Session 5. Understanding Compound Returns

Most of us under-rate the importance of time in producing differences in wealth.  The first object of this session is to clarify the way time interacts with differences in rates of return to produce differences in wealth that grow exponentially.

Compare two bank accounts.  One pays 3% annual interest, and the other 4%.  If you put a dollar in each, what will happen over time?  The exhibit below shows the result.  It assumes continuous compounding, but would look very similar if interest were paid at monthly increments.  We see two exponential growth curves.  At first, there is not much difference.  But after fifty years, the wealth attained at a 4% annual return is about 65% greater than that attained at 3%.

The second chart shows the same bank account balances plotted on a logarithmic vertical scale. That is, equal intervals on the vertical scale represent equal ratios.  It shows that a constant rate of return (exponential growth) gives a straight line in wealth when plotted on a logarithmic scale.  Thus, time is multiplied by the growth rate to give the log of wealth.  This is just another way of representing exponential growth.  This chart also shows the ratio of the contents of the 4% bank account to the 3% account.  It is clear that this ratio also grows exponentially.

The takeaway here is that even very small differences in return can become quite important if they persist over long periods of time. Just as it is important to start saving early, it is important to try for higher returns early.  And the improvement in return need not be large to have quite significant results over the long-term.  This is why it is very important for the investor to pay attention to what seem to be small differences in fees, trading costs, and not-so-small differences in effective tax rates.

Sadly, once you make sure that you have done a good job in keeping fees, trading costs, and effective tax rates down, the main way to increase return is to bear more risk.  However...Too much risk will actually reduce the typical result of compounding returns over long periods, even as it increases the expected return for a single period.

Mathematically trained statisticians will at first object to this statement, because it is true that the expected, or average, result of a compound return is the same as the compounding of the expected result for a single period.  Once you know expected return for each period, risk has no further role to play in determining expected wealth.  The key insight, however, is that the average compound result in a risky process will be distorted upward from what most investors receive.  A much better indicator of what you as an investor are likely to attain is given by the median compounded wealth.  And this quantity is very much influenced by volatility of returns.

Consider an example where returns are volatile but known, so that we can do an easy back of the envelope calculation.  You start with a dollar.  The first period you receive a 100% return, so that your wealth becomes two dollars.  The second period, you receive a 50% loss, and your wealth shrinks back to a single dollar.   This sequence can be repeated indefinitely, and your money will not grow.  What is going on?  The losses reduce the capital base from which the gains must grow!

Now make that initial dollar a silver dollar, a coin to be flipped.  If it comes up heads, your stake doubles.  If it comes up tails, your stake is cut in half.  Your average or expected return in a single period is 25%.  What happens if you compound this process by flipping the coin 10 times?  The median result will be an equal number of heads and tails, landing you right back where you started.  That is, your median result will be 0%, even though for a single period the expected return is 25%!  There is also a quite large probability that you will lose money.  The expected or average wealth will be far higher than the median, because it includes the approximately one-in-a-thousand chance of getting ten heads in a row, but this may not be very relevant.

The fact that volatility reduces median compound results is one of the reasons why other investors will pay you to bear risk.  The other is that they cannot cope with large losses, or shortfalls, along the way.  Estimating the size of these effects in a given situation requires some more mathematics.  For now, it may be enough to remember that you should be skeptical when people say that in order to earn higher returns you must take more risk.  The truth of that statement is contingent on whether you are already facing so much risk that more will actually cause your median long-term prospects to deteriorate.

The key concept for this session is that your results will depend on compounding, which is determined by time, expected return each period, and the volatility of that return.  Patience combined with modest improvements in returns of moderate risk is greatly rewarded.  Large returns combined with high volatility can seriously reduce your median results.

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