In this exercise, we examine the effect of combining investmentswith positively correlated risks, negatively correlated risks, anduncorrelated risks. A firm is considering a portfolio of assets.The
portfolio is comprised of two assets, which we will call ''A\" and\"B.\" Let X denote the annual rate of return from asset A in thefollowing year, and let Y denote the annual rate of return fromasset B in the following year. Suppose that
E(X) = 0.15 and E(Y) = 0.20,
SD(X) = 0.05 and SD(Y) = 0.06,
and CORR(X, Y) = 0.30.
(a) What is the expected return of investing 50% of the portfolioin asset A and 50% of the portfolio in asset B? What is thestandard deviation of this return?
(b) Replace CORR(X, Y) = 0.30 by CORR(X, Y) = 0.60 and answer thequestions in part (a). Do the same for CORR(X, Y) = 0.60, 0.30, and0.0.
(c) (Spreadsheet Exercise). Use a spreadsheet to perform thefollowing analysis. Suppose that the fraction of the portfolio thatis invested in asset B is f, and so the fraction of the portfoliothat is invested in asset A is (1 f). Letting f vary from f = 0.0to f = 1.0 in increments of 5% (that is, f = 0.0, 0.05, 0.10, 0.15,. . . ), compute the mean and the standard deviation of the annualrate of return of the portfolio (using the original data for theproblem). Notice that the expected return of the portfolio varies(linearly) from 0.15 to 0.20, and the standard deviation of thereturn varies (non-linearly) from 0.05 to 0.06. Construct a chartplotting the standard deviation as a function of the expectedreturn.
(d) (Spreadsheet Exercise). Perform the same analysis as in part(c) with CORR (X, Y) = 0.30 replaced by CORR(X, Y) = 0.60, 0.0,0.30, and 0.60.