Investors usually assign part of their funds to asset managers, who are given the task of beating a benchmark. Asset managers usually face a constraint on maximum tracking error volatility (TEV), which is imposed by the risk management office. In the mean-variance space, Jorion, in his 2003 paper "Portfolio optimization with tracking-error constraints", shows that this constraint determines an ellipse containing all admissible portfolios. However, many admissible portfolios have problems in mean-variance terms, for example, because of an overly high variance. To overcome this problem, Jorion also fixes a constraint on variance, while, in their 2008 paper "Active portfolio management with benchmarking: adding a value-at-risk constraint", Alexander and Baptista fix a constraint on value-at-risk (VaR). In this paper, I determine an optimal value for a set of limits composed of the lower limit on TEV, the upper limit on TEV and the upper limit on VaR. To fix the upper limit on VaR, I use the TEV constrained efficient frontier developed in Palomba and Riccetti's 2013 paper "Asset management with TEV and VAR constraints: the constrained efficient frontiers", which is the set of portfolios that is on Jorion's ellipse and not dominated from the mean-variance perspective. In particular, I develop a strategy to impose on asset managers a set of portfolios that contains as many TEV constrained efficient portfolios and as few inefficient portfolios as possible. Moreover, I show that a limit on maximum VaR is usually better than a limit on maximum variance.
Many investors assign part of their wealth to asset managers (eg, mutual funds). These investors choose a benchmark but also ask for active management strategies in order to beat it. The active management approach is very different from the passive one, in which the index fund or the exchanged traded fund (ETF) replicates the benchmark’s composition and performance. Instead, active management creates two requirements: (i) keeping the risk of the portfolio (relatively) close to that of the selected benchmark, and (ii) beating the benchmark and maximizing the investors’ utility moving away from the benchmark.
To meet the first requirement, the risk management office usually imposes a maximum value on tracking error volatility (TEV). However, Roll (1992) shows that such portfolios are generally suboptimal, because they do not belong to the mean–variance frontier (MVF; Markowitz (1952)) and are overly risky.11However, investors often face additional sources of risk, such as those arising from labor income and real estate. These risks are usually called “background risk” (see, for example, Gollier 2001). In the presence of background risk, the optimal portfolio is often mean–variance inefficient (Flavin and Yamashita 2002; Baptista 2008). Indeed, the presence of background risk makes individuals less willing to bear other risks (see, for example, Gollier 2001). Therefore, even if TEV optimization is suboptimal for an investor with mean–variance preferences when background risk is absent, TEV optimization can be optimal for the investor when this risk is present. Baptista (2008) provides conditions under which the use of a mean–TEV objective function by the portfolio manager is optimal from the perspective of an investor who faces background risk. Indeed, the so-called mean–TEV frontier (MTF), that is, the set of portfolios with a given expected return and the smallest TEV, is usually far from the efficient frontier. A portfolio’s efficiency loss is defined as the portfolio’s variance minus the variance of the portfolio on the MVF with the same expected return.22Jorion (2002) provides empirical evidence, based on US stock-based funds, that portfolios subject to a TEV limit end up with a variance higher than their benchmarks.
Some methods try to mitigate the portfolio efficiency loss by imposing limits on the amount of risk that asset managers can take, using different measures of risk: Roll (1992) imposes a limit on beta; Jorion (2003) adds a limit on portfolio variance in a framework with a given TEV (); Alexander and Baptista (2008) impose a limit on value-at-risk (VaR); and Alexander and Baptista (2010) propose a target on ex ante alpha, while still minimizing TEV.
In Palomba and Riccetti (2012), the authors analyze all the interactions among some of the proposed portfolio frontiers. They find an area in the mean–variance space, delimited by the fixed VaR–TEV frontier (FVTF), in which an asset manager can choose portfolios that simultaneously satisfy both TEV and VaR limits and present a trade-off between minimizing relative risk (measured with TEV) and minimizing absolute risk (measured with overall portfolio variance) for any given level of expected return. Moreover, Palomba and Riccetti (2013) draw attention to the mean–variance efficient subset of Jorion’s ellipse (2003), determining the so-called efficient constrained TEV frontier (ECTF). For a more detailed review of these papers, see Section 2.
Concerning the second requirement reported above, Riccetti (2012) proposes applying a lower limit on TEV to force the asset manager to have an active strategy. Indeed, the recent growth of ETFs33At the end of the first 2016 quarter, global ETF industry assets reached a new record of over US$3 trillion (http://www.etf.com). ETFs often take advantage of lower fees, better liquidity and better disclosure compared with mutual funds. has increased the importance of beating the benchmark with an active management focused on the maximization of the utility and, thus, on the reduction of the absolute risk with a constraint on the relative risk, as in Jorion (2003). Indeed, investors (especially those who ask for mutual funds with active management strategies) are not interested in minimizing relative risk, but are interested in maximizing their utility, which is a function of portfolio mean, variance and tracking error, as explained by Chow (1995). This author affirms that, due to the uncertainty of absolute returns, investors may decide to compare the performance against a benchmark, but they are “still concerned with the prospect of losing money”; therefore, they “seek portfolios with high return, low standard deviation and low tracking error”. This idea is supported by empirical evidence: most practitioners use both total and relative risk measures.
Therefore, in this paper, I will propose a consistent set of limits that risk management offices should fix in order to help asset managers achieve the two requirements of investors who choose a mutual fund with an active management strategy.
First of all, I will use the lower limit on TEV developed in Riccetti (2012). Beside the lower limit on TEV, I will set a range of optimum values for the upper limit on TEV, using some TEV properties found by Jorion (2003); I will also focus on the ECTF of Palomba and Riccetti (2013) in order to determine the optimum range for the upper limit on the VaR constraint, with two purposes:
to reduce the number of efficient constrained TEV portfolios that cannot lie among feasible portfolios if other constraints are used;
to reduce the amount of feasible portfolios far from the ECTF that an asset manager can choose.
The proposed framework can be applied irrespective of the method used to estimate the required inputs (the expected return vector and variance–covariance matrix), even if, obviously, the method exploited is very relevant for the out-of-sample performance of the allocation. For a good review of the different methods used to estimate expected returns and the variance–covariance matrix, see, for example, DeMiguel et al (2009).
This analysis is significant both for practitioners and for academics. In the former group, it can be suitable for a risk management office and, indirectly, for the allocation chosen by the asset managers. However, in this paper, I only define useful values for some risk management constraints. I do not select a single “optimal” allocation; that is the task of the asset managers, who can freely choose among the portfolios that comply with the constraints. Therefore, as already said, this paper is also indirectly relevant for asset managers, because a good set of constraints can help asset managers to achieve a good performance, but this performance depends on the exact portfolio choice (which is also due to many other factors, such as the goodness of the estimates of the inputs, as explained above). For a numerical practical application, see Section 5.
Instead, this paper contributes to the academic literature because it performs a clear and explicit comparison of variance and VaR limits in order to select which one should be chosen in various contexts. It also adds a range of optimal values for TEV and VaR constraints, linking the limits in order to keep them compatible.
This paper proceeds as follows. Section 2 reviews the approaches of Jorion (2003), Alexander and Baptista (2008) and Palomba and Riccetti (2013). Section 3 explains how lower and upper limits on TEV should be set, while Section 4 explains how a risk management office should set the upper limit on the VaR constraint, given the upper limit on TEV. Section 5 illustrates the results with a simple practical application. Section 6 shows the analytical solutions of the proposed limits. Finally, Section 7 concludes.
2 Review of some portfolio frontiers
In this section, I review in a nontechnical way (for analytical details, see Sections 6.1, 6.2 and 6.3) some portfolio frontiers that have been proposed in the literature, and which will be useful in this paper. The analysis is conducted in variance–expected return space (), but all figures are presented in the usual standard deviation–expected return space (). I assume that asset returns have a multivariate normal distribution, and that short sales are allowed. I refer to Palomba and Riccetti (2012) for a deeper analytical analysis of these frontiers.
I analyze the choice of an asset manager who faces an optimization with a constraint on maximum TEV. So, feasible portfolios for the asset manager are inside an elliptical (for relatively low TEV values) area that contains the benchmark, as shown by Jorion (2003).44The area is elliptical if it does not “touch” the mean–variance frontier. Moreover, Jorion (2003) fixes the TEV value and not an upper limit on TEV. Fixing the TEV, Jorion (2003) shows extreme cases in which the required TEV is so high that the benchmark is outside the area delimited by the portfolios with the required TEV. However, this is not the case for the maximum TEV analysis. I also call this ellipse the constrained TEV frontier (CTF), as in Palomba and Riccetti (2012).
Moreover, Jorion (2003) shows that, with an upper limit on TEV, in order to avoid overly risky portfolios it is reasonable to choose the portfolio on the CTF at the same level of variance as the benchmark (obviously, the one with the highest expected return, in the upper part of the CTF). This portfolio forces the overall risk (the variance) to be at the same level as the benchmark, and it does not usually have a return that is much smaller than the portfolio on the CTF with the highest expected return, because the top part of the ellipse can be rather flat. Further, Jorion (2003) also characterizes conditions under which this constraint is most useful (for instance, if the benchmark is relatively inefficient).
In Alexander and Baptista (2008), the authors insert a VaR constraint into the framework of Roll (1992). In the () space, the VaR constraint is represented by a line with intercept , where is the VaR limit, and slope , where is the critical value obtained from the inverse cumulative distribution of a standardized normal with confidence level . Portfolios that satisfy the VaR constraint lie on or above the line.
Alexander and Baptista (2008) improve the MTF with the constrained mean–TEV frontier (CMTF): a portfolio is on the CMTF if it satisfies the VaR constraint and there is no other portfolio with the same expected return and a smaller TEV. In Figure 1, CMTF is composed of segment , arc and segment . Then, CMTF reduces the efficiency loss compared with MTF: for a given expected return, CMTF dominates MTF in mean–variance terms, that is, portfolios on CMTF have a smaller or equal variance (graphically, portfolios on CMTF are at the left of portfolios on MTF or, at most, are the same). Moreover, Alexander and Baptista (2008) provide conditions for the VaR limit, such that the portfolio on the CMTF with a given expected return is also on the CTF determined by a given TEV value.
Even if the CMTF can select (for a given expected return) the same portfolio determined by Jorion (2003) (for a given level of TEV and standard deviation), the two frontiers are not directly related, and intersections could not exist; indeed, the VaR limit is not related to the benchmark (and even the benchmark can be out of the portfolios that satisfy the VaR limit), and the ellipse could not contain portfolios that satisfy the VaR limit. Palomba and Riccetti (2012) widely discuss all interactions between the cited frontiers. In the current paper, I select the limits on VaR and TEV in a unified framework that is also able to prevent the risk of two incompatible limits.
Palomba and Riccetti (2012) also emphasize that a VaR limit can improve the portfolio allocation because the constrained optimal portfolios are closer to the MVF, but this could have a cost in terms of TEV; indeed, the constrained optimal portfolios can be further away from the MTF. The authors face this issue in the presence of a TEV limit too. Indeed, they introduce the concept of “fixed VaR–TEV frontier” (FVTF): this is composed of the lines that delimit the area of feasible portfolios for the asset manager (which satisfy both VaR and maximum TEV constraints), and that cannot have, for any given level of expected return, other portfolios with both a smaller variance and a smaller TEV. In practice, this area is inside the ellipse, at the left of the CMTF (in Figure 1, among the left arc on the ellipse, segment , arc and segment ). Portfolios inside FVTF face a trade-off between absolute risk (variance) reduction and relative risk (TEV) reduction; indeed, moving toward the left, the variance is reduced and the TEV is increased, and vice versa. A similar issue holds for the limit on variance.
Palomba and Riccetti (2013) focus on reducing the absolute risk, because, in a classical mean–variance context, it influences the investors’ utility, which is not directly related to TEV. For this reason, the authors introduce the concept of “efficient constrained TEV frontier” (ECTF), defined as the set of portfolios that are on the “constrained TEV frontier” (the ellipse) and are not dominated on the mean–variance criterion.
In the following figures (see, for instance, Figure 2), the ECTF is composed of portfolios on the upper arc , where
is the portfolio on the ellipse that has the minimum variance,
is the intersection between the ellipse and the MTF, that is, the portfolio on the ellipse with the highest expected return.
The ECTF definition is unchanged in spite of the slope of the ellipse, and it is also almost unchanged in spite of the level of the upper limit on TEV. Indeed, the ECTF is always the mean–variance efficient part of the set of portfolios feasible after the TEV constraint imposition. Obviously, the ECTF is not an arc on the ellipse in extreme cases, such as the upper limit on TEV being equal to zero (in which case the only admissible portfolio is the benchmark), or the upper limit on TEV being above a threshold , at which the ellipse is tangent to the Markowitz frontier.
3 Lower and upper limit on TEV
The topic of setting a lower limit on TEV has already been dealt with by Riccetti (2012), who analytically determined the TEV that gives the asset manager the possibility of obtaining an excess return over the benchmark equal to the management fees required from the investors. Indeed, if the asset manager exactly duplicated the benchmark , the investor would obtain a return that would be equal to the benchmark’s return less the fees, with a portfolio that would present the same standard deviation of the benchmark. This could be a relatively inefficient strategy if the investor could replace the passive asset manager with an ETF that would require a lower fee. Therefore, as already explained in the introduction, the large growth of ETFs has increased the importance of beating the benchmark with an active portfolio allocation. Then, the risk management office has to force the asset manager to perform an active strategy to beat the benchmark, imposing a lower limit on TEV.55Riccetti (2012) also faces some peculiar cases, such as very high fees or a benchmark on (or very near) the Markowitz efficient frontier; in the latter case, choosing an ETF that reproduces the benchmark (or a mutual fund with a passive management strategy) is obviously best.
The choice of the upper limit on TEV is free to be made by the risk management office, given the characteristics with which the mutual fund wants to comply, in order to attract investors. However, I can give some insights in order to establish a reasonable range for this constraint.
I distinguish two cases, depending on whether the expected return of the benchmark is above or below the expected return of the global minimum variance portfolio on the Markowitz efficient frontier, which will be called .
In the first case, that is, if the expected return of the benchmark is above the expected return of the global minimum variance portfolio , the upper limit on TEV should not be above a value , found by Jorion (2003), at which portfolio coincides with portfolio . An upper limit on TEV above this value could be dangerous; indeed, Jorion (2003) shows that portfolios with a high TEV are on a line that moves to the right when the TEV increases. In order to reach a very high TEV, the asset manager has to choose a portfolio with a very high overall variance. Obviously, the asset manager is not forced to use all the TEV, but the risk management office has to prevent this possible event. When increases above , the ECTF only enlarges with portfolios that are on the Markowitz frontier with a high variance level; at the same time, a large number of very risky dominated portfolios enter the feasible set of the asset manager. Instead, below a TEV increase makes the ECTF improve, reaching portfolios ever closer to the Markowitz frontier till the TEV value at which the ellipse has a contact point with the Markowitz efficient frontier (analytically found by Jorion (2003)); moreover, between this value and , ECTF touches additional portfolios on the efficient frontier both with lower and higher variance compared with the portfolio at the tangency point.
If is a reasonable maximum for the upper limit on TEV, I now want to set a minimum in order to have a range for the upper limit on the TEV constraint. This minimum should be above the lower limit on TEV for an amount large enough to give asset managers a nonempty and not-too-narrow set of feasible portfolios. Therefore,
In the opposite case, that is, if , the maximum of the range should be enlarged. Indeed, the benchmark could be very inefficient, because it could be far from the efficient part of the Markowitz frontier, that is, the part with a return above . Then, the asset manager can largely improve the benchmark performance with a very active strategy that requires a lot of “bets”; this moves the managed portfolio allocation away from the benchmark, with a consequent relatively high TEV. Therefore, the further the benchmark is from portfolio , the more the maximum of the range should be enlarged. Subsequently, should be multiplied for a number that is an increasing function of both and .
Given that the minimum of the range is unchanged, the range is now
In both cases, if the minimum of the range is above a reasonable level for the maximum of the range, this is a signal that the fees (which determine the value of ) are probably too high.
4 Upper limit on value-at-risk in Jorion’s framework
Given the upper limit on TEV (which determines the positions of portfolios and ), I can set an upper limit on VaR. Palomba and Riccetti (2013) insert the VaR limit in the framework of Jorion (2003). Indeed, instead of constraining the standard deviation to be no more than the standard deviation of a reference portfolio (for instance, the benchmark’s variance as in Jorion (2003)), the VaR could be constrained to be no more than the VaR of the reference portfolio . Palomba and Riccetti (2013) determine the subsection of the ECTF that is available for asset managers after the imposition of the limit on standard deviation or VaR. Moreover, they analyze all the possible relationships among these frontiers, showing under which analytical conditions the VaR or standard deviation cut a different subset of efficient constrained TEV portfolios.
Starting from that analysis, I want to determine an optimal range for the VaR limit. This optimal range has to leave as many efficient constrained TEV portfolios in the set of feasible portfolios for the asset manager as possible and, at the same time, reduce (as much as possible) the number of portfolios that are not efficient. In this discussion, I will also show that the decision to set a limit on the portfolio variance, as in Jorion (2003), is usually less efficient than the decision to set the VaR constraint in the optimal range that I will determine. I will explain all these concepts graphically; I refer to Section 6 for the analytical framework.
First of all, I have to distinguish two cases.
The case in which the VaR of portfolio , , is lower than the VaR of portfolio , , which is usual for the commonly employed confidence level values used for the VaR (above 90%): ;
The opposite and peculiar case in which .
When the VaR of portfolio is lower than the VaR of portfolio ,
the upper limit on VaR has to be set above and below : ;
the previous limit on VaR is “better” than a limit on variance, that is, it can leave more efficient constrained TEV portfolios in the set of feasible portfolios for the asset manager and, at the same time, reduce the number of inefficient ones.
The reason for setting as the minimum value of the range for the upper limit on VaR is that a VaR below this threshold cuts some portfolios near from the ECTF (the arc in Figure 2) or cannot intersect the ellipse, leaving the asset manager with an empty set of admissible portfolios. In other words, does not give asset managers the possibility of working and choosing the part of the efficient subsection of the ellipse with the lowest standard deviation (arc ).
The maximum value of the range for the upper limit on VaR has to be set below or equal to . If I fix the VaR limit at the level of portfolio , that is, the portfolio with maximum expected return of the ellipse, the feasible set of portfolios includes all the efficient constrained portfolios, as shown in Figure 3. Setting a VaR above this value only adds, in the feasible set, portfolios dominated by the portfolios on the ECTF.
Regarding the second issue stated above, it is straightforward to show graphically that the constraint on VaR is “better” than the constraint on portfolio variance. Indeed, the VaR constraint eliminates from the admissible portfolios the inefficient ones in the area among segment , segment and arc in Figure 3. The reason for this improvement is that the VaR constraint is always less inclined than the vertical variance constraint.
However, can sometimes represent quite a high level of risk compared with the risk of a reference portfolio (usually the benchmark). Then, I can distinguish two cases:
the VaR of the benchmark portfolio is above the VaR of portfolio , that is, ;
the VaR of the benchmark portfolio is below the VaR of portfolio , that is, .
The first case does not present the previously explained problem; the maximum level of admitted VaR is even below the VaR of the reference portfolio.
The second case is represented in Figure 4. It is easy to show that a limit on VaR is, again, better than a limit on variance. Suppose, for instance, that the upper limit on variance or the upper limit on VaR is set at the benchmark (portfolio ) level. Admissible portfolios for the asset manager are the following:
portfolios in the left area, between segment and arc , if the risk management office sets an upper limit on TEV and a variance constraint at the benchmark level;
portfolios in the left area, between segment and arc , if the risk management office sets an upper limit on TEV and a VaR constraint at the benchmark level.
A variance constraint cuts off from the feasible portfolios those on arc that are efficient constrained. Instead, a VaR constraint saves efficient constrained portfolios on arc , while it cuts off inefficient ones in the area among segment , segment and arc .
This result is obtained with a positive slope of the ellipse, but it could be generalized to the cases of horizontal or negative slope.
The upper limit on VaR has to be set in the following way: . The use of VaR is preferable to the use of variance with regard to constraining the absolute risk, because it can eliminate several portfolios that are surely not optimal (in the low part of the ellipse) from the feasible set, while inserting portfolios that could be of interest in mean–variance terms. Moreover, in order to give asset managers a not-too-narrow feasible set of portfolios, the upper limit on VaR should be set as follows (see Section 6.6.3):
if , then the optimal choice for risk management is to couple the upper limit on TEV with an upper limit on VaR set to pass through : ;
if , then the optimal choice for risk management is to couple the upper limit on TEV with an upper limit on VaR set at the same level as the benchmark: .
In this way,
compared with Jorion (2003), I improve the set of feasible portfolios, substituting the constraint on variance with the constraint on VaR (reducing or avoiding the removal of efficient constrained portfolios and, at the same time, reducing the number of inefficient portfolios).
This very special case occurs when the slope of the VaR straight line is flatter than the slope of the straight line that passes through points and .66This case occurs if the confidence level used to determine VaR is sufficiently low. Moreover, if the confidence level is not high enough, it could happen that the intercept of the VaR straight line is on the positive side of the -axis. Here, it is not VaR that is a loss by definition. To avoid this possible problem, I define VaR as the “worst expected return” case with probability .
However, this is the only situation that can really create a trade-off between the variance constraint and VaR constraint in terms of efficient constrained portfolios. For example, in Figure 5, the VaR line that passes through the benchmark, compared with the constraint on variance, adds to the feasible set all portfolios between and ; however, it cuts the efficient constrained portfolios with the lowest variance, which are in the arc . Moreover, the VaR constraint cuts off some inefficient portfolios with low variance (under segment and at the left of segment ), but it adds some inefficient portfolios with high variance (above and at the right of ).
It is possible to set the VaR to pass through in order to avoid extremely risky portfolios, but the feasible set could be quite small (between segment and arc in Figure 6); however, the less risky efficient constrained TEV portfolios (arc ) are excluded from the feasible portfolios.
To summarize, in this case, the choice between VaR and variance constraint, as well as the level of the constraint, is a free choice for the risk management office; however, the choice of VaR implies a riskier set of feasible portfolios in variance terms.
5 An empirical application
A short numerical application is reported in order to show the practical simplicity of the method illustrated above.
I use a small portfolio composed of eight of the largest eurozone banking groups: Banco Bilbao Vizcaya Argentaria SA (BBVA), Banco Santander SA, BNP Paribas SA, Crédit Agricole SA, Deutsche Bank AG, ING Group NV, Société Générale SA and UniCredit SpA. The chosen benchmark is the STOXX Europe 600 Banks (which is composed of a subset of forty-eight banking groups included in the STOXX Europe 600 Index). I use daily returns (for simplicity, continuously computed as ) of the last two years (2014 and 2015), with 506 observations for each time series. In the considered period, the eurozone banking sector shows no trend, with a flat evolution, as shown by the benchmark; this presents a daily average return equal to and a median return equal to . However, in the same period, many banks show a high return volatility. Table 1 reports more detailed summary statistics.
I also assume that the fees required by the asset manager are equal to a fixed yearly (daily ), and that the risk management office computes the VaR with a confidence interval .
First of all, I compute the minimum variance portfolio feasible with the eight stocks included in the portfolio. Portfolio presents a daily return equal to , with a standard deviation of . Following Riccetti (2012) (in particular, applying the simplified equation (6.5)), I find that the daily TEV should be above .
Now, I can compute the range for the upper limit on TEV. I observe that the expected return of the benchmark is above the expected return of the global minimum variance portfolio , therefore the upper limit on TEV should not be above the value , found by Jorion (2003), at which portfolio coincides with portfolio . This maximum (equation (6.6)) is equal to . Therefore, the upper limit on TEV should be in the following range: . As already said, is a free choice for the risk management office, given the characteristics with which the mutual fund wants to comply. In this case, I decide to set the upper limit on TEV to be in the middle of the previous range, that is, .
After setting , it is possible to compute the characteristics of the relevant portfolios and , reported in Table 2 together with the benchmark. For mean and variance equations, see Table 4; the VaR can be computed with (6.2) or (6.8) and (6.9); the Sharpe ratio is computed assuming a daily risk-free rate equal to zero (therefore, it is simply the expected return divided by the standard deviation); and the information ratio is calculated as the expected return of the portfolio less the expected return of the benchmark, divided by the TEV.
In order to define the range for the upper limit on VaR, I have to check the VaR of portfolios and . In this case, ; therefore, the minimum of the range for the upper limit on VaR is , and the maximum of the range for the upper limit on VaR is , that is, . Inside this range, observing that , a possible choice for the upper limit on VaR is . Graphically, this case is similar to that reported in Figure 4. Moreover, as already explained, in this case an upper limit on VaR “dominates” an upper limit on standard deviation/variance.
6 Analytical solutions of the limits
To analytically describe all the selected limits, as is usual in the literature, some notation has to be provided.
First of all, the TEV measures the volatility of the deviations of active portfolio returns from the benchmark portfolio returns. Using the following notation,
(vector of benchmark asset weights),
(vector of portfolio asset weights),
(vector of deviations from benchmark),
(variance–covariance matrix of asset returns),
the tracking error variance , also called tracking error volatility (TEV), can be written as
Alexander and Baptista (2008) insert a VaR constraint, which is the maximum loss at a given confidence level that the portfolio can suffer over a period of time. In the usual mean–variance framework, it is assumed that asset returns have a multivariate normal distribution. Then, the portfolio VaR at the confidence level (with ) is
where , is the standard normal cumulative distribution function, is the standard deviation of portfolio returns and is its expected return.77Some academics, such as Maspero and Saita (2005), and some practitioners define as relative VaR (RVaR) the opposite of what I call VaR. Inserting a constraint on maximum VaR in (6.2), I find that
which represents, in the () space, a straight line with intercept and slope . Portfolios that satisfy the VaR constraint lie to the left/above the half-plane generated by the line represented by (6.3).
|Upper limit on TEV|
|Upper limit on VaR, case|
6.3 Efficient constrained TEV frontier (ECTF)
Palomba and Riccetti (2013) define the efficient constrained TEV frontier (ECTF) as the set of portfolios that are on the constrained TEV frontier (that is, on the ellipse if the TEV is not too large) and are not dominated in mean–variance terms. The ECTF is composed of portfolios on the small arc on the ellipse. To describe the two portfolios and , further notation has to be provided: given assets, the -dimensional column vector contains the expected returns (while the squared matrix represents the variance–covariance matrix, as already explained). The following constants are defined: , , and , where is an -dimensional column vector in which each element is 1.
The minimum variance portfolio of the mean–variance frontier has expected return and variance .
All these values are independent from managers’ strategies because they are derived exclusively from the available data.
The benchmark can be defined as the reference portfolio with expected return and variance .
Last, I define
determines the slope of the ellipse: if is positive, the slope is positive, and vice versa.
Now I can determine portfolios and . , which has expected return and variance , is the portfolio of the ellipse that has the minimum variance.
, which has expected return and variance , is the intersection between the ellipse and the MTF, that is, the portfolio on the ellipse with the highest return.
|: ellipse portfolio with minimum variance|
|: ellipse portfolio with maximum return|
6.4 Lower limit on TEV
Riccetti (2012) deals with the topic of analytically determining the minimum TEV that must be requested for asset managers to be considered as actively managing. In particular, the paper requires the minimum effort to beat the benchmark covering the fees requested from the investors. In other words, Riccetti (2012) derives the lower limit on TEV that gives the asset manager the possibility of obtaining a gain equal to the requested fees.
Defining “com” as the amount of fees paid by the investors to the asset manager, the equations that a risk management office has to apply to understand if the asset manager has done enough activity to beat the benchmark are the following.
If , the lower limit on tracking error variance is
Otherwise (that is, if ), the lower limit on tracking error variance is
Even if (6.5) should only be used with an inefficient benchmark (negative , that is, ), as shown by Riccetti (2012), it is much simpler than (6.4). In addition, it always implies a lower limit on TEV smaller than that obtained by (6.4), so it can be set as a lower limit for tracking error variance.
6.5 Upper limit on TEV
6.5.1 Upper limit on TEV: maximum of the range
The upper limit on TEV should not be above a value , where is the value at which portfolio coincides with (the global minimum variance portfolio on the Markowitz efficient frontier), found by Jorion (2003). Jorion analytically derives this value, finding
Further, should be a decreasing function of and an increasing function of . In particular, should be set at 1 if , and above 1 when .
6.5.2 Upper limit on TEV: minimum of the range
The minimum of the range for the upper limit on TEV has to be above the lower limit on TEV defined in Section 6.4.
Therefore, the upper limit on TEV should be in the following range:
6.6 Upper limit on VaR: the case
6.6.1 Upper limit on VaR: minimum of the range
To fix a VaR constraint smaller than causes the loss of the efficient constrained TEV portfolios with the smallest variance. This is due to a trade-off between VaR and standard deviation of returns for the portfolios included in arc , where is the tangency point between the VaR straight line and the CTF (see Palomba and Riccetti (2012) to determine portfolio ). Among these portfolios, presents the minimum standard deviation and the maximum VaR, while has the maximum standard deviation and the minimum VaR; all portfolios inside this arc face a trade-off between VaR and variance reduction. For a discussion on this trade-off, see Palomba and Riccetti (2013) or Alexander (2009).88Substituting the concept of mean–variance efficiency with the concept of mean–VaR efficiency (Alexander 2009), this problem is obviously overcome. The limit for setting the minimum acceptable VaR is , even if in this extreme case the only feasible portfolio is , which is a too-restricted and practically unreasonable space for asset managers’ choices.
Therefore, the minimum of the range for the upper limit on VaR is
6.6.2 Upper limit on VaR: maximum of the range
In general, the maximum of the range for the upper limit on VaR is :
6.6.3 Upper limit on VaR: a possible choice
As explained in Section 4, when selecting the upper limit on VaR, two cases are of interest.
In this case, the optimal VaR level is reported in (6.9), that is, the one that passes through the portfolio with the highest return of the ellipse (). It makes all efficient constrained TEV portfolios feasible, and, compared with the variance constraint at the level, this constraint always cuts off some inefficient portfolios in the lower part of the ellipse.
means that the value of the VaR constraint for the VaR straight line that passes through the benchmark is higher than the value for the VaR straight line that passes through the portfolio with the highest return of the ellipse (portfolio ). The former is equal to
while the latter is in (6.9).
So, the first is higher than the second if
As explained in Palomba and Riccetti (2013), a necessary and sufficient condition for is that the slope of the straight line that passes through and is negative or vertical, or is positive and higher than (that is, ).
This is contrary to the previous case. In this case, it may be better to set the upper limit on VaR at the level (6.10).
6.7 Upper limit on VaR: the case
As explained in Palomba and Riccetti (2013), a necessary and sufficient condition for is that the slope of the VaR straight line is flatter than the slope of the straight line that passes through points and . Since the straight line passing through and has the slope
where and are the standard errors associated with portfolios and , then when .
With a further decomposition of the equation, I also obtain
In this very peculiar case, the risk management office might prefer a limit on variance to a limit on VaR.
Table 5 summarizes the proposed constraints.
|Upper limit on TEV (range)|
|Upper limit on VaR,|
|Upper limit on VaR,|
Asset managers that follow active management strategies usually face an upper limit on the value of TEV; this is imposed by the risk management office to avoid asset managers holding a portfolio that is much riskier than the selected benchmark. In the mean–variance space, Jorion (2003) shows that this constraint determines an ellipse that contains all the admissible portfolios. However, many feasible portfolios are not a good choice for investors in mean–variance terms. For example, many of these portfolios are overly risky. To overcome this problem, Jorion (2003) adds to the TEV constraint a variance constraint, while Alexander and Baptista (2008) add a VaR constraint.
In this context, Palomba and Riccetti (2013) define “efficient constrained TEV frontier” (ECTF) as the set of portfolios that is on the “constrained TEV frontier” (Jorion’s ellipse) and not dominated in mean–variance terms.
In this paper, I choose a strategy that imposes on asset managers a set of (constrained TEV) feasible portfolios, which contains as many efficient constrained portfolios and, at the same time, as few inefficient portfolios as possible. With this aim, the choice of setting a VaR constraint in the Jorion framework is better than the choice of setting a variance constraint; in other words, a good choice for risk management is to fix an upper limit on both TEV and VaR. In particular, the VaR can be chosen in this way:
if the value of the VaR constraint that passes through the benchmark is lower than that which passes through the portfolio with the highest expected return of the ellipse, then a reasonable choice is to set the VaR at the same level as the benchmark’s VaR;
if the value of the VaR constraint that passes through the benchmark is higher than (or equal to) that which passes through the portfolio with the highest expected return of the ellipse, then the optimal VaR is set so that the straight line passes through the portfolio with the highest expected return of the ellipse.
Indeed, on the one hand, the variance constraint keeps several inefficient portfolios in the feasible area in the lower part of the ellipse, while the VaR constraint cuts part of them. On the other hand, the VaR constraint can add some efficient constrained portfolios that a variance constraint might cut. Moreover, in this way, I link the VaR constraint to the TEV constraint; thus, I solve the Alexander and Baptista (2008) problem of a VaR not related to the benchmark.
The only peculiar exception, in which this strategy could be a non-optimal way of fixing constraints for a risk-averse risk management office, is the case in which the slope of the VaR straight line is flatter than the slope of the straight line that passes through points and .
The choice of the optimal VaR limit derives from the choice of the upper limit on TEV. Using a result obtained by Jorion (2003) and the lower limit on TEV calculated by Riccetti (2012), I also determine a range in which to set the upper limit on TEV.
Moreover, as already said, I recall the lower limit on TEV established by Riccetti (2012) to force asset managers to have an active strategy.
All in all, I determine a consistent set of limits that risk management offices should impose on asset managers, composed of
a lower limit on TEV,
an upper limit on TEV,
an upper limit on VaR.
Further developments of this analysis could be the goal of future research. First, this framework can be extended to methodologies, such as the Black–Litterman model, commonly used in the mutual fund industry. Moreover, the most important advancements are (i) studying the combination of TEV, VaR and variance constraints, with constraints on portfolio weights (see, for example, Bajeux-Besnainou et al 2011) as short sales prohibition, and (ii) studying more complex utility functions that also consider higher moments (such as skewness and kurtosis) of the returns distribution.
Declaration of interest
The author reports no conflicts of interest. The author alone is responsible for the content and writing of the paper.
I am grateful to an anonymous referee for helpful comments and suggestions, to Giulio Palomba and to participants in the “ADEIMF 2014 Summer Conference” (organized at Università Cattolica del Sacro Cuore, Milan, September 5–6, 2014) and, in particular, to Paolo Cucurachi.
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