Determine the AV of 13 annual deposits of $1,429 one year after the last deposit, at 2.10% effective
I got confused when calculating AV in this question.
Apparently the correct formula would be $1,429cdot 1.021 cdot frac{1.021^{13} - 1}{1.021 - 1}$. Or at least, this equals $21.511.03, which is supposed to be the correct answer. But when I was thinking of geometric series, don't we have $AV = 1,429 cdot 1.021 cdot (1 + 1.021 + ... + 1.021^{13}) = 1,429cdot 1.021 frac{1.021^{14} - 1}{1.021 - 1}$.
I just can't figure out why the formula with 14 is wrong. Any help would much be appreciated! Thank you :)
finance actuarial-science
asked Jan 4 at 2:04
quietkidquietkid
254
add a comment |
I got confused when calculating AV in this question.
Apparently the correct formula would be $1,429cdot 1.021 cdot frac{1.021^{13} - 1}{1.021 - 1}$. Or at least, this equals $21.511.03, which is supposed to be the correct answer. But when I was thinking of geometric series, don't we have $AV = 1,429 cdot 1.021 cdot (1 + 1.021 + ... + 1.021^{13}) = 1,429cdot 1.021 frac{1.021^{14} - 1}{1.021 - 1}$.
I just can't figure out why the formula with 14 is wrong. Any help would much be appreciated! Thank you :)
finance actuarial-science
asked Jan 4 at 2:04
quietkidquietkid
254
add a comment |
I got confused when calculating AV in this question.
Apparently the correct formula would be $1,429cdot 1.021 cdot frac{1.021^{13} - 1}{1.021 - 1}$. Or at least, this equals $21.511.03, which is supposed to be the correct answer. But when I was thinking of geometric series, don't we have $AV = 1,429 cdot 1.021 cdot (1 + 1.021 + ... + 1.021^{13}) = 1,429cdot 1.021 frac{1.021^{14} - 1}{1.021 - 1}$.
I just can't figure out why the formula with 14 is wrong. Any help would much be appreciated! Thank you :)
finance actuarial-science
asked Jan 4 at 2:04
quietkidquietkid
254
I got confused when calculating AV in this question.
Apparently the correct formula would be $1,429cdot 1.021 cdot frac{1.021^{13} - 1}{1.021 - 1}$. Or at least, this equals $21.511.03, which is supposed to be the correct answer. But when I was thinking of geometric series, don't we have $AV = 1,429 cdot 1.021 cdot (1 + 1.021 + ... + 1.021^{13}) = 1,429cdot 1.021 frac{1.021^{14} - 1}{1.021 - 1}$.
I just can't figure out why the formula with 14 is wrong. Any help would much be appreciated! Thank you :)
finance actuarial-science
finance actuarial-science
asked Jan 4 at 2:04
quietkidquietkid
254
asked Jan 4 at 2:04
quietkidquietkid
254
asked Jan 4 at 2:04
quietkidquietkid
254
asked Jan 4 at 2:04
quietkidquietkid
254
asked Jan 4 at 2:04
quietkidquietkid
254
254
add a comment |
add a comment |
1 Answer
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There are only $13$ deposits. Your sum clearly contains $14$ terms, which is why it cannot be correct.
When we write the cash flow for the accumulated value, the final ($13^{rm th}$) deposit has had $1$ year to accumulate value. This much should be obvious. Then the $12^{rm th}$ deposit has had $2$ years, and so forth, so that the first deposit has had $13$ years to accumulate value. Consequently, the accumulated value of this first payment must be $1429(1.021)^{13}$. So not only do you have an extra deposit, you also have an extra year of interest accrued. The correct cash flow written out is
$$1429 left( (1.021)^{13} + (1.021)^{12} + cdots + (1.021)^2 + (1.021) right).$$ After factoring out the common factor of $1.021$, and applying the formula for the sum of a geometric series, we get the claimed answer.
answered Jan 4 at 2:09
heropupheropup
62.6k66099
Okay! Thank you :) that makes sense
– quietkid
Jan 4 at 3:02
add a comment |
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1 Answer
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There are only $13$ deposits. Your sum clearly contains $14$ terms, which is why it cannot be correct.
When we write the cash flow for the accumulated value, the final ($13^{rm th}$) deposit has had $1$ year to accumulate value. This much should be obvious. Then the $12^{rm th}$ deposit has had $2$ years, and so forth, so that the first deposit has had $13$ years to accumulate value. Consequently, the accumulated value of this first payment must be $1429(1.021)^{13}$. So not only do you have an extra deposit, you also have an extra year of interest accrued. The correct cash flow written out is
$$1429 left( (1.021)^{13} + (1.021)^{12} + cdots + (1.021)^2 + (1.021) right).$$ After factoring out the common factor of $1.021$, and applying the formula for the sum of a geometric series, we get the claimed answer.
answered Jan 4 at 2:09
heropupheropup
62.6k66099
Okay! Thank you :) that makes sense
– quietkid
Jan 4 at 3:02
add a comment |
There are only $13$ deposits. Your sum clearly contains $14$ terms, which is why it cannot be correct.
When we write the cash flow for the accumulated value, the final ($13^{rm th}$) deposit has had $1$ year to accumulate value. This much should be obvious. Then the $12^{rm th}$ deposit has had $2$ years, and so forth, so that the first deposit has had $13$ years to accumulate value. Consequently, the accumulated value of this first payment must be $1429(1.021)^{13}$. So not only do you have an extra deposit, you also have an extra year of interest accrued. The correct cash flow written out is
$$1429 left( (1.021)^{13} + (1.021)^{12} + cdots + (1.021)^2 + (1.021) right).$$ After factoring out the common factor of $1.021$, and applying the formula for the sum of a geometric series, we get the claimed answer.
answered Jan 4 at 2:09
heropupheropup
62.6k66099
Okay! Thank you :) that makes sense
– quietkid
Jan 4 at 3:02
add a comment |
There are only $13$ deposits. Your sum clearly contains $14$ terms, which is why it cannot be correct.
When we write the cash flow for the accumulated value, the final ($13^{rm th}$) deposit has had $1$ year to accumulate value. This much should be obvious. Then the $12^{rm th}$ deposit has had $2$ years, and so forth, so that the first deposit has had $13$ years to accumulate value. Consequently, the accumulated value of this first payment must be $1429(1.021)^{13}$. So not only do you have an extra deposit, you also have an extra year of interest accrued. The correct cash flow written out is
$$1429 left( (1.021)^{13} + (1.021)^{12} + cdots + (1.021)^2 + (1.021) right).$$ After factoring out the common factor of $1.021$, and applying the formula for the sum of a geometric series, we get the claimed answer.
answered Jan 4 at 2:09
heropupheropup
62.6k66099
There are only $13$ deposits. Your sum clearly contains $14$ terms, which is why it cannot be correct.
When we write the cash flow for the accumulated value, the final ($13^{rm th}$) deposit has had $1$ year to accumulate value. This much should be obvious. Then the $12^{rm th}$ deposit has had $2$ years, and so forth, so that the first deposit has had $13$ years to accumulate value. Consequently, the accumulated value of this first payment must be $1429(1.021)^{13}$. So not only do you have an extra deposit, you also have an extra year of interest accrued. The correct cash flow written out is
$$1429 left( (1.021)^{13} + (1.021)^{12} + cdots + (1.021)^2 + (1.021) right).$$ After factoring out the common factor of $1.021$, and applying the formula for the sum of a geometric series, we get the claimed answer.
answered Jan 4 at 2:09
heropupheropup
62.6k66099
answered Jan 4 at 2:09
heropupheropup
62.6k66099
answered Jan 4 at 2:09
heropupheropup
62.6k66099
answered Jan 4 at 2:09
heropupheropup
62.6k66099
62.6k66099
Okay! Thank you :) that makes sense
– quietkid
Jan 4 at 3:02
add a comment |
Okay! Thank you :) that makes sense
– quietkid
Jan 4 at 3:02
Okay! Thank you :) that makes sense
– quietkid
Jan 4 at 3:02
Okay! Thank you :) that makes sense
– quietkid
Jan 4 at 3:02
add a comment |
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