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 ≈ 7.35, σx ≈ 6.59, μy ≈ 13.16, σy ≈ 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 μw and risk
σw.
μw =
σw =
(c) Repeat part (b) if w = 0.3x +
0.7y.
μw =
σw =
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
Previously, you studied linear combinations of independent random variables. What happens if the variables are not...
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, +...
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