Are you about to handcuff your information ratio? Received: 14th March, 2006

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1 Are you about to handcuff your information ratio? Received: 14th March, 2006 Renato Staub is a senior asset allocation and risk management analyst for UBS Global Asset management. He joined UBS in 1996 as a quantitative analyst. He has published articles in the Journal of Portfolio Management, thejournal of Investment Management, the Journal of Alternative Investments, and the Journal of Investing. In addition, he has spoken at conferences, including the Q-Group, Barra, Risk Waters Group, the Society of Quantitative Analysis (New York), and the Quantitative Work Alliance for Applied Finance, Education & Wisdom (Chicago). He holds a PhD in finance. Asset Allocation & Currency, Global Investment Solutions, UBS Global Asset Management, One North Wacker Drive, Chicago, IL 60606, USA. Tel: þ ; Fax: þ ; renato.staub@ubs.com Abstract It is well known that short constraints impair portfolio efficiency, and most examinations provide evidence through constrained optimisations. But we think it is paramount to understand first how information must be translated to be in compliance with the fundamental law of active management (Flam). Hence, we simulate first signals and translate them into positions as suggested by Flam. Next, when hitting a constraint, we examine two separate effects that an optimiser does not distinguish: cutting off and subsequent reallocation. Both impair efficiency. Journal of Asset Management (2007) 7, doi: /palgrave.jam Keywords: active management, alpha, constraints, information ratio, long-short investing, security selection Introduction The fundamental law of active management (Flam), a cornerstone of modern finance, states that the information ratio from active management increases with the square root of breadth available to the manager. 1 Although this sounds straightforward, several important issues must be dealt with when putting Flam into practice. For instance, long short investing, Flam s twin brother, may be impaired through short constraints. It is our objective to answer the following questions: How many independent bets are available to a manager? What is the allocation rule for exploiting information as suggested by Flam? That is, how is information translated? What are the implications if the translation is restricted? Our examination starts with what comes first, the signals. Unlike other examinations, we do not simulate alphas directly, as alpha is the result of a translation, and inferior translation impairs alpha. Therefore, it is paramount to understand how information must be translated. In contrast to most studies, we do not optimise, which is why our results are easily replicable. In spite of contrary claims, it is not necessarily the case in practice that optimised results are optimal. Weak optimisers or poor problem specification may trap the result in a local optimum or even move away from any local or absolute optimum. All a reader usually can do is either believe in optimal allocations and their implications or not. While it is not our objective to argue this, we want to understand what is feasible under certain circumstances and what is not. If the optimiser hits a bound, efficiency is impaired as we cannot invest as much in an asset as desired. But there is a second effect through subsequent investing into another, less 358 Journal of Asset Management Vol. 7, 5, & 2007 Palgrave Macmillan Ltd, $3

2 Are you about to handcuff your information ratio? attractive asset that does not touch the bound. We want to capture these effects separately, as there are strong implications from this. Our examination is done in the context of the capital asset pricing model (CAPM), that is we only allow for market bets and bets on idiosyncratic 2 security returns, but not for industry bets or bets on statistical factors, since our point can be made without them. Further, we answer on the basis of theoretical considerations and, where necessary, suitable simulations. After a brief literature overview, we distinguish first between security bets and the market bet, and then examine how many independent bets a manager can really make. Next, we derive the allocation rule as implied by Flam for a single security world and discuss its structure. Then, we extend this rule to a multiple-stock world, and in the following sections, we employ some constraints as typically used in portfolio management. It is our objective to demonstrate to what extent the information ratio can be impaired by statistically unsubstantiated constraints. Finally, in the last section, we summarise the conclusions. Literature Grinold (1989) states that the value added through active management is based on two parameters, the manager s skill and the breadth available to him or her. Later, Jacobs and Levy (1993) portray long short strategies as simply being long a gain and short a loss, leading to double alpha. Michaud (1993), however, thinks long short investing is overstated, as he believes that the unsystematic risks of the long and short portfolio are significantly correlated. Arnott and Leinweber (1994), in turn, and Brush (1997) defy this view. Staying away from this debate, Grinold (1994) shows how idiosyncratic information is best translated into an alpha-generating process. And finally, Grinold and Kahn (2000) as well as Clarke et al. (2002) identify significant efficiency losses due to various constraints. Security bets vs market bet According to CAPM, a security s excess return, that is, its return over cash, is a function of the market s excess return security s exposure to the market and security s independent return. That is: m ¼ b ret m þ e ð1þ where m is the security s excess return, ret m is the market s excess return, b is the security s exposure to the market, and e is the security s idiosyncratic return. From Equation (1), we infer two kinds of bets that a manager can play: a bet on the market and bets on the idiosyncratic returns of securities. Market bet: Managers can go long or short the market. If a portfolio has a beta of 1.13 and is perfectly correlated with the market, the portfolio has a single bet on the market. Since all portfolio positions are proportionally increased vs the policy, there is no security bet. Security bets: Based on his or her expectations, the manager overweights selected securities and underweights certain others. But betting on an individual security entails both a bet on the security s residual return and a bet on the market through the security s beta. As a result, a number of security bets usually incorporate a market bet as well. On the other hand, the portfolio s resulting market over- or underexposure can easily be offset by going short or long via futures. The idea of treating the market bet independently from the security bets through the use of futures is called separation. Without the existence of derivatives (futures, swaps, etc), the market bet and the security bets would be inextricably linked. & 2007 Palgrave Macmillan Ltd, $3 Vol. 7, 5, Journal of Asset Management 359

3 Staub A common perception is that the overall correlation level between securities of a universe determines the amount of independent bets available: that is, the higher the average correlation, the smaller the number of independent bets. But as long as no security can be perfectly replicated by one or several other securities, this perception is wrong. The average correlation level by itself does not say anything about the available amount of independent bets. And since independence of bets is a crucial issue, we must elaborate on this. Assume, for instance, a universe of 100 stocks, where all stocks have for simplicity s sake identical mutual correlations of R (Table 1). A principal components analysis (PCA) tells us how much of the entire correlation structure is explained by the most powerful factor and how much by all remaining factors. While the most powerful factor turns out to explain per cent of the entire correlation structure in case of R ¼ 0.40, the other 99 factors explain 0.60 per cent each. On the other hand, the most powerful factor increases to per cent for R ¼ 0.80, while the other 99 factors decrease to 0.20 per cent. That is, even in a case of R being large, all factors still do explain some volatility, as long as every stock has some idiosyncratic risk and hence cannot be replicated. Therefore, an n n matrix usually entails n bets. The average correlation size drives the share of explanation of the market factor vs the idiosyncratic risks. On the other hand, when a security can be perfectly replicated by one or several other securities, the amount of independent bets is smaller than n. 3 In this case, a PCA would reveal one or more void factors. The most extreme situation is a correlation matrix with all correlations being 1, in which case a PCA reveals one factor only, explaining the entire correlation structure. Clearly, that is extreme and limited to theory. The following figure plots the information ratio as a function of capability and breadth according to Flam, that is: pffiffi IR ¼ r n ð2þ where IR is the resulting information ratio of the portfolio, r the capability (ie the correlation between a security s predicted and realised idiosyncratic return), and n the breadth. (Figure 1) Of course, the portion of explanation from the security s systematic return m sys ¼ bret m ð1aþ vs the portion of explanation from its idiosyncratic return m id ¼ e ð1bþ is not meaningless. If the security s systematic return dominates the total return, and if at r= 0.02 r= 0.01 r= 0.03 Number of Securities r= 0.04 r= 0.05 Figure 1 Information ratio as a function of the size of the stock universe and the capability Table 1 Hypothetical correlation matrix for our stock universe Stock 1 Stock 2? Stock 99 Stock 100 Stock 1 1 R R R Stock 2 R 1 R R Stock 99 R R 1 R Stock 100 R R R Journal of Asset Management Vol. 7, 5, & 2007 Palgrave Macmillan Ltd, $3

4 Are you about to handcuff your information ratio? the same time the systematic return s error margin increases with its portion of the explanation, it may prove difficult to accurately isolate bets on idiosyncratic security returns through market hedging. Exploiting a single security bet At this point, we must determine how to best exploit a single stock s return signal. To that end, assume: There is a stock with idiosyncratic return, e, only (ie its beta equals zero). There is a signal, s. In practice, we may have it as the stock s expected return beyond its systematic return. The stock and the signal have a normal distribution with identical standard deviations. The correlation between the stock and the signal, r, is positive. You are invested 100 per cent in cash. In response to the signal you either buy the stock, or you short it. Since the signal is positively correlated with the stock, you go long (short) the stock if the signal is positive (negative); but by how much? The stock s expected idiosyncratic return, given the signal at time t, s t,is 4 E½e t js t Š¼bs t ¼ r s t ð3þ where b refers to the relationship between the stock and its signal. We call (3) the conditional idiosyncratic return, as it is the return that we expect, given the signal. In Appendix A, we derive the optimal amount to invest in the stock, given the signal, and the resulting information ratio. 5 It turns out that the optimal information ratio is achieved if the amount invested in the stock is proportional to its signal, that is w ¼ C s ð4aþ where C is a constant. Let s call this proportional allocation rule (PAR): The optimal amount to be invested in the stock is proportional to its signal. The reason for this is that the resulting information ratio turns out to be a linear function of the expected value of the product w s, which increases with the correlation between w and s. And if w is proportional to s, the correlation is perfect. The average expected information ratio from PAR turns out to be 6 r IR ¼ pffiffiffiffiffiffiffiffiffiffiffiffi r ð5aþ Apparently, Flam is an approximation (E), and it is only valid in cases with a small r, but this is a weak assumption, as a consistent correlation higher than 0.05 proves unrealistic. As an alternative approach, for reasons of comparison, we consider going long (short) the entire security, if the signal is positive (negative), that is w ¼ signðsþ ð4bþ We call this incremental allocation rule (IAR). The average expected information ratio from IAR turns out to be r IR ¼ pffiffiffiffiffiffiffiffiffiffiffiffi EN r EN ð5bþ where EN is the average absolute deviation of a standard normal distribution. And since EN equals 0.80, that is less than 1, the information ratio for IAR turns out to be smaller than for PAR. 7 While C is a constant with no impact on the information ratio, it does have practical relevance in that it determines the level of aggressiveness, that is the target tracking error (TE). To determine a particular TE, the theory introduces a negative utility that is usually set proportional to the square of the residual risk. The optimum is found as the tangential point between a utility function and the efficient frontier of active management, which is a straight line (Figure 2). 8 Although useful as a didactical tool, utility functions prove not particularly helpful in practice, as true utility cannot be measured. & 2007 Palgrave Macmillan Ltd, $3 Vol. 7, 5, Journal of Asset Management 361

5 Staub Residual Return Utility Functions Residual Risk Optimal Allocation of Residual Risk Figure 2 Optimal level of aggressiveness, as determined by a utility function Further, since the signal, s, is as much negative as positive in terms of both frequency and scale, we conclude that stock selection will not be efficient for portfolios with constrained negative positions. The most familiar constraint limits underweights to the size of the corresponding benchmark positions. Because the size of bets increases with the level of aggressiveness, higher aggressiveness results in hitting the constraints more often and results in a bigger efficiency loss. To sum up this section, we identified the required allocation rule for achieving the information ratio as suggested by Flam. The according rule, PAR, requires the amount invested in a stock to be proportional to its signal. Further, the information ratio resulting from this rule reveals that the standard Flam formula is an approximation. Multiple security bets Next, it is our objective to determine the efficiency of stock portfolios, unconstrained and constrained. As Grinold and Kahn point out, the impact of allocation constraints can usually not be represented by closed-form solutions, as they are inequality constraints. 9 But closed-form solutions do exist, at least for trivial constraints. In all other instances, the alternative way is a simulation. Therefore, we derive first the efficiency of unconstrained and trivially constrained portfolios. In the next step, we simulate portfolios with non-trivial constraints. We assume: A universe of n stocks each stock has a residual risk, s, of 25 per cent; n signals, that is, one signal per stock and a correlation of 0.05 between each stock and its signal. We simulate an active portfolio, most likely it is limited in some form by its benchmark positions, unless it is a hedge fund with a cash benchmark. If we refer to short positions, this denotes an active portfolio. We start with a ten-stock active portfolio and gradually increase it to a 500-stock active portfolio. For all portfolios, we run 1 m simulations. While a bunch of security bets usually results in a market bet, this is of no concern as we can hedge the market exposure with futures. Unconstrained portfolios The efficiency of an unconstrained portfolio can be derived on the basis of Equations (2), (5a), and (5b). PAR long/short The portfolio s alpha increases proportionally to each stock added, while the idiosyncratic stock risks aggregate proportionally to the square root (only) of each stock added. Therefore, the resulting information ratio grows proportionally to the square root of the number of stocks, that is, rffiffiffiffiffiffiffiffiffiffiffiffi n IR ¼ r ð6aþ IAR long/short If we go long (short) the entire stock in case of a positive (negative) signal, the efficiency is, as in the single stock case, reduced by a factor EN vs PAR, that is rffiffiffiffiffiffiffiffiffiffiffiffiffiffi n IR ¼ r EN ð6bþ where EN equals Journal of Asset Management Vol. 7, 5, & 2007 Palgrave Macmillan Ltd, $3

6 Are you about to handcuff your information ratio? Portfolios with zero constraints By zero constraint we mean that the active portfolio must not have short positions. The reason to distinguish between zero constraints and other constraints is that zero constraints are trivial in that they allow a closed-form solution. In practice, zero constraints are usually not an issue, as each strategy underweight means a negative position in the active portfolio. Zero constraints, however, are valuable information, as they mark the lower bound of efficiency. PAR long only Being prevented from making short bets means that only half of all available bets can be played, p ffiffi which reduces efficiency by a factor 2. Hence: rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n IR ¼ r 2ð ð7aþ Þ IAR long only p ffiffi Again, efficiency is reduced by a factor 2, that is rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n IR ¼ r 2ð EN ð7bþ Þ The following figure shows the information ratios for PAR Long/Short and PAR Long only. As we see, these two curves shape a range that, under certain conditions, cannot be left. The efficiency function of constrained portfolios will be positioned within this range. (Figure 3) Short selling constraints Let us assume a benchmark with n stocks where all stocks have identical weights, 1/n. Further, assume no strategy underweight is bigger than its underlying benchmark position, that is, stocks may not be shorted in the strategy portfolio. In other words, minimum positions in the active portfolio are PAR Long/Short PAR Long only 500 Figure 3 Information ratios for unconstrained portfolios and portfolios with zero constraints (r=0.05) constrained as follows: w 1 ð8þ n That is, the bigger the benchmark universe, the bigger n and hence the tighter the short selling constraint in the strategy portfolio. Further, in a more aggressive case, it is more likely to hit the constraints, as this implies larger target positions. Hence, being more aggressive is being more constraint sensitive, and we expect, if everything else remains unchanged, efficiency to decrease in response to an increased benchmark universe. The following figure shows how often the short constraints become binding based on a simulation generating 50 m positions (Figure 4). As mentioned, in case of non-trivial constraints, the solution must be found with simulations. We simulate constraints (8) for different levels of aggressiveness. Thereby, we do not reallocate cut-off exposures to other securities. 11 The resulting efficiency is presented in Figure 5. As no short bet is entirely removed, the resulting efficiency must be bigger than for PAR Long only, that is it remains, as expected, within the range. Efficiency turns out to be particularly sensitive for a large benchmark universe combined with a high tracking error (TE). While a 100-stock universe combined with a 2 per cent TE is subject to an insignificant efficiency drop only due to a short constraint, the impact for a 500-stock universe combined with an 8 per & 2007 Palgrave Macmillan Ltd, $3 Vol. 7, 5, Journal of Asset Management 363

7 Staub Percentage of Binding Constraints 50% 40% 30% 20% 10% 0% Figure 4 Amount of binding short constraints TE = 2% TE = 4% TE = 6% TE = 8% Resulting Tracking Error (RTE) 8% 6% 4% 2% 0% Figure 6 error TE = 2% TE = 4% TE = 6% TE = 8% Resulting tracking error vs target tracking PAR Long/Short PAR Long only TE=2% TE=4% TE=6% TE=8% PAR Long/Short PAR Long only TE = 2% TE = 4% TE = 6% TE = 8% Figure 5 Efficiency of portfolios with short constraints Figure 7 Efficiency, given a full investment constraint cent TE is much bigger; efficiency moves far south, quite close to the lower bound of the range. A measure for the tightness of the constraints, that is for how often they are binding, is the comparison between TE and the resulting tracking error (RTE). The tighter the constraints, the bigger the downward deviation of RTE vs TE, as presented in Figure 6. Enforcing a bigger RTE would require a higher TE than actually targeted. On the other hand, this would further drag efficiency through additional cut-offs. Next, we implement a full investment constraint, which means a balance between long bets and short bets. Or in other words, the long bets are financed by the short bets. We do this by rescaling the long positions such that their sum equals the sum of the short positions. (Figure 7) Because this additional constraint reduces the sum of all long bets, the portfolio alpha decreases further, but this does not Resulting Tracking Error (RTE) 8% 6% 4% 2% 0% TE = 2% TE = 4% TE = 6% TE = 8% Figure 8 Resulting tracking error vs target tracking error in case of a full investment constraint necessarily imply an efficiency decrease. Rather, the enforced balance between long and short exposures is going to dominate the information ratio through a significantly reduced risk. The net effect is a higher efficiency at a lower alpha. As Figure 8 demonstrates, RTE is not only reduced, but in a case of high aggressiveness, it is massively reduced, resulting in an efficiency increase vs portfolios with no full investment constraint. 364 Journal of Asset Management Vol. 7, 5, & 2007 Palgrave Macmillan Ltd, $3

8 Are you about to handcuff your information ratio? Overall, we observe that the standard short constraints alone reduce efficiency perceptibly, although the reduction is not as big as one might expect. Depending on the size of TE, the resulting efficiency is in a range of per cent compared to the efficiency of the unconstrained portfolio. In particular, if full investment constraints enforce a balanced risk, the net impact is mild. To sum up, although the portfolio alpha may be reduced significantly due to possibly tight short constraints, this is not the case for the resulting information ratio. In reference to our title, the resulting information ratio is handcuffed, but the handcuffs are still quite loose. Reallocation What about reallocation of cut-off exposures? In order to answer this question, it is important to know that a binding constraint may impair efficiency in two ways. First, a target position cannot be (fully) established as suggested. And secondly, a subsequent reallocation assigns some other stocks a different exposure than they deserve. Hence, cutting off without re-allocation is more efficient. Let us support this with an example. There are three stocks. Their signals predict residual returns of 8, 1, and 1 per cent, and an efficient allocation of 8, 1, and 1 per cent results in an information ratio of Next, assuming a short constraint of 6 per cent, and hence setting weights of 6, 1, and 1 per cent, results in a reduced information ratio of But if we reallocate the cut-off exposure by setting weights of 6, 2, and 2 per cent, the information ratio drops even further to Obviously, a subsequent reallocation can harm efficiency much more than the cut-off itself. So far, we have not dealt with optimisers, as we could make our point without them. Before reallocation forces us to think about % 0.80% 0.60% 0.40% 0.20% % -0.20% -0.40% Figure 9 First 100 positions, based on a simulation, given an 8 per cent TE, a benchmark universe of stocks and a full investment constraint optimisers, let us consider Figure 9 above. It shows the first 100 positions, produced by a simulation run, given an 8 per cent TE, a benchmark universe of 500 stocks, and a full investment constraint (Figure 9). While the long positions are unconstrained, the short positions are capped at 0.2 per cent (ie, 1/500). As a result of the high level of aggressiveness, they look almost identical to an IAR allocation, while the long portfolio is fully consistent with PAR. Since almost all positions of the short portfolio equal the lower bound, this means that we cannot add or increase negative bets. And at the same time, it prevents us from adding or increasing long bets as well due to the full investment constraint. In such a situation, impairing the diversification of the long portfolio to achieve the target tracking error is all an optimiser can do. That is, the optimiser is forced to reduce efficiency by the tracking error constraint. As it is often difficult to follow an optimiser in detail, in particular because of its tendency to reshuffle portfolios massively for little return, we do not run an optimiser. Another reason is that most optimisers cannot, in spite of their claims, cope with a benchmark universe of 500 stocks; definitely not if it is combined with a target tracking error as a non-linear equality constraint. Instead, we want to assess what the optimiser can achieve at best and at worst when it comes to reallocation of cut-off exposure. & 2007 Palgrave Macmillan Ltd, $3 Vol. 7, 5, Journal of Asset Management 365

9 Staub Exposure Feasible Reallocation (1) Short Exposure before Reallocation (2) Cut-off Exposure (3) Max. Short Exposure (4) Best Case Worst Case Figure 10 Reallocation potential for a portfolio with a TE of 4 per cent. It turns out to be a humped function To that end, we first examine the amount of feasible reallocation, which is determined by two factors. First, we cannot reallocate more than what has been cut off in aggregate; the according amount (function 3 in Figure 10) increases with the size of the benchmark universe. And secondly, reallocation is limited tothegapbetweenthecap((4)infigure10) and the aggregated short exposure after cutting-off but before reallocation ((2) in Figure 10). The gap decreases with the size of the benchmark universe. The amount of feasible reallocation equals the smaller of (i) the total amount cut and (ii) the free room under the cap. It happens to be a humped function that levels off toward the right end. Figure 10 represents this for a TE of 4 per cent. The next observation is that reallocated cut-off exposure cannot provide more alpha per exposure than all the exposure that has not been cut. Otherwise, it would have been allocated differently in the first place. Hence, we assume in the most optimistic case that the reallocated exposure provides the same amount of alpha per exposure as does all the exposure that has not been cut. This allows us to calculate an improved alpha after reallocation. The following figures show the best and worst case for each of our four standard TE scenarios. In the best case, we assume the return equals the improved alpha risk equals TE Figure 11 Best and worst reallocation scenario with a TE of 2 per cent Best Case Worst Case Figure 12 Best and worst reallocation scenario with a TE of 4 per cent Best Case Worst Case Figure 13 Best and worst reallocation scenario with a TE of 6 per cent and in the worst case we assume the return equals alpha risk equals TE Figures show the information ratios resulting from this analysis. Obviously, enforcing TE may result in a massive efficiency loss. In particular, when a high level of aggressiveness combines with tight short constraints as a result of a large 366 Journal of Asset Management Vol. 7, 5, & 2007 Palgrave Macmillan Ltd, $3

10 Are you about to handcuff your information ratio? Best Case Worst Case Target Tracking Error 800% 600% 400% 200% TE = 2% TE = 4% TE = 6% TE = 8% Figure 14 Best and worst reallocation scenario with a TE of 8 per cent 0% Figure Resulting turnover benchmark universe, reallocation causes an outright collapse of the information ratio, since the optimiser has nowhere to go, in which case then ends up adding noise. Only in cases with a moderate TE, is the resulting information ratio acceptable, but that does not come as a surprise, as this is consistent with the smallest difference between TE and RTE. Overall, short constraints combined with reallocation and enforcing a high TE can do much harm. In reference to our title analogy, not only does this imply the existence of handcuffs, but they are tight. Turnover constraints A sole information coefficient does not reveal information about the underlying betting frequency. That is, a security s annualised information ratio may be the result of a single bet per year or four bets per year at a quarterly information ratio that equals half the annual information ratio. 12 Ultimately, as stated by Grinold and Kahn s casino analogy, it depends on how often the game is played. 13 This, however, determines turnover. In this section, we assume annual bets. Provided a position is changed once a year on average, the expected turnover is 14 pffiffiffiffiffi 2n TO ¼ TE EN s ð9þ 125% 100% 75% 50% 25% 0% TE = 2% TE = 4% TE = 6% TE = 8% 500 Figure 16 Implications of a 200 per cent turnover constraint. That is, it is proportional to the tracking error; inversely proportional to the residual stock risk; and proportional to the square root of the universe size. Figure 15 shows the turnover as a function of TE and the universe size. Apparently, turnover may get quite large, even for small TEs. Hence, it is the most crucial parameter directly controlled by the manager. In practice, a binding turnover constraint means that you are prevented from making as many bets as you could. In a scenario of a 500-stock universe and a 4 per cent TE, a turnover of 200 per cent is already achieved with 122 bets. Assuming an information coefficient of 0.05, this suggests an information ratio of 0.37 as compared to 1.12 that could be achieved in case of no turnover constraint (Figure 16). Table 2 summarises the resulting numbers of bets and information ratios for our four & 2007 Palgrave Macmillan Ltd, $3 Vol. 7, 5, Journal of Asset Management 367

11 Staub Table 2 examined levels of aggressiveness in case of a 200 per cent turnover constraint. If the bound becomes a binding constraint, TE and information ratio are related as follows: IR TE ¼ constant ð10þ In other words, alpha is capped by the turnover constraint, and we should try to run the TE as implied by the turnover cap rate. Running a higher TE provides no additional return but increases risk and is a waste of risk budget. Clearly, turnover constraints have the potential to harm efficiency significantly. In short, there is no doubt that unappropriately high turnover constraints imply handcuffs, but the question is how many of them. Combined with short constraints and enforced reallocation, the legs are cuffed as well and there is no way to run. Summary Impact of a 200% turnover constraint Tracking error (%) Bets Information ratio In a market with n securities, there are n bets available, unless some security can be replicated. For achieving an information ratio as implied by Flam, these bets are implemented by following a specific allocation rule. This rule requires the security bets to be proportional to their signals. In practice, the short positions of active portfolios are usually constrained by the size of the positions in the underlying benchmark universe. Binding short constraints combined with a full investment constraint, however, do not impair efficiency significantly, if we do not reallocate. The reason is that both alpha and risk are reduced. On the other hand, if the cut-off exposure is reallocated, the resulting efficiency may be hampered, in particular when an aggressive strategy combines with a large benchmark universe. Reallocation means evading an alpha opportunity and subsequent reallocation is only a partial solution. The alpha opportunity is where it is, and constraints do not change that. It is like unloading some heavy freight from a powerful truck to a weak truck; the latter does not get any stronger just because it was selected as a substitute. Finally, turnover constraints combined with a high level of aggressiveness prove equally undermining. When combined with tight short constraints, opportunities are simply limited. Most opportunity is simply ruled out and enforced reallocation cannot change this. Tight constraints, combined with high aggressiveness, are a contradiction and hence an obstruction. Just like handcuffs. 15 Notes 1. Breadth is defined as the amount of independent bets. 2. Idiosyncratic returns are defined as uncorrelated with the market return and uncorrelated among themselves. 3. In formal terms, this means there are linear dependencies. 4. Since both the stock and the signal have the same standard deviation, there is b ¼ r. 5. See Appendix A.2 Derivation of PAR. 6. Defined as the average expected idiosyncratic return, divided by the average expected idiosyncratic risk. 7. See Appendix A.2 Derivation of Flam. 8. See for instance, Grinold and Kahn (1999), p. 96f. 9. Grinold and Kahn (2000), p. 45. While equality constraints require the constraint to be met exactly, inequality constraints should not be exceeded. 10. For the derivation of EN, see Appendix A.2 Derivation of PAR. 11. If the unconstrained exposure is e and the constraint is c, then the cut-off exposure equals e c. We will deal with reallocation of cut-off exposure in the section Reallocation. 12. Thereby, we assume that alpha increases proportionally to time and risk with the square root. 13. Grinold and Kahn (1999), p For the derivation of formula (9), see Appendix A.3 Derivation of the Turnover. Turnover in our formula is defined as the total amount of dollars bought and sold. 15. Equation (A.9) does not represent a trivial problem, since in usual calculations, allocations are considered deterministic and returns stochastic. However, here we 368 Journal of Asset Management Vol. 7, 5, & 2007 Palgrave Macmillan Ltd, $3

12 Are you about to handcuff your information ratio? deal with the product of two stochastic variables. While Goodman provides the solution to this problem, and the formula is quite complicate, it does not prove relevant to us. The reason is that the correlation between the two factors is only 0.05, that is close to zero. σ =1 References Arnott, R. D. and Leinweber, D. J. (1994) Long/Short Strategies Reassessed, Financial Analysts Journal, 50(5), Brush, J. S. (1997) Comparisons and Combinations of Long and Long/Short Strategies? Financial Analysts Journal, 53(3), Clarke, R., de Silva, H. and Thorley, S. (2002) Portfolio Constraints and the Fundamental Law of Active Management, Financial Analysts Journal, 58(5), Goodman, L. A. (1960) On the Exact Variance of Products, Journal of American Statistical Association, 55(252), Grinold, R. C. (1989) The Fundamental Law of Active Management, The Journal of Portfolio Management, 15(3), Grinold, R. C. (1994) Alpha is Volatility Times IC Times Score, The Journal of Portfolio Management, 20(4), Grinold, R. C. and Kahn, R. (1999) Active Portfolio Management, Probus Publications, Chicago. Grinold, R. C. and Kahn, R. N. (2000) The Efficiency Gains of Long Short Investing, Financial Analysts Journal, 56(6), Jacobs, B. I. and Levy, K. N. (1993) Long/Short Equity Investing, The Journal of Portfolio Management, 20(1), Michaud, R. O. (1993) Are Long/Short Equity Strategies Superior? Financial Analysts Journal, 49(6), APPENDIX A.1 Derivation of the average absolute deviation For a standard normal distribution, the absolute average deviation equals the average of all absolute values of the distribution. That is, we can take the average of each side of the distribution and add their absolute values (Figure A.1). In formal terms, the average absolute deviation, EN, is EN ¼ E½absðxÞŠ ¼2 Z 1 0 x f x dx¼ 0:80 ða:1þ Figure A.1 A.2 Derivation of PAR Assume a security has idiosyncratic risk only. Further, there is a signal, s. Both are distributed normally with identical standard deviations, s. The correlation between the security and the signal is r. Given the signal at time t, s t, the expected idiosyncratic risk is E½e t =s t Š¼bs t ¼ rs t ða:2þ and the conditional idiosyncratic variance is V ¼ s 2 ð Þ ða:3þ If you invest the amount w into the security, the instantaneous information ratio at time t, defined as the expected idiosyncratic return divided by the idiosyncratic risk is IR t ¼ w t E½e t =s t Š pffiffiffiffiffiffiffiffiffi wt 2V r ¼ p s ffiffiffiffiffiffiffiffiffiffiffiffi 0 -Average = 0.40 Average = 0.40 Average absolute deviation w t s t jw t j ða:4þ It can assume two different values only, that is IR t ¼ r s t p s ffiffiffiffiffiffiffiffiffiffiffiffi ða:4 0 Þ where the positive (negative) value results if the signs of w and s t are identical (opposite). But as investments are made over some time horizon, the instantaneous information ratio is not meaningful in practice. The expected average information ratio, defined as the & 2007 Palgrave Macmillan Ltd, $3 Vol. 7, 5, Journal of Asset Management 369

13 Staub expected idiosyncratic return over time divided by the expected idiosyncratic risk over time, is Ew ½ ejsš IR ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi stdðw e Þ r p s ffiffiffiffiffiffiffiffiffiffiffiffi E½w sš s w ða:4 00 Þ Its maximum is achieved if w and s are perfectly correlated. That is, if w is proportional to s w ¼ C s ða:5aþ where C is a constant. In this case, we get r IR PAR ¼ pffiffiffiffiffiffiffiffiffiffiffiffi ða:4aþ On the other hand, in cases of incremental allocation, that is w ¼ signðsþ ða:5bþ the information ratio turns out to be r IR IAR ¼ pffiffiffiffiffiffiffiffiffiffiffiffi EN ða:4bþ A.3 Derivation of the turnover According to our assumptions, the signal is normally distributed with an expected value of 0. PAR requires an allocation proportional to the signal. Hence, and Dw ¼ w t1 w t0 s Dw ¼ p ffiffi 2 sw ða:6þ ða:7þ and since there are n stocks and turnover equals the expected absolute value of s Dw,we get p TO ¼ ffiffi 2 sw EN n ða:8þ Further, there is pffiffi TE s w s i n ða:9þ Finally, if we resolve (A.9) for s w and plug it into (A.8), we get pffiffiffiffiffi 2n TE EN TO ða:8 0 Þ s 370 Journal of Asset Management Vol. 7, 5, & 2007 Palgrave Macmillan Ltd, $3