Starting from basic portfolio management, this work provides one with: (1) .... 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 ...
DERIVATIVE AND QUANTITATIVE RESEARCH
In Search of the Perfect Hedge Underlying Ideas in Active Portfolio Decomposition, Hedge Portfolio Selection & Alternative Hedging Methods Anthony Seymour, Florence Chikurunhe and Emlyn Flint
1
SUMMARY This report attempts to answer the question: What underlying portfolio should one use to hedge an active fund? In order to do so properly, one first needs to have a complete understanding of the underlying sources of risk and reward in the fund. To build this understanding, we consider three different decompositions for active return and tracking error respectively. These decompositions focus on different aspects of the portfolio, allowing one to precisely quantify and thus manage the risk and reward sources in the fund. Additionally, one also needs to be able to select a basket of assets that will most accurately replicate those sources of risk and reward whilst simultaneously complying with real-world market constraints. To achieve this, we describe a general mixed integer programming framework that allows one to create a constrained minimum tracking error basket. Finally, one also needs to understand the extent to which hedge mismatch affects the level of protection afforded by a given hedge. This intuition is found by studying how the effectiveness of an index hedge decreases with tracking error, where ‘effectiveness‘ is measured in terms of the change in downside risk measures. Motivated by these three elements, we introduce several alternative hedging methods for the fund manager to implement a better hedge for their active portfolio. In this report, we focus specifically on the use of long-only and long/short custom basket options as a means of creating an appropriate portfolio hedge.
3
CONTENTS Summary 1 1.
Introduction
4
2.
Active Management Fundamentals
4
2.1
An Active Share Interlude
6
2.2
Simulating Active Portfolios
7
3.
Selecting an Appropriate Hedge or Tracking Portfolio 3.1
3.2
9
Return & Risk Decompositions
10
3.1.1 Active Return Decomposition
10
3.1.2 Tracking Error Decomposition
12
A Mixed Integer Programming Approach for Selecting the Hedge Portfolio
14
4.
Hedge Mismatch Error using Standard Index Options
16
5.
Alternative Hedging Methods
18
5.1
Basket Option Pricing, Volatility Skews & Correlation Sensitivity
18
5.2
An Introduction to Long/Short Basket Options
20
6.
Conclusion
References
22 23
4
1.
IN SEARCH OF THE PERFECT HEDGE
INTRODUCTION
In previous Peregrine Securities research, we have studied several aspects of what the ‘optimal’ hedge is for a given portfolio and, more importantly, tried to provide a systematic approach for finding that optimal hedge. We have considered optimisation of the derivative structure used to hedge – futures, put spreads, collars, etc – as well as the characteristics within a given structure – strike and term – across varying market conditions. We have also looked at the more general question of when one should or should not be hedging at all. In this report, we tackle a different aspect of the hedging problem: What is the optimal underlying – as opposed to optimal structure – that one should use to hedge an active fund? As with all questions of this nature, the perfect answer will likely never be found (and if it is, it is not likely to remain perfect for long). Be that as it may, this has not dissuaded us from attempting to provide at least a general solution approach to the question of finding the perfect hedge. The local equity derivative market has become increasingly varied, owing to the advent of the South African Futures Exchange (SAFEX) Can-Do option platform, which allows one to list and thus exchange-trade a myriad of exotic options. In particular, listed options on long-only and long/short custom baskets are now readily available to fund managers as potential hedging instruments. This has led to an explosion of choice in the hedging candidate underlyings available to the fund manager. While many of these candidate underlyings will neither be applicable nor perhaps tradable, there will still be a large number of them that can and should be considered. The goal of this report is to show how one can approach the question of finding the optimal hedge underlying for a given active portfolio. Starting from basic portfolio management, this work provides one with: (1) methods for decomposing and quantifying the sources risk and reward within a portfolio, (2) a framework for selecting the subset of assets that mimic those identified sources, (3) the knowledge to know when the basic index hedge is and is not sufficient and (4) a number of alternative methods for implementing the most appropriate hedge for the portfolio. A lot of the work in this report is not only limited to the hedging problem but can be applied to a wide variety of issues in portfolio management. We trust that this work will not only be of interest and use to participants in the derivative space but also to all participants in the greater investment management field. The remainder of this report is structured as follows. Section 2 outlines the fundamental tenets of active portfolio management and provides a brief discussion on the relationship between tracking error and active share. Based on these fundamentals, a framework is developed which allows one to simulate realistic active portfolios via a constrained brute-force algorithm. Section 3 introduces several novel active portfolio decompositions, which allows one to precisely quantify and therefore manage the risk and reward contributions per active bet. Furthermore, a mixed integer programming approach is presented, which allows one to find the subset of stocks that will most closely replicate a portfolio’s future performance while simultaneously complying with real-world market constraints. Section 4 uses the simulated portfolios from Section 2 to quantify the effect of mismatch error when using standard index options to hedge increasingly active portfolios. This is done by considering how downside risk measures of the hedged returns change with increasing portfolio tracking error. Motivated by these findings, Section 5 suggests several alternative hedging methods for active portfolios that provide significantly greater levels of protection. Technical pricing issues are discussed and alternative hedge examples consisting of a long-only and a long/short basket option respectively are given. Finally, Section 6 provides a brief conclusion.
2.
ACTIVE MANAGEMENT FUNDAMENTALS
At its core, active management is about making decisions: when to buy or sell any given asset and in what quantity. These decisions are made in order to add value to a passive benchmark, be it a nominated index or cash-based rate. In this setting, ‘value’ is usually defined in two ways. The first is by achieving a positive return, or alpha, over and above the nominated benchmark at an acceptable level of risk. The second is by achieving a specified target return at a lower level of risk than that of comparable passive market products.
5
In both cases, the strength of any active decision should be measured by how much value it generates for the fund, conditional on the market and fund constraints faced by the manager at the time. In order to do this rigorously, practitioners generally adopt the framework first articulated by Grinold (1989) and subsequently generalised by Clarke et al (2002) and De Silva et al (2006): the fundamental law of active management, or FLOAM. This framework provides the quantitative links between the interconnected areas of signal generation, portfolio construction and underlying market conditions. In doing so, it describes a holistic approach for active portfolio management, both descriptive and prescriptive. In this report, our focus is on a single aspect of the portfolio construction area: hedging active equity portfolios efficiently. That being said, we will borrow heavily from the FLOAM framework during our analysis in order to generate realistic active equity portfolios, decompose active return and risk, and find a suitable hedging portfolio. Let us start by introducing some general concepts and notation which we use throughout. Assume that there are N stocks in the underlying investment universe and that the excess-to-cash return on any stock in a given period, Ri, is governed by a simple one-factor model:
Rit = βi RMt + rit ,
(1)
where RM is the excess return on the market, βi is the sensitivity to the market return and ri is the residual stock return. The first term above represents the systematic component of the stock return and the second represents the idiosyncratic component. The alpha of stock i at time t is then defined as
αit = 𝔼(rit ).
(2)
Dropping the time subscript for simplicity, portfolio returns are calculated as the weighted sum of underlying stock returns, Rp = ∑wpi Ri , and similarly for a nominated benchmark, Rb = ∑wbi Ri . Active portfolio management is generally concerned with active rather than absolute return and risk though. Given our framework above, we define the relationship between relative returns, ΔR ≡ Rp – Rb , and active returns, Ra as
ΔR = (βp – βb )R M + Ra
(3)
where βp = ∑wpi βi and βb = ∑wb i βi are the market betas of the portfolio and benchmark respectively. In a similar manner, the relationship between relative risk or tracking error, TE, and active risk, sa, is given by
TE2 = (βp – βb )2σM2 + σ a2 ,
(4)
where sM is the market volatility. If one considers the special case where the benchmark portfolio is the market portfolio and the portfolio beta is equal to one, then the relative return is equivalent to the active return and the tracking error is equivalent to the active risk. This is a common simplification used by practitioners and it is one which we will use throughout the remainder of the report. Relative returns can also be defined in terms of active weights, portfolio and the benchmark,
ΔR =
N
∑
wa, i.e. the differences in weight of each stock in the
(wpi – wbi)Ri =
i=1
N
∑w i=1
R .
ai i
(5)
Assuming that both the portfolio and benchmark weights sum to one, the sum of the active weights must be zero by construction. Therefore, one can separate the active portfolio – and thus also the active portfolio return – into an active long portfolio and an active short portfolio of equal weight. Figure 1 illustrates this weight decomposition by graphing benchmark, portfolio, active long and active short weights for a 28-stock portfolio with 3% tracking error to the Top40 Index.
6
IN SEARCH OF THE PERFECT HEDGE
FIGURE 1: WEIGHT DECOMPOSITION OF AN EQUITY PORTFOLIO RELATIVE TO ITS BENCHMARK INDEX 15%
PORTFOLIO WEIGHT
10%
5%
0%
-5% 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42
STOCKS IN UNIVERSE Fund
Active Shorts
Active Longs
Benchmark
From Equation 5, we can also redefine tracking error as
N
TE2 =
N
∑ ∑w i=1 j=1
wajσij ,
ai
(6)
where σij is the covariance of stock i and stock j. In all of the equations above, one can either calculate realised (ex-post) values from historical return series’, or expected (exante) values by using forward-looking risk and return estimates. An important distinction between these two calculations pointed out by Satchell & Hwang (2001) is that portfolio and benchmark weights vary during ex-post calculations, while they are assumed to be fixed during ex-ante estimation. Because of this, ex-post and ex-ante estimates are actually not directly comparable. This is also why ex-post tracking error over a given period is always higher than the ex-ante tracking error estimated at the start of the same period.
2.1 An Active Share Interlude Another measure of benchmark deviation based purely on active weights which has garnered considerable interest of late is Active Share (Cremers & Petajisto, 2009),
1 AS = 2
N
∑|w
|
ai
(7)
i=1
Active Share is bounded between 0 and 1 and is equal to the size of the active long and active short portfolios respectively. A value of 0 represents an index tracking portfolio, while a value of 1 implies that the portfolio only holds nonbenchmark stocks. The interest from managers and consultants alike in Active Share is driven by two features. Firstly, it is considerably simpler to calculate than tracking error and secondly, Cremers & Petajisto (2009) assert that Active Share predicts fund performance by showing that the highest Active Share funds in their tested sample significantly and consistently outperformed their respective benchmarks, and vice versa for the lowest Active Share funds. This combination of simplicity and prediction is indeed compelling.
7
However, several practitioners and academics have recently questioned the validity of using Active Share in a predictive sense (Cohen et al (2014), Schlanger et al (2012)) and have shown that prior reported results suggesting such a conclusion may simply have been driven by the strong correlation between Active Share and the benchmark type (Frazzini et al, 2015). These authors suggest that while positive Active Share is obviously a necessary condition to perform differently to a benchmark, it is by no means a sufficient condition for achieving outperformance. In fact, Frazzini et al (2015) contend that Active Share “is as likely to correlate positively with performance as it is to correlate negatively.” Thus, while active share is clearly a good measure of the amount of active bets that a manager is taking, it may not directly infer any knowledge about the skill underlying the bets taken. Practitioners should therefore remain wary of conflating Active Share with future manager performance. Interestingly, and perhaps of more practical use, Sapra & Hunjan (2013) derive an exact relationship between tracking error and Active Share using the FLOAM framework:
X TE2 = (βp – βb)2 σM2 + AS 2 x +
2π _ 2 σ , N a
(8)
_
where σ a2 is the average residual return variance of all stocks in the universe. Tracking error variance is thus a linear function of squared Active Share, conditional on the given levels of systematic and residual variance. It is clear from Equation 8 that if a fund takes large systematic bets relative to stock-specific bets, then Active Share will play a small role in the total tracking error of the fund.1 However, if one again considers the special case of using the market as the benchmark as well as a portfolio with unit beta, then tracking error variance is directly proportional to squared Active Share. Specifically, the sensitivity of tracking _ error to active share increases for smaller portfolios (smaller ) and during times of greater market dispersion (larger σ a2).
2.2 Simulating Active Portfolios In order to properly analyse the effect of mismatch error when using index options to hedge increasingly active portfolios, one first needs to construct a comprehensive range of increasingly active portfolios. However, creating random realistic active portfolios is not a trivial exercise. While a common approach is to follow the generalised alpha generation process of De Silva et al (2006) based on the market covariance and an assumption of manager skill, this is more suited to finding optimal active weights given a specific portfolio objective. Rather than optimal portfolios, we want to create the total range of possible active portfolios that meet a given set of fund constraints. Therefore, instead of using the alpha signal generation process, we focus directly on active weight generation. In particular, De Silva et al (2006) show that under the FLOAM framework, the optimal unconstrained active weights are normally distributed with zero mean and variance proportional to the active portfolio risk. Using this as a starting point, we generate realistic active portfolios from the following constrained brute-force algorithm: 1. Select an appropriate benchmark (Top40 or Swix40), stock universe (Top40 or Top40 + MidCap stocks) and covariance matrix for the chosen universe. 2. Specify the tracking error target, the allowable cardinality (i.e. number of stocks) range and the maximum individual weight for the active portfolio. 3. Generate active weights from a normal distribution for a randomly selected number of stocks. The mean of the weight distribution fluctuates randomly around zero while the variance of the weight distribution is a function of the specified tracking error target. 4. Scale the generated random weights to match the specified tracking error target within a given tolerance, while obeying the maximum weight and portfolio budget constraints. 5. Repeat steps 3 and 4 until a random portfolio is generated which obeys the given constraints. 6. Repeat steps 3 to 5 to generate a large number of portfolios for a given tracking error target. 7. Repeat steps 2 to 6 across a range of tracking error targets.
1
While we have defined the systematic component above in terms of a single market factor, it is a fairly trivial exercise to show that a similar expression holds when using a multi-factor return model.
8
IN SEARCH OF THE PERFECT HEDGE
TABLE 1: ACTIVE PORTFOLIO SIMULATION VARIABLES & SPECIFIED RANGES Variables
Allowed Values & Ranges
Active Weight
wa = wp – wb ~ N(μa,σ a2)
μa ∈ [-0.025,0.025]
{
0.015 ∀ TE < 0.03 σ a = TE/2 otherwise. Benchmark
Bmk = {Top40,Swix40}
Tracking Error Target
TE ∈ [0.002, 0.004, … , 0.1]
Tracking Error Tolerance
∈ = | 0.025 * TE |
Maximum Weight
wp ≤ 0.15
Cardinality Range
KT40 ∈ [15,42], KT100 ∈ [25,100]
Covariance Estimate
Σ=
No. Portfolios per TE
K = 500
{
Σcurrent Σquiet Σturbulent
Table 1 details all variables within the active weight generation process along with allowed values and ranges for each variable respectively. For the purposes of this report, we limit ourselves to a tracking error range of 0% - 10%, and a stock selection universe of either the top 40 or the top 100 stocks by full market capitalisation on the JSE. The Top40 and Swix40 indices are the only allowable benchmarks so that tracking error targets are specified relative to the available index hedging instruments. The Current covariance matrix is estimated from the most recent three years of daily data, while Turbulent and Quiet covariance estimates are calculated from daily return data back to 1995, partitioned into two regimes based on the South African Turbulence Index.2 Return histories of varying lengths are dealt with by using an initial pairwise correlation and volatility calculation linked with a subsequent correlation eigenvalue filtering function in order to create valid covariance estimates for each regime. As it turns out, the Current and Quiet regimes are very similar in terms of underlying market conditions. Therefore, only the Current and Turbulent regimes are used in the simulations. Note that the Cardinality Range and Maximum Weight limits given in Table 1 are ‘soft’ constraints in that they may be violated due to the nature of our random generation process. That being said, the number and size of these violations is generally quite small and thus of little concern. Figure 2 gives the distributional output from a simulation of 500 active portfolios per TE level from a universe of Top40 stocks using Turbulent market conditions.
2
The SA Turbulence Index is calculated as the Mahalanobis distance using daily historical total return data from the ten South African sector indices. Smoothed daily turbulence scores are calculated and used to classify the market as either turbulent or quiet based on the percentile of the turbulence score. Please see Flint, Seymour & Chikurunhe (2014) for information on the construction of the Turbulence Index.
9
FIGURE 2: SIMULATED ACTIVE PORTFOLIO DISTRIBUTION PERCENTILES ACROSS TRACKING ERROR TARGET RANGE A: Actual Tracking Error 0.025
10%
0.25
0.5
0.75
B: Active Share 0.975
0.025
0.7
0.25
0.5
C: Number of Stocks
0.75
0.975
0.6
8%
0.4
4%
0.3
2%
4%
6%
8%
10%
15 0%
0%
0.5
0.75
2%
4%
6%
8%
10%
0.025
14%
0.25
0.5
0.75
-8%
12%
6%
10%
4%
6%
8%
TRACKING ERROR TARGET
10%
10%
0.25
0.5
0.75
0.975
6%
0% 2%
8%
8%
2%
-12%
0.025
16%
8%
4%
-10%
6%
14%
10%
-6%
4%
F: Maximum Portfolio Weight 0.975
12%
-4%
2%
TRACKING ERROR TARGET
D: Maximum Active Long
0.975
-2%
0%
0%
TRACKING ERROR TARGET
E: Maximum Active Short 0.25
0.975
20
TRACKING ERROR TARGET
0.025
0.75
25
0 0%
0.5
30
0.1
0%
0.25
35
0.2
2%
45 40
0.5
6%
0.025
0%
2%
4%
6%
8%
TRACKING ERROR TARGET
10%
0%
2%
4%
6%
8%
10%
TRACKING ERROR TARGET
Obviously, in order to achieve a low tracking error of 0.2% - the left-most point – one needs to essentially hold the index; this is clearly evident in all the panels of Figure 2. As one increases the tracking error target, the range of possible active portfolios increases in kind. Given our use of the simplifying market benchmark and zero active beta assumptions, one observes that median Active Share increases quadratically with tracking error, in line with the relationship shown by Sapra & Hunjan (2013). Dispersion in the cardinality and weight distributions respectively also largely increases with tracking error although skewness depends on the statistic under review. The maximum number of constituents and the maximum active short weight are bounded by the choice of benchmark and universe respectively. This results in clearly visible distribution limits in Panels C and E. In contrast, additional (soft) bound of +15% are imposed for the maximum portfolio weight, and therefore also for the maximum active long weight. Overall, a randomly generated sample of 500 portfolios for each of the 50 tracking error targets – a total of 25 000 portfolios – gives one relatively smooth and realistic active fund test distributions. Using the constrained brute force algorithm outlined above, we generate similar active funds datasets as portrayed in Figure 2 for each of the eight parameter combinations of benchmark, stock universe and covariance estimate.
3.
SELECTING AN APPROPRIATE HEDGE OR TRACKING PORTFOLIO
A significant proportion of recent Peregrine Securities research has focussed on issues related to the construction of portfolios containing hedging derivatives. Central to this work has been the development of a simulation and optimisation platform which allows one to consider and thus optimise portfolios consisting of multiple underlying asset classes as well as vanilla and exotic derivatives based on a wide range of risk, return and utility criteria. Using this platform, Seymour, Chikurunhe & Flint (2012) outline a systematic approach for determining the optimal equity hedge for a given portfolio in terms of the type of hedging structure (futures, puts, collars, etc.), the characteristics of the structure (strike and term) and the proportion of hedged equity in the portfolio.3
3
Seymour, Chikurunhe & Flint (2015) have since extended this framework to allow fund managers to calculate the optimal currency hedge – futures or optionality – for a given multi-asset portfolio in the presence of non-constant volatility and correlation.
10
IN SEARCH OF THE PERFECT HEDGE
While this approach includes flexibility on several important aspects of the chosen hedging instrument, it does not necessarily account for the fact that active (i.e. non-index) portfolios will always be imperfectly hedged by available index derivatives. To address this, Seymour, Flint & Chikurunhe (2014) showed how fund managers can limit possible negative return contributions from their active positions by using single stock derivative overlays. Furthermore, they introduced the idea of using option structures on a custom basket of stocks to more accurately hedge out active portfolio risk. Using a high TE and a low TE active portfolio, Seymour et al (2014) showed that downside risk was reduced more by using the appropriate basket hedge rather than the approximate index hedge and that this reduction was significantly larger in the case of the high TE portfolio. In this section, we consider the problem of selecting the most appropriate tracking portfolio under real-world trading constraints for a given active portfolio. To be clear then, the ‘tracking’ context used in this section is slightly different to that given in Section 2, which discussed how well an active portfolio tracks its benchmark index. Here, we are looking at finding a subset of the underlying stock universe which tracks the active portfolio, and thus provides one with a more appropriate hedge underlying than the standard index underlying. To emphasis this difference, we will refer to this as the hedge portfolio. In the Seymour et al (2014) example, the two portfolios analysed only comprised the largest 10 stocks in the Swix40. The use of these portfolios was a simplifying assumption that made the choice of hedging portfolio straightforward (use the portfolio itself as the underlying basket) and thus expedited the main objective of their analysis. However, in many cases the choice of hedge portfolio is not straightforward. Real-world portfolios generally have far more than ten holdings, some of which could be fairly illiquid. This makes it difficult for market makers to write derivatives directly on the complete underlying portfolio because of their inability to accurately hedge out their underlying risk for the illiquid constituents. Therefore, in order to create a tradable hedge portfolio, one needs to be cognisant of how the total portfolio nominal and portfolio weights affect the liquidity requirements necessary to hedge out the delta in the underlying stocks. As a rule of thumb, the maximum nominal per stock shouldn’t exceed 25% of the respective average daily volumes traded. Under current market conditions, this translates into a maximum rand nominal of between R50m and R100m. These constraints are general and hold even if one only considers the more liquid, large-cap counters. In essence then, one needs to find the subset of stocks that will most closely replicate the portfolio’s future performance while simultaneously complying with the weight and cardinality constraints imposed by current market conditions. It turns out that there are many ways to solve this problem and we will consider one such method in Section 3.2. Before describing this method though, it is worth first considering the related and more general issue of risk and return decomposition.
3.1 Return & Risk Decompositions Portfolio decomposition is all about understanding what factors a portfolio is exposed to and in what amount. This allows one to understand exactly how each portfolio component affects the whole and thus pinpoint exactly which components are most important. Portfolios can be decomposed in many different ways. In Section 2, we have already extensively discussed weight decomposition relative to a benchmark, illustrated in Figure 1. Here we will consider two further decompositions; namely, active return and tracking error decomposition.4
3.1.1 Active Return Decomposition As described in Section 1, the return of a portfolio is equal to the sum of the constituent stock returns weighted by their respective portfolio weights. Similarly, the relative return – taken as the active return under our simplifying assumptions – is equal to the sum of the constituent returns weighted by their respective active weights. But as Macqueen (2011) points out, this is actually just one of many ways of defining active return: N
∑w
ΔR =
i=1 N
R
ai i
N
∑
∑
= wpi (Ri – Rb ) = wpi ri
=
i=1 N
∑w i=1
4
(9)
i=1
r
ai i
By active return decomposition, we are not referring to the standard performance attribution methodology of Brinson, Hood & Beebower (1985). Rather than focus on their allocation, selection and interaction effects, we consider return decomposition on a stock level.
11
Equation 9 formulates active return in three ways: firstly using active weights and total stock returns, secondly using portfolio weights and residual – or active – stock returns, and thirdly using active weights and active stock returns. Each formulation emphasises different aspects of active portfolio return and while the first is definitely the industry standard, it is likely not optimal in all situations. As an example, consider the 28-stock active portfolio shown in Figure 1. Using expected return estimates from the Current regime, the portfolio has an expected return is 21.71% compared to the benchmark’s 19.81%, giving an expected active return of 1.91%. Using the three active return decompositions given in Equation 9, we calculate stock return contributions and display the different active return contribution paths in Figure 3. The contribution path is a means of visually displaying the size and sign of each stock’s contribution – as per the size and colour of each bar – but in a cumulative manner so that the ending level of the right-most bar represents the sum of all stock contributions; in this case the portfolio’s active return. FIGURE 3: RETURN CONTRIBUTION PATHS FOR DIFFERENT ACTIVE RETURN DECOMPOSITIONS Form 1: Active Weights & Total Returns
4%
ACTIVE RETURN CONTRIBUTION
3% 2% 1% 0% -1% -2% -3% -4% -5% 1
6
11
16
21
PORTFOLIO STOCKS
26
31
36
31
36
31
36
41
Form 2: Portfolio Weights & Residual Returns
4%
ACTIVE RETURN CONTRIBUTION
3% 2% 1% 0% -1% -2% -3% -4% -5% 1
6
11
16
21
PORTFOLIO STOCKS
26
Form 3: Active Weights & Residual Returns
4%
ACTIVE RETURN CONTRIBUTION
3% 2% 1% 0% -1% -2% -3% -4% -5% 1
6
11
16
21
PORTFOLIO STOCKS
26
41
Clearly, all three formulations, denoted Forms 1, 2 and 3 respectively, give the same portfolio active return of 1.91% – see the matching right-hand end points – but the individual stock contributions are significantly different. And this is not simply a scaling issue; stocks with positive contributions in one decomposition can have negative contributions in another. For instance, the return contribution from Stock 6 is negative in the first decomposition, zero in the second and positive in the third.
12
IN SEARCH OF THE PERFECT HEDGE
Because Form 1 uses full stock returns but only active weights, the contribution path highlights how effective all stock selection decisions were relative to the benchmark. Therefore, positive contributions come from either overweighting positive performers or underweighting/excluding negative performers. The former can be thought of as an ‘explicit’ effect, with the latter being an ‘implicit’ effect. In contrast, Form 2 uses full portfolio weights but only active stock returns. This means that the contribution path ignores all non-portfolio stocks – and therefore all implicit contributions as well – highlighting only how effective the manager was at taking positive active bets. Finally, Form 3 uses both active weights and active returns. Because of this, the total range of the contribution path is smaller than those given in Forms 1 and 2. Positive contributions are now harder to attain because both active weight and active return need to be positive. Macqueen (2011) contends that this decomposition is the best and most intuitive because it combines the explicit and implicit active weight effects with the effectiveness of the manager at selecting only positive active bets. The path thus highlights the true contributions of each decision relative to the benchmark weights as well as the underlying return opportunity set.
3.1.2 Tracking Error Decomposition From the active return formulations given above, one can also create different tracking error decompositions
TE = 2
N
N
∑∑w i = 1j = 1 N N
ai
wajsij
∑∑
∼
= wpiwpjs ij
=
(10)
i = 1j = 1 N N
∑∑w w s∼ , i = 1j = 1
ai aj
ij
∼
where s ij is covariance of the active returns on asset i and asset j. This ‘active covariance’ can be calculated in a straightforward manner from the full return covariance matrix, the given benchmark weights and the observation that a stock’s active return, ri , can be written as the linear sum of all full stock returns:
ri = Ri – Rb = (1 – wbi)Ri –
N
∑w
j≠1
bj
Rj .
Because covariance is a bilinear function, the ‘active covariance’ between stocks 1 and 2, function of the full covariance matrix:
∼= s 12
N
N
∑∑w w
i = 1j = 1
1i
(11)
∼ , can be defined as a s 12
s ,
2j ij
(12)
where w.i are taken as the stock-specific weight terms given in Equation 11. Active variance and active correlation are then calculated as usual but using the active covariance as a starting point. What is interesting to note is that a change to a single estimate in the full covariance matrix affects all active covariance estimates to some extent. The three tracking error decompositions above are again equivalent on the portfolio level (at 3.05%) but different on the individual stock level. Figure 4 shows the respective tracking error contribution paths under the three different formulations.5 As a base comparison, realise that equal contributions from all stocks would be graphed as a straight line ending at 3.05%.
5
Meucci (2009) develops a similar tracking error concentration curve as above but calculated using principal portfolios rather than stock positions.
13
Similarly to active return, each tracking error decomposition emphasises different risk aspects of a manager’s active bets, making each contribution path useful in its own right. Form 2 – using portfolio weights and active covariance – ignores non-portfolio stocks and is significantly different from its counterparts. In contrast, Forms 1 and 3 look very similar to each other at first glance, although there are noticeable differences in some contribution values. That being said, these two decompositions react differently to changes in underlying volatilities and correlations. Both decompositions use active weights, meaning that explicit and implicit bets are included in the contributions. However, because Form 3 uses the active covariance matrix, the effect of a single stock volatility or correlation change feeds through to all active covariance estimates, which leads to more stable tracking error contributions. FIGURE 4: TRACKING ERROR CONTRIBUTION PATHS FOR DIFFERENT TRACKING ERROR DECOMPOSITIONS Form 1: Active Weights & Total Returns
TE CONTRIBUTION
3.5% 3.0% 2.5% 2.0% 1.5% 1.0% 0.5% 0.0% 1
6
11
21
PORTFOLIO STOCKS
26
31
36
41
31
36
41
31
36
41
Form 2: Portfolio Weights & Residual Returns
3.5%
TE CONTRIBUTION
16
3.0% 2.5% 2.0% 1.5% 1.0% 0.5% 0.0% 1
6
11
21
PORTFOLIO STOCKS
26
Form 3: Active Weights & Residual Returns
3.5%
TE CONTRIBUTION
16
3.0% 2.5% 2.0% 1.5% 1.0% 0.5% 0.0% 1
6
11
16
21
PORTFOLIO STOCKS
26
In summary, the active return and tracking error decompositions displayed here allow one to precisely quantify and therefore manage the risk and reward contributions per active bet. This can either be done by repositioning the stock portfolio to target a specific profile or by using derivatives to remove undesirable risk and enhance desirable upside potential.
14
IN SEARCH OF THE PERFECT HEDGE
3.2 A Mixed Integer Programming Approach for Selecting the Hedge Portfolio Let us reiterate the problem posed in the introduction of Section 3: how to select an appropriate hedge portfolio for a given active portfolio. While this problem has not been officially addressed in the derivative hedging literature, it has been extensively studied from an index tracking perspective. In this formulation, Beasley (2013) defines the goal of index tracking as identifying the best subset of stocks to hold, as well as their appropriate weightings, in order to replicate the future performance of that index over a given investment horizon. This is nearly exactly the same goal that hedgers have when trying to select the optimal hedging portfolio, except what is needing to be replicated now is the active portfolio rather than the index. Given this similarity, it would seem obvious to consider the approaches used in the index tracking space. One such approach which has gained popularity with the advent of increased computing power and readily available solvers is mixed integer programming. A mixed integer program is one in which some variables are continuous while others take on integer values. This is ideal for setting up a problem in which one chooses a small subset of stocks from the available universe – the integer variables – and then searches for the set of weights – the continuous variables – which minimises an objective function under given a set of constraints. The choice of objective function defines the problem either as linear or nonlinear. In general mixed integer programs can be quite hard to solve unless one can formulate the problem in a very particular way. Thankfully, one can do just this for index tracking problems. Below, we discuss a mixed integer linear programming (MILP) and a mixed integer quadratic programming (MIQP) approach for selecting the hedge portfolio which can be solved fairly easily – albeit slowly – with freely available optimisation toolboxes and heuristic solvers. All results below are produced using the YALMIP toolbox for Matlab (Lofberg, 2004) in conjunction with MOSEK optimisation software. Canakgoz & Beasley (2008) propose an MILP formulation of the index tracking problem which includes transaction costs, a limit on the number of stocks – i.e. a cardinality constraint – and a limit on the total transaction costs incurred – i.e. a turnover constraint. They show that it is possible to view index tracking from a regression standpoint. That is, if a perfect tracking portfolio was regressed against the index, then one would expect to find a beta of 1 and an alpha of 0. Using this insight, Canakgoz & Beasley set up a two-stage MILP formulation that initially solves for a portfolio with unit beta to the index and then solves for a portfolio with zero alpha to the index while maintaining the optimised beta found in the first stage. The major advantage of this approach is that the problem remains linear, meaning that it is computationally easy to solve even for indices with thousands of constituents. As an example, Canakgoz & Beasley show that under realistic transaction cost limits, a portfolio with as few as 70 stocks can be timeously found that replicates the Russell 3000 with a high degree of precision. An alternative approach to that proposed above is to find the constrained portfolio that minimises some return dispersion measure to the index. The most common such measure is the classical tracking error, although many alternative deviation statistics have been proposed in the literature. Cesarone et al (2014) and Xu, Lu & Xu (2015) outline such a MIQP formulation which includes cardinality and weight range constraints. And while this formulation is more difficult to solve than the linear regression MILP problem, due to its special structure it can still be solved for portfolios with several hundreds of variables. In this work, we showcase the MIQP framework because of its ties to the more commonly used tracking error measure, and formulate the MIQP hedge portfolio problem as follows:
minimise s.t.
N
N
∑∑(x – w i = 1j = 1 N
i
pi
)(xj – wpj)sij
∑x = 1
(budget constraint)
∑z = K
(cardinality constraint)
i=1 N i=1
i
i
(13)
l ≤ xi ≤ u, i = 1, ... N (weight constraint) zi ∈ [0,1], i = 1, ... N The xi are the hedge portfolio stock weights, K is the allowed number of stocks, l and u are the lower and upper weight bounds respectively and zi is a Boolean vector identifying which stocks are included in the hedge portfolio. This
15
formulation is fairly general and can easily be modified to consider alternative objectives such as minimum variance or equal-weight optimised hedge portfolios.6 To illustrate the effectiveness of the MIQP approach, consider the problem of finding an optimal Top40 hedge portfolio for varying hedge portfolio sizes. No weight range limits are imposed apart from the usual long-only constraint. Using the formulation above, we can solve this problem to any level of tolerance. Figure 5 graphs hedge portfolio tracking error against number of stocks calculated under Current and Turbulent market conditions respectively, and Figure 6 displays the corresponding hedge portfolio weights calculated under Current market conditions. FIGURE 5: TRACKING ERROR CURVE FOR VARYING HEDGE PORTFOLIO SIZES IN CURRENT AND TURBULENT MARKETS 14% 12%
VOLATILITY
10% 8% 6% 4% 2% 0%
0
5
10
15
20
25
NUMBER OF STOCKS Current
30
35
40
Turbulent
Assume that an optimal hedge portfolio is defined as the smallest subset of stocks that has a tracking error of 1% or lower. Under this assumption, one needs 23 stocks to accurately hedge the Top40 under Current conditions and 27 stocks under Turbulent conditions. Out of interest, if one changed the objective to minimise portfolio variance rather than tracking error, one would only require 10 stocks to achieve this goal under both regimes. FIGURE 6: PORTFOLIO WEIGHTS FOR VARYING HEDGE PORTFOLIO SIZES IN CURRENT MARKETS 1
HEDGE PORTFOLIO WEIGHTS
0,9 0,8 0,7 0,6 0,5 0,4 0,3 0,2 0,1 0
0
5
10
15
20
25
NUMBER OF STOCKS
30
35
40
Although the example shown above may seem somewhat contrived, it is actually a good test of the effectiveness of the mixed integer programming approach. The Top40 Index is arguably more concentrated than most active portfolios would be, making it more difficult to track. Furthermore, testing under Turbulent market conditions – characterised by higher volatilities and correlations – also showcases the robustness of the MIQP approach above. The results are generally very stable across the stock range and, although not shown here, have been tested successfully on active portfolios of up to 100 stocks.
6
One can also include a return constraint in the above program if wanted. An example of such an MIQP problem would be enhanced indexation, where the goal is to achieve a fixed positive excess return relative to the benchmark (Beasley, 2013).
16
4.
IN SEARCH OF THE PERFECT HEDGE
HEDGE MISMATCH ERROR USING STANDARD INDEX OPTIONS
While tracking error provides a well understood measure of divergence between a portfolio and its benchmark index, it is difficult to translate this directly into a measure of hedge mismatch. As discussed in previous Peregrine research (Seymour et al, 2014), one approach to achieve this translation is to compare how the hedged return distribution changes as one considers increasingly active portfolios. In particular, by examining the differences in the risk measures – standard and downside – of the hedged portfolios, one can quantify the relationship between tracking error and hedge mismatch. Knowing this helps one to decide whether a simple index hedge will provide sufficient protection for a given active portfolio. Using the active portfolio simulation framework from Section 2 benchmarked to the Top40, we compare distributional statistics from thousands of active portfolios hedged firstly with a three month 100 Top40 outright put and secondly with a 100-90 Top40 put spread. Hedge distributions are calculated using both Current and Turbulent risk and return assumptions in order to give an indication of how the underlying market conditions affect the severity of the hedge mismatch. We first consider the outright put. Figure 7 displays the three month volatility, Value-at-Risk (VaR) and probability of achieving a negative quarterly return (NegProb) for the outright put hedged portfolios. FIGURE 7: THREE MONTH RISK PERCENTILES FOR OUTRIGHT PUT HEDGED PORTFOLIOS VERSUS TRACKING ERROR Value-at-Risk
Volatility 12%
0.025
0.25
0.5
0.75
0.975
CURRENT REGIME
11%
-2%
0.025
0.75
Pr[Neg Return] 0.975
9%
60%
0.025
0.25
0.5
0.75
0.975
50%
-6%
40%
-8%
8%
30%
-10%
7%
20%
-12%
6% 5%
10%
-14% 1% 2% 3% 4% 5% 6% 7% 8% 9% 10%
1% 2% 3% 4% 5% 6% 7% 8% 9% 10%
1% 2% 3% 4% 5% 6% 7% 8% 9% 10%
TRACKING ERROR
TRACKING ERROR
TRACKING ERROR
Volatility 0.025
0.25
0.5
Value-at-Risk 0.75
0.975
11%
TURBULENT REGIME
0.5
-4%
10%
12%
0.25
-2%
0.025
0.25
0.5
0.75
Pr[Neg Return] 0.975
-4%
10%
0.25
0.5
0.75
0.975
50%
-6%
9%
60%
0.025
40%
-8%
8%
30%
-10%
7%
20%
-12%
6% 5%
10%
-14% 1% 2% 3% 4% 5% 6% 7% 8% 9% 10%
1% 2% 3% 4% 5% 6% 7% 8% 9% 10%
1% 2% 3% 4% 5% 6% 7% 8% 9% 10%
TRACKING ERROR
TRACKING ERROR
TRACKING ERROR
While we do see an increase in Current regime median volatility and volatility dispersion across the full tracking error range, this effect is quite small. Median volatility increases by 1% over a 10% TE range and the 95% volatility range even at the highest tracking error is still only 1.6%. In comparison, median volatility is much higher under the Turbulent regime but remains constant across tracking error. As expected though, volatility dispersion increases significantly, leading to a maximum 95% range of 3.3%. Note that even for a 10% tracking error, the lowest Turbulent volatility percentile remains above the highest Current percentile, emphasising the effect of underlying regime on hedge outcome. In contrast to volatility, which is a symmetric measure of risk, downside risk measures give one a better indication of the effectiveness of a given hedge. Looking at the VaR and NegProb panels, it is clear to see the waning protection that an index option provides for active portfolios. In the Current regime, median VaR decreases sharply from -2.8% to -9.8% across the tracking error range, with the worst case VaR – the ‘VaR-of-VaR’ if you will – being as low as -11.9%. The
17
magnitude of these VaR numbers would likely be too large most managers, especially given the choice of an outright put as the hedging instrument. A similarly worrying picture is shown for Current NegProb, which starts at a median 21% probability of achieving a negative quarterly return for an index-like portfolio and increases to a 41% probability for very active portfolios. There is also a sharp increase in the right skew of the NegProb distribution. The lower percentile starts at 17%, decreases to a low of 12% at 7% tracking error and then increases thereafter to end at 19%. In comparison, the upper NegProb percentile increases sharply and consistently from 25% to a staggering 57%. This is due to the combination of portfolio composition affecting expected portfolio return and higher tracking error affecting hedge efficacy. Finally, note that all NegProb values are increased slightly due to the cost of the outright put. The Turbulent downside risk measures provide an interesting comparison. When markets are turbulent - highly volatile, down-trending and strongly correlated – the median VaR and NegProb are significantly worse for an index-like portfolio, starting at -9.4% and 28% respectively for 1% tracking error. However, in comparison to the Current risk measures, which show significantly declining hedge efficacy, median Turbulent VaR only falls to -11.2% and median NegProb remains constant at 28%. VaR dispersion does increase more in Turbulent markets, with the VaR-of-VaR now reaching a low of -14.1%. Interestingly though, NegProb dispersion is only half that seen in Current markets and the distribution remains symmetric for any tracking error. This is because the general market down-trend increases the initial median NegProb, while the increased market volatilities and correlations decrease NegProb dispersion and skewness. FIGURE 8: THREE MONTH RISK PERCENTILES FOR PUT SPREAD HEDGED PORTFOLIOS VERSUS TRACKING ERROR Value-at-Risk
Volatility 13%
0.025
0.25
0.5
0.75
0.975
CURRENT REGIME
12% 11% 10% 9% 8% 7% 6% 5%
-2%
0.025
0.75
Pr[Neg Return] 0.975
60% 50%
-6%
40%
-8%
30%
-10%
20%
-12%
10%
-14%
0.025
0.25
0.5
0.75
0.975
0%
1% 2% 3% 4% 5% 6% 7% 8% 9% 10%
1% 2% 3% 4% 5% 6% 7% 8% 9% 10%
1% 2% 3% 4% 5% 6% 7% 8% 9% 10%
TRACKING ERROR
TRACKING ERROR
TRACKING ERROR
0.025
0.25
0.5
Value-at-Risk 0.75
0.975
12%
TURBULENT REGIME
0.5
-4%
Volatility 13%
0.25
11% 10% 9% 8% 7% 6% 5%
-2%
0.025
0.25
0.5
0.75
Pr[Neg Return] 0.975
60%
-4%
50%
-6%
40%
-8%
30%
-10%
20%
-12%
10%
0.025
0.25
0.5
0.75
0.975
0%
-14% 1% 2% 3% 4% 5% 6% 7% 8% 9% 10%
1% 2% 3% 4% 5% 6% 7% 8% 9% 10%
1% 2% 3% 4% 5% 6% 7% 8% 9% 10%
TRACKING ERROR
TRACKING ERROR
TRACKING ERROR
Results for the put spread hedged portfolios are given in Figure 8. In most panels, the patterns and value ranges are very similar to the outright put risk statistics. However, there are some differences which are worth highlighting. Firstly, given that the put spread only provides protection over the 90-100 index range, the absolute values of the Turbulent risk statistics are considerably higher than those shown for the outright put. Median volatility, VaR and NegProb percentiles remain fixed around 11%, -17% and 35% respectively. Secondly, put spread VaR distributions are slightly lower for indexlike portfolios than their outright put counterparts but are slightly higher for very active portfolios. Thirdly, put spread NegProb distributions are also slightly lower – a function of the lower relative cost of the put spread – and show less dispersion than the counterpart distributions given in Figure 7 – a function of the limited protection range.
18
5.
IN SEARCH OF THE PERFECT HEDGE
ALTERNATIVE HEDGING METHODS
The most important finding of Section 4 is that, in some instances, index options may not provide sufficient protection for active equity portfolios. If this is the case, then one needs to consider alternative hedging methods that do give one the required level of protection. Focussing on a higher level discussion of hedge appropriateness, one can classify the available hedge methods as follows: 1. 2. 3. 4.
Index-only hedge – the status quo Index hedge plus single stock overlays Custom long-only basket hedge Index hedge plus active long/short basket hedge
Note that this list is given in decreasing order of hedge mismatch but increasing order of complexity. An index-only hedge is simple to implement but leaves the active component of the portfolio unhedged, as shown above. Assuming that options are available on the underlying stocks, an alternative hedge method would be to augment the index-only hedge with single stock overlays. In particular, buy downside protection (i.e. puts) for long active positions and buy upside participation (i.e. calls) for short active positions. Because these positions are supplementary, one can tailor the type and size of hedge based on the respective conviction levels of the underlying bets. That being said, if the goal is to achieve maximum protection for the total portfolio, one is still subject to pricing mismatch between the index and single stock positions as well as trade constraints in the single stock derivative space. The third alternative above attempts to address this by directly considering a hedge on a custom basket of stocks.7 The level of hedge mismatch is precisely defined by the choice of custom basket. Although one does not have the same trade constraints as in the single stock derivative space, remember that all custom basket options are subject to the nominal and liquidity constraints discussed in Section 3. The fourth alternative has only recently become available to the market but is arguably most in line with the FLOAM framework.8 It is the combination of an index hedge plus a custom basket hedge directly on the active long/short portfolio. As with the custom long-only basket hedge, the level of hedge mismatch is completely controllable by the portfolio manager. That being said, it is definitely the most complex of the hedge alternatives in terms of pricing and thus would surely carry with it an additional pricing premium. Obviously, the optimal hedge method will depend on a number of factors, including the level of portfolio activeness, the risk and return objectives and limits of the manager, and the prevailing market conditions at the time of hedging. Seymour et al (2014) discuss several of these issues in the context of single stock overlays and custom long-only basket hedges. In this work, we will consider select aspects of the long-only basket hedge method as well as a general overview of the index plus active long/short basket hedge method.
5.1 Basket Option Pricing, Volatility Skews & Correlation Sensitivity A difficulty associated with pricing basket options is deriving an applicable volatility skew. As in previous work, we consider a simple method commonly used by dispersion trading practitioners (Deng (2008), Avellaneda (2009)) that constructs a basket skew from available single stock and index skews.9 Portfolio volatility is a function of the underlying weights, volatilities and correlations of the underlying constituents. If one considers a traded index as the portfolio then all the underlying weights become fixed. Furthermore, if one uses existing implied volatilities as estimates for the respective index and stock volatility parameters, then the only remaining unknowns are the correlations. Using basic algebra, one can replace the pairwise correlations with a single average correlation parameter and rearrange the − , implied by the current index weights and implied volatilities, s ^: equation to solve for this average correlation, ρ
7 8 9
Such a basket hedge is readily available in the form of SAFEX Can-Do derivatives. While it has been possible to price long/short basket options via Monte Carlo simulation for decades, only recently has the SAFEX trade system allowed one to list a basket option which can have a negative value. While more sophisticated (and thus complex) methods do exist for pricing basket options - see Borovkova et al (2012) and Venkatramanan & Alexander (2011) – the additional accuracy in the calculated volatility skew is not necessary for our illustrative purposes. Furthermore, basket pricing quotes across market makers will generally vary far more than for comparative index quotes making the quest for additional theoretical precision of questionable value.
19
σ 2p
=
N
∑ i=1
ρ− (K,T )
wi σ 2i +
N
N
∑∑w w σ σ ρ
2
i=1 j=1
i j i j
ij
s^ 2l (K,T ) – Σwis^ 2i (K,T ) _____________________ = 2ΣΣwiwjs^ i (K,T )s^ j (K,T )
(14)
Note that is a function of strike and term because the underlying implied volatilities are functions of strike and term. The Chicago Board Options Exchange (CBOE) uses Equation 14 with at the money options to publish two implied average correlation indices – short-term and long-term – for the S&P 500 Index. According to the CBOE, these indices offer insight into the amount of forward-looking diversification priced into index options relative to individual single stock options. With an estimate of the average implied correlation surface calculated from available market indices – such as the Top40 and Swix40 – one can construct a volatility skew for any custom basket by combining this with the known single stock implied volatilities skews. This construction method can be deemed market-consistent because the constructed volatility skew of an index-tracking portfolio will exactly match the traded index volatility skew. In other words, the model has been calibrated to match the only basket options that actively trade in the market. As we have shown in prior work (Flint, Seymour & Chikurunhe, 2015), the largest contribution to total index volatility generally comes from correlation rather than outright volatilities. However, as we have also recently discussed, correlations are uncertain and can vary significantly over the life of the option. From a pricing perspective then, this means that market makers will either add a correlation risk premium to a given theoretical price, or will use a higher correlation assumption when pricing via simulation. To build intuition on how the price of a basket option changes with implied correlation – this sensitivity is sometimes called ‘rega’ or ‘correlation delta’ – consider an at the money put option written on the current Top40 basket. Figure 9 graphs option premium against implied correlation shifts for options terms of up to a year. There is a significant positive relationship between option premium and implied correlation which becomes stronger with term. Premium is also very slightly concave across changes in implied correlation, indicating a negligible (negative) second-order effect. FIGURE 9: AT THE MONEY BASKET PUT OPTION PREMIUM ACROSS IMPLIED CORRELATION SHIFTS 9%
OPTION PREMIUM
8% 7% 6% 5% 4% 3% 2% 1%
-0.3
-0.2
-0.1
3 Month Put
0
-0.1
-0.2
CHANGE IN IMPLIED CORRELATION 6 Month Put
9 Month Put
-0.3
12 Month Put
Although Figure 9 gives an indication of how correlation affects a single option strike across multiple terms, it does not consider how correlation sensitivity changes across strikes. To analyse this, we construct a measure of correlation sensitivity by calculating the percentage change in option premium for a 1% shift in implied correlation. Figure 10 graphs this correlation sensitivity across moneyness (strike over spot) for option terms of up to a year. Correlation sensitivity peaks around the 100-110% moneyness range depending on option term and decays as one goes further out the money, particularly on the right-hand side. The increase in correlation sensitivity with respect to term is now also clearly evident.
20
IN SEARCH OF THE PERFECT HEDGE
FIGURE 10: BASKET PUT OPTION SENSITIVITY TO CORRELATION ACROSS MONEYNESS
% PRICE CHANGE FOR A 1% CORRELATION SHIFT
8% 7% 6% 5% 4% 3% 2% 1% 0%
75%
80%
85%
90%
95%
100%
105%
110%
115%
120%
125%
MONEYNESS 3 Month
6 Month
9 Month
12 Month
Having an understanding of how correlation affects basket option pricing provides one with useful intuition of how much diversification potential is being assumed in the basket. Given that the example chosen above reproduces the index skew for zero implied correlation shifts, it also gives an approximate indication of how a basket option’s price is likely to differ from readily available index pricing, based on the correlation between the basket and index.
5.2 An Introduction to Long/Short Basket Options As discussed throughout this report, the most intuitive return decomposition of an active portfolio is into an index component and an active component, which itself can be further decomposed into an active long component and an active short component of equal size. An index hedge removes the index component of the portfolio, leaving only the active long and active short components. The greatest relative risk then faced by the manager is that the gain from the active long component is offset by a loss in the active short component. In this case, the most elegant hedging instrument would be one which pays out the difference in value between the active long portfolio and the active short portfolio, thus ensuring a return greater than or equal to the index. Such an instrument exists and is called an outperformance option. This is because one is taking a view on the relative outperformance of one asset versus another. In the classical form, the payoff is simply a function of the return differential between two assets. However, if one extends this basic payoff to include a strike level K, then this new instrument is referred to as a spread option and has a terminal payoff of [ω (S1 – S2 –K)]+, ω = 1 where for a call option ω = –1 and for a put option. Outperformance options are then simply defined as spread options with zero strike. Another way to think of a spread option though is as a long/short basket comprising only two assets with weights of +100% and -100% respectively. Using this alternative definition, the link between spread options and basket options on the long/short active components becomes clear. If one thinks of the active long and active short components as two separate assets, then a simple method for pricing and understanding the multi-asset long/short active basket option is to re-write it as a two-asset spread option. A well-known closed-form approximation for pricing European spread options on futures is given by Kirk (1996).10 As an illustrative example, consider a 25-stock portfolio with a tracking error of 5% to the Top40. The portfolio has an active share of 48.3%, equivalent to the nominal size of the active long and active short portfolios respectively. The active long portfolio consists of 23 stocks and the active short portfolio comprises 19 stocks. Volatility skews are created for each portfolio using the method discussed in Section 5.1 and the implied average correlation skew is used as a measure of the correlation between the two active portfolios. In order to use Kirk’s approximation, one needs to impose a strike convention for choosing the appropriate active volatility values from the complete skew. In this example, we set the selected strikes from each active skew to be equal to the given spread option strike.
10 See Venkatramanan & Alexander (2011) for a more accurate – and complex – alternative pricing method.
21
FIGURE 11: GREEKS FOR A THREE MONTH SPREAD OPTION PUT WRITTEN ON THE ACTIVE LONG AND ACTIVE SHORT COMPONENTS FROM A 5% TRACKING ERROR PORTFOLIO Delta
1 0.5 0 -0.5 -1.0 75
80
85
90
95
100
STRIKE
105
110
115
120
125
105
110
115
120
125
105
110
115
120
125
105
110
115
120
125
Gamma & Cross Gamma
0.06 0.04 0.02 0 -0.02 -0.04 -0.06
75
80
85
90
95
100
STRIKE Vega
0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0
75
80
85
90
95
100
STRIKE Correlation Delta
0 -0.005 -0.1 -0.015 -0.02 -0.025
75
80
85
90
95
Active Long
100
STRIKE Active Short
Cross-Gamma
22
IN SEARCH OF THE PERFECT HEDGE
Figure 11 gives a selection of greeks for the three month spread option put – equivalent to the three month active long/ short basket put – graphed against spread option strike. Studying the greeks – and comparing them to more common vanilla option greeks – helps one build the necessary intuition about how long/short basket options behave with respect to their underlying assets, volatilities and correlations. Note that because the spread option is written on two underlyings, there are generally two values for each option greek. The put option delta values are intuitively negative and positive for the long and short portfolio respectively, and are very similar to the deltas of vanilla put and call options. The portfolio gammas are both positive and also display a similarity to the vanilla gamma profile, spiking just above the 100 strike level. Note that the longer tail is seen on the left-hand side rather than on the right as for vanilla options. An additional multi-asset greek is cross gamma, measuring the change in delta of one portfolio with respect to a change in the underlying value of the other portfolio. The cross gamma for spread options is negative as one would expect and of similar magnitude to the individual gammas. Vega is larger for the active short portfolio, indicating a higher volatility skew particularly on the left-hand side. Finally, the correlation delta vertically mirrors the three month profile given in Figure 10 for the long-only basket. Again, this is to be expected given that high correlation between the two active portfolios would cause the return spread to decline.
6.
CONCLUSION
In this work, we have shown how one can approach the question of finding the optimal hedge portfolio for a given active fund. Starting from the fundamental law of active management, we introduced a framework which allowed us to conduct analysis on simulated realistic active portfolios in order to build intuition as to how hedge mismatch error affects the level of protection afforded by a given hedge. We showed that for typical market conditions, hedge effectiveness improves dramatically when using a hedge portfolio that more accurately reflects the underlying portfolio. This has clear consequences for using generic index options to hedge highly active portfolios. We also showcased several active return and tracking error decompositions that allow one to precisely quantify and thus manage the sources of risk and rewards within a given portfolio. Building on this knowledge, we discussed a mixed integer quadratic programming formulation that enabled us to search across a large investment universe in order to find the subset of stocks that would most closely replicate a given portfolio’s future performance, whilst simultaneously complying with realistic market constraints. Motivated by our index hedge mismatch findings as well as our success in creating appropriate hedging baskets, we suggested several alternative hedging methods for active portfolios that can provide significantly greater levels of protection than the generic index hedge. From these alternatives, we focussed on long-only basket hedging and long/ short basket hedging. Several technical pricing issues were discussed and the reader was introduced to the long/short basket portfolio in order to build intuition about the usefulness and behaviour of such instruments.
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DERIVATIVES DEALING Gavin Betty +27 11 722 7506 Roberto Pharo +27 11 722 7504
EQUITY DEALING Warren Chapman +27 11 722 7516 Paul Tighy +27 11 722 7521
RESEARCH Anthony Seymour +27 11 722 7549 Florence Chikurunhe +27 11 722 7551 Emlyn Flint +27 11 722 7556
DERIVATIVES STRUCTURING AND CONSULTING Kobus Esterhuysen +27 11 722 7572 Edru Ochse +27 11 722 7570
