Precious metals

From 2006-2009, the value of precious metals changed rapidly. The data in the following table represent the total rate of return (in percentage) for platinum, gold, and silver from 2006 through 2009: Year Platinum Gold Silver 2009 62.7 25.0 56.8 2008 -41.3 4.3 -26.9 2007 36.9 31.9 14.4 2006 15.9 23.2 46.1 a. Compute the geometric mean rate of return per year for platinum, gold, and silver from 2006 through 2009. b. What conclusions can you reach concerning the geometric mean rates of the three precious metals?

Correct answer:

r₁ =  1.1095
r₂ =  1.2065
r₃ =  1.1765

Step-by-step explanation:

$$\begin{gathered} p_1=100\%+62.7\%=1+\frac{62.7}{100}=1.627 \\ {p}_2=100\%+-41.3\%=1+\frac{-41.3}{100}=0.587 \\ {p}_3=100\%+36.9\%=1+\frac{36.9}{100}=1.369 \\ {p}_4=100\%+15.9\%=1+\frac{15.9}{100}=1.159 \\ {r}_1=\sqrt[4]{p_1\cdot p_2\cdot p_3\cdot p_4}=\sqrt[4]{1.627\cdot 0.587\cdot 1.369\cdot 1.159}=1.1095 \end{gathered}$$

$$\begin{gathered} g_1=100\%+25.0\%=1+\frac{25.0}{100}=1.25 \\ {g}_2=100\%+4.3\%=1+\frac{4.3}{100}=1.043 \\ {g}_3=100\%+31.9\%=1+\frac{31.9}{100}=1.319 \\ {g}_4=100\%+23.2\%=1+\frac{23.2}{100}=1.232 \\ {r}_2=\sqrt[4]{g_1\cdot g_2\cdot g_3\cdot g_4}=\sqrt[4]{1.25\cdot 1.043\cdot 1.319\cdot 1.232}=1.2065 \end{gathered}$$

$$\begin{gathered} s_1=100\%+56.8\%=1+\frac{56.8}{100}=1.568 \\ {s}_2=100\%+-26.9\%=1+\frac{-26.9}{100}=0.731 \\ {s}_3=100\%+14.4\%=1+\frac{14.4}{100}=1.144 \\ {s}_4=100\%+46.1\%=1+\frac{46.1}{100}=1.461 \\ {r}_3=\sqrt[4]{s_1\cdot s_2\cdot s_3\cdot s_4}=\sqrt[4]{1.568\cdot 0.731\cdot 1.144\cdot 1.461}\doteq 1.1765 \\ {r}_1\approx r_2\approx r_3 \end{gathered}$$

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