Question 315530: Some people look forward to retiring from work by the age of 65. Compare the amounts at age 65 that would result from making an annual deposit of $1000 starting at age 20, or from making an annual deposit of $3000 starting at age 50, to an RRSP that earns 6% interest per annum, compounded annually. What are the final amounts for both situations and state which one is more profitable.
I know how to get the answer the only thing I'm iffy about is the amount of years.
For Option 1, is it (65-20=45 years) of saving or 46 years? Same thing with Option 2, is it (65-50=15 years) or 16 years? I'd really appreciate it if someone could help me out. Thanks in advance.
Answer by Edwin McCravy(20056)   (Show Source):
You can put this solution on YOUR website!
I may be using a different formula from the one you were taught but it will
amount to the same, especially in the number of years.
Suppose we look first at the case of making an annual deposit of $1000 starting at age 20 and making the last deposit at age 64.
The $1000 deposit made at age 64 will have drawn interest for 1 year, and will be worth 1000(1.06)^1
The $1000 deposit made at age 63 will have drawn interest for 2 years, and will be worth 1000(1.06)^2
The $1000 deposit made at age 62 will have drawn interest for 3 years, and will be worth 1000(1.06)^3
... (We observe that the number of years the deposit has drawn interest is always 65 minus the age) ...
The $1000 deposit made at age 22 will have drawn interest for 43 years, and will
be worth 1000(1.06)^43
The $1000 deposit made at age 21 will have drawn interest for 44 years, and will
be worth 1000(1.06)^44
The $1000 deposit made at age 20 will have drawn interest for 42 years, and will
be worth 1000(1.06)^45
So we have the geometric series:
1000(1.06) + 1000(1.06)^2 + ... + 1000(1.06)^45 =
1000(1.06 + 1.06^2 + 1.06^3 + ... + 1.06^45)
So the series in the parentheses is:
Therefore 1000 times that is $225508.12, which is the amount
at age 65.
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Next we look first at the case of making an annual deposit of $3000 starting at age 50 and making the last deposit at age 64.
The $3000 deposit made at age 64 will have drawn interest for 1 year, and will be worth 3000(1.06)^1
The $3000 deposit made at age 63 will have drawn interest for 2 years, and will be worth 3000(1.06)^2
The $3000 deposit made at age 62 will have drawn interest for 3 years, and will be worth 3000(1.06)^3
... (We observe again that the number of years is always 65 minus the age) ...
The $3000 deposit made at age 52 will have drawn interest for 43 years, and will
be worth 3000(1.06)^13
The $3000 deposit made at age 51 will have drawn interest for 44 years, and will
be worth 3000(1.06)^14
The $3000 deposit made at age 50 will have drawn interest for 42 years, and will
be worth 3000(1.06)^15
So we have the geometric series:
3000(1.06) + 3000(1.06)^2 + ... + 3000(1.06)^15 =
3000(1.06 + 1.06^2 + 1.06^3 + ... + 1.06^15)
So the series in the parentheses is:
Therefore 3000 times that is $74017.58, which is the amount
at age 65.
Edwin
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