Portfolio Construction Introduction Implementation is the efficient translation of research into portfolios. Implementation is not glamorous, but it is important. Good implementation can't help poor research, but poor implementation can foil good research. A manager with excellent information and faulty implementation can snatch defeat from the jaws of victory. Implementation includes both portfolio construction, the subject of this chapter (and to some extent the next chapter), and trading, a subject of Chap. 16. This chapter will take a manager's investment constraints (e.g., no short sales) as given and build the best possible portfolio subject to those limitations. It will assume the standard objective: maximizing active returns minus an active risk penalty. The next chapter will focus specifically on the very standard no short sales constraint and its surprisingly significant impact. This chapter will also take transactions costs as just an input to the portfolio construction problem. Chapter 16 will focus more on how to estimate transactions costs and methods for reducing them. Portfolio construction requires several inputs: the current portfolio, alphas, covariance estimates, transactions cost estimates, and an active risk aversion. Of these inputs, we can measure only the current portfolio with near certainty. The alphas, covariances, and transactions cost estimates are all subject to error. The alphas are often unreasonable and subject to hidden biases. The covariances and transactions costs are noisy estimates; we hope that they are
Page 378 unbiased, but we know that they are not measured with certainty. Even risk aversion is not certain. Most active managers will have a target level of active risk that we must make consistent with an active risk aversion. Implementation schemes must address two questions. First, what portfolio would we choose given inputs (alpha, covariance, active risk aversion, and transactions costs) known without error? Second, what procedures can we use to make the portfolio construction process robust in the presence of unreasonable and noisy inputs? How do you handle perfect data, and how do you handle less than perfect data? How to handle perfect data is the easier dilemna. With no transactions costs, the goal is to maximize value added within any limitations on the manager's behavior imposed by the client. Transactions costs make the problem more difficult. We must be careful to compare transactions costs incurred at a point in time with returns and risk realized over a period of time. This chapter will mainly focus on the second question, how to handle less than perfect data. Many of the procedures used in portfolio construction are, in fact, indirect methods of coping with noisy data. With that point of view, we hope to make portfolio construction more efficient by directly attacking the problem of imperfect or ''noisy" inputs. Several points emerge in this chapter: • Implementation schemes are, in part, safeguards against poor research. • With alpha analysis, the alphas can be adjusted so that they are in line with the manager's desires for risk control and anticipated sources of value added. • Portfolio construction techniques include screening, stratified sampling, linear programming, and quadratic programming. Given sufficiently accurate risk estimates, the quadratic programming technique most consistently achieves high value added. • For most active institutional portfolio managers, building portfolios using alternative risk measures greatly increases the effort (and the chance of error) without greatly affecting the result.
Page 379 • Managers running separate accounts for multiple clients can control dispersion, but cannot eliminate it. Let's start with the relationship between the most important input, alpha, and the output, the revised portfolio. Alphas and Portfolio Construction Active management should be easy with the right alphas. Sometimes it isn't. Most active managers construct portfolios subject to certain constraints, agreed upon with the client. For example, most institutional portfolio managers do not take short positions and limit the amount of cash in the portfolio. Others may restrict asset coverage because of requirements concerning liquidity, selfdealing, and so on. These limits can make the portfolio less efficient, but they are hard to avoid. Managers often add their own restrictions to the process. A manager may require that the portfolio be neutral across economic sectors or industries. The manager may limit individual stock positions to ensure diversification of the active bets. The manager may want to avoid any position based on a forecast of the benchmark portfolio's performance. Managers often use such restrictions to make portfolio construction more robust. There is another way to reach the same final portfolio: simply adjust the inputs. We can always replace a very sophisticated (i.e., complicated) portfolio construction procedure that leads to active holdings , active risk , and an ex ante information ratio IR with a direct unconstrained mean/variance optimization using a modified set of alphas and the appropriate level of risk aversion.1 The modified alphas are 1The simple procedure maximizes . The first-order conditions for this problem are . Equations (14.1) and (14.2) ensure that hPA will satisfy the first-order conditions. Note that we are explicitly focusing portfolio construction on active return and risk, instead of residual return and risk. Without benchmark timing, these perspectives are identical.
Page 380 and the appropriate active risk aversion is Table 14.1 illustrates this for Major Market Index stocks as of December 1992. We assign each stock an alpha (chosen randomly in this example), and first run an unconstrained optimization of risk-adjusted active return (relative to the Major Market Index) using an active risk aversion of 0.0833. Table 14.1 shows the result. The unconstrained optimization sells American Express and Coca-Cola short, and invests almost 18 percent of the portfolio in 3M. We then add constraints; we disallow short sales and require that portfolio holdings cannot exceed benchmark holdings by more than TABLE 14.1 Stock Index Weight Alpha Optimal Holding Constrained Optimal Holding Modified Alpha American Express 2.28% –3.44% –0.54% 0.00% –1.14% AT&T 4.68% 1.38% 6.39% 6.18% 0.30% Chevron 6.37% 0.56% 7.41% 7.05% 0.11% Coca-Cola 3.84% –2.93% –2.22% 0.00% –0.78% Disney 3.94% 1.77% 5.79% 5.85% 0.60% Dow Chemical 5.25% 0.36% 5.78% 6.07% 0.22% DuPont 4.32% –1.50% 1.54% 1.67% –0.65% Eastman Kodak 3.72% 0.81% 4.07% 4.22% 0.14% Exxon 5.60% –0.10% 4.57% 4.39% –0.19% General Electric 7.84% –2.80% 0.53% 0.92% –1.10% General Motors 2.96% –2.50% 1.93% 1.96% –0.52% IBM 4.62% –2.44% 3.24% 3.54% –0.51% International Paper 6.11% –0.37% 5.73% 6.15% 0.01% Johnson & Johnson 4.63% 2.34% 7.67% 7.71% 0.66% McDonalds 4.47% 0.86% 5.07% 4.98% 0.14% Merck 3.98% 0.80% 4.72% 4.78% 0.20% 3M 9.23% 3.98% 17.95% 14.23% 0.91% Philip Morris 7.07% 0.71% 7.82% 7.81% 0.12% Procter & Gamble 4.92% 1.83% 6.99% 6.96% 0.44% Sears 4.17% 0.69% 5.57% 5.54% 0.35%
Page 381 5 percent. This result is also displayed in Table 14.1. The optimal portfolio no longer holds American Express or Coca-Cola at all, and the holding of 3M moves to exactly 5 percent above the benchmark holding. The other positions also adjust. This constrained optimization corresponds to an unconstrained optimization using the same active risk aversion of 0.0833 and the modified alphas displayed in the last column of Table 14.1. We derive these using Eqs. (14.1) and (14.2). These modified alphas are pulled in toward zero relative to the original alphas, as we would expect, since the constraints moved the optimal portfolio closer to the benchmark. The original alphas have a standard deviation of 2.00 percent, while the modified alphas have a standard deviation of 0.57 percent. We can replace any portfolio construction process, regardless of its sophistication, by a process that first refines the alphas and then uses a simple unconstrained mean/variance optimization to determine the active positions. This is not an argument against complicated implementation schemes. It simply focuses our attention on a reason for the complexity. If the implementation scheme is, in part, a safeguard against unrealistic or unreasonable inputs, perhaps we can, more fruitfully, address this problem directly. A direct attack calls for either refining the alphas (preprocessing) or designing implementation procedures that explicitly recognize the procedure's role as an "input moderator." The next section discusses preprocessing of alphas. Alpha Analysis We can greatly simplify the implementation procedure if we ensure that our alphas are consistent with our beliefs and goals. Here we will outline some procedures for refining alphas that can simplify the implementation procedure, and explicitly link our refinement in the alphas to the desired properties of the resulting portfolios. We begin with the standard data screening procedures of scaling and trimming.2 2Because of their simplicity, we treat scaling and trimming first. However, when we implement alpha analysis, we impose scaling and trimming as the final step in the process.
Page 382 Scale the Alphas Alphas have a natural structure, as we discussed in the forecasting rule of thumb in Chap. 10: α = volatility · IC · score. This structure includes a natural scale for the alphas. We expect the information coefficient (IC) and residual risk (volatility) for a set of alphas to be approximately constant, with the score having mean 0 and standard deviation 1 across the set. Hence the alphas should have mean 0 and standard deviation, or scale, of Std{α} ~ volatility · IC.3 An information coefficient of 0.05 and a typical residual risk of 30 percent would lead to an alpha scale of 1.5 percent. In this case, the mean alpha would be 0, with roughly two-thirds of the stocks having alphas between –1.5 percent and +1.5 percent and roughly 5 percent of the stocks having alphas larger than +3.0 percent or less than –3.0 percent. In Table 14.1, the original alphas have a standard deviation of 2.00 percent and the modified alphas have a standard deviation of 0.57 percent. This implies that the constraints in that example effectively shrank the IC by 62 percent, a significant reduction. There is value in noting this explicitly, rather than hiding it under a rug of optimizer constraints. The scale of the alphas will depend on the information coefficient of the manager. If the alphas input to portfolio construction do not have the proper scale, then rescale them. Trim Alpha Outliers The second refinement of the alphas is to trim extreme values. Very large positive or negative alphas can have undue influence. Closely examine all stocks with alphas greater in magnitude than, say, three times the scale of the alphas. A detailed analysis may show that some of these alphas depend upon questionable data and should 3There is a related approach to determining the correct scale that uses the information ratio instead of the information coefficient. This approach calculates the information ratio implied by the alphas and scales them, if necessary, to match the manager's ex ante information ratio. The information ratio implied by the alphas is We can calculate this quickly by running an optimization with unrestricted cash holdings, no constraints, no limitations on asset holdings, and an active risk aversion of 0.5. The optimal active portfolio is and the optimal portfolio alpha is (IR0)2. If IR is the desired ex ante information ratio, we can rescale the alphas by a factor (IR/IR0).
Page 383 be ignored (set to zero), while others may appear genuine. Pull in these remaining genuine alphas to three times scale in magnitude. A second and more extreme approach to trimming alphas is to force4 them into a normal distribution with benchmark alpha equal to 0 and the required scale factor. Such an approach is extreme because it typically utilizes only the ranking information in the alphas and ignores the size of the alphas. After such a transformation, you must recheck benchmark neutrality and scaling. Neutralization Beyond scaling and trimming, we can remove biases or undesirable bets from our alphas. We call this process neutralization. It has implications, not surprisingly, in terms of both alphas and portfolios. Benchmark neutralization means that the benchmark has 0 alpha. If our initial alphas imply an alpha for the benchmark, the neutralization process recenters the alphas to remove the benchmark alpha. From the portfolio perspective, benchmark neutralization means that the optimal portfolio will have a beta of 1, i.e., the portfolio will not make any bet on the benchmark. Neutralization is a sophisticated procedure, but it isn't uniquely defined. As the technical appendix will demonstrate, we can achieve even benchmark neutrality in more than one way. This is easy to see from the portfolio perspective: We can choose many different portfolios to hedge out any active beta. As a general principle, we should consider a priori how to neutralize our alphas. The choices will include benchmark, cash, industry, and factor neutralization. Do our alphas contain any information distinguishing one industry from another? If not, then industry-neutralize. The a priori approach works better than simply trying all possibilities and choosing the best performer. 4Suppose that hB,n is the benchmark weight for asset n. Assume for convenience that the assets are ordered so that α1 ≤ α2 ≤ α3, etc. Then define p1 = 0.5 · hB,1 and for n ≥ 2, pn = pn – 1 + 0.5 · (hB,n – 1 + hB,n). We have 0 < p1 < p2 < · · · < pN – 1 < pN < 1. Find the normal variate zn that satisfies pn = Φ{zn}, where Φ is the cumulative normal distribution. We can use the z variables as alphas, after adjustments for location and scale.
Page 384 Benchmark- and Cash-Neutral Alphas The first and simplest neutralization is to make the alphas benchmark-neutral. By definition, the benchmark portfolio has 0 alpha, although the benchmark may experience exceptional return. Setting the benchmark alpha to 0 ensures that the alphas are benchmark-neutral and avoids benchmark timing. In the same spirit, we may also want to make the alphas cash-neutral; i.e., the alphas will not lead to any active cash position. It is possible (see Exercise 11 in the technical appendix) to make the alphas both cash- and benchmark-neutral. Table 14.2 displays the modified alphas from Table 14.1 and shows how they change when we make them benchmark-neutral. In this example, the benchmark alpha is only 1.6 basis points, so TABLE 14.2 Stock Beta Modified Alpha Modified BenchmarkNeutral Alpha American Express 1.21 –1.14% –1.16% AT&T 0.96 0.30% 0.29% Chevron 0.46 0.11% 0.10% Coca Cola 0.96 –0.78% –0.79% Disney 1.23 0.60% 0.58% Dow Chemical 1.13 0.22% 0.20% DuPont 1.09 –0.65% –0.67% Eastman Kodak 0.60 0.14% 0.13% Exxon 0.46 –0.19% –0.20% General Electric 1.30 –1.10% –1.12% General Motors 0.90 –0.52% –0.53% IBM 0.64 –0.51% –0.52% International Paper 1.18 0.01% –0.01% Johnson & Johnson 1.13 0.66% 0.64% McDonalds 1.06 0.14% 0.12% Merck 1.06 0.20% 0.18% 3M 0.74 0.91% 0.90% Philip Morris 0.94 0.12% 0.10% Procter & Gamble 1.00 0.44% 0.42% Sears 1.05 0.35% 0.33%
Page 385 subtracting βn · αB from each modified alpha does not change the alpha very much. We have shifted the alpha of the benchmark Major Market Index from 1.6 basis points to 0. This small change in alpha is consistent with the observation that the optimal portfolio before benchmark neutralizing had a beta very close to 1. Risk-Factor-Neutral Alphas The multiple-factor approach to portfolio analysis separates return along several dimensions. A manager can identify each of those dimensions as either a source of risk or a source of value added. By this definition, the manager does not have any ability to forecast the risk factors. Therefore, he or she should neutralize the alphas against the risk factors. The neutralized alphas will include only information on the factors the manager can forecast, along with specific asset information. Once neutralized, the alphas of the risk factors will be 0. For example, a manager can ensure that her portfolios contain no active bets on industries or on a size factor. Here is one simple approach to making alphas industry-neutral: Calculate the (capitalization-weighted) alpha for each industry, then subtract the industry average alpha from each alpha in that industry. The technical appendix presents a more detailed account of alpha analysis in the context of a multiple-factor model. We can modify the alphas to achieve desired active common-factor positions and to isolate the part of the alpha that does not influence the common-factor positions. Transactions Costs Up to this point, the struggle has been between alpha and active risk. Any klutz can juggle two rubber chickens. The juggling becomes complicated when the third chicken enters the performance. In portfolio construction, that third rubber chicken is transactions costs, the cost of moving from one portfolio to another. It has been said that accurate estimation of transactions costs is just as important as accurate forecasts of exceptional return. That is an over-
Page 386 statement,5 but it does point out the crucial role transactions costs play. In addition to complicating the portfolio construction problem, transactions costs have their own inherent difficulties. We will see that transactions costs force greater precision on our estimates of alpha. We will also confront the complication of comparing transactions costs at a point in time with returns and risk which occur over an investment horizon. The more difficult issues of what determines transactions costs, how to measure them, and how to avoid them, we postpone until Chap. 16. When we consider only alphas and active risk in the portfolio construction process, we can offset any problem in setting the scale of the alphas by increasing or decreasing the active risk aversion. Finding the correct trade-off between alpha and active risk is a one-dimensional problem. By turning a single knob, we can find the right balance. Transactions costs make this a two-dimensional problem. The trade-off between alpha and active risk remains, but now there is a new trade-off between the alpha and the transactions costs. We therefore must be precise in our choice of scale, to correctly trade off between the hypothetical alphas and the inevitable transactions costs. The objective in portfolio construction is to maximize risk-adjusted annual active return. Rebalancing incurs transactions costs at that point in time. To contrast transactions costs incurred at that time with alphas and active risk expected over the next year requires a rule to allocate the transactions costs over the one-year period. We must amortize the transactions costs to compare them to the annual rate of gain from the alpha and the annual rate of loss from the active risk. The rate of amortization will depend on the anticipated holding period. An example will illustrate this point. We will assume perfect certainty and a risk free rate of zero; and we will start and end invested in cash. Stock 1's current price is $100. The price of stock 1 will increase to $102 in the next 6 months and then remain at 5Perfect information regarding returns is much more valuable than perfect information regarding transactions costs. The returns are much less certain than the transactions costs. Accurate estimation of returns reduces uncertainty much more than accurate estimation of transactions costs.
Page 387 $102. Stock 2's current price is also $100. The price of stock 2 will increase to $108 over the next 24 months and then remain at $108. The cost of buying and selling each stock is $0.75. The annual alpha for both stock 1 and stock 2 is 4 percent. To contrast the two situations more clearly, let's assume that in 6 months, and again in 12 months and in 18 months, we can find another stock like stock 1. The sequence of 6-month purchases of stock 1 and its successors will each net a $2.00 profit before transactions costs. There will be transactions costs (recall that we start and end with cash) of $0.75, $1.50, $1.50, $1.50, and $0.75 at 0, 6, 12, 18, and 24 months, respectively. The total trading cost is $6, the gain on the shares is $8, the profit over 2 years is $2, and the annual percentage return is 1 percent. With stock 2, over the 2-year period we will incur costs of $0.75 at 0 and 24 months. The total cost is $1.50, the gain is $8, the profit is $6.50, and the annual percentage return is 3.25 percent. With the series of stock 1 trades, we realize an annual alpha of 4 percent and an annualized transactions cost of 3 percent. With the single deal in stock 2, we realize an annual alpha of 4 percent and an annualized transactions cost of 0.75 percent. For a 6-month holding period, we double the round-trip transactions cost to get the annual transactions cost, and for a 24-month holding period, we halve the round-trip transactions cost to get the annual transactions cost. There's a general rule here: The annualized transactions cost is the round-trip cost divided by the holding period in years.