An investment portfolio consists of three funds: Fund Alpha, Fund Beta, and Fund Gamma. Th… — GMAT

An investment portfolio consists of three funds: Fund Alpha, Fund Beta, and Fund Gamma. The table below presents the capital committed (in millions of dollars) and the annual yield percentage for each fund during the past fiscal year:

Fund	Capital Committed ($ millions)	Annual Yield (%)
Fund Alpha	40	8%
Fund Beta	60	12%
Fund Gamma	$c$	$y\%$

What was the overall annual yield percentage for the combined portfolio of all three funds?

(1) The capital committed to Fund Gamma, $c$ , was equal to the total capital committed to Fund Alpha and Fund Beta combined.
(2) The annual yield percentage for Fund Gamma, $y\%$ , was equal to the weighted average annual yield percentage of Fund Alpha and Fund Beta combined.

A
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.Answer
C
BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
D
EACH statement ALONE is sufficient.
E
Statements (1) and (2) TOGETHER are NOT sufficient.

Answer

Statement (2) ALONE is sufficient to answer the question, but statement (1) alone is not sufficient.

The option stating that Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient is correct. Fund Alpha and Fund Beta combined have a weighted average yield of

\frac{40(8\%) + 60(12\%)}{100} = 10.4\%

. Statement (2) tells us that Fund Gamma's yield is also

10.4\%

. Combining any two groups with identical average yields results in an overall average yield equal to

10.4\%

, regardless of the amount of capital

c

in Fund Gamma. Statement (1) provides only the capital amount

c = 100

, leaving the yield

y

unknown and thus insufficient.

Step-by-Step Solution

Formulate the algebraic expression for the overall portfolio yield.

Overall Portfolio Yield =

\frac{40(8\%) + 60(12\%) + c(y\%)}{40 + 60 + c} = \frac{3.2 + 7.2 + \frac{cy}{100}}{100 + c} = \frac{10.4 + \frac{cy}{100}}{100 + c}

The total annual return is the sum of returns from each fund, divided by total capital committed.

Calculate the combined weighted average yield of Fund Alpha and Fund Beta.

Sub-portfolio yield =

\frac{40(8) + 60(12)}{40 + 60} = \frac{320 + 720}{100} = 10.4\%

Determining the performance of the known portion of the portfolio simplifies statement evaluation.

Evaluate Statement (1) independently:

c = 40 + 60 = 100

Overall Yield =

\frac{10.4 + y}{200} \times 100\% = \frac{10.4 + y}{2}\%

. Since

y

is unknown, this value is not unique.

Without knowing Fund Gamma's yield rate

y\%

, the capital amount alone does not provide a definitive overall yield.

Evaluate Statement (2) independently:

y\% = 10.4\%

Overall Yield =

\frac{100(10.4\%) + c(10.4\%)}{100 + c} = \frac{10.4\%(100 + c)}{100 + c} = 10.4\%

When a new component is added to a group with a value equal to the group's current average, the overall average remains unchanged regardless of the size (

c

) of the new component.

Key Concept

Weighted Average Invariance in Data Sufficiency

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