## 08 May 2005

### Procrastinating apartment

You should know by now: I am not an economist.

In two weeks I worked 0 minutes for my first open source project. That's because I'm so good at procrastinating. Let me tell you the latest innovation I made in this domain. Instead of watching a movie with my girlfriend, programming, playing table tennis, solving a puzzle, etc. I spent time thinking about how credits for buying an apartment work. Talk about wasted time!

Anyway, in order to keep wasted time to a minimum I'd like not to repeat the experience so I'm recording my findings here. (or maybe I'm just prolonging the procrastination state?)

The computations done here are rough estimates done on an idealized model. The monthly rate you pay to the bank is a constant R. The (monthly) interest is also a constant d-1. Even with these constraints the equations you get convey pretty useful information. To help intuition I'll use figures from an offer of BCR.

First of all the money you get from the bank is A = R (1-1/d^n) / (d-1). Here n is the number of months you pay. This number has an upper limit given by the bank. A typical value is 300. The interest is fixed by the bank: you have no choice here. You can pay the bank each month how much you like, but when the bank computes how much money it can give you it uses an upper bound on R. A typical value is half your net salary. Another constraint that is not captured in the equation above is that the money you pay in advance' B must satisfy B/A > th, where th is a threshold fixed by the bank. A typical value is 0.2.

You can think about it in these terms. You have B EUR available and a salary. Then, by making a credit you can buy an apartment of up to A+B EUR. The value you get from the bank depends linearly on your salary. The value you pay up front must be big enough.

Let's explore a bit more the dependency of the money you can get from a bank on the three factors which appear in the equation above.

The dependency A(R) is linear. That means that if you have a salary twice as large then the bank will give you twice as much. For the BCR EUR credit the relationship is: A = 57 S. In other words the bank will give you at most 60 times your net salary.

The dependency A(n) is... exponential. This means that if you want to increase the ammount you get by a constant then you'll have multiply the period you have to give money to the bank by a constant. The function converges to A=R/(d-1) when n goes to infinity.

The dependency A(d) is also exponential... but the other way around. I.e. a constant decrease of the interest can multiply by a constant the money you can get from the bank. When the interest goes to infinity the amount of money you can get from the bank tends to 0. When the interest is 0 the amount of money you can get is A=Rn`.

In terms of profitability there is of course no question: you have to pay up front as much as you can and pay back the bank as quick as you can. If you need to balance these somehow opposing forces then favour giving more up-front money.