Investment?

Hi there. If you’re like me, you feel like you want to kick yourself in the butt for not having invested into your own future. Life, when you’re young, just feels too good to think about any of that. And do you really have any extra cash that you could save? $50. Would it even matter?

It’s never too late. So why not start today?

The big idea is of course compounding. There are many great shortcomings of the human race, but according to physicist Dr. Albert Bartlett:

The greatest shortcoming of the human race is our inability to understand the exponential function.

Investment Scenario Calculator

Let’s see how far you can get with saving a little extra every year (10%).

Here are a few sliders to play around with. There are dollar amounts (I live in New Zealand, so don’t read too much into the numbers. 1 NZD today is about USD 0.58, or EUR 0.50) and there are rates. And there are years (otherwise there wouldn’t be much compounding, would there?)

You can start with zero capital, or with no savings and let only your invested capital compound.

If you know by how much your annual income would grow, you can do change that rate, too.

I’ve also included Uncertainty. It increases or decreases the rates by that much, so we can check out worst and best case scenarios. (The numbers in brackets show the values for worst- and best-case).

There is a line called Retirement Income. This is an inflation-adjusted annual lump sum you would receive after tax, if you were to move all your wealth into a term deposit and just live off the interest rates.

The last two lines are sensitivities. I talk about them down below.

OK. I’ll leave you to it. Happy investing!!

Worst Middle Best

Technical Details

This is the basic formula for everything above.

\[K_{nom} = K_0 (1+r)^N + s \cdot Y_1 \cdot \frac{(1+r)^N - (1+g)^N}{r - g}\]

The nominal captial is quite large. That’s because we haven’t adjusted it for inflation. Dividing the nominal capital by the total inflation over N years, gives us the real capital, in terms of today’s purchasing power.

\[K_{real} = K_{nom} / (1+\pi)^N\]

I was also curious about how tiny adjustments in behaviour might lead to bigger savings. There are two things we can control: savings rate and extra years.

Mathematically, the sensitivities can be calculated using derivates. Taking the derivative of \(K_{nom}\) (or \(K_{real}\)) with respect to \(s\) gives us the sensitivity to changing the savings rate.

\[\frac{dK_{nom}}{ds} = Y_1 \cdot \frac{(1+r)^N - (1+g)^N}{r - g}\]

You’ve probably seen that some sliders don’t make a difference in the +1% Savings Impact. Now, you can look for yourself why that is. (Yes. LOOK AT THE FORMULA!!!)

Astonishingly, the savings rate itself doesn’t make a difference. Why? Because every additional 1 percentage point of savings buys you the same chunk of lifetime wealth, regardless of whether you’re already saving 2% or 40%.

In real life, you also don’t have much control about \(Y_1\), \(r\), \(g\); all variables that determine the sensitivity. But you have control of \(N\), the time you let your wealth accumulate.

Let’s look at the sensitivity of saving for another year.

\[\frac{dK_{nom}}{dN} = K_0 (1+r)^N \ln(1+r) + s \cdot Y_1 \cdot \frac{ (1+r)^N \ln(1+r) - (1+g)^N \ln(1+g) }{r - g}\]

I know, there is a lot going on here. But we can break it up and look into the two terms of the above sum.

The first term is related to your starting capital compounding over the years (\(K_0 (1+r)^N\)) and it will simply grow some more for another year. If you dial the slider of your year-1 income down to 0, you can see that the +1 Year Impact is the Real Capital times the Investment Return Rate. An extra year is valuable because it applies your return rate to your entire accumulated financial history.

The second term is related to your savings. It’s a bit too complicated to explain it in simple terms (and there is a minus in the numerator of the fraction). But you can think of it as growth on past savings (\((1+r)^N \ln(1+r)\)) that is adjusted for income growth (\(-(1+g)^N \ln(1+g)\)).

To sum up all of the details about the sensitivities, just keep in mind

Savings rate controls how much fuel you add, but time controls how long it burns.