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We study consumption and investment decisions given realistic time-varying constraints on borrowing. We first consider the case where borrowing is constrained by a maximum debt-to-income ratio. We then consider collateral borrowing with a maximum loan-to-value ratio. The resulting implications for optimal policies differ considerably from those obtained in the existing literature based on fixed borrowing limits but are consistent with those documented in the empirical literature.

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We investigate a portfolio selection problem involving an agent’s realistic housing service choice, where the agent not only has to choose the size of house to live in, but also has to select between renting and purchasing a house. Adopting a dynamic programming approach, we derive a closed-form solution to obtain the optimal policies for the consumption, investment, housing service, and purchasing time for a house. We also present various numerical demonstrations showing the impacts of parameters in the financial and housing markets and the agent’s preference, which visually show the economic implications of our model. Our model makes a significant contribution because it is a pioneering model for the optimal time to purchase a house, which has not been investigated in depth in existing mathematical portfolio optimization models.

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