Editing Modern portfolio theory (section)

===Risk and expected return===
{{unreferenced section|date=April 2021}}

MPT assumes that investors are [[risk averse]], meaning that given two portfolios that offer the same expected return, investors will prefer the less risky one.  Thus, an investor will take on increased risk only if compensated by higher expected returns. Conversely, an investor who wants higher expected returns must accept more risk. The exact trade-off will not be the same for all investors. Different investors will evaluate the trade-off differently based on individual risk aversion characteristics.  The implication is that a [[Rationality|rational]] investor will not invest in a portfolio if a second portfolio exists with a more favorable [[risk-return spectrum|risk vs expected return profile]] — i.e., if for that level of risk an alternative portfolio exists that has better expected returns.

Under the model:
*Portfolio return is the [[linear combination|proportion-weighted combination]] of the constituent assets' returns.
*Portfolio return volatility <math>\sigma_p</math> is a function of the [[correlation]]s ''ρ''<sub>ij</sub> of the component assets, for all asset pairs (''i'', ''j''). The volatility gives insight into the risk which is associated with the investment. The higher the volatility, the higher the risk.

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In general:

*Expected return:
:<math> \operatorname{E}(R_p) = \sum_i w_i \operatorname{E}(R_i) \quad </math>

:where <math>R_p</math> is the return on the portfolio, <math> R_i </math> is the return on asset ''i'' and <math> w_i </math> is the weighting of component asset <math> i </math> (that is, the proportion of asset "i" in the portfolio, so that <math>\sum_i w_i = 1</math>).

*Portfolio return variance:
:<math> \sigma_p^2 = \sum_i w_i^2 \sigma_{i}^2 + \sum_i \sum_{j \neq i} w_i w_j \sigma_i \sigma_j \rho_{ij} </math>,
:where <math> \sigma_{i} </math> is the (sample) standard deviation of the periodic returns on an asset ''i'', and <math>\rho_{ij}</math> is the [[correlation coefficient]] between the returns on assets ''i'' and ''j''. Alternatively the expression can be written as:

:<math> \sigma_p^2 = \sum_i \sum_j w_i w_j \sigma_i \sigma_j \rho_{ij} </math>,
:where <math> \rho_{ij} = 1 </math> for <math> i =  j </math> , or

:<math> \sigma_p^2 = \sum_i \sum_j w_i w_j \sigma_{ij} </math>,
:where <math> \sigma_{ij} = \sigma_i \sigma_j \rho_{ij} </math> is the (sample) covariance of the periodic returns on the two assets, or alternatively denoted as <math> \sigma(i,j) </math>, <math> \text{cov}_{ij} </math> or <math> \text{cov}(i,j) </math>.

*Portfolio return volatility (standard deviation):
:<math>  \sigma_p = \sqrt {\sigma_p^2} </math>

For a '''two-asset''' portfolio:
*Portfolio expected return: <math> \operatorname{E}(R_p) =  w_A \operatorname{E}(R_A) +
w_B \operatorname{E}(R_B) = w_A \operatorname{E}(R_A) + (1 - w_A) \operatorname{E}(R_B). </math>
*Portfolio variance: <math> \sigma_p^2  = w_A^2 \sigma_A^2  + w_B^2 \sigma_B^2 + 2w_Aw_B  \sigma_{A} \sigma_{B} \rho_{AB}</math>

For a '''three-asset''' portfolio:
*Portfolio expected return: <math> \operatorname{E}(R_p) = w_A \operatorname{E}(R_A) + w_B \operatorname{E}(R_B) + w_C \operatorname{E}(R_C) </math>
*Portfolio variance: <math> \sigma_p^2  = w_A^2 \sigma_A^2  + w_B^2 \sigma_B^2 + w_C^2 \sigma_C^2 + 2w_Aw_B  \sigma_{A} \sigma_{B} \rho_{AB}
+ 2w_Aw_C  \sigma_{A} \sigma_{C} \rho_{AC} + 2w_Bw_C  \sigma_{B} \sigma_{C} \rho_{BC}</math>

The algebra can be much simplified by expressing the quantities involved in matrix notation.<ref>David Lando and Rolf Poulsen's lecture notes, Chapter 9, "Portfolio theory" [http://www2.imm.dtu.dk/courses/02724/general_information/RolfNotes/ifnotes.pdf]</ref> Arrange the returns of N risky assets in an <math> N\times 1</math> vector <math> R</math>, where the first element is the return of the first asset, the second element of the second asset, and so on. Arrange their expected returns in a column vector <math> \mu </math>, and their variances and covariances in a [[covariance matrix]] <math>\Sigma</math>. Consider a portfolio of risky assets whose weights in each of the N risky assets is given by the corresponding element of the weight vector <math> w</math>. Then:

*Portfolio expected return:  <math> w'\mu</math>
and
*Portfolio variance: <math>w'\Sigma w</math>

For the case where there is investment in a riskfree asset with return <math>R_f</math>, the weights of the weight vector do not sum to 1, and the portfolio expected return becomes <math> w'\mu+(1-w'1)R_f</math>. The expression for the portfolio variance is unchanged.

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