Answer:
Annual investment= $2,855.71
Explanation:
First, we will determine the future value of the investment of Twin 1 at the end of the firsts 10 years.
Twin 1:
Annual investment= $1,500
Number of periods= 10 years
Interest rate= 7%
FV= {A*[(1+i)^n-1]}/i
A= annual deposit
FV= {1,500*[(1.07^10) - 1]} / 0.07
FV= $20,724.67
Now, the value of the account of Twin 1 after 32 years (65 - 33), if he leaves the money to gain interest:
FV= PV*(1+i)^n
FV= 20,724.67*(1.07^32)
FV= $180,621.11
Finally, the annual deposit that Twin 2 must make to equal the amount earned by Twin 1:
FV= {A*[(1+i)^n-1]}/i
A= annual deposit
Isolating A:
A= (FV*i)/{[(1+i)^n]-1}
A= (180,621.11*0.07) / [(1.07^25) - 1]
A= $2,855.71
Twin 2 must make an annual deposit of $2,855.71 to match the amount earned by Twin 1, which is the annual investment.
First, we'll calculate the future value of Twin 1's investment at the conclusion of the first ten years.
[tex]\text{Twin 1}:\\\text{Annual investment}= $1,500\\\text{Number of periods= 10 years}\\\text{Interest rate= 7} \text{percent}\\FV= {A\text{x}[(1+i)^n-1]}/i\\\text{A= annual deposit}FV= {1,500 \text{x} [(1.07^{10} ) - 1]} / 0.07FV= $20,724.67[/tex]
The following is the worth of Twin 1's account after 32 years (65 - 33), assuming he leaves the money to earn interest:
[tex]\text{FV= PV} \text{x}(1+i)^n\\FV= 20,724.67\text { x }(1.07^{32})\\FV= 180,621.11[/tex]
Finally, Twin 2 must make an annual deposit equivalent to the amount generated by Twin 1:
[tex]\text{FV}= {\text{A} \text{x}{[(1+i)^n-1]}/\text{i}\\\text{A= annual deposit}[/tex]
[tex]\text{Isolating A}:\\A= (FV \text{x} i)/{[(1+i)^n]-1}\\A= (180,621.11 \text{x} 0.07) / [(1.07^{25} ) - 1]\\A= 2,855.71[/tex]
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