:In this paper, we consider the equity premium puzzle under a general utility function. We derive that
the optimal strategy under a general utility function approximate the optimal strategy under the special utility
function. This result posed in the present paper can be regarded as a generalization of the work by Gong and
Zou [13]
Bid and Ask Prices Tailored to Traders' Risk Aversion and Gain Propension: a ...Waqas Tariq
Risky asset bid and ask prices “tailored” to the risk-aversion and the gain-propension of the traders are set up. They are calculated through the principle of the Extended Gini premium, a standard method used in non-life insurance. Explicit formulae for the most common stochastic distributions of risky returns, are calculated. Sufficient and necessary conditions for successful trading are also discussed.
Modeling+pricing+strategies+using+game+theory+and+support+vector+machinesMuhammad Akbar Khan
This document discusses modeling pricing strategies in the credit industry using game theory and support vector machines (SVMs). It proposes a model where pricing is determined through a game-theoretic framework modeling competition between companies. Demand is estimated at the customer-level using SVMs to incorporate a large number of variables. This detailed demand estimation is then used in a market-level game theoretic model to understand the strategic interactions between competitors. The model allows companies to gain insights into customer behavior and competitor strategies to inform their own pricing decisions.
Affine Term Structure Model with Stochastic Market Price of RiskSwati Mital
- The document proposes a new affine term structure model that combines principal components analysis with a stochastic market price of risk.
- Principal components provide useful information about yield curves and only three components explain over 95% of yield variation.
- Previous models linked risk premium deterministically to return-predicting factors like slope, but this could result in unrealistic risk premium levels.
- The new model introduces an additional state variable to capture the stochastic market price of risk and break the deterministic link between risk premium and return-predicting factors.
This blog is a place by Apoorva javadekar to discuss various aspects of life. But in specific, it discusses Economics, Mathematics, Politics and sports events.
I like to write about recent economic policies, Socio - political issues, reviews about new books, cricket and fitness. This blog is a good place to take a look at least once a week.
This document summarizes a research paper that examines the optimal investment, consumption, and life insurance selection problem for a wage earner. The problem is modeled using a financial market with one risk-free asset and one risky jump-diffusion asset, along with an insurance market composed of multiple life insurance companies. The goal is to maximize the wage earner's expected utility from consumption during life, wealth at retirement or death, by choosing an optimal investment, consumption, and insurance strategy. The authors use dynamic programming to characterize the optimal solution and prove existence and uniqueness of a solution to the associated nonlinear Hamilton-Jacobi-Bellman equation.
Effects of dividends on common stock prices the nepalese evidencePankajKunwar3
presentation on effects of dividends on common stock prices in the nepelese evidence paper was was published by prof. rade shyam pradhan in the the excellence
The document discusses calculating price returns and total returns for individual securities using IBM stock as an example. Price returns are calculated based solely on the closing stock price, while total returns include both capital gains from price changes as well as cash flows like dividends that are reinvested. Using IBM data from Yahoo Finance, the document demonstrates calculating daily price returns and total returns over 2011-2013. It notes that adjusted closing prices from Yahoo Finance incorporate stock splits and dividend payments, making them preferable to use for calculating total returns.
Dear students get fully solved assignments
Send your semester & Specialization name to our mail id :
help.mbaassignments@gmail.com
or
call us at : 08263069601
Bid and Ask Prices Tailored to Traders' Risk Aversion and Gain Propension: a ...Waqas Tariq
Risky asset bid and ask prices “tailored” to the risk-aversion and the gain-propension of the traders are set up. They are calculated through the principle of the Extended Gini premium, a standard method used in non-life insurance. Explicit formulae for the most common stochastic distributions of risky returns, are calculated. Sufficient and necessary conditions for successful trading are also discussed.
Modeling+pricing+strategies+using+game+theory+and+support+vector+machinesMuhammad Akbar Khan
This document discusses modeling pricing strategies in the credit industry using game theory and support vector machines (SVMs). It proposes a model where pricing is determined through a game-theoretic framework modeling competition between companies. Demand is estimated at the customer-level using SVMs to incorporate a large number of variables. This detailed demand estimation is then used in a market-level game theoretic model to understand the strategic interactions between competitors. The model allows companies to gain insights into customer behavior and competitor strategies to inform their own pricing decisions.
Affine Term Structure Model with Stochastic Market Price of RiskSwati Mital
- The document proposes a new affine term structure model that combines principal components analysis with a stochastic market price of risk.
- Principal components provide useful information about yield curves and only three components explain over 95% of yield variation.
- Previous models linked risk premium deterministically to return-predicting factors like slope, but this could result in unrealistic risk premium levels.
- The new model introduces an additional state variable to capture the stochastic market price of risk and break the deterministic link between risk premium and return-predicting factors.
This blog is a place by Apoorva javadekar to discuss various aspects of life. But in specific, it discusses Economics, Mathematics, Politics and sports events.
I like to write about recent economic policies, Socio - political issues, reviews about new books, cricket and fitness. This blog is a good place to take a look at least once a week.
This document summarizes a research paper that examines the optimal investment, consumption, and life insurance selection problem for a wage earner. The problem is modeled using a financial market with one risk-free asset and one risky jump-diffusion asset, along with an insurance market composed of multiple life insurance companies. The goal is to maximize the wage earner's expected utility from consumption during life, wealth at retirement or death, by choosing an optimal investment, consumption, and insurance strategy. The authors use dynamic programming to characterize the optimal solution and prove existence and uniqueness of a solution to the associated nonlinear Hamilton-Jacobi-Bellman equation.
Effects of dividends on common stock prices the nepalese evidencePankajKunwar3
presentation on effects of dividends on common stock prices in the nepelese evidence paper was was published by prof. rade shyam pradhan in the the excellence
The document discusses calculating price returns and total returns for individual securities using IBM stock as an example. Price returns are calculated based solely on the closing stock price, while total returns include both capital gains from price changes as well as cash flows like dividends that are reinvested. Using IBM data from Yahoo Finance, the document demonstrates calculating daily price returns and total returns over 2011-2013. It notes that adjusted closing prices from Yahoo Finance incorporate stock splits and dividend payments, making them preferable to use for calculating total returns.
Dear students get fully solved assignments
Send your semester & Specialization name to our mail id :
help.mbaassignments@gmail.com
or
call us at : 08263069601
This document provides a critical review of the 1996 paper "The Conditional CAPM and the Cross-Section of Expected Returns" by Jagannathan and Wang. The review summarizes the key findings of the original paper, which showed that conditional CAPM can explain the cross-sectional variation in stock returns better than static CAPM. However, the review also notes some limitations in the assumptions around time-varying betas and use of R-squared. Overall, it evaluates the original paper as influential but also discusses subsequent research that built on its findings or identified weaknesses.
11.sales forecast by taking the concept of blue ocean theory using mat lab pr...Alexander Decker
This document summarizes a journal article about using Blue Ocean Theory and Matlab programming to forecast sales. It discusses key concepts from Blue Ocean Theory like red oceans representing existing competitive industries and blue oceans representing new market space. It also covers managerial analysis techniques like break-even analysis, capital asset pricing models, and calculating profit. The conclusion states that introducing new growing sectors using Blue Ocean Theory and Matlab programming allows assessing break-even points, profits/losses, and capital asset prices to determine how products will impact market trends.
Marek Weretka, the principal investigator in the INFERENCE project was a seminar speaker at the Peking University in Shenzhen (China) during The Second PHBS Workshop in Macroeconomics and Finance.
The document discusses pricing interest rate derivatives using the one factor Hull-White short rate model. It begins with an introduction to short rate models and the Hull-White model specifically. It describes how the Hull-White model can be calibrated to market prices by relating its parameter θ to the market term structure. The document then discusses implementing the Hull-White model using trinomial trees and pricing constant maturity swaps.
Peter Warken - Effective Pricing of Cliquet Options - Masters thesis 122015Peter Warken
This thesis examines pricing methods for cliquet options, a type of path-dependent option linked to equity indices. The author develops a semi-closed form pricing formula for cliquet options in a Black-Scholes market and compares it to Monte Carlo simulation. The impact of stochastic volatility and interest rates on pricing is also examined. Numerical experiments illustrate how market parameters affect cliquet option prices. The pricing approach is also applied to similar products like sum cap contracts.
In this paper, the black-litterman model is introduced to quantify investor’s views, then we expanded
the safety-first portfolio model under the case that the distribution of risk assets return is ambiguous. When
short-selling of risk-free assets is allowed, the model is transformed into a second-order cone optimization
problem with investor views. The ambiguity set parameters are calibrated through programming
- The document analyzes forecasting volatility for the MSCI Emerging Markets Index using a Stochastic Volatility model solved with Kalman Filtering. It derives the Stochastic Differential Equations for the model and puts them into State Space form solved with a Kalman Filter.
- Descriptive statistics on the daily returns of the MSCI Emerging Markets Index ETF from 2011-2016 show a mean close to 0, standard deviation of 0.01428, negative skewness, and kurtosis close to a normal distribution. The model will be evaluated against a GARCH model.
The document describes a fuzzy portfolio optimization model using trapezoidal possibility distributions to account for uncertainty in asset returns. The model formulates the portfolio selection problem as a mathematical optimization that maximizes expected return minus risk. Lagrange multipliers and Karush-Kuhn-Tucker conditions are used to derive the optimal solution. Real stock market data is used to provide a numerical example.
This document discusses problem solving with multiple interdependent criteria. It begins by describing how objectives in multi-objective programming problems are typically assumed to be independent, but can actually be interdependent through supportive or inhibitory feedbacks from other objectives. It then provides an example of a negotiation problem between a buyer and seller that involves a third party (government) with multiple objectives that influence and are influenced by the negotiation process, demonstrating interdependent objectives. The document goes on to describe a framework for modeling additive linear interdependencies between objective functions in multi-objective programming problems.
1) The analyst critically reviewed management projections provided for an acquisition target company called Innovation Models.
2) Several projections were deemed unrealistic, including constant high revenue growth, increasing gross margins, and decreasing capex.
3) The analyst made adjustments to projections to reflect more stable 3% revenue growth
This document discusses an optimal inventory model that considers the effect of income and loan availability elasticity on demand. It introduces a concept of elasticity of demand (E1) as the proportionate change in demand due to a proportionate change in income and loan availability. The paper develops models to determine the optimal special order quantity under two conditions: when remnant inventory is zero and when it is finite. It derives equations to find the optimal order quantity and maximum profit based on the elasticity of demand and other factors like supplier price reductions, buyer discounts, income changes, and loan availability. The models aim to help stockists optimize their ordering policies and margins.
A review on various optimization techniques of resource provisioning in cloud...IJECEIAES
Cloud computing is the provision of IT resources (IaaS) on-demand using a pay as you go model over the internet.It is a broad and deep platform that helps customers builds sophisticated, scalable applications. To get the full benefits, research on a wide range of topics is needed. While resource over-provisioning can cost users more than necessary, resource under provisioning hurts the application performance. The cost effectiveness of cloud computing highly depends on how well the customer can optimize the cost of renting resources (VMs) from cloud providers. The issue of resource provisioning optimization from cloud-consumer potential is a complicated optimization issue, which includes much uncertainty parameters. There is a much research avenue available for solving this problem as it is in the real-world. Here, in this paper we provide details about various optimization techniques for resource provisioning.
Welfare effects of fiscal policy in reforming the pension systemGRAPE
Most reforms of the pension systems imply substantial redistribution between cohorts and within a cohort. Fiscal policy, which accompanies these changes may counteract or reinforce this redistribution. Moreover, the literature has argued that the insurance motive implicit in some pension systems plays a major role in determining the welfare eects of the reform: reforms otherwise improving welfare become detrimental to welfare once insurance motive is internalized. We show that this result is not universal, i.e. there exists a variety of scal closures which yield welfare gains and political support for a pension system reform. In an OLG model with uncertainty, we analyze two sets of fiscal adjustments: fiscally neutral adjustments in the pension system (via contribution rate or replacement rate) and balancing pension system by a combination of taxes and/or public debt. We find that fiscally neutral pension system reforms are more likely to yield welfare gains. Many adjustments obtain sufficient political support despite yielding aggregate welfare losses and vice versa. Furthermore, we point to fiscal closures which attenuate and reinforce the relevance of the insurance motive in determining the welfare effects.
Income and substitution effects of estate taxationwarawut ruankham
This file is created for educational purpose not for sale.
Creator of this presentation is highly appreciated the Authors:James R. Hines Jr.
Source: The American Economic Review, Vol. 103, No. 3,
Papers and proceedings of
the one hundred twenty-fifth annual meeting of the American economic ASSOCIATION (MAY 2013), pp. 484-488
The document describes using gradient boosted trees to predict store sales for a competition on Kaggle. Key points:
- Gradient boosted trees were used to build an ensemble of decision trees that could accurately predict store sales based on historical data.
- Hyperparameters like learning rate, tree depth, and regularization were tuned using Bayesian optimization to improve prediction accuracy.
- External data on weather and search trends was incorporated to provide additional context.
- Feature engineering techniques like mean encoding and lagged variables were applied to better handle time series and categorical data.
- The model was evaluated using a 2014 validation set to mimic the 2015 test set and minimize overfitting. Cross validation was used to select the best model.
The document examines optimal operating strategies for income-producing properties to maximize net present value. It establishes a framework where rental rate and operating expenses are set to optimize return on investment, given a demand curve for the property. With a Cobb-Douglas demand curve, closed-form solutions are derived for optimal income ratio, operating expense ratio, and vacancy rate. Comparative statics provide insight into optimal building size and rehabilitation decisions. An empirical case study demonstrates how the model applies in practice.
Increasing the retirement age is analyzed under different pension systems (defined benefit, notional defined contribution, funded defined contribution) using an overlapping generations model. The results show:
1) Increasing the retirement age is efficient and leads to welfare gains for all generations under all systems.
2) Labor supply increases at the aggregate level, though individual labor supply may decrease for those close to retirement.
3) Capital per worker decreases primarily due to reduced precautionary savings rather than labor supply adjustments.
This document introduces the key concepts in econometrics and financial econometrics. It defines econometrics as the application of statistical and mathematical techniques to economic and financial problems. Some examples of problems that can be solved using econometrics are testing market efficiency, modeling volatility, and forecasting correlations. The document discusses the different types of data used in econometrics, including time series, cross-sectional, and panel data. It also covers important financial concepts like returns, deflating nominal values for inflation, and the differences between classical and Bayesian statistical approaches.
Estimating Financial Frictions under LearningGRAPE
The paper studies the implication of initial beliefs and associated confidence under adaptive learning. We first illustrate how prior beliefs determine learning dynamics and the evolution of endogenous variables in a small DSGE model with credit-constrained agents, in which rational expectations are replaced by constant-gain adaptive learning. We then examine how discretionary experimenting with new macroeconomic policies is affected by expectations that agents have in relation to these policies. More specifically, we show that a newly introduced macro-prudential policy that aims at making leverage counter-cyclical can lead to substantial increase in fluctuations under learning, when the economy is hit by financial shocks, if beliefs reflect imperfect information about the policy experiment.
In this paper, we consider an AAI with two types of insurance business with p-thinning dependent
claims risk, diversify claims risk by purchasing proportional reinsurance, and invest in a stock with Heston
model price process, a risk-free bond, and a credit bond in the financial market with the objective of maximizing
the expectation of the terminal wealth index effect, and construct the wealth process of AAI as well as the the
model of robust optimal reinsurance-investment problem is obtained, using dynamic programming, the HJB
equation to obtain the pre-default and post-default reinsurance-investment strategies and the display expression
of the value function, respectively, and the sensitivity of the model parameters is analyzed through numerical
experiments to obtain a realistic economic interpretation. The model as well as the results in this paper are a
generalization and extension of the results of existing studies.
Optimal Portfolio Selection for a Defined Contribution Pension Fund with Retu...BRNSS Publication Hub
This paper investigates the optimal investment strategies for a defined contribution pension fund with return clauses of premiums with interest under the mean-variance criterion. Using the actuarial symbol, we formalize the problem as a continuous time mean-variance stochastic optimal control. The pension fund manager considers investments in risk and risk-free assets to increase the remaining accumulated funds to meet the retirement needs of the remaining members. Using the variational inequalities methods, we established an optimized problem from the extended Hamilton–Jacobi–Bellman Equations and solved the optimized problem to obtain the optimal investment strategies for both risk-free and risky assets and also the efficient frontier of the pension member. Furthermore, we evaluated analytically and numerically the effect of various parameters of the optimal investment strategies on it. We observed that the optimal investment strategy for the risky asset decreases with an increase in the risk-averse level, initial wealth, and the predetermined interest rate.
This document provides a critical review of the 1996 paper "The Conditional CAPM and the Cross-Section of Expected Returns" by Jagannathan and Wang. The review summarizes the key findings of the original paper, which showed that conditional CAPM can explain the cross-sectional variation in stock returns better than static CAPM. However, the review also notes some limitations in the assumptions around time-varying betas and use of R-squared. Overall, it evaluates the original paper as influential but also discusses subsequent research that built on its findings or identified weaknesses.
11.sales forecast by taking the concept of blue ocean theory using mat lab pr...Alexander Decker
This document summarizes a journal article about using Blue Ocean Theory and Matlab programming to forecast sales. It discusses key concepts from Blue Ocean Theory like red oceans representing existing competitive industries and blue oceans representing new market space. It also covers managerial analysis techniques like break-even analysis, capital asset pricing models, and calculating profit. The conclusion states that introducing new growing sectors using Blue Ocean Theory and Matlab programming allows assessing break-even points, profits/losses, and capital asset prices to determine how products will impact market trends.
Marek Weretka, the principal investigator in the INFERENCE project was a seminar speaker at the Peking University in Shenzhen (China) during The Second PHBS Workshop in Macroeconomics and Finance.
The document discusses pricing interest rate derivatives using the one factor Hull-White short rate model. It begins with an introduction to short rate models and the Hull-White model specifically. It describes how the Hull-White model can be calibrated to market prices by relating its parameter θ to the market term structure. The document then discusses implementing the Hull-White model using trinomial trees and pricing constant maturity swaps.
Peter Warken - Effective Pricing of Cliquet Options - Masters thesis 122015Peter Warken
This thesis examines pricing methods for cliquet options, a type of path-dependent option linked to equity indices. The author develops a semi-closed form pricing formula for cliquet options in a Black-Scholes market and compares it to Monte Carlo simulation. The impact of stochastic volatility and interest rates on pricing is also examined. Numerical experiments illustrate how market parameters affect cliquet option prices. The pricing approach is also applied to similar products like sum cap contracts.
In this paper, the black-litterman model is introduced to quantify investor’s views, then we expanded
the safety-first portfolio model under the case that the distribution of risk assets return is ambiguous. When
short-selling of risk-free assets is allowed, the model is transformed into a second-order cone optimization
problem with investor views. The ambiguity set parameters are calibrated through programming
- The document analyzes forecasting volatility for the MSCI Emerging Markets Index using a Stochastic Volatility model solved with Kalman Filtering. It derives the Stochastic Differential Equations for the model and puts them into State Space form solved with a Kalman Filter.
- Descriptive statistics on the daily returns of the MSCI Emerging Markets Index ETF from 2011-2016 show a mean close to 0, standard deviation of 0.01428, negative skewness, and kurtosis close to a normal distribution. The model will be evaluated against a GARCH model.
The document describes a fuzzy portfolio optimization model using trapezoidal possibility distributions to account for uncertainty in asset returns. The model formulates the portfolio selection problem as a mathematical optimization that maximizes expected return minus risk. Lagrange multipliers and Karush-Kuhn-Tucker conditions are used to derive the optimal solution. Real stock market data is used to provide a numerical example.
This document discusses problem solving with multiple interdependent criteria. It begins by describing how objectives in multi-objective programming problems are typically assumed to be independent, but can actually be interdependent through supportive or inhibitory feedbacks from other objectives. It then provides an example of a negotiation problem between a buyer and seller that involves a third party (government) with multiple objectives that influence and are influenced by the negotiation process, demonstrating interdependent objectives. The document goes on to describe a framework for modeling additive linear interdependencies between objective functions in multi-objective programming problems.
1) The analyst critically reviewed management projections provided for an acquisition target company called Innovation Models.
2) Several projections were deemed unrealistic, including constant high revenue growth, increasing gross margins, and decreasing capex.
3) The analyst made adjustments to projections to reflect more stable 3% revenue growth
This document discusses an optimal inventory model that considers the effect of income and loan availability elasticity on demand. It introduces a concept of elasticity of demand (E1) as the proportionate change in demand due to a proportionate change in income and loan availability. The paper develops models to determine the optimal special order quantity under two conditions: when remnant inventory is zero and when it is finite. It derives equations to find the optimal order quantity and maximum profit based on the elasticity of demand and other factors like supplier price reductions, buyer discounts, income changes, and loan availability. The models aim to help stockists optimize their ordering policies and margins.
A review on various optimization techniques of resource provisioning in cloud...IJECEIAES
Cloud computing is the provision of IT resources (IaaS) on-demand using a pay as you go model over the internet.It is a broad and deep platform that helps customers builds sophisticated, scalable applications. To get the full benefits, research on a wide range of topics is needed. While resource over-provisioning can cost users more than necessary, resource under provisioning hurts the application performance. The cost effectiveness of cloud computing highly depends on how well the customer can optimize the cost of renting resources (VMs) from cloud providers. The issue of resource provisioning optimization from cloud-consumer potential is a complicated optimization issue, which includes much uncertainty parameters. There is a much research avenue available for solving this problem as it is in the real-world. Here, in this paper we provide details about various optimization techniques for resource provisioning.
Welfare effects of fiscal policy in reforming the pension systemGRAPE
Most reforms of the pension systems imply substantial redistribution between cohorts and within a cohort. Fiscal policy, which accompanies these changes may counteract or reinforce this redistribution. Moreover, the literature has argued that the insurance motive implicit in some pension systems plays a major role in determining the welfare eects of the reform: reforms otherwise improving welfare become detrimental to welfare once insurance motive is internalized. We show that this result is not universal, i.e. there exists a variety of scal closures which yield welfare gains and political support for a pension system reform. In an OLG model with uncertainty, we analyze two sets of fiscal adjustments: fiscally neutral adjustments in the pension system (via contribution rate or replacement rate) and balancing pension system by a combination of taxes and/or public debt. We find that fiscally neutral pension system reforms are more likely to yield welfare gains. Many adjustments obtain sufficient political support despite yielding aggregate welfare losses and vice versa. Furthermore, we point to fiscal closures which attenuate and reinforce the relevance of the insurance motive in determining the welfare effects.
Income and substitution effects of estate taxationwarawut ruankham
This file is created for educational purpose not for sale.
Creator of this presentation is highly appreciated the Authors:James R. Hines Jr.
Source: The American Economic Review, Vol. 103, No. 3,
Papers and proceedings of
the one hundred twenty-fifth annual meeting of the American economic ASSOCIATION (MAY 2013), pp. 484-488
The document describes using gradient boosted trees to predict store sales for a competition on Kaggle. Key points:
- Gradient boosted trees were used to build an ensemble of decision trees that could accurately predict store sales based on historical data.
- Hyperparameters like learning rate, tree depth, and regularization were tuned using Bayesian optimization to improve prediction accuracy.
- External data on weather and search trends was incorporated to provide additional context.
- Feature engineering techniques like mean encoding and lagged variables were applied to better handle time series and categorical data.
- The model was evaluated using a 2014 validation set to mimic the 2015 test set and minimize overfitting. Cross validation was used to select the best model.
The document examines optimal operating strategies for income-producing properties to maximize net present value. It establishes a framework where rental rate and operating expenses are set to optimize return on investment, given a demand curve for the property. With a Cobb-Douglas demand curve, closed-form solutions are derived for optimal income ratio, operating expense ratio, and vacancy rate. Comparative statics provide insight into optimal building size and rehabilitation decisions. An empirical case study demonstrates how the model applies in practice.
Increasing the retirement age is analyzed under different pension systems (defined benefit, notional defined contribution, funded defined contribution) using an overlapping generations model. The results show:
1) Increasing the retirement age is efficient and leads to welfare gains for all generations under all systems.
2) Labor supply increases at the aggregate level, though individual labor supply may decrease for those close to retirement.
3) Capital per worker decreases primarily due to reduced precautionary savings rather than labor supply adjustments.
This document introduces the key concepts in econometrics and financial econometrics. It defines econometrics as the application of statistical and mathematical techniques to economic and financial problems. Some examples of problems that can be solved using econometrics are testing market efficiency, modeling volatility, and forecasting correlations. The document discusses the different types of data used in econometrics, including time series, cross-sectional, and panel data. It also covers important financial concepts like returns, deflating nominal values for inflation, and the differences between classical and Bayesian statistical approaches.
Estimating Financial Frictions under LearningGRAPE
The paper studies the implication of initial beliefs and associated confidence under adaptive learning. We first illustrate how prior beliefs determine learning dynamics and the evolution of endogenous variables in a small DSGE model with credit-constrained agents, in which rational expectations are replaced by constant-gain adaptive learning. We then examine how discretionary experimenting with new macroeconomic policies is affected by expectations that agents have in relation to these policies. More specifically, we show that a newly introduced macro-prudential policy that aims at making leverage counter-cyclical can lead to substantial increase in fluctuations under learning, when the economy is hit by financial shocks, if beliefs reflect imperfect information about the policy experiment.
In this paper, we consider an AAI with two types of insurance business with p-thinning dependent
claims risk, diversify claims risk by purchasing proportional reinsurance, and invest in a stock with Heston
model price process, a risk-free bond, and a credit bond in the financial market with the objective of maximizing
the expectation of the terminal wealth index effect, and construct the wealth process of AAI as well as the the
model of robust optimal reinsurance-investment problem is obtained, using dynamic programming, the HJB
equation to obtain the pre-default and post-default reinsurance-investment strategies and the display expression
of the value function, respectively, and the sensitivity of the model parameters is analyzed through numerical
experiments to obtain a realistic economic interpretation. The model as well as the results in this paper are a
generalization and extension of the results of existing studies.
Optimal Portfolio Selection for a Defined Contribution Pension Fund with Retu...BRNSS Publication Hub
This paper investigates the optimal investment strategies for a defined contribution pension fund with return clauses of premiums with interest under the mean-variance criterion. Using the actuarial symbol, we formalize the problem as a continuous time mean-variance stochastic optimal control. The pension fund manager considers investments in risk and risk-free assets to increase the remaining accumulated funds to meet the retirement needs of the remaining members. Using the variational inequalities methods, we established an optimized problem from the extended Hamilton–Jacobi–Bellman Equations and solved the optimized problem to obtain the optimal investment strategies for both risk-free and risky assets and also the efficient frontier of the pension member. Furthermore, we evaluated analytically and numerically the effect of various parameters of the optimal investment strategies on it. We observed that the optimal investment strategy for the risky asset decreases with an increase in the risk-averse level, initial wealth, and the predetermined interest rate.
This document summarizes a research article that investigates optimal investment strategies for a defined contribution pension fund with return clauses for premiums paid with a predetermined interest rate under a mean-variance framework. The researchers develop a continuous-time stochastic optimal control model to formalize the problem. They obtain the optimal investment strategies for risk-free and risky assets by solving the extended Hamilton-Jacobi-Bellman equations using variational inequality methods. The optimal strategies are affected by parameters like risk aversion, initial wealth, and interest rate.
This document presents research on optimal portfolio selection for a defined contribution pension fund. It considers investments in risk-free and risky assets to maximize the remaining accumulated funds for retirement. Using variational inequality methods, the researchers establish an optimized problem from the Hamilton-Jacobi-Bellman equations and solve it to obtain the optimal investment strategies. They find that the optimal investment in the risky asset decreases with increases in risk aversion, initial wealth, and the predetermined interest rate. The researchers also evaluate the effect of various parameters on the optimal investment strategies analytically and numerically.
1) This document examines optimal taxation and policy selection within a rational expectations framework using dynamic models with capital.
2) It finds that standard control theory techniques cannot be used to solve for the optimal taxation policy over time, as the optimal policy would be time inconsistent - what was optimal in the past would not be optimal now.
3) It develops a novel recursive method to overcome this difficulty by determining both the constraint set and value function recursively, even though there is doubt the optimal policy could actually be implemented due to its time inconsistency. The optimal policy still provides a benchmark to evaluate alternative time consistent policies.
GEORGE MASON UNIVERSITYSchool of ManagementEMBA 703 Financia.docxbudbarber38650
GEORGE MASON UNIVERSITY
School of Management
EMBA 703 Financial Markets
Dr. Hanweck
Final Examination
Fall 2013
NAME: ___________________________________
G-code: _____________________________
Answer all questions. Place your answer to each question on a separate sheet of paper. Please write your name on the top left corner of each page. Document your answers and show your work. Read each question carefully and answer all parts. Try and answer something on each question. Your guess may turn out to be correct. The number in parentheses is the point weight for the question. Attach the exam to your answers.
(15)
1.(a)
Discuss various measures of capital market efficiency and how efficient capital markets contribute to the efficiency in the market for goods and services (including productive capital). As part of your discussion, consider the implications of the fact that the bulk of trading in capital markets is in outstanding securities and analyze the meaning of the terms "depth," "breadth," and "resiliency" as descriptions of capital markets. Include in your discussion the types of legislative and regulatory reforms that might be or have recently been instituted in order to improve the efficiency of capital markets and the role of "insider trading" and the SEC as they affect market efficiency.
(b)
Compare money and capital markets and identify the major issuers of securities in the different markets and the difference among the various types of securities within and between each of the markets. Within your discussion of the money markets include a consideration of the role of the Federal Reserve System (Fed) and the banking system as they interact through required reserve maintenance, needs for liquidity and monetary policy actions by the Fed. Consider in your analysis the types and significance of the links between the money and capital markets via the term structure of interest rates, issuers of debt and equity and the presence of interest rate and credit risk derivatives.
(10)
2. There are a number of theories of the term structure of interest rates including the unbiased expectations hypothesis, preferred habitat hypothesis, and market segmentation hypothesis. Discuss the implications of the unbiased expectations hypothesis within the context of the following problem. Problem 1: For a two year, default free, zero coupon security, compute its yield to maturity and draw the respective yield curves assuming two different expectations of inflation employing the Fisher Effect and the data below: (a) 4 percent one year from now, and (b) 2 percent one year from now. In addition, define and compute the implied forward yield on a one year security one year from now, assuming the current two year yield is 6.0 percent. Discuss the assumptions underlying this calculation and how it can be used to evaluate the implied forward yield on a 1-year loan, next year. (c) Wh.
Optimal Policy Response to Booms and Busts in Credit and Asset PricesSambit Mukherjee
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The Equity Premium Puzzle undera General Utility Function
1. International Journal of Business Marketing and Management (IJBMM)
Volume 5 Issue 3 March 2020, P.P. 37-46
ISSN: 2456-4559
www.ijbmm.com
International Journal of Business Marketing and Management (IJBMM) Page 37
The Equity Premium Puzzle undera General Utility Function
Yulin Zhou,Peibiao Zhao
School of Science, Nanjing University of Science and Technology,
Nanjing 210094, Jiangsu, P. R. China
Abstract:In this paper, we consider the equity premium puzzle under a general utility function. We derive that
the optimal strategy under a general utility function approximate the optimal strategy under the special utility
function. This result posed in the present paper can be regarded as a generalization of the work by Gong and
Zou [13]
JEL classification numbers: D9;G1;E1
Keywords:The equity premium puzzle; Asset pricing; The spirit of capitalism; Stochastic growth model.
I. Introduction
The equity premium puzzle was first put forward by MrMehra and Prescott [1] in 1985, through an
analysis of the American historical data over the past more than a century, they found that the return rate of
stocks is 7.9%, and the return rate of the corresponding risk-free securities is only 1%, the premium is 6.9%.
Since Mr. Mehra and Prescott proposed the puzzle of equity premium in 1985, this puzzle has attracted
a lot of scholars’ attention. Equity premium is the compensation for stock investors to bear stock risks. The
mystery of equity premium reveals a contradiction, that is, there is a big gap between the traditional asset
pricing theory and the actual return and price of assets, so it is of great significance to study equity premium.
Measuring and explaining equity premium is one of the most challenging research work in asset pricing theory
of modern financial economics. The equity premium puzzle was usually explained by finding the
inconsistencies with realities from the theoretical model, or by finding the causes and solutions to the equity
premium puzzle from the empirical aspects. We mainly focus on the first aspect to study the equity premium
puzzle.
Benartzi and Thaler [2], Barberis, Huang and Santos [3] explained the equity premium puzzle with the
prospect theory; while Camerer and Weber [4], Maenhout [5] explained it by the Ellsberg Paradox theory.
What’s more, Constantinides, Donaldson and Mehra [6] studied the asset pricing with limitations on borrowing
to explain the equity premium puzzle. McGrattan and Prescott [7] explained the equity premium by the change
of individual income tax rate. Heaton and Lucas [8, 9] researched the equity premium based on the infinite-
horizon model. Dai and Li [10] explained the equity premium under the consideration of investors’ general risk
aversion. Brad Barber and Odean [11] thought that the equity premium puzzle by virtue of the return rate of
stocks under the markets with various costs. Rietz [12] explained the equity premium by introducing the small
probability events which caused a decrease in consumption. Using the stochastic optimal control theory, Gong
and Zou [13] studied the equity premium puzzle based on the domestic output process satisfying a geometric
brownian motion, and derived the explicit solutions to the growth rates of consumption and savings and
equilibrium on all assets under the original special utility function.
Most of the classic equity premium is based on a specific utility function, which gives a reasonable
explanation of equity premium through the optimal strategy conversion in the sense of equilibrium. A natural
question is whether we can also study the equity premium by analogy under the general utility function?
Based on this consideration, this paper extends the utility function to the general utility function to
study the equity premium problem. Using the stochastic optimal control theory and the approach theory, this
paper studies and gives a reasonable explanation of equity premium.
2. The Equity Premium Puzzle Under A General Utility Function
International Journal of Business Marketing and Management (IJBMM) Page 38
In this paper, we first study an investment strategy problem based on the domestic output process
satisfying geometric Brownian motion, and solve the optimal investment strategy of investors in the sense of
equilibrium market. It is proved that the optimal strategy under the general utility function can approach the
optimal strategy under the special utility function;Secondly, with the help of the above optimal strategy, this
paper calculates the expected return level of financial assets, and further obtains the corresponding premium
level. Therefore, we can get the corresponding equity premium level by tracking the optimal strategy of
investors. In other words, we prove that the level of equity premium under general utility function approaches to
that under special utility.Finally, this paper gives an example of Equity Premium Analysis and sensitivity
analysis. It is proved that the existence of the spirit of capitalism leads to the difference between the return rate
of risk-free assets and that of risk-free assets. After that, this paper uses the established theoretical model to
estimate the premium level of China’s A-share market.
II. Equity Premium Puzzle Model
Consider a financial market consisting of two assets:the government bond B, and the capital stock K.
The representative agent’s wealth Wthas a representation
W(t) = B(t) + K(t).
Define
,
then we have 𝑛 𝐵 + 𝑛 𝐾 = 1, where nBand nKshow the proportion of the wealth invested on the bond and the
capital
We next assume that the output Y and the government expenditure G are modelled by stochastic
differential equations as follows [14, 15]:
𝑑𝑌 = 𝛼𝐾𝑑𝑡 + 𝛼𝐾𝑑𝑦, (2.1)
𝑑𝐺 = 𝑔𝛼𝐾𝑑𝑡 + 𝛼𝐾𝑑𝑧, (2.2)
whereα is the marginal physical product of the capital stock K, 𝑑𝑦 is temporally independent, normally
distributed, and satisfies
;
besides, g is the percentage of government expenditure accounting for output,𝑑𝑧is temporally independent,
normally distributed, and satisfies
.
If the inflation rate is stochastic, then the return on the government bond B will also satisfy a stochastic process.
In the period of time dt, the stochastic real rate of the return on the bond B, 𝑑𝑅 𝐵 is assumed to be
𝑑𝑅 𝐵 = 𝑟𝐵 𝑑𝑡 + 𝑑𝑢 𝐵. (2.3)
where the macroeconomic equilibrium determines 𝑟𝐵and 𝑑𝑢 𝐵endogenously.
As for the second asset, the capital stock K, with Equation (2.1), the stochastic real rate of return on capital
can be given by
. (2.4)
Without loss of generality, the taxes are levied according to the capital income and the consumption c, i.e.
𝑑𝑇 = (𝜏𝑟𝐾 𝐾 + 𝜏 𝑐 𝑐)𝑑𝑡 + 𝜏′
𝐾𝑑𝑢 𝐾 = (𝜏𝛼𝐾 + 𝜏 𝑐 𝑐𝑡)𝑑𝑡 + 𝜏′
𝛼𝐾𝑑𝑦 (2.5)
whereτ, τ are the tax rates on the deterministic component and the stochastic component of the capital income,
respectively, and 𝜏 𝑐is the tax rate on the consumption.
In the absence of lump-sum taxation, government budget constraint can be described as
3. The Equity Premium Puzzle Under A General Utility Function
International Journal of Business Marketing and Management (IJBMM) Page 39
𝑑𝐵 = 𝐵𝑑𝑅 𝐵 + 𝑑𝐺 − 𝑑𝑇
A balanced product market requires
𝑑𝐾 = 𝑑𝑌 − 𝑐𝑑𝑡 − 𝑑𝐺, (2.6)
Substituting (2.1),(2.2),(2.3) and (2.5) into (2.6), we have
𝑑𝐾
𝐾
= [𝛼(1 − 𝑔) −
𝑐
𝑛 𝐾 𝑊
]𝑑𝑡 + 𝛼(𝑑𝑦 − 𝑑𝑧) = 𝜙𝑑𝑡 + 𝛼(𝑑𝑦 − 𝑑𝑧), (2.7)
With the assumptions above, the representative agent’s goal is to maximize his expected utility subject to the
budget constraint by choosing the ratio of consumption to wealth,
𝑐
𝑊
, and the proportions of investment, 𝑛 𝐵and
𝑛 𝐾. Then the problem can be described as
𝑚𝑎𝑥𝐸 ∫ 𝑢
∞
0
(𝑐𝑡, 𝑊𝑡)𝑒−𝛽𝑡
𝑑𝑡
s.t.
𝑑𝑊𝑡 = (𝑛 𝐵 𝑊𝑡 𝑟𝐵 + 𝑛 𝐾 𝑊𝑡(1 − 𝜏)𝑟𝐾 − (1 + 𝜏 𝑐)𝑐𝑡)𝑑𝑡 + 𝑊𝑡 𝑑𝜔. (2.8)
𝑛 𝐵 + 𝑛 𝐾 = 1 (2.9)
Where 𝛽is the time discount rate, 𝑢(𝑐, 𝑊) is the utility function with consumption c, 𝑑𝜔 = 𝑛 𝐵 𝑑𝑢 𝐵 + 𝑛 𝐾(1 −
𝜏′
)𝑑𝑢 𝐾.
This chapter mainly establishes and solves a domestic production process based on geometric
Brownian motion.The optimal strategy of portfolio.First of all, according to the government’s income and
expenditure balance.In this way, the equilibrium market investment group is obtained question. Then, according
to the principle of dynamic programming,the problem of portfolio optimization is solved in the sense of
equilibrium market.The investment strategies of investors and a series of dynamic descriptions of the
equilibrium system of the market are obtained.
III. Optimal solutions to the stochastic optimal control problem
To solve the agent’s optimization problem above, we define the value function
By the dynamic programming maximal principle, we can get the following Bellman equation
= 0 (3.1)
where 𝜌 = 𝑛 𝐵 𝑟𝐵 + 𝑛 𝐾(1 − 𝜏)𝑟𝐾, and it is the expected after-tax return on total property. Taking differentiation
of the Bellman equation with respect to c, 𝑛 𝐵and 𝑛 𝐾respectively, then we have
Proposition 3.1.The first-order conditions for the optimization problem can be written as follows
𝜕𝑢(𝑐,𝑊)
𝑐
= (1 + 𝜏 𝑐)𝑉 𝑊, (3.2)
(𝑟𝐵 𝑉 𝑊 𝑊 − 𝜂)𝑑𝑡 + 𝑐𝑜𝑣(𝑑𝜔, 𝑑𝑢 𝐵)𝑉 𝑊𝑊 𝑊2
= 0, (3.3)
((1 − 𝜏)𝑟𝐾 𝑉 𝑊 𝑊 − 𝜂)𝑑𝑡 + 𝑐𝑜𝑣(𝑑𝜔, (1 − 𝜏′
)𝑑𝑢 𝐾)𝑉 𝑊𝑊 𝑊2
= 0, (3.4)
𝑛 𝐵 + 𝑛 𝐾 = 1, (3.5)
whereη is the Lagrangian multiplier with the portfolio selection constraint (2.9).
4. The Equity Premium Puzzle Under A General Utility Function
International Journal of Business Marketing and Management (IJBMM) Page 40
Furthermore, with equation (2.7), we have
Proposition 3.2.The equilibrium system of the economy can be summarized as
Proposition 3.3.The normal fluctuation component of the stochastic return of bonds, duB, and the total wealth,
dw, are decided by the following formulas
] (3.6)
𝑑𝜔 = 𝛼(𝑑𝑦 − 𝑑𝑧) (3.7)
Proof.According to the inter-temporal invariance of portfolio shares [17], we have
(3.8)
i.e., all the real assets grow at a common stochastic rate. Combining with (2.7), (2.8), (3.6) and (3.8), we get
Noticing the fact 𝑛 𝐵 + 𝑛 𝐾 = 1, we will get
This ends the Proof of Proposition 3.3.
These two equations enable us to compute all the necessary covariances and variances in the full equilibrium
system.
With Proposition 3.3, we have
Noticing equations (3.3) and (3.4), we can get
Proposition 3.4.The mean return on bonds is
] (3.9)
and the stochastic growth rate of the economy is
5. The Equity Premium Puzzle Under A General Utility Function
International Journal of Business Marketing and Management (IJBMM) Page 41
(3.10)
where 𝜌 = 𝑛 𝐵 𝑟𝐵 + 𝑛 𝐾(1 − 𝜏)𝑟𝐾.
With Proposition 3.4, we now have our main theorem of this section
Theorem 3.5.The explicit solutions of the economic system are
(3.11)
𝑛 𝐵 = 1 − 𝑛 𝐾. (3.12)
Proof.Notice the conditions
.
Combining Equations (3.9) and (3.10) to solve nK, we can easily get (3.11).
Next we specify the utility function as in Bakshi and Chen [18]
(3.13)
whereγ >0, and γ ≥ 1 when λ ≥ 0, and λ <0 otherwise.
Under the form of the utility function in (3.13), we have
Proposition 3.6.The first-order optimal conditions are
𝑐
𝑊
=
𝛽+
1
2
𝜎 𝜔
2 (1−𝛾−𝜆)(𝛾+𝜆)−𝜌(1−𝛾−𝜆)
(𝛾+𝜏 𝑐)(1−𝛾−𝜆)/(1−𝛾)
(3.14)
(𝑟𝐵 −
𝜂
𝛿(1−𝛾−𝜆)𝑊1−𝛾−𝜆)𝑑𝑡 = (𝛾 + 𝜆)𝑐𝑜𝑣(𝑑𝜔, 𝑑𝑢 𝐵) (3.15)
(𝑟𝐾(1 − 𝜏) −
𝜂
𝛿(1−𝛾−𝜆)𝑊1−𝛾−𝜆)𝑑𝑡 = (𝛾 + 𝜆)(1 − 𝜏′
)𝑐𝑜𝑣(𝑑𝜔, 𝑑𝑢 𝐾) (3.16)
and Equation (3.5). whereη is the Lagrangian multiplier with the portfolio selection constraint (2.9), and
𝜌 = 𝑛 𝐵 𝑟𝐵 + 𝑛 𝐾(1 − 𝜏)𝑟𝐾, (3.17)
𝜔 = 𝑛 𝐵 𝑑𝑢 𝐵 + 𝑛 𝐾(1 − 𝜏′
)𝑑𝑢 𝐾, (3.18)
𝜎 𝜔
2
= 𝑛 𝐵
2
𝜎 𝐵
2
+ 𝑛 𝐾
2
(1 − 𝜏′
)2
𝜎 𝐾
2
+ 2𝑛 𝐵 + 𝑛 𝐾(1 − 𝜏′
)𝜎 𝐵𝐾, (3.19)
With Proposition 3.3, (3.15) and (3.16), we have
Proposition 3.7.Under the form of the utility function in (3.14), the mean return on bonds and the stochastic
growth rate of the economy are
] (3.20)
6. The Equity Premium Puzzle Under A General Utility Function
International Journal of Business Marketing and Management (IJBMM) Page 42
(3.21)
Similar to Theorem 3.5, we have
Theorem 3.8.Under the form of the utility function in (3.14), the explicit solutions of the economic system
are
(3.22)
(3.23)
(3.24)
where𝑛 𝐾
∗
and 𝑛 𝐵
∗
are the optimal proportions of investment under the utility function (3.13).
IV. Approximation of strategy under general utility
As we discussed in the previous sections, for the equity premium puzzle with utility function u(c,W) =
, the optimal strategy is to invest a constant proportion of wealth in the risk asset. It is in general
difficult to find the optimal strategy for a general utility, however, if the utility displays the behavior of a power
utility when the level of wealth is very high, then the following Merton strategy can still approximately achieve
the optimal value, irrespective of the initial wealth level, as long as the investment horizon is sufficient long, i.e.
t → ∞.
Assume that the investor consumes at a fixed proportion of wealth if the investment horizon is
sufficient long, that is, lim𝑡→∞
𝑐
𝑊
= 𝑘, where 0 < k <1 is defined as (3.22). We therefore need to estimate the
difference between the optimal strategy under special utility function nKand the optimal strategy under general
utility function , that is
(4.1)
where 𝐴(𝑊) = [𝛼(𝜏 − 𝑔) + (1 + 𝜏 𝑐)𝑘 +
𝛼2 𝑊𝑉 𝑊𝑊
𝑉 𝑊
(𝜏′
𝜎 𝑦
2
+ 𝜎𝑧
2
)][𝛼(𝜏 − 𝑔) + (1 + 𝜏 𝑐)𝑘 − 𝛼2
(𝛾 + 𝜆)(𝜏′
𝜎 𝑦
2
+
𝜎𝑧
2
)] > 0
It’s easy to deduce that if lim𝑡→∞ 𝑊𝑉 𝑊𝑊 + (𝛾 + 𝜆)𝑉 𝑊 → 0, then we have lim 𝑡→∞ 𝑛 𝐾 − 𝑛 𝐾
∗
→ 0.
The next result is referred to Bian and Zheng [18]
Lemma 4.1.Let w ∈C1,2
(R+ × R) be a solution to equation
) (4.2)
Let 𝜓(𝑥) = 𝑒 𝛼𝑥
𝜙(𝑥), with constant α >0. Assume that ψ ∈C1
(R), and
(4.3)
for some constants K ≥ 1 and q <1. Then we have, for x ≥ 0,
) (4.4)
Where𝐿0(𝑡) = 𝑒 𝛼2 𝑎2 𝑡
+ 𝑒(𝛼−𝑞)2 𝑎2 𝑡
. Furthermore, we have
(4.5)
7. The Equity Premium Puzzle Under A General Utility Function
International Journal of Business Marketing and Management (IJBMM) Page 43
where the convergence is uniform for 𝑡 ∈ [𝑡0, 𝑡1] with any 0 < 𝑡0 < 𝑡1.
Let 𝑤(𝑡, 𝑥) = 𝑒−(𝛼𝑥+𝛽𝑡)
𝑣(𝑡, 𝑒−𝑥
). Then w is a solution to (4.2) with the initial condition 𝑤(0, 𝑥) = 𝜙(𝑥) =
𝑒−𝛼𝑥
𝑣(𝑒−𝑥
), thus we have
Lemma 4.2.Assume that u ∈C1
(R) and satisfies
lim
𝑦→∞
𝑢′(𝑦)
𝑦−𝛾−𝜆 = 1, 𝑦𝑢′
(𝑦) ≤ {
𝐾𝑒1−𝛾−𝜆
, 𝑦 ≤ 1
𝐾, 𝑦 ≥ 1
(4.6)
Then we have that for𝑦 ≥ 1,
, (4.7)
Furthermore
(4.8)
where the convergence is uniform f or 𝑡 ∈ [𝑡0, 𝑡1] with any 0 < 𝑡0 < 𝑡1.
The next result shows the approximation of strategy under general utility.
Theorem 4.3.Assume that and
(4.9)
holds, where C ∈R. Then we have, for W >0
(4.10)
Proof.According to (4.7) and (4.8), we get
.
For any fixed 𝜖 > 0, there is 𝛿 = 𝛿(𝜖) > 0, such that
. (4.11)
Define
,
for∀(𝑡, 𝑊) ∈ [𝑡0, 𝑡1] × 𝑅+with any 𝑡1 > 𝑡0. Then w satisfies the following equation and boundary conditions
{
𝜕𝑤
𝜕𝑡
+
1
2
𝜎 𝜔
2
𝑊2
𝑤 𝑊𝑊 + (𝜙 + 𝜎 𝜔
2
𝑊)𝑤 𝑊 + 𝜙𝑤 = 𝜙𝜖, (𝑡, 𝑊) ∈ (𝑡0, 𝑡1) × 𝑅+,
𝑤(𝑡0, 𝑊) > 0, 𝑊 ∈ 𝑅+,
𝑙𝑖𝑚𝑖𝑛𝑓 𝑊→0 𝑤(𝑡, 𝑊) > 0, 𝑙𝑖𝑚𝑖𝑛𝑓 𝑊→∞ 𝑤(𝑡, 𝑊) > 0, 𝑡 ∈ [𝑡0, 𝑡1].
(4.12)
By the maximum principle, we conclude that w(t,W) >0 for all (t,y) ∈ [t0,t1] × R+, which implies
𝑊𝑉 𝑊𝑊(𝑡, 𝑊) + (𝛾 + 𝜆)𝑉 𝑊(𝑡, 𝑊) ≤ 𝜖(𝑊−𝛾−𝜆
𝑒 𝜉(𝑡−𝑡0)
+ 1) + 𝛿𝑒−𝑟(−𝛾−𝜆)(𝑡−𝑡0)
(4.13)
Let t → ∞ and 0, we have
lim
𝑡→∞
𝑊𝑉 𝑊𝑊(𝑡, 𝑊) + (𝛾 + 𝜆)𝑉 𝑊(𝑡, 𝑊) = 0 (4.14)
By (4.1), we can arrive equation (4.10). This ends the Proof of Theorem 4.3.
Example 4.4.Define
8. The Equity Premium Puzzle Under A General Utility Function
International Journal of Business Marketing and Management (IJBMM) Page 44
A simple calculus gives that the utility functions above satisfies the conditions of Theorem 4.3 with γ = 3 and λ =
2, and (4.10) holds under this utility function.
On the basis of the optimal investment strategy in the equilibrium market obtained in the previous
chapter, this chapter proves that under a kind of general utility function which is connected with the original
power utility function, we can also get the investment strategy similar to the original portfolio optimization
problem. In other words, under this kind of generalized utility function, the optimal strategy is to invest the fixed
proportion of wealth in risk assets and risk-free assets respectively. On the premise of these assumptions, we use
the optimal investment strategy to estimate the expected rate of return of bonds and equity in the market,
describe the risk-free rate of return, and calculate the level of equity premium by changing the energy, so that we
can connect the approximation of the optimal strategy with the approximation of equity premium in the
following, which can explain the mystery of equity premium under the general utility function.
V. The explaination of the equity premium puzzle
We will explain the equity premium puzzle in this section. With the investor’s optimal strategy, we can
calculate the equity premium that the investor accepts. For simplicity, we set consumption tax,τc, to zero. With
the optimal strategy that we got in the previous sections, the return rate on the market portfolio Q = nBW+
nKWcan be represented as
Then we have
Propositin 5.1.The equilibrium asset-pricing relationships are
𝑟𝐵 −
𝜂
𝑊𝑉 𝑊
= 𝛽 𝐵(𝑟𝑄 −
𝜂
𝑊𝑉 𝑊
) (5.1)
(1 − 𝜏)𝑟𝐾 −
𝜂
𝑊𝑉 𝑊
= 𝛽 − 𝐾(𝑟𝑄 −
𝜂
𝑊𝑉 𝑊
) (5.2)
where
Proof.From Proposition 3.1.and equation (3.11),we have
𝜂
𝑊𝑉 𝑊
= 𝛼(1 − 𝜏) +
𝛼2
𝑊𝑉 𝑊𝑊
𝑉 𝑊
(1 − 𝜏′
)𝜎 𝑦
2
Since
𝑟𝑄 = 𝑟𝐵 𝑛 𝐵 + 𝑟𝐾(1 − 𝜏)𝑛 𝐾 = 𝛼(1 − 𝜏) −
𝛼2
𝑊𝑉 𝑊𝑊
𝑉 𝑊
[𝜏′
𝜎 𝑦
2
+ 𝜎𝑧
2
]
we get
𝑟𝑄 −
𝜂
𝑊𝑉 𝑊
= −
𝛼2
𝑊𝑉 𝑊𝑊
𝑉 𝑊
[𝜎 𝑦
2
+ 𝜎𝑧
2
]
and using Proposition 3.1. ,we can come to the conclusion.
Again, can be regarded as the risk-free return.The equation (5.1) and (5.2) shows that the returns
on government bonds and capital are given by the familiar consumption-based capital asset pricing model with
rQas the return on the market portfolio.
9. The Equity Premium Puzzle Under A General Utility Function
International Journal of Business Marketing and Management (IJBMM) Page 45
From the conclusions in Section 4, we know that as long as the investment horizon is sufficient long,
the equations in Propositin 5.1. can converge to:
Corollary 5.2.
𝑟𝐵 −
𝜂
𝛿(1−𝛾−𝜆)𝑊1−𝛾−𝜆
= 𝛽 𝐵(𝑟𝑄 −
𝜂
𝛿(1−𝛾−𝜆)𝑊1−𝛾−𝜆
)(5.3)
(1 − 𝜏)𝑟 𝐾 −
𝜂
𝛿(1−𝛾−𝜆)𝑊1−𝛾−𝜆 = 𝛽 𝐾(𝑟𝑄 −
𝜂
𝛿(1−𝛾−𝜆)𝑊1−𝛾−𝜆) (5.4)
where
We define 𝑟 𝐵 and 𝑟 𝐾as the return rate on the government bond and the capital stock respectively in the
absence of the spirit of capitalism, i.e. λ = 0. Like Gong and Zou [13],we can get
Propositin 5.3.If the investment horizon is sufficient long , and λ >0, then we have
Propositin 5.3 indicates that the existence of the spirit of capitalism makes the gap between the risky assets
and risk-free assets enlarged in the long term, and this explains the equity premium puzzle in Mehra and
Prescott [1] partially.
Remark 5.4.Considering the difference caused by the spirit of capitalism in original papers( 𝑟𝐵 −
𝜂
𝛿(1−𝛾−𝜆)𝑊1−𝛾−𝜆 − [𝑟 𝐵 −
𝜂
𝛿(1−𝛾)𝑊1−𝛾]),we can get the same difference when𝑢(𝑐, 𝑊) =
𝑐1−𝛾
1−𝛾
𝑊−𝜆
. Since we
study the problem under a wider range of utility functions, our result is obviously better than the original.
Example 5.5.Next, we will use our model to estimate the premium in Chinese market. From Proposition
3.1. and equation (3.11), we know that under the utility function 𝑢(𝑐, 𝑊) =
𝑐1−𝛾
1−𝛾
𝑊−𝜆
, the premiums of the
capital stock and the government bond are calculated as follows:
(5.5)
] (5.6)
Now we choose A-share data in Chinese market directly from Shanghai Stock Exchange as the research object.
Substituting Chinese market data into formula (5.5),(5.6) and choosing γ = 3 ,λ = 0, we then estimate the
investor’s acceptable premiums of bonds and A-share in Chinese market as table 1.
year 2011 2012 2013 2014 2015 2016 2017 2018
bonds 3.70% 5.57% 5.14% 4.02% 2.72% 2.88% 2.43% 4.39%
A-shares 4.22% 6.63% 5.85% 5.10% 3.53% 3.68% 3.01% 4.78%
table1 :Investors’ acceptable equity premium in the past eight years when λ = 0
Similarly, if we select γ = 3 ,λ= 1, we have that the investor’s acceptable premiums of bonds and A-share in
Chinese market are shown in the following table 2.
year 2011 2012 2013 2014 2015 2016 2017 2018
10. The Equity Premium Puzzle Under A General Utility Function
International Journal of Business Marketing and Management (IJBMM) Page 46
bonds 3.80% 5.66% 5.53% 4.26% 2.89% 3.21% 2.79% 4.98%
A-shares 4.52% 6.83% 5.99% 5.31% 3.83% 3.91% 3.31% 5.57%
table2 :Investors’ acceptable equity premium in the past eight years when λ = 1
This calculation result shows that the spirit of capitalism makes the gap between the risky assets and
risk-free assets enlarged, which therefore confirms Propositin 5.3.
VI. Conclusion
In this paper, similar to the work by Gong and Zou [13], we study the equity premium puzzle based on
the domestic output process. We further study the approximation of the optimal strategy. We proved that the
optimal strategy under general utility functions satisfying certain conditions approximate the strategy under the
original special utility function. Then we explained the equity premium puzzle.
VII. Acknowledgement
This work is supported by NNSF of China(No.11871275; No.11371294)
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