(Old SOA sample problem) Kathryn deposits $100$ into an account at the beginning of each $4$-year period for $40$ years. The account credits interest at an annual effective interest rate of $i$. The accumulated amount in the account at the end of $40$ years is $X$, which is $5$ times the accumulated amount in the account at the end of $20$ years.
My attempt: Let $j$ denote the $4$-yr annual effective rate. $$ A(40) = X = 100 s_{10|j\%} $$ $$ A(20) = X/5 = 100 s_{5|j\%} $$
Using the formula $s_{n|j\%} = \dfrac{(1+j)^n-1}{j}$, we have
$$ \frac{(1+j)^{10} - 1}{(1+j)^5 - 1} = 5$$
Let $x = (1+j)^5$. Then we have $x^2-5x+4 = 0$, which implies $x = 4$, which leads to $(1+j) = 4^{1/5}$. So, $$ X= \dfrac{100[(1+j)^{10}-1]}{j} \approx \dfrac{100[16-1]}{0.319} \approx 4694$$
$4694$ is one of the options, but not the correct one. I found this question, but it just confused me even more. Can someone please let me know why my solution is incorrect? Thanks!
Your solution is incorrect because you are using the formula for the accumulated value of an annuity-immediate, rather than an annuity-due. Payments are made at the beginning of each period, not at the end. Therefore, your solution fails to calculate the accumulated value at the correct point in time; instead, your answer $X = 4694$ represents the accumulated value at the beginning of year $17$ just after the fifth payment is made, and you need to accrue for another four years at a rate of $j = 4^{1/5} - 1 \approx 0.319508$ to obtain the correct value. You can see that if you simply multiply your answer by $1+j$, you get the correct answer of $6195$.