The incredible story of the expected that is not to be expected

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)ⁿ, 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%) 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.

A small bootstrap experiment will show the improbability of the expected value. In the bootstrap I created 10.000 sequences of 20 annual returns. Each sequence is created by randomly selecting a return from the 1993-2012 period which is placed back in the sample again, repeated 20 times. The summary results of the bootstrap are summarize in the table and graph

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.

As you can see from the graph, the compounding of returns causes the distribution of the future values to become skewed.  There are fewer values in excess of the average future value, but they exceed it with a greater amount than the amount by which the below average values fall short of the average value.  So next time you are offered an investment opportunity or new pension plan be sure to realize that the expected value is an inflated estimate of the future value, the median future value is more realistic as you can expect to achieve or exceed it half the time. The expected is to be expected in that case. Besides the now standard disclaimer that past performance is not necessary indicative for future results, banks should also remark the results shown are an overestimation of the expected results.