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 μx and μy, variances σ2x and σ2y, and population correlation coefficient ρ (the Greek letter rho). Let a and b be any constants and let w = ax + by for the following formula.

μw = aμx + bμy

σ2w = a2σ2x + b2σ2y + 2abσxσyρ

In this formula, r is the population correlation coefficient, theoretically computed using the population of all (x, y) data pairs. The expression σxσyρ is called the covariance of x and y. If x and y are independent, then ρ = 0 and the formula for σ2w 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.

μx ≈ 7.34, σx ≈ 6.56, μy ≈ 13.17, σy ≈ 18.59, ρ ≈ 0.424

(b) Suppose you decide to put 60% of your investment in bonds and 40% in real estate. This means you will use a weighted average w = 0.6x + 0.4y. Estimate your expected percentage return μ_w and risk σ_w.
μ_w =
σ_w =

(c) Repeat part (b) if w = 0.4x + 0.6y.
μ_w =
σ_w =

Answer 1