# Precious metals

In 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, an 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?

Result

r1 =  1.11
r2 =  1.206
r3 =  1.176

#### Solution:

$p_{1}=62.7 \%=1 + \dfrac{ 62.7 }{ 100 }=1.627 \ \\ p_{2}=-41.3 \%=1 + \dfrac{ -41.3 }{ 100 }=0.587 \ \\ p_{3}=36.9 \%=1 + \dfrac{ 36.9 }{ 100 }=1.369 \ \\ p_{4}=15.9 \%=1 + \dfrac{ 15.9 }{ 100 }=1.159 \ \\ \ \\ r_{1}=\sqrt[4]{ p_{1} \cdot \ p_{2} \cdot \ p_{3} \cdot \ p_{4}}=\sqrt[4]{ 1.627 \cdot \ 0.587 \cdot \ 1.369 \cdot \ 1.159} \doteq 1.1095 \doteq 1.11$
$g_{1}=25.0 \%=1 + \dfrac{ 25.0 }{ 100 }=1.25 \ \\ g_{2}=4.3 \%=1 + \dfrac{ 4.3 }{ 100 }=1.043 \ \\ g_{3}=31.9 \%=1 + \dfrac{ 31.9 }{ 100 }=1.319 \ \\ g_{4}=23.2 \%=1 + \dfrac{ 23.2 }{ 100 }=1.232 \ \\ \ \\ r_{2}=\sqrt[4]{ g_{1} \cdot \ g_{2} \cdot \ g_{3} \cdot \ g_{4}}=\sqrt[4]{ 1.25 \cdot \ 1.043 \cdot \ 1.319 \cdot \ 1.232} \doteq 1.2065 \doteq 1.206$
$s_{1}=56.8 \%=1 + \dfrac{ 56.8 }{ 100 }=1.568 \ \\ s_{2}=-26.9 \%=1 + \dfrac{ -26.9 }{ 100 }=0.731 \ \\ s_{3}=14.4 \%=1 + \dfrac{ 14.4 }{ 100 }=1.144 \ \\ s_{4}=46.1 \%=1 + \dfrac{ 46.1 }{ 100 }=1.461 \ \\ \ \\ r_{3}=\sqrt[4]{ s_{1} \cdot \ s_{2} \cdot \ s_{3} \cdot \ s_{4}}=\sqrt[4]{ 1.568 \cdot \ 0.731 \cdot \ 1.144 \cdot \ 1.461} \doteq 1.1765 \doteq 1.176 \ \\ \ \\ r_{1} \approx r_{2} \approx r_{3}$

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Leave us a comment of this math problem and its solution (i.e. if it is still somewhat unclear...):

Showing 1 comment:
Dr Math
So all metals have approx the same rate of price growth per year over time

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