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 $x_0$ dollars on one date and then later sell it for $x_1$ dollars. We call the ratio: $$R=\frac{x_1}{x_0}$$ the return on the asset. The rate of return on the asset is given by : $$r=\frac{x_1-x_0}{x_0}=R-1$$ Therefore, $$x_1=R{x}_{0}and{x}_1=(1+r)x_0$$ 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

InvestmentAmountExpected ReturnweightStandard Deviation(square root of variance)A$100,0005%25%7%B$300,00010%75%14%Portfolio$400,000$25\%\ast 5\%+75\%\ast 10\%=8.75\%$$\sqrt{(25{\%}^2\ast 7{\%}^2)+(75{\%}^2\ast 14{\%}^2)+(2\ast 25\%\ast 7\%\ast 75\%\ast 14\%\ast 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