9.5 Summary
In Chapter 1, we saw that it is very hard to quantify risk. R&S (1970) establish-several equivalent definitions of risk (and variance is not one of them!) that enable us to define one variable as “more risky” than another for equal means distributions. According to these definitions, risk is not quantified, but ranking investments by their risk is enabled.
Where Ef(x) ≠ EG(x), R&S’s definition still holds and it is possible to establish two sets of mixes of the random variables with the riskless asset such that Gβ is more risky than Fα. Finally, the addition of a DARA assumption may enable the ranking of investments by their risk in Ud when it is impossible to do so in U2: While G is riskier than F in U2 if G = F + MPS, G will be riskier than F in Ud if G = F + MPSA where MPSA denotes mean preserving spread antispread.
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References
Rothschild, M. and J.E. Stiglitz, 1970, “Increasing Risk: I. A Definition, Journal of Economic Theory, 1970, pp. 225–343.
Leshno, M., Levy, H. and Spector, Y., “A Comment on Rothschild and Stiglitz’s Increasing Risk: I. A Definition,” Journal of Economic Theory, 1997, pp. 223–228.
Levy, H., 1977, “The Definition of Risk: An Extension,” Journal of Economic Theory, 14, 1977, pp. 232–234.
Kroll, Y., Leshno, M., Levy, H., and Spector, Y., “Increasing Risk, Decreasing Absolute Risk Aversion and Diversification,” Journal of Mathematical Economics, 24, 1995, pp. 537–556.
See Leshno, M., Levy, H., and Spector, Y., “Increasing Risk, Decreasing Absolute Risk Aversion and Diversification,” Journal of Mathematical Economics, 24, 1995, pp. 537–556 footnote 4.
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(2006). Stochastic Dominance and Risk Measures. In: Stochastic Dominance. Studies in Risk and Uncertainty, vol 12. Springer, Boston, MA . https://doi.org/10.1007/0-387-29311-6_9
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