There is an classic SOA problem I found confused about. College tuition is 6000 for the current school year, payable in full at the beginning of the school year. College tuition will grow at an annual rate of 6%. A parent sets up a college savings fund earning interest at an annual effective rate of 7%. The parent deposits 750 at the beginning of each school year for 18 years, with the first deposit made at the beginning of the current school year. Immediately following the 18th deposit, the parent pays tuition for the 18th school year from the fund. The amount of money needed, in addition to the balance in the fund, to pay tuition at the beginning of the 19th school year is X. Calculate X.
When calculating the balance immediately after the last payment, I am confused. I got $750{\ddot{S}_{17}}$, but the answer is $750 S_{17}$. The reason for this is an annuity immediate is that, the evaluation is done at the time the last payment is made. But it seems to me that it is an annuity due because the payment is made at the beginning of each period, which accords to the studying manual. So, how do we determine it is an annuity due or annuity immediate?
$\require{enclose}$ Let's look at the accumulated value of the fund when $X$ is paid, which is at the beginning of year $19$. The payments of $750$ into the fund occur at times $t = 0, 1, 2, \ldots, 17$, corresponding to the beginning of years $1, 2, \ldots, 18$. So $X$ is paid at time $t = 18$.
Consequently, the accumulated value of the fund is $$750\left((1+i)^{18} + (1+i)^{17} + \cdots + (1+i)\right) = 750 \ddot s_{\enclose{actuarial}{18} i}, \tag{1}$$ where $i = 0.07$ is the interest rate of the fund. This, plus the amount $X$ paid, must equal the present value of the total tuition paid at time $t = 18$. Note that there are two tuition payments: one at $t = 17$ immediately after the last contribution to the fund, and a second one at time $t = 18$ at which the additional payment of $X$ is made.
At $t = 17$, the nominal amount of tuition is $6000(1+j)^{17}$ where $j = 0.06$ is the annual rate of tuition increase. But this is paid one year before the valuation point, so the present value of this payment at time $t = 18$ is $$6000(1+j)^{17}(1+i).$$ The next payment is $6000(1+j)^{18}$ and there is no adjustment for the time value. So we must have $$750 \ddot s_{\enclose{actuarial}{18}i} + X = 6000(1+j)^{17} \left((1+i) + (1+j)\right). \tag{2}$$ Solving gives $X = 7129.41$.
To check, we calculate the fund's value at time $t = 17$ before any tuition is paid:
$$750 s_{\enclose{actuarial}{18}i} \approx 25499.27438.$$
Then the tuition of $6000(1+j)^{17} \approx 16156.6367$ is paid, leaving a balance of $9342.6377$ in the fund. The next year, when tuition is due, it has accumulated an additional $7\%$ interest, so it is $9996.6223$. The tuition is now $6000(1+j)^{18} \approx 17126.0349$, so the shortfall is $7129.41$.
Notice how we can express the equation of value with either $\ddot s$ or $s$. Which one we use merely depends on the choice of valuation point. Equation $(2)$ corresponds to time $t = 18$ when $X$ is paid. But if we choose $t = 17$, one year before $X$ is paid, then the equation of value becomes
$$750 s_{\enclose{actuarial}{18}i} + Xv = 6000(1+j)^{17}\left(1 + (1+j)v\right), \tag{3}$$ where $v = 1/(1+i)$ is the present value discount factor for the fund. The reason is because at $t = 17$, neither $X$ nor the second year of tuition has been paid. It occurs one year into the future, so we must discount these transactions by one year of the fund's interest rate.
As for $s_{\enclose{actuarial}{17}}$, this makes no sense. There are $18$ payments into the fund. Only accumulating $17$ of those payments will not work no matter which valuation point is chosen. The last payment of $750$ must be included elsewhere in the equation of value.
The takeaway here is that you would be best served by not fixating on whether the equation of value should be written with an annuity-immediate, or annuity-due. Whichever is applicable depends on the valuation point. What is more important is to conceptualize the cash flows, and set up the equation of value from the cash flow after it has been written out.