understanding dynamic mean variance asset allocation abstract we provide a new portfolio decomposition formula that sheds light on the economics of portfolio choice for investors following the mean variance mv criterion we show that the number of components of a dynamic portfolio strategy can be reduced to two the first is preference free and hedges the risk of a discount bond maturing at the investor s horizon while the second hedges the time variation in pseudo relative risk tolerance both components entail strong horizon effects in the dynamic asset allocation as a result of time varying risk tolerance and investment opportunity sets we also provide closed form solutions for the optimal portfolio strategy in the presence of market return predictability the model parameters are estimated over the period 1963 to 2012 for the u s market we show that i intertemporal hedging can be very large ii the mv criterion hugely understates the true extent of risk aversion for high values of the risk aversion parameter and the more so the shorter the investment horizon and iii the efficient frontiers seem problematic for investment horizons shorter than one year but satisfactory for large horizons overall adopting the mv model leads to acceptable results for medium and long term investors endowed with medium or high risk tolerance but to very problematic ones otherwise

hedge fund seeding via fees for seed swaps under idiosyncratic risk we develop a dynamic valuation model of the hedge fund seeding business by solving the consumption and portfolio choice problem for a risk averse manager who launches a hedge fund through a seeding vehicle this vehicle i e fees for seed swap specifies that a strategic partner seeder provides a critical amount of capital in exchange for participation in the funds revenue our results indicate that the new swap not only solves the serious problem of widespread financing constraints for new and early stage funds esfs managers but can be highly beneficial to both the manager and the seeder if structured properly

the effect of r d services offshoring on skilled employment firm evidence the offshoring of high tech services has greatly increased in recent years with consequences for firms demand for skilled employment in firms this paper specifically analyses the relationship between r d offshoring and the demand for r d employment using firm level data for spanish manufacturing and services companies during the period 2004 11 estimating different specifications with panel data techniques we find that this association is statistically positive in particular for services firms a 1 percentage point increase in r d offshoring raises the demand for researchers by about 11 per cent

an asset allocation puzzle comment should the proportion of risky assets in the risky part of an investor s portfolio depend on the investor s risk aversion according to basic financial theory in particular the mutual fund separation theorem with a riskless asset the answer is no the theorem states that rational investors should divide their assets between a riskless asset and a risky mutual fund the composition of which is the same for all investors risk aversion affects only the allocation between the riskless asset and the fund however niko canner et al 1997 cmw hereafter observed that popular investment advice does not conform to this theory they reported the stocks bonds and cash allocations recommended by four advisors for conservative moderate and aggressive investors as shown in table 1 which is reproduced from cmw the advisors recommend a bond stock ratio that varies directly with risk aversion for example fidelity recommends a bond stock ratio of 1 50 for a conservative more riskaverse investor a ratio of 1 00 for a moderate less risk averse investor and a ratio of 0 46 for an aggressive still less risk averse investor the inconsistency between such advice and the separation theorem is called an asset allocation puzzle by cmw they attempted to solve the puzzle by relaxing key assumptions in the theory but finally reached a negative conclusion although we cannot rule out the possibility that popular advice is consistent with some model of rational behavior we have so far been unable to find such a model p 181 however they suggested that consideration of intertemporal trading might help resolve the puzzle in the present paper we provide theoretical support for the popular advice the two key insights are that the investor s horizon may exceed the maturity of the cash asset and that the investor rebalances the portfolio as time passes if the investor s horizon exceeds the maturity of cash which might be a money market security with maturity of one to six months then cash is not the riskless asset as is commonly assumed in the basic theory in a theory allowing portfolio rebalancing as opposed to a buy and hold framework it is not unreasonable to assume that the investor can synthesize a riskless asset a zero coupon bond maturing at the horizon using a bond fund and cash then bonds will be both in the synthetic riskless asset and in the risky mutual fund and we show that in this case the theoretical bond stock ratio varies directly with risk aversion for any hyperbolic absolute risk aversion hara investor as an example of the type of results that a specific model can produce we provide a continuous time model with closed form solutions which produces theoretical bond stock ratios similar to the popular advice the present paper is organized as follows in the next section we analyze the popular advice in terms of the theory of mutual fund separation of david cass and joseph e stiglitz 1970 we show that this theory is relevant both in static and dynamic frameworks and use it to analyze the popular advice in complete and incomplete markets in section ii we analyze the popular advice in the context of robert c merton s 1971 continuous time statement of mutualfund separation and present an illustrative model in which a crra investor makes continuous time portfolio decisions under interest rate and stock price uncertainty in section iii numerical results are compared with the popular advice section iv is a conclusion bajeux besnainou department of finance school of business and public management george washington university 2023 g street nw washington dc 20052 jordan national economic research associates 1255 23rd street nw washington dc 20037 portait cnam and essec finance chair cnam 2 rue conte paris france this research was supported by a grant from the institute for quantitative investment research we thank two anonymous referees for their comments 1 hara functions include quadratic utility which is one way of justifying mean variance preferences and constant relative risk aversion crra utility both quadratic and crra utility were considered in the cmw analysis

a note on merton s optimum consumption and portfolio rules in a continuous time model in the paper optimum consumption and portfolio rules in a continuous time model by r c merton j econ theory 3 1971 373 413 solutions obtained in cases when marginal utility at zero consumption is finite are not feasible while they do satisfy the hamilton jacobi bellman equations they do not represent appropriate value functions because the boundary behavior near zero wealth is not satisfactorily dealt with in this note we specify the boundary behavior and characterize optimal solutions journal of economic literature classification numbers 022 213

risk aversion in a dynamic asset allocation experiment we conduct a controlled laboratory experiment in the spirit of merton 1971 in which subjects dynamically choose their portfolio allocation between a risk free and risky asset using the optimal allocation of an investor with hyperbolic absolute risk aversion hara utility we fit the experimental choices to characterize the risk profile of our participants despite substantial heterogeneity decreasing absolute risk aversion and increasing relative risk aversion are the predominant types we also find some evidence of increased risk taking after a gain finally the session level risk attitudes show a different profile than the individual descriptions of risk attitudes

optimal portfolio policies with borrowing and shortsale constraints abstract we characterize optimal intertemporal portfolio policies for investors with crra utility facing either a borrowing constraint or shortsale restrictions or both the optimal constrained portfolios are identified as optimal unconstrained portfolios for a higher riskless rate or for a subset of the risky assets or for a combination of the two settings our characterization is based on duality results in cvitanic and karatzas 1992 annals of applied probability 2 767 818 for optimal portfolio investment when portfolio values are more generally constrained to a closed convex nonempty subset of r n

acreage decisions under risk the case of corn and soybeans an acreage supply response model is developed under expected utility maximization the resulting framework is used to specify and estimate a system of risk responsive acreage equations for corn and soybeans in the united states particular attention is given to the truncation effects of government price supports on the distribution of corn and soybean prices also a wealth variable is included in the acreage equations the empirical results indicate that risk and wealth variables play an important role in cornsoybean acreage decisions the analysis also shows that cross commodity risk reduction is important in acreage allocation decisions

producer surplus and risk this paper examines the welfare economics of producer behavior under risk aversion hicksian marshallian and money equivalent measures are explored it is found that under decreasing absolute risk aversion compensating variation is less than the ordinary marshallian surplus which is less than the money equivalent measure under constant absolute risk aversion all measures coincide finally bounds on compensating and equivalent variations using ordinary producer surplus in a manner analogous to willige approach in consumer theory under certainty are studied similar results hold mutatis mutandis for input demands

dynamic portfolio choice with parameter uncertainty and the economic value of analysts recommendations we derive a closed form solution for the optimal portfolio of a non myopic utility maximizer who has incomplete information about the alphas or abnormal returns of risky securities we show that the hedging component induced by learning about the expected return can be a substantial part of the demand using our methodology we perform an ex ante empirical exercise which shows that the utility gains resulting from optimal allocation are substantial in general especially for long horizons and an ex post empirical exercise which shows that analysts recommendations are not very useful

optimal portfolio management with american capital guarantee abstract the aim of the paper is to investigate finite horizon portfolio strategies which maximize a utility criterion when a constraint is imposed on a terminal date european guarantee or on every intermediate date american guarantee we prove the optimality of the option based portfolio insurance method for both european and american cases when an expected crra utility function is maximized the american obpi is fully described in a black scholes environment as well as in the more general case of complete markets using the gittins index techniques developed by el karoui and karatzas 1995 optimality results are extended to general utility functions

pricing stock and bond options when the default free rate is stochastic we derive formulas for the valuation of call options on stocks and bonds when the defaultfree rate is stochastic the formulas highlight the role of the correlation between the unanticipated returns on the underlying security and the changes in the short term rate in determining the options value our numerical analysis indicates that option prices predicted by the proposed formula differ from those predicted by the black and scholes formula when this correlation is relatively large and the short term rate s instantaneous variance is relatively large as well moreover the proposed formula predicts higher lower stock option prices than those predicted by the black and scholes formula for correlation values that are lower higher than some positive critical value

who should buy long term bonds according to conventional wisdom long term bonds are appropriate for conservative long term investors this paper develops a model of optimal consumption and portfolio choice for infinite lived investors with recursive utility who face stochastic interest rates solves the model using an approximate analytical method and evaluates conventional wisdom as risk aversion increases the myopic component of risky asset demand disappears but the intertemporal hedging component does not conservative investors hold assets to hedge the risk that real interest rates will decline long term inflation indexed bonds are most suitable for this purpose but nominal bonds may also be used if inflation risk is low

an isomorphism between asset pricing models with and without linear habit formation we show an isomorphism between optimal portfolio selection or competitive equilibrium models with utilities incorporating linear habit formation and corresponding models without habit formation the isomorphism can be used to mechanically transform known solutions not involving habit formation to corresponding solutions with habit formation for example the constantinides 1990 and ingersoll 1992 solutions are mechanically obtained from the familiar merton solutions for the additive utility case without recourse to a bellman equation or first order conditions more generally recent solutions to portfolio selection problems with recursive utility and a stochastic investment opportunity set are readily transformed to novel solutions of corresponding problems with utility that combines recursivity with habit formation the methodology also applies in the context of hindy huang kreps 1992 preferences where our isomorphism shows that the solution obtained by hindy and huang 1993 can be mechanically transformed to dybvig s 1995 solution to the optimal consumption investment problem with consumption ratcheting this article presents a general method for solving asset pricing or portfolio selection models involving linear habit formation of the type studied by sundaresan 1989 constantinides 1990 detemple and zapatero 1991 ingersoll 1992 chapman 1998 and others the basic idea we pursue is that linear habit formation can be thought of as a redefinition of what constitutes consumption to include not only the current consumption rate but also a fictitious possibly negative consumption rate derived from past actual consumption by pricing out this fictitious consumption rate correctly the economy with linear habit formation can be mechanically transformed to an equivalent economy without habit formation this analysis simplifies and unifies existing results but also generates novel solutions for example together with the results of schroder and skiadas 1997 1999 our method produces optimal lifetime consumption and portfolio policies with preferences that combine recursive utility with habit formation under a stochastic

the role of risk aversion and intertemporal substitution in dynamic consumption portfolio choice with recursive utility abstract our objective is to understand how risk aversion and elasticity of intertemporal substitution under recursive utility affect dynamic consumption portfolio decisions for a three date model with a stochastic interest rate we obtain an analytic solution for the optimal policies we find that in general consumption and portfolio decisions depend on both risk aversion and elasticity of intertemporal substitution the size of risk aversion relative to unity determines the sign of the intertemporal hedging portfolio while elasticity of intertemporal substitution affects only its magnitude the portfolio weight is independent of elasticity of intertemporal substitution only for the case of a constant investment opportunity set

the numeraire portfolio abstract a portfolio formed from a given list of assets is defined as a numeraire portfolio for the list if a it is self financing b its value is always positive and c zero is always the best conditional forecast of the numeraire dominated rate of return of every asset on the list the numeraire portfolio exists if and only if there are no profit opportunities from trading assets on the list for a sample list of heterogeneous assets nyse size quintile portfolios corporate bonds and short term bills numeraire dominated returns are similar to market model forecast errors and as abnormal return measures clearly dominate market adjusted returns

continuous time methods in finance a review and an assessment i survey and assess the development of continuous time methods in finance during the last 30 years the subperiod 1969 to 1980 saw a dizzying pace of development with seminal ideas in derivatives securities pricing term structure theory asset pricing and optimal consumption and portfolio choices during the period 1981 to 1999 the theory has been extended and modified to better explain empirical regularities in various subfields of finance this latter subperiod has seen significant progress in econometric theory computational and estimation methods to test and implement continuous time models capital market frictions and bargaining issues are being increasingly incorporated in continuous time theory the roots of modern continuous time methods in finance can be traced back to the seminal contributions of merton 1969 1971 1973b in the late 1960s and early 1970s merton 1969 pioneered the use of continuous time modeling in financial economics by formulating the intertemporal consumption and portfolio choice problem of an investor in a stochastic dynamic programming setting merton 1973b also showed how such a framework can be used to develop equilibrium asset pricing implications thereby significantly extending the asset pricing theory to richer dynamic settings and expanding the scope of applications of continuous time methods to study problems in financial economics 1 within a span of about 30 years from the publication of merton s inf luential papers continuous time methods have become an integral part of financial economics indeed in certain core areas in finance such as e g asset pricing derivatives valuation term structure theory and portfolio selection continuoustime methods have proved to be the most attractive way to conduct research and gain economic intuition the continuous time approach in these areas has produced models with a rich variety of testable implications the econometric theory for testing continuous time models has made rapid strides in the last decade and has thus kept pace with the impressive progress on the theoretical front one hopes that the actual empirical investigations and estimation using the new procedures will follow suit soon

a monte carlo method for optimal portfolios this paper proposes a new simulation based approach for optimal portfolio allocation in realistic environments with complex dynamics for the state variables and large numbers of factors and assets a first illustration involves a choice between equity and cash with nonlinear interest rate and market price of risk dynamics intertemporal hedging demands significantly increase the demand for stocks and exhibit low volatility we then analyze settings where stock returns are also predicted by dividend yields and where investors have wealth dependent relative risk aversion large scale problems with many assets including the nasdaq sp500 bonds and cash are also examined the question of optimal portfolio allocation has been of long standing interest for academics and practitioners in finance while the mean variance analysis of markowitz 1952 is still commonly used among portfolio managers it has been well understood since merton 1971 that long term investors would prefer portfolios that include hedging components to protect against fluctuations in their investment opportunities prompted by the seminal papers of merton 1969 1971 and samuelson 1969 studies have explored various aspects of the dynamic portfolio problem when asset prices follow diffusion processes e g richard 1975 this literature has relied for the most part on a dynamic programming approach to the problem more recent contributions by pliska 1986 karatzas lehoczky and shreve 1987 and cox and huang 1989 have proposed an alternative resolution method based on martingale techniques in the context of this approach an optimal portfolio formula was derived by ocone and karatzas 1991 this expression involves expectations of random variables depending on j6erme b detemple is at boston university school of management and cirano rene

economic behavior under temporal uncertainty a considerable amount of research has focused on the effects of uncertainty on economic behavior in the absence of complete risk markets private risk bearing is widely believed to influence economic decisions arrow 1 sandmo 23 chavas 4 dalal 6 and others have investigated the role of uncertainty in resource allocation for example sandmo 23 has proposed a theory of the firm under risk he showed how risk and risk aversion can have a negative effect on production decisions

dynamic hedging and extreme asset co movements the paper investigates the portfolio allocation effects of increased asset co movements during market downturns we develop a model for the stock price process that allows for increased and asymmetric dependence between extreme return realizations we isolate the portfolio hedging demands that arise due to extreme co movements and find a substantial shift of the portfolio holdings toward the risk free asset we demonstrate that accounting for dependence between extreme events in portfolio decisions leads to significant economic gains that stem primarily from intertemporal hedging motives these findings are robust along alternative modeling assumptions of extreme co movements and conditional correlation

nontraded asset valuation with portfolio constraints a binomial approach we provide a simple binomial framework to value american style derivatives subject to trading restrictions the optimal investment of liquid wealth is solved simultaneously with the early exercise decision of the nontraded derivative noshort sales constraints on the underlying asset manifest themselves in the form of an implicit dividend yield in the risk neutralized process for the underlying asset one consequence is that american call options may be optimally exercised prior to maturity even when the underlying asset pays no dividends applications to executive stock options eso are presented it is shown that the value of an eso could be substantially lower than that computed using the black scholes model we also analyze nontraded payoffs based on a price that is imperfectly correlated with the price of a traded asset the economics of asset pricing when one or more of the assets in the opportunity set are either subject to trading restrictions or entirely nontraded is a matter of great interest viewed from a practical perspective we have several important examples of such assets that are subject to trading restrictions pensions which represent perhaps the most significant of assets held by individual households are subject to trading restrictions it is typically the case that assets in pensions are not available for immediate consumption borrowing against pension assets is subject to significant direct and indirect costs by way of taxes and early withdrawal penalties human capital is another example housing investment is also illiquid and subject to significant transaction costs together pensions human capital and housing constitute a substantial part of a typical household s assets the significance of such nontraded assets for risk premia has already been noted by bewley 1982 there are other circumstances where lack of unrestricted trading plays an important role executive compensation plans usually take the this article was presented at boston university we would like to thank the editor bernard dumas and two anonymous referees for very helpful comments we also thank nalin kulatilaka and seminar participants for useful suggestions ganlin chang fei guan and carlton osakwe provided valuable research assistance j detemple acknowledges support from the social sciences and humanities research

optimal investment and consumption when allowing terminal debt we analyze a dynamic optimization problem which involves the consumption and investment of an investor with constant relative risk aversion for consumption but with a risk aversion for final wealth which does not necessarily imply that terminal wealth must always be positive we require risk aversion for terminal wealth to be positive but not monotone there is a point of maximal risk aversion at zero wealth and the investor may continue to consume when wealth is negative using dual optimization methods we can derive explicit solutions and we find that the optimal solution differs in a fundamental way from the case where risk aversion is monotone it turns out that the optimal consumption function is convex and concave at different wealth levels and that the optimal investment strategy may no longer be monotone as a function of the remaining time to invest and consume

willingness to pay measures and expectations an experiment in studies of the willingness to pay for various environmental commodities services it is typically explicitly or implicitly assumed that the respondent has perfect has perfect foresight this paper which is based on a questionnaire completed by examining if and how the value of a hunting permit is affected by the hunter s expectations regarding the outcome of the hunting in particular the willingness to pay is estimated as a function of expected values as well as variances of different variables using a question which reflects an extension of jensen s inequality to the multivariate case as well as measures of downside risk it is shown that risk attitudes are important determinants of the willingness to pay for a hunting permit

optimal consumption and portfolio policies when asset prices follow a diffusion process abstract we consider a consumption portfolio problem in continuous time under uncertainty a martingale technique is employed to characterize optimal consumption portfolio policies when there exist nonnegativity constraints on consumption and on final wealth we also provide a way to compute and verify optimal policies our verification theorem for optimal policies involves a linear partial differential equation unlike the nonlinear partial differential equation of dynamic programming the relationship between our approach and dynamic programming is discussed we demonstrate our technique by explicitly computing optimal policies in a series of examples in particular we solve the optimal consumption portfolio problem for hyperbolic absolute risk aversion utility functions when the asset prices follow a geometric brownian motion the optimal policies in this case are no longer linear when nonnegativity constraints on consumption and on final wealth are included by these examples one can see that our approach is much easier than the dynamic programming approach

refutable implications of the firm model under risk curvature properties of the indirect utility function are shown to be necessary and sufficient for refutable behavioural postulates in the form of comparative static results reciprocity relations and restrictions on output and input responses for firm models under risk these postulates are independent of onerous restrictions on risk preference technology or random variable distribution as an exposition the complete set of refutable implications for a multi output firm operating under price risk is derived at the data means these implications are not rejected when they are tested using firm level data but both risk neutrality and constant absolute risk aversion are rejected at individual observations one implication is rejected almost as frequently as risk neutrality and constant absolute risk aversion

the equilibrium allocation of diffusive and jump risks with heterogeneous agents abstract we study a two agent pure exchange equilibrium subject to both nondiversifiable diffusive and jump risks agents can trade in a financial market consisting of a stock market a money market and an insurance market for jump risk heterogeneity is introduced through different levels of relative risk aversion in the framework of standard expected utility we find the surprising result that the less risk averse agent purchases insurance contracts against jump risk from the more risk averse agent this equilibrium allocation is linked to the non linear wealth sharing rule in such an economy and preserves the wealth effects studied by dumas rev financ stud 2 1989 157 in the case of pure diffusive risk since the benchmark economy with homogeneous agents generates no excess uncertainty in the stock market we study the effect on excess volatility and excess jump size solely due to different levels of relative risk aversion we observe 3 excess uncertainty in jump sizes for a reasonable specification of economic fundamentals

valuation optimal asset allocation and retirement incentives of pension plans we provide a framework in which we link the valuation and asset allocation policies of defined benefits plans with the lifetime marginal productivity schedule of the worker and the pension plan formula in turn we examine the retirement policies that are implied by the primitives of the model and the value of pension obligations our model provides an explicit valuation formula for a stylized defined benefits plan the optimal asset allocation policies consist of the replicating portfolio of the pension liabilities and the growth optimum portfolio independent of the pension liabilities we show that the worker will retire when the ratio of pension benefits to current wages reaches a critical value which depends on the parameters of the pension plan and the discount rate using numerical techniques we analyze the feedback effect of retirement policies on the valuation of plans and on the asset allocation decisions article published by oxford university press on behalf of the society for financial studies in its journal the review of financial studies

a model of capital and crises we develop a model in which the capital of the intermediary sector plays a critical role in determining asset prices the model is cast within a dynamic general equilibrium economy and the role for intermediation is derived endogenously based on optimal contracting considerations low intermediary capital reduces the risk bearing capacity of the marginal investor we show how this force helps to explain patterns during financial crises the model replicates the observed rise during crises in sharpe ratios conditional volatility correlation in price movements of assets held by the intermediary sector and fall in riskless interest rates

optimal risk sharing with effort and project choice we consider first best risk sharing problems in which the agent can control both the drift effort choice and the volatility of the underlying process project selection in a model of delegated portfolio management it is optimal to compensate the manager with an option type payoff where the functional form of the option is obtained as a solution to an ordinary differential equation in the general case the optimal contract is a fixed point of a functional that connects the agent s and the principal s maximization problems we apply martingale duality methods familiar from optimal consumption investment problems

on equilibrium asset price processes in this article we derive necessary and sufficient conditions that must be satisfied by equilibrium asset price processes in apure exchange economy we examine a world in which asset prices follow a diffusion process asset markets are dynamically complete all investors maximize their state independent expected utility of consumption at some future date and investors have nonrandom exogenous income we show that it is necessary and suffcient that the coefficients of an equilibrium diffusion price process satisfy a partial differential equation and a boundary condition we also examine how the dynamics of asset prices are related to the shape of the representative investor s utility function through the boundary condition for example in a constant volatility economy the expected instantaneous return of the market portfolio is mean reverting if and only if the relative risk aversion of the representative investor is decreasing in terminal wealth

implications of the sharpe ratio as a performance measure in multi period settings we study efiects of using sharpe ratio as a performance measure for compensating money managers in a dynamic and frictionless market setting first we demonstrate that with such a performance measure the manager s focus on the short horizon performance is detrimental to the investor s long horizon performance numerical experiments illustrate that when returns are iid the performance loss is signiflcant even when the investor s investment horizon is not much longer than the manager s when expected returns are mean reverting the performance loss is exacerbated second we show that sharpe ratio maximization strategies tend to increase decrease the risk in the later part of the optimization period after a bad good performance in the earlier part of the optimization period as illustrated by a simulation exercise this prediction is in agreement with empirical observations and it presents a rational expectations alternative to the prevalent tournament theory explanation

precautionary savings and the permanent income hypothesis this paper derives the explicit solution of a dynamic stochastic optimal consumption problem for infinitely lived agents whose preferences exhibit in the presence of non diversifiable labour income uncertainty a constant elasticity of intertemporal substitution and constant absolute risk aversion the constancy of the elasticity of intertemporal substitution which implies that marginal utility at zero consumption is infinite guarantees that the non negativity constraint on consumption is never binding along the optimal path the assumption of constant absolute risk aversion allows an explicit computation of human wealth and provides a simple representation of the precautionary savings motive

a theory of dual labor markets with application to industrial policy discrimination and keynesian unemployment this paper develops a model of dual labor markets based on employers need to motivate workers in order to elicit effort from their workers employers may find it optimal to pay more than the going wage this changes fundamentally the character of labor markets the model is applied to a wide range of labor market phenomena it provides a coherent framework for understanding the claims of industrial policy advocates it also can provide the basis for a theory of occupational segregation and discrimination that will not be eroded by market forces finally the model provides the basis for a theory of involuntary unemployment

dynamic asset allocation with mean variance preferences and a solvency constraint abstract the aim of this article is to analyze the portfolio strategies that are mean variance efficient when continuous rebalancing is allowed and a solvency constraint wealth positiveness is imposed at each point in time under very general assumptions ito processes for the security price dynamics those efficient strategies are identified as synthetic put options on the particular inefficient portfolio yielding the return with minimal second moment this general characterization allows us to derive a closed form formula for the optimal portfolio weights when the investment opportunity set is constant the optimal weights are explicit functions of the current market index the formula is then used to simulate dynamically efficient strategies for different horizons and targets and to describe the time pattern of the allocation to stocks versus bonds

dynamic asset allocation in a mean variance framework the aim of this article is to analyze the portfolio strategies that are mean variance efficient when continuous rebalancing is allowed between the current date 0 and the horizon t under very general assumptions when a zero coupon bond of maturity t exists the dynamic efficient frontier is a straight line the slope of which is explicitly characterized every dynamic mean variance efficient strategy can be viewed as buy and hold combinations of two funds the zero coupon bond of maturity t and a continuously rebalanced portfolio an appropriate dynamic strategy defining the latter is explicitly derived for two particular price processes and comparisons of the efficient frontiers static versus dynamic are provided in these cases

equilibrium prices in the presence of delegated portfolio management abstract this paper analyzes the asset pricing implications of commonly used portfolio management contracts linking the compensation of fund managers to the excess return of the managed portfolio over a benchmark portfolio the contract parameters the extent of delegation and equilibrium prices are all determined endogenously within the model we consider symmetric fulcrum performance fees distort the allocation of managed portfolios in a way that induces a significant and unambiguous positive effect on the prices of the assets included in the benchmark and a negative effect on the sharpe ratios asymmetric performance fees have more complex effects on equilibrium prices and sharpe ratios with the signs of these effects fluctuating stochastically over time in response to variations in the funds excess performance

an intertemporal general equilibrium asset pricing model the case of diffusion information this paper provides sufficient conditions for the equilibrium price system and a vector of exogenously specified state variable processes to form a diffusion process in a pure exchange economy the conditions involve smoothness of agents utility functions and certain nice properties of the aggregate endowment process and the dividend processes of traded assets in place of the dynamic programming a martingale representation technique is utilized to characterize equilibrium portfolio policies this technique is useful even when there does not exist a finite dimensional markov structure in the economy and thus the markovian stochastic dynamic programming is not applicable a gents are shown to hold certain hedging mutual funds and the riskless asset in contrast to earlier results the market portfolio does not have a special role in hedging since the markets are dynamically complete when there exists a finite dimensional markov system in the economy the dimension of the hedging demand identified through the markovian dynamic programming may be much larger than that identified by the martingale method copyright 1987 by the econometric society

risk aversion and yield uncertainty in duality models of production a mean variance approach a duality model of production is developed that endogenizes stochastic output decisions under risk aversion the model incorporates both yield and price uncertainty and nonlinear mean variance risk preferences this permits modeling of the firm s expected output supply and input demand decisions under risk and also in principle the firm s output variance decisions the duality model is tractable for empirical research

two trees with heterogeneous beliefs spillover effect of disagreement in a model where investors disagree about the fundamentals of two stocks the state price density depends on investor disagreements for both stocks especially the larger stock this implies that disagreement among investors in a large firm has a spillover effect on the pricing of other stocks owned by these investors the pricing effects of investor disagreements crucially depend on the average belief biases empirical findings support the novel model prediction of a disagreement spillover effect and help reconcile some mixed evidence in the literature

optimal consumption wealth relationships derived by consumer intertemporal profit maximisation abstract this paper proposes an artificial intertemporal consumer profit maximisation problem that is simple to solve the optimal consumption function which is consistent with intertemporal utility maximisation is then generated directly from a specification of the consumer s instantaneous profit function

on dynamic decision making to meet consumption targets we investigate a dynamic decision model that facilitates a target oriented decision maker in regulating her risky consumption based on her desired target consumption level in every period in a finite planning horizon we focus on dynamic operational decision problems of a firm where risky cash flows are being resolved over time the firm can finance consumption by borrowing or saving to attain prescribed consumption targets over time to evaluate the ability of the consumption in meeting respective targets we propose the consumption shortfall risk csr criterion which has salient properties of attainment content starvation aversion subadditivity and positive homogeneity we show that if borrowing and saving are unrestricted and their interest rates are common the optimal policy that minimizes the csr criterion is to finance consumption at the target level for all periods except the last for general convex dynamic decision problems the optimal policies correspond to those that maximize an additive

production location decision and free entry oligopoly abstract this paper presents a simple oligopolistic model of location in which all firms in an imperfectly competitive industry are assumed to produce a homogeneous output and make cournot conjectures about their rivals input and location decisions within this framework this paper points out that the key difference between oligopoly and monopoly is the following in the case of monopoly a positive shift in demand requires that a monopolist increase his output but in the case of oligopoly an increase in industry output may or may not increase output per firm because of the number of firms at free entry equilibrium is not constant it is shown that in an oligopoly model of location that allows free entry the demand condition plays an important role on the determination of industrial location

dynamic asset allocation for stocks bonds and cash closed form solutions for hara optimal portfolios are obtained in a dynamic portfolio optimization model in three assets stocks bonds and cash in a vasicek type model of stochastic interest rates with correlated stock prices the hara is a buy and hold combination of a zero coupon bond with maturity matching the investor s horizon and a crra mutual fund this simple characterization facilitates insights about investor behavior over time and provides explanations on the rational use of convex versus concave investment strategies the model illuminates clearly the role of the different market parameters and relative risk aversion in portfolio strategies

equilibrium open interest abstract this paper analyses what determines an individual investor s risk sharing demand for options and aggregating across investors what the equilibrium demand for options we find that agents trade options to achieve their desired skewness specifically we find that portfolio holdings boil down to a three fund separation theorem that includes a so called skewness portfolio that agents like to attain our analysis indicates also however that the common risk sharing setup used for option demand and pricing is incompatible with a stylized fact about open interest across strikes

dynamic asset allocation portfolio decomposition formula and applications a new decomposition of the optimal portfolio in dynamic models with von neumann morgenstern preferences and ito prices is established the formula rests on a change of numeraire that uses pure discount bonds as units of account the dynamic hedging demand has two components the first hedge insures against fluctuations in an optimally designed bond with a maturity date matching the investor s horizon the second hedge immunizes against fluctuations in the market price of risk in the bond numeraire various applications are examined new results concerning the behavior of extremely risk averse individuals the demand for bonds and its long horizon limit and the optimal portfolio in incomplete markets are derived the author 2009 published by oxford university press on behalf of the society for financial studies all rights reserved for permissions please email journals permissions oxfordjournals org oxford university press

an exact bond option formula this paper derives a closed form solution for european options on pure discount bonds assuming a mean reverting gaussian interest rate model as in vasicek 8 the formula is extended to european options on discount bond portfolios

consumption and portfolio choice under internal multiplicative habit formation this paper explores the optimal consumption and investment behavior of an individual who derives utility from the ratio between his consumption and an endogenous habit we obtain closed form policies under general utility functionals and stochastic investment opportunities by developing a nontrivial linearization to the budget constraint this enables us to explicitly characterize how habit formation affects the marginal propensity to consume and optimal stock bond investments we also show that in a setting that combines habit formation with epstein zin utility consumption no longer grows at unrealistically high rates at high ages and investments in risky assets decrease

optimal hedging in a dynamic futures market with a nonnegativity constraint on wealth abstract this paper examines the issue of optimal hedging demands for futures contracts from an investor who cannot freely trade his portfolio of primitive assets in the context of lognormal rather than normal returns and of a constant absolute risk aversion utility function in this context the nonnegativity constraint on wealth is binding and the optimal hedging demands are not identical with those encountered in the bulk of the literature which has largely overlooked this problem negative results concerning the derivation of equilibrium in the futures markets and the computation of the open interests then follow

a minimum variance result in continuous trading portfolio optimization the problem of minimizing the variance of discounted wealth at the end of a fixed period is solved when the expectation of terminal wealth is constrained to a specified investment goal the results are obtained in a continuous trading framework under the assumption that the funds can be exchanged between a riskless bond and a stock whose discounted price is described by a geometric brownian motion transaction costs are ignored i e the market is frictionless and unlimited borrowing is permitted at the same rate as the return on the bond typically the optimal trading policy under the above assumptions involves a highly leveraged investment in the stock in the early stages followed by an accumulation of the bond in the later stages numerical results are provided as an illustration of the theory

on the optimality of path dependent structured funds the cost of standardization abstract this paper examines the suitability of an important class of standard financial structured products namely those whose performances are based on smoothing the return of a given risky underlying asset while providing a guarantee at maturity using various assumptions about the customers attitudes towards risk we show that such standardized products are not optimal even if the financial market volatility is constant as a by product we provide in particular the optimal portfolio value in the regret rejoice framework to go further with the notion of aversion of getting a return smaller than the risk free one using the notion of compensating variation we determine for the first time the monetary losses of providing these standardized products instead of the optimal ones to the customers we show that these monetary losses can be very significant when the volatility of the risky asset is stochastic from the operational point of view such results highly suggest to trade on the volatility index vix and or to introduce derivatives written on it when selling standardized funds in order to better meet investors needs and preferences

some stronger measures of risk aversion in the small and the large with applications the arrow pratt measures of risk aversion for von neumann morgenstern utility functions have become workhorses for analyzing problems in the microeconomics of uncertainty they have been used to characterize the qualitative properties of demand in insurance and asset markets to examine the properties of risk taking in taxation models and to study the interaction between risk and life cycle savings problems to name just a few applications equally importantly they have generated the linear risk tolerance class of utility functions which has provided canonical examples in such diverse areas as portfolio theory and the theory of teams despite these successes there have been a number of areas for which the results have been weaker than hoped it is natural to use the risk aversion measures to compare the behavior of individuals in risky choice situations for example consider the individual portfolio choice problem in a two asset world with a riskless asset and a risky asset if individual a has a uniformly higher arrow pratt coefficient of risk aversion than individual b then b will always choose a portfolio combination with more wealth invested in the risky asset but suppose that both assets are risky now there is no obvious sense in which the more risk averse individual can be said to hold a less risky portfolio but it seems strange that such a simple alteration should destroy the analytics which support the basic intuition similarly consider the basic insurance problem if one individual a is more risk averse than another b in the arrow pratt sense it follows that a will pay a larger premium to insure against a random loss than will b typically though an individual evaluates partial rather than total insurance that is only some gambles can be insured against and others must be retained in this case even when the gambles which are retained are independent from those which are insured it is no longer true that the individual whose arrow pratt measure of risk aversion is higher will pay a larger insurance premium the situation is no better when we consider comparative statics exercises for a single individual decreasing absolute risk aversion in the sense of arrow and

on the theory of the competitive firm under uncertainty when initial wealth is random since sandmo s seminal paper on the theory of the competitive firm under output price uncertainty there has been considerable research on this topic 2 5 6 10 11 14 in all these studies it is assumed that in the absence of complete contingent markets output price risk is the only source of uncertainty for the firm however this may not be a realistic assumption in particular as argued by kihlstrom et al 7 it appears reasonable to consider the case where some of the initial wealth of the firm is held in risky assets in that case initial wealth is uncertain and should be modelled as a random variable the objective of this paper is to investigate the introduction of random initial wealth in sandmo s theory of the firm under price uncertainty in sandmo s model a number of important comparative static results e g the conditions under which the supply function is upward sloping 14 69 or under which increased price uncertainty reduces supply 5 depends critically on the sign of a particular expression stated in lemma 2 in ishii 5 while sandmo 14 69 shows this expression is non negative under arrow pratt decreasing absolute risk aversion it can be verified that his proof no longer holds when initial wealth is random therefore the comparative static analysis conducted by sandmo 14 batra and ullah 2 and ishii 5 is no longer valid and gives ambiguous results under initial wealth uncertainty the reason is that the arrow pratt measures of risk aversion 1 12 are too weak for making comparisons in situations where there is more than one source of uncertainty 13 then what can be deduced from the theory of the competitive firm in the realistic case where both initial wealth and output price are uncertain we will show here that meaningful results on the behavior of the competitiv firm under uncertainty can be obtained provided that the arrow pratt measures of risk aversion 1 12 are replaced by the ross measures of risk aversion 13 in particular a comparative static analysis of the production decisions using ross characterization of risk behavior is presented a number of new results are derived on the properties of risk responsive supply and input demand functions under both price and initial wealth uncertainty

does option compensation increase managerial risk appetite this paper solves the dynamic investment problem of a risk averse manager compensated with a call option on the assets he controls under the manager s optimal policy the option ends up either deep in or deep out of the money as the asset value goes to zero volatility goes to infinity however the option compensation does not strictly lead to greater risk seeking sometimes the manager s optimal volatility is less with the option than it would be if he were trading his own account furthermore giving the manager more options causes him to reduce volatility managers with convex compensation schemes play an important role in financial markets this paper solves for the optimal dynamic investment policy for a risk averse manager paid with a call option on the assets he controls the paper focuses on how the option compensation impacts the manager s appetite for risk when he cannot hedge the option position on one hand the convexity of the option makes the manager shun payoffs that are likely to be near the money under the optimal policy the manager either significantly outperforms his benchmark or else incurs severe losses furthermore in examples of optimal trading strategies asset volatility goes to infinity as asset value goes to zero yet option compensation does not strictly lead to greater risk seeking as asset value grows large or if the evaluation date is far away the manager moderates asset risk for example if the manager has constant relative risk aversion crra asset volatility converges to the merton constant as asset value goes to infinity in some situations the manager actually chooses a lower asset volatility than he would if he were investing on his own because the leverage inherent in his option magnifies his exposure to the asset volatility in addition with all constant or decreasing absolute risk averse utility functions from the hyperbolic absolute risk averse hara class giving the manager more options causes him to reduce asset volatility in the crra case for example the manager targets a fixed volatility for his personal

optimal contracts in a continuous time delegated portfolio management problem this article studies the contracting problem between an individual investor and a professional portfolio manager in a continuous time principal agent framework optimal contracts are obtained in closed form these contracts are of a symmetric form and suggest that a portfolio manager should receive a fixed fee a fraction of the total assets under management plus a bonus or a penalty depending upon the portfolio s excess return relative to a benchmark portfolio the appropriate benchmark portfolio is an active index that contains risky assets where the number of shares invested in each asset can vary over time rather than a passive index in which the number of shares invested in each asset remains constant over time fund managers compensation schemes in a delegated portfolio management setting have been of special interest to both academics and practitioners academic interest stems in part from the rapid growth of managed funds over the last two decades gompers and metrick 1998 report that by december 1996 mutual funds pension funds and other financial intermediaries held discretional control over more than half of the u s equity market it is thus of importance to study the impact of institutional trading on asset prices and to integrate into one model both asset pricing and delegated portfolio management compensation as advocated by brennan 1993 allen and santomero 1997 and allen 2001 it is also important to understand the relationship between a manager s trading strategies and his implicit incentives or career concerns see e g brown harlow and starks 1996 brown goetzmann and park 1997 chevalier and ellison 1999 and chen and pennacchi 2000 these concerns include internal promotion from a small to a large fund higher outside offers and the flow of money through the fund under management 1 the importance of fund manager compensation is also underscored by congressional passage in 1970 of the amendment to

a general equilibrium model of international portfolio choice we investigate in a two country general equilibrium model whether a bias in consumption towards domestic goods will necessarily lead to a preference for domestic securities we develop a model where investors are constrained to consume only from their domestic capital stock and where it is costly to transfer capital across countries in this model investors less risk averse than an investor with log utility bias their portfolios towards domestic assets investors more risk averse than log however prefer foreign assets thus this model suggests that it is unlikely that the portfolios observed empirically can be explained by the high proportion of domestic goods in total consumption we investigate in a two country general equilbrium model whether a bias in consumption toward domestic goods will necessarily lead to a preference for domestic assets in an investor s optimal portfolio we constrain investors to consume only from their domestic capital stock and introduce a proportional cost for transferring goods from one country this cost gives rise to endogenous deviations from the law of one price lop thus the home and foreign investors do not hold identical portfolios we show that an investor s optimal portfolio is biased towards domestic equity only if she is less risk averse than an investor with log utility investors with relative risk aversion greater than one exhibit a preference for the foreign asset this is because the exchange rate derived endogenously is negatively correlated with the return on the foreign asset and therefore the translated return on the foreign stock is less risky than that on the domestic stock thus the results of this model suggest that it is unlikely that the preference for domestic assets that is observed empirically can be explained by the high proportion of domestic goods in total consumption there exists a considerable body of literature documenting the gains from international diversification grubel 1968 was the first to propose that international diversification allows investors to attain lower return variances than those achievable by diversifying domestically this proposition relies on the correlation between stock indices across countries being significantly less

winners and losers a micro level analysis of international outsourcing and wages our paper investigates the link between international outsourcing and wages utilizing a large household panel and combining it with industry level information on industries outsourcing activities from input output tables this approach avoids prob lems such as aggregation bias potential endogeneity bias and poor skill definitions that commonly hamper industry level studies we find that outsourcing has had a marked impact on wages applying two alternative skill classifications we find evidence that a 1 percentage point increase in outsourcing reduced the wage for workers in the lowest skill categories by up to 1 5 while it increased wages for high skilled workers by up to 2 6 this result is robust to a number of different specifications jel classification f16 j31 gagnants et perdants une analyse micro economique de la sous traitance internationale et des salaires ce memoire etudie le lien entre la sous traitance internationale et les salaires a l aide de donnees d un grand panel de menages qu on combine avec des donnees au niveau des industries sur les activites de sous traitance de celles qui en font obtenues des tableaux interindustriels cette approche evite les problemes qui habituellement polluent les etudes au niveau de l industrie biais d agregation biais d endogeneite potentielle mauvaises definitions des habiletes on decouvre que la sous traitance a un effet sur les salaires a l aide de differentes formes de classification des habiletes il appert qu un point de pourcentage d augmentation dans la sous traitance reduit les salaires des travailleurs dans les categories d habilete les plus basses de 1 5 alors qu il accroit les salaires de travailleurs specialises de 2 6 ce resultat est robuste et avere meme quand on utilise diverses specifications

optimal investment with minimum performance constraints abstract we consider the portfolio problem of an investor whose wealth is constrained to be at least as large as that generated by investment in a stochastic benchmark portfolio using standard option pricing results the optimal portfolio policy of a hara utility investor is derived explicitly this policy is shown to be equivalent at any point in time to the investor s optimal unconstrained policy when he has contracted to paying out a proportion of the value of the benchmark portfolio at the terminal date this proportion which lies between zero and one is smaller the more likely it is that the investor will strictly outperform the benchmark over the investment horizon

semi analytical solutions for dynamic portfolio choice in jump diffusion models and the optimal bond stock mix this paper studies the optimal portfolio selection problem in jump diffusion models where an investor has a hara utility function and there are potentially a large number of assets and state variables more specifically we incorporate jumps into both stock returns and state variables and then derive semi analytical solutions for the optimal portfolio policy up to solving a set of ordinary differential equations to greatly facilitate economic insights and empirical applications of jump diffusion models to examine the effect of jump risk on investors behavior we apply our results to the bond stock mix problem and particularly revisit the bond stock ratio puzzle in jump diffusion models our results cast new light on this puzzle that unlike pure diffusion models it cannot be rationalized by the hedging demand assumption due to the presence of jumps in stock returns

innovation and wage effects of international outsourcing abstract we investigate the effects of increased outsourcing of production to a low wage country such international outsourcing lowers the marginal cost of production and thus increases profits creating greater incentives for innovation a reduction in the resource requirement in adapting technology relative to improving products or an expansion in the portion of production that can be outsourced generates a greater extent of international outsourcing a lower relative wage and a faster rate of innovation an increase in production taxes in the north production subsidies in the south or a subsidy to adapting technologies has similar effects

offshoring and wage inequality using occupational licensing as a shifter of offshoring costs changes in information and communication technologies have increased the offshorability of tasks as routine tasks pol antras luis garicano and esteban rossi hansberg 2006 and lower offshoring cost tasks gene grossman and esteban rossi hansberg 2008 move abroad production processes become increasingly fragmented recent empirical papers e g mary amiti and shang jin wei 2005 runjuan liu and daniel trefler 2008 rosario crino forthcoming have aimed to illuminate the extent to which this is the case and the consequences of such changes a finding of this literature is that offshoring hurts disproportionately lower skill occupations and those occupations that are more tradable however this literature has been hampered by the difficulty of finding a good proxy for offshoring costs to measure the effect of offshoring on workers one would ideally want to increase offshorability of some tasks and not others and then compare the resulting changes in wages and employment in the occupations performing the tasks that are offshored with the ones that are not instead the literature has relied on proxies that are more or less directly linked to the skill content of the occupation and that as a result confound the effect of production technology and of offshoring cost there exist some partial exceptions such as j bradford jensen and lori g kletzer 2005 who argue that if a service is offshoring and wage inequality using occupational licensing as a shifter of offshoring costs

optimal investment with s shaped utility and trading and value at risk constraints an application to defined contribution pension plan abstract in this paper we investigate an optimal investment problem under loss aversion s shaped utility and with trading and value at risk var constraints faced by a defined contribution dc pension fund manager we apply the concavification and dual control method to solve the problem and derive the closed form representation of the optimal terminal wealth in terms of a controlled dual state variable we propose a simple and effective algorithm for computing the initial dual state value the lagrange multiplier and the optimal terminal wealth theoretical and numerical results show that the var constraint can significantly impact the distribution of the optimal terminal wealth and may greatly reduce the risk of losses in bad economic states due to loss aversion

constrained choices under risk there has recently been a revival of interest in the implications of rationing for economic theory extending the work of samuelson 21 tobin and houthakker 24 and pollak 16 recent papers by bruno 3 howard 8 latham 13 and neary and roberts 15 have explored the theory of producer and consumer behavior under quantity rationing these developments have provided useful hypotheses in the investigation of economic behavior for example they have stimulated a reexamination of macroeconomic disequilibrium however previous work seems to have focused on economic models in the absence of uncertainty yet given that production decisions are often made before output prices are known and in the absence of complete contingent markets sandmo 22 batra and ullah 2 and others have developed the theory of the competitive firm under price uncertainty under the expected utility maximization hypothesis such a theory provides useful information about the influence of changing economic conditions or of risk behavior on production decisions but because of the presence of short run adjustment costs leading to short run input fixity or because of regulatory or institutional constraints quantity rationing often influences production decisions in order to evaluate how rationing influences such decisions there is a need to develop a conceptual model of constrained choices of the firm under risk the objective of this paper is to explore the implications of quantity rationing in the context of production theory under price uncertainty by introducing risk and risk behavior it extends the classical analysis of quantity constrained production decisions 3 21 the implications of quantity constraints for risk responsive choice functions are derived using the concepts of compensated choice functions and virtual prices it is shown that expected price effects can be decomposed into substitution effects and wealth effects the substitution effects measure the effects of a change in expected price on the compensated choice functions the compensation being made through a change in initial wealth holding expected utility constant the wealth effects characterize risk preferences since a non zero wealth effect reflects a departure from an arrow pratt constant absolute risk aversion utility function the influence of rationing on the properties of ordinary as well as compensated choice functions is analyzed in details by deriving the implications of quantity constraints for firm behavior under uncertainty the paper presents a number of useful results concerning the influence of risk or technology on economic behavior under rationing for example a

non addictive habits optimal consumption portfolio policies abstract we formulate a model of preferences with non addictive habits where consumption is required to be non negative at all times but can fall below a standard of living index that aggregates past consumption we study the consumption portfolio problem taking account of the non negativity constraint on consumption and provide a constructive proof for the existence of an optimal policy on a finite time horizon 0 t we show that the consumption constraint binds up to an endogenous stopping time 0 t after which it remains slack until t a decomposition of constrained consumption involving an asian average strike capped call option is demonstrated

market timing with option implied distributions a forward looking approach we address the empirical implementation of the static asset allocation problem by developing a forward looking approach that uses information from market option prices to this end we extract constant maturity s p 500 implied distributions and transform them to the corresponding risk adjusted ones then we form optimal portfolios consisting of a risky and a risk free asset and evaluate their out of sample performance we find that the use of risk adjusted implied distributions times the market and makes the investor better off than if she uses historical returns distributions to calculate her optimal strategy the results hold under a number of evaluation metrics and utility functions and carry through even when transaction costs are taken into account not surprisingly the reported market timing ability deteriorated during the recent subprime crisis an extension of the approach to a dynamic asset allocation setting is also presented this paper was accepted by wei xiong finance

pricing interest rate derivative securities this article shows that the one state variableinterest rate models of vasicek 1977 and cox ingersoll and ross 1985b can be extended sothat they are consistent with both the current termstructure of interest rates and either the currentvolatilities of all spot interest rates or the currentvolatilities of all forward interest rates theextended vasicek model is shown to be very tracta ble analytically the article compares option pricesobtained using the extended vasicek model withthose obtained using a number of other models