Investing in Stocks and Bonds in the Long Run

Abstract

In recent research on asset allocation attention has been paid to time variations in the expected return on stocks. In particular, how does the long horizon variation in expected returns influence the inter-temporal hedging demand for stocks. Unfortunately, this problem for investors leads to a complex mathematical problem. As a result, the solution to the investor's problem must be approximated. Up to now researchers have used low order approximations without knowing the amount of approximation error in their solutions. This issue is vitally import to investment managers given the possible variation of expected returns between bull and bear markets. In our previous research CCCH (2005), CCH (2006, 2007a, 2007b, 2007c, 2007d) on asset pricing models, we have shown that financial economic problems can be solved using analytic methods. Analytic methods allow one to represent the solution to financial problems with Taylor polynomials. This solutions can be quickly calculated on a standard {\it PC}. In addition, it is possible to find the radius of convergence for a given financial problem, which allows one to find bounds on the error in the polynomial approximation. For example CCH (2007c) solve Campbell and Cochrane's (1999) asset pricing model with a $260^{th}$ order Taylor polynomial in less than ten seconds. In this case the error is less than one in a billion dollars for dividend growth in the range of plus or minus $20\%$ per month around the historic average dividend growth. Thus, it is feasible to quickly and accurately solve financial problems. While the investor's problem of allocation between stocks and bonds is mathematically more complex than the asset pricing problems, we have been able to use the analytic method to quickly and accurately solve this problem as well. To capture the non-linear behavior of the optimal investment decision we find that the order of approximation must be substantially higher than currently used by researchers. The purpose of this research project is to document the characteristics of this solution over ranges of expected returns on stock which would be observed over standard fluctuations in the stock market. In carrying out this project we will use state of the art statistical procedures to document the long-run predictability of stock returns. This analysis would help us place confidence bounds on the optimal inter-temporal hedging demand for stocks. Thus, the purpose of this project is to improve investment management techniques over bull and bear markets.