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Difficulty: Very hardFractions, Decimals, and Percents Arithmetic

At the start of a fiscal year, a sovereign wealth fund allocated its capital into three asset classes: Stocks, Bonds, and Commodities. Exactly 0.400.40 of the total capital was allocated to Stocks. Of the remaining capital, 13\frac{1}{3} was allocated to Bonds and the rest was allocated to Commodities. Over the course of the fiscal year, the value of Stocks increased by 15%15\% and the value of Bonds decreased by 10%10\%. If the total value of the fund increased by 5%5\% by the end of the fiscal year, by what percent did the value of Commodities change?

Answer: 2.5 %

Answer

The value of Commodities increased by 2.5%2.5\%.
By setting the initial total capital to VV, the initial values are Stocks = 0.40V0.40V, Bonds = 13(0.60V)=0.20V\frac{1}{3}(0.60V) = 0.20V, and Commodities = 0.60V−0.20V=0.40V0.60V - 0.20V = 0.40V. Applying the percentage changes gives final values of Stocks = 0.46V0.46V, Bonds = 0.18V0.18V, and Total Fund = 1.05V1.05V. The final Commodities value is 1.05V−0.64V=0.41V1.05V - 0.64V = 0.41V. The percent change in Commodities is 0.41V−0.40V0.40V×100%=2.5%\frac{0.41V - 0.40V}{0.40V} \times 100\% = 2.5\%.

Step-by-Step Solution

1
Determine initial asset allocations as proportions of total capital VV
Stocks (S0S_0) = 0.40V0.40V, Bonds (B0B_0) = 0.20V0.20V, Commodities (C0C_0) = 0.40V0.40V
Stocks take 0.40V0.40V, leaving 0.60V0.60V. Bonds take 13\frac{1}{3} of 0.60V=0.20V0.60V = 0.20V, leaving 0.40V0.40V for Commodities.
2
Compute final values for Stocks, Bonds, and Total Fund after percentage changes
S1=0.46VS_1 = 0.46V, B1=0.18VB_1 = 0.18V, and Total Fund V1=1.05VV_1 = 1.05V
Stocks increase by 15%15\% (0.40V×1.15=0.46V0.40V \times 1.15 = 0.46V), Bonds decrease by 10%10\% (0.20V×0.90=0.18V0.20V \times 0.90 = 0.18V), and the total fund grows by 5%5\% (1.05V1.05V).
3
Calculate the end-of-year value of Commodities
C1=0.41VC_1 = 0.41V
Subtract final Stocks and Bonds from total fund: C1=1.05V−(0.46V+0.18V)=0.41VC_1 = 1.05V - (0.46V + 0.18V) = 0.41V.
4
Calculate the percent change of Commodities relative to its initial value
2.5%2.5\% increase
Percent change =0.41V−0.40V0.40V×100%=0.010.40×100%=2.5%= \frac{0.41V - 0.40V}{0.40V} \times 100\% = \frac{0.01}{0.40} \times 100\% = 2.5\%.

Key Concept

Weighted percentage change and fractional portion modeling
Estimated Time:1m 40s
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Topics

Fractions, Decimals, and Percents Arithmetic

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