Abstract
We have seen that the MEUC is the optimal investment criterion. If there is full information on preferences (e.g., U(w) = log (w)), we simply calculate EU(w) of all the competing investments and choose the one with the highest expected utility. In such a case, we arrive at a complete ordering of the investments under consideration: there will be one investment which is better than (or equal to) all of the other available investments. Moreover, with a complete ordering, we can order the investments from best to worst. Generally, however, we have only partial information on preferences (e.g., risk aversion) and, therefore, we arrive only at a partial ordering of the available investments. Stochastic dominance rules as well as other investment rules (e.g., the mean-variance rule) employ partial information on the investor’s preferences or the random variables (returns) and, therefore, they produce only partial ordering.
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Notes
- 1.
Hanoch, G. and H. Levy, “The Efficiency Analysis of Choices Involving Risk ,” Review of Economic Studies, 36, 1969, pp. 335–346.
- 2.
Tesfatsion, L., “Stochastic Dominance and the Maximization of Expected Utility,” Review of Economic Studies, 43, 1976, pp. 301–315.
- 3.
Utility is defined in terms of terminal wealth. Hence, for $1 investment, a rate of return of −5 % implies a terminal wealth of $0.95. In most examples we will use the rates of return rather than terminal wealth without affecting the analysis. However, in some cases, we need to adhere to terminal wealth. For example, if the utility function is U(x) = log x then it is not defined for x = −5 % but it is defined for x = 0.95. We will elaborate on this issue as we proceed.
- 4.
See R. Ibbotson and Associate, Stocks, Bonds, Bills and Inflation. (Chicago, IL; Ibbotson Associate various yearbooks).
- 5.
The function U0(x) is not differentiable at x = x0. However, for this specific function, we can always obtain a differentiable utility function that is arbitrarily as close as one wishes to U0.
- 6.
We can shift to terminal wealth by adding $1 to all figures of x without affecting the result of this example. See also footnote 3.
- 7.
Milton Friedman and Leonard J. Savage, “The Utility Analysis of Choices Involving Risk ,” The Journal of Political Economy, LVI, No. 4, August 1948, pp. 279–304.
- 8.
See Kahneman, Daniel and Tversky, Amos, “Prospect Theory : An Analysis of Decision Under Risk ,” Econometrica, Vol 47, 1979, pp. 263–291.
- 9.
See F.D. Arditti, “Rate and the Required Return on Equity,” Journal of Finance, 22, 1967. pp. 19–36.
- 10.
See K.J. Arrow, Aspects of the Theory of Risk - Bearings, Helsenki, Yrjö Jahnssonin Säätiö, 1965
- 11.
See J.W. Pratt, “Risk Aversion in the Small and in the Large,” Econometrica, 32, 1964, pp. 122–136.
- 12.
Another U0(x) that can be employed in the necessity proof is a linear utility function in most of the range with a small range x0 ≤ x ≤ x0 + ε at which U‴ > 0. Thus, this function is close to the linear function (for sufficiently small ε) which can be used to prove that Ef(x) ≥ E G(x) is a necessary condition for TSD of F over G.
- 13.
See J.S. Hammond III, “Simplifying the Choice Between Uncertain Prospects where Preference is Nonlinear, Management Science, 20, 1974, pp. 1047–1072.
- 14.
See R.G. Vickson, “Stochastic Dominance Tests for Decreasing Absolute Risk Aversion . I. Discrete Random Variables,” Management Science, 21, 1975, pp. 1438–1446 and “Stochastic Dominance Tests for Decreasing Absolute Risk Aversion II: General Random Variables,” Management Science, 23, 1977, pp. 478–489.
- 15.
Friedman, M. and L.J. Savage, “The utility analysis of choices involving risk ,” The Journal of Political Economics, 56, 1948, pp. 279–304.
- 16.
Markowitz, H.M., “The Utility of Wealth,” The Journal of Political Economy, 60, 1952b, pp. 151–158.
- 17.
Kahneman, D. and A. Tversky, “Prospect Theory : An Analysis of Decision Under Risk ,” Econometrica, 47, 1979, pp. 263–291.
- 18.
One can also establish a similar utility function which is differential in all points (see footnote 5).
- 19.
it can be shown that the situation FDG by SSD and GDF by \( \overline{SSD} \), is possible only if E F (x) = E G (x), as the linear utility function is a borderline between risk -seeking and risk averse utility function s.
- 20.
Actually, this follows from the properties of Riemann - Stieltjes integral. However, we provide here the detailed proof.
- 21.
Note that G may not start at x0 = 0 or may not end at xn+1 = b. In such a case, simply add these two values with zero probability to obtain the same formulation as for F. In such a case, we may have p(x0) = 0 and p(xn+1) = 0 but this does not change the generality of the above proof.
- 22.
Hanoch, G. and H. Levy, “The Efficiency Analysis of Choices Involving Risk ,” Review of Economic Studies, 36, pp. 335–346, 1969.
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Levy, H. (2016). Stochastic Dominance Decision Rules. In: Stochastic Dominance. Springer, Cham. https://doi.org/10.1007/978-3-319-21708-6_3
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