An investment fund allocated its total capital into three distinct asset classes: Class X,… — GMAT

An investment fund allocated its total capital into three distinct asset classes: Class X, Class Y, and Class Z. Class X yielded an annual return of $12\%$ , Class Y yielded an annual return of $18\%$ , and Class Z yielded an annual return of $6\%$ . The capital invested in Class X was exactly twice the capital invested in Class Y. If the overall weighted annual return for the entire fund was $11\%$ , what percentage of the total capital was invested in Class Z?

Answer: 37.5 %

Answer

37.5%

Using the given relationship

W_X = 2W_Y

and total weight constraint

W_X + W_Y + W_Z = 1

, Class Z's weight can be written as

1 - 3W_Y

. Plugging these into the overall weighted return formula yields

12(2W_Y) + 18W_Y + 6(1 - 3W_Y) = 11

, which solves to

W_Y = 5/24

. Substituting back gives

W_Z = 9/24 = 37.5\%

Step-by-Step Solution

Define fractional weight variables for each asset class.

Let

W_X

W_Y

, and

W_Z

be the proportions of total capital invested in Class X, Class Y, and Class Z, such that

W_X + W_Y + W_Z = 1

Establishing a standard framework for weighted average calculation.

Express

W_X

and

W_Z

in terms of

W_Y

W_X = 2W_Y

, so

2W_Y + W_Y + W_Z = 1 \implies W_Z = 1 - 3W_Y

Reducing the number of unknown variables to one.

Formulate the weighted average equation using component returns.

12W_X + 18W_Y + 6W_Z = 11 \implies 12(2W_Y) + 18W_Y + 6(1 - 3W_Y) = 11

The fund's overall return equals the weighted sum of individual returns.

Solve the algebraic equation for

W_Y

24W_Y + 18W_Y + 6 - 18W_Y = 11 \implies 24W_Y + 6 = 11 \implies 24W_Y = 5 \implies W_Y = \frac{5}{24}

Determining the exact weight of Class Y.

Calculate the weight percentage for Class Z.

W_Z = 1 - 3\left(\frac{5}{24}\right) = 1 - \frac{15}{24} = \frac{9}{24} = \frac{3}{8} = 0.375 \text{ or } 37.5\%

Evaluating the target component weight requested by the question.

Key Concept

Weighted Average in Combined Financial Assets

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