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 σ^{2}_{x} and σ^{2}_{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.

In this formula, *r* is the population correlation coefficient, theoretically computed using the population of all (*x*, *y*) data pairs. The expression σ_{x}σ_{y}ρ is called the *covariance* of *x* and *y*. If *x* and *y* are independent, then ρ = 0 and the formula for σ^{2}* _{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

μ_{x} ≈ 7.34, σ_{x} ≈ 6.55, μ_{y} ≈ 13.19, σ_{y} ≈ 18.57, ρ ≈ 0.424

(a) Do you think the variables*x* and *y* are independent? Explain your answer.

(b) Suppose you decide to put 75% of your investment in bonds and 25% in real estate. This means you will use a weighted average*w* = 0.75*x* + 0.25*y*. Estimate your expected percentage return μ_{w} and risk σ_{w}.

μ_{w} =

σ_{w} =

(c) Repeat part (b) if*w* = 0.25*x* + 0.75*y*.

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* = 0.75x + 0.25y produces higher return with lower risk as measured by σ_{w}.
*w* = 0.25x + 0.75y produces higher return with lower risk as measured by σ_{w}.
*w* = 0.25x + 0.75y produces higher return with greater risk as measured by σ_{w}.
*w* = 0.75x + 0.25y produces higher return with greater risk as measured by σ_{w}.

(a) Do you think the variables

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

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

No. Interest rate probably affects both investment returns.

Yes. Interest rate probably affects both investment returns.

(b) Suppose you decide to put 75% of your investment in bonds and 25% in real estate. This means you will use a weighted average

μ

σ

(c) Repeat part (b) if

μ

σ

(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

Both investments produce the same return with the same risk as measured by σ_{w}.

Step 1

Hey, since there are multiple subpart questions posted, we will answer first three subparts. If you want any specific subpart to be answered, then please submit that subpart only or specify the subpart number in your message

Step 2

**(a) Independence of two variables:**

Here, *x* represent annual percentage return on the Vanguard Total Bond Index Fund and *y* represent annual percentage return on the Fidelity Real Estate Investment Fund.

If *x *and* y* are independent, ρ = 0 and σ^{2}* _{w}* reduces to the appropriate formula.

The variables *x *and* y *are not independent here. Because, the interest rate probably affects both investment returns. The weight of *w* = a*x* + b*y* is different from *w* = b*x* + a*y.*

Therefore, the correct answer is **No. Interest rate probably affects both investment returns.**

Step 3

**(b) Estimation of expected percentage return µ w and risk σw if w = 0.75x + 0.25y:**

It is given that the weighted average *w* = 0.75*x* + 0.25*y*. The estimation of expected percentage return µ*w* is **8....**

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