and volumes. That helps describe the dependence of market-based price autocorrelation between times t and t-{\tau} on statistical moments and correlations between trade values and volumes. That highlights the impact of the randomness of the size of market deals on price statistical moments and autocorrelation. Statistical moments and correlations of market trade values and volumes are assessed by conventional frequency-based probabilities. The distinctions between market-based price autocorrelation and autocorrelation that are assessed by the frequency-based probability analysis of price time series reveal the different approaches to the definitions of price probabilities. To forecast market-based price autocorrelation, one should predict the statistical moments and correlations of trade values and volumes., Comment: 13 pages

The economic and financial variables of economic agents determine macroeconomic variables. Current models consider agents' variables that are determined by the sums of values and volumes of agents' trades during some time interval {\Delta}. We call them first-order economic variables. We describe how the volatilities and correlations of market trade values and volumes determine price volatility. We argue that such a link requests consideration of agents' economic variables of the second order that are composed of sums of squares of agents' transactions during {\Delta}. Almost any variable of the first order should be complemented by its second-order pair. Respectively, the sums of agents' second-order variables introduce macroeconomic variables of the second order. The description of the first- and second-order macroeconomic variables establishes the subject of second-order economic theory. We highlight that the complexity of second-order economic theory essentially restricts any hopes for precise predictions of price probability and, at best, could provide estimates of price volatility. That limits the predictions of price probability to Gauss's approximations only., Comment: 13 pages

We consider the consumption-based asset pricing model, derive a new modified basic pricing equation, and present its successive approximations using the Taylor series expansions of the investor's utility during the averaging time interval. For linear and quadratic Taylor approximations, we derive new expressions for the mean price, mean payoff, volatility, skewness, and the asset's amount that define the maximum of the investor's utility. We discuss the market-based origin of price probability. We use volume weighted average price (VWAP) as a market-based average price and introduce market-based price volatility. The use of VWAP results in zero correlations between the price p and trade volume U. We derive a correlation between price p and squares of trade volume {U^2} and between squares of price {p^2} and volume {U^2}. To predict market-based price volatility, one should forecast the 2-d statistical moments of the market trade values and volumes at the same horizon T., Comment: 20 pages

We consider economic obstacles that limit the reliability and accuracy of value-at-risk (VaR). Investors who manage large market transactions should take into account the impact of the randomness of large trade volumes on predictions of price probability and VaR assessments. We introduce market-based probabilities of price and return that depend on the randomness of market trade values and volumes. Contrary to them, the conventional frequency-based price probability describes the case of constant trade volumes. We derive the dependence of market-based price volatility on the volatilities and correlation of trade values and volumes. In the coming years, that will limit the accuracy of price probability predictions to Gaussian approximations, and even the forecasts of market-based price volatility will be inaccurate and highly uncertain., Comment: 11 pages

We suggest use continuous numerical risk grades [0,1] of R for a single risk or the unit cube in Rn for n risks as the economic domain. We consider risk ratings of economic agents as their coordinates in the economic domain. Economic activity of agents, economic or other factors change agents risk ratings and that cause motion of agents in the economic domain. Aggregations of variables and transactions of individual agents in small volume of economic domain establish the continuous economic media approximation that describes collective variables, transactions and their flows in the economic domain as functions of risk coordinates. Any economic variable A(t,x) defines mean risk XA(t) as risk weighted by economic variable A(t,x). Collective flows of economic variables in bounded economic domain fluctuate from secure to risky area and back. These fluctuations of flows cause time oscillations of macroeconomic variables A(t) and their mean risks XA(t) in economic domain and are the origin of any business and credit cycles. We derive equations that describe evolution of collective variables, transactions and their flows in the economic domain. As illustration we present simple self-consistent equations of supply-demand cycles that describe fluctuations of supply, demand and their mean risks., Comment: 28 pages

We introduce the price probability measure {\eta}(p;t) that defines the mean price p(1;t), mean square price p(2;t), price volatility {\sigma}p2(t)and all price n-th statistical moments p(n;t) as ratio of sums of n-th degree values C(n;t) and volumes U(n;t) of market trades aggregated during certain time interval {\Delta}. The definition of the mean price p(1;t) coincides with definition of the volume weighted average price (VWAP) introduced at least 30 years ago. We show that price volatility {\sigma}p2(t) forecasting requires modeling evolution of the sums of second-degree values C(2;t) and volumes U(2;t). We call this model as second-order economic theory. We use numerical continuous risk ratings as ground for risk assessment of economic agents and distribute agents by risk ratings as coordinates. We introduce continuous economic media approximation of squares of values and volumes of agents trades and their flows aggregated during time interval {\Delta}. We take into account expectations that govern agents trades and introduce aggregated expectations alike to aggregated trades. We derive equations for continuous economic media approximation on the second-degree trades. In the linear approximation we derive mean square price p(2;t) and volatility {\sigma}p2(t) disturbances as functions of the first and second-degree trades disturbances. Description of each next n-th price statistical moment p(n;t) with respect to the unit price measure {\eta}(p;t) depends on sums of n-th degree values C(n;t) and volumes U(n;t) of market trades and hence requires development of the corresponding n-th order economic theory., Comment: 32 pages

We consider the randomness of market trade as the origin of price and return stochasticity. We look at time series of trade values and volumes as random variables during the averaging interval {\Delta} and describe the dependences of market-based volatilities of price and return on the volatilities and correlations of market trade values and volumes. We describe the market-based origin of the lower boundaries of the accuracy of macroeconomic variables and consider, as an example, the accuracy of macroeconomic investments. We highlight that current macroeconomic models describe relations between the 1st order variables determined by sums of trade values or volumes. To predict market-based volatilities of price, return, and volatilities of macroeconomic variables, one should develop econometric methodologies, collect data, and elaborate macroeconomic theories of the 2nd order that model the mutual dependence of the 1st and 2nd order economic variables. The absence of macroeconomic theories of the 2nd order means no economic basis for predictions of market-based volatilities of price and return, as well as volatilities of any macroeconomic variables. In turn, that limits the accuracy of forecasting probabilities of price, return, and the accuracy of macroeconomic variables in the best case by Gaussian distributions., Comment: 16 pages

This paper considers the asset price p as relations C=pV between the value C and the volume V of the executed transactions and studies the consequences of this definition for the option pricing equations. We show that the classical BSM model implicitly assumes that value C and volume V of transactions follow identical Brownian processes. Violation of this identity leads to 2-dimensional BSM-like equation with two constant volatilities. We show that agents expectations those approve execution of transactions can further increase the dimension of the BSM model. We study the case when agents expectations may depend on the option price data and show that such assumption can lead to the nonlinear BSM-like equations. We reconsider the Heston stochastic volatility model for the price determined by the value and the volume and derive 3-dimensional BSM-like model with stochastic value volatility and constant volume volatility. Variety of the BSM-like equations states the problem of reasonable balance between the accuracy and the complexity of the option pricing equations., Comment: 16 pages

This paper describes asset price and return disturbances as result of relations between transactions and multiple kinds of expectations. We show that disturbances of expectations can cause fluctuations of trade volume, price and return. We model price disturbances for transactions made under all types of expectations as weighted sum of partial price and trade volume disturbances for transactions made under separate kinds of expectations. Relations on price allow present return as weighted sum of partial return and trade volume "return" for transactions made under separate expectations. Dependence of price disturbances on trade volume disturbances as well as dependence of return on trade volume "return" cause dependence of volatility and statistical distributions of price and return on statistical properties of trade volume disturbances and trade volume "return" respectively., Comment: 23 pages

Current business cycle theory is an application of the general equilibrium theory. This paper presents the business cycle model without using general equilibrium framework. We treat agents risk assessments as their coordinates x on economic space and establish distribution of all economic agents by their risk coordinates. We suggest aggregation of agents and their variables by scales large to compare with risk scales of single agents and small to compare with economic domain on economic space. Such model is alike to transition from kinetic description of multi-particle system to hydrodynamic approximation. Aggregates of agents extensive variables with risk coordinate x determine macro variables as functions of x alike to hydrodynamic variables. Economic and financial transactions between agents define evolution of their variables. Aggregation of transactions between agents with risk coordinates x and y determine macro transactions as functions of x and y and define evolution of macro variables at points x and y. We describe evolution and interactions between macro transactions by hydrodynamic-like system of economic equations. We show that business cycles are described as consequence of the system of economic equations on macro transactions. As example we describe Credit transactions CL(tax,y) that provide Loans from Creditors at point x to Borrowers at point y and Loan-Repayment transactions LR(t,x,y) that describe repayments from Borrowers at point y to Creditors at point x. We use hydrodynamic-like economic equations and derive from them the system of ordinary differential equations that describe business cycle fluctuations of macro Credits C(t) and macro Loan-Repayments LR(t) of the entire economics. The nature of business cycle fluctuations is explained as oscillations of "mean risk" of economic variables on bounded economic domain of economic space., Comment: 31 pages

This paper presents hydrodynamic-like model of business cycles aggregate fluctuations of economic and financial variables. We model macroeconomics as ensemble of economic agents on economic space and agent's risk ratings play role of their coordinates. Sum of economic variables of agents with coordinate x define macroeconomic variables as functions of time and coordinates x. We describe evolution and interactions between macro variables on economic space by hydrodynamic-like equations. Integral of macro variables over economic space defines aggregate economic or financial variables as functions of time t only. Hydrodynamic-like equations define fluctuations of aggregate variables. Motion of agents from low risk to high risk area and back define the origin for repeated fluctuations of aggregate variables. Economic or financial variables on economic space may define statistical moments like mean risk, mean square risk and higher. Fluctuations of statistical moments describe phases of financial and economic cycles. As example we present a simple model relations between Assets and Revenue-on-Assets and derive hydrodynamic-like equations that describe evolution and interaction between these variables. Hydrodynamic-like equations permit derive systems of ordinary differential equations that describe fluctuations of aggregate Assets, Assets mean risks and Assets mean square risks. Our approach allows describe business cycle aggregate fluctuations induced by interactions between any number of economic or financial variables., Comment: 31 pages

This paper describes surface-like waves of macroeconomic Credits-Loans transactions on economic space. We use agent's risk ratings as their coordinates and describe evolution of macro variables by transactions between agents. Aggregations of agent's variables with risk coordinates x on economic space define macro variables as function of x. Aggregations of transactions between agents at point x and y determine functions of two variables (x,y) on economic space. As example we study Credits transactions provided from agents at point x to agents at point y and thus amount of Loans received by agents at point y from agents at point x at moment t during time term dt. We model evolution of macro transactions by hydrodynamic-like equations. Agents fill macro domain on economic space that is bounded by minimum risk ratings of most secure and maximum risk ratings of most risky agents. Economic and financial shocks can disturb steady borders of macro domain and cause perturbations of transactions. Such disturbances can generate waves that can propagate along risk borders alike to surface waves in fluids. As example, we describe simple model interactions between two transactions by hydrodynamic like equations in a closed form. We introduce notions of "macro accelerations" and their potentials that establish steady state distributions of transactions on economic space. For this model in linear approximation we describe surface-like waves and show that perturbations induced by surface-like waves can exponentially grow up inside macro domain and induce macro instabilities in a low risk area. Description of possible steady state distributions of transactions and surface-like waves on economic space might be important for macro modeling and policy-making., Comment: 20 pages

This paper models macro financial variables alike to financial fluids with local interactions and describes surface-like waves of Investment and Profits. We regard macro-finance as ensemble of economic agents and use their risk ratings as coordinates on economic space. Aggregations of agent's financial variables with risk coordinates x on economic space define macro financial variables as function of x. We describe evolution and interactions between macro financial variables alike to financial fluids by hydrodynamic-like equations. Minimum and maximum risk grades define most secure and most risky agents respectively. That determines borders of macro-finance domain that is filled by economic agents. Perturbations of agent's risk coordinates near risk borders of macro domain cause disturbances of macro financial variables like Investment and Profits. Such disturbances can generate waves that propagate along risk borders. These waves may exponentially amplify perturbations inside of macro domain and impact financial sustainability. We study simple model Investment and Profits and describe linear approximation of steady state distributions of Investment and Profits on macro-finance domain that fulfill dreams of Investors: "more risks-more Profits". We describe Investment and Profits waves on risk border of economic space alike to surface waves in fluids. We present simple examples that specify waves as possible origin of time fluctuations of macro financial variables. Description of possible steady state distributions of macro financial variables and financial risk waves on economic space could help for better policy-making and managing sustainable macro-finance., Comment: 21 pages

We present macroeconomic model that describes evolution of macroeconomic variables and macroeconomic waves on economic space. Risk ratings of economic agents play role of their coordinates on economic space. Aggregation of economic variables like Assets and Investment, Credits and Loans of economic agents at point x define corresponding macroeconomic variables as functions of time t and coordinates x on economic space. Evolution of macroeconomic variables is determined by economic and financial transactions between economic agents. Such transactions can occur between economic agents with any coordinates x and y and that reflect non-local "action-at-a-distance" character of internal macroeconomic interactions. For instance, Buy-Sell transactions between points x and y on economic space define dynamics of Assets at point x and Investment at point y. Aggregates of transactions between economic agents at point x and y on economic space define economic fields as functions of two coordinates. To describe dynamics of economic fields on economic space we derive hydrodynamic-like equations. For simple models of interactions between economic fields we derive hydrodynamic-like equations in a closed form and obtain wave equations for their perturbations. Economic field waves propagate on economic space and their amplitudes can grow up as exponent in time and may disturb economic stability. Diversities of macroeconomic and financial waves on economic space in simple models uncover importance of wave processes for macroeconomic modeling and forecasting., Comment: 24 pages. arXiv admin note: text overlap with arXiv:1701.06625

This paper presents macroeconomic model that is based on parallels between macroeconomic multi-agent systems and multi-particle systems. We use risk ratings of economic agents as their coordinates on economic space. Aggregates of economic or financial variables like Investment, Assets, Demand, Credits and etc. of economic agents near point x define corresponding macroeconomic variables as functions of time t and coordinates x on economic space. Parallels between multi-agent and multi-particle systems on economic space allow describe transition from economic kinetic-like to economic hydrodynamic-like approximation and derive macroeconomic hydrodynamic-like equations on economic space. Economic or financial transactions between economic agents determine evolution of macroeconomic variables This paper describes local macroeconomic approximation that takes into account transactions between economic agents with coordinates near same point x on economic space only and describes interaction between macroeconomic variables by linear differential operators. For simple model of interaction between macroeconomic variables as Demand on Investment and Interest Rate we derive hydrodynamic-like equations in a closed form. For perturbations of these macroeconomic variables we derive macroeconomic wave equations. Macroeconomic waves on economic space can propagate with exponential growth of amplitude and cause irregular time fluctuations of macroeconomic variables or induce economic crises., Comment: 22 pages