Housing affordability is the main policy challenge for most large cities in the world. Zoning changes, rent control, housing vouchers, and tax credits are the main levers employed by policymakers. How effective are they at combatting the affordability crisis? We build a dynamic stochastic spatial equilibrium model to evaluate the effect of these policies on the well-being of its citizens. The model endogenizes house prices, rents, construction, labor supply, output, income and wealth inequality, the location decisions of households within the city as well as inter-city migration. Its main novel features are risk, risk aversion, and incomplete risk-sharing. We calibrate the model to the New York MSA. Housing affordability policies carry substantial insurance value but affect aggregate housing and labor supply and cause misallocation in labor and housing markets. Housing affordability policies that enhance access to this insurance, especially for the neediest households create substantial net welfare gains.