Comparing Lump-Sum and Staggered Investment Contributions

# Periodic investments with compound interest: where's the mistake?


# Periodic investments with compound interest: where's the mistake?












Consider two investment strategies:

Every year, I have a quantity $I_a$ to invest. There is a financial object that gives an anual return of $r$, that is, after a year it transforms $I_a \mapsto rI_a$. Say, there are two ways I could invest it over a period of $N$ years:

A) I can invest the entirety of $I_a$ at the start of every year. After $N$ years this gives $$ I_N^A = I_a + I_a r + \cdots + I_a r^N = I_a (1+r+\cdots+r^N)=I_a\left(\frac{r^N-1}{r-1} \right). $$

B) Alternatively, I could make the bad decision to invest $I_a$ monthly, that is invest $I_a/12$ every month. How does $r$ act monthly? Well, it has to be $r_m = r^{1/12}$ so that multiplied $12$ times it yields $r_m^{12}=r$. So just like in option A), after $N$ years this gives $$ I_N^B = \frac{I_a}{12} (1 + r_m + \cdots + r_m^{12N})= \frac{I_a}{12} \left(\frac{r_m^{12N}-1}{r_m-1}\right)=\frac{I_a}{12} \left(\frac{r^{N}-1}{r_m-1}\right) $$ where I used in the last equation $r_m^{12}=r$.

Which one is better? Well, $$ \frac{I_N^A}{I_N^B}=12\left(\frac{r_m-1}{r-1}\right) = 12\left(\frac{r^{1/12}-1}{r-1}\right) $$ given $r>1$ there exists a choice of $r$ for which $I_N^B > I_N^A$.

To test this, if I plug SP500 numbers ($r=1.07$) over 10 years we have $\frac{I_N^A}{I_N^B}\approx 0.969282...$.

Where did I make a mistake?

TL;DR: The mistake was that $$ (1+r+\cdots+r^N) \neq \frac{r^N-1}{r-1}. $$ I made a mistake in my derivation, here's a correct derivation:

Let $$ S_N = 1+r+\cdots+r^N, $$ then $$ S_{N+1} = rS_N + 1 \quad \text{and} \quad S_{N+1} = S_N + r^{\color{red}{N+1}}. $$ Hence, $$ rS_N + 1 = S_N +r^{\color{red}{N+1}} \implies S_N=\frac{r^{\color{red}{N+1}}-1}{r-1} $$ As a sanity check, note that in the limit $r\rightarrow 1$, by L'Hopital rule, $$ \lim_{r\rightarrow 1} S_N = N+1. $$

Note: in the answer bellow they asume you start with $r$ and not with $1$ as I stated. Regardless, you can derive their formula using the above derivation by noting that $$ r+\cdots+r^N=r(1+\cdots+r^{N-1})=rS_{N-1}=r\frac{r^N-1}{r-1}, $$ but it is not what I asked.

## Answer by Kermittfrog (score 3, accepted)

https://quant.stackexchange.com/a/79764

The reasoning - without any math - is as follows.

At the beginning of each year, you get a cash amount $I_a$ that you can either invest instantly, so that it will yield $I_a*r$ over the next period, or you can spread the cash out over the next 12 months and invest a monthly fraction $1/12\times I_a$. In the meantime, the uninvested cash will not attract any interest - Hence, for positive returns $r>1$, it is not advisable to spread the investment (i.e. postpone returns) in your example.

Assuming you receive the cash amount at the start of each year and decide to split the investment into $M\geq 1$ equal parts per year. If returns are accumulated at the end of the respective period (e.g. annually, monthly...), the total return is

$$ I_{N,M}^A=\frac{A}{M}r^{\frac{1}{M}\times MN}+\frac{A}{M}r^{\frac{1}{M}\times MN-1}+\cdots+\frac{A}{M}r^{\frac{1}{M}}=\frac{A}{M}r^{\frac{1}{M}}\frac{r^{\frac{1}{M}\times MN}-1}{r^{\frac{1}{M}}-1} $$