Question

An investor has $100 to invest and two investments between which to divide it. If she invests the entire amount in the first investment, her return will be X while if she invests the entire amount in the second investment her return will by Y. Both X and Y have mean $9 and standard deviation $3.7. The correlation between x and y is 0.3. find the mean return and risk if she invests $31 in the first investment and $69 in the second.

Answer 1

Answer:

The mean return is $9 and the risk is $3.1.

Step-by-step explanation:

It is provided that on investing an amount of $100 in the two investments the return will be X and Y.

Given:

$E(X)=E(Y)=\ 9\\SD(X)=SD(Y)=\ 3.7\\r(X, Y)=0.3$

It is also provided that the investor invested $31 and $69 in the first and second investment respectively.

The return equation will be:

$R=0.31X+0.69Y$

Compute the expected value of return as follows:

$E(R)=E(0.31X+0.69Y)\\=0.31E(X)+0.69E(Y)\\=(0.31\times 9)+(0.69\times9)\\=\ 9$

Thus, the mean return is $9.

Compute the risk as follows:

Risk = SD (0.31X + 0.69Y)

$=\sqrt{V(0.31X)+V(0.69Y)+2Cov(0.31X,0.69Y)}\\=\sqrt{0.31^{2}V(X)+0.69^{2}V(Y)+(2\times031\times0.69\times r(X,Y)\times SD(X)\times SD(Y))}\\=\sqrt{(0.31^{2}\times3.7^{2})+(0.69^{2}\times3.7^{2})+(2\times031\times0.69\times r(X,Y)\times SD(X)\times SD(Y))}\\=\sqrt{9.5903926}\\=3.09\\\approx3.1$Thus, the risk is $3.1.