Active portfolio managers must simultaneously maximise excess returns (over benchmarks), limit risk and observe constraints on, for example, tracking errors (TRs), betas and asset weights.
Determining the range of possible risk and returns attainable by such constrained portfolios is of interest to active portfolio managers. Weight restrictions reduce the range of achievable returns. This work demonstrates the magnitude of these reductions.
This research installs and augments an approach that ascertains the effect on a TR (active) constrained portfolio in absolute risk–return space. The effects are displayed in risk–return space, demonstrating the impact on such constraints.
A theoretical approach to plot the constant TR frontier was used. Theoretical and quantitative analytical approaches to establish changes in the constant TR frontier on a simulated (but highly stylistic) market portfolios were also employed.
Considerable reduction is observed in possible investable portfolios, even for limited asset weight restrictions. This effect is amplified if multiple restrictions are imposed simultaneously, driven by both a reduced area in risk–return space enclosed by the constant TR frontier and changes in the frontier longaxis slope.
The change in the longaxis slope sign is also a feature of changing economic conditions, thereby acting as an early warning signal with associated ramifications for asset managers.
The combined effects on active portfolio performance of TR and asset weight constraints have not been investigated and demonstrated before.
Passive investment attempts to match the performance of certain market indices rather than attempting to outperform these through specific stock assessment and selection. Active investment managers invest in funds whose constituents’ worth are independently assessed, whereas their passive counterparts construct portfolios of market indices in the proportion they are held in the specific index and rebalance these proportions as the market changes.
The goal of active management is to beat the market or outperform agencydetermined benchmarks. These benchmarks are often inefficient, comprising assets with arbitrarily defined upper and lower limits as a result of partial securitisation and free float restrictions. Active investing is generally costlier than passive investing (research analysts and portfolio managers require compensation, and frequent trading also incurs costs), and many active managers do not beat the index after expenses are accounted for. Active managers must also comply with strict tracking error (TE) (the variance of the difference between portfolio and benchmark returns) ceilings, where punitive penalties for noncompliance can be severe (Riccetti
The competition between advocates of active and passive management has, however, recently (2018) intensified. Historically, passive investing has taken preference globally; however, recent evidence identifies a change in this trend (Torr
The success of active portfolio managers is measured relative to a benchmark. The components of these benchmarks are frequently defined by agents who may know little about asset allocation and whose principal interest is often to limit risk through the imposition of strict weight allocation ranges. Asset allocation weights can depend on tax considerations, geographical restrictions (such as foreign exchange limits) or simply agent preferences. Active portfolio components can be under or overweighted relative to the benchmark, but the overall weights can also be negative, that is short positions. These constraints coupled with strict TE limits generally inhibit fund manager performance.
This article contributes by investigating the effect of imposing longonly portfolio component weights on active portfolios subject to TE constraints. In addition, the effect of asset weight ranges on such portfolios is also explored and new constrained frontiers are established as a result. Properties of such frontiers are of considerable interest to active fund managers and investors.
The remainder of this article proceeds as follows: the section ‘Literature review’ explores the relevant literature governing
For the fund to outperform the benchmark, the generation of a positively expected TE is implied (Roll
Roll (
Maxwell et al. (
Maxwell and Van Vuuren (
This catalogue of work involving TEconstrained portfolios – although comprehensive – ignores a fundamental reality of active portfolios: mandated constraints on constituent asset weights. Work that covers this important aspect of active portfolios is limited.
Ammann and Zimmermann (
In this article, we use a stylised example of realistic market data like that employed by BajeuxBesnainou et al. (
To establish the methodologies required for various frontiers, some definitions are first required. This section proceeds by introducing and describing the relevant variables and algebraic components. The mathematics governing the generation of the efficient frontier is then set out, followed by the algebra that defines the TE frontier and then the
Active fund managers are tasked with outperforming specified benchmarks, and the active asset positions they take may or may not be benchmark components (depending on the mandate governing the fund). The algebra required to derive the relevant investment strategy weights uses the same underlying variables, matrices and matrix notation, as defined below.
the vector of benchmark weights for a sample of 

the vector of deviations from the benchmark  
the vector of portfolio weights  
the vector of expected returns  
the vector of benchmark component volatilities  
the benchmark correlation matrix  
the covariance matrix of asset returns and  
the riskfree rate. 
Net short sales
Expected returns and variances are expressed in matrix notation as:
the expected benchmark return  
the variance of benchmark return  
the expected excess return and  
the TE variance (i.e. 
The active portfolio expected return and variance is given
Merton (
Note that deviations from the benchmark are represented by
Minimise
The locus of points in return–risk space is the efficient frontier, not subject to sales constraints, that is shortselling of assets is permitted (as shown in
Efficient frontier and tracking error frontier in mean–standard deviation space. The diamond marker indicates the benchmark on the tracking error frontier.
Maximise
The benchmark – also indicated in
Maximise
The vector of deviations from the benchmark
Jorion (
Constant (unconstrained) tracking error frontier for
We display the entire TE ellipse to demonstrate the domain of all possible TEconstrained portfolios, even the inefficient ones. In our experience, although portfolio managers do not consider portfolios with high risk, given the same return, sometimes the vagaries of market conditions render their portfolios inefficient. Note that each risk–return combination within a TE ellipse (even the benchmark risk and return) has the same TE. The benchmark’s risk–return combination may be achieved via many different constituent weight combinations – not just the unique benchmark configuration. If the benchmark’s risk–return coordinate is within the TE ellipse, a combination of constituent weights exists, which leads to the same risk–return benchmark coordinate. The asset weights will nevertheless be sufficiently different from the benchmark weights to warrant a
The locus of the constant TE ellipse in return–risk space under increasing TEs is informative.
At first localised on the benchmark portfolio position, the ellipse expands (long and short axes increase) as TE increases until the left end coincides with the efficient frontier’s turning point region. Increasing TE still further drags the ellipse back to the right and increasing TE further shifts the frontier so far to the right that the benchmark is eventually excluded, that is it lies outside the ellipse. In such cases, the TE is sufficiently loose to permit a high enough level of risk between the portfolio and the benchmark as to exclude benchmark constituents entirely.
Here we use the results of BajeuxBesnainou et al. (
Consider an inequality constraint
Stylised input data.
Assets  Domestic 
Foreign 


Benchmark weights (%)  14.3  14.3  14.3  14.3  14.3  14.3  14.3 
Annual return (%)  16.7  16.0  14.9  15.8  14.5  12.0  15.3 
Annual volatility (%)  19.0  27.0  22.0  26.0  21.0  27.0  26.0 
Correlation matrix  1  0.3  0.3  0.3  0.3  0.2  0.2 
0.3  1  0.3  0.3  0.3  0.2  0.2  
0.3  0.3  1  0.3  0.3  0.2  0.2  
0.3  0.3  0.3  1  0.3  0.2  0.2  
0.3  0.3  0.3  0.3  1  0.2  0.2  
0.2  0.2  0.2  0.2  0.2  1  0.2  
0.2  0.2  0.2  0.2  0.2  0.2  1 
BajeuxBesnainou et al. (
The objective is to maximise
The loci of the inequality weightconstrained constant TE frontier – as constraints change – are shown in
Unconstrained (long and short positions permissible) constant tracking error frontier for
The data comprised simulated realistic weights, returns, volatilities and correlations for a small, standardised benchmark comprising equal weights in seven assets (five domestic:
We also tested portfolios comprising assets with higher correlations, with correlations <
Portfolio constituents were derived from the universe of investable assets (i.e. domestic and foreign components). Note that the assets which constitute the portfolio in the following examples could be asset
This article followed all ethical standards for research without direct contact with human or animal subjects.
By using the stylised example set out in the section ‘Research design’, comprising domestic and foreign assets (
A common restriction on component weights is that longonly positions are permitted, i.e.
Some results are presented in
The grey dashed line shows the unconstrained constant TE frontier. This is the locus of points in return–risk space defined by the maximum and minimum obtainable annual returns at each level of risk subject to a portfolio TE constraint only (i.e. weights can be positive or negative).
Imposing the inequality
The slope of the ellipse’s long axis also decreases at the asset weight restrictions increase. For the unconstrained constant TE, the slope is positive (and
Sign of slopes of long axes for unconstrained constant
For increasing constraints on
The inclusion of a riskfree asset in the portfolio was also considered. Inverting a variance covariance matrix (VCV) with one (or more) component = 0%, however, results in divisions by 0 and associated intractable mathematical problems, so a small – but nonzero risk – ‘risk free’ security was included.
Allowing for short selling did not change the results either. When the weight restrictions were imposed, portfolio weights were
We also considered multiple constraints, for example, the imposition of weight restrictions on both domestic and foreign securities simultaneously. The universe of possible portfolios shrinks faster – with increasing restrictions – than that observed for restrictions on only domestic assets. Portfolio constraints in the form of weight restrictions reduce the portfolio choice considerably to the point of vanishing relevance as these constraints become more limiting as shown in
Unconstrained (long and short positions permissible) constant tracking error frontier for
Imposing weights constraints reduces the number of investable portfolios because of an overall reduction in TE ellipse region as well as a telescoped efficient arc brought about by a negative longaxis slope (
Impact of weight constraints on investable portfolios. Note the truncated axes.
The Sharpe ratio was calculated for increasing domestic asset weight restrictions between 80% ≤
Sharpe ratios versus annual portfolio risk for
It is interesting to note that, although the
The IR is given by:
BajeuxBesnainou et al. (
Both IR and Sharpe ratios diminish with increasing restrictions. Unconstrained portfolios occupy the largest area in risk–return space, and this space diminishes as constraints are added. The more restrictive the constraint, the smaller the potential investment area and the lower the performance ratio attainable. The IR diminishes to zero (and can become negative for more severe constraints on asset weights) because at these high weight constraints, the excess return over the benchmark approaches 0% (see
Optimising portfolios’ risk and return originated with Markowitz (
This work – by using stylised market data for simulations – developed the constant TE frontier subject to asset weight constraints and established the region in mean–variance space of possible risk–return coordinates for increasingly restrictive boundaries. Unconstrained (i.e. long and short absolute positions permitted) portfolios subject to a TE occupied the largest possible area in risk–return space. Each subsequent restriction diminished this area by shortening both the long and short axes of the constant TE ellipse. This shortening was asymmetrical: the left end of the long axis and the bottom end of the short axis were fixed for increasing restrictions on asset components – thus, these constraints reduce the maximum returns attainable whilst reducing the risk. The range of possible investable portfolios (i.e. the region in risk–return space enclosed by the constant TE frontier) shrinks rapidly and considerably with increasing severity of restrictions, even for relatively small constraints. Combining multiple constraints, such as restrictions on both domestic and foreign weights, amplifies this reduction. The change in the slope of the constant TE frontier’s long axis reduces the range of investable portfolios by shrinking the range of the efficient portfolio set.
At sufficiently ‘severe’ constraints, the benchmark lies outside the realm of possible portfolios. This means that for suitably restrictive constraints, the region of possible risk–return combinations does not embrace a sufficiently large area to include the benchmark. The benchmark and portfolio are considerably different, and acceptable risk–return coordinates for the former do not apply to the latter.
This work used stylised simulation inputs – future work could consider real portfolio data evolving over time to establish how the constant TE frontier changes in crisis periods when also constrained by asset weights.
Normal distributions of returns are assumed in the mean–variance framework; this restrictive assumption could be relaxed in ongoing studies to consider other return distributions, including fattailed ones such as Students’
The authors have declared that no competing interest exists.
Both authors contributed equally to this work.
This research received no specific grant from any funding agency in the public, commercial or notforprofit sectors.
Data sharing is not applicable to this article as no empirical data were used – only simulated data.
The views and opinions expressed in this article are those of the authors and do not necessarily reflect the official policy or position of any affiliated agency of the authors.
Unconstrained because there are no restrictions on the values of the relative asset weights,
The TE frontier is the locus of risk–maximum return points for many different TEs. The constant TE frontier is the locus of points, which encloses all possible risk–return combinations for a single TE. The maximum return obtainable for a given TE on the constant TE frontier is, by definition, also a point on the TE frontier – hence their intersection at the maximum return.
Such a security is not unrealistic.