In: Statistics and Probability
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 ≈
7.33, σx
≈
6.56, μy
≈
13.16, σy
≈ 18.56, ρ ≈
0.424
(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.No. Interest rate probably affects both investment returns. No. Interest rates probably has no effect on the investment returns.Yes. Interest rate probably affects both investment returns.
(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 μw and risk
σw.
μw =
σw =
(c) Repeat part (b) if w = 0.4x +
0.6y.
μw =
σw =
(d) Compare your results in parts (b) and (c). Which investment has
the higher expected return? Which has the greater risk as measured
by σw?
w = 0.4x + 0.6y produces higher return with greater risk as measured by σw.w = 0.6x + 0.4y produces higher return with greater risk as measured by σw. w = 0.4x + 0.6y produces higher return with lower risk as measured by σw.w = 0.6x + 0.4y produces higher return with lower risk as measured by σw.Both investments produce the same return with the same risk as measured by σw.