Question

An investor plans to divide 200,000 between two investments. The first yields a certain profit of 10% while the second yields a profit with expected value of 18% and a standard deviation of 6%. If the investor divides the money equally between these two investments, find the mean and standard deviation of the total profit

Answer 1

Linear combinations of Random Variables:

Let us consider a two random variable X and Y. let us define the linear combination of X and Y as follows.

$W = aX-bY$

Here, a and b are constants.

The mean or expected value of W is

$μ_W = a \times μ_X -b \times μ_Y$

The variance for W is as follows.

$σ_W^2 = a^2 \times σ_X^2 + b^2 \times σ_Y^2 -2 \times a \times b \times ρ(X,Y) \times σ_X \times σ_Y$

We are working on an investor plans. Let us define two random variables X and Y. X represent the first investment and Y represents second investments

$X = N(μ_X=0.1, σ_X^2 = 0) \\ Y = N(μ_Y=0.18, σ_Y^2 = 0.06)$

We are interested in computing the mean and variance of the total profit.

$P = 100000X + 100000Y$

We compute the mean and standard deviation of P by using previous formula

$a = 100000 \\ b = 100000 \\ μ_P = a \times μ_X + b \times μ_Y \\ = 100000 \times 0.1 + 100000 \times 0.18 \\ = 28000$

The mean of total profit is 28000.

$σ_W^2 = a^2 \times σ_X^2 + b^2 \times σ_Y^2 -2 \times a \times b \times ρ(X,Y) \times σ_X \times σ_Y \\ = (100000)^2 \times 0 + (100000)^2 \times (0.06)^2 -2 \times 100000 \times 100000 \times (0) \times 0.06 \\ = 36000000$

We know that the standard deviation is the positive square root of the variance, thus we have

$σ = \sqrt{36000000} \\ = 6000$

The standard deviation of total profit is 6000.