## The ultimate equation

It is often said that Albert Einstein described compound interest as the eighth wonder of the world. I doubt he said it but whoever attributed it to him has a point.

What if I told if you save half of what you earn, you can retire in 15 years. That's right, after 15 years work you're done and you never have to work again.

Does this require some crazy investment strategy? No.

Does it require any special skill? Nothing other than being disciplined with making the contributions.

All you have to do is save.

### How does it work?

Suppose we could earn 5% on our savings per year. This interest is paid at the end of the year.

Getting this interest rate is hard to do in a cash ISA right now, but can be obtained relatively easily by buying corporate bonds. More on that later.

We then define x as being your annual salary. If you're saving half your salary, you will be saving 0.5 of this x per year.

Calculating how much money you would have at the end of 15 years is pretty straightforward.

If we put in 0.5x of your salary on the first year and left it for 15 years, we'd end up with a final balance of:

1.0515(0.5x)

Here the 1.05 figure represents the interest rate of 5%. We raise it to the 15th power because we're going to compound that interest for 15 years.

The next year, we add another 0.5x. However, we now have 14 years left until 15 years are up. So the total at the end of 15 years is:

1.0515(0.5x) + 1.0514(0.5x)

That is the capital and interest earned on the 1st payment when held for 15 years added to the capital and interest earned on the second payment, held for 14 years.

If you're struggling to see why this is true, imagine that the money is held in two separate bank accounts. If the money is left in one for 15 years and the other for 14, then the first would get 15 years of interest and the second 14 years.

Clearly, the total earned is the sum of the two balances. Well, given that both accounts attract the same interest rate it must be the case that even if both payments were held in the same bank account the amount earned would be the same.

We can use the same argument to think about all fifteen deposits:

1.0515(0.5x) + 1.0514(0.5x) + 1.0513(0.5x) + 1.0512(0.5x) + 1.0511(0.5x) + 1.0510(0.5x) + 1.059(0.5x) + 1.058(0.5x) + 1.057(0.5x) + 1.056(0.5x) + 1.055(0.5x) + 1.054(0.5x) + 1.053(0.5x) + 1.052(0.5x) + 1.05(0.5x)

We can factor out the 0.5x term to give the following sum:

0.5x(1.0515 + 1.0514 + 1.0513 + 1.0512 + 1.0511 + 1.0510 + 1.059+ 1.058 + 1.057 + 1.056 + 1.055 + 1.054 + 1.053 + 1.052 + 1.05)

If you know your maths, you'll realise that the sum in the brackets looks like the sum of a geometric series:

a(1-rn/1-r)

In our case, a is equal to 1 and r is equal to 1.05, which is our interest rate.

Substituting the numbers we get:

0.5x(1-1.0515/1-1.05) = 0.5x(21.578) = 10.7892x

So we have 10.7 years' salary saved. This is interesting in itself, even if we earned no interest we can live on this sum for 20 years. This is because we already assumed you were saving half your salary; it must necessarily be the case that you only need half of it.

How long will it last if we invest it? Suppose we're able to invest with a return of 5% per annum. How much do we make?

Interest earned: 1.05(10.7892x) - 10.7892x = 11.34x - 10.8x = 0.5454x

So you're now in a position where you're earning enough interest to cover half of your annual salary. As I mentioned earlier, this must cover your expenses since you're saving half your salary. At this point, you can retire.

### How do I get 5% on my savings?

At the moment, if you put your money in a savings account at a bank you will make very little interest. However, if you invest your money you can make a 5% return relatively easily.

It worth noting at this point that I am not a financial adviser and that if you decide to invest in stocks and shares, you do so at your own risk. Frankly, you'd be a bit mad to take investment advice of a random guy on the Internet anyway. And like all investments, past performance is no indication of future returns.

That said, you don't really need to do anything terribly sophisticated. A garden variety tracker fund will probably do it.

For example, this Vanguard Life Strategy fund is averaging a 7% return while being pretty conservatively invested. You will probably need the additional 2% growth to deal with charges and inflation - so overall expecting a real return of 5% from this fund is probably reasonable.

### Conclusions

It is possible to retire very early if you're willing to sacrifice half your salary to do it. This is the discipline. There are many companies / people trying to convince you to part with your money. If you can make the sacrifice, if you can show the discipline then you can literally get your life back.

We're only here once! Why spend most of the useful part of your life in work?

1. 2016-08-07 21:41:00 GMT
2. #Finance
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