Longer time periods make interest compound, and growth exponential. However, at least in in the case of stock market investments, they also bring increased uncertaintainty.

The graph at the bottom of this page represents the nett effect of this growth and uncertainty. It combines the continualy-steepening upwards curve representing the exponential growth of the most probable portfolio value, with the bell-curve representing the probability-density of a normal distribution. Its component parts are illustrated in the paragraphs below…

Historical
evidence shows that the world’s stock markets, have generated a positive
exponential growth over the last 150 years. Taking an estimate of this growth
rate to be 7% per annum (*in inflation-adjusted or real-terms, and assuming
dividends re-invested*), and ignoring for a moment volatility, the value of an
average £10,000 portfolio would change as shown below…

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Unfortunately, history also shows that there is a significant uncertainty in stock market values. Approximating this with a random walk with normal distribution (I.e unskewed), gives a central or most probable value, with less probable values lying on each side.

Just to show the general shape, the height of the “bell curve” on the right, is proportional to the probability of values more negative or positive than its central zero value, based on a standard deviation of 1.

The volatility of the FTSE 100, however, is centred on its current value (of 5,736 on 2012-11-13) and, averaged over the past year, has had a standard deviation of 15.3% of its value. (http://www.ftse.co.uk/Indices/UK_Indices/Downloads/UKX.pdf )

As time progresses, the probability distribution widens. In the chart below, the bell curve, has therefore been re-drawn for each year after year 0. Each curve has then been turned on its side and aligned with the portfolio value scale on the left. For clarity, each curve has also been mirror imaged, the mirrors placed together, and the area “under” the curves filled to form a solid bar. What would have been a smooth curve has then been approximated to by drawing a succession of 10 rectangles or rectangular shoulders inside it. The width of each rectangle or shoulder is still proportional to (or slightly less than) the probability density for that value of portfolio. However, this has allowed each of the 10 keys on the right to be marked with the probability of acheving a portfolio value which can be read off as the height of the corresponding shoulder.

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As can be seen, volatility increases rapidly with time in the early years, but increases less quickly after about year 10. This is most easily explained by the fact that the year 1 portfolio value includes the full annual volatility, whereas for subsequent years, there is an increasing opportunity for changes to be reversed and their net effects therefore cancelled.

This is the opposite of the growth of the most probable value, which increases slowly at first, and then speeds up in later years. As a result, as can be seen from the red portions of the bars there is a 1%-or-over chance of net loss, for years 1 - to year 11. It is this possibility of loss that is suggested justifies the high return from equities which is graphed here…

The percentage growth is calculated from 1 - (1 + 7%)^Years

The percentage volatility is calculated from (15.3%)*(Years)^0.5

FTSE 100 is a price index, whereas the graph above is meant to represent “Total Returns” including dividends. The FTSE therefore understates the performance of the average portfolio, whereas the graph above should not.

The 5 year period from 2007 to 2012, with zero FTSE 100 total return, had an 11.5% chance of occurring. Presumably the next 5 years has a similar possibility of no growth.

The effect of even 0.5% annual portfolio costs is both cumulative and compound, and over long periods can result in large, even the majority, of the value indicated in the graph above being transferred from the investor to the fund manager.

Many FTSE100 shares have a volatility twice that of the index, which knocks the graph above into a cocked hat. This shows that at least under the shorter time scales, it is vital to diversify risk across perhaps a dozen shares and sectors, and perhaps choose those shares which will give the portfolio the minimum volatility (but that justifies another web page).

The simple
normal distribution graphed here ignores the excess of down side volatility (E.g.
*events such as Black
Monday*). The graph may not therefore apply to portfolios subject to
margin-calls or which for any other reason have to be sold on drastic market
falls.

At least one
commentator has said that historical estimates of world stock market returns have
been dominated by the US stock market. Since this has survived whereas some others
have not, the calculated 7% return takes little account of such possibilities. The
graph above may therefore only apply in counties, and for periods of time,
where there are no such events (*However this may not have much affect on the
relative desirability of stock market investments, since other investments
could also be expected to perish in such circumstances*).

At least one
commentator has analysed the components which made up the last period of high stock
market returns (E.g. 1974 – 1999), and concluded that the major component (*increasing
P/E levels*) should not be expected to be repeated. The 7% central returns
expectation may therefore only apply to investments made when P/E levels are
reasonable (Nov 2012 FTSE 100 P/E = 11.05% http://ftse.co.uk/objects/csv_to_table.jsp?infoCode=NGUK&theseFilters=&csvAll=&theseColumns=MCwxLDIsMyw0LDU=&tableTitle=FTSE%20UK%20Index%20Series%20Values
).

Click to download the Excel 2007 spreadsheet which generated these graphs

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