This is an excerpt from the draft of an article I wrote in the late 1990’s which was published in The Journal of Portfolio Management, Jarrod W .  Wilcox, “Investing at the Edge”, Spring 1998, Vol. 24, No. 3: pp. 9–21.  The article  described the advantages of investing in emerging markets because they were at that time rather independent of one another, and the advantages of diversification as a means of accelerating risky growth were not well understood.  In the excerpt, I use a simpler example involving flipping coins to show that diversification is not just a means of reducing anxiety, but can actually lead to a faster growth rate of your funds.

A COIN FLIPPING ECONOMY

From elementary statistics, we know that the expected value of a product of two random, independent variables equals the product of their expected values.  Then the ratio of expected terminal value to initial value of a multi-period, independently distributed, investment process is simply the product of the expected returns, plus one, of each step.  Since diversification among investments of equal expected returns can not change the expected return of the portfolio for single periods, it also can not, therefore, change the mean terminal value after multiple periods.

However, mathematically, expected compound return is not generally equal to the “return” calculated from the expected terminal value. Because of the multiplicative nature of successive returns, the distribution of possible multi-period outcomes will include some very high payoffs with very small probabilities.  Under edge investment conditions, the median terminal value will be far below the mean terminal value.  As we noted in reviewing Hakansson’s capital growth work, over many periods, the median terminal value is determined by mean compound return.

For most real-world investors, but especially professional investors, improving the odds of beating median returns will be an important consideration even if it does not result in improved mean terminal value or in the average return calculated from it.

We will use a coin-flipping example to see how dramatic the difference can be between these two concepts.  Suppose you flip a fair coin once a year.  If the first outcome is heads, your capital, which is initialized at 1, doubles.  If tails, your capital is halved.  The game continues for three years.  The top half of Figure 1 shows the mean terminal value T after one, two and three periods. After the third period there is  a one-eighth chance that wealth is 8, three-eighths chance that wealth is 2, three-eighths chance that wealth is 0.5, and a one-eighth chance that wealth is 0.125.  The mean terminal value is 0.125*8 + 0.375*2 + 0.375*0.5 + 0.125*0.125, or 1.9531.  Taking the cube root, we get 1.25.  Thus, the average return calculated from mean T is 25%.  On the other hand, we have a one-eighth chance that our return per period is 100%, a three-eighth’s chance that our return per period is 26%, a three-eighth’s chance that our return per period is -20.6% and a one-eighth’s chance our return is -50% per period.  The mean compound return is therefore 0.125*100% + 0.375*26% + 0.375*-20.6% + 0.125*-50%, or only 8.3%.

The average return calculated from mean terminal value is a constant 25% no matter how many periods.  However, in the single-coin case the expected compound return begins at 25% and declines to only 8% as we increase our horizon to three periods.

Now consider a second case, the same game with two coins, each of which receives half of your capital each period.  The odds at each step are a 25% chance to double your capital, a 50% chance to increase by a quarter, and a 25% chance of losing half.  The bottom of Figure 1 shows the result.  Again, the return calculated from expected terminal value is a steady 25%.  Again, expected compound return declines with increasing horizons.  But it erodes far more slowly than in the single-coin case, declining only to 16% rather than 8% over three periods.

To get a picture of what happens with more periods and more diversification among coins at each period, we resort to computer simulation.  Figure 2 compares  mean compound return to the average return calculated from the median terminal value for 2000 sequences of 12 coin flips using a single coin.  Note that as the number of periods increases, both returns asymptotically approach zero.  This is not surprising since if there is an equal number of heads and tails in a sequence, the terminal value equals the starting value.  On the other hand, the average return based on the theoretical mean terminal value is a constant 25% no matter how many periods we consider.  It diverges very far from the typical results indicated by the median. Figure 3 shows an analogous result for two coins,  There are 1000 sequences of 12 periods represented, each combining the result of flipping two coins at a time with equal capital invested in each.  In this case, the mean compound return asymptotically approaches about 12%.  We will see how to estimate the asymptote shortly.  Diversification has no effect on the compound return calculated from the mean terminal value.  However, it has dramatically increased both the mean compound return and the return calculated from the median terminal value over longer periods. Consider a universe of portfolio managers all of whom flip one coin.  You have the advantage of being able to join the game with two coins.  Figure 4 shows a new simulation, again with 2000 sequences of coins, comparing your median rank with the median and top quartile of the single-coin competitors.  Your expected rank after about 5 periods will be substantially above median and not far from top quartile.  This happens even though you have exactly the same expected terminal value and exactly the same average return calculated from it as your single-coin competitors.  The more coins you can flip per period, the better will be your long-term expected rank.  For you, diversification return is very real. Let’s investigate further.  Compound return is obtained by raising e to the power given by the mean natural log return and subtracting 1.  The mean of log returns is: where  is the arithmetic return in period t and there are n periods.  Each period’s log returns summed in the expression can be expanded by a Taylor series around the expected value of  r, which, following Markowitz, we label E. (See also Booth and Fama .)  We can express each period’s log return as follows: In order for the series to converge, expected r-E must be less than 1+E.  We can usually meet this requirement by choosing a measurement period sufficiently short.  Then take the expectation of both sides of Equation 3, assuming finite expectations, and you will have a good approximation of Expression (2), the expected compound return.

Taking this expectation, we discover the approximate expected compound return in log terms: where V(r) is the variance of r.   If we substitute in equation(3) the conventional definitions for skewness S(r) and kurtosis K(r), then a convenient form is shown as equation(5). Equation (5) provides immediate understanding of the facts that variance of portfolio arithmetic return produces a drag on portfolio growth, that positive skewness is desirable, and that kurtosis is undesirable.  It explains in part the instinctive preference most investors have for less variance.  It also may explain preferences for downside protection to produce positive skewness, and finally for avoidance of super risk through kurtosis (fat tails) that may unduly threaten the investor’s capital base.

Equation (5) also gives us an intuition for the impact of combining assets in a diversified portfolio.  Diversification is an operator that produces a simple weighted average portfolio E but has a generally reducing effect on portfolio V, S and K.

Consider two investments identical in average arithmetic return E and variance V, with uncorrelated returns. How will the compound return of an equal-weighted portfolio of the two behave?  Ignoring higher moments, their individual expected log returns will be ln(1+E)-V/2(1+E).   However, if we put them together in a portfolio, they will produce an asset with identical E but a V only half as great as before.  Thus the expected rate of compound return will increase by V/4(1+E).  To realize the full potential of this improvement, we would have to continue to rebalance to maintain equal proportions.  However, there will be a substantial benefit for considerable time even without rebalancing until one asset grows to several times the size of the other.

Now we are ready to use Equation (5) to estimate the compound returns to be expected from various coin-flipping scenarios.  Figure 5 shows its right-hand side ingredients for the case of one coin, two coins and 17 coins, based on exact probability.  We also look at a case based on 250 simulations of  17 coins.  Here, the initial capital invested is distributed according to the initial weight of 17 countries in the International Finance Corporation’s Global Composite emerging market index as of December 31, 1984.  In the simulation, the returns of each coin are reinvested in that coin, without rebalancing.

We see first from the figure that the primary impact of increasing diversification is to reduce the drag from variance.   For example, the 17-coin portfolios have an expected log return increment of  .17 higher than the single-coin alternative from this factor.  There is also an improvement in log return of .03 from reduced kurtosis.  The 0.20 improvement in log return yields a 23% higher expected compound return.  (There is no impact through skew because a coin flip has a symmetric probability distribution.)  Second, we see that the simulated 17-coin capitalization-weighted index behaves as though it had substantially less diversification, earning only an 18% mean compound return.   It has almost five times the variance drag (.054 versus .011) of the equal-weighted and rebalanced  example.