Annuities, Perpetuities, Choosing a Comparison Date - Solution Verification

Question

I am working on the following question:

• A man turns $40$ today and wishes to provide supplemental retirement income of $3000$ at the beginning of each month starting on his $65$-th birthday.
• Starting today, he makes monthly contributions of $X$ to a fund for $25$ years.
• The fund earns an annual nominal interest rate of $8 \%$ compounded monthly.
• On his $65$-th birthday, each $1000$ of the fund will provide $9.65$ of income at the beginning of each month starting immediately and continuing as long as he survives.
• Calculate $X$.

I am getting two different answers depending on which comparison date I pick to accumulate and discount. I set up my time diagram to denote end-of-months: for $t=0$ through $t=299$, I know that the amount $X$ is to be deposited, and from $t=300$ onwards, the amount $3000$ needs to be paid into a perpetuity.

Now, it makes the most sense to me to choose $t=299$ as the comparison date. With this comparison date, we get

$$X s_{300\rceil 8/12\%} = \frac{3000\cdot 1000}{9.65}$$

This did not lead to the correct value of $X$. Then I found this similar question, and it appears that the solution in this linked post uses $t=300$ as the comparison date. With this new comparison date, we get

$$X \ddot s_{300\rceil 8/12\%} = \frac{3000\cdot 1000}{9.65}$$

which leads to the correct value of $X$ (because the value of the perpetuity does not change, I believe).

My question is: why is the solution incorrect when we pick $t=299$ as the comparison date? Theoretically, we should be able to pick any comparison date to get to the correct answer as long as we accumulate/discount cash flows accurately.

Note: The solution in the linked post is not correct according to official SOA solutions, but I think that the solution in the linked post is correct and there is a problem with the official SOA solution.

Edit: Official SOA solution:

To receive 3000 per month at age $65$ the fund must accumulate to $3000 (1000/9.65) = 310,880.83$. The equation of value is $310,880.83 = X \ddot s_{300\rceil 0.08/12} = 957.36657X \implies 324.72$.

Answer 1

If you use 299 months it means you start using the account (withdrawing 3000) before the end of the month 300, therefore loosing the interest which is added at the last day of the month 300.

If you were to repay a debt then you should do it before the day the interest accumulates. This is why you are allowed to make payments any day until the due date, without triggering a recalculation of the mortgage. However, if you save your on money is best to first accumulate the interest before withdrawing.

You didn’t ask for the full solution. You might want to skip the rest.

1. For the person to be able to live indefinitely off the accumulated savings, the amount saved must be $$\frac{3000}{9.65}\cdot 1000$$
2. This saved amount will not accumulate any further interest since it pays a monthly dividend. At the moment of death, the savings will still be in the account to be inherited.
3. Now to calculate the monthly payment needed to reach the savings in 25 years, at the given interest rate, the equation is: $$X\left(1+\frac{0.08}{12}\right )^{300}= \frac{3000}{9.65}\cdot 1000$$