Abstract
We consider decision-making under risk in which random events affect the value of the portfolio multiplicatively, rather than additively. In this case, a higher variability in the rate of return not only is associated with a higher risk, a bad property, but also engenders a higher expected return, a good property. As a result, certain expected utility maximizing investors, namely those with the lowest risk aversion, will prefer some portfolios with higher variances in the rate of return over others with lower ones. This is demonstrated, and implications are considered.
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Notes
Here we measure variability by means of σ2, the variance of logarithms. The variance and Gini coefficient for the terminal wealth distribution are monotonic functions of σ2.
Note that CARA implies increasing relative risk aversion (IRRA).
Note that \( \lim_{a \to 0} {{U^{a} (x)} \mathord{\left/ {\vphantom {{U^{a} (x)} a}} \right. \kern-0pt} a} = x + U_{0} (x) \): the risk neutral utility function \( U(x) = x \) is in essence common to both families.
The value of the limiting CARA parameter a 0 is immaterial for drawing this conclusion. However, if we set \( y = 1 - U^{a} (w) = \exp \left\{ { - aw} \right\} \), so that \( \ell n\left[ { - \ell n\left[ y \right]} \right] = \ell n\left[ a \right] + \ell n\left[ w \right] \sim \) \( N\left( {\mu + \ell n\left[ a \right],\sigma^{2} } \right) \), we see that y follows the ‘double lognormal distribution,’ introduced by Meinhold and Singpurwalla (1987), whose mean can be approximated as \( E\left[ y \right] = 1 - E\left[ {U^{a} (w)} \right] \simeq\) \( \left( {1 - \sigma^{2} \exp \left\{ {\mu + \ell n\left[ a \right]} \right\}} \right)^{{ - \sigma^{ - 2} }} \) given our parameter values, so long as \( \sigma^{2} < {1 \mathord{\left/ {\vphantom {1 {\exp \left\{ {\mu + \ell n\left[ a \right]} \right\}}}} \right. \kern-0pt} {\exp \left\{ {\mu + \ell n\left[ a \right]} \right\}}} \) (ibid., page 106); and some manipulation then shows that a necessary condition for \( \frac{\partial }{\partial \nu }E\left[ {U^{a} (w)} \right] > 0 \), so that a higher variability is preferred, is that \( a < \exp \left\{ { - 1 - \mu } \right\} \), i.e. \( a_{0} \le \exp \left\{ { - 1 - \mu } \right\}. \)
The groups comprise values below and above the given quantile.
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Acknowledgments
We wish to thank Haim Levy and Jack Meyer for helpful comments on an earlier draft of this paper, and a perceptive referee whose systematic and astute comments have helped us to improve the presentation of our ideas.
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Yitzhaki, S., Lambert, P.J. Is higher variance necessarily bad for investment?. Rev Quant Finan Acc 43, 855–860 (2014). https://doi.org/10.1007/s11156-013-0395-3
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DOI: https://doi.org/10.1007/s11156-013-0395-3