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Difficulty: HardPercentages, Percent Change, and Interest

A venture capital fund allocated an initial sum of money between Portfolio Alpha and Portfolio Beta, with Portfolio Alpha receiving 60%60\% of the total sum and Portfolio Beta receiving the remaining 40%40\%. During the first year, the value of Portfolio Alpha increased by 20%20\%, while the value of Portfolio Beta decreased by 15%15\%. During the second year, the value of Portfolio Alpha decreased by 10%10\% relative to its value at the end of the first year, while the value of Portfolio Beta increased by 25%25\% relative to its value at the end of the first year. By what percent did the total combined value of the two portfolios increase from the initial allocation to the end of the second year?

Answer: 7.3 %

Answer

The total combined value of the two portfolios increased by 7.3%.
To find the net percent change over the two-year period, track each portfolio's value year by year. Assuming an initial combined total of 100,PortfolioAlphabeginsat100, Portfolio Alpha begins at 60 and Portfolio Beta at 40.BytheendofYear1,Alphaincreasesby2040. By the end of Year 1, Alpha increases by 20% to 72 (60×1.2060 \times 1.20), while Beta decreases by 15% to 34(34 ( 40 \times 0.85 ).InYear2,Alphadecreasesby10). In Year 2, Alpha decreases by 10% relative to its Year 1 ending value, becoming 64.80 (72×0.9072 \times 0.90), while Beta increases by 25% relative to its Year 1 ending value, becoming 42.50(42.50 ( 34 \times 1.25 ).ThetotalcombinedvalueattheendofYear2is). The total combined value at the end of Year 2 is 64.80 + 42.50=42.50 = 107.30. Relative to the initial $100 allocation, this represents a net increase of 7.3%.

Step-by-Step Solution

1
Assign a convenient base value for the initial total investment.
Assume an initial total allocation of 100.PortfolioAlphastartswith100. Portfolio Alpha starts with 60 and Portfolio Beta starts with $40.
Percent changes are proportional and scale-invariant, making $100 a simple base value for calculation.
2
Compute the value of each portfolio at the end of Year 1.
Portfolio Alpha = 60×(1+0.20)=60 \times (1 + 0.20) = 72. Portfolio Beta = 40×(1−0.15)=40 \times (1 - 0.15) = 34.
Apply Year 1 growth (+20%) and loss (-15%) to their respective starting funds.
3
Compute the value of each portfolio at the end of Year 2.
Portfolio Alpha = 72×(1−0.10)=72 \times (1 - 0.10) = 64.80. Portfolio Beta = 34×(1+0.25)=34 \times (1 + 0.25) = 42.50.
Apply Year 2 changes (-10% and +25%) to the Year 1 ending values rather than the original principal.
4
Find the final total combined value and calculate the overall net percentage increase.
Combined Year 2 value = 64.80+64.80 + 42.50 = $107.30. Net percentage increase = \frac{107.30 - 100}{100} \times 100\% = 7.3\%.
Compare the final combined sum of 107.30totheinitialcombinedsumof107.30 to the initial combined sum of 100.

Key Concept

Successive percent changes with shifting base values across multiple assets
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Topics

Percentages, Percent Change, and Interest

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