Browse Month

# 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.

# ScheduledExecutorServices

I briefly touched on this in my first post, but I find that there are quite a few things which I aim to carry out (at least / at most) N times per time interval T. This is described by John Sonmez in Soft Skills as a quota. You may have been able to infer this from the frequency of posts here – I’ve been aiming to write at least 1 blog post per week.

On another note, scheduled executor services are a part of the Java concurrency libraries (introduced in Java 5). They allow clients to submit one-shot tasks wrapped in callables (possibly with some delay before execution), as well as tasks with a fixed rate (more similar to a quota, though not quite – mainly because quotas are concerned with getting things done, while executor services are concerned with enabling tasks to run; also because a scheduled executor service won’t start concurrent tasks). Clients can also specify tasks with a fixed delay; this differs from a fixed rate in that the countdown to the next execution starts after the current task has completed.

If one assumes that the tasks complete relatively quickly, then quotas are, in a way, less restrictive than scheduled executor services; they give flexibility as to when in the time period T the N events occur. This is especially important for tasks that require 2-way synchronization (for me, that largely involves spending time with friends) – it would be even more so for barriers involving multiple people though I haven’t actually planned any such arrangements.

A downside of this is that it delays the decision of deciding when in each period T the events should be scheduled; it’s arguably simpler to schedule them at the same point in each period. Also, if one follows the letter of the quota, then this can lead to very uneven intervals between occurrences – for syncing up with friends, blog posts and quite a few other things, while this certainly isn’t bad it’s also less than ideal (imagine if I had a “weekly sync” with a friend, but actually spoke to them once every 2 weeks at 23.35 on Sunday for 20 minutes, and then again at 00.05 on Monday for 20 minutes). I find that a good way around this is to normally target the same point, but allow for flexibility; I’m not sure you can readily do this in ScheduledExecutorService (you’d have to cancel the old task and reschedule a new one with the correct delay, I think).

The diagram above more succinctly illustrates the difference between the timing semantics for the various things I’ve described. More often than not, at least for meeting up and/or syncing with people, the pattern is closer to the second or fourth above (i.e. with random variation, plus the occasional additional occurrence; perhaps the occasional missed occurrence as well, though I didn’t include that in the diagram).

Another way I’ve modeled this in the past is on the concept of a failure detector in distributed systems. Servers/subsystems can arrange periodic heartbeats, suspecting them of failure if a heartbeat is not received (and “un-suspecting” them if one is subsequently received). Though because of the aforementioned flexibility, a pattern that conforms to a quota could result in a heartbeat interval of 2T. I guess the idea I had previously was I didn’t want to lose contact with friends, bearing in mind that I was still in Imperial at that time and thus would quite naturally and easily meet many of my friends in lectures or in labs – on the other hand, I’m the only person from my class to go to Palantir (at least at this point). I find using a system based on quotas is certainly much easier for me to manage, as well.

# Up All Night

“Knew we would crash at the speed that we were going
Didn’t care if the explosion ruined me…”
– Charlie Puth, “Dangerously”

The quote above is from a song that I’ve been listening to a fair bit recently, and I’ve picked up on those two lines although in a different context (as you might expect, the original song is concerned with a reaction to a breakup). I’ve been thinking about how my work practices could work in the longer term and what would be sustainable. Nonetheless, hearing those two lines makes me think of deep surges; some of the most short-term of these could perhaps take the form of all-nighters.

I’ve been fairly lucky in that I haven’t had to pull many all-nighters for quite some time. I think I only did this once for MCMAS-Dynamic (during the report-writing stage; generally given the technical complexity of the work I don’t think it would have made sense), and I don’t think I did one during the third year group project. I also remember having executed one during second year when revising for the exams, though that was thankfully well before said exam period. There have been several hackathons, of course, as well as other occasional personal surges but generally I find that I perform best if I have adequate sleep, and even in the relatively short run I’d be better off doing three say 15-hour days, punctuated by relatively normal sleep (well, as normal as that can be given such a schedule) than plugging away in a continuous stretch.

Anyway, besides the Charlie Puth song I’m also writing about this now because I voluntarily did one this week, though for a rather different reason: watching the US presidential election. I had a couple tabs open with various election newsfeeds and a couple watching market futures and GBPUSD. On hindsight I’m not sure exactly why I did it since it was pretty apparent midway through (I think around 2-3 am in London time) that things were going Trump’s way, and I wasn’t trading through the night (by the time markets opened in the morning there wasn’t too much of a cheap-buying opportunity). That’s a subject for another post, though.

I think the negative effects of sleep deprivation are well-documented; I’m not sure exactly why I pulled the all-nighter for the MCMAS-Dynamic report (probably wanted to rush something out for a supervisor meeting the next day), but I do distinctly remember that the two or three pages that I cranked out, while probably not bad per se fell particularly far short of my quality standards in a later proofread. The problem I’m trying to address with an all-nighter involves not having enough time to deal with a short-run (typically next-day) requirement, and in less extreme cases it’s not the only solution; where possible, I’d also like to try an alternative of waking up abnormally early to work on the issue. Understandably, there are risks that one might fail to actually wake up early, though I think this can be mitigated with suitable (read: loud and highly dissonant) alarms.

However, there are cases where I find this to be the best solution anyway. Some of this might involve external time constraints (for example, if it involves live following of current events – the aforementioned US election is one, or the recent World Series if one’s so inclined; examples from software engineering could include firing off long-running performance or integration tests, or meeting sudden customer requirements). Also, for suitably short time spans this is likely to be an optimal or near-optimal solution (even then, a 1.5 hour nap could potentially be useful in such cases). I think another useful factor to bear in mind would be the activities planned for next day (an exam or interview would be very bad, for instance).

Once the decision to forego sleep has been made, I usually don’t find the direct implementation of all-nighters to be too bad, perhaps because for things to have reached that point there must have been a compelling reason. Typically, by then the outcome-oriented side of me takes over and decides that it would be a night of crushing things (though it doesn’t always calculate the costs appropriately).

I think for me at least the most challenging part of this is managing its costs the next day. I personally don’t perform well if I haven’t had enough sleep, and there’s also a risk of overcorrection (that is, sleeping too early, which messes with the sleep schedule for the next few days). I guess caffeine can be deployed to some extent to address this, though I’ve been on the wrong side of that as well. I find that removing access to a bed at least until only a few hours before one’s normal bedtime can help as well – in fact, staying outside is probably even better (I can sleep on a chair if I’m at home).

In summary, it’s a very useful tool in my experience, and there are circumstances where it might be necessary or optimal, but generally speaking where possible this should be avoided.

# Tracking Times at Imperial

The graph you see above reflects the number of hours I spent on “work” each week, from the week starting 5th October 2015 (I finished my internship at Palantir on the 2nd of October) up to the week starting 5th September 2016 (I started full-time at Palantir on the 12th). Obviously, this includes all time spent explicitly on academic work, at Palantir (social events do not count) as well as time spent tutoring (inclusive of marking and preparing the problem sheets). There’s more to it, though – I place work in quotation marks, as there are quite a number of activities that people might not classify as work that I do count towards the total, such as having 1:1 syncs with people and personal reviews.

I’ve annotated the significant peaks and troughs on the graph with some of the events that had taken place around then that contributed to why I worked so much (or so little). You’ll see that I’ve shaded the part of the graph above 70 on the y-axis in red; for me at least, I think I instinctively start feeling some degree of push-back at that point (and I’ve been cautioned that 70 is already pretty far on this).

Typically, when I look at a graph, I try and identify things that I find to stand out as unusual, and then seek explanation for them. Initially, what does stand out to me is the relative lack of height of the peak labelled (6), the weeks leading up to the end of the Final Project; I would have expected something quite a bit more. I’d attribute this to the sheer cognitive difficulty of the final stages of said project; I remember finding that I would be drained very, very quickly when working on it. I guess for the final project I worked at it pretty consistently over the year, so there was no need for a massive surge at the end as well.

I notice extremely sharp drop-offs (A) and (B) after the end of term 1 and 2, yet no such drop-off exists after term 3. Perhaps, this is a signal that the 58.95 or 61.25 averages in those terms are too harsh (summer term was a relatively tame 52.33), and this does already factor in exam week or week 11, which tends to be less intense as I need to conserve my energy for the examinations themselves. I tend to think of week 8 or 9 as the busiest week in each term, owing to exam revision, and this pattern is reflected in peak (5), but seems absent in term 1 which, in fact, exhibits a convex sequence. This might feed back into the earlier point about considering 70 hours a week as a dangerous point to be insufficiently prudent; there is a crash even after a series of weeks in the 60s.

Although I recompiled this graph recently, I first performed the labelling in September just before I started at Palantir. Nonetheless, looking back at it about two months later, one of the labels stands out to me, that being (A) perhaps because in and of itself it does not seem to give a proper explanation of why the trough (or peak, in the other cases) was there. I did return to Singapore to spend time with my family and a few friends, but it wasn’t really the case that I did very much on that front – in fact, a fair chunk of my time in Singapore went towards MCMAS* (which explains that mid-50 spike, which is actually the week starting 28th December) and Fallout 4 (which, of course, did not count).

It’s heartening for me at least to see that I have a fair degree of intrinsic motivation, as shown by the red line. Over the roughly two months, I managed to work on MCMAS-Dynamic, program extensions for Keep Talking and Nobody Explodes (and, in doing so, revisit programming in C#), set up this website, complete a full retrospective round of the first and second year examinations and learn more about personal finance and investment.

Clearly, the data may be analysed as a time series; in fact, I have bucketed this graph into weekly aggregates, but I do have data down to a daily granularity. An alternative way of handling the daily data could have been to compute a simple or exponential moving average of the data, though I don’t really like doing this at a daily granularity is because of very clear seasonality (in particular, I tend to work the most on Tuesdays, or Wednesdays part-time at Palantir; I work the least on Thursdays and Sundays).

I only did this in year 4; it would be interesting to see how the time profile would have looked like in previous years and how similar it might have been, though the different structure of each academic year would probably have made the series look somewhat different (for example, the beginning of the third term in year 3, very near the (B) trough, would have been a very high peak as I aggressively ramped up for the industrial placement at Palantir).

This was a rather interesting exercise, even without looking into the distribution of time across modules and/or activities. If that is factored in, there are other interesting patterns; for example, the amount of time spent on MCMAS, which involved considerable surges after each set of exams was dealt with, and the time spent on 1:1s and syncs, which was generally a few hours per week but had a 12.5 blip in the final week of the summer term (before people dispersed) and a 10 somewhere later on (maximum likelihood guess would be meeting three of the guys I regularly sync with nowadays for meals on three distinct occasions).

I haven’t been doing this as rigorously ever since I started at Palantir, mainly because the initial impetus behind this initiative was actually understanding which modules I was spending a disproportionate amount of time on (Computing for Optimal Decisions, Software Reliability), and also because I had ready access to my personal email accounts (I tracked the data using Google Calendar); also, my work is far more reactive to changes that may happen because some high-priority issue appeared. Perhaps using it for spare time could be useful, but then the administrative overhead is much larger relative to the time actually being tracked – to a point where I’m not sure it’s worthwhile.