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 μ_x and μ_y, variances σ²_x and σ²_y, 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.

μ_w = aμ_x + bμ_y
σ²_w = a²σ²_x + b²σ²_y + 2abσ_xσ_yρ

In this formula, r is the population correlation coefficient, theoretically computed using the population of all (x, y) data pairs. The expression σ_xσ_yρ is called the covariance of x and y. If x and y are independent, then ρ = 0 and the formula for σ²_w 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.

μ_x ≈ 7.35,      σ_x ≈ 6.59,      μ_y ≈ 13.16,      σ_y ≈ 18.56,      ρ ≈ 0.421

(b) Suppose you decide to put 70% of your investment in bonds and 30% in real estate. This means you will use a weighted average w = 0.7x + 0.3y. Estimate your expected percentage return μ_w and risk σ_w.
μ_w =
σ_w =

(c) Repeat part (b) if w = 0.3x + 0.7y.
μ_w =
σ_w =

Answer 1

The values provided in the questions are as follows

μx ≈ 7.35,      σx ≈ 6.59,      μy ≈ 13.16,      σy ≈ 18.56,      ρ ≈ 0.421

First we have to find part (b) is as follows

In part (b) we decide to put 70% of your investment in bonds and 30% in real estate

This means we will use a weighted average w = 0.7x + 0.3y.......(1)

We estimate the expected percentage return μw and risk σw.

w = 0.7x + 0.3y

μw = 0.7 *μx + 0.3* μy

= (0.7 * 7.35) + (0.3 * 13.16)

μw = 9.093............(2)

First we find σw2 and then find σw

σw2 = 0.72 *  σx2 + 0.32 *σy2 + (2 * 0.7 * 0.3 * σx*σy* ρ)

= (0.49 * 6.592) + (0.09*18.562) + ( 2 * 0.7 *0.3 * 6.59 * 18.56 * 0.421)

σw2 = 21.2798 + 31.0026 + 21.6269

σw2 = 73.9093

σw =

σw = 8.5971-----------(3)

c) In this part the equation is

w = 0.3x + 0.7y.............(4)

μw = 0.3 *μx + 0.7* μy

= (0.3 * 7.35) + (0.7 * 13.16)

μw = 11.417..............(5)

we find σw2 and then find σw

σw2 = 0.32 *  σx2 + 0.72 *σy2 + (2 * 0.7 * 0.3 * σx*σy* ρ)

= (0.09 * 6.592) + (0.49*18.562) + ( 2 * 0.7 *0.3 * 6.59 * 18.56 * 0.421)

= 3.9085 + 168.7920 + 21.6269

σw2 = 194.3274

σw =

σw = 13.9401...............(6)

Summary : -

(b) μw = 9.093

σw = 8.5971

(c) μw = 11.417

  σw = 13.9401