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| :''[[Mean-variance analysis]] redirects here. For mean-variance portfolio theory, see [[Modern portfolio theory]] or [[Mutual fund separation theorem]].''
| | Hello from Australia. I'm glad to came across you. My first name is Heath. <br>I live in a small town called Mount Napier in western Australia.<br>I was also born in Mount Napier 31 years ago. Married in December 2011. I'm working at the the office.<br><br>My homepage :: [http://minecraftplayforfreeonline.net/profile/184452/qtwma Transfering to mountain bike sizing.] |
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| In [[decision theory]], [[economics]], and [[finance]], a '''two-moment decision model''' is a model that [[Positive economics|describes]] or [[Normative economics|prescribes]] the process of making decisions in a context in which the decision-maker is faced with [[random variable]]s whose realizations cannot be known in advance, and in which choices are made based on knowledge of two [[Moment (mathematics)|moments]] of those random variables. The two moments are almost always the mean—that is, the [[expected value]], which is the first moment about zero—and the [[variance]], which is the second moment about the mean (or the [[standard deviation]], which is the square root of the variance).
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| The most well-known two-moment decision model is that of [[modern portfolio theory]], which gives rise to the decision portion of the [[Capital Asset Pricing Model]]; these employ '''mean-variance analysis''', and focus on the mean and variance of a portfolio's final value.
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| ==Two-moment models and expected utility maximization==
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| Suppose that all relevant random variables are in the same [[location-scale family]], meaning that the distribution of every random variable is the same as the distribution of some linear transformation of any other random variable. Then for any [[von Neumann–Morgenstern utility function]], using a mean-variance decision framework is consistent with [[Expected utility hypothesis|expected utility]] maximization,<ref>{{cite journal |last=Mayshar |first=J. |title=A note on Feldstein's criticism of mean-variance analysis |journal=[[Review of Economic Studies]] |volume=45 |year=1978 |issue=1 |pages=197–199 |doi= |jstor=2297094 }}</ref><ref>{{cite book |last=Sinn |first=H.-W. |authorlink=Hans-Werner Sinn |title=Economic Decisions under Uncertainty |edition=Second English |year=1983 |publisher=North-Holland |location=Amsterdam |isbn=0444863877 }}</ref> as illustrated in example 1:
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| ''Example'' 1:<ref name=Meyer/><ref name=Tobin/><ref name=Mueller/><ref name=Thorn/><ref name=Williams/><ref name=bookx/><ref name=Tobin2/><ref name=Laidler/> Let there be one risky asset with random return ''r'', and one riskfree asset with known return ''r''<sub>''f''</sub>, and let an investor's initial wealth be ''w''<sub>0</sub>. If the amount ''q'', the choice variable, is to be invested in the risky asset and the amount ''w''<sub>0</sub> – ''q'' is to be invested in the safe asset, then contingent on ''q'' the investor's random final wealth will be ''w'' = (''w''<sub>0</sub> – ''q'')''r''<sub>''f''</sub> + ''qr''. Then for any choice of ''q'', ''w'' is distributed as a location-scale transformation of ''r''. If we define random variable ''x'' as equal in distribution to <math> \tfrac{w-\mu_w}{\sigma_w},</math> then ''w'' is equal in distribution to <math> \mu_w + \sigma_w x </math>, where ''μ'' represents an expected value and σ represents a random variable's [[standard deviation]] (the square root of its second moment). Thus we can write expected utility in terms of two moments of ''w'':
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| :<math>\operatorname{E}u(w)=\int_{- \infty} ^ \infty \! u(\mu_w+ \sigma _w x)f(x) \, dx \equiv v(\mu_w, \sigma_w),</math>
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| where ''u'' is the [[von Neumann–Morgenstern utility theorem|von Neumann–Morgenstern utility function]], ''f'' is the [[density function]] of ''x'', and ''v'' is the derived mean-standard deviation choice function, which depends in form on the density function ''f''. The von Neumann–Morgenstern utility function is assumed to be increasing, implying that more wealth is preferred to less, and it is assumed to be concave, which is the same as assuming that the individual is [[risk aversion|risk averse]].
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| It can be shown that the partial derivative of ''v'' with respect to ''μ<sub>w</sub>'' is positive, and the partial derivative of ''v'' with respect to σ<sub>''w''</sub> is negative; thus more expected wealth is always liked, and more risk (as measured by the standard deviation of wealth) is always disliked. A mean-standard deviation [[indifference curve]] is defined as the locus of points (''σ''<sub>''w''</sub>, ''μ''<sub>''w''</sub>) with ''σ''<sub>''w''</sub> plotted horizontally, such that E''u''(''w'') has the same value at all points on the locus. Then the derivatives of ''v'' imply that every indifference curve is upward sloped: that is, along any indifference curve ''dμ<sub>w</sub>'' / ''d''σ<sub>''w''</sub> > 0. Moreover, it can be shown<ref name=Meyer/> that all such indifference curves are convex: along any indifference curve, ''d''<sup>2</sup>μ<sub>w</sub> / ''d''(σ<sub>''w''</sub>)<sup>2</sup> > 0.
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| ''Example'' 2: The portfolio analysis in example 1 can be generalized. If there are ''n'' risky assets instead of just one, and if their returns are [[elliptical distribution|jointly elliptically distributed]], then all portfolios can be characterized completely by their mean and variance—that is, any two portfolios with identical mean and variance of portfolio return have identical distributions of portfolio return—and all possible portfolios have return distributions that are location-scale-related to each other.<ref>{{cite journal |last=Chamberlain |first=G. |title=A characterization of the distributions that imply mean-variance utility functions |journal=[[Journal of Economic Theory]] |volume=29 |year=1983 |issue=1 |pages=185–201 |doi=10.1016/0022-0531(83)90129-1 }}</ref><ref>{{cite journal |last=Owen |first=J. |last2=Rabinovitch |first2=R. |title=On the class of elliptical distributions and their applications to the theory of portfolio choice |journal=[[Journal of Finance]] |volume=38 |year=1983 |issue=3 |pages=745–752 |doi= |jstor=2328079 }}</ref> Thus portfolio optimization can be implemented using a two-moment decision model.
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| ''Example'' 3: Suppose that a [[Perfect competition|price-taking]], [[risk aversion|risk-averse]] firm must commit to producing a quantity of output ''q'' before observing the market realization ''p'' of the product's price.<ref>{{cite journal |last=Sandmo |first=Agnar |title=On the theory of the competitive firm under price uncertainty |journal=[[American Economic Review]] |volume=61 |year=1971 |issue=1 |pages=65–73 |doi= |jstor=1910541 }}</ref> Its decision problem is to choose ''q'' so as to maximize the expected utility of profit:
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| :Maximize E''u''(''pq'' – ''c''(''q'') – ''g''),
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| where E is the [[expected value]] operator, ''u'' is the firm's utility function, ''c'' is its [[Cost curve|variable cost function]], and ''g'' is its [[fixed cost]]. All possible distributions of the firm's random revenue ''pq'', based on all possible choices of ''q'', are location-scale related; so the decision problem can be framed in terms of the expected value and variance of revenue.
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| ==Non-expected-utility decision making==
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| If the decision-maker is [[Generalized expected utility|not an expected utility maximizer]], decision-making can still be framed in terms of the mean and variance of a random variable if all alternative distributions for an unpredictable outcome are location-scale transformations of each other.<ref>Bar-Shira, Z., and Finkelshtain, I., "Two-moments decision models and utility-representable preferences," ''Journal of Economic Behavior and Organization'' 38, 1999, 237-244. See also Mitchell, Douglas W., and Gelles, Gregory M., "Two-moments decision models and utility-representable preferences: A comment on Bar-Shira and Finkelshtain, vol. 49, 2002, 423-427.</ref>
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| ==See also==
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| *[[Decision theory]]
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| *[[Intertemporal portfolio choice]]
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| *[[Microeconomics]]
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| ==References==
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| {{Reflist|refs=
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| <ref name=Meyer>{{cite journal |last=Meyer |first=Jack |title=Two-moment decision models and expected utility maximization |journal=[[American Economic Review]] |volume=77 |year=1987 |issue=3 |pages=421–430 |doi= |jstor=1804104 }}</ref>
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| <ref name=Tobin>{{cite journal |last=Tobin |first=J. |title=Liquidity preference as behavior towards risk |journal=[[Review of Economic Studies]] |volume=25 |issue=1 |year=1958 |pages=65–86 }} </ref>
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| <ref name=Mueller> {{cite book |editor-first=M. G. |editor-last=Mueller |title=Readings in Macroeconomics |publisher=Holt, Rinehart & Winston |year=1966 |pages=65–86 }} </ref>
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| <ref name=Thorn> {{cite book |editor-first=Richard S. |editor-last=Thorn |title=Monetary Theory and Policy |location= |publisher=Random House |year=1966 |pages=172–191 }} </ref>
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| <ref name=Williams> {{cite book |editor-first=H. R. |editor-last=Williams |editor2-first=J. D. |editor2-last=Huffnagle |title=Macroeconomic Theory |publisher=Appleton-Century-Crofts |year=1969 |pages=299–324 }} </ref>
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| <ref name=bookx> {{cite book |first1=J.|last1=Tobin|title=Essays in Economics: Macroeconomics |volume=1 |chapter=Chapter 15: Liquidity Preference as Behavior towards Risk |year=1971|isbn= 0262200627|publisher=MIT Press }}</ref>
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| <ref name=Tobin2>Tobin, J.; Hester, D. eds. (1967) ''Risk Aversion and Portfolio Choice'', Cowles Monograph No. 19, John Wiley & Sons {{pn|date=April 2013}} </ref>
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| <ref name=Laidler>David Laidler, ed. (1999) ''The Foundations of Monetary Economics, Vol. 1'', Edward Elgar Publishing Ltd. {{pn|date=April 2013}}</ref>
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| }}
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| {{DEFAULTSORT:Two-Moment Decision Models}}
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| [[Category:Decision theory]]
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| [[Category:Risk]]
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| [[Category:Mathematical finance]]
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Hello from Australia. I'm glad to came across you. My first name is Heath.
I live in a small town called Mount Napier in western Australia.
I was also born in Mount Napier 31 years ago. Married in December 2011. I'm working at the the office.
My homepage :: Transfering to mountain bike sizing.