Here’s a calculation that I found pretty surprising. Have a guess as to its meaning:

If you guessed something relating to savings, you’d be on the right track; the denominator would probably have clued you in to something along those lines, since the is the marginal tax that higher rate earners in the UK would face (as of the 2016/17 tax year). You might have figured out then, that the fraction on top probably refers to some form of a time savings expressed in minutes. The is probably a bit more puzzling especially if you’re not from London; from the previous observations it’s the cost of taking some convenience that saves nine minutes. That’s right; you might recognise that as the price of a single-trip Tube fare.

Putting that together, that computation is actually the effective hourly gross rate I’m being paid to walk to my office as opposed to taking the Tube. Specifically, I’m comparing (a) taking the Tube, which takes about 15 minutes plus a single trip fare of 2.40, and (b) walking, which takes 24 minutes. We’re thus looking at a net rate of 2.40/9 minutes. Of course, I pay for the Tube using post tax money, so we need to factor that in, and to get an hourly rate we scale that linearly.

Now of course this doesn’t mean I’ll never take the Tube to work again, or even that I’ll take it much less often – the 9 minute difference can be important (say there’s a high priority issue going on, or I’m about to be late for a meeting), and walking can be pretty unpleasant (if I’m very tired, it’s late, or it’s too cold; that said, I’ve done this in the current ~0 degree weather and found it fine, at least). Probably, doing this in the mornings would generally be more reasonable and sustainable as I’m pretty tired by the end of the day. Saving one trip even occasionally is enough of a win, I’d say, and getting some time to clear my head in the mornings is probably not too bad as well.

A question could be why I don’t use a Travelcard for the computations instead. The weekly one for zones 1-2 costs  and since I largely stay within zone 1 and 2 we can assume that that’s all I’ll need (I’ll ignore the odd trip to Heathrow). I think or is probably about the number of rides I’d take in a week if I used it aggressively (two a day, maybe three or four on the weekends). We can re-run the numbers with , which means a trip costs . Our final answer is which is still pretty solid (if you work 42 hours a week that’s a gross annual income of just over ). Anyway, as it turns out because I use contactless, if I happen to have a week where I do travel a lot I’ll effectively be using one.

Now let’s have a look at the monthly or annual tickets. These cost for monthly, or for annual. Assuming that we make trips a day on average (taking the middle of the estimates above), and a month as days, with the monthly card the cost of a trip falls to and with the annual card it falls to . The “gross wage” calculations become and , which while not as high are still solid amounts. an hour corresponds to about just over in gross income assuming a 42 hour week, which would put you in about the 78th percentile of earners according to the ONS data for 2013-2014. I guess this will probably be a bit lower now, with inflation/wage growth, but still decent.

Assuming usage is high, it seems that the annual card might be the way to go. However, an issue is that I sometimes need to travel overseas for work potentially a fair chunk (let’s say personal plus work travel adds up to two months a year), so the annual one is quite definitely a non-starter. Multiply the price of a trip by to factor that in, and you end up with a per-trip figure that’s higher than the monthly card. I have been getting the monthly card in the past, especially when I was a student and it only cost , partly because walking to Imperial was going to take a very long time (that route yields savings more on the order of minutes). Note that while losing a card is a legitimate concern, you can usually recover the travel passes using the TfL website.

(N.B. This idea was conceived independent of the currently ongoing Tube strike, though in a sense it’s a bit of a bonus that things aren’t affected.)

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 and lends out your money at ; 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 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 and really we should be expecting 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 days, but
• From 1 July to 31 December you have days!

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

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

which is notably less than the 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.