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# Modern portfolio theory

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• Introduced by Harry Markowitz in the 1950s
• Cov_11, cov_22, …, cov_nn = var_1, var_2, …, var_n
• Source: www.investingdaily.com
• ### Modern portfolio theory

1. 1. Modern Portfolio Theory Trading Software and ProgrammingAcedo  Fabia  Reyes  Sorbito  Vidamo
2. 2. Modern Portfolio Theory • Markowitz provides the tools for identifying the portfolio which give the highest return for a particular level of risk (mean-variance portfolio theory) • Total risk of the portfolio can be reduced by diversification – this can be achieved by investing in assets that have low positive correlation, or better still, a Harry Markowitz negative correlation
3. 3. Markowitz model assumptions • Investors consider each investment alternative as being presented by a probability distribution of 1 expected returns over some holding period. • Investors maximize one-period expected utility, and their utility curves demonstrate 2 diminishing marginal utility of wealth. • Investors estimate the risk of the portfolio on the basis of the variability of expected returns. 3
4. 4. Markowitz model assumptions • Investors base decisions solely on expected return and risk, so their utility curves are a function of expected return and the expected variance (or standard deviation) 4 of returns only. • For a given level of risk, investors prefer higher returns to lower returns. Similarly, for a given level of expected 5 returns, investors prefer less risk to more risk. Under these assumptions, a single asset or portfolio of assets is considered to be efficient if no other asset or portfolio of assets offers higher expected return with the same (or lower) risk, or lower risk with the same (or higher) expected return.
5. 5. Mean variance optimizationFind the portfolio that that minimizes variancefor a given level of return, or maximizes returnfor a given level of riskBasic philosophy: don’t put all your eggs in onebasket! Key assumption: returns are normallydistributed
6. 6. Mean variance optimization Expected Standard return of deviation of each asset returns of each asset Correlation of returns between assets The Efficient Frontier
7. 7. General Formulas Given: Wi is the percent of the portfolio in asset i Ri is the possible rate of return for asset i E(Ri) is the expected rate of return for asset i Pi is the probability of the possible rate of return Ri n Expected Return E RPort Wi E Ri i 1 nMeasure of Risk 2 2 Variance Pi Ri E Ri i 1 n 2 Standard Deviation P Ri i E Ri i 1
8. 8. General FormulasGiven: Ri is the possible rate of return for asset i E(Ri) is the expected rate of return for asset i σi is the standard deviation in rates of return for asset i Covariance (2 asset portfolio Covij E Ri E Ri Rj E Rj i and j ) Covij rij Correlation i j
9. 9. General FormulasGiven: wi is the weights of the individual assets in the portfolio σ2i is the variance of the rate of return for asset i Covij is covariance bet the rates of return for assets i and j, where Covij = rij σi σj n n n 2 2 2Portfolio Variance port w i i wi w j Covij i 1 i 1 j 1 n n n PortfolioStandard Deviation port wi2 i 2 wi w j Covij i 1 i 1 j 1
10. 10. Optimization problem cov11 cov12  cov1n cov21 cov22  cov2 n V     covn1 covn 2  covnn
11. 11. Efficient frontier
12. 12. CriticismsEstimates relies on historical return dataand probability, assumption of normality Assumes all investors are rational, risk averse, maximize utility Assumes markets are efficient: assets are fairly valued or correctly priced No consideration for transaction cost or taxes
13. 13. References• Bodie, Kane and Marcus. Essentials of Investments, Eight Edition• Brown,K. and Reilly, F. Investment Analysis and Portfolio Management• Elton, E. and Gruber, M. Modern Portfolio Theory, 1950 to Date. Working Paper Series 1997.• Wilmott, P. Paul FAQs Quantitative Finance• www.investingdaily.com• Berkelaar, Arjan. Portfolio Optimization powerpoint presentation