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This paper studies a consumption-portfolio problem where money enters the agent's utility function. We solve the corresponding Hamilton-Jacobi-Bellman equation and provide closed-form solutions for the optimal consumption and portfolio strategy both in an infinite- and finite-horizon setting. For the infinite-horizon problem, the optimal stock demand is one particular root of a polynomial. In the finite-horizon case, the optimal stock demand is given by the inverse of the solution to an ordinary differential equation that can be solved explicitly. We also prove verification results showing that the solution to the Bellman equation is indeed the value function of the problem. From an economic point of view, we find that in the finite-horizon case the optimal stock demand is typically decreasing in age, which is in line with rules of thumb given by financial advisers and also with recent empirical evidence.
This paper studies a consumption-portfolio problem where money enters the agent's utility function. We solve the corresponding Hamilton-Jacobi-Bellman equation and provide closed-form solutions for the optimal consumption and portfolio strategy both in an infinite- and finite-horizon setting. For the infinite-horizon problem, the optimal stock demand is one particular root of a polynomial. In the finite-horizon case, the optimal stock demand is given by the inverse of the solution to an ordinary differential equation that can be solved explicitly. We also prove verification results showing that the solution to the Bellman equation is indeed the value function of the problem. From an economic point of view, we find that in the finite-horizon case the optimal stock demand is typically decreasing in age, which is in line with rules of thumb given by financial advisers and also with recent empirical evidence.
We show that the net corporate payout yield predicts both the stock market index and house prices and that the log home rent-price ratio predicts both house prices and labor income growth. We incorporate the predictability in a rich life-cycle model of household decisions involving consumption of both perishable goods and housing services, stochastic and unspanned labor income, stochastic house prices, home renting and owning, stock investments, and portfolio constraints. We find that households can significantly improve their welfare by optimally conditioning decisions on the predictors. For a modestly risk-averse agent with a 35-year working period and a 15-year retirement period, the present value of the higher average life-time consumption amounts to roughly $179,000 (assuming both an initial wealth and an initial annual income of $20,000), and the certainty equivalent gain is around 5.5% of total wealth (financial wealth plus human capital). Furthermore, every cohort of agents in our model would have benefited from applying predictor-conditional strategies along the realized time series over our 1960-2010 data period.