moorfield wrote:The table TJH posted on the
Admiral thread has got me thinking - can one model (roughly) how much income a HYP loses every few years, on the average, due to dividend cuts? And does that have any useful practical application, for example in helping to "size" a cash reserve, or set a limit on income concentration?
To start with, one question comes to mind, which is beginning to bug me...
If each of my N holdings cuts its dividend by P% once every T years, on the average, how much income can I expect to lose overall every few (say 3) years? (Assuming for simplicity each holding initially produces equal income and is not increasing its dividend).
That's making the assumptions that some dividends are being cut, that all other dividends are not being raised, and that you're looking at the situation over years and not just an isolated single year. Together, those assumptions imply that in the absence of further investment (including dividend reinvestment), the HYP's dividend income is bound to fall, or at best remain static if no company's P% chance of a cut happens in a particular year. That totally fails to match the real-world observation that the dividend income of a HYP frequently does rise year-on-year... So any conclusions one draws from those assumptions aren't going to be useful in practice.
In short, it's a good principle to simplify to make analysis easier - but there's a second part to that principle: don't
over-simplify. In particular, a simplification that leads to a pretty obvious, major failure to match reality is an over-simplification.
Changing your assumptions to eliminate that failure to match the reality of HYPs' actual income growth figures, I'll change your question to "
If each of my N holdings cuts its dividend by L% once every T years, on the average, and grows its dividend by G% in the other years, what do I expect to happen to its income overall over a period of a few (say 3) years? (Assuming for simplicity each holding initially produces equal income.)". (Note that this isn't actually dodging your original question: you can always substitute L = P and G = 0 in what follows to get answers to that original question. It's just saying that those particular answers are clearly useless, and you need to substitute other values of L and G to have a hope of getting useful answers.)
That question can be answered fairly easily in terms of statistical expectations. First look at an individual holding, producing £I/N income initially, where the total initial income of the HYP is £I. Each year, it has a 1/T chance of cutting its income by a factor of (1-L/100) and a (T-1)/T chance of growing it by a factor of (1+G/100). So its statistically-expected income at the end of the first year is (1/T) * (1-L/100) * (£I/N) + ((T-1)/T) * (1+G/100) * (£I/N) = (1 + (G-(G+L)/T)/100) * (£I/N). I.e. its income is statistically expected to grow by a factor of 1 + (G-(G+L)/T)/100 in the year (which some might understand more easily as expecting G% growth of the holding, but with a 1/T chance of missing out on that G% growth and instead suffering L% shrinkage). The statistically-expected income from each of the N holdings is multiplied by that same factor, and so the statistically-expected total income of the HYP also grows by that same factor, to (1 + (G-(G+L)/T)/100) * £I.
That same argument applies equally well if I just assume that holding 1 initially produces £I_1 income, holding 2 £I_2 income, holding 3 £I_3 income, etc, up to holding N producing £I_N income, with the total income of the HYP being £I = £I_1 + £I_2 + £I_3 + ... + £I_N. Each component of that sum is statistically expected to be multiplied by the same factor 1 + (G-(G+L)/T)/100 in the year, and so the sum is statistically expected to be multiplied by the same factor. I.e. we don't actually need the assumption of equal initial incomes to come to that conclusion. And without going into detailed formulae, it's quite easy mathematically to cater for more that just the two possibilities of +G% growth and -L% shrinkage: one can assume any statistical distribution of positive and negative growth rates, calculate its statistically-expected growth rate E%, and conclude that at the end of year 1, the statistically-expected total income of the HYP grows by E%. (In the above 2-outcome +G%/-L% case, the calculation of the statistically-expected growth rate is E% = (1/T) * (-L%) + ((T-1)/T) * (+G%) = (-L + (T-1)*G) / (100T) = (-L + TG - G) / (100T) = (TG - (G+L)) / (100T) = (G-(G+L)/T)%.)
So far, so good - we can deal with any initial distribution of the HYP's income between its holdings and any statistical distribution of the single-year dividend growth rates. But problems start to emerge when we look at year two and later years... The main problem is whether we assume that the year-2 dividend-growth-rate distribution is independent of the year-1 dividend-growth-rate distribution. If we do, then every holding is again statistically expected to increase its dividend by E% in year 2, and that will result in the total portfolio income also again being statistically expected to rise by E%. At the opposite extreme, if we assume that the year-2 dividend-growth-rate distribution is totally dependent on the year-1 dividend-growth-rate distribution, in the sense that every company raises or shrinks its dividend by the same percentage as it did in year 1, the total portfolio income will be statistically expected to rise by
more than E% in year 2.
A 2-outcome example to illustrate this, with numbers chosen to keep the arithmetic reasonably simple: assume that each company has a 1-in-5 chance of delivering a 29% dividend cut each year, and otherwise it raises its dividend by 11%. So in the above terms, T=5, G=11 and L= 29, resulting in the statistically expected dividend growth rate being 11% - (11%+29%)/5 = 11% - 8% = 3%. Assume also that we start with a 25-share HYP, with each holding initially producing £100 income for a £2,500 portfolio total. If the portfolio behaves as statistically expected in year 1, five of its holdings cut their income levels to £71 and the remaining twenty raise theirs to £111, producing total portfolio income of 5*£71 + 20*£111 = £2575, a 3% rise. If it again behaves as statistically expected in year 2,
independently of year 1, one of the already-cut holdings cuts again by 29%, to £50.41, while the other four follow the cut with an 11% raise, to £78.81, and four of the already-raised holdings cut by 29%, also to £78.81, while the other sixteen raise by 11% again, to £123.21. That results in a total portfolio income of £50.41 + 8*£78.81 + 16*£123.21 = £2,652.25, which is a 3% increase on £2575. On the other hand, if the year 2 behaviour is totally dependent on the year 1 behaviour, all five already-cut holdings cut again, to £50.41, and all twenty already-raised holdings raise again, to £123.21, and the portfolio total is 5*£50.41 + 20*£123.21 = £2,716.25, a 5.49% rise on £2,575. Carrying the example forward in terms of exactly the correct numbers of holdings behaving as statistically expected would involve starting with unrealistically-large HYPs for the statistically-independent calculations - 125 holdings to start with to get to year 3, 625 to get to year 4, 3,125 to get to year 5, etc, but the statistically-expected total dividend incomes can be calculated, and will continue to show 3% year-on-year increases, to £2,731.82, £2,813.77, £2,898.19, etc. And the totally-dependent cases can easily be calculated on the basis of behaving entirely as total dependence says they will after the first year's outcomes: year 3 produces 5*£35.79 + 20*£136.76 = 2,914.15, year 4 produces 5*£25.41 + 20*£151.81 = £3,163.25, year 5 produces 5*£18.04 + 20*£168.51 = £3460.40, etc, so that the year-on-year increases are 7.29%, 8.55%, 9.39%, etc. (In the long run, they'll get to 11%, as the five multiply-cut holdings become totally insignificant and so the twenty multiply-raised holdings completely dominate the portfolio income's growth rate.)
Time for another reality check: In reality, far too many HYP holdings have historical dividend growth patterns like
AstraZeneca,
BP,
Carillion,
Pennon and
Tesco (to name just a few), in which long periods of fairly consistent dividend growth (or non-growth!) occur, punctuated by shifts to a different rate (sometimes 0%) and/or by dividend cuts (and the only one of those we can be confident will not have another such punctuation is Carillion - its last such punctuation was a full stop...). That's not consistent with successive years' dividend growth rates being independent of each other, nor with them being totally dependent on each other. So it lies somewhere between the two above extremes of independence and total dependence - but where? That can make a big difference: in the above example, being closer to the 3% long-term dividend growth rate for years being independent of each other or the 11% growth rate for them being totally dependent on each other in the above examples can make a massive difference over a decade or two...
So basically, to get a
useful answer to your questions, you need not only to get a good understanding of the statistical distribution of HYP companies' year-on-year dividend growth rates, but also of how independent those growth-rate distributions are of each other in successive years. And that doesn't necessarily just mean in
two successive years - my guess is that if you had a good idea of the joint statistical distribution of those growth rates in one year and the next, and produced randomly-generated dividend records that matched that distribution and with years otherwise independent of each other, they would look more like actual company dividend records but still be very perceptibly different from them, and those perceptible differences could still make a significant difference to a HYP's income-growth prospects. You would probably want joint distributions over more years - but how many more, I don't know! And there are distinct problems getting enough real-world data to provide any confidence that a proposed multi-year joint distribution matches the real world at all well...
In short, the easily-obtained models of the statistics of the percentage changes companies make to their dividends fail to match reality in ways that can make big differences to one's expectations about what happens to portfolio income, and models of those statistics that are good enough not to make such big differences look to be at best hard work to obtain, and quite likely impossible to obtain due to shortage of real-world data. So I wish you well in your endeavours, but I'm afraid I don't rate your chances of coming up with something useful very highly...
One final note is that I've generally said "statistically expected" above rather than just "expected". The reason for that is that for some distributions, what is statistically expected can differ markedly from what is expected in the ordinary sense of the word. A pretty extreme (and totally unrealistic) example of that: suppose a company guaranteed to either raise its dividend by 70% or cut it by 50% each year, with equal chances of each and with every year's change independent of those for all previous years. If you started with a holding giving you £100 income, after one year you would have an income of £50 or £170, each with probability 50% - statistically expected income £110, 50% chance of ending up with less income than you started with. After two years, you would have income of £25, £85 or £289, with probabilities 25%, 50% and 25% respectively - statistically expected income £121, 75% chance of ending up with less income than you started with. After 3 years, you would have income of £12.50, £42.50, £144.50 or £491.30, with probabilities 12.5%, 37.5%, 37.5% and 12.5% respectively - statistically expected income £133.10, 50% chance of ending up with less income than you started with. And so it goes on - the following table summarises the results up to 16 years:
As the years go by, the statistically-expected income can be expected to rise steadily at 10% per year, but the expectation in the normal sense of the word is that the income is more and more likely to be less than its £100 starting level - not a steadily-increasing chance, but it generally drifts upwards, and it will eventually reach a point beyond which it never drops below 60% again (I believe that point is 9 years), then a later point beyond which it never drops below 70% again, then a later point beyond which it never drops below 80% again, then a later point beyond which it never drops below 85% again, etc. Those two statements are compatible with each other because the statistically-expected income becomes more and more concentrated in smaller and smaller chances of bigger and bigger income levels.
As I said, that example is pretty extreme and totally unrealistic - but there is a lesson about reality in it. That lesson is that the expectation in the ordinary sense, i.e. an income level that you're reasonably certain to achieve, is generally less than the statistical expectation, i.e. a weighted average of the income levels you might achieve, weighted by the chances that you'll achieve them. How much less basically depends on how varied the plausible outcomes are: the more varied those outcomes are, the greater the reduction. It takes a pretty ridiculous level of variability of the outcomes like the example's 50%-each chances of a 50% cut and a 70% rise to create a reduction from a statistically-expected very healthy growth rate and a normal-sense expectation of shrinking to arbitrarily low levels, but lesser levels of the same effect do occur and are the basic reason behind HYPers' general preference for a decent level of diversification, for shares with fairly steady rates of dividend growth rather than stop-start dividend growth, etc.
Gengulphus
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