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


  • Phillip Monin

    () (Office of Financial Research)

  • Thaleia Zariphopoulou

    () (The University of Texas at Austin)


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.

Suggested Citation

  • Phillip Monin & Thaleia Zariphopoulou, 2014. "On the Optimal Wealth Process in a Log-Normal Market: Applications to Risk Management," Staff Discussion Papers 14-01, Office of Financial Research, US Department of the Treasury.
  • Handle: RePEc:ofr:discus:14-01

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    References listed on IDEAS

    1. Campbell, Rachel & Huisman, Ronald & Koedijk, Kees, 2001. "Optimal portfolio selection in a Value-at-Risk framework," Journal of Banking & Finance, Elsevier, vol. 25(9), pages 1789-1804, September.
    2. William F. Sharpe, 1964. "Capital Asset Prices: A Theory Of Market Equilibrium Under Conditions Of Risk," Journal of Finance, American Finance Association, vol. 19(3), pages 425-442, September.
    3. Jianming Xia, 2008. "Risk Aversion and Portfolio Selection in a Continuous-Time Model," Papers 0805.0618,, revised Dec 2011.
    4. Carlo Acerbi & Dirk Tasche, 2002. "Expected Shortfall: A Natural Coherent Alternative to Value at Risk," Economic Notes, Banca Monte dei Paschi di Siena SpA, vol. 31(2), pages 379-388, July.
    5. J. Dhaene & S. Vanduffel & M. Goovaerts, 2007. "Comonotonicity," Review of Business and Economic Literature, KU Leuven, Faculty of Economics and Business (FEB), Review of Business and Economic Literature, vol. 0(2), pages 265-278.
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    Cited by:

    1. Ljudmila A. Bordag, 2019. "Portfolio optimization in the case of an exponential utility function and in the presence of an illiquid asset," Papers 1910.07417,, revised May 2020.
    2. Wai Mun Fong, 2018. "Synthetic growth stocks," Journal of Asset Management, Palgrave Macmillan, vol. 19(3), pages 162-168, May.

    More about this item


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

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