MODERN PORTFOLIO THEORY MATH-MATICAL MODEL In some sense the mathematical derivation below is MPT, although the basic concepts behind the model have also been very influential.! This section develops the classic MPT model. There have been many extensions since.! RISK & EXPECTED RETURN - 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 be the same for all investors, but different investors will evaluate the trade-off differently based on individual risk aversion characteristics. The implication is that a rational investor will not invest in a portfolio if a second portfolio exists with a more favorable risk-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 proportion-weighted combination of the constituent assets returns. Portfolio volatility is a function of the correlations ρij of the component assets, for all asset pairs (i, j). In general: Expected return: \operatorname{E}(R_p) = \sum_i w_i \operatorname{E}(R_i) \quad where R_p is the return on the portfolio, R_i is the return on asset i and w_i is the weighting of component asset i (that is, the proportion of asset i in the portfolio). Portfolio return variance: \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}, where \rho_{ij} is the correlation coefficient between the returns on assets i and j. Alternatively the expression can be written as: \sigma_p^2 = \sum_i \sum_j w_i w_j \sigma_i \sigma_j \rho_{ij} , where \rho_{ij} = 1 for i=j. Portfolio return volatility (standard deviation): \sigma_p = \sqrt {\sigma_p^2} For a two asset portfolio: Portfolio return: \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). Portfolio variance: \sigma_p^2 = w_A^2 \sigma_A^2 + w_B^2 \sigma_B^2 + 2w_Aw_B \sigma_{A} \sigma_{B} \rho_{AB} For a three asset portfolio: Portfolio return: w_A \operatorname{E}(R_A) + w_B \operatorname{E}(R_B) + w_C \operatorname{E}(R_C) Portfolio variance: \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} DIVERSIFICATION - An investor can reduce portfolio risk simply by holding combinations of instruments that are not perfectly positively correlated (correlation coefficient -1 \le \rho_{ij}< 1). In other words, investors can reduce their exposure to individual asset risk by holding a diversified portfolio of assets. Diversification may allow for the same portfolio expected return with reduced risk. These ideas have been started with Markowitz and then reinforced by other economists and mathematicians such as Andrew Brennan who have expressed ideas in the limitation of variance through portfolio theory.! If all the asset pairs have correlations of 0—they are perfectly uncorrelated—the portfolios return variance is the sum over all assets of the square of the fraction held in the asset times the assets return variance and the portfolio standard deviation is the square root of this sum.! HAVE A NICE DAY ,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,, FRIENDS !
Posted on: Sat, 27 Dec 2014 13:17:46 +0000
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