Stochastic Income Research Articles

New empirical evidence on income convergence

This paper aims at providing further evidence on stochastic income convergence for a group of 30 countries over a century long period, from 1900 to 2008, utilizing novel econometric methods for determining structural breaks and unit roots. Recent studies have noted the problem of low power and size distortions in unit root tests, and spurious rejections that plague unit root tests which treat structural breaks asymmetrically. The proposed methodology used in this paper enables robust detection and estimation of breaks, and reliable inference regarding the presence of unit roots conditional on the presence or absence of breaks. We find that the periods of War, the Great Depression and oil price shocks to be possible reasons for the occurrence of breaks. The unit root test results indicate limited evidence of stochastic income convergence. We find none of the current upper-middle-income and lower-middle-income countries to show convergence, while high-income countries such as Australia, Canada, the Netherlands, Norway, Portugal and USA show signs of convergence toward the world average.

Analysis of ruin measures for two classes of risk processes with stochastic income

In this paper, we consider the ruin measures for two classes of risk processes. We assume that the claim number processes are independent Poisson and generalized Erlang(n) processes, respectively. Historically, it has been assumed that the premium size is a constant. In this contribution, the premium income arrival process is a Poisson process. In this framework, both the integro-differential equation and the Laplace transform for the expected discounted penalty function are established. Explicit expressions for the expected discounted penalty function are derived when the claim amount distributions belong to the rational family. Finally, Numerical examples are

Assessing economic convergence in the EU: is there a perspective for the ‘cohesion countries’?

This article analyses the stochastic income convergence within the EU-15. The empirical analysis uses per capita GDP, in PPP and in constant prices of 2005 for the period 1950 to 2010. Apart from the traditional DF type tests we also account for possible structural changes. In this direction, we employ the Zivot-Andrews (1992) and the Lee-Strazicich (1999, 2003) testing procedures, for one and two breaks, endogenously determined. Furthermore, we apply the Carlino and Mills (1993) methodology proposed for the detection of β-convergence. The overall evidence supports the existence of two discrete clubs, the first by the ‘cohesion countries’ (Portugal, Ireland, Greece and Spain) and the second by the remaining members. In particular, there is a clear evidence of convergence within each club, whereas between clubs there is a luck of catching-up effects. Furthermore, investigation of correlation between relative per capita GDP of each country and several factors that are often identified as growth stimulants, namely Total Factor Productivity, FDI, investment and openness confirm, with the exception of Greece, a strong association between these factors and the convergence process. However, progress in the convergence has not been uniform across countries and over time, reflecting the specific interactions between domestic and international factors and their impact on the convergence process of individual countries.

Incomplete Continuous-Time Securities Markets with Stochastic Income Volatility

We derive closed-form solutions for the equilibrium interest rate and market price of risk processes in an incomplete continuous-time market with uncertainty generated by Brownian motions. The economy has a finite number of heterogeneous exponential utility investors, who receive partially unspanned income and can trade continuously. Countercyclical stochastic income volatility generates a countercyclical equilibrium market price of risk process and a procyclical equilibrium interest rate process. We show that when the investors’ unspanned income volatility is countercyclical, the resulting equilibrium displays both lower interest rates and higher risk premia compared with the equilibrium in an otherwise identical complete market.

An Empirical Model of Tax Convexity and Self-Employment

Do progressive marginal income tax rates discourage self-employment? We assume risk neutrality to construct an implicit surtax on stochastic income relative to steady income, arising from a convex tax schedule. It is computed as part of a structural probit model with earnings equations and a tax simulator. The tax convexity variable and the net-of-tax income difference between self- and paid employment have the predicted signs and high levels of statistical significance for the probability of self-employment. A simulated flat tax reform suggests the tax effects are small.

Optimal investment-consumption-insurance with random parameters

This paper discusses an optimal investment, consumption, and life insurance purchase problem for a wage earner in a complete market with Brownian information. Specifically, we assume that the parameters governing the market model and the wage earner, including the interest rate, appreciation rate, volatility, force of mortality, premium-insurance ratio, income and discount rate, are all random processes adapted to the Brownian motion filtration. Our modeling framework is very general, which allows these random parameters to be unbounded, non-Markovian functionals of the underlying Brownian motion. Suppose that the wage earner’s preference is described by a power utility. The wage earner’s problem is then to choose an optimal investment-consumption-insurance strategy so as to maximize the expected, discounted utilities from intertemporal consumption, legacy and terminal wealth over an uncertain lifetime horizon. We use a novel approach, which combines the Hamilton–Jacobi–Bellman equation and backward stochastic differential equation (BSDE) to solve this problem. In general, we give explicit expressions for the optimal investment-consumption-insurance strategy and the value function in terms of the solutions to two BSDEs. To illustrate our results, we provide closed-form solutions to the problem with stochastic income, stochastic mortality, and stochastic appreciation rate, respectively.

Sovereign Risk and Procyclical Fiscal Policy in Emerging Market Economies

This study develops a dynamic stochastic general equilibrium model to account for the differences in fiscal policy stance over the business cycle between developed and emerging market countries, and, in particular, for the volatile and procyclical government consumption and transfer payment in emerging market countries. Two models with and without default option in sovereign borrowings replicate the contrasting cyclical behaviors indicating that the default option is responsible for procyclical fiscal policy. Further, augmented model with third-party bailouts, together with the stochastic trend income process, successfully predicts high volatilities of fiscal expenditures. These imply that procyclical fiscal policy, entailed by default option, may exacerbate the business cycle in emerging market countries.

Sovereign risk and procyclical fiscal policy in emerging market economies

This study develops a dynamic stochastic general equilibrium model to account for the differences in fiscal policy stance over the business cycle between developed and emerging market countries, and, in particular, for the volatile and procyclical government consumption and transfer payment in emerging market countries. Two models with and without default option in sovereign borrowings replicate the contrasting cyclical behaviors indicating that the default option is responsible for procyclical fiscal policy. Further, augmented model with third-party bailouts, together with the stochastic trend income process, successfully predicts high volatilities of fiscal expenditures. These imply that procyclical fiscal policy, entailed by default option, may exacerbate the business cycle in emerging market countries.

Expected discounted penalty function of Markov-Modulated, Constant barrier and Stochastic income model

This article considers a Markov -modulated risk process with stochastic premium income and a constant dividend barrier. We derive the integro-differential equations satisfied by the expected discounted penalty function regulated by an external environment. Applications of the integral equations are given to be the discounted expectation of the deficit at ruin. Explicit solution of the expected deficit at ruin for the model is obtained for exponential distribution. Finally in two state model, numerical example illustrates the effect of the different parameters.

Asset allocation for a DC pension fund with stochastic income and mortality risk: A multi-period mean–variance framework

This paper investigates an asset allocation problem for defined contribution pension funds with stochastic income and mortality risk under a multi-period mean–variance framework. Different from most studies in the literature where the expected utility is maximized or the risk measured by the quadratic mean deviation is minimized, we consider synthetically both to enhance the return and to control the risk by the mean–variance criterion. First, we obtain the analytical expressions for the efficient investment strategy and the efficient frontier by adopting the Lagrange dual theory, the state variable transformation technique and the stochastic optimal control method. Then, we discuss some special cases under our model. Finally, a numerical example is presented to illustrate the results obtained in this paper.

Mean-Variance Portfolio Selection with a Stochastic Cash Flow in a Markov-switching Jump–Diffusion Market

This paper considers a non-self-financing mean-variance portfolio selection problem in which the stock price and the stochastic cash flow follow a Markov-modulated Levy process and a Markov-modulated Brownian motion with drift, respectively. The stochastic cash flow can be explained as the stochastic income or liability of the investors during the investment process. The existence of optimal solutions is analyzed, and the optimal strategy and the efficient frontier are derived in closed-form by the Lagrange multiplier technique and the LQ (Linear Quadratic) technique.

On the Expected Discounted Penalty Function for a Markov Regime-Switching Insurance Risk Model with Stochastic Premium Income

We consider a Markovian regime-switching risk model (also called the Markov-modulated risk model) with stochastic premium income, in which the premium income and the claim occurrence are driven by the Markovian regime-switching process. The purpose of this paper is to study the integral equations satisfied by the expected discounted penalty function. In particular, the discount interest force process is also regulated by the Markovian regime-switching process. Applications of the integral equations are given to be the Laplace transform of the time of ruin, the deficit at ruin, and the surplus immediately before ruin occurs. For exponential distribution, the explicit expressions for these quantities are obtained. Finally, a numerical example is also given to illustrate the effect of the related parameters on these quantities.

Randomized Dividends in a Discrete Insurance Risk Model with Stochastic Premium Income

The compound binomial insurance risk model is extended to the case where the premium income process, based on a binomial process, is no longer a constant premium rate of 1 per period and insurer pays a dividend of 1 with a probabilityq0when the surplus is greater than or equal to a nonnegative integerb. The recursion formulas for expected discounted penalty function are derived. As applications, we present the recursion formulas for the ruin probability, the probability function of the surplus prior to the ruin time, and the severity of ruin. Finally, numerical example is also given to illustrate the effect of the related parameters on the ruin probability.

Characterization of a risk sharing contract with one-sided commitment

In this paper I provide a stopping-time-based solution to a long-term contracting problem between a risk-neutral principal and a risk-averse agent. The agent faces a stochastic income stream and cannot commit to the long-term contracting relationship. To compute the optimal contract, I also design an algorithm that is more efficient than value-function iteration.

Lifetime investment and consumption using a defined-contribution pension scheme

During the accumulation phase of a defined-contribution pension scheme, a scheme member invests part of their stochastic income in a portfolio of a stock and a bond in order to build up sufficient funds for retirement. It is assumed that the remainder of their salary pre-retirement is consumed, an annuity is purchased at retirement, and the stock allocation and consumption pre-retirement maximise the total expected lifetime consumption using a CARA utility function. Perfect correlation between the scheme member's income and the stock price leads to analytical expressions for the controls for a general income model. If the correlation is imperfect then analytical controls are found for two particular stochastic income models.

Stochastic National Income

How can national income be adjusted to indicate welfare improvement if the future is uncertain? The present paper extends the definition of national income to stochastic settings on the basis of discounted utilitarian welfare function. Real interest rate of consumption is redefined so that real national income can be interpreted as the expected present value of real interest on future national consumption. A stochastic one-good model is used to illustrate the application of the stochastic real national income. It turns out that under uncertainty real national income may be decreasing even though captial stock is constant or increasing over time.

Incomplete Continuous-Time Securities Markets with Stochastic Income Volatility

We derive closed-form solutions for the equilibrium interest rate and market price of risk processes in an incomplete continuous-time market with uncertainty generated by Brownian motions. The economy has a finite number of heterogeneous exponential utility investors, who receive partially unspanned income and can trade continuously. Countercyclical stochastic income volatility generates a countercyclical equilibrium market price of risk process and a procyclical equilibrium interest rate process, and we show that when the investors' unspanned income volatility is countercyclical, the resulting equilibrium displays both lower interest rates and higher risk premia compared to the equilibrium in an otherwise identical complete market.

The expected discounted penalty function under a renewal risk model with stochastic income

In this paper, we consider a renewal risk model with stochastic premiums income. We assume that the premium number process and the claim number process are a Poisson process and a generalized Erlang ( n) processes, respectively. When the individual stochastic premium sizes are exponentially distributed, the Laplace transform and a defective renewal equation for the Gerber–Shiu discounted penalty function are obtained. Furthermore, the discounted joint distribution of the surplus just before ruin and the deficit at ruin is given. When the claim size distributions belong to the rational family, the explicit expression of the Gerber–Shiu discounted penalty function is derived. Finally, a specific example is provided.

The Risky Steady State

We propose a simple quantitative method to linearize around the risky steady state of a small open economy. Unlike when the deterministic steady state is used, the net foreign asset position is well defined. We allow for stochastic income and stochastic interest rate.

Are Heuristics Better than Theory if Market Crashes are Possible?

Financial advisers often recommend that investors should place (100 − age) % of their wealth in a well-diversified equity portfolio. Another popular approach is based on the complete markets model with CRRA utility and i.i.d. returns which implies a constant portfolio share invested in stocks. I solve a calibrated life-cycle model with stochastic labor income and rare disasters to investigate the welfare costs of constraining portfolio allocations to mimic these heuristics. When there is a possibility of a stock-market crash accompanied by a macroeconomic contraction, the simple rules of thumb outperform the more sophisticated dynamic programming rule that ignores the risk of rare disasters. The annual utility loss of following the complete markets solution amounts to only 0.5% relative to the optimal allocation. The (100-age)% rule also generates reasonably small welfare losses. In contrast, ignoring disasters can impose a cost as high as 5%.