Flying a rocket into space takes a lot of fuel, a huge amount in fact. The reason for this is that every kilogram on board needs to be lifted up against the Earth’s gravity, and that takes energy. Rockets and their payloads tend to be quite heavy, and the Earth’s gravity is quite strong; these two factors combine to mean that it takes a lot of energy, a huge amount in fact.

When the rocket is part of the way up, some of the fuel will now have been burnt, and so one of the many fuel tanks will now be empty. Given that the rocket has no further use of an empty fuel tank, and that it is literally weighing the rocket down, in a multistage rocket it will be jettisoned.

Thinking about it another way, the rocket’s objective is to put the payload into space, but that takes fuel, and you can’t carry fuel without fuel tanks; so we need fuel tanks, but they add weight, which means more fuel. The fuel tanks are essentially a necessary evil, but we can mitigate that by jettisoning them as soon as they become empty.

If you haven’t followed either of those explanations, there’s more detail of the maths behind this here.

But what does all of this have to do with buying a house? Well, the property ladder is just like a multistage rocket but in reverse!

### Mortgage costs

Let’s look at the marginal cost of adding an extra £ 1 to a 25-year mortgage with 3% APR compound interest.

Because we’re analysing the marginal cost, we assume that the mortgage is paid off in precisely the same way that it would have been, and so 25 years later everything has been paid off apart from the extra £ 1 that we added at the beginning. However, the interest on that extra £ 1 has been accumulating so that the debt is now:

£ 1 * 1.03 ^ 25 = £ 2.09

However, we’re currently assuming that the interest rate will remain at 3% for 25 years, which is perhaps a little over-optimistic :) Nobody knows precisely what’ll happen to it, but let’s pick some numbers out of thin air. Let’s assume that it’s 3% for 5 years, then 6% for 5 years, then 9% for 5 years, and then 3% again for 5 years, followed by 6% for 5 years:

£ 1 * 1.03 ^ 10 * 1.06 ^ 10 * 1.09 ^ 5 = £ 3.70

Now this seems high, but bear in mind that it’s an amount of money in 2039. That’s way in the future — at that point we’ll know how Y2038 compared to the Y2K bug… — and a lot of inflation will have occurred. Expressing it in today’s money, assuming an inflation rate of 2.5% per year:

£ 1 * 1.03 ^ 10 * 1.06 ^ 10 * 1.09 ^ 5 / 1.025 ^ 25 = £ 2.00

### Property ladder

Consider two strategies:

Firstly, buying the house that we really want straight away. With this strategy, we accumulate interest on the whole amount from day 1.

Secondly, buying a cheaper house, and living there for 10 years, before upgrading to the house that we originally really wanted. With this strategy, for the first 10 years, we’re only accumulating interest on the lower debt of the cheaper house, then, later on, we take on the debt associated with the more expensive house, and at that point, start to accumulate interest on it too.

The second strategy is just like the multistage rocket in reverse. Just as we only carried the fuel tanks for the minimum amount of time that they were needed, we’ve used the same trick with mortgage debt, and only carried it for the minimum amount of time that it’s needed.

### In the real world…

However, in the real world it’s not as simple as all that. To start with, house prices tend to go up, and we haven’t taken account of that; next up, moving from our starter home to the place we really wanted is not zero cost: there’s estate agent fees, legal fees, stamp duty, etc…

With the first strategy, we pay interest to the bank on the whole amount.

If we assume that all houses prices go up at 3.5% per year (regardless of their current value), then with the second strategy, we still pay interest on the smaller amount to the bank, but for the difference between the amounts, instead of paying interest on that, we instead pay the owner, because of the higher purchase price when we come to buy it.

Again, let’s think of what happens to our marginal £ 1 for the first 10 years where the strategies differ (again, expressed in today’s money):

First strategy: £ 1 * 1.03 ^ 5 * 1.06 ^ 5 / 1.025 ^ 10 = £ 1.21

Second strategy: £ 1 * 1.035 ^ 10 / 1.025 ^ 10 = £ 1.10

So far, the second strategy is winning, by 11p per marginal £ 1.

To make it more concrete, let’s assume that there are 80,000 of these marginal £ 1s, i.e. the difference between the two houses that we buy under the second strategy is £ 80,000. Therefore, so far, the second strategy is winning by £ 8,800.

However, there are several costs with the second strategy when we move house: estate agent fees, legal fees, stamp duty, etc… which might be around £ 10,000. This not only wipes out its lead, but firmly tips the balance in the favour of the first strategy.

### Conclusion

I picked assumptions out of the air to write this blog post. That we lived in our first house for 10 years before moving, an inflation rate of 2.5% per year, the housing market increasing at 3.5% per year, interest rates rising from 3% to 6%, that the difference between our first and second house was £ 80,000, and moving costs of £ 10,000.

If instead, houses prices rise not by 3.5% per year, but by 3% per year, then instead of losing by £ 1,200 the second strategy actually wins by £ 3,000.

Separately, if interest rates end up not at 6% but at 7%, then instead of losing by £ 1,200 the second strategy actually wins by nearly £ 3,500.

Basically, you can prove anything with statistics :) And, as always, to get the answer that you’re looking for, simply tweak the assumptions… Or ask Captain Hindsight!

At the end of the day, it’s not clear which strategy to choose, and in any case, the bank may well force the decision on you, by limiting the amount of money that they’re prepared to lend you, which will make the second strategy the only viable one.