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Abstract:
This paper presents the "multifractal model of asset returns" ("MMAR"), based upon the pioneering research into multifractal measures by Mandelbrot (1972, 1974). The multifractal model incorporates two elements of Mandelbrot's past research that are now well known in finance. First, the MMAR contains long-tails, as in Mandelbrot (1963), which focused on Levy-stable distributions. In contrast to Mandelbrot (1963), this model does not necessarily imply infinite variance. Second, the model contains long-dependence, the characteristic feature of fractional Brownian Motion (FBM), introduced by Mandelbrot and van Ness (1968). In contrast to FBM, the multifractal model displays long dependence in the absolute value of price increments, while price increments themselves can be uncorrelated. As such, the MMAR is an alternative to ARCH-type representations that have been the focus of empirical research on the distribution of prices for the past fifteen years. The distinguishing feature of the multifractal model is multiscaling of the return distribution's moments under time-rescalings. We define multiscaling, show how to generate processes with this property, and discuss how these processes differ from the standard processes of continuous-time finance. The multifractal model implies certain empirical regularities, which are investigated in a companion paper.

Abstract:
The Multifractal Model of Asset Returns (See Mandelbrot, Fisher, and Calvet, 1997 ) proposes a class of multifractal processes for the modelling of financial returns. In that paper, multifractal processes are defined by a scaling law for moments of the processes' increments over finite time intervals. In the present paper, we discuss the local behavior of multifractal processes. We employ local Holder exponents, a fundamental concept in real analysis that describes the local scaling properties of a realized path at any point in time. In contrast with the standard models of continuous time finance, multifractal processes contain a multiplicity of local Holder exponents within any finite time interval. We characterize the distribution of Holder exponents by the multifractal spectrum of the process. For a broad class of multifractal processes, this distribution can be obtained by an application of Cramer's Large Deviation Theory. In an alternative interpretation, the multifractal spectrum describes the fractal dimension of the set of points having a given local Holder exponent. Finally, we show how to obtain processes with varied spectra. This allows the applied researcher to relate an empirical estimate of the multifractal spectrum back to a particular construction of the stochastic process.

Abstract:
This paper presents the first empirical investigation of the Multifractal Model of Asset Returns ("MMAR"). The MMAR, developed in Mandelbrot, Fisher, and Calvet (1997) (See Mandelbrot, Fisher, and Calvet, 1997 at the following URL: http://papers.ssrn.com/sol3/papers.cfm?abstract_id=78588 ), is an alternative to ARCH-type representations for modelling temporal heterogeneity in financial returns. Typically, researchers introduce temporal heterogeneity through time-varying conditional second moments in a discrete time framework or time-varying volatility in a continuous time framework. Multifractality introduces a new source of heterogeneity through time-varying local regularity in the price path. The concept of local Holder exponent describes local regularity. Multifractal processes bridge the gap between locally Gaussian (Ito) diffusions and jump-diffusions by allowing a multiplicity of Holder exponents. This paper investigates multifractality in Deutschemark / US Dollar currency exchange rates. After finding evidence of multifractal scaling, we show how to estimate the multifractal spectrum via the Legendre transform. The scaling laws found in the data are replicated in simulations. Further simulation experiments test whether alternative representations, such as FIGARCH, are likely to replicate the multifractal signature of the Deutschemark / US Dollar data. On the basis of this evidence, the MMAR hypothesis appears more likely. Overall, the MMAR is quite successful in uncovering a previously unseen empirical regularity. Additionally, the model generates realistic sample paths and opens the door to new theoretical and applied approaches to asset pricing and risk valuation. We conclude by advocating further empirical study of multifractality in financial data, along with more intensive study of estimation techniques and inference procedures.

Abstract:
This paper shows that the precautionary motive, combined with asset incompleteness, is a major source of volatility and indeterminacy in financial markets. Price fluctuations originate from agents' efforts to insure themselves through time by borrowing and lending instead of shifting income across states of nature by trading in risky assets. A high interest rate at a future date reduces the potential for future consumption smoothing across time via borrowing; this leads to a strong precautionary motive and a low interest rate in the current period. The negative feedback between future and current interest rates generates fluctuations. This logic is developed in SPEC, a CARA-normal model with many periods, risky time-dependent endowments, and an endogenous interest rate. Unlike existing frameworks, SPEC allows us to analyze the effect of financial structure on temporal fluctuations along a given path. In equilibrium, individual consumption is random, but the macro variables are deterministic and vary through time. When there is an intermediate level of market incompleteness and sufficient investor impatience, fluctuations in the real interest rate can be large, even though the aggregate endowment is constant. SPEC has a unique equilibrium under a finite horizon; on the other hand, with a finite number of infinitely-lived agents, there exists a robust continuum of equilibria that are neither bubbles nor sunspots.

Abstract:
We introduce a Ramsey growth model with incomplete markets, decentralized production, and idiosyncratic technological risk. The combination of uninsurable shocks with the precautionary motive can slow down capital accumulation or give rise to persistent fluctuations even when agents are very patient and technology is strictly convex. The model generates closed-form for the equilibrium dynamics under a finite or infinite horizon. Multiple steady states and poverty traps can arise from the endogeneity of the interest rate instead of the usual wealth effect. Depending on the economy's parameters, the local dynamics around a steady state are locally unique, totally unstable or locally undetermined, and the equilibrium path can be attracted to a limit cycle. In calibrated examples, financial incompleteness substantially slows down convergence to the steady state and thus increases the persistence of aggregate shocks.

Idiosyncratic Risk, Precautionary Motive, Endogenous Fluctuations.

Abstract:
This paper theoretically investigates the pricing effects of financial innovation in an economy with endogenous participation and heterogeneous income risks. The introduction of non-redundant assets can endogenously modify the participation set, reduce the covariance between dividends and participants consumption and thus lead to lower risk premia. This mechanism is demonstrated in a tractable exchange economy with a finite number of macroeconomic factors. Agents can freely borrow and lend, but must pay a fixed entry cost to invest in risky assets. Security prices and the participation structure are jointly determined in equilibrium. The model is consistent with several features of financial markets over the past few decades: substantial financial innovation; a sharp increase in investor participation; improved risk management practices; a slight increase in interest rates; and a reduction in risk premia.

Endogenous participation, Epstein-Zin utility, financial innovation, incomplete markets, multiple risk factors, spanning

Abstract:
This paper theoretically investigates the pricing effects of financial innovation in an economy with endogenous participation and heterogeneous income risks. The introduction of non-redundant assets can endogenously modify the participation set, reduce the covariance between dividends and participants consumption and thus lead to lower risk premia. This mechanism is demonstrated in a tractable exchange economy with a finite number of macroeconomic factors. Agents can freely borrow and lend, but must pay a fixed entry cost to invest in risky assets. Security prices and the participation structure are jointly determined in equilibrium. The model is consistent with several features of financial markets over the past few decades: substantial financial innovation; a sharp increase in investor participation; improved risk management practices; a slight increase in interest rates; and a reduction in risk premia.

Endogenous participation, Epstein-Zin utility, financial innovation, incomplete markets, multiple risk factors, spanning

Abstract:
This paper investigates the pricing effects of financial innovation in an economy with endogenous participation and heterogeneous income risks. The introduction of non-redundant assets endogenously modifies the participation set, reduces the covariance between dividends and participants' consumption and thus leads to lower risk premia. In multisector economies, financial innovation spreads across markets through the diversified portfolio of new entrants, and has rich effects on the cross-section of expected returns. The price changes can also lead some investors to leave the markets and give rise to non-degenerate forms of participation turnover. The model is consistent with several features of financial markets over the past few decades: substantial innovation; higher participation; significant turnover in investor composition; improved risk management practices; a slight increase in interest rates; and a reduction in risk premia.

Abstract:
This paper proposes that the introduction of non-redundant assets can endogenously modify trader participation in financial markets, which can lead to a lower market premium and a higher interest rate. We demonstrate this mechanism in a tractable exchange economy with endogenous participation. Investors receive heterogeneous random incomes determined by a finite number of macroeconomic factors. They can freely borrow and lend, but must pay a fixed entry cost to invest in risky assets. Security prices and the participation structure are jointly determined in equilibrium. The model reconciles a number of features that have characterized financial markets in the past three decades: substantial financial innovation; a sharp increase in investor participation; improved risk management practices; an increase in interest rates; and a reduction in the risk premium.

Endogenous Participation, Epstein-Zin Utility, Financial Innovation, Incomplete Markets, Multiple Risk Factors, Risk Premium, Spanning

Abstract:
Inspired by the recent literature on aggregation theory, this paper introduces HITS, a semiparametric model of consumer demand that allows for diversity in tastes. The strong variation of budget shares observed across income strata can arise from two economic factors: the individual income effect, and taste differences between poor and rich households. Consumer expenditure surveys that report repeated cross-sections do not permit the direct measurement of these two effects, and the paper solves this difficulty by developing a new microeconometric framework. We model consumer demand by a class of Nearly Ideal Demand Systems parameterized by a unique taste parameter. Linear heterogeneity allows GMM estimation of the structural coefficients on an aggregate time series, and the joint density of spending and tastes is recovered from cross-sections by a nonparametric procedure involving a deconvolution. We develop an asymptotic theory and demonstrate the accuracy of the algorithm by Monte Carlo and bootstrap simulations. The model is estimated on four size groups using the British Family Expenditure Survey (1968-98). We report a strong correlation between income and tastes, which explains most of the observed variation of budget shares with income. Unlike some earlier models, this new approach is consistent with both the yearly cross-sections of individual choices and the dynamics of aggregate shares.

Abstract:
We propose a discrete-time stochastic volatility model in which regime-switching serves three purposes. First, changes in regimes capture low frequency variations, which is their traditional role. Second, they specify intermediate frequency dynamics that are usually assigned to smooth autoregressive processes. Finally, high frequency switches generate substantial outliers. Thus, a single mechanism captures three important features of the data that are typically addressed as distinct phenomena in the literature. Maximum likelihood estimation is developed and shown to perform well in finite sample. We estimate on exchange rate data a version of the process with four parameters and more than a thousand states. The estimated model compares favor-ably to earlier specifications both in- and out-of-sample. Multifractal forecasts slightly improve on GARCH(1,1) at daily and weekly inter-vals, and provide considerable gains in accuracy at horizons of 10 to 50 days.

Abstract:
We propose a discrete-time stochastic volatility model in which regime-switching serves three purposes. First, changes in regimes capture low frequency variations, which is their traditional role. Second, they specify intermediate frequency dynamics that are usually assigned to smooth autoregressive processes. Finally, high frequency switches generate substantial outliers. Thus, a single mechanism captures three important features of the data that are typically addressed as distinct phenomena in the literature. Maximum likelihood estimation is developed and shown to perform well in finite sample. We estimate on exchange rate data a version of the process with four parameters and more than a thousand states. The estimated model compares favorably to earlier specifications both in- and out-of-sample. Multifractal forecasts slightly improve on GARCH(1,1) at daily and weekly intervals, and provide considerable gains in accuracy at horizons of 10 to 50 days.

Forecasting, long memory, Markov regime-switching, maximum likelihood estimation, scaling, stochastic volatility, time deformation, volatility component, Vuong test

Abstract:
We propose a discrete-time stochastic volatility model in which regime-switching serves three purposes. First, changes in regimes capture low frequency variations, which is their traditional role. Second, they specify intermediate frequency dynamics that are usually assigned to smooth autoregressive processes. Finally, high frequency switches generate substantial outliers. Thus, a single mechanism captures three important features of the data that are typically addressed as distinct phenomena in the literature. Maximum likelihood estimation is developed and shown to perform well in finite sample. We estimate on exchange rate data a version of the process with four parameters and more than a thousand states. The estimated model compares favorably to earlier specifications both in- and out-of-sample. Multifractal forecasts slightly improve on GARCH(1,1) at daily and weekly intervals, and provide considerable gains in accuracy at horizons of 10 to 50 days.

Forecasting, long memory, Markov regime-switching, maximum likelihood estimation, scaling, stochastic volatility, time deformation, volatility component, Vuong test

Abstract:
We propose a discrete-time stochastic volatility model in which regime switching serves three purposes. First, changes in regimes capture low frequency variations, which is their traditional role. Second, they specify intermediate frequency dynamics that are usually assigned to smooth autoregressive processes. Finally, high frequency switches generate substantial outliers. Thus, a single mechanism captures three important features of the data that are typically addressed as distinct phenomena in the literature. Maximum likelihood estimation is developed and shown to perform well in finite sample. We estimate on exchange rate data a version of the process with four parameters and more than a thousand states. The estimated model compares favorably to earlier specifications both in- and out-of-sample. Multifractal forecasts slightly improve on GARCH(1,1) at daily and weekly intervals, and provide considerable gains in accuracy at horizons of 10 to 50 days.

Abstract:
We propose a discrete-time stochastic volatility model in which regime-switching serves three purposes. First, changes in regimes capture low frequency variations, which is their traditional role. Second, they specify intermediate frequency dynamics that are usually assigned to smooth autoregressive processes. Finally, high frequency switches generate substantial outliers. Thus, a single mechanism captures three important features of the data that are typically addressed as distinct phenomena in the literature. Maximum likelihood estimation is developed and shown to perform well in finite sample. We estimate on exchange rate data a version of the process with four parameters and more than a thousand states. The estimated model compares favorably to earlier specifications both in- and out-of-sample. Multifractal forecasts slightly improve on GARCH(1,1) at daily and weekly intervals, and provide considerable gains in accuracy at horizons of 10 to 50 days.

Forecasting, Long Memory, Markov Regime-switching, Maximum Likelihood Estimation, Scaling, Stochastic Volatility, Time Deformation, Volatility Component, Vuong Test

Abstract:
This paper investigates the dynamics of individual portfolios in a unique dataset containing the disaggregated wealth and income of all households in Sweden. Between 1999 and 2002, stockmarket participation slightly increased but the average share of risky assets in the financial portfolio of participants fell moderately, implying little aggregate rebalancing in response to the decline in risky asset prices during this period. We show that these aggregate results conceal strong household-level evidence of active rebalancing, which on average offsets about one half of idiosyncratic passive variations in the risky asset share. Sophisticated households with greater education, wealth, and income, and holding better diversified portfolios, tend to rebalance more aggressively. We also study the decisions to enter and exit risky financial markets. More sophisticated households are more likely to enter, and less likely to exit. Portfolio characteristics and performance also influence exit decisions. Households with poorly diversified portfolios and poor returns on their mutual funds are more likely to exit; however, consistent with the literature on the disposition effect, households with poor returns on their directly held stocks are less likely to exit.

Asset allocation, disposition effect, diversification, participation, portfolio rebalancing

Abstract:
We introduce a neoclassical growth economy with idiosyncratic production risk and incomplete markets. Each agent is an entrepreneur operating her own neoclassical technology with her own capital stock. The general equilibrium is characterized in closed form. Idiosyncratic production shocks introduce a risk premium on private equity and reduce the demand for investment. The steady state is characterized by a lower capital stock due to entrepreneurial risk and a lower interest rate due to precautionary savings as compared to complete markets. The private equity premium is endogenously countercyclical: The anticipation of low savings and high interest rates in the future feed back to high risk premia and low investment in the present. Countercyclicality in risk taking slows down convergence to the steady state and amplifies the magnitude and persistence of the business cycle. These results, which contrast sharply with those obtained in Bewley models, highlight the macroeconomic significance of missing markets in production and investment risk.

Abstract:
We introduce a neoclassical growth economy with idiosyncratic production risk and incomplete markets. Each agent is an entrepreneur operating her own neoclassical technology with her own capital stock. The general equilibrium is characterized in closed form. Idiosyncratic production shocks introduce a risk premium on private equity and reduce the demand for investment. The steady state is characterized by a lower capital stock due to entrepreneurial risk and a lower interest rate due to precautionary savings as compared to complete markets. The private equity premium is endogenously countercyclical: the anticipation of low savings and high interest rates in the future feed back to high risk premia and low investment in the present. Countercyclicality in risk taking slows down convergence to the steady state and amplifies the magnitude and persistence of the business cycle. These results, which contrast sharply with those obtained in Bewley models, highlight the macroeconomic significance of missing markets in production and investment risk.

Incomplete Markets, Entrepreneurial Risk, Investment, Growth, Fluctuations, Precautionary Savings

Abstract:
We implement a multifrequency volatility decomposition of three exchange rates and show that components with similar durations are strongly correlated across series. This motivates a bivariate extension of the Markov-Switching Multifractal (MSM) introduced in Calvet and Fisher (2001, 2004). Bivariate MSM is a stochastic volatility model with a closed-form likelihood. Estimation can proceed by ML for state spaces of moderate size, and by simulated likelihood via a particle filter in high-dimensional cases. We estimate the model and confirm its main assumptions in likelihood ratio tests. Bivariate MSM compares favorably to a standard multivariate GARCH both in- and out-of-sample. We extend the model to multivariate settings with a potentially large number of assets by proposing a parsimonious multifrequency factor structure.

Multivariate MSM, comovement, maximum likelihood, particle filter, Markov-switching, stochastic volatility, multifrequency volatility decomposition, value at risk, quantile forecasts

Abstract:
This paper investigates the efficiency of household investment decisions in a unique dataset containing the disaggregated wealth and income of the entire population of Sweden. The analysis focuses on two main sources of inefficiency in the financial portfolio: underdiversification of risky assets ("down") and nonparticipation in risky asset markets ("out"). We find that while a few households are very poorly diversified, the cost of diversification mistakes is quite modest for most of the population. For instance, a majority of participating Swedish households are sufficiently diversified internationally to outperform the Sharpe ratio of their domestic stock market. We document that households with greater financial sophistication tend to invest more efficiently but also more aggressively, so the welfare cost of portfolio inefficiency tends to be greater for these households. The welfare cost of nonparticipation is smaller by almost one half when we take account of the fact that nonparticipants would be unlikely to invest efficiently if they participated in risky asset markets.

Institutional subscribers to the NBER working paper series, and residents of developing countries may download this paper without additional charge at www.nber.org.

Abstract:
This paper investigates the efficiency of household investment decisions in a unique dataset containing the disaggregated wealth and income of the entire population of Sweden. The analysis focuses on two main sources of inefficiency in the financial portfolio: underdiversification of risky assets ("down") and nonparticipation in risky asset markets ("out"). We find that while a few households are very poorly diversified, the cost of diversification mistakes is quite modest for most of the population. For instance, a majority of participating Swedish households are sufficiently diversified internationally to outperform the Sharpe ratio of their domestic stock market. We document that households with greater financial sophistication tend to invest more efficiently but also more aggressively, so the welfare cost of portfolio inefficiency tends to be greater for these households. The welfare cost of nonparticipation is smaller by almost one half when we take account of the fact that nonparticipants would be unlikely to invest efficiently if they participated in risky asset markets.

Asset allocation, diversification, familiarity, participation

Abstract:
We investigate a neoclassical economy with heterogeneous agents, convex technologies and idiosyncratic production risk. Combined with precautionary savings, investment risk generates rich effects that do not arise in the presence of pure endowment risk. Under a finite horizon, multiple growth paths and endogenous fluctuations can exist even when agents are very patient. In infinite-horizon economies, multiple steady states may arise from the endogeneity of risktaking and interest rates instead of the usual wealth effects. Depending on the economy`s parameters, the local dynamics around a steady state are locally unique, totally unstable or locally undetermined, and the equilibrium path can be attracted to a limit cycle. The model generates closed-form expressions for the equilibrium dynamics and easily extends to a variety of environments, including heterogeneous capital types and multiple sectors.

Abstract:
We investigate a neoclassical economy with heterogeneous agents, convex technologies and idiosyncratic production risk. Combined with precautionary savings, investment risk generates rich effects that do not arise in the presence of pure endowment risk. Under a finite horizon, multiple growth paths and endogenous fluctuations can exist even when agents are very patient. In infinite-horizon economies, multiple steady states may arise from the endogeneity of risk-taking and interest rates instead of the usual wealth effects. Depending on the economy's parameters, the local dynamics around a steady state are locally unique, totally unstable or locally undetermined, and the equilibrium path can be attracted to a limit cycle. The model generates closed-form expressions for the equilibrium dynamics and easily extends to a variety of environments, including heterogeneous capital types and multiple sectors.

Entrepreneurial Risk, Precautionary Motive, Endogenous Fluctuations, Poverty Traps

Abstract:
Recent research documents that aggregate stock prices are driven by shocks with persistence levels ranging from daily intervals to several decades. Building on these insights, we introduce a parsimonious equilibrium model in which regime-shifts of heterogeneous durations affect the volatility of dividend news. We estimate tightly parameterized specifications with up to 256 discrete states on daily U.S. equity returns. The multifrequency equilibrium has significantly higher likelihood than the classic Campbell and Hentschel (1992) specification, while generating volatility feedback effects 6 to 12 times larger. We show in an extension that Bayesian learning about stochastic volatility is faster for bad states than good states, providing a novel source of endogenous skewness that complements the "uncertainty" channel considered in previous literature (e.g., Veronesi, 1999). Furthermore, signal precision induces a tradeoff between skewness and kurtosis, and economies with intermediate investor information best match the data.

Institutional subscribers to the NBER working paper series, and residents of developing countries may download this paper without additional charge at www.nber.org.

Abstract:
Recent research documents that aggregate stock prices are driven by shocks with persistence levels ranging from daily intervals to several decades. Building on these insights, we introduce a parsimonious equilibrium model in which regime-shifts of heterogeneous durations affect the volatility of dividend news. We estimate tightly parameterized specifications with up to 256 discrete states on daily U.S. equity returns. The multifrequency equilibrium has significantly higher likelihood than the classic Campbell and Hentschel (1992) specification, while generating volatility feedback effects 6 to 12 times larger. We show in an extension that Bayesian learning about stochastic volatility is faster for bad states than good states, providing a novel source of endogenous skewness that complements the uncertainty channel considered in previous literature (e.g., Veronesi, 1999). Furthermore, signal precision induces a tradeoff between skewness and kurtosis, and economies with intermediate investor information best match the data.

Multifrequency news, volatility feedback, learning, information quality, endogenous skewness and kurtosis, Epstein-Zin utility

Abstract:
This paper proposes that equilibrium valuation is a powerful method to generate endogenous jumps in asset prices, which provides a structural alternative to traditional reduced-form specifications with exogenous discontinuities. We specify an economy with continuous consumption and dividend paths, in which endogenous price jumps originate from the market impact of regime-switches in the drifts and volatilities of fundamentals. We parsimoniously incorporate shocks of heterogeneous durations in consumption and dividends while keeping constant the number of parameters. Equilibrium valuation creates an endogenous relation between a shock`s persistence and the magnitude of the induced price jump. As the number of frequencies driving fundamentals goes to infinity, the price process converges to a novel stochastic process, which we call a multifractal jump-diffusion.

Abstract:
This paper proposes that equilibrium valuation is a powerful method to generate endogenous jumps in asset prices, which provides a structural alternative to traditional reduced-form specifications with exogenous discontinuities. We specify an economy with continuous consumption and dividend paths, in which endogenous price jumps originate from the market impact of regime-switches in the drifts and volatilities of fundamentals. We parsimoniously incorporate shocks of heterogeneous durations in consumption and dividends while keeping constant the number of parameters. Equilibrium valuation creates an endogenous relation between a shock's persistence and the magnitude of the induced price jump. As the number of frequencies driving fundamentals goes to infinity, the price process converges to a novel stochastic process, which we call a multifractal jump-diffusion.

Endogenous jumps, general equilibrium, Markov regime-switching, multifrequency, fat tails, stochastic volatility, time deformation, volatility component

Abstract:
This paper develops analytical methods to forecast the distribution of future returns for a new continuous-time process, the Poisson multifractal. Out model captures the thick tails and volatility persistence exhibited by many financial time series. We assume that the forecaster knows the true generating process with certainty, but only observes past returns. The challenge in this environment is long memory and the corresponding infinite dimension of the state space. We show that a discretized version of the model has a finite state space, which allows an analytical solution to the conditioning problem. Further, the discrete model converges to the continuous-time model as time scale goes to zero, so that forecasts are consistent. The methodology is implemented on simulated data calibrated to the Deutschemark/US Dollar exchange rate. Applying these results to option pricing, we find that the model captures both volatility smiles and long-memory in the term structure of implied volatilities.

Forecasting, Implied Volatility, Long Memory, Multifractal Model of Asset Returns, Option Pricing, Poisson Multifractal, Trading Time, Volatility Smile

Abstract:
This paper investigates the Multifractal Model of Asset Returns, a continuous-time process that incorporates the thick tails and volatility persistence exhibited by many financial time series. The model is constructed by compounding a Brownian Motion with a multifractal time-deformation process. Return moments scale as a power law of the time horizon, a property confirmed for Deutschemark / U.S. Dollar exchange rates and several equity series. The model implies semi-martingale prices and thus precludes arbitrage in a standard two-asset economy. Volatility has long-memory, and the highest finite moment of returns can have any value greater than two. The local variability of the process is characterized by a renormalized probability density of local Hölder exponents. Unlike standard models, multifractal paths contain a multiplicity of these exponents within any time interval. We develop an estimation method, and infer a parsimonious generating mechanism for the exchange rate series. Simulated samples replicate the moment-scaling found in the data.

Multifractal Model of Asset Returns, Compound Stochastic Process, Time Deformation, Scaling, Self-Similarity, Multifractal Spectrum, Stochastic Volatility

Abstract:
This paper constructs an index of financial sophistication that, in comprehensive data on Swedish households, best explains a set of three investment mistakes: underdiversification, risky share inertia, and the tendency to sell winning stocks and hold losing stocks (the disposition effect). The index of financial sophistication increases strongly with financial wealth and household size, and to a lesser extent with education and proxies for financial experience. The index is strongly positively correlated with the share of risky assets held by a household.

Institutional subscribers to the NBER working paper series, and residents of developing countries may download this paper without additional charge at www.nber.org.

Abstract:
We propose a discrete-time stochastic volatility model in which regime switching serves three purposes. First, changes in regimes capture low-frequency variations. Second, they specify intermediate-frequency dynamics usually assigned to smooth autoregressive transitions. Finally, high-frequency switches generate substantial outliers. Thus a single mechanism captures three features that are typically viewed as distinct in the literature. Maximum-likelihood estimation is developed and performs well in finite samples. Using exchange rates, we estimate a version of the process with four parameters and more than a thousand states. The multifractal outperforms GARCH, MS-GARCH, and FIGARCH in- and out-of-sample. Considerable gains in forecasting accuracy are obtained at horizons of 10 to 50 days.

forecasting, long memory, Markov-switching multifractal (MSM), closed-form likelihood, scaling, stochastic volatility, volatility component, Vuong test

Abstract:
This paper investigates the dynamics of individual portfolios in a unique dataset containing the disaggregated wealth of all households in Sweden. Between 1999 and 2002, we observe little aggregate rebalancing in the financial portfolio of participants. These patterns conceal strong household-level evidence of active rebalancing, which on average offsets about one half of idiosyncratic passive variations in the risky asset share. Wealthy, educated investors with better diversified portfolios tend to rebalance more actively. We find some evidence that households rebalance towards a higher risky share as they become richer. We also study the decisions to trade individual assets. Households are more likely to fully sell directly held stocks if those stocks have performed well, and more likely to exit direct stockholding if their stock portfolios have performed well; but these relationships are much weaker for mutual funds, a pattern which is consistent with previous research on the disposition effect among direct stockholders and performance sensitivity among mutual fund investors. When households continue to hold individual assets, however, they rebalance both stocks and mutual funds to offset about one sixth of the passive variations in individual asset shares. Households rebalance primarily by adjusting purchases of risky assets if their risky portfolios have performed poorly, and by adjusting both fund purchases and full sales of stocks if their risky portfolios have performed well. Finally, the tendency for households to fully sell winning stocks is weaker for wealthy investors with diversified portfolios of individual stocks.

Abstract:
We implement a multifrequency volatility decomposition of three exchange rates and show that components with similar durations are strongly correlated across series. This motivates a bivariate extension of the Markov-Switching Multifractal (MSM) introduced in Calvet and Fisher (2001, 2004). Bivariate MSM is a stochastic volatility model with a closed-form likelihood. Estimation can proceed by ML for state spaces of moderate size, and by simulated likelihood via a particle filter in high-dimensional cases. We estimate the model and confirm its main assumptions in likelihood ratio tests. Bivariate MSM compares favorably to a standard multivariate GARCH both in- and out-of-sample. We extend the model to multivariate settings with a potentially large number of assets by proposing a parsimonious multifrequency factor structure.

Abstract:
This paper investigates the efficiency of household investment decisions using comprehensive disaggregated Swedish data. We consider two main sources of inefficiency: underdiversification ("down") and nonparticipation in risky asset markets ("out"). While a few households are very poorly diversified, most Swedish households outperform the Sharpe ratio of their domestic stock index through international diversification. Financially sophisticated households invest more efficiently but also more aggressively, and overall they incur higher return losses from underdiversification. The return cost of nonparticipation is smaller by almost one-half when we take account of the fact that nonparticipants would unlikely be inefficient investors.