Consider the list of annual rates of growth of an olive tree expressed in percen
ID: 3357842 • Letter: C
Question
Consider the list of annual rates of growth of an olive tree expressed in percentiles. 5; 4:5; 4:3; 5:6; 3:6 What would be the size of an olive tree which was 1 meter tall after 5 years of these growth rates ? (hint: after the first year the height of the tree would be 1:05 meters.) Express the growth rates in multiplicative factors. That is, the rst year the height of the tree is multiplied by 1.05 because it grows by 5%. Take the geometric mean of the multiplicative growth rates. Why is that mean appropriate for this situation ? 1
Explanation / Answer
The size of the tree after 5 years can be expressed as
H*(1+G1)*(1+G2)*(1+G3)*(1+G4)*(1+G5)
where, G1, G2,... are growth rate sin year 1,2, .. respectively
=100*(1+.05)*(1+.045)*(1+.043)*(1+.056)*(1+.036)= 125.202 metres.
It's like the hieght of tree which is there at the end of 1st year H*(1+G1) which gets grown by G2. Similar to the case of compound interest.Hence, the multiplicative factors and that is why geometric mean( a CAGR) which will give the similar results
So basically, we can compute a geometric mean which is equivalent to
(1+GM)^5=(1+G1)*(1+G2)*(1+G3)*(1+G4)*(1+G5)=1.25202
=>(1+GM)=(1.25202)^.2
=>1+ GM=1.04597
=>GM=.04597=4.597%
=>(1+GM)=(1.25202)^.2
=>1+ GM=1.04597
=>GM=.04597=4.597%