انتخاب پرتفوی چرخه عمر : نقش ناهمگونی های تحت تنوع بخشی

In life-cycle portfolio choice models it is standard to assume that all agents invest in a diversified stock market index. In contrast recent empirical evidence, summarized in Campbell [2006. Household finance. Journal of Finance 61, 1553–1604] suggests that households’ financial portfolios are under-diversified and that there is substantial heterogeneity in diversification. In the present paper I examine the effects of heterogeneous under-diversification in a life-cycle portfolio choice model with uninsurable uncertain earnings and fixed per-period participation costs. The analysis of the model shows that realistically calibrated under-diversification gives an important contribution to the explanation of two key facts of households’ portfolio allocation: the moderate stock market participation rate and the moderate stock share for participants

Traditional life-cycle portfolio choice models with intermediate consumption and uninsurable labor income have typically explored investors’ decisions about how to allocate wealth between a risk-free and a risky asset. The assumption common to these models is that all agents face the same risky asset that can be interpreted as a stock index fund. This assumption is contradicted by abundant empirical evidence that documents that households invest in a limited number of individual stocks or mutual funds thus facing substantial idiosyncratic risk on their equity investment.1 The empirical evidence also suggests that more financially sophisticated households, defined by greater education and wealth hold better diversified equity portfolios. In the present paper I explore the effects of portfolio under-diversification on household life-cycle asset allocation. I find that this so far overlooked feature of households’ investment strategy substantially improves the ability of the model to rationalize two key empirical facts: the low stock market participation rate in the population and the moderate stock share for market participants. The model presented here is standard in most respects. It is characterized by finitely lived households that go through the stages of working life and retirement. During working life they face idiosyncratic earnings uncertainty around a deterministic hump-shaped trend. In retirement they face constant and progressive social security benefits. They solve an optimal consumption-saving problem and make an asset allocation decision from a menu of financial assets. Asset demand is subject to a borrowing and short sale constraint. Payment of a one time initial entry cost is needed to gain access to the risky asset market.2 The key departure from the traditional framework is the assumption that there are two mutually exclusive risky financial assets with the same mean but different standard deviation of returns. The two assets are meant to capture in a stylized way the idea of a well diversified and a poorly diversified stock portfolio. On top of the initial entry cost, investors must pay a fixed cost in each period in which they want to participate in the stock market and this cost is higher for the risky asset with lower standard deviation of returns. The main result of this research is that allowing for under-diversification of households’ stock portfolios provides an explanation to two key empirical observations: the low stock market participation rate and the moderate portfolio stock share for participants.3 The intuition for this result is the following: the increased volatility of the low cost risky asset implies that the optimal share conditional on investing in it is lower. Agents with low wealth do not find it optimal to pay the larger cost needed to buy the well diversified stock portfolio and buy the poorly diversified one, thus lowering the average conditional share in the population. At the same time both the reduced optimal share and the increased variance of returns decrease the benefits of participation for these agents. This, in the presence of the initial entry cost, deters part of them from participating altogether, thus helping to reduce the average participation rate as well. The interesting finding of the quantitative analysis of the model is that the amount of heterogeneity in the volatility of individual stock portfolios needed to rationalize participation rates and conditional stock shares falls well within the available empirical evidence. Previous attempts at rationalizing low participation rates and conditional shares had focused on background risk and/or risk aversion. As it is well explained in Gomes and Michaelides (2005) this line of attack carries an implicit tension: increasing risk aversion or background risk reduces the portfolio share of stock but increases wealth accumulation as the precautionary motive for saving is strengthened, thus increasing participation. Gomes and Michaelides (2005) resolved the issue by assuming heterogeneity in risk aversion but at the cost that endogenously stock market participants are the more risk averse individuals which seems to contradict survey evidence that stock market participants tend to be more willing to take financial risk.4 The present model by focusing on the risk properties of the stock investment itself rather than on risk aversion or background risk avoids the above mentioned contradiction. Another appealing feature of this explanation emerges when the progressive social security system is considered. As it is shown in the paper, when replacement ratios are progressive and under the standard assumption of investment in a common stock index fund, conditional stock shares are larger for households with lower permanent income, in contrast to the empirical evidence. In the present model wealthier households endogenously choose the stock portfolio with lower variance. This increases the share they optimally invest in the risky asset relative to poorer households, restoring the positive relationship between permanent income and conditional portfolio stock shares. Given the key importance of heterogeneous stock portfolio diversification for the results of this paper it is important to insure that such an assumption is well motivated. This is the case both from an empirical and a theoretical perspective. Empirical work on household portfolio diversification has traditionally relied either on survey data or on administrative records from brokerage houses or retirement plans. One example of the first type of studies is Polkovnichenko (2005). The author, using the SCF, finds that the median share of directly held stocks for equity holders declines with wealth except at the top of the distribution and that the number of directly held stocks increases from 1 in the bottom quintile to 15 in the top quintile of the distribution. Similarly, surveys about household stock market behavior conducted by the Investment Company Institute (1999 and 2002) show that the median number of individual stocks held is 2 for direct stock holders with less than 25 000$ of financial wealth and 8 for those with more than 500 000$; a similar pattern is observed when looking at stock mutual funds or at both types of equities jointly. The survey based studies, even though representative of the whole population and the whole household financial portfolio, have the limitation that they only allow to know the number of stocks or mutual funds held but not the variance of the risky portfolio, a more accurate measure of diversification since it captures also the correlation structure of stocks. In order to overcome this limitation other authors have used administrative records that also have the advantage of being subject to less reporting error. An example of this research is Goetzmann and Kumar (2008) who used the records of a large brokerage house and found large differences both in the number of stocks held and in estimated portfolio variances and even larger differences in risk-adjusted returns. As Bilias et al. (2008a) pointed out though, the behavior of investors with brokerage accounts is not representative of the whole population. Beside that, studies based on administrative records also do not cover the entire household financial portfolio. The limitations of both approaches have been overcome in recent research by Calvet et al., 2007 and Calvet et al., 2009. The authors exploit a data-set collected by the Swedish statistical agency that covers the whole population and has information at the individual asset level, thus allowing to compute household portfolio performance. They find that idiosyncratic volatility is an important part of portfolio volatility and that there is a wide dispersion in both portfolio volatility and return losses caused by under-diversified investment. They also find that larger wealth and better education predict improved diversification of risky portfolios although the total dollar cost of under-diversification is higher for better diversified households due to more aggressive investment strategies. From a theoretical perspective under-diversification arises in a variety of contexts. Brennan (1975) showed in a static model that in the presence of a fixed cost of transacting in each security the optimal number of risky securities in a portfolio is increasing in wealth hence the total variance of portfolio return is declining. More recently Van Nieuwerburgh and Veldkamp (2008) showed that in a model where agents allocate capacity to acquire information about securities the optimal portfolio implies investment in a diversified fund and at the same time in a small number of correlated securities. More capacity leads to larger investment in the under-diversified portfolio, however, because of the information acquired the resulting portfolio displays better performance. Finally Polkovnichenko (2005) shows that investment in an under-diversified portfolio of stocks also arises under a class of preferences known as rank dependent utility although in this case there is no clear relationship between under-diversification and observable households’ characteristics. The papers mentioned above focus on the choice of assets in households’ financial portfolio. They do that at the cost of simplifying the study of their real actions.5 The present paper follows a complementary approach: it focuses on those real actions and how they interact with under-diversification of financial portfolios to derive predictions that can be quantitatively contrasted with the data. In doing so it simplifies the study of how the stock portfolio is constructed and why agents may choose not to diversify. The rest of the paper is organized as follows. In Section 2 I present the description of the life-cycle model. In Section 3 I describe the choice of parameters, in Section 4 I describe the results of the quantitative analysis and in Section 5 I present some brief conclusions. Finally an Appendix describes the numerical methods used in the solution of the model.

In the present paper I have considered an extension of the basic life-cycle asset allocation model that allows for heterogeneous under-diversification. This was obtained in a stylized way by assuming the existence of two mutually exclusive stock portfolios with different return variances and different associated fixed participation cost. The result that emerged is that under-diversification of households’ stock portfolios provides a quantitatively plausible explanation for two key empirical facts: the moderate stock market participation rates and portfolio share of stocks for participants. Moreover, in line with the empirical evidence, endogenously agents with greater permanent income and/or wealth invested in the more diversified stock. This allowed the model to reverse the pattern of portfolio stock shares by permanent income induced by the progressive social security formula bringing the model in line with the data along this dimension as well. The hope is to have convinced the reader that under-diversification of stock investment is a key feature to understand households’ life-cycle asset allocation. Heterogeneous under-diversification was obtained here in a stylized way; the natural next step is to integrate the basic life-cycle portfolio choice model with a more sophisticated theory of household portfolio diversification. Given the huge computational effort needed to solve such a model and the fact that there is not yet a consensus on which theory best explains under-diversification of stock portfolios, this research line is left for the future.