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# An investor has \$100 to invest, and two investments ISBN: 9780073401331 38

## Solution for problem 23E Chapter 2.6

Statistics for Engineers and Scientists | 4th Edition

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Problem 23E

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 be Y. Both X and Y have mean \$6 and standard deviation (risk) \$3. The correlation between X and Y is 0.3.

a. Express the return in terms of X and Y if she invests \$30 in the first investment and \$70 in the second.

b. Find the mean return and the risk if she invests \$30 in the first investment and \$70 in the second.

c. Find the mean return and the risk, in terms of K, if she invests \$K in the first investment and \$(100- K) in the second.

d. Find the value of K that minimizes the risk in part (c).

e. Prove that the value of K that minimizes the risk in part (c) is the same for any correlation ρX,Y with ρX,Y ≠ 1.

Step-by-Step Solution:

Step 1 of 6:

Given that, an investor has \$100 to invest, and two investments between which to divide it.

Let X will be if she invests the entire amount in the first investment.

Let Y will be if she invests the entire amount in the second investment.

Here we have, both X and Y have mean \$6 and standard deviation(risk) \$3, and the correlation between X and Y is 0.3.

Step 2 of 8:

a). Here we need to express the return in terms of X and Y if she invests \$30 in the first investment and \$70 in the second.

That is,

The total return (R) = R = 0.3X + 0.7Y

Therefore, the return in terms of X and Y if she invests \$30 in the first investment and \$70 in the second is 0.3X + 0.7Y.

Step 3 of 6:

b). To find the mean return and the risk, if she invests \$30 in the first investment and \$70 in the second.

That is, (since R = 0.3X + 0.7Y)

= = (0.3)(6) + (0.7)(6)

= 1.8 + 4.2 = 6.

Then,

The formula for risk is

Portfolio of standard deviation  =

Where, are portfolio weights, are variances and is the covariance.

The risk is = Where, = (since correlation = 0.3 and standard deviation =3)

= (0.3)(3)(3)

= 2.7.

Therefore,  = = 2.5207

Therefore, the mean return and the risk, if she invests \$30 in the first investment and \$70 in the second is = 6 and 2.52

Step 4 of 6

Step 5 of 6

##### ISBN: 9780073401331

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