Looking at the "Junior Centenar" program

This is going to be a financial analysis of a special investing instrument created by Romania. In 2018, celebrating 100 years after the most important event in its history, and thinking of the children, the government created a special program – link is only in Romanian, sorry about that – where parents can invest a minimum of 1200 RON – the Romanian currency – every year (computed as a minimum of 100 each month, but parents can also deposit everything once a year) to an account that would be locked until the kid gets to 18 years. If the minimum is reached, then the Romanian government would also deposit 600 for that year. Sounds good at first glance, but there are caveats.

First, the interest rate is set at 3% every year and does not change. In 2018, the inflation rate was 4.6%, in 2022 it was 13.8% and in 2023 it was 10.4%. This already means that the interest would not catch up with inflation in a lot of these years. In fact, since 2018, the only year with inflation less than 3% was 2020 (when it was 2.6%).

Next, the interest rate only applies to the sum that the parent has deposited. The 600 deposited by the state every year is just a flat sum, never compounded.

A method to estimate the value of the account at maturity is to look at every year \(y \in \{y_0, \ldots 17\}\) before the kid turns 18 (where \(y_0\) is the age of the kid when the parents start contributing). If the parents deposited a value \(s_y\) in that year, then we get a contribution of \(s_y \times 1.03 ^ {18 - y}\) from the interest (which compounds) and 600 more if \(s_y \ge 1200\) (no compounding).

For simplicity, let’s assume, for now, that parents always contribute the same sum \(s\), which is at least 1200, to get the state bonus. Then, the value of the account is

\[V = \sum_{y=y_0}^{17}\left(1.03^{18-y}s + 600\right)\]

Expanding and using the geometric sum formula we have

\[V = 600(18-y_0) + \frac{1.03}{0.03}s\left(1.03^{18-y_0} - 1\right)\]

Which reduces to

\[V = \frac{103}{3}s\left(1.03^{18-y_0} - 1\right) - 600y_0 + 10800\]

There is an exponential term here, which would mean that starting later would decrease the value of the account more. Or, in other words, the earlier you start, the exponentially higher the value of the account is.

In this plot, I used \(s=1200\) (the minimum) as green and \(s=2400\) as blue. Starting as early as possible makes the first option gets to nearly 40k (39740), whereas the other one gets to 68680. This sounds great, right?

Well, first, observe that to reach 68680, the parent has to deposit 2400 for 18 years, so they get a return of 58.98%. But, if the parent deposits just the minimum 1200, the return is 84.98%, much higher. If the deposits start later, the formula for the return would be

\[\eta = \frac{V}{(18-y_0)s} = \frac{600}{s} + \frac{103}{3}\left(1.03^{18-y_0} - 1\right)\]

Since \(s\) only shows up as a denominator for the sum that the state deposits into the account, this proves that it is in the interest of the parent to deposit just the bare minimum (1200), to maximize the returns. Because the interest doesn’t apply to the bonus.

But, we are not done. Let’s look at the inflation. Let’s assume a constant rate \(r\) of inflation for all the years that the account is being fed. A sum of \(V\) after \(18 - y_0\) years would be now worth only \(\frac{V}{\left(1+r\right)^{18-y_0}}\). If the parent decides to open an account now for the \(18 - y_0\) years, then they would give up \(s\left(18 - y_0\right)\) in today’s money. To be worth it, this sum needs to much smaller than the fraction from backporting the future sum.

\[s(18-y_0) \ll \frac{V}{\left(1+r\right)^{18-y_0}}\]

Which is the same as

\[\left(1+r\right)^{18-y_0} \ll \eta\]

So, a parent should only invest if the inflation rate is much much smaller than the expected return ratio over the entire period of the account. But, remember that the interest rate was set to a value that was nearly always less than the inflation rate.

So, this investment account is not worthwhile, except if other investment objects cannot be used. I used so many words for such a negative conclusion :|