理解Mean-Variance Portfolio Theory In MPT

Markowitz Mean-Variance Portfolio Theory

An investment instrument that can be bought and sold is often called an asset.

Suppose we purchase an asset for x0x_0x0 dollars on one date and then later sell it for x1x_1x1 dollars. We call the ratio:
R=x1x0R=\frac{x_1}{x_0} R=x0x1
the return on the asset. The rate of return on the asset is given by :
r=x1−x0x0=R−1r=\frac{x_1-x_0}{x_0}=R-1 r=x0x1−x0=R−1
Therefore,
x1=Rx0andx1=(1+r)x0x_1=Rx_0 \; and \; x_1=(1+r)x_0 x1=Rx0andx1=(1+r)x0
Somethinds it is possible to sell and asset we do not own. This is called short selling. On your account asset sheet, the short sale appears as a negative number associated with the shorted asset, this number is not denominated in dollars, but rather in the number of stocks shorted.
Mean-Variance Analysis

Mean-Variance analysis is the process of weighting risk, expressed as variance, against expected return. Investors use mean-variance analysis to make decisions about which financial instruments to invest in, based on how much risk they are willing to take on in exchange for different levels of reward. Mean-variance analysis allows investors to find the biggest reward at a given level of risk or the least risk at a given level of return.

Mean-variance analysis is one part of modern portfolio theory, which assumes that investors will make rational decisions about investments if they have complete information. One assumption is that investors want low risk and high reward.

There are two main parts of mean-variance analysis:
- variance
  
  Variance is a number that represents how varied or spread out the numbers in a set are.
  
  For example, variance may tell how spread out the returns of a specific security are on a daily or weekly basis.
- expected return
  
  The expected return is a probability expressing the estimated return of the investment in the security.
If two different securities have the same expected return, but one has lower variance, the one with lower variance is the better pick.

Similarly, if two different securities have approximately the same variance, the one with the higher return is the better pick.

In modern portfolio theory, an investor would choose different securities to invest in with different levels of variance and expected return.

Sample Mean-Variance Analysis

Investment	Amount	Expected Return	weight	Standard Deviation (square root of variance)
A	$100,000	5%	25%	7%
B	$300,000	10%	75%	14%
Portfolio	$400,000	25%∗5%+75%∗10%=8.75%25\%5\% + 75\%10\% = 8.75\%25%∗5%+75%∗10%=8.75%		(25%2∗7%2)+(75%2∗14%2)+(2∗25%∗7%∗75%∗14%∗0.65)=11.71%\sqrt{(25\%^27\%^2)+(75\%^214\%^2)+(225\%7\%75\%14\%*0.65)}=11.71\%(25%2∗7%2)+(75%2∗14%2)+(2∗25%∗7%∗75%∗14%∗0.65)=11.71%

The correlation between the two investments is 0.65

References

Investopedia : Mean-Variance Analysis
THE MEAN-VARIANCE MODEL, Zdenek Konfrst, Czech Technical University
Markowitz Mean-Variance Portfolio Theory

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理解Mean-Variance Portfolio Theory In MPT

Markowitz Mean-Variance Portfolio Theory

Mean-Variance Analysis

variance

expected return

Sample Mean-Variance Analysis

References

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