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The method used in selecting the most desirable portfolio involves the use of indifference curves. These curves represent an investor's preferences for risk and return. It can be drawn on a twodimensional graph, where the horizontal axis usually indicates risk as measured by variance or standard deviation and the vertical axis indicates reward as mesured by expected return. Using variance as relevant risk measure comes from Markowitz's paper and is always used in practice, although other possibilities have been considered (see [#!Rothschild:Risk!#].)
This definition gives us the following properties, assuming we have a 'rational investor'^{1}:
 All portfolios that lie on the same indifference curve are equally desirable to the investor (even though they have different expected returns and variance.) An obvious implication is that indifference curves do not intersect.
 An investor will find any portfolio that is lying on an indifference curve that is "further northwest" to be more desirable than any portfolio lying on an indifference curve that is "not as far northwest."
But how are the indifference curves shaped?
Generally it is assumed that investors are risk averse, which means that the investor will choose the portfolio with the smaller variance given the same return. Risk averse investors will not want to take fair gambles (where the expected payoff is zero).
These two assumptions of nonsatiation and risk aversion cause indifference curves to be positively sloped and convex.
Figure:
A high, moderately and slightly risk averse indifference curves.

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Magnus Bjornsson
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