Investing at the Edge

INVESTING AT THE EDGE

Jarrod W. Wilcox

DRAFT

September 4, 1997

INTRODUCTION

We will refer to the center as the universe of investments in which capitalization-weighted index funds are most successful, for example the S&P 500 Index of large US stocks. In the center, passive investors should invest in a market capitalization index. Here, we can reasonably judge active investors by their deviation from this standard. On the other hand, the edge consists of those assets in which investing in capitalization-weighted indices is not a particularly good strategy. Emerging market stocks, venture capital, commodity futures and certain derivative assets appear to fall into this universe. Here the passive investor should not capitalization-weight index. Here the active investor should not necessarily keep residual risk small versus an index. We will demonstrate the principles behind this assertion by examples.

The body of this paper has three parts:

1. Brief review of theory;

2. Hypothetical coin-flipping economy;

3. Case study of emerging markets.

We close with suggestions for practical edge investing and for further research.

I. BRIEF REVIEW OF THEORY

Normative Portfolio Choice

Harry Markowitz [1952, 1959] proposed a quadratic approximation to utility maximization for portfolio selection. One should invest in a portfolio that maximizes expected portfolio return less a linear function of its variance. A Markowitz efficient portfolio gives maximum expected return for a given level of variance.

This simple criterion gives an excellent approximation to maximizing single-period utility for a wide range of plausible utility functions and return probability assumptions. It is clear from his writings that Markowitz also visualized his model as an elegant single-period approximation to a multi-period problem of capital growth for typical investing situations. An entire chapter in his 1959 book deals with estimating long-run return. He first expressed return in log form, that is, the natural logarithm of one plus the fractional return. He then demonstrated that expected compound return, based on expected log return, is a declining function of variance in single-period arithmetic return. The desire to increase average compound return provides one underlying reason for investor aversion to arithmetic return variance, and a justification for limiting arithmetic variance even for the investor seeking maximum return.

Expected portfolio arithmetic (percentage) return is simply the weighted expected percentage return of individual securities. One calculates the percentage portfolio risk by pre- and post-multiplying the percentage return covariance matrix with these weights. Such simple connections between security and portfolio characteristics are not available if one seeks to control portfolio log return. Though the latter problem might have been of greater concern to the investor, it is mathematically more difficult. The arithmetic version comprises a quadratic programming problem that one could solve efficiently in the 1950’s before the advent of today’s cheap computing power.

Descriptive Market Equilibrium

Sharpe [1964] outlined a single-period model of market equilibrium for capital asset prices (CAPM). It incorporated two ideas on the rise. The first was the Markowitz mean-variance optimizing investor. The second was the efficient market, efficient in the sense that it embodies new information in prices so swiftly that they appeared to follow random walks. He also assumed, rather heroically, that individuals were identical in expectations, investment horizon and access to available securities. He assumed, as well, that each could borrow and lend at will at the same interest rate, the risk-free rate. Since his model was essentially one of static equilibrium, he omitted all the dynamic details of taxes, transaction costs, changing preferences and changing availability of security returns.

What he found was that under these conditions arbitrage among risky securities and a risk-free asset would produce a striking and intuitive result. First, the market of risky securities as a capitalization-weighted whole would be efficient in the Markowitz sense of best expected mean return for a given expected variance. Second, the expected return for an individual security would be the risk-free rate plus the product of the expected regression slope (beta) of the stock’s return versus the expected excess return for the market. This provided a “capital market line” linking risk and return.

Support for these two empirical predictions within equity markets has proved elusive. Nevertheless, two practices emerged as a direct result of the CAPM that have had great benefits for investing at the center. The first is the operation of highly satisfactory capitalization-weighted index funds (particularly those based on the S&P 500). The second is rigorous evaluation of professional investors relative to these index funds. Sharpe [1981] later argued that portfolio managers attempting to outperform index funds could be given an objective function in which it is costly to incur variance of residual return (portfolio return less index return) rather than the actual total risk achieved. Experience has shown this heuristic is often good advice for investing at the center.

However, as noted by Roll [1992] and Wilcox [1994], economizing residual risk discourages the search for portfolios lower in total risk and higher in return than the index. This will be dysfunctional when the active investor has valid information. The same is true when the index is not Markowitz efficient even for a passive investor, as we will argue is the case for edge markets.

In the remainder of the paper, we will focus on two ideas that better adapt these theories to edge investing: segmented markets and capital growth theory.

Segmented Markets

Segmentation is a broad idea that takes many guises. Merton [1987] proposed a single-period model, similar to the CAPM, but simplified so that all investors have the same risk aversion. Underlying his securities he posits firms with cash-flow payoffs based on a common factor plus specific risk There is a riskless security and a market-like security with a payoff based on the common factor. However, either lack of information or other reasons not measured as risk prevents some investors from investing in some firms.

In this model, the market portfolio is not mean-variance efficient, even for those investors with no restrictions. Also, expected returns differ from the “capital market line” of the CAPM. They include a new factor proportional to the product of the firm’s specific cash flow risk and the ratio of each firm’s size to the wealth of its available investors. Merton’s insight is that segmentation reduces the diversifiability of specific risk. The latter must therefore be rewarded with higher expected return in proportion to the relative scarcity of investor capital that has access to it. We might label this return increment “segmentation return.”

Capital Growth Theory

Hakansson [1971] published an important and controversial paper arguing the benefits of maximizing expected compound return in multi-period investment. In doing so, he went well beyond where Markowitz left this topic. Let T be the ratio of final terminal value to initial value and n be the number of periods in a series of independent returns. The following inequality is obvious whenever returns are uncertain:

(1)

The expression on the left corresponds to one plus the expected compound return. It depends not only positively on single period mean percentage return but also negatively on its variance and higher statistical moments. The expression on the right is one plus the average return that can be determined using only single-period expected returns and the resulting expected terminal value.

Hakansson showed the inadequacy in some multi-period contexts of a sequence of Markowitz mean-variance choices. (Consider what will eventually happen to the venture capitalist who invests in only one venture at a time if there is a small probability of losing one’s entire investment each period. Ultimately, an unlucky hit will ruin the investor, despite modest variance in percentage return for a single period.) He showed that maximizing expected compound return produced intuitively more satisfying results and he argued that it induced a plausible logarithmic utility curve for final terminal wealth. He also noted that if log returns satisfy the requirements of the Central Limit Theorem, their average will approach a normal distribution as the sequence grows longer. Thus, in the long run, the mean compound return (expressed in log form) will tend to be normally distributed.

The mean of a normal distribution is also an estimate of its median. Order statistics such as the median are not affected by monotonic change of variable as we take the antilog of the expression to produce terminal wealth. Therefore, improving expected compound return improves expected median terminal wealth. Diversification of identical but independent risks cannot increase mean terminal wealth, but as it increases mean compound return it will increase median terminal wealth. [1]

Merton and Samuelson [1972] argued that Hakansson had erred by going further to assert that maximizing expected compound return also maximizes a wide range of utility functions for long-run terminal wealth. Their rebuttal, together with others’ conclusions that the mean-variance approximation was adequate for typical portfolios of common stocks, appears to have discouraged the widespread use of capital growth theory for investing. This was unfortunate. Whatever the merits of Hakansson’s arguments concerning utility, his observation of the power of diversification to improve expected median wealth is powerful. Also, quoting Merton [1973]:

“The ‘growth optimum’ model of Hakansson can be formulated as an equilibrium model although it is consistent with expected utility maximization only if all investors have logarithmic utility functions. However, Roll has shown that the model fits the data about as well as the capital asset pricing model.”

This approach to portfolio selection is still alive, though not widely known. Michaud [1981] reviewed it favorably with the intent of bridging the gap to the CAPM. More recently, MacLean, Ziemba and Blazenko [1992] drew on the approach to analyze strategies both for investing in the turn of the year effect and for various gambling games. Booth and Fama [1992] reintroduced to investment practitioners the idea of increasing return through diversification. They relied on the same Taylor-series expansion used by Markowitz to illustrate how the variance of the portfolio arithmetic return reduces expected compound return. Understanding this relationship, to which we now turn, is key to understanding strategies for above-median performance in edge markets.

II. HYPOTHETICAL COIN-FLIPPING ECONOMY

We employ a coin-flipping example to illustrate without extraneous detail the impact of capital growth theory in edge market investing. Then we will combine this idea with segmentation to study the real-world case example of emerging markets.

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, as we saw in Inequality (1), 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. We use it as an example of Inequality (1). 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:

(2)

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 [1992].) We can express each period’s log return as follows:

ln(1+r) ln(1+E) + - + - … (3)

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:

Expected ln(1+E) -

+ Expected- Expected (4)

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). [2]

Expected ln(1+E) - + - (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. [3] 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.

III. CASE STUDY OF EMERGING MARKETS

Emerging markets are typical edge investments. Individual emerging markets are volatile and little correlated. The collection of countries included in the emerging market asset class appear segmented, both with each other and with the central world markets. [4]

The International Finance Corporation Composite provides a history of emerging market total returns based on capitalization-weighting. [5] Figure 6 illustrates that its risk-return pattern has been greatly inferior to that of an index equal-weighted by country. After investigating the significance of this observation, we will use segmentation analysis and capital growth theory to understand this phenomenon.

Source: International Finance Corporation .

The figure shows wealth indices of two emerging market portfolios based on monthly cumulative total return to a dollar-based investor. The lower dotted line represents the capitalization-weighted IFC Composite. The upper solid line represents an equal-weighted index constructed as follows. Each December 31, the countries then included in the Composite are equal-weighted. Their intervening weights are determined by their total return as represented by IFC’s monthly individual country indices until the end of the next year. We treat each country as a single security, with the return reported by the IFC for that individual country’s index. No allowance is made for varying difficulty for foreigners investing in each of the countries included and no allowance is made for taxes or trading costs. However, since the average turnover is only 17% a year, even including new country additions to the index, the trading cost that could be netted out is modest – not much more than 1% per year.

The period shown is the entire live history of the IFC Composite since inception of the index, beginning at the end of 1984, and running through 1996. The vertical scale in the figure is logarithmic, to give a true picture of volatility and rate of return over time. The compound rate of return for the market-capitalization-weighted index was a mere14.8%, while that for the equal-weighted index was an astonishing 33.5%. The realized risk was actually higher for the Composite. Standard deviation of returns was 29.4% for the IFC Composite as compared with 27.4% for the equal-weighted index.

One or more successive observations on one side of the series median define a “run.” For both equal weighting and capitalization weighted returns, there were 8 runs. This is nearly mid-way between the possible extremes of 2 and 12. The runs test establishes sufficient independence of each year’s observations. There were only 2 out of 12 years when the capitalization-weighted index had the greater return. The binomial probability of such few “wins” happening if the true odds favored the Composite market index each year is only about 1.9%. A paired t-test on the difference in mean percentage return being positive in favor of the Composite also rejects this hypothesis at a significance level of 1.9%. On the other hand, the difference in risk is not statistically significant. However, one can note that the inferiority of the Composite in this respect is not based just on an outlier, but also on a slightly greater interquartile range.

The market price of risk may show the degree of inferiority of the combination of risk and return offered by the market index Composite as compared to the equal-weighted strategy. Suppose investors require 6% of return for every 20% of their annual standard deviation. From our paired t-test on the difference in means there appears to have been a 91% chance that the underlying return sacrifice to be expected from investing in the Composite was greater than 6%. But even as little as 6% would require that the Composite’s annual expected standard deviation be 20% lower. This is quite implausible given our observation of a standard deviation 2% higher than for equal weighting. Consequently, even a very risk averse global investor would have rationally preferred to mix a risk-free investment such as Treasury bills with an equal-weighted emerging markets strategy rather than use the capitalization-weighted Composite index.

Segmentation in Emerging Markets

It is commonplace today that institutional portfolios hold far fewer emerging market securities than optimal asset allocation studies would indicate, given the relatively low correlations of emerging market securities with developed world equities. Data available to global investors for stock investment in emerging markets, for example, I/B/E/S earnings estimates, is of more recent vintage, and is sparser, than for developed markets. Trading costs are substantially higher in emerging markets. An examination of the country returns in the Appendix shows very low correlation of individual country emerging market returns.

Because the country list is growing through time, we will focus on the returns of the 17 countries present since inception at the end of 1984. A principal component analysis of their return series indicates that the eigenvalue of only one principal component exceeds unity, and it accounts for only 2.28/17.00, or 13%, of the total equal-weighted variance of the sample. Correlations with developed markets, not shown here, are also low. Emerging market risk is overwhelmingly specific rather than global.

Merton’s paper suggests that we look for differences in investor access as evidence for segmentation that would convey additional returns for specific risk and render the market index inefficient. Figure 7 compares the ratio of GDP to market capitalization for emerging markets versus developed countries in 1984. (Taiwan GDP is not published by the IMF, our source for GDP.) These are rough numbers, not only because IFC indices and MSCI indices were used for total market capitalization, but because several other important factors besides investor access affect the ratios. For example, the GDP/CAP ratios are high in many emerging markets because of state-owned industries. Nevertheless, it is obvious that the emerging markets have the earmarks of not being exposed to much global ownership..

Figure 8 shows the correlations with average arithmetic return and average compound return over the 12 year period. Although we have only 16 observations, the results are intriguing.

First, the relationships between arithmetic mean and either arithmetic variance or arithmetic standard deviation are overwhelmingly positive, with correlations of 0.69 and 0.79 respectively. The stronger role of realized standard deviation versus realized variance may well stem from its superiority as a more robust indicator of prior expectations. It clearly does not reflect “beta” since almost all the risk is specific. Second, the ratio of GDP to market capitalization is also positively correlated with return. Third, however, the highest correlate with compound return is with the product of GDP/CAP and the realized standard deviation of return. Given the lack of prior return data from which to construct a model of expected risk, I take this evidence of segmentation return to be at least modest confirmation of Merton’s model. It is an important ingredient in understanding why the IFC Composite has been inferior to an equal-weighted index.

Capital Growth in Emerging Markets

If the market is segmented, and volatilities are high, the opportunity for the global investor to take advantage of specific risk over multiple periods through improving expected compound return is high. It extends well beyond the improvement in single-period expected percentage return envisaged in Merton’s model. In Figure 9, we use Equation 5 to build up a picture of the components of expected compound return for a prototypical single emerging market. We use as our robustly estimated prototype the median arithmetic mean, standard deviation, skew and kurtosis from the sample of annual returns for each of the 17 countries. Note the role of less developed high GDP/CAP, high variance countries such as Argentina and the Philippines in building up the typical high mean, high variance statistics of the prototype.

The collection of countries has a high median mean arithmetic return of 34%. The opportunity to achieve nearly this result by eliminating variance drag of about 9% is enhanced by making full use of diversification.

Based on our earlier principal components analysis, we ought to be able to achieve a portfolio variance roughly estimated (assuming average variances) at only 2/17 (systematic) plus 1/17 (diversified specific) of its median component .09. This would eliminate all but a bit under .02 in log terms. Diversification also could eliminate the skew and kurtosis effects. The result would be a log return of 0.28, or in percentage terms, an improvement in compound return from the 23% of the prototype individual country to about 32% for a prototype equal-weighted strategy.

The bottom two panels in Figure 9 show the actual equal-weighted and Composite compound results and their ingredient mean, variance, skew and kurtosis of single-period percentage returns. These indices include additional countries during the later years, but the effect is not material to our comparison.

The equal-weighted index shown in Figure 6 had a log return contribution from the mean of 0.31, a drag from variance of only .02, and near zero impact of skewness and kurtosis individually. The four terms of our Taylor series give an estimate of 0.29, with a compound arithmetic return of 33.4%, very close to the actual figure of 33.5% and rather similar to the hypothetical prototype return of 32% of the first panel.

On the other hand, the IFC Composite begins with a much lower mean arithmetic return, in log terms only 0.18. The biggest culprit was Malaysia, perhaps not coincidentally the country with lowest GDP/CAP ratio and the second lowest volatility. However, the overall pattern of negligible weights for the high variance, high GDP/CAP countries such as Argentina and the Philippines also played a big role in the weaker level of initial mean return. Let’s move to the next term in the Taylor series. Although the Composite began with ingredients lower in individual volatility, its inferior diversification more than offset this advantage. Investing in the Composite reduced variance drag only to .03, higher than for the equal-weighted index. Finally, as with the equal-weighted index, skewness and kurtosis played no role in determining compound return of the diversified portfolio. The result was a compound return of only about 15%.

Emerging Markets in Summary

Nowhere have we argued that equal-weighting is optimal in edge markets in general or emerging markets in particular. We do argue that the superior twelve-year performance of an equal-weighted index over the market index of emerging markets is not simply a chance result. The return patterns most often seen in practice will reflect expected median terminal wealth, not mean terminal wealth. This will be determined by expected compound return. The latter has been far higher for the equal-weighted emerging market index for two reasons.

First, mean single-period returns of lower capitalization countries appear much higher than would be justified to a global investor by their high local variance and kurtosis. A combination of segmentation for less-developed countries and lack of reaction to, or anticipation of, the impact of diversification on expected compound returns may be the reason. Second, the capitalization-weighted scheme more than offsets its initial advantage in lower component risk by poor diversification of these risks. Again, the lack of an integrated market equilibrium is suggested.

SUGGESTIONS FOR PRACTICE AND FURTHER RESEARCH

We have illustrated with emerging markets how to look for exploitable segmentation return and opportunities to improve mean compound return. However, investors should be alert for the symptoms of edge investing in many situations. Whenever there is restricted investor access, whether self or externally imposed, combined with high specific risk at the disaggregate level, investors may be able to exploit segmentation return. Venture capital and commodity futures, and possibly small stocks at a global level, come to mind. These conditions will also generate the potential for improving mean compound return through capital growth analysis.

Capital growth analysis is also called for whenever there is high variance, skew, or kurtosis at the aggregate portfolio level. This holds not only for venture capital and commodity futures, but for many other situations. These include investing on margin and in portfolio insurance or other derivative strategies for obtaining positive skew on the total portfolio return. Seeking positive skewness is not just a matter of protecting agents or of satisfying atavistic intuition. When it can be obtained at reasonable cost, positive skewness will have a positive impact on expected compound return.

Market capitalization-weighted indices are suspect in edge markets. Also in such cases, the practice of managing a portfolio so as to keep tracking error low versus a market index will generally be harmful. Pension funds, consultants and investment managers would particularly benefit from changing practice here.

Quantitatively oriented investment managers frequently adapt Markowitz mean-variance optimization by substituting tracking error variance for total return variance. This has merit in center markets, both for offsetting unrealistic return expectations and for maintaining a specialization mandate near some benchmark. These goals can still be achieved in marginal edge markets by partially returning to the original Markowitz framework using total return variance. Simply substitute a mixture of cash and the original index benchmark for the benchmark, and optimize expected return versus the resulting hybrid residual risk.

A possible next step would be a portfolio construction tool that constructs an efficient frontier of portfolio mean log returns versus variance of log returns. Such an “optimizer” does not yet appear to be advertised as commercially available, although academic examples have been reported. Whether large scale problem-solving in this way is practical is not yet clear, but surely worthy of investigation.

There are many open issues for research in edge investing, and even more generally in examining the role of segmentation and capital growth ideas in markets that mostly satisfy the assumptions of the center. Here are a few examples.

What is the potential for earning segmentation or diversification returns in specific fields such as venture capital, commodity futures, global small stocks, and even global real estate?

Given today’s cost of put options or dynamic hedging strategies for the S&P 500, can they enhance expected compound return, even if they do not improve mean terminal wealth? Can currency hedging to increase positive skewness on total international returns be justified in terms of expected compound return?

Some studies of US stocks use individual stock regression while some aggregate by quantile portfolios to study small capitalization effects. However, the act of aggregating stocks in a subportfolio may produce diversification return that improves the median return likely to be observed. How can the results of such studies be made more comparable?

What would one conclude from a dynamic equilibrium market model that incorporated segmentation, a preference for expected compound return and an aversion to compound return variance?

APPENDIX

Source: International Finance Corporation

NOTES

[1] I am grateful to my colleagues Peter Rathjens and John Capeci for pointing out to me the key role played by expected mean terminal wealth in mainstream financial theory, and to Dick Crowell for his overall comments.

[2] We refer here to the kurtosis definition that is 3 for a normal distribution.

[3] I first saw the “variance drag” label used by Thomas Messmore.

[4] Several of the figures in this section are drawn from Wilcox [1997]. However, the interpretation here is considerably broader.

[5] The emerging market data for this paper were taken from International Finance Corporation [1997]. Gross Domestic Products for Figure 7 are from the International Monetary Fund [1992]. Developed country market capitalizations for Figure 7 were taken from Morgan Stanley Capital International Perspective back issues.

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