On the Optimal Wealth Process in a Log-normal Market: Applications to Risk Management

This paper presents a model that ties investors’ risk preferences, which are key determinants of the allocation of credit and risk in the financial system, to the measures used in risk management of their investments. Such models provide a direct link between risk management and the dynamics of the financial system.


The theory of portfolio choice holds that investors balance risk and reward in their investment decisions. We explore the relationship between investors’ attitudes towards taking risk and their objectives for managing the risk they take on. Working in a classical theoretical model, we calculate the distribution and density functions of an investor’s optimal wealth process and prove new mathematical results for these functions under general risk preferences. By applying our results to a constant relative risk aversion investor who has a targeted value at risk or expected shortfall at a given future time, we are able to infer the investor’s risk preferences and prescribe how to invest to achieve the desired goal. Then, drawing analogies to the option greeks, we define and derive closed-form expressions for “portfolio greeks,” which measure the sensitivities of an investor’s optimal wealth to changes in the cumulative excess stock return, time, and market parameters. Like option greeks, portfolio greeks can be used in the risk management of investors’ portfolios.

Keywords: expected utility; Merton problem; value at risk (VaR); expected shortfall; portfolio greeks