Optimal allocation-consumption problem for a portfolio with an illiquid asset

• During financial crises investors manage portfolios with low liquidity, where the paper-value of an asset differs from the price proposed by the buyer. We consider an optimization problem for a portfolio with an illiquid, a risky and a risk-free asset. We work in the Merton's optimal consumption framework with continuous time. The liquid part of the investment is described by a standard Black-Scholes market. The illiquid asset is sold at a random moment with prescribed distribution and generates additional liquid wealth dependent on its paper-value. The investor has a hyperbolic absolute risk aversion also denoted as HARA-type utility function, in particular, the logarithmic utility function as a limit case. We study two different distributions of the liquidation time of the illiquid asset - a classical exponential distribution and a more practically relevant Weibull distribution. Under certain conditions we show the smoothness of the viscosity solution and obtain closed formulae relevant for numerics.

Ämnesord

NATURVETENSKAP  -- Matematik (hsv//swe)

NATURAL SCIENCES  -- Mathematics (hsv//eng)

Nyckelord

91G10

35Q93

49L25

91G80

49L20

portfolio optimization

random income

viscosity solutions

illiquidity

Mathematics

91G10

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