estimation and control of transactions costs. For the remainder of this chapter, we will assume that we know the cost for each anticipated trade. Practical Details Before proceeding further in our analysis of portfolio construction, we should review some practical details concerning this process. First, how do we choose a risk aversion parameter? We briefly discussed this problem in Chap. 5. There we found an optimality relationship between the information ratio, the risk
Page 388 aversion, and the optimal active risk. Repeating that result here, translated from residual to active return and risk, The point is that we have more intuition about our information ratio and our desired amount of active risk. Hence, we can use Eq. (14.3) to back out an appropriate risk aversion. If our information ratio is 0.5, and we desire 5 percent active risk, we should choose an active risk aversion of 0.05. Note that we must be careful to verify that our optimizer is using percents and not decimals. A second practical matter concerns aversion to specific as opposed to common-factor risk. Several commercial optimizers utilize this decomposition of risk to allow differing aversions to these different sources of risk: An obvious reaction here is, ''Risk is risk, why would I want to avoid one source of risk more than another?" This is a useful sentiment to keep in mind, but there are at least two reasons to consider implementing a higher aversion to specific risk. First, since specific risk arises from bets on specific assets, a high aversion to specific risk reduces bets on any one stock. In particular, this will reduce the size of your bets on the (to be determined) biggest losers. Second, for managers of multiple portfolios, aversion to specific risk can help reduce dispersion. This will push all those portfolios toward holding the same names. The final practical details we will cover here concern alpha coverage. First, what happens if we forecast returns on stocks that are not in the benchmark? We can always handle that by expanding the benchmark to include those stocks, albeit with zero weight. This keeps stock n in the benchmark, but with no weight in determining the benchmark return or risk. Any position in stock n will be an active position, with active risk correctly handled. What about the related problem, a lack of forecast returns for stocks in the benchmark? Chapter 11 provided a sophisticated approach to inferring alphas for some factors, based on the alphas for other factors. We could apply the same approach in this case. For stock-specific alphas, we can use the following approach.
Page 389 Let N1 represent the collection of stocks with forecasts, and N0 the stocks without forecasts. The value-weighted fraction of stocks with forecasts is The average alpha for group N1 is To round out the set of forecasts, set for stocks in N1 and for stocks in N0. These alphas are benchmark-neutral. Moreover, the stocks we did not cover will have a zero, and therefore neutral, forecast. Portfolio Revisions How often should you revise your portfolio? Whenever you receive new information. That's the short answer. If a manager knows how to make the correct trade-off between expected active return, active risk, and transactions costs, frequent revision will not present a problem. If the manager has human failings, and is not sure of his or her ability to correctly specify the alphas, the active risk, and the transactions costs, then the manager may resort to less frequent revision as a safeguard. Consider the unfortunate manager who underestimates transactions costs, makes large changes in alpha estimates very frequently, and revises his portfolio daily. This manager will churn the portfolio and suffer higher than expected transactions costs and lower than expected alpha. A crude but effective cure is to revise the portfolio less frequently. More generally, even with accurate transactions costs estimates, as the horizon of the forecast alphas decreases, we expect them to contain larger amounts of noise. The returns themselves become noisier with shorter horizons. Rebalancing for very short horizons would involve frequent reactions to noise, not signal. But the transactions costs stay the same, whether we are reacting to signal or noise.
Page 390 This trade-off between alpha, risk, and costs is difficult to analyze because of the inherent importance of the horizon. We expect to realize the alpha over some horizon. We must therefore amortize the transactions costs over that horizon. We can capture the impact of new information, and decide whether to trade, by comparing the marginal contribution to value added for stock n, MCVAn, to the transactions costs. The marginal contribution to value added shows how value added, as measured by risk-adjusted alpha, changes as the holding of the stock is increases, with an offsetting decrease in the cash position. As our holding in stock n increases, αn measures the effect on portfolio alpha. The change in value added also depends upon the impact (at the margin) on active risk of adding more of stock n. The stock's marginal contribution to active risk, MCARn, measures the rate at which active risk changes as we add more of stock n. The loss in value added due to changes in the level of active risk will be proportional to MCARn. Stock n's marginal contribution to value added depends on its alpha and marginal contribution to active risk, in particular: Let PCn be the purchase cost and SCn the sales cost for stock n. For purposes of illustration, we take PCn = 0.50 percent and SCn = 0.75 percent. If the current portfolio is optimal,6 then the marginal contribution to value added for stock n should be less than the purchase cost. If it exceeded the purchase cost, say at 0.80 percent, then a purchase of stock n would yield a net benefit of 0.80 percent – 0.50 percent = 0.30 percent. Similarly the marginal contribution to value added must be greater than the negative of the sales cost. If it were – 1.30 percent, then we could decrease our holding of stock n and save 1.30 percent at the margin. The cost would be the 0.75 percent transactions cost, for a net benefit of 1.30 percent – 0.75 percent = 0.55 percent. 6Assuming no limitations on holdings, no limitations on the cash position, and no additional constraints. Aficionados will realize that this analysis becomes more complicated, but not essentially different, if we include these additional constraints.
Page 391 This observation allows us to put a band around the alpha for each stock. As long as the alpha stays within that band, the portfolio will remain optimal, and we should not react to new information. The bandwidth is the total of the sale plus purchase costs, 0.50 percent + 0.75 percent = 1.25 percent in our example. If we just purchased a stock, its marginal contribution to value added will equal its purchase cost. We are at the upper end of the band. Any increase in alpha would lead to further purchases. The alpha would have to decrease by 1.25 percent before we would consider selling the stock. The situation before new information arrives is or, using Eq. (14.7), This analysis has simplified the problem by subsuming the amortization horizon into the costs SC and PC. To fully treat the issue of when to rebalance requires analyzing the dynamic problem involving alphas, risks, and costs over time. There are some useful results from this general treatment, in the very simple case of one or two assets. Leland (1996) solves the asset allocation problem of rebalancing around an optimal stock/bond allocation. Let's assume that the optimal allocation is 60/40. Assuming linear transactions costs and a utility function penalizing active variance (relative to the optimal allocation) and transactions costs over time, Leland shows that the optimal strategy involves a no-trade region around the 60/40 allocation. If the portfolio moves outside that region, the optimal strategy is to trade back to the boundary. Trading only to the boundary, not to the target allocation, cuts the turnover and transactions costs roughly in half, with effectively no change in risk over time. The size of the notrade region depends on the transactions costs, the risk aversion, and the expected return and risk of stocks and bonds. Obviously, changing the size of the notrade region will change the turnover for the strategy. This result concerns a problem that is much simpler than our general active portfolio management problem: The solved problem
Page 392 Figure 14.1 After-cost information ratio for various half-lives. is one-dimensional and does not involve the flow of information (the target allocation is static). Still, it is useful in motivating rebalancing rules driven not purely by the passage of time (e.g., monthly or quarterly rebalancing), but rather by the portfolio's falling outside certain boundaries. Another approach to the dynamic problem utilizes information horizon analysis, introduced in Chap. 13. Here we apply trading rules like Eq. (14.9) in the dynamic case of trading one position only, over an indefinite future,7 with information characterized by an information horizon. Figure 14.1 shows how the after-cost information ratio declines as a function of both the (one-way) cost and the half-life of the signals. Two effects are at work. First, when we trade, we pay the costs. Second, and more subtle, the transactions costs makes us less eager; we lose by intimidation. Techniques for Portfolio Construction There are as many techniques for portfolio construction as there are managers. Each manager adds a special twist. Despite this personalized nature of portfolio construction techniques, there are 7There is a pleasant symmetry in this approach. Conventional portfolio optimization considers lots of assets in a one-period framework; we are considering one-asset (position) in a multiple-period framework.
Page 393 four generic classes of procedures that cover the vast majority of institutional portfolio management applications:8 • Screens • Stratification • Linear programming • Quadratic programming Before we examine these procedures in depth, we should recall our criteria. We are interested in high alpha, low active risk, and low transactions costs. Our figure of merit is value added less transactions costs: We will see how each of these procedures deals with these three aspects of portfolio construction. Screens Screens are simple. Here is a screen recipe for building a portfolio from scratch: 1. Rank the stocks by alpha. 2. Choose the first 50 stocks (for example). 3. Equal-weight (or capitalization-weight) the stocks. We can also use screens for rebalancing. Suppose we have alphas on 200 stocks (the followed list). Divide the stocks into three categories: the top 40, the next 60, and the remaining 100. Put any stock in the top 40 on the buy list, any stock in the bottom 100 on the sell list, and any stock in the middle 60 on the hold list. Starting with the current 50-stock portfolio, buy any stocks that are on the buy list but not in the portfolio. Then sell any assets that are in 8The techniques we review successfully handle monthly or quarterly rebalancing of portfolios of up to 1000 assets and asset universes that can exceed 10,000 for international investing. Later, we will discuss nonlinear programming and stochastic optimization, whose applications are generally limited to asset allocation schemes involving few (less than 25) asset classes and long planning horizons.
Page 394 the portfolio and on the sell list. We can adjust the numbers 40, 60, and 100 to regulate turnover. Screens have several attractive features. There is beauty in simplicity. The screen is easy to understand, with a clear link between cause (membership on a buy, sell, or hold list) and effect (membership in the portfolio). The screen is easy to computerize; it might be that mythical computer project that can be completed in two days! The screen is robust. Notice that it depends solely on ranking. Wild estimates of positive or negative alphas will not alter the result. The screen enhances alphas by concentrating the portfolio in the high-alpha stocks. It strives for risk control by including a sufficient number of stocks (50 in the example) and by weighting them to avoid concentration in any single stock. Transactions costs are limited by controlling turnover through judicious choice of the size of the buy, sell, and hold lists. Screens also have several shortcomings. They ignore all information in the alphas apart from the rankings. They do not protect against biases in the alphas. If all the utility stocks happen to be low in the alpha rankings, the portfolio will not include any utility stocks. Risk control is fragmentary at best. In our consulting experience, we have come across portfolios produced by screens that were considerably more risky than their managers had imagined. In spite of these significant shortcomings, screens are a very popular portfolio construction technique. Stratification Stratification is glorified screening. The term stratification comes from statistics. In statistics, stratification guards against sample bias by making sure that the sample population is representative of the total population as it is broken down into distinct subpopulations. The term is used very loosely in portfolio construction. When a portfolio manager says he uses stratified sampling, he wants the listener to (1) be impressed and (2) ask no further questions. The key to stratification is splitting the list of followed stocks into categories. These categories are generally exclusive. The idea is to obtain risk control by making sure that the portfolio has a representative holding in each category. As a typical example, let's
Page 395 suppose that we classify stocks into 10 economic sectors and also classify the stocks in each sector by size: big, medium, and small. Thus, we classify all stocks into 30 categories based on economic sector and size. We also know the benchmark weight in each of the 30 categories. To construct a portfolio, we mimic the screening exercise within each category. We rank the stocks by alpha and place them into buy, hold, and sell groups within each category in a way that will keep the turnover reasonable. We then weight the stocks so that the portfolio's weight in each category matches the benchmark's weight in that category. Stratification ensures that the portfolio matches the benchmark along these important dimensions. The stratification scheme has the same benefits as screening, plus some. It is robust. Improving upon screening, it ignores any biases in the alphas across categories. It is somewhat transparent and easy to code. It has the same mechanism as screening for controlling turnover. Stratification retains some of the shortcomings of a screen. It ignores some information, and does not consider slightly over-weighting one category and underweighting another. Often, little substantive research underlies the selection of the categories, and so risk control is rudimentary. Chosen well, the categories can lead to reasonable risk control. If some important risk dimensions are excluded, risk control will fail. Linear Programming A linear program (LP) is space-age stratification. The linear programming approach9 characterizes stocks along dimensions of risk, e.g., industry, size, volatility, and beta. The linear program does not require that these dimensions distinctly and exclusively partition the stocks. We can characterize stocks along all of these dimensions. The linear program will then attempt to build portfolios that are reasonably close to the benchmark portfolio in all of the dimensions used for risk control. 9A linear program is a useful tool for a variety of portfolio management applications. The application described here is but one of those applications.
Page 396 It is also possible to set up a linear program with explicit transactions costs, a limit on turnover, and upper and lower position limits on each stock. The objective of the linear program is to maximize the portfolio's alpha less transactions costs, while remaining close to the benchmark portfolio in the risk control dimensions. The linear program takes all the information about alpha into account and controls risk by keeping the characteristics of the portfolio close to the characteristics of the benchmark. However, the linear program has difficulty producing portfolios with a prespecified number of stocks. Also, the riskcontrol characteristics should not work at cross purposes with the alphas. For example, if the alphas tell you to shade the portfolio toward smaller stocks at some times and toward larger stocks at other times, you should not control risk on the size dimension. Quadratic Programming Quadratic programming (QP) is the ultimate10 in portfolio construction. The quadratic program explicitly considers each of the three elements in our figure of merit: alpha, risk, and transactions costs. In addition, since a quadratic program includes a linear program as a special case, it can include all the constraints and limitations one finds in a linear program. This should be the best of all worlds. Alas, nothing is perfect. One of the main themes of this chapter is dealing with less than perfect data. The quadratic program requires a great many more inputs than the other portfolio construction techniques. More inputs mean more noise. Does the benefit of explicitly considering risk outweigh the cost of introducing additional noise? A universe of 500 stocks will require 500 volatility estimates and 124,750 correlation estimates.11 There are ample opportunities to make mistakes. It is a fear of garbage in, garbage out that deters managers from using a quadratic program. 10Given our criterion of portfolio alpha minus a penalty for active risk and less transactions costs. 11Chapter 3 discusses how to accurately approach this problem.
Page 397 This fear is warranted. A lack of precision in the estimate of correlations is an inconvenience in the ordinary estimation of portfolio risk. For the most part, the estimation errors will cancel out. It is an obstacle in optimization. In optimization, the portfolio is selected to, among other things, have a low level of active risk. Because the optimizer tries to lower active risk, it will take advantage of opportunities that appear in the noisy estimates of covariance but are not present in reality. An example can illustrate the point. Suppose we consider a simple cash versus market trade-off. Let ζ be the actual volatility of the market and σ our perceived volatility. If VA* is the optimal value added that we can obtain with the correct risk estimate ζ, then the loss12 we obtain with the estimate σ is Figure 14.2 shows the percentage loss, Loss/VA*, as a function of the estimated market risk, assuming that the true market risk is 17 Figure 14.2 Estimated market volatility. 12The technical appendix derives a more general version of this result.
Page 398 percent. In this example, market volatility estimates within 1 percent of the true market volatility will not hurt value added very much, but as estimation error begin to exceed 3 percent, the effect on value added becomes significant, especially if the error is an underestimate of volatility. In fact, an underestimate of 12 percent market volatility (5 percent below the ''true" volatility) leads to a negative value added. There are two lessons here. The first is that errors in the estimates of covariance lead to inefficient implementation. The second, which is more positive and, indeed, more important, is that it is vital to have good estimates of covariance. Rather than abandon the attempt, try to do a good job. Tests of Portfolio Construction Methods We can test the effectiveness of these portfolio construction procedures by putting them on an equal footing and judging the performance of their outputs. In this case, we will input identical alphas to four procedures, described below, and ignore transactions costs.13 The alphas are great. They include the actual returns to the 500 stocks in the S&P 500 over the next year plus noise, combined so that the correlation of the alphas with the returns (the information coefficient) is 0.1. The fundamental law of active management therefore predicts14 an information ratio of 2.24. So not only will we feed the same alphas into each portfolio construction method, but we know what the final result should be. The four portfolio construction techniques are Screen I. Take the N stocks with the highest alphas and equal-weight them. Use N = 50, 100, and 150 for low, medium, and high risk aversion, respectively. Screen II. Take the N stocks with the highest alphas and capitalization-weight them. Use N = 50, 100, and 150 for low, medium, and high risk aversion, respectively. 13For more details, see Muller (1993). We ignore transactions costs to simplify the test. 14The information coefficient of 0.1 and the breadth of 500 leads to .
Page 399 Strat. Take the J stocks with the highest alphas in each of the BARRA 55 industry categories. Use J = 1, 2, and 3 for low, medium, and high risk aversion portfolios, which will have 55, 110, and 165 stocks, respectively. QP. Choose portfolios which maximize value added, assuming low, medium, and high risk aversion parameters. Use full investment and no short sales constraints, and constrain each position to constitute no more than 10 percent of the entire portfolio. Portfolios were constructed in January 1984 and rebalanced in January 1985, January 1986, and May 1987, with portfolio performance tracked over the subsequent year. Table 14.3 contains the results. Table 14.3 displays each portfolio's ex post information ratio. In this test, the quadratic programming approach clearly led to consistently the highest ex post information ratios. On average, TABLE 14.3 Date Risk Aversion Screen I Screen II Strat QP High 1.10 1.30 0.63 2.16 January 1984 Medium 0.95 2.24 0.64 1.89 Low 0.73 1.31 0.69 1.75 High 0.78 1.47 1.98 0.98 January 1985 Medium 0.74 –0.53 1.29 1.68 Low 0.50 –0.15 0.83 1.49 High 1.17 0.91 0.69 2.08 January 1986 Medium 0.69 0.98 0.33 2.29 Low 0.60 0.99 0.51 2.51 High 1.43 2.04 2.82 2.14 May 1987 Medium 1.01 1.48 2.60 1.76 Low 0.66 1.17 2.17 1.82 Average 0.86 1.10 1.27 1.88 Standard deviation 0.27 0.79 0.89 0.40 Maximum 1.43 2.24 2.82 2.51 Minimum 0.50 –0.53 0.33 0.98 Source: Peter Muller, "Empirical Tests of Biases in Equity Portfolio Optimization," in Financial Optimization, edited by Stavros A. Zenios (Cambridge: Cambridge University Press, 1993), Table 4-4.
Page 400 it surpassed all the other techniques, and it exhibited consistent performance around that average. A stratified portfolio had the single highest ex post information ratio, but no consistency over time. The screening methods in general do not methodically control for risk, and Table 14.3 shows that one of the screened portfolios even experienced negative returns during one period. Recall that the ex ante target for the information ratio was 2.24. None of the methods achieved that target, although the quadratic program came closest on average. Part of the reason for the shortfall is the constraints imposed on the optimizer. We calculated the target information ratio ignoring constraints. As we have seen, constraints can effectively reduce the information coefficient and hence the information ratio. Alternatives to Mean/Variance Optimization In Chap. 3, we discussed alternatives to standard deviation as risk measurements. These included semivariance, downside risk, and shortfall probability. We reviewed the alternatives and chose standard deviation as the best overall risk measure. We return to the issue again here, since our portfolio construction objective expresses our utility, which may in fact depend on alternative measures of risk. But as two research efforts show, even if your personal preferences depend on alternative risk measures, mean/variance analysis will produce equivalent or better portfolios. We present the research conclusions here, and cite the works in the bibliography. Kahn and Stefek (1996) focus on the forward-looking nature of portfolio construction. The utility function includes forecasts of future risk. Mean/variance analysis, as typically applied in asset selection, relies on sophisticated modeling techniques to accurately forecast risk. Chapter 3 discusses in detail both the advantages of structural risk models and their superior performance. Forecasting of alternative risk measures must rely on historical returns–based analysis. Kahn and Stefek show that higher moments of asset and asset class return distributions exhibit very little predictability, especially where it is important for portfolio construction. Return kurtosis is predictable, in the sense that most return distributions exhibit positive kurtosis ("fat tails") most of the time.
Page 401 The ranking of assets or asset classes by kurtosis exhibits very little predictability. The only exception is options, where return asymmetries are engineered into the payoff pattern. The empirical result is that most alternative risk forecasts reduce to a standard deviation forecast plus noise, with even the standard deviation forecast based only on history. According to this research, even investors with preferences defined by alternative risk measures are better served by mean/variance analysis.15 Grinold (1999) takes a different approach to the same problem, in the specific case of asset allocation. First, he adjusts returns-based analysis to the institutional context: benchmark-aware investing with typical portfolios close to the benchmark. This is the same approach we have applied to mean/variance analysis in this book. Then he compares mean/variance and returns-based analysis, assuming that the benchmark holds no options and that all options are fairly priced. The result is that portfolios constructed using returns-based analysis are very close to mean/variance portfolios, although they require much more effort to construct. Furthermore, managers using this approach very seldom buy options. If options are fairly priced relative to the underlying asset class, then optimization will pursue the alphas directly through the asset class, not indirectly through the options. So Kahn and Stefek argue the asset selection case for mean/variance, and Grinold argues the asset allocation case for mean/variance. Furthermore, Grinold shows why institutional investors, with their aversion to benchmark risk, will seldom purchase options—the only type of asset requiring analysis beyond mean/variance. As a final observation, though, some active institutional investors do buy options. We argue that they do so typically to evade restrictions on leverage or short selling, or because of liquidity 15The case of investors in options and dynamic strategies like portfolio insurance is a bit trickier, but also handled in the paper. There the conclusion is to apply mean/variance analysis to the active asset selection strategy, and to overlay an options-based strategy based on alternative risk measures. But see Grinold (1999), who shows that under reasonable assumptions, even with alternative risk measures, most institutional investors will not use such strategies.
Page 402 concerns. Only in the case of currency options do we see much evidence of investors choosing options explicitly for their distributions. Many managers have a great aversion to currency losses, and options can provide downside protection. We still advocate using mean/variance analysis generally and, if necessary, treating currency options as a special case. Dispersion Dispersion plagues every manager running separate accounts for multiple clients. Each account sees the same alphas, benchmark, and investment process. The cash flows and history differ, however, and the portfolios are not identical. Hence, portfolio returns are not identical. We will define dispersion as the difference between the maximum return and minimum return for these separate account portfolios. If the holdings in each account are identical, dispersion will disappear. If transactions costs were zero, dispersion would disappear. Dispersion is a measure of how an individual client's portfolio may differ from the manager's reported composite returns. Dispersion is, at the least, a client support problem for investment managers. In practice, dispersion can be enormous. We once observed five investors in a particular manager's strategy, in separate accounts, incur dispersion of 23 percent over a year. The manager's overall dispersion may have been even larger. This was just the dispersion involving these five clients. In another case, with another manager, one client outperformed the S&P 500 by 15 percent while another underperformed by 9 percent, in the same year. At that level, dispersion is much more than a client support problem. We can classify dispersion by its various sources. The first type of dispersion is client-driven. Portfolios differ because individual clients impose different constraints. One pension fund may restrict investment in its company stock. Another may not allow the use of futures contracts. These client-initiated constraints lead to dispersion, but they are completely beyond the manager's control. But managers can control other forms of dispersion. Often, dispersion arises through a lack of attention. Separate accounts
Page 403 exhibit different betas and different factor exposures through lack of attention. Managers should control this form of dispersion. On the other hand, separate accounts with the same factor exposures and betas can still exhibit dispersion because of owning different assets. Often the cost of holding exactly the same assets in each account will exceed any benefit from reducing dispersion. In fact, because of transactions costs, some dispersion is optimal. If transactions costs were zero, rebalancing all the separate accounts so that they hold exactly the same assets in the same proportions would have no cost. Dispersion would disappear, at no cost to investors. With transactions costs, however, managers can achieve zero dispersion only with increased transactions costs. Managers should reduce dispersion only until further reduction would substantially lower returns on average because much higher transactions costs would be incurred. Example To understand dispersion better, let's look at a concrete example. In this example, the manager runs an existing portfolio and receives cash to form a new portfolio investing in the same strategy. So at one point in time, the manager is both rebalancing the existing portfolio and constructing the new portfolio. The rebalanced portfolio holdings will reflect both new and old information. With zero transactions costs, the manager would rebalance to the new optimum. Given an existing portfolio, though, he rebalances only where the new information more than overcomes the transactions costs, as in Eq. (14.9). This trade-off does not affect the new portfolio in the same way. The manager starts from cash, and while he would still like to minimize transactions costs, he assumes a fairly high transactions cost for the initial portfolio construction. For this example, we'll assume that the new portfolio he builds is optimal and reflects entirely the manager's new information. Clearly there will be dispersion between the existing portfolio and the new portfolio. There are two methods by which the manager could reduce dispersion to zero. He could invest the new portfolio in the rebalanced existing portfolio. This sacrifices returns, since the new portfolio will reflect both new and old information
Page 404 instead of just new information. The other choice is to invest the composite in the new optimum. But this would require paying excess transactions costs. By treating the existing portfolio and the new portfolio separately, the manager accepts some level of dispersion in order to achieve higher average returns. Furthermore, he can hope that this dispersion will decrease over time. Characterizing Dispersion We will now perform some static analysis to understand the causes of dispersion. First, consider dispersion caused by different betas or factor exposures. If the separate account betas range from 0.9 to 1.1 and the market return is 35 percent one year, then the dispersion would be 7 percent based just on the differing betas. This range of betas is quite large for an efficient, quantitatively run optimal process, and yet it doesn't come close to explaining some of the extreme war stories. Now let's consider static analysis of managed dispersion—where the manager has matched factor exposures but not assets across all accounts—to try to understand the magnitude of the effect. In this simple model, we will consider N portfolios, all equally weighted with identical factor exposures. Each portfolio contains 100 stocks, and out of that 100 stocks, M stocks appear in all the portfolios and 100 – M stocks are unique to the particular portfolio. Furthermore, every stock has identical specific risk of 20 percent. Figure 14.3 displays the results, assuming normal distributions. We can use the model to show that dispersion will depend on the number of stocks the portfolios have in common, the overall levels of specific risk, and the overall number of portfolios under management. Managing Dispersion We have seen how some level of dispersion is optimal and have discussed why dispersion arises. The next question is whether dispersion decreases over time: Do dispersed portfolios converge, and how fast? In general, convergence will depend on the type
Page 405 Figure 14.3 Dispersion: 100 stock portfolios. of alphas in the strategy, the transactions costs, and possibly the portfolio construction methodology. If alphas and risk stay absolutely constant over time, then dispersion will never disappear. There will always be a transactions cost barrier. An exact matching of portfolios will never pay. Furthermore, we can show (see the technical appendix) that the remaining tracking error is bounded based on the transactions costs and the manager's risk aversion: where TC measures the cost of trading from the initial portfolio to the zero transactions cost optimal portfolio (which we will refer to as portfolio Q), and we are measuring tracking error and risk aversion relative to portfolio Q. With very high risk aversion, all portfolios must be close to one another. But the higher the transactions costs, the more tracking error there is. Given intermediate risk aversion of λA = 0.10 and round-trip transactions costs of 2 percent, and assuming that moving from the initial portfolio to portfolio Q in-
Page 406 volves 10 percent turnover, Eq. (14.12) implies tracking error of 1 percent. Since tracking error is bounded, dispersion is also bounded. Dispersion is proportional to tracking error, with the constant of proportionality dependent on the number of portfolios being managed: where this constant of proportionality involves the inverse of the cumulative normal distribution function Φ, and ψ is the tracking error of each portfolio relative to the composite (see the technical appendix for details). Figure 14.3 displays this function. For a given tracking error, more portfolios lead to more dispersion because more portfolios will further probe the extremes of the return distribution. If the alphas and risk vary over time—the usual case—then convergence will occur. We can show that with changing alphas and risk each period, the portfolios will either maintain or, more typically, decrease the amount of dispersion. Over time, the process inexorably leads to convergence, because each separate account portfolio is chasing the same moving target. These general arguments do not, however, imply any particular time scale. As an empirical example, we looked at five U.S. equity portfolios in a strategy with alphas based on book-to-price ratios and stock-specific alphas. Roughly two-thirds of the strategy's value came from the book-to-price factor tilt, with one-third arising from the stock-specific alphas. We started these five portfolios in January 1992 with 100 names in each portfolio, but not the same 100 names in each portfolio. Each portfolio had roughly 3 percent tracking error relative to the S&P 500. We analyzed the initial level of dispersion and then looked at how that changed over time. We used a consistent alpha generation process and standard mean/variance optimization with uniform transactions costs. To understand convergence and transactions costs, we looked at behavior as we changed the overall level of transactions costs. What we found was a steady decrease in average tracking error (relative to the composite) and dispersion, with the smallest dispersion exhibited when we assumed the lowest transactions costs. Figure 14.4 displays the results. So even though our starting portfolios
Page 407 Figure 14.4 Convergence. (Courtesy of BARRA.)
Page 408 differed, they steadily converged over a roughly 5-year period. In real-life situations, client-initiated constraints and client-specific cash flows will act to keep separate accounts from converging. One final question is whether we can increase convergence by changing our portfolio construction technology. In particular, what if we used dual-benchmark optimization? Instead of penalizing only active risk relative to the benchmark, we would also penalize active risk relative to either the composite portfolio or the optimum calculated ignoring transactions costs. Dual-benchmark optimization can clearly reduce dispersion, but only at an undesirable price. Dualbenchmark optimization simply introduces the trade-off we analyzed earlier: dispersion versus return. Unless you are willing to give up return in order to lower dispersion, do not implement the dual-benchmark optimization approach to managing dispersion. Summary The theme of this chapter has been portfolio construction in a less than perfect world. We have taken the goals of the portfolio manager as given. The manager wants the highest possible after-cost value added. The before-cost value added is the portfolio's alpha less a penalty for active variance. The costs are for the transactions needed to maintain the portfolio's alpha. Understanding and achieving this goal requires data on alphas, covariances between stock returns, and estimates of transactions costs. Alpha inputs are often unrealistic and biased. Covariances and transactions costs are measured imperfectly. In this less than perfect environment, the standard reaction is to compensate for flawed inputs by regulating the outputs of the portfolio construction process: placing limits on active stock positions, limiting turnover, and constraining holdings in certain categories of stocks to match the benchmark holdings. These are valid approaches, as long as we recognize that their purpose is to compensate for faulty inputs. We prefer a direct attack on the causes. Treat flaws in the alpha inputs with alpha analysis: Remove biases, trim outlandish values, and scale alphas in line with expectations for value added. This strengthens the link between research and portfolio construction. Then seek out the best possible
Page 409 estimates of risk and transactions costs. As appropriate, use a powerful portfolio construction tool with as few added constraints as possible. Near the end of the chapter, we returned to the topic of alternative risk measures and alternatives to mean/variance optimization. For most active institutional managers (especially those who do not invest in options and optionlike dynamic strategies such as portfolio insurance), alternatives to mean/variance analysis greatly complicate portfolio construction without improving results. At the stock selection level, results may be much worse. Finally, we analyzed the very practical issue of dispersion among separately managed accounts. We saw that managers can control dispersion—especially that driven by differing factor exposures—but should not reduce it to zero. Problems 1. Table 14.1 shows both alphas used in a constrained optimization and the modified alphas which, in an unconstrained optimization, lead to the same holdings. Comparing these two sets of alphas can help in estimating the loss in value added caused by the constraints. How? What is that loss in this example? The next chapter will discuss this in more detail. 2. Discuss how restrictions on short sales are both a barrier to a manager's effective use of information and a safeguard against poor information. 3. Lisa is a value manager who chooses stocks based on their price/earnings ratios. What biases would you expect to see in her alphas? How should she construct portfolios based on these alphas, in order to bet only on specific asset returns? 4. You are a benchmark timer who in backtests can add 50 basis points of risk-adjusted value added. You forecast 14 percent benchmark volatility, the recent average, but unfortunately benchmark volatility turns out to be 17 percent. How much value can you add, given this misestimation of benchmark volatility?
Page 410 5. You manage 20 separate accounts using the same investment process. Each portfolio holds about 100 names, with 90 names appearing in all the accounts and 10 names unique to the particular account. Roughly how much dispersion should you expect to see? References Chopra, Vijay K., and William T. Ziemba. ''The Effects of Errors in Means, Variances, and Covariances on Optimal Portfolio Choice." Journal of Portfolio Management, vol. 19, no. 2, 1993, pp. 6–11. Connor, Gregory, and Hayne Leland. "Cash Management for Index Tracking."Financial Analysts Journal, vol. 51, no. 6, 1995, pp. 75–80. Grinold, Richard C. "The Information Horizon." Journal of Portfolio Management, vol. 24, no. 1, 1997, pp. 57–67. ———. "Mean-Variance and Scenario-Based Approaches to Portfolio Selection." Journal of Portfolio Management, vol. 25, no. 2, Winter 1999, pp. 10–22. Jorion, Philippe. "Portfolio Optimization in Practice." Financial Analysts Journal, vol. 48, no. 1, 1992, pp. 68–74. Kahn, Ronald N. "Managing Dispersion." BARRA Equity Research Seminar, Pebble Beach, Calif., June 1997. Kahn, Ronald N., and Daniel Stefek. "Heat, Light, and Downside Risk." BARRA Preprint, December 1996. Leland, Hayne. Optimal Asset Rebalancing in the Presence of Transactions Costs. University of California Research Program in Finance, Publication 261, October 1996. Michaud, Richard. "The Markowitz Optimization Enigma: Is 'Optimized' Optimal?" Financial Analysts Journal, vol. 45, no. 1, 1989, pp. 31–42. Muller, Peter. "Empirical Tests of Biases in Equity Portfolio Optimization." In Financial Optimization, edited by Stavros A. Zenios (Cambridge: Cambridge University Press, 1993), pp. 80– 98. Rohweder, Herold C. "Implementing Stock Selection Ideas: Does Tracking Error Optimization Do Any Good?" Journal of Portfolio Management, vol. 24, no. 3, 1998, pp. 49–59. Rudd, Andrew. "Optimal Selection of Passive Portfolios." Financial Management, vol. 9, no. 1, 1980, pp. 57–66. Rudd, Andrew, and Barr Rosenberg. "Realistic Portfolio Optimization." TIMS Study in the Management Sciences, vol. 11, 1979, pp. 21–46. Stevens, Guy V.G. "On the Inverse of the Covariance Matrix in Portfolio Analysis." Journal of Finance, vol. 53, no. 5, 1998, pp. 1821–1827. Technical Appendix This appendix covers three topics: alpha analysis, in particular how to neutralize alphas against various factor biases; the loss of value
Page 411 added due to errors in the estimated covariance matrix; and dispersion. Alpha Analysis Our goal in this section is to separate alphas into common-factor components and specific components, and correspondingly to define active portfolio positions arising from these distinct components. This will involve considerable algebra, but the result will allow us to carefully control our alphas: to neutralize particular factor alphas and even to input factor alphas designed to achieve target active factor exposures. We will analyze stock alphas in the context of a multiple-factor model with N stocks and K factors: We make the usual assumptions that b and u are uncorrelated, and that the components of u are uncorrelated. The model has the covariance structure The unconstrained portfolio construction procedure leads to an active position determined by the first-order conditions: where is the active factor position. We now want to separate α into a common-factor component αCF and a specific component αSP, and similarly separate into common-factor holdings hCF and specific holdings hSP. We will eventually see that each component will separately satisfy Eq. (14A.3), with the common-factor alpha component leading to the common-factor active positions and the specific alpha component leading to the specific active positions and containing zero active factor positions. Equation (14A.3) uniquely determines the optimal active holdings and optimal active factor exposures . However, it does not uniquely determine a separation of into hCF and hSP: There are an infinite number of portfolios with active factor exposures
Page 412 . We can uniquely define the separation if we stipulate that hCF is the minimum-risk portfolio with factor exposures . Let H be the K by N matrix with factor portfolios as rows. The matrix H is Note that H · X = I; each factor portfolio has unit exposure to its factor and zero exposure to all the other factors. Also remember that each factor portfolio has minimum risk, given its factor exposures. Then, starting with the uniquely defined and our definition of hCF as the minimum-risk portfolio with these factor exposures, we find Knowing that hCF and hSP add up to and applying additional algebra, we find that Exercise 1 at the end of this appendix asks the reader to demonstrate Eq. (14A.5), i.e., that, as defined, hCF does lead to the minimum-risk portfolio with factor exposures . Exercise 2 asks the reader to demonstrate that hSP does not contain any common-factor exposures, i.e., that XT · hSP = 0. The separation of the optimal holdings into common-factor and specific holdings has been the hard part of our task. With this behind us, the easier part is to separate α into a common-factor component αCF and a specific component αSP. These are Notice that H · α is the alpha of each of the K factor portfolios. Call αF = H · α the factor alphas. Then X · αF maps those factor alphas back onto the assets. So we have separated both the optimal active holdings and the alphas into common-factor and specific pieces. We can also
Page 413 check the correspondence between common-factor alphas and common-factor holdings, and between specific alphas and specific holdings. According to Eq. (14A.3), the optimal active common-factor exposures are Exercises 4, 5, and 6 help to show that just using αCF will lead to active holdings hCF and commonfactor exposures and that αSP will lead to active holdings hSP and common-factor exposures of zero. How can we use this glut of algebra? Suppose we believe that our alphas contain valuable information only for specific stock selection. Instead of relying on any factor forecasts inadvertently contained in our alphas, we have defined a target active position for the common factors. To achieve this target, we can replace the original alpha α with α', where The first term on the right-hand side of Eq. (14A.10) replaces αCF and will result in an active common-factor exposure of . The second term on the right-hand side of Eq. (14A.10) does not affect the common-factor positions and preserves the pure stock selection information from the original set of alphas. The columns of the N by K matrix V · HT are of particular interest. Column k of V · HT leads to an active factor position that is positive for factor k and zero for all other factors. This insight can provide us with pinpoint control over the factors. Let's say we partition the factors so that x = {y,z}, where we are willing to take an active position on the y factors and no active position on the z factors. Then we could set where is the y component of . This would result in no active position on the z factors. If we wanted no active positions on any of the factors, we could set xPAT = 0.
Page 414 Optimization and Data Errors We now turn to the second topic of this technical appendix: the erosion in value added due to inaccuracies in our estimated covariance matrix. Consider a situation where the true covariance matrix is V, but the manager uses a covariance matrix U V. To simplify matters, assume that the manager imposes no constraints. Using Eq. (14A.3), the manager then will choose optimal active positions Using V and , the value added (risk-adjusted return) for the manager will be If the manager had known the true covariance matrix, the active position would have been with value added The loss in value added is just the difference between and . Using Eqs. (14A.12) and (14A.14), this becomes where we have defined the loss as a positive quantity (i.e., the amount lost). You can see that if U = V, this term becomes zero. Dispersion Here we derive two results stated in the main text. First, we derive a bound on tracking error in the presence of transactions costs. Start with initial portfolio I and zero transactions cost optimal portfolio Q, which we will treat as a benchmark. To find an optimal
Page 415 solution, portfolio P, we will trade off tracking error relative to portfolio Q against the transactions costs of moving from portfolio I to portfolio P: where PC is a vector of purchase costs, SC is a vector of sell costs, the inner maximum functions look element by element to determine whether we are purchasing or selling, and we have defined the active portfolio relative to portfolio Q. This optimization includes alphas implicitly, in portfolio Q. At optimality, where the elements of PC' match those of PC only if at optimality hPA(n) > hIA(n) and are zero otherwise. We similarly define the elements of SC' based on whether we are selling the asset. If we multiply Eq. (14A.18) by , we find Now focus on an asset n that we purchased, hP(n) > hI(n). For this asset, we expect hQ(n) ≥ hP(n), i.e., if it were not for transactions costs, we would have purchased even more and moved all the way from hI(n) to hQ(n). Therefore, and, once again for purchased assets, Similar arguments hold for sold assets. If we define then Eqs. (14A.20), (14A.21), and (14A.19) imply This is the result in the main text. The other dispersion result we want to derive here concerns the expected dispersion for N portfolios, each with tracking error
Page 416 of ψ relative to a composite portfolio. Assuming that active returns rPA relative to the composite are independent and normally distributed with mean 0 and standard deviation ψ, the probability of observing an active return less than some rPA,max is The probability of observing N independent active returns, each less than rPA,max, is We can therefore solve for the expected (median) rPA,max as Assuming symmetry, we will find a similar result for the expected minimum. Hence as reported in the main text. Exercises 1. Show that the minimum-risk portfolio with factor exposures xp is given by hP = HT · xP, where H is defined in Eq. (14A.4). Recall that a portfolio is diversified with respect to the factor model (X,F,Δ), diversified for short, if it has minimum risk among all portfolios with the same factor exposures. This result says that all diversified portfolios are made up of a weighted combination of the factor portfolios. 2. Show that the optimal specific asset holdings hSP, defined in Eq. (14A.6), have zero exposure to all the factors, i.e., XT · hSP = 0.
Page 417 3. Establish the following: If the benchmark portfolio is diversified and the alphas are benchmarkneutral, then both αCF and αSP are benchmark-neutral. 4. Establish the identities Hint: Recall Exercise 5 in the technical appendix of Chap. 3. 5. Establish the identity 6. Show that the common-factor component of alpha leads to the common-factor holdings, i.e., 2 · λA · V · hCF = αCF, and that the specific component of alpha leads to the specific holdings, i.e., 2 · λA · V · hSP = αSP. This implies the identities 7. This exercise invokes regression to separate the components of alpha. Show that we can calculate the factor alphas αF using a weighted regression, α = X · αF + ∈, with weights inversely proportional to specific variance. The residual of this regression, i.e., ∈, will equal αSP. 8. Suppose we wish to constrain the common-factor exposures to satisfy Q · x = p, where Q is a J by K matrix of rank J. This could constrain some factor exposures and leave others unconstrained. Let p* be the result using the original alpha and an unconstrained optimization, i.e., . Show that the revised alpha will result in a portfolio that satisfies the constraints.
Page 418 9. Consider the optimization Maximize {hT · α – λA · hT · V · h} subject to the inequality constraints b ≤ A · h ≤ d Show that any modified α+, where α+ satisfies 2 · λA · b ≤ A · V-1 · α+ ≤ 2 · λA · d will produce a portfolio that satisfies the inequality constraints. How would you choose α+? 10. Input alphas are cash-neutral if they lead to an active cash position of zero. Show that alphas are cash-neutral if and only if , where hC is the fully invested portfolio with minimum risk. 11. To make the alphas both benchmark- and cash-neutral, modify them as follows: α+ = α – CB · V · hB – CC · V · hC Choose the constants CB and CC to ensure benchmark neutrality, , and cash neutrality, . Why? Applications Exercises For these exercises, you will need alphas from a dividend discount model for all MMI stocks. (Alternatively, you could use alphas from some other valuation model, but it would be useful to have some intuition for these sources of alphas.) 1. Generate the unconstrained optimal portfolio using moderate active risk aversion of λA = 0.10 and the CAPMMI as benchmark. What is the optimal portfolio beta? What are the factor exposures of the optimal portfolio? Discuss any concerns over these factor exposures. 2. Now industry-neutralize the alphas and reoptimize. What are the new factor exposures? Compare the benefits of this portfolio to the previous optimal portfolio. How would you justify an argument that the first portfolio should outperform the second?
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