Leasehold Mathematics

The Beeb has decided to spend the day explaining why you don’t want to buy a leasehold property.

Recently I picked up a book on the history of housing, which remarked (almost in so many words) that the point of leasehold was that you owned the land, got a house built on it and leased it out for the life expectancy of the house. After the lease was up you got a tatty old house back which you knocked down (or rather your descendants knocked down) and replaced with a new, up-to-date house, maintaining the quality of the area. (And as all the houses would be on the same leasehold they could be knocked down simultaneously.)

This worked fine until someone came up with the idea of lease extensions, with the result that all these cheaply built Georgian leasehold houses remain highly desirable properties on short leases. Despite houses now having a higher life expectancy (it having been realised that a cheap Georgian 5-storey terrace does not fall down after 80 years) leaseholds remain a peculiarly popular way of dealing with perfectly ordinary houses which at first glance look like freehold.

And, of course, they are handy for freehold houses turned into several flats (as not all the flats can own the freehold and the landlord would probably rather it if none of them did).

Which brings us to this week’s sob story. This chap has bought a leasehold flat.

Its lease lasts for 190 years and the lease cost doubles every ten years (and is, of course, on top of the £150k that he paid for the right to live in the flat and pay this lovely lease).

It starts at £250 per annum.

Readers of Murderous Maths (or indeed anyone with a reasonable grasp of arithmetic) will rapidly realise this may be going to get messy (this chap obviously wasn’t brought up on Murderous Maths and Asterix the Gaul) – the total rent he will have paid over the lease is seen here totted up by decade end:

  1. £2,500 (some people pay this in monthly rent);
  2. £7,500 (a slightly excessive annual rent for a flat – but remember this is a total divided over 20 years);
  3. £17,500 (about what I paid in total rent over four years for a flat, but divided over 30 years);
  4. £37,500;
  5. £77,500;
  6. £157,500 (bit more than my mortgage, spread out over 60 years instead of 25);
  7. £317,500;
  8. £637,500;
  9. £1,277,500 (which would currently buy the landlord a technology-stuffed self-powered railway carriage – after 90 years of patient totting-up);
  10. £2,557,500;
  11. £5,117,500;
  12. £10,237,500;
  13. £20,477,500;
  14. £40,957,500;
  15. £81,917,500;
  16. £163,837,500;
  17. £327,677,500;
  18. £655,357,500;
  19. £1,310,717,500 (which will then allow the landlord to spend the total takings by paying the premium for a moderately prosperous intercity rail franchise, and makes an average annual rent of about £6.9million on a flat).

The formula for working out how much will have been paid in total by the end of each decade is something to the effect of:

(2,500 x 2^) – 2,500 

^ represents multiplying the 2 by the power of the decade number (there should really be a little elevated x instead, but that takes mucking around with HTML superscript codes). So for the 7th decade you multiply 2,500 by 2 to the power of 7 (or 2 x 2 x 2 x 2 x 2 x 2 x 2) (and get 320,000) then subtract 2,500 (which is 317,500).

Put the 2,500 back on again and divide by 2 to work out the amount to be paid over that decade – thusly:

(2,500 x 2^) / 2

or indeed

2,500 x 2^-1

Which for the 7th decade is £160,000, or £16,000 per annum (£250 x 2 x 2 x 2 x 2 x 2 x 2 – ignoring the last x 2 in the upper of these two formulas because it’s cancelled out by dividing by 2 – and aren’t formulas neat?)

This formula is very handy for working out how much you’ll have to pay in ground rent for a flat on a leasehold arrangement like this one if you stay there for ten years.

On the other hand, this sort of contract – with its annual rent in the last decade of £250 x 2^18, or £65,536,000 per annum – is a rather nifty bet against inflation.

(If somewhat overkill. £250 in 1826 would officially now be worth (2016 prices) £22,756.58.)

tottenham-house-1-jpgThis is a house, set in several hundred acres of landscaped countryside and possibly actually worth £6.9million (though not per annum).


Light Maths

The next post on this blog will be rather heavy, so here’s something lighter in a similarly mathematical vein to get people in the mood:


Find “x”.

(Assuming that my maths is good enough for it to work…)