My friend and I are theorizing whether or not it is always better to make a one-time payment into a fonds, stock title or whatever instead of a savings plan IF the total interest over a period of time is positive. Example:
Here, we have a total positive interest of 3.5% on average.
Now how to prove it? Here is my approach:
$$(I): R_{oneTime} = X \cdot(1+i)^n$$ $$(II): R_{multiple = \frac{X}{n}\cdot\sum_{k=1}^{n}(1+i)^k}$$
where I substitute $(1+r) = z > 1$ (remember the average interest ought to be positive) for convenience and $i$ is the interest-rate, $X$ is the initial payment and $n$ is the number of years.
I use the geometric series formula to convert: $$(II)': R_{multiple} = \frac{X}{n}\cdot\frac{z-z^{n+1}}{1-z}$$
By setting $(I) \stackrel{?}{>} (II)'$ I get: $$z^n \stackrel{?}{>}\frac{1}{n}\cdot\frac{z-z^{n+1}}{1-z}$$
and that's about it. I tried various approaches to prove that the relation holds (or not) for all $n > 1$ and $z>1$ but I cannot get a result.
What do you suggest for a solution?
Let's look at a simple case. Say we have a fund that pays an interest rate of $i$ in the first year and $j$ in the second. Now if we invest $2$ at the beginning of the first year, the accumulated value at the end of two years is $$A = 2(1+i)(1+j).$$ If instead we invest $1$ at the beginning of each year, then the accumulated value at the end of two years is $$B = 1(1+i)(1+j) + 1(1+j).$$ We are interested in comparing these values for different rates of interest $i, j$. Note that $A = B$ implies $$2(1+i)(1+j) = (1+i)(2+j),$$ or $2(1+j) = 2+j$, or $2j = j$. This means $j = 0$ would give you equality; i.e., there is no interest in the second year. This is true irrespective of the interest in the first year. Otherwise, $2j > j$ for any other positive interest rate $j > 0$, hence the cash flow corresponding to $A$ will accumulate greater value. But since we could have negative returns, $j < 0$ implies $B$ will accumulate greater value.
It follows that in the general case, your answer depends on the interest rates in each period, and the order in which they occur. Of course, if the rate is fixed per compounding period, then it becomes obvious that the earlier the funds are deposited, the more interest accumulates.