### Hamilton

Very late, I listened to the Hamilton cast album. My take is that it is good, and below I list some of my favourite bits (numbers are track numbers).

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Spoilers ahead. The Internet has no shortage of Star Wars opinions, but Star Wars is fun to think and write about even if there's no-one reading.

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(There's no point to this story – it's just an unusual pair of dreams for me.)

A while ago – a year-ish perhaps, but it wasn't so remarkable at the time that I noted it down anywhere – I had a dream, possibly a recurring one, that I'd gone back to university. Studying for a coursework Master's, I was enrolled in a geostatistics course, the sort of course that I was lecturing in real life. And I was*totally* overwhelmed by it. I couldn't keep up with the assignments, and as the weeks went by, it became clearer and clearer to me that I was going to fail. I dropped out, my self-confidence crushed, having lost what is perhaps my best skill, namely passing maths-heavy university courses.

Those feelings of total inadequacy passed soon after waking up, and I hadn't thought back to those dreams since long after waking up after the last of them. Then last night I dreamt that I'd taken on tutoring a third-year maths course in neural networks, despite only knowing about one sixth of the lecture material. (I'm working through a book on the subject over at my .com.) I was sure that I'd be able to learn the content well enough as the course progressed to teach it.

I can't remember if I was awake or still dreaming when I recalled my dreamt coursework failure of a year ago, though given the strength of the emotions involved, it was surely while still dreaming. It was as though, by becoming a tutor for material that I didn't yet know and being sure that I'd do a good job at it, I'd redeemed myself for giving up on the geostats course. A small moment of strongly felt triumph, a heavy weight lifted from my shoulders. I woke up happy.

A while ago – a year-ish perhaps, but it wasn't so remarkable at the time that I noted it down anywhere – I had a dream, possibly a recurring one, that I'd gone back to university. Studying for a coursework Master's, I was enrolled in a geostatistics course, the sort of course that I was lecturing in real life. And I was

Those feelings of total inadequacy passed soon after waking up, and I hadn't thought back to those dreams since long after waking up after the last of them. Then last night I dreamt that I'd taken on tutoring a third-year maths course in neural networks, despite only knowing about one sixth of the lecture material. (I'm working through a book on the subject over at my .com.) I was sure that I'd be able to learn the content well enough as the course progressed to teach it.

I can't remember if I was awake or still dreaming when I recalled my dreamt coursework failure of a year ago, though given the strength of the emotions involved, it was surely while still dreaming. It was as though, by becoming a tutor for material that I didn't yet know and being sure that I'd do a good job at it, I'd redeemed myself for giving up on the geostats course. A small moment of strongly felt triumph, a heavy weight lifted from my shoulders. I woke up happy.

On a recent trip to Brisbane I was having dinner with a group of old friends, and I realised that we had become incredibly boring people who talked about superannuation, the stock market, and so forth. In this post, I set out some notes on the taxation of partially franked dividends; I think it is relevant to at least two people, author included.

(I worked this algebra out today, not financial advice, may only apply to Australia, may not even be correct though I think it is, etc.)

You own shares in a company. The company pays out a dividend to its shareholders,*without having paid company tax on it first*. In this case, we call the dividend *unfranked*, and you pay tax on it according to your marginal income tax rate (probably 32.5% or 37%). If your marginal income tax rate is t_{i}, and the dividend is D, then you'll owe the tax office t_{i}*D.

The more complicated case is if the company pays its company tax on the dividend before giving it to shareholders. If the company tax has been paid in full, then the dividend you get is*fully franked*. The key points are:

Algebra should make this clearer (if not, there are heaps of explanations on Google). Let t_{c} be the company tax rate (30%). Let D_{f} be the franked dividend, i.e., what you receive. Let D_{g} be the grossed-up dividend, i.e., what the tax is being imposed on. We have

(1 - t_{c})*D_{g} = D_{f}.

More generally, the dividend may be only partially franked. Let f be the fraction of the dividend which is fully franked. Then

(1 - f*t_{c})*D_{g} = D_{f}. (*)

The total amount of tax the ATO wants is t_{i}*D_{g}. The company's already paid f*t_{c}*D_{g}, so you owe

tax_owing = t_{i}*D_{g} - f*t_{c}*D_{g} = D_{g}*(t_{i} - f*t_{c}).

(If this quantity is negative, then the tax office owes you money, and this can either turn into a tax refund or offset some other tax.)

It is more useful to use (*) to work out how much tax you owe as a function of the dividend that you receive:

tax_owing = D_{f} * (t_{i} - f*t_{c}) / (1 - f*t_{c}).

Plugging some numbers in: if the dividend is fully franked, then f=1. Say t_{i} = 37.5%, and t_{c} = 30%. Then you owe the ATO (37.5% - 30%) / (1 - 30%) = 10.7% of the dividend you receive.

In the last couple of years, Vanguard's VHY fund has been giving distributions (which I gather is called a different term to 'dividend' because the distribution comprises lots of individual dividends from all the companies in the fund) around 70% franked. Plugging in f = 0.7, we get (37.5% - 0.7*30%) / (1 - 0.7*30%) = 20.9%. So about a fifth of the distribution you get will go to the tax office.

(This remains the case even if you set up an automatic re-investment plan; the ATO treats it as though you received the income in the form of partially franked dividends, then bought more shares with it. The income is taxed.)

We can use this to estimate the effective returns from a fund. From the above link, VHY's growth since inception has been an annualised average of 13% p.a., comprising 7.1% p.a. growth in the unit price and 5.9% p.a. in distributions. Since about a fifth of the latter is eaten up by tax, the effective return has been a bit under 12% p.a. rather than 13%.

My eyeballing of the table in that PDF file suggests that taking a percentage point off the returns is a decent rule of thumb for working this out, at least if you're in my tax bracket. The forecast growth is 8%, so I'll interpret that as around 7%. (And remember that as a 95% confidence interval, the forecast for a single year is more like (8 +/- 25)%.)

(I worked this algebra out today, not financial advice, may only apply to Australia, may not even be correct though I think it is, etc.)

You own shares in a company. The company pays out a dividend to its shareholders,

The more complicated case is if the company pays its company tax on the dividend before giving it to shareholders. If the company tax has been paid in full, then the dividend you get is

- The ATO taxes, at your marginal income tax rate, the imputed full value of the dividend
*before the company tax was paid on it*(the grossed-up dividend). - You get credits for the tax that the company has already paid on the grossed-up dividend, so that the tax isn't paid twice.

Algebra should make this clearer (if not, there are heaps of explanations on Google). Let t

(1 - t

More generally, the dividend may be only partially franked. Let f be the fraction of the dividend which is fully franked. Then

(1 - f*t

The total amount of tax the ATO wants is t

tax_owing = t

(If this quantity is negative, then the tax office owes you money, and this can either turn into a tax refund or offset some other tax.)

It is more useful to use (*) to work out how much tax you owe as a function of the dividend that you receive:

tax_owing = D

Plugging some numbers in: if the dividend is fully franked, then f=1. Say t

In the last couple of years, Vanguard's VHY fund has been giving distributions (which I gather is called a different term to 'dividend' because the distribution comprises lots of individual dividends from all the companies in the fund) around 70% franked. Plugging in f = 0.7, we get (37.5% - 0.7*30%) / (1 - 0.7*30%) = 20.9%. So about a fifth of the distribution you get will go to the tax office.

(This remains the case even if you set up an automatic re-investment plan; the ATO treats it as though you received the income in the form of partially franked dividends, then bought more shares with it. The income is taxed.)

We can use this to estimate the effective returns from a fund. From the above link, VHY's growth since inception has been an annualised average of 13% p.a., comprising 7.1% p.a. growth in the unit price and 5.9% p.a. in distributions. Since about a fifth of the latter is eaten up by tax, the effective return has been a bit under 12% p.a. rather than 13%.

My eyeballing of the table in that PDF file suggests that taking a percentage point off the returns is a decent rule of thumb for working this out, at least if you're in my tax bracket. The forecast growth is 8%, so I'll interpret that as around 7%. (And remember that as a 95% confidence interval, the forecast for a single year is more like (8 +/- 25)%.)

I've been learning a tiny little bit of music theory – major scales and chords and so on – and I would like to change everyone's use of relative notation.

The usual way of writing relative notes in a scale is 1, 2, 3, 4, 5, 6, 7. In C major, this would correspond to C, D, E, F, G, A, B. Chords are usually written in Roman numerals, with capital letters for major chords and lower-case letters for minor chords: I, ii, iii, IV, V, vi, vii^{o} for C, Dm, Em, F, G, Am, Bdim.

The first problem I see with this is when we want to describe secondary chords like V/V, "five of five": we temporarily go to the major scale of the 5, and extract the V chord from that scale. In C major, the V chord is a G major; in the G major scale, the 5 is a D, so the V/V is a D major chord.

I expect that people who work with these things regularly work these out as quickly as I can do my times tables, hopping between scales with ease. But I would like to work things out in terms of modular arithmetic. In the case above, things appear to work out: there are eight notes in an octave, 5 + 5 = 10, and 10 (mod 8) = 2, and D is the 2 in the C scale.

But this breaks down if we want the V/ii chord: 5 + 2 = 7, but the ii is D, and the 5 in the D scale is an A, not a B.

The mathematically-inclined may have already spotted at least one of the mistakes in the above reasoning. There are seven different notes in the scale, so we should be working modulo 7, not modulo 8. The second mistake is with the notation: the scale should start with zero, not 1.

To do the calculation properly, we need to first subtract 1 after doing the addition. So, V/V is 5 + 5 - 1 (mod 7) = 2. And V/ii is 2 + 5 - 1 (mod 7) = 6. It works.

Having to subtract the 1 is really annoying though, and the special case of ending on a 7 (e.g., V/iii which becomes zero mod 7) needs to be handled. A better scale would be

0, 1, 2, 3, 4, 5, 6

with chords

O, i, ii, III, IV, v, vi^{o}.

Then the normal-notation V/V becomes a IV/IV, and can be calculated as 4 + 4 (mod 7) = 1, the D major chord. And a normal-notation V/ii becomes a IV/i, and can be calculated as 1 + 4 (mod 7) = 5, the A major chord.

This is much cleaner, and the only minor issue is a roman numeral for the zero , which I wrote above as a letter O.

Taking into account how little music theory I know, I figure my proposal is somewhere about as optimistic as my suggestion for question marks.

The usual way of writing relative notes in a scale is 1, 2, 3, 4, 5, 6, 7. In C major, this would correspond to C, D, E, F, G, A, B. Chords are usually written in Roman numerals, with capital letters for major chords and lower-case letters for minor chords: I, ii, iii, IV, V, vi, vii

The first problem I see with this is when we want to describe secondary chords like V/V, "five of five": we temporarily go to the major scale of the 5, and extract the V chord from that scale. In C major, the V chord is a G major; in the G major scale, the 5 is a D, so the V/V is a D major chord.

I expect that people who work with these things regularly work these out as quickly as I can do my times tables, hopping between scales with ease. But I would like to work things out in terms of modular arithmetic. In the case above, things appear to work out: there are eight notes in an octave, 5 + 5 = 10, and 10 (mod 8) = 2, and D is the 2 in the C scale.

But this breaks down if we want the V/ii chord: 5 + 2 = 7, but the ii is D, and the 5 in the D scale is an A, not a B.

The mathematically-inclined may have already spotted at least one of the mistakes in the above reasoning. There are seven different notes in the scale, so we should be working modulo 7, not modulo 8. The second mistake is with the notation: the scale should start with zero, not 1.

To do the calculation properly, we need to first subtract 1 after doing the addition. So, V/V is 5 + 5 - 1 (mod 7) = 2. And V/ii is 2 + 5 - 1 (mod 7) = 6. It works.

Having to subtract the 1 is really annoying though, and the special case of ending on a 7 (e.g., V/iii which becomes zero mod 7) needs to be handled. A better scale would be

0, 1, 2, 3, 4, 5, 6

with chords

O, i, ii, III, IV, v, vi

Then the normal-notation V/V becomes a IV/IV, and can be calculated as 4 + 4 (mod 7) = 1, the D major chord. And a normal-notation V/ii becomes a IV/i, and can be calculated as 1 + 4 (mod 7) = 5, the A major chord.

This is much cleaner, and the only minor issue is a roman numeral for the zero , which I wrote above as a letter O.

Taking into account how little music theory I know, I figure my proposal is somewhere about as optimistic as my suggestion for question marks.