Reference no: EM131834567

A man invests a total of N dollars in a group of n securities, whose rates of return (interest rates) are independent random variables X₁, X₂, .........., X_n, respectively, with means i₁, i₂, ... , in and variances σ_l², σ₂², ... , a_n², respectively. If the man invests N_j dollars in the jth security, then his return in dollars on this particular portfolio is a random variable R given by R = N₁X₁ + N₂X₂ + ... + N_nX_n. Let the standard deviation σ[R] of R be used as a measure of the risk involved in selecting a given portfolio of securities. In particular, let us consider the problem of distributing investments of 5500 dollars between two securities, one of which has a rate of return Xv with mean 6 % and standard deviation 1 %, whereas the other has a rate of return X₂ with mean 15% and standard deviation 10%. (i) If it is desired to hold the risk to a minimum, what amounts N₁ and N₂ should be invested in the respective securities? What is the mean and variance of the return from this portfolio? (ii) What is the amount of risk that must be taken in order to achieve a portfolio whose mean return is equal to 400 dollars? (iii) By means of Chebyshev's inequality, find an interval, symmetric about 400 dollars, that, with probability greater than 75 %, will contain the return R from the portfolio with a mean return E[R] = 400 dollars. Would you be justified in assuming that the return R is approximately normally distributed?