Determine the AV of 13 annual deposits of $1,429 one year after the last deposit, at 2.10% effective

Question

I got confused when calculating AV in this question. Apparently the correct formula would be $1,429\cdot 1.021 \cdot \frac{1.021^{13} - 1}{1.021 - 1}$. 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 $AV = 1,429 \cdot 1.021 \cdot (1 + 1.021 + ... + 1.021^{13}) = 1,429\cdot 1.021 \frac{1.021^{14} - 1}{1.021 - 1}$. I just can't figure out why the formula with 14 is wrong. Any help would much be appreciated! Thank you :)

Answer 1

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.