1. combined variance = variance of square A * A of investment ratio+variance of square B * B of investment ratio +2 * investment ratio A * investment ratio B * standard deviation A * standard deviation b * correlation coefficient of a and B.
=x^2*0.3^2+( 1-x)^2*0.25^2+2x( 1-x)*0.3*0.25*(- 1)
X is the investment ratio of A, and 1-x is of course the investment ratio of B. 。
Find the minimum variance, take the first derivative of X to make it equal to 0, and calculate x=5/ 1 1 (parabolic principle can be used without derivative).
Put x back into the formula for calculating variance and get the minimum variance =0.
2. In the same way, the difference is that A and B are completely irrelevant, and the correlation coefficient =0.
Expected return and variance are used to measure the return and risk of a single security. A portfolio consists of a certain number of individual securities, each of which occupies a certain proportion. We can also regard the portfolio as a kind of securities, and the return and risk of the portfolio can also be measured by expected return and variance. However, the expected return and variance of a portfolio can be expressed by the expected return and variance of a single security composed of it.
Returns and risks of two securities portfolios
There are two kinds of securities, A and B. An investor invests a sum of money in securities A according to the ratio of X and B according to the ratio of Y, and x+y= 1, which means that the investor has a portfolio P. If the yield of securities A and B is A and Q:
Q=ax+by
The weight in the portfolio can be negative, such as x 1. Investors do not know the exact values of x and y when making investment decisions, so x and y should be random variables, and the simplified description of their distribution is their expected values and variances. The expected return e and return variance of portfolio p are: E=xa+yb.
Variance = square of x × variance of securities A+square of y × variance of securities B +2xy× standard deviation of securities A × standard deviation of securities B × correlation coefficient of securities portfolio.
Where: the standard deviation of security A × the standard deviation of security B × the correlation coefficient-covariance of portfolio, which is denoted as COV(A, b).