Once in a while I get an invitation from my bank to discuss
my personal financial situation. Most of the time I ignore these invitations because
I know their objective, it’s like this. After reviewing my investments, pension
and mortgage the lovely lady (most of the time it is a lady, don’t know why)
suggests to change my investment portfolio for better performance and puts a
brochure in front of me of a new investment opportunity. Most prominent part of the brochure is a bar
chart with the expected return of the investment compared with a couple of
other investments (from other banks) and of course the new opportunity does
best. But can I expect this expected
return? Kahneman taught me not to trust my intuition in this, so this time I
decide to go home and do some analytical experiments and decide based on the
facts.

Suppose it is suggested to invest in the Dutch AEX index, an
index composed of Dutch companies that are traded on the Amsterdam stock
exchange. In the table below you find the annual returns over the past 30
years.

If I had invested €1,000 at the start of 1993 and reinvested
the returns each year, the investment would have grown to €2,642 at the end of
2012 with an average annual return of 8.38%. The future value of the investment
the bank is suggesting is calculated using the average annual return of the
past and compounding it over a 20 year period. In a formula:

*I**· (1 + r)*, with n = 20, I = the amount invested and r= the expected annual return. Following this logic and assuming that returns from the past are representative for the future a €1,000 investment now would result in €1,000 · (1 + 8.38%)^{n}^{ 20}*= €4,999 in 20 years**.*Wait a minute, how does this compare to the €2,642 we calculated earlier*?*The €4,999 is nearly twice as much. How come that the investments are expected to do better than the past? My experiment will show why and also that the future value of the investment is improbable because it will occur less than half of the time. So what the bank is telling me to expect is not to be expected.
As you can see the expected value indeed is larger (€4,985)
compared the realized value of €2,642 but also that in nearly 70% of the time the
future value will be lower than average. Also note that the spread between the
minimum value and the average is much smaller than between the average and the
maximum value, the distribution of the future values is skewed. Cause of all this
is the effect of compounding, which is key in explaining why the expected is
not to be expected.

The expected value of the investment is derived by
compounding the initial investment with the

**arithmetic**average of the returns. In our case 8.38% so theoretically a future value of €4,999 results. However, the median value of the investment, the value for which there is a 50% chance that we will either fail or exceed, is derived by compounding the €1,000 using the**geometric**average return (or constant rate of return). The geometric average return for 1993-2012 equals 4.98% therefore the theoretical median value equals €2,642.