A recent much-quoted article in the Financial

Times under the heading ‘Longer time hori-

zon â€does not reduce riskâ€’ provides the

ideal platform from which to explain the difference between the actuarial and the unthinking statistical approaches. In that article Zvi Bodie, professor of finance and economics at Boston University, is quoted as saying ‘There is a very common, almost universal, belief, for example, that the longer your time horizon, the less risky to hold stocks But professors of finance will tell you this is a fallacy’.

Both these dogmatic statements should make the actuary a little suspicious. Those who are versed in finance will know that there are a number of textbooks and papers, some of which have been written by eminent professors of finance, which say no such thing. But then we know that, long after Rutherford, there were physicists still teaching that the atom was indivisible. So, perhaps, the authors of these textbooks and papers have not kept pace with recent research. Or is it possible that they were not impressed with the statistics which lay behind that research?

Equity and bond return comparisons

In order to support his thesis, Bodie explains that the likelihood of stocks delivering a better return than risk-free government bonds is about 60% over one year but rises to 95% over 25 years. How does he arrive at these statistics? He suggests that we examine the performance over a number of years, write the stockmarket returns for every one of those years on a piece of paper, and then put them in a hat. Then we should pick 25 out at random, but one at a time and being careful to return the one which we have drawn to the hat before picking again. To quote the professor again:

‘The likely result is that in 95% of cases your 25-year return will be higher than the return on bonds. But there will be 5% where the results will be bad in some cases spectacularly bad.’

Some doubters may wish to repeat Bodie’s experiment, but I am sure that they would find his figures accurate. Actuaries might question why it was necessary to use a boot-strapping process when so much crude data already existed. Why not use actual 25-year returns?

Actuarial techniques demand that construction of the model which best fits the data should commence with an analysis of all the available data, and only if the data were insufficient would it be permissible to use artificial statistical techniques in order to fabricate additional data. Let us, therefore, examine the hypothesis using real rather than fabricated data.

The CSFB 2001 Equity Gilt Study provides tables showing both nominal and real UK equity and gilt returns for the years 1869 to 2001. From these observations covering 130 years, 107 separate 25-year periods can be obtained. This constitutes an actual sample sufficiently large to test the accuracy of Professor Bodie’s assertion. If the assertion is correct, we should expect to find that, in at least five of those 25-year periods, the return on gilts had exceeded the return on equities.

In no single one of those 107 25-year periods did this happen. Expected five, actual zero not conclusive but certainly lending support to the rejection of Bodie’s assertion. Work carried out by Professor Clarkson on US, Japanese, and French data, as yet unpublished, shows similar results.

Supposed inflation hedging

But the supposed failure of equity returns to outperform bond returns over the long term is but one prong of Professor Bodie’s attack on the cult of the equity. The second prong of his attack states that the only assets that can be relied upon as a hedge against inflation over the long term are index-linked securities. Further examination of the CSFB 2001 Equity Gilt Study data lends a modicum of support to this view. Twice out of the 107 25-year periods the return on equities failed to match inflation. Those two periods ended in 1920 and 1921. However, extending the first period by two years to 1922 and the second by one year to 1922, equities once again are seen to have outperformed inflation. In the context of a pension fund it must be asked whether 25-year returns are really more significant than 26- or 27-year returns.

The paucity of data relating to actual returns from index-linked bonds makes it difficult to state, with any accuracy, what the expected real annual rate of return on a 25-year index-linked bond might be. The current market expectation of the real rate of return on 25-year index-linked gilts is 2.3%. In eight of the 107 25-year periods, those ending between 1916 and 1923 inclusive, the real rate of return on equities fell below 2.3%. From which it may be surmised that index-linked bonds have an 8% chance of providing a better 25-year return than equities. Against that, however, the mean real rate of return on equities over all the 107 25-year periods was 5.9%, or more than double the expected real rate of return on index-linked gilts.

The reader must judge whether the evidence supports Professor Bodie’s second assertion that ‘Inflation-linked bonds are the only assets which can be relied on as a hedge against inflation’.

Boot-strapping at work

Since the evidence which Professor Bodie adduced in support of first assertion is clearly erroneous, one may ask where he went wrong. Rather than use real 25-year data, he constructed his own data by boot-strapping a series of one-year returns. However, the logic underpinning boot-strapping demands that the underlying data are mutually independent. There is, however, a weight of evidence to show that equity and bond returns for any single year are not independent of the equity and bond returns for the immediate prior or succeeding years. In this context boot-strapping is not a valid process.

Let me end this article by posing a problem which is more amenable to boot-strapping. Consider the movement of a rod, fixed at one end, which has been observed, over a short period of time during which there has been a total of 19 successive 60? movements. Ten of these movements have been to the right and nine to the left. We want to know how frequently the rod could be expected to make a complete rotation about its axis in either direction. Of course, as any statistician will confirm, a mere 19 successive movements is too small a sample for a conclusion. As the statistician who has been invited to assist, the reader may wish to boot-strap in order to replicate sufficient data to provide a realistic estimate. What is the probable frequency?

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