# Interest on the Interest

I don’t remember the early years of my education that well. I do remember that maths was consistently my favourite subject back in primary school, though I wasn’t particularly good at it.

Anyway, it was around year 4 (so I was about 10 years old) when I started to take a bit more interest in personal finance. I’m not sure why this happened (I don’t remember young Jeremy being very interested in material things, and although the dot-com crash was in 2001 I’m not sure I knew about it at all back then!). I think at the time I viewed the stock market as very speculative (clearly hadn’t heard of mutual funds or ETFs); the childish me probably saw it as an “adult thing” to do as well (to be fair, if manually picking stocks that’s probably a reasonable view). I was thus more focused on what the older me would recognise as fixed-income investments.

However, in any case, I had saved some money from birthdays and the Lunar New Year, and given that I wasn’t going to be using it immediately I thought it would be good to put it to work. I was vaguely aware of how banks worked, at least as far as retail banking was concerned (i.e. the bank takes your deposit at rate $x$ and lends out your money at $y > x$; the delta is for the service of matching depositors and borrowers). I was also aware of other schemes such as fixed deposits and other types of savings accounts. Interest rates at the time were about 2 to 3 percent, and knowing little else I thought that was not too bad for a start; my account at the time had an annual equivalent rate of 3%.

I remember looking through my bank statements then, and noticing that interest was paid twice a year, at the end of June and December. It didn’t take long for me to figure out that the December figure was bigger, at least partially because it was calculated including the interest from June. I then started wondering what would happen if the interest payments were made monthly, daily … or even billions of times per second. With some research I learned about continuous compounding; even if you were able to do this compounding at 3% infinitely often you’d still “only” get a rate of $e^{0.03} - 1 = 0.0305$ for your efforts.

However, the figures didn’t tally up with my calculations for a long while. I remember initially wondering why I wasn’t paid exactly 1.5% on each payment. Nevertheless, by then I had some familiarity with exponents, and I realised that $1.015^2 = 1.030025 > 1.03$ and really we should be expecting $\sqrt{1.03} - 1 = 0.01489$ each time, rather than 1.5 percent. Still, this didn’t square up with the figures (it was getting down to cents, but still). I let the matter rest at the time, since it was broadly correct. Also, I noticed that the June payments tended to be a little small, the December payments a little too big – so I thought it averaged out in the end (which it did – that’s the point of an AER!).

Anyway, 15 years later I found the reason why, as part of prep work for a reading group I’m doing with Stan. I’m surprised I didn’t think about it back then especially given the observation about June and December payments (at the time, I made the oversimplifying abstraction that the payments were made “every six months”). The key is that the interest was calculated using what is known as an act/365 daycount which factors in the actual number of days for the period you were earning interest, and the first “half” of the year is shorter than the second “half”! Consider that in a non-leap year:

• From 1 January to 30 June you have $3 \times 31 + 2 \times 30 + 28 = 181$ days, but
• From 1 July to 31 December you have $365 - 181 = 184$ days!

With this, we can calculate how much should actually be paid each time. We need to solve

$\dfrac{181}{365} r + \dfrac{184}{365} r \left( 1 + \dfrac{181}{365} r \right) = 0.03 \leadsto r \approx 0.0297783$

And so for the January-June period, on a \$1 investment you would expect interest of

$\dfrac{181}{365} r \approx 0.0147668$

which is notably less than the $0.01489$ figure that we have treating each month to be the same length.

Note that a wide variety of daycount conventions are used, depending on which financial instruments are concerned! There is the 30/360 daycount, where every month is treated as 30 days and the year as having 360 days, which makes month-level abstractions valid but becomes unpleasant when you go below that; you also have the act/360 which like act/365 seems computationally nice. There’s also act/act (used for US treasury debt, notably), which guarantees identical value per day within a period at the expense of dealing annoyingly with leap years and/or the fact that the number of days in a year is odd, and many further variants of what I’ve discussed so far as well including a few particularly nasty ones that scale on business days as opposed to calendar days.