Question

Previously, you studied linear combinations of independent random variables. What happens if the variables are not independent? A lot of mathematics can be used to prove the following: Let x and y be random variables with means Wy and Wyr variances o2, and y, and population correlation coefficient p (the Greek letter rho). Let a and b be any constants and let w = ax + by for the following formula. Ww=aux + buy 2 = 3202 + b2x2, + 2abog, In this formula, r is the population correlation coefficient, theoretically computed using the population of all (x,y) data pairs. The expression qp is called the covariance of x and y. If x and y are independent, then p = 0 and the formula for a reduces to the appropriate formula for independent variables. In most real-world applications the population parameters are not known, so we use sample estimates with the understanding that our conclusions are also estimates. Do you have to be rich to invest in bonds and real estate? No, mutual fund shares are available to you even if you aren't rich. Let x represent annual percentage return (after expenses) on the Vanguard Total Bond Index Fund, and let y represent annual percentage return on the Fidelity Real Estate Investment Fund. Over a long period of time, we have the following population estimates. u 7.33 06.58, 13.18, 18.55, p0.425 (a) Do you think the variables x and y are independent? Explain your answer. Yes. Interest rates probably has no effect on the investment returns. No. Interest rate probably affects both investment returns. No. Interest rates probably has no effect on the investment returns. Yes. Interest rate probably affects both investment returns. (b) Suppose you decide to put 65% of your investment in bonds and 35% in real estate. This means you will use a weighted average w = 0.65x + 0.35y. Estimate your expected percentage return w and risk ow. M = 9.3775 0 = 84.0484 X (c) Repeat part (b) if w = 0.35x + 0.65y. 4 = 11.1325 174.290 X (d) Compare your results in parts (b) and (c). Which investment has the higher expected return? Which has the greater risk as measured by ow?

Answer 1

b) σ²w =0.65^2*6.58^2+0.35^2*18.55^2+2*0.65*0.35*6.58*18.55*0.425= 84.0484

σw = √84.048 = 9.1678

c) σ²w =0.35^2*6.58^2+0.65^2*18.55^2+2*0.65*0.35*6.58*18.55*0.425= 174.2903

σw = √174.2903 = 13.2019

d) higher expected return = part c)

greater risk =part c)

investment of 35% in bonds and 65% in real estate