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 ≈ 7.33,      σ_x ≈ 6.58,      μ_y ≈ 13.18,      σ_y ≈ 18.56,      ρ ≈ 0.422

(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.

Yes. Interest rate probably affects both investment returns.    

No. Interest rates probably has no effect on the investment returns.

No. Interest rate probably affects both investment returns.

(b) Suppose you decide to put 55% of your investment in bonds and 45% in real estate. This means you will use a weighted average w = 0.55x + 0.45y. Estimate your expected percentage return μ_w and risk σ_w.
μ_w =  
σ_w =  

(c) Repeat part (b) if w = 0.45x + 0.55y.
μ_w =  
σ_w =  
(d) Compare your results in parts (b) and (c). Which investment has the higher expected return? Which has the greater risk as measured by σ_w?

w = 0.45x + 0.55y produces higher return with greater risk as measured by σ_w.

w = 0.45x + 0.55y produces higher return with lower risk as measured by σ_w.  

  w = 0.55x + 0.45y produces higher return with lower risk as measured by σ_w.

Both investments produce the same return with the same risk as measured by σ_w.

w = 0.55x + 0.45y produces higher return with greater risk as measured by σ_w

Answer 1

Solution

Back-up Theory

Given,

If a and b be any constants and w = ax + by ………………………………………………………… (1)

then,

μ_w = aμ_x + bμ_y …………………………………………………………………………………....……. (2)
σ²_w = a²σ²_x + b²σ²_y + 2abσ_xσ_yρ…………………………………………………………..…………. (3)

Now, to work out the solution,

Also given,

μ_x ≈ 7.33, μ_y ≈ 13.18 ……………………………………………………………………....…………..(4)     

σ_x ≈ 6.58,   σ_y ≈ 18.56 ……………………………………………………………………..…………..(5)     

ρ ≈ 0.422 ………………..…………………………………………………………………..…………..(6)

Part (a)

Vide (5), correlation coefficient is not zero => x and y are NOT independent.

And answer option is the last Option Answer 1

Part (b)

Vide (1), here a = 0.55 and b = 0.45. So, vide (2) and (4),

μ_w = (0.55 x 7.33) + (0.45 x 13.18)

= 9.9625 Answer 2

Vide (1), here a = 0.55 and b = 0.45. So, vide (3) (5) and (6),

σ²_w  = (0.55² x 6.58²) + (0.45² x 18.56²) + (2 x 0.55 x 0.45 x 6.58 x 18.56)

= 108..3640

Hence, σ_w = sqrt(108..3640)

= 10.41 Answer 3

Part (c)

Vide (1), here a = 0.45 and b = 0.55. So, vide (2) and (4),

μ_w = (0.45 x 7.33) + (0.55 x 13.18)

= 10.5475 Answer 4

Vide (1), here a = 0.45 and b = 0.55. So, vide (3) (5) and (6),

σ²_w  = (0.45² x 6.58²) + (0.55² x 18.56²) + (2 x 0.55 x 0.45 x 6.58 x 18.56)

= 138..481

Hence, σ_w = sqrt(138..481)

= 11.77 Answer 5

Part (d)

Comparison

w = 0.45x + 0.55y produces higher return with greater risk as measured by σw. Answer 6

DONE