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Chapter 6. Mean-Variance Portfolio Theory || Part 2 본문

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Chapter 6. Mean-Variance Portfolio Theory || Part 2

incastle 2020. 5. 16. 15:09

6.2 Random Variables

Random Returns

  • Frequently the amount of money to be obtained when selling an asset is uncertain at the time of purchase

  • In that case, return is random and can be described in probabilistic terms

Random Variable

Expected Value

  • Expected value of a random variable x is the average value obtained by regarding the probabilities as frequencies

  • Terms mean or mean value are often used for the expected value

Variance

  • Measure of the degree of possible deviation from the mean

Standard Deviation

  • We frequently use the square root of the varianceàStandard deviation

 

Independent variables : Outcome probabilities for one variable do not depend on the outcome of the other

 

Covariance

  • When considering two or more random variables, their mutual dependence can be summarized conveniently by their covariance

  • Covariance can also be expressed as

 

Variance of a Sum

 

 

6.4 Portfolio Mean and Variance

Mean Return of a Portfolio

  • Rate of return of the portfolio in terms of the return of the individual returns is

r은 random rate, w는 투자 weight

  • We can take the expected value of both sides to get (using linearity)

  • Expected rate of return of the portfolio is found by taking the weighted sum of the individual expected rate of returns

 

Variance of Portfolio Return

  • Variance of the rate of return of a portfolio can be found by performing a straight forward calculation

만약 2개가 아니고 3개라면

Diversification

  • Portfolios with only a few assets may be subject to a high degree of risk, represented by a relatively large variance

  • Risk of the return of a portfolio can be reduced by including additional assets in the portfolioe => Diversification

  • Effect of diversification can be explained by the mean-variance framework
    • >> By using variance for measuring risk

  • In general, diversification may reduce the overall expected return while reducing the variance

  • Diversification is not desirable without an understanding of its influence on both the mean and variance of return

  • Mean-variance approach helps understand the trade-off between mean and variance

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