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An Introduction to Modern Portfolio Theory (MPT) in Portfolio Optimization
In every investment decision on stock, as a rational investor, the attention is always aimed at the return of the investment. Investors will choose investments in stocks that give the highest expected return (return). However, in real life investments, each selection of stock will contain an element of uncertainty, so investors should consider the risk factor [1]. This leads to the combination of multiple stocks and the weight of holding it will result in different expected returns and different risk associated with the investors.
As shown in the Figure 1 above, consider an investor portfolio holding that select can be purchased with X amount of capital in a set of stocks for example, Real Estate Investment Trust (REIT) set of stocks. With the same amount of capital X, Portfolio B contribute to a lower expected return and higher risk bare by the investor and it can be optimize to have a lower risk but similar expected return (Portfolio C) or a higher expected return with a same risk (Portfolio A) by changing the weight of holding in each stock in the REIT stock sets according to the preference of investor. A simpler mathematical model that illustrate the relationship is as below:
Max Z₁ = A₁X₁ + A₂X₂ + A₃X₃ + A₄X₄….+ AₙXₙ
Where:
Aₙ = Amount of expected return for particular stock n
Xₙ = Weight in portfolio holding stock n.
Z₁ = Expected return of portfolio
Or
Min Z₂ = A₁X₁ + A₂X₂ + A₃X₃ + A₄X₄….+ AₙXₙ
Where:
Aₙ = Amount of risk* associated for particular stock n
Xₙ = Weight in portfolio holding stock n
Z₂ = Risk associated with the portfolio
*Risk associated absolute deviation of the rate of return, and not the standard deviation as illustrated in the Markowitz Model
The relationship of above mathematics is linear, however, in real world situations…
