تخصیص دارایی پویا با درآمد اتفاقی و نرخ بهره

We solve for optimal portfolios when interest rates and labor income are stochastic with the expected income growth being affine in the short-term interest rate in order to encompass business cycle variations in wages. Our calibration based on the Panel Study of Income Dynamics (PSID) data supports this relation with substantial variation across individuals in the slope of this affine function. The slope is crucial for the valuation and riskiness of human capital and for the optimal stock/bond/cash allocation both in an unconstrained complete market and in an incomplete market with liquidity and short-sales constraints.

It is well-documented in the theoretical asset allocation literature that the inclusion of labor income has dramatic effects on the optimal long-term portfolio choice of individual investors. Several studies, e.g. Heaton and Lucas (1997) and Cocco, Gomes, and Maenhout (2005), conclude that for an empirically reasonable insignificant correlation between labor income shocks and stock market shocks, the labor income stream is a substitute for an investment in the risk-free asset so that the financial wealth should be directed to stocks (typically, significantly levered, if possible). However, as these studies are cast in a setting where interest rates are assumed constant, they cannot distinguish short-term risk-free assets (cash deposits) from long-term risk-free assets (Treasury bonds). In order to investigate when human capital resembles a long-term bond and when it resembles cash and to assess the implications for the optimal stock/bond/cash portfolio choice, we set up, calibrate, and solve a specific model with stochastic interest rates and with a stochastic labor income that can be instantaneously correlated with interest rates, bond prices, and stock prices. A special and important feature of our model is that the expected labor income growth rate is an affine function of the real short-term interest rate in order to encompass business cycle variations in wages, bonuses, and layoffs. Our calibration of the model based on PSID income data supports such a relation with a substantial variation across individuals in the business cycle sensitivity of income, i.e., the slope of the relation between expected income growth and the short-term interest rate. We demonstrate that this slope is crucial for the valuation and riskiness of the human capital and, consequently, for the optimal stock/bond/cash allocation. If the expected labor income growth is non-cyclical (zero slope), the human capital substitutes a long-term coupon bond. In that case, the optimal unconstrained investment of the financial wealth involves a large long position in stocks and significant borrowing, and will typically still involve a long position in long-term bonds for speculation and intertemporal hedging purposes. If the income is counter-cyclical (negative slope), the human capital is equivalent to a levered position in a long-term bond, and a smaller (larger) share of the financial wealth should be allocated to bonds (cash). If the income is pro-cyclical (positive slope) and the slope is exactly equal to one, the human capital will substitute for cash only. If the slope is higher than one, the human capital is like having a short position in a long-term bond and more than 100% in cash. If the slope is between zero and one, the human capital is equivalent to a moderate long position in cash and in a long-term bond. The optimal weights of the long-term bond and cash in the financial portfolio are thus highly dependent on the business cycle variations of labor income. Throughout the paper we consider investors with time-additive power utility of consumption and terminal wealth. The dynamics of labor income, interest rates, bond prices, and stock prices are modeled by diffusion processes. First we derive a closed-form solution for the optimal consumption and investment decisions under the simplifying assumptions of no unspanned labor income risk and no portfolio constraints. While these assumptions are clearly questionable, the closed-form solution allows us to develop intuition of the economic forces at play and to understand the effect of the business cycle variations in income growth in an idealized setting. Next we allow for unspanned labor income risk and impose borrowing constraints and short-sales constraints, in which case we solve the utility maximization problem by a numerical dynamic programming technique. Our extensive numerical analysis based on the calibrated model shows that the intuition from the unconstrained, complete market version of the problem carries over to the constrained, incomplete market setting. Although the quantitative effects of the business cycle variations in income growth are dampened, the slope of the relation between expected income growth and the short-term interest rate remains an important parameter for the optimal consumption and investment decisions and, in particular, for the relative allocation between cash and long-term bonds. We illustrate the impact of this slope on the investment behavior of individuals with various levels of education using the life-cycle income profiles estimated from PSID income data, thereby generalizing the model and insights of Cocco, Gomes, and Maenhout (2005). Let us briefly review the relevant literature for this study. As first noted by Merton (1971), long-term investors will generally hedge stochastic variations in the investment opportunity set. Stochastic interest rates are an important source of shifts in investment opportunities, and the effect of interest rate uncertainty on the optimal strategies of an investor without labor income is by now relatively well-studied. Sørensen (1999) and Brennan and Xia (2000) consider interest rate dynamics as in the Vasicek (1977) model and assume complete financial markets and constant market prices of both interest rate risk and stock market risk. They find that the optimal investment strategy of an investor with power utility of terminal wealth only is a simple combination of the mean-variance optimal portfolio, i.e., the optimal portfolio assuming investment opportunities do not change, and the zero-coupon bond maturing at the end of the investment horizon. Other studies of dynamic portfolio choice with uncertain interest rates include Brennan, Schwartz, and Lagnado (1997), Campbell and Viceira (2001), Deelstra, Grasselli, and Koehl (2000), Munk and Sørensen (2004), Sangvinatsos and Wachter (2005), and Liu (2007). None of these papers take into account a labor income stream of the investor, although labor income is the main source of funds for most individuals. On the other hand, several papers discuss how the presence of a labor income process affects the consumption and investment decisions of individual investors in an environment of constant investment opportunities. A deterministic income stream is equivalent to an implicit investment in the risk-free asset and, hence, it is optimal to invest a higher fraction of financial wealth in the risky assets than in the no-income case; cf., e.g., Hakansson (1970) and Merton (1971). With stochastic income, but fully hedgeable income risk, the optimal unconstrained strategies can be deduced from the optimal strategies without labor income, cf. Bodie, Merton, and Samuelson (1992): given the risk structure of human capital, the financial investment is determined in order to obtain the desired overall risk exposure. Since the human capital of long-term investors is often very large compared to financial wealth, labor income has dramatic effects on their optimal portfolios. Duffie, Fleming, Soner, and Zariphopoulou (1997), Koo (1998), and Munk (2000) study (mostly by use of numerical methods) the valuation of income and the optimal consumption and investment strategies of an infinite-horizon, liquidity constrained power utility investor with non-spanned income risk. The presence of liquidity constraints can significantly decrease the individual's implicit valuation of the future income stream and, hence, dampen the quantitative effects of income on portfolio choice. Other recent papers on consumption and portfolio choice with stochastic income include He and Pagès (1993), Heaton and Lucas (1997), Viceira (2001), Constantinides, Donaldson, and Mehra (2002), and Cocco, Gomes, and Maenhout (2005). Besides working with constant investment opportunities, the concrete models with stochastic income in these papers assume a single risky asset, interpreted as the stock market index. Since different risky assets will have different correlations with the labor income of a given individual, this assumption is not without loss of generality. We allow for multiple risky assets (the stock market index and bonds) and a link between labor income and investment opportunities. Lynch and Tan (2009) consider a model where the stock market dividend yield predicts both stock market returns and the expected growth (and potentially also the volatility) of labor income leading to a negative hedging demand for stocks that partially offsets the high speculative stock demand. We model the business cycle sensitivity of income growth through the interest rate level instead of the dividend yield. While their model apparently includes time-varying interest rates, they do not allow for investments in bonds, only in cash and the stock market. Benzoni, Collin-Dufresne, and Goldstein (2007) postulate a long-run cointegration between labor income and stock market dividends and show that such a relation can substantially reduce optimal stock holdings for sufficiently risk-averse long-term investors. In contrast to these two papers, we focus on the joint implications of stochastic interest rates and labor income for the valuation of human capital and for the stock/bond/cash allocation. The model of Koijen, Nijman, and Werker (2009) includes both stochastic interest rates and labor income, but the income process is assumed to have constant expected growth and constant volatility so it cannot capture business cycle variations in income. On the other hand, they allow for time-variation in bond risk premiums and inflation. Finally, Van Hemert (2009) studies a rich life-cycle model with both stochastic income and interest rates, but he also disregards business cycle variations in income and he focuses on the optimal mortgage choice for homeowners. The rest of the paper is organized as follows. In Section 2 we set up the general model of the financial market, specify the preferences and income of the individual, and calibrate the model to data. Section 3 focuses on the special case with unconstrained investment strategies and either locally risk-free income or spanned income risk so that we obtain closed-form solutions allowing us to understand the economic forces at play. Section 4 explains the numerical solution technique applied to the case with unspanned income uncertainty and relevant portfolio constraints and shows and discusses the results for the calibrated model. Section 5 gives some concluding remarks. The appendices contain proofs of propositions and detailed descriptions of the calibration procedure and the numerical solution technique.

This paper has demonstrated that the relative allocation to stocks, bonds, and cash is significantly affected by the presence of uncertain labor income and the variations in labor income over the business cycle. Our model allows the expected labor income growth rate to be an affine function of the real short-term interest rate in order to encompass such business cycle variations, and our calibration using PSID income data supports such a relation with a substantial variation across individuals in the slope of that affine function. Our analysis demonstrates the importance of this slope for the interest rate risk inherent in the human capital and thus, for the allocation of financial wealth to different asset classes. We provide a closed-form solution for the optimal consumption and investment strategies of an unconstrained power utility investor for the case where the labor income contains no unspanned risk. Here, the asset allocation implications of the labor income variations over the business cycle become clear. If the expected labor income growth is non-cyclical (zero slope), the human capital substitutes a long-term coupon bond so that the optimal financial investment is tilted away from long-term bonds and more towards cash. If the income is pro-cyclical with a slope exactly equal to one, the human capital will substitute for cash only and financial investments should involve more bonds and less cash. When the slope is between zero and one, the human capital is like a mix of cash and the long-term bond. When the slope is above one, human capital is like a levered position in the bond, and when the slope is negative, the human capital is like short-selling the bond and depositing the proceeds at the short-term risk-free rate with the obvious consequences for the financial investments in bonds and cash. Given the natural path of human capital over the life-cycle, the quantitative impact on the optimal financial investments is typically large for young individuals and then decreases until retirement. In the unconstrained case, the optimal stock position is only affected due to the fact that the valuation of the human capital depends on this slope parameter. For the more realistic case with unspanned labor income risk and liquidity and short-sales constraints, our extensive numerical results show that the slope of the relation between income growth and the interest rate level remains an important parameter for the optimal investment decisions. Naturally, the quantitative impact of the features of labor income is smaller when constraints on the use of future income are imposed, but there are still substantial differences in the optimal investment strategies of individuals who earn income with different business cycle sensitivity but are otherwise identical. Due to short-sales constraints, this sensitivity will now have a bigger effect on the optimal stock investments than in the unconstrained case, but the sensitivity is still mainly important for the allocation between cash and long-term bonds. The results of the present paper highlight the need for precise information on the labor income risk characteristics when formulating asset allocation advice to households. Identification of these characteristics for different occupational categories would be an interesting empirical project. On the modeling side, our analysis can be extended in several interesting directions albeit with added computational complexity. One extension would be to allow the investor to invest in residential real estate that can serve as collateral for a mortgage loan, thus avoiding a strict no borrowing constraint. This would basically call for a combination of our model with a set-up like that considered by Yao and Zhang (2005) and Van Hemert (2009). Another extension is to allow for several stocks. Presumably a larger fraction of the income rate variations can be hedged using multiple risky assets, but little is known about the typical correlations between a labor income stream and individual stocks and, hence, it is unclear how large a fraction of the risk of a typical labor income stream that can be hedged in the financial markets.12