An investment fund allocates its total portfolio assets among three asset classes: stocks,… — GRE General Test

An investment fund allocates its total portfolio assets among three asset classes: stocks, bonds, and real estate. Initially, $\frac{5}{12}$ of the total portfolio is invested in stocks and $\frac{1}{3}$ of the total portfolio is invested in bonds, with the remaining fraction invested in real estate. If $\frac{2}{5}$ of the stock portfolio and $\frac{1}{4}$ of the bond portfolio are subsequently liquidated, what fraction of the fund's remaining total portfolio is invested in real estate?

A
$\frac{1}{4}$
B
$\frac{3}{10}$
$\frac{1}{3}$ Answer
D
$\frac{4}{9}$
E
$\frac{5}{12}$

Answer

The fraction of the fund's remaining total portfolio invested in real estate is

\frac{1}{3}

Let the original total portfolio be

T

. Initially, stocks are

\frac{5}{12}T

, bonds are

\frac{4}{12}T

, and real estate is

1 - \frac{9}{12}T = \frac{3}{12}T

. After liquidating

\frac{2}{5}

of stocks and

\frac{1}{4}

of bonds, the remaining stock value is

\frac{3}{5} \times \frac{5}{12}T = \frac{3}{12}T

and the remaining bond value is

\frac{3}{4} \times \frac{4}{12}T = \frac{3}{12}T

. Real estate remains

\frac{3}{12}T

. The new total portfolio is

\frac{3}{12}T + \frac{3}{12}T + \frac{3}{12}T = \frac{9}{12}T = \frac{3}{4}T

. Therefore, real estate represents

\frac{3/12 T}{9/12 T} = \frac{3}{9} = \frac{1}{3}

of the remaining total portfolio.

Step-by-Step Solution

Determine the initial fraction invested in real estate.

Real estate accounts for

1 - \left(\frac{5}{12} + \frac{1}{3}\right) = 1 - \frac{9}{12} = \frac{3}{12} = \frac{1}{4}

of the original total portfolio.

The sum of all three asset fractions must equal 1.

Calculate the remaining fractions for stocks and bonds after liquidation.

Remaining stocks =

\left(1 - \frac{2}{5}\right) \times \frac{5}{12} = \frac{3}{5} \times \frac{5}{12} = \frac{3}{12}

. Remaining bonds =

\left(1 - \frac{1}{4}\right) \times \frac{1}{3} = \frac{3}{4} \times \frac{4}{12} = \frac{3}{12}

Multiplying the unliquidated fraction by each asset's initial fraction gives its remaining fraction relative to the original total.

Calculate the total remaining portfolio fraction.

Total remaining portfolio =

\frac{3}{12} \text{ (stocks)} + \frac{3}{12} \text{ (bonds)} + \frac{3}{12} \text{ (real estate)} = \frac{9}{12} = \frac{3}{4}

of the original total portfolio.

Sum the remaining portions of all three asset categories.

Compute the ratio of real estate to the remaining total portfolio.

\frac{\text{Real Estate}}{\text{Remaining Total}} = \frac{\frac{1}{4}}{\frac{3}{4}} = \frac{1}{3}.

The question asks for the fraction relative to the new reduced total portfolio.

Key Concept

Sequential Fraction Allocation and Reduced Base Ratios

Estimated Time:1m 30s

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