At the beginning of Year 1, a commercial real estate portfolio consisted of two properties… — GRE General Test

At the beginning of Year 1, a commercial real estate portfolio consisted of two properties, Property A and Property B, where the value of Property A was $50\%$ greater than the value of Property B. During Year 1, the value of Property A increased by $20\%$ while the value of Property B decreased by $20\%$ . During Year 2, the value of Property A decreased by $x\%$ while the value of Property B increased by $x\%$ . If the combined value of the two properties at the end of Year 2 was equal to the combined value of the two properties at the beginning of Year 1, what is the value of $x$ ?

Cevap: 10

Cevap

Tracking base value changes across both properties yields a Year 1 ending total of

1.8B + 0.8B = 2.6B

. Applying Year 2 percentage adjustments gives a total value of

1.8B(1 - 0.01x) + 0.8B(1 + 0.01x) = 2.6B - 0.01Bx

. Setting this equal to the initial portfolio value of

2.5B

gives

0.1B = 0.01Bx

, which solves to

x = 10

Adım Adım Çözüm

Define initial values for both properties relative to a single variable at the beginning of Year 1

Property B value =

B

, Property A value =

1.5B

, and total portfolio value =

2.5B

Setting Property B as

B

allows all subsequent values and combined sums to be expressed cleanly in terms of

B

Calculate individual property values at the end of Year 1 after applying respective

+20\%

and

-20\%

changes

Property A value =

1.5B \times 1.20 = 1.8B

; Property B value =

B \times 0.80 = 0.8B

Year 1 percentage changes must be calculated using each property's initial starting base.

Formulate algebraic expressions for the property values at the end of Year 2

Total Year 2 end value =

1.8B \left(1 - \frac{x}{100}\right) + 0.8B \left(1 + \frac{x}{100}\right) = 2.6B - 0.01Bx

Year 2 percentage shifts apply to the new updated base values (

1.8B

and

0.8B

) obtained at the end of Year 1.

Set the Year 2 end total value equal to the Year 1 start total value and solve for

x

2.6B - 0.01Bx = 2.5B \implies 0.1B = 0.01Bx \implies x = 10

Equating the two expressions eliminates

B

from both sides, leaving a linear equation in

x

Anahtar Kavram

Successive Percentage Changes and Base Shifts across Multiple Portfolio Assets

Soru

Cevap

Adım Adım Çözüm

Anahtar Kavram

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