Estimating consumption and portfolio decisions in an economy with jump and volatility risk

This paper provides a framework to study an agent's optimal dynamic consumption and portfolio decisions in an incomplete market with jump and volatility risk.; The historical stock return data indicate the expected excess return over approximately risk-free securities has been about 6% and its standard deviation has been about 15%. With these numbers, the optimal portfolio rule derived by Merton (1969) (hereafter, Merton's formula) implies that a consumer with only financial wealth and logarithmic utility should borrow and hold a position in stocks equal to about 200% of his or her financial wealth. Evidence on portfolio holdings suggests that there are few consumers who make choices of this type. Merton's formula is based on the assumption that stock prices follow a geometric Brownian motion (GBM). There is a large body of evidence on stochastic volatility that contradicts this assumption. Moreover, there is some evidence that a stochastic process that allows for jumps in the process of stock prices better fits the data.; In this paper we examine how these alternative assumptions on the process that governs stock prices affect consumption and portfolio decisions.{09}In particular, could it be that the 200% number that comes from Merton's formula is too large if one takes into account stochastic volatility and jumps? We also examine how well we can estimate the optimal portfolio choice of a consumer, given the statistical uncertainty about the stochastic process that describes stock returns.; We find that incorporating jump and volatility risk has a substantial impact on the optimal portfolio weight so that the estimated portfolio weight under a stochastic volatility jump-diffusion (SVJD) process with a relative risk aversion (RRA) coefficient of 1.5 is 83% as opposed to about 149% in the case of the GBM. The statistical uncertainty about these estimates is also significant. At an RRA of 1.5, the 5%–95% ranges of the estimated portfolio weight distributions are 63%–100% for the SVJD process and 88%–352% for the GBM process. Allowing for jump and volatility risk and accounting for the statistical uncertainty in the estimates make data on portfolio choice seem a lot less puzzling than Merton's formula suggests.