The effect of downside risk reduction on UK equity portfolios included with Managed Futures Funds

International Review of Financial Analysis 18 (2009) 303–310

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International Review of Financial Analysis

The effect of downside risk reduction on UK equity portfolios included with Managed Futures Funds Kai-Hong Tee Loughborough University, Business School Loughborough, Leicestershire LE11 3TU, United Kingdom

a r t i c l e

i n f o

Article history: Received 10 September 2008 Received in revised form 16 September 2009 Accepted 17 September 2009 Available online 27 September 2009 JEL classification: G15 G11 Keywords: Downside risk Value-at-risk Lower partial moment Managed Futures Portfolio diversification

a b s t r a c t The concept of asymmetric risk estimation has become more widely applied in risk management in recent years with the increased use of Value-at-risk (VaR) methodologies. This paper uses the n-degree lower partial moment (LPM) models, of which VaR is a special case, to empirically analyse the effect of downside risk reduction on UK portfolio diversification and returns. Data on Managed Futures Funds are used to replicate the increasingly popular preference of investors for including hedge funds and fund-of-funds type investments in the UK equity portfolios. The result indicates, however that the potential benefits of fund diversification may deteriorate following reductions in downside risk tolerance levels. These results appear to reinforce the importance of risk (tolerance) perception, particularly downside risk, when making decisions to include Managed Futures Funds in UK equity portfolios as the empirical analysis suggests that this could negatively affect portfolio returns. © 2009 Elsevier Inc. All rights reserved.

1. Introduction Academic and practitioner interest in asymmetric risk analysis, in particular relating to the lower partial moment (thereafter, LPM) and the development of practical applications of Value-at-risk1 (thereafter, VaR) methodologies, has greatly increased in recent years. For example, research by Danielsson, Jorgesen, Sarma, and De Vries (2006) and Hyung and de Vries (2005) have related VaR to the lower partial moments of return distributions. The initial academic interest in LPM can in fact be traced back to Markowitz (1952) seminal paper on portfolio diversification. However, due to the combination of computational costs and the success of his mean-variance framework, Markowitz's insights into the LPM were largely ignored over the subsequent 40 years. With the development of information technology and the limitations of the mean-variance framework becoming more apparent, these constraints no longer apply and hence interest in developing LPM methods has greatly increased. Even so, to date this work has tended not to focus on how the LPM can flexibly capture varying degrees of risk tolerance and their implications in respect of portfolio allocation problems, which is the primary focus of this paper. The purpose of the current paper is to first review and discuss the risk measures related to LPM, its development and the relationship to the currently used VaR model and second to empirically evaluate from a UK investor perspective the practical implications in terms of portfolio

1

E-mail address: [email protected]. See Jorion (2001) for an overview of Value-at-Risk concepts and applications.

1057-5219/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.irfa.2009.09.007

performance. The empirical evaluation of these issues from a UK investor perspective provides the first indication regarding how LPM can be utilised to affect downside risk reduction of portfolio returns and diversification. The paper is structured as follows. In Section 2, the paper reviews the literature dealing with the rationale, structure and development of the LPM model. Section 3 discusses the empirical objective of the study and the data and research method used. Section 4 presents and discusses the main findings, and Section 5 summarises the results and discusses their implications. 2. Literature review 2.1. Variance and below-target-returns variation as risk measures Since the publication of Markowitz's (1952) seminal paper on portfolio diversification, there have been numerous subsequent studies on portfolio selection and performance, the overwhelming majority of which have focused exclusively upon the first two moments of return distributions: the mean and variance. The concept of downside risk was first systematically analysed by Markowitz (1959) where he recognises that analyses based on variance assume that investors are equally anxious to eliminate both extremes of the return distribution. Markowitz (1959) suggested however that this does not accurately reflect investor preferences for minimising possible losses and that, therefore, analyses based on the semi-variance, which assumes that investors' primary decision criterion is on reducing losses below-target mean returns, could

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provide a more accurate model of investor decision making. By concentrating on minimising portfolio losses below some target mean returns, this type of analysis produces portfolio allocations that minimise the probability of below-target means returns.2 According to Nawrocki (1999), Markowitz (1959) provides two suggestions for measuring downside risk: a semi-variance computed from the mean return or below-mean semi-variance (SVm) and a semi-variance computed from a target return or below-target semivariance (SVt). The two measures compute variance using the returns below the mean return (SVm) or below a target return (SVt). Since only a subset of the return distribution is used, Markowitz called them partial or semi-variances and their computation is as follows: SVm =

SVt =

1 k 2 ∑ ½Maxð0; ðE−RT Þ k i=1 1 k 2 ∑ ½Maxð0; ðt−RT Þ k i=1

below  mean semi variance

ð1Þ

below  target semi variance

ð2Þ

where RT is the asset return during time period T, k is the number of observations, t is the target rate of return and E is the expected mean return of the asset being considered. Max indicates that the formula will square the greater of the two values, 0, or (t − RT). Nawrocki (1999) and Harlow (1991) discuss the development and research of both below-target and below-mean semi-variances and emphasize that one of the most enduring and related ideas involves focusing on the tail of the relevant distribution of returns, i.e., the returns below some specific threshold level or target rate. Risk measures of this type are referred to as “lower partial moments” (LPM) because only the left-hand tail (i.e., probability of underachieving a threshold return) of the return distribution is used in calculating risk. LPM may sometimes reveal the extent3 of skewness, but it cannot be identified as the “third moment” (skewness) since skewness4 assumes variance as the primary risk measure while LPM assumes variation of below-target return as the risk measure. 2.2. Lower partial moment and the relation to Value-at-risk Nawrocki (1999) observes that the research and subsequent development of downside risk measures and LPM only really progressed following the publication of the Bawa (1975) and Fishburn 2 However, due to the complexity and the costs involved in the computation of semivariance analyses, especially so when such analysis can only be undertaken iteratively, Markowitz (1959) chose not to pursue this line of inquiry. He rejected the semi-variance as the preferred risk measure and concentrated instead on his now famous mean-variance approach to portfolio theory. Even so, Markowitz (1959, p. 194) commented that the superiority of variance with respect to computational and other costs, convenience and familiarity does not, and may not in the future, preclude the use of semi-variance. 3 Skewness measures the concentration of return distributions around the mean values. LPM, however, measures the deviations of return below a certain target rate, which may not necessarily be the mean value. If the target rate is the mean value, then the idea of LPM will be similar to that of positive skewness. In this case, however, computationally both LPM and skewness are entirely different arising from their assumptions about their underlying risk measures. 4 To illustrate their differences, consider a portfolio selection problem with skewness that adopts the Polynomial Goal programming (PGP) method for optimisation, see Lai (1991), Chunhachinda, Dandapani, Hamid, and Prakash (1997) and Prakash, Chang, and Pactwa (2003) for more details. In constructing the optimisation, the standard statistical moment of distributions, where investors exhibit a preference for higher values of odd moments (mean return, skewness) and a dislike for higher values of the even moments (variance, kurtosis) (see Scott and Horvath 1980), are incorporated. Here, multiple objectives related to the three moments are defined, i.e., to maximize expected rate of return, minimise variance and maximize skewness and solved by PGF. Unlike the LPM method, the optimisation algorithm of PGP solved the portfolio selection problem (with skewness) assuming variance as a risk measure. In this case, skewness, together with the other two moments, is used to reflect the attitude towards both the upper and the lower parts of the distribution. In the case of LPM, the optimisation algorithm solved the portfolio selection problem by the minimisation of the variation below the assets' return target level, which is the definition of risk measure.

(1977) studies which described the LPM as below-target risk in terms of risk tolerance. Given an investor risk tolerance value n, the general measure, the lower partial moment, was defined as follows.

LPMðn; tÞ =

1 k n ∑ ½Maxð0; ðt−RT Þ k i=1

ð3Þ

where k is the number of observations, t is the target return,5 RT is the return for the asset during time period T and n is the degree of the lower partial moment. It is the “n” value that differentiates the LPM from the Semi-variance models (in Eqs. (1) and (2)), which only restricts n to 2. The value of n is viewed as the “weights” that are placed on the tolerance for the below-target variation. The higher the “n” values, the more the investor is risk averse with respect to belowtarget variation. Eq. (3) implies that investors are not likely to be risk averse throughout the entire return distribution and show only risk-averse behaviour based on the target return, since the target return should differentiate and determine the preferred gain and the corresponding risk tolerance of the investors (see, Fishburn, 1977 for additional details). If the target return is the mean value, as in Eq. (1), then the utility measure inherent in the mean-LPM analysis (n > 1) exhibits asymmetric pattern, i.e. risk averse on downside risk and risk neutral on upside returns, implying a skewness preference of the investors, and that the higher the degree n, the greater will be the skewness preference. Bawa (1975) defines LPM as a general family of below-target risk measures, one of which is the below-target mean semi-variance, that was discussed in Markowitz (1959) and described by Eq. (1). Fishburn (1977) regards this as simply a special case and argues that the flexible n-degree LPM allows different values of “n” to be approximated, which implies a variety of attitudes towards the risks of falling below a certain target level of returns. According to Fishburn (1977), n < 1 is where investors seek additional risk to a portfolio; where n > 1 investors are risk averse to below-target returns. Fishburn (1977) and Nawrocki (1992) argue that the LPM algorithm is general enough for it to be tailored to the utility function of individual investors. Conceptually at least, an n-degree LPM algorithm such as Eq. (3) should provide scope for Stochastic Dominance analysis given that the Second degree stochastic dominance (SSD) also includes all LPM utility functions where n > 1. Besides, there are also no restrictive assumptions about the probability distribution of security rates of return6 underlying the n-degrees LPM model. Guthoff, Pfingsten, and Wolf (1997) explain how Value-at-risk (VaR) can be transformed into the LPM at n = 0.7 Comparing the various risk measures, Kaplanski and Kroll (2001) note that VaR can be differentiated from the Fishburn n-degree risk measures. However, like the other below-target-returns risk measures, the VaR measure accounts for risk as being below a fixed reference point. VaR, in this case, is different from Fishburn's n-degree measurement of risk because the latter weighs all the results below a fixed reference point t. However, VaR measures risk or the maximum potential loss assuming this loss has a confidence interval of 1 − P (where P is defined as one of the lower quantiles of the distribution of returns that is only exceeded by a certain percentage such as 1%, 5%, or 10%). Hence, VaR considers risk as one potential loss with a cumulative probability of occurrence of 1 − P, while ignoring both larger and smaller potential losses, involving a target rate. 5 The target value is normally assumed to be “zero”. Depending on how target rate is to be defined, alternatively, risk free rate or mean value can also be used as target return. 6 This means, despite the distributional characteristics or the probability distribution of the security returns, they are transformed to capture the upside and downside returns by the LPM optimisation algorithm in equation (3). 7 Appendix 1 further illustrates this point.

K.-H. Tee / International Review of Financial Analysis 18 (2009) 303–310

While Guthoff et al. (1997) have explained how VaR is related to a special case of LPM at n = 0, this nevertheless reveals the relatively restrictive scope of VaR in explaining risk tolerance levels. However, it supports the argument of Fishburn (1977) that LPM provides a general model encompassing utility functions of various forms and patterns. Indeed, it is shown analytically in the Appendix that VaR is part of the LPM family and that LPM at n = 0 assumes a normal distribution. Thus at n = 0, normality is imposed on the distributional pattern of securities returns. The scope and practicability of VaR therefore become somewhat limited when compared to the n-degree LPM models. These conditions are even more restrictive applying to assets whose returns are skewed, as is the case in our empirical study, which is the primary reason why it is only appropriate to consider LPM of n ≥ 1 in our portfolio analysis.8 2.3. Lower partial moment and the relation to the co-lower partial moment Research in the area of applying LPM models to asset allocation problems is fairly limited, and especially so when including the degree of risk tolerance in the allocation problem. Nawrocki (1992) provides one of the few examples of research in this area. He investigates two topics in relation to LPM theory: namely, the size and composition of portfolios selected by an n-degree LPM algorithm, and the effect of varying downside risk tolerance on the performance of investment portfolios. Nawrocki (1991) describes a methodology for modelling co-lower partial moment (CLPM). CLPM incorporates the relationship or interactions of the underlying two assets' lower partial moment. This involves, firstly, the calculation of the semi deviation (SD) as follows:  SDni =

1 m n ∑ ½Maxð0; h−Rit Þ m t =1

1 = n

ð5Þ

where h is the target return, m is the number of observations, n is the LPM degree, which is non-negative and SDni is the semi deviation for security i for the period n. SD is included in the CLPM9 computations as follows: CLPM = ðSDni ÞðSDnj Þðrij Þ

ð6Þ

where r is the correlation coefficient between securities i and j, SDni and SDnj are the semi deviations for security i and j for the period n. This paper adopts the methodology for modelling CLPM used in Nawrocki (1991) and Nawrocki (1992). Similar applications of the methodology can also be found in recent research such as Moreno, Marco, and Olmeda (2005). Section 3 provides a more detailed discussion of the methodology adopted by this paper. 3. An empirical study 3.1. Data The aim of this paper is to conduct an empirical study using the ndegree LPM models for asset allocation problems in a manner similar to that of Nawrocki (1992). However, in this paper the time period of the empirical study is updated and covers the period from 199910 to 8 Additionally, Value-at-Risk (VaR) or more precisely LPM at n = 0 that defines the maximum potential loss to an investment with a pre-specified confidence level, also has risk coherence issue when solved in a portfolio optimisation problem. Addressing these issues is beyond the scope of this paper. See Acerbi and Tasche (2002) and Artzner et al. (1999) for more discussion and details. 9 This method used to compute the CLPM is also known as the Symmetrical CLPM approach. According to Nawrocki (1991), Elton, Gruber, and Urich (1978) provide the motivation for using the Symmetrical CLPM. They show that a simple algorithm like this can provide better forecast than a complex optimal algorithm. The Nawrocki (1991)'s approach to formulate the symmetrical CLPM gives positive semi-definite matrix, which is an important property for solving an optimisation problem. 10 The reason to start from 1999 is because France and Germany begin using euros from 1999. This helps reduce problems on currency conversion.

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2006. In this study, both equity data and Managed Futures Funds11 data are used to reflect the increasingly popular investment strategies that include fund-of-funds type investments (such as fund of hedge funds) in equity portfolios. Furthermore, as Managed Futures often exhibit positively skewed12 distributional characteristics, this suits the application of the n-degree LPM models13 very well, an application that is also rarely seen in the academic literature. Our study uses the monthly MSCI stock return data for: the USA, Japan, Germany, France, Switzerland, Canada and the United Kingdom, accessible from DataStream International and are value weighted.14 We convert returns into UK£ using a one-month forward currency rate.15 We include six Managed Futures Indexes — also known as the Commodity Trading Advisers (CTAs) indices — in the UK investor portfolios. These are the currency CTA, discretionary CTA, diversified CTA, finance CTA, equity CTA and Systematic CTA indexes. The source of the data is from CISDM (The Center for International Securities and Derivatives Markets, see cisdm.som.umass.edu). Table 1 describes the strategies used by CTAs. All Managed Futures and MSCI Stock Indexes data are used in the allocation process. Table 2A,B and C shows the summary statistics for the full period, and the in sample and the out of sample periods. The value of LPM, which measures the average monthly below-target variations, is computed assuming the target rate to be 0%. 16 Table 2A shows that the LPM values are quite similar for the CTA indexes, ranging from about 0.23% to 0.26%. For the stock indexes, it ranges from 0.08% to 0.3%. The table also shows that most of the stock indexes are negatively skewed, while most CTA indexes are positive skewed. Table 2B shows that most CTA indexes have relatively lower LPM, from 0.06% to 0.11%. Among them, the diversified CTA, finance CTA and the Systematic CTA indexes are significantly and positively skewed. However the stock indexes have higher LPM values, ranging from about 0.18% to 0.43%. 3.2. Methodology The minimum variance portfolio analysis is considered as an appropriate benchmark because of its general use in portfolio theory

11 A survey by Eurohedge (see www.eurohedge.com), a trade publication for the European Hedge fund community, shows an annual mid-year (i.e. as at 30th June 2004) total of $216 billion of assets under management by the European hedge fund community, an increase of over 70% from $125 billion at the end of June 2003 and more than 25% above the $168 billion estimated to have been invested at the start of the year, January 2004. Managed Futures funds are also a subset of the hedge fund industry and the survey provides a breakdown of the $216 billion assets under managements, by the type of trading strategies adopted by the hedge funds. The volume of assets under management that were classified as ‘Managed Futures’ strategies was $20.3 billion as at July 2004, an increase from the $12.7 billion invested as at July 2003 and the $16.2 billion invested at the beginning of January 2004. The Eurohedge research also shows that, out of the $216 billion assets under management by the hedge fund community, more than 50% of the managers are domiciled or based in London. London, therefore, remains, by far the dominant centre for European hedge fund activity, accounting for more than 75% of the European total assets under management. The huge growth of the Managed Futures industry in Europe over the past years has possibly benefited from the more established, Managed Futures industry in the United States. 12 This is well documented in the literatures. See for example, Lamm (2005) and Lamm (2003). In our paper, all managed futures exhibit positive, but not significant, skewness for the full period, but there is significant positive skewness in some managed futures' returns in the in-sample and out of sample. See Table 2. 13 See Footnote 6. 14 The countries selected are the same as those in Eun and Resnick (1988). It is in the currencies of these countries that the UK investors can hedge currency risk via a welldeveloped forward market. 15 It is more realistic to report findings in UK£ returns rather than in foreign currency returns since this is from the UK investors' perspectives. This is particularly so for the UK institutional investors, who know more about the UK currency forward contracts. Using the currency forward contracts has the advantage of potentially reducing the variability of assets' returns, which is in the investors' favour. See Eun and Resnick (1988) for more discussions. 16 See Footnote 5.

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Table 1 Commodity Trading Advisor (CTAs) investment styles and descriptions. Commodity Trading Advisors Commodity Trading Advisors are specialised traders in the futures and option markets, whose instruments traded are often treated as asset classes. These traded derivatives are normally managed as a fund, and consisting of investments in the financial futures market, commodity futures market and some over-the-counter (OTC) derivatives contacts, such as the forward and options contracts. The following descriptions of the various CTAs are those that are used in this study. The data used in this study are the monthly Net Asset Values (NAV) of the assets under management of the respective CTAs, classified by the investment style, and weighted according to the value of each CTA's assets under management. Investment style

Brief descriptions and the main derivatives instruments invested

Currency CTAs1 Financial CTAs1 Diversified CTAs1 Discretionary CTAs2

Trade mainly on futures, forwards and options on currencies Trade futures, forwards, and options on fixed-income instruments Trade futures, forwards, and options on all types of commodities and financial instruments Trade on most derivative instruments, except the key in this case is that the advisor may or may not follow the signals being generated by the trading ‘system’, unlike most other CTAs. Discretionary CTA, apart from using computer software programs to follow price trends and perform quantitative analysis, also forecasts prices by analysis of supply and demand factors and other market information Trade mainly on a wide variety of OTC and exchange traded equity index futures and options. Trade a wide variety of OTC and exchange traded forward, futures and options markets, except that they often adopt a pre-determined systematic trading model and involved, for example, momentum or contrarian strategy in their models.

Equity CTAs3 Systematic CTAs3 1 2 3

Source: Edwards and Caglayan, (2001). Source: Epstein (1992), pg 125. Source: The CISDM Website.

applications. Therefore, the minimum variance model alongside the minimum portfolio LPM models is used in the investigation and their findings compared. The following presents the minimum variance portfolio formulations n

n

Minimize ∑ ∑ σij xi xj : i=1 j=1

Subject to:

n

∑ rj xj ≥μ n

ð7Þ

∑ xj = 1 0≤xj ≤1; j = 1; 2; ::::; n

Subject to: n

∑ rj xj ≥μ n

∑ xj = 1 0≤xj ≤1; j = 1; 2; ::::; n where µ is the portfolio expected rate of returns, rj is the expected return of security j, σij is the expected covariance between returns of index i and of index j, xj is the proportion invested in asset j. We use the n-degree portfolio LPM algorithm to model the portfolio downside risk presented in Nawrocki (1991) and Moreno et al (2005). The following presents the minimum n-degree portfolio LPM formulations n

n

LPM = ∑ ∑ xi xj CLPM ij i=1 j=1

ð8Þ

where, CLPMij = LPMi when i = j CLPMij = CLPMji when i≠j: The investor is assumed to be risk averse below the target (returns) variation and the objective function, which includes the above mentioned n-degree portfolio LPM, is as follows: n

n

Minimize ∑ ∑ xi xj CLPMij : i=1 j=1

ð9Þ

where µ is the expected rate of return for the portfolio, rj is the expected return of security j,17 σij is the expected covariance between returns of index i and of index j, xj is the proportion invested in asset j. The minimisation function considers the co-lower partial moment (CLPM).18 This implies that the lower partial moment for the portfolio is minimised taking into account the relationship of the lower partial moments of the underlying portfolio asset returns. The model's allocation is therefore based on assets with the lowest interacted lower partial moment values in the portfolios. We assume that short selling is prohibited throughout the analysis. The analysis involved optimising the objective functions (8) and (9). For objective function (9), the degree of n, ranging from 1 to 4,19 is used in the minimisation of the portfolio LPM, with n = 1 exhibiting the “most tolerant” for below-target variation and n = 4 as the “least tolerant” in that regard. The weights for the portfolios are derived using the constraints for the respective objective functions. The data inputs are returns in UK£ for all Managed Futures and stock market indexes. This study analyses the performance consequences arising from including these market indexes assets in UK investment portfolios from 1999 to 2006. The estimation interval is 4 years, from 1999 to 2002. The out of sample testing periods are from 2003 to 2006. The holding period is 4 years and the holding period returns are used to compare portfolio returns generated from objective functions (7) and (9).

17 The expected portfolio returns, µ, is based on the expected returns of the individual assets and their respective weightings. The expected returns of the individual asset are calculated based on the in-sample mean returns, from 1999 to 2002. 18 The CLPM takes a symmetrical form as defined in Eq. (6). See Footnote 9. 19 We only use degree of n ≥ 1 as explained in Section 2.2, but restricted value of n from 1 to 4, follows that of Nawrocki (1992).

K.-H. Tee / International Review of Financial Analysis 18 (2009) 303–310

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Table 2 Descriptive statistics summary for the returns of the MSCI and CTA Indexes. MSCI stock/Managed Futures Indexes

Skew

LPM

Kurtosis

Jarque–Bera (JB)

0.3076 ⁎⁎⁎0.8386 − 0.0769 0.0190 ⁎⁎⁎− 0.9339 − 0.3734 ⁎⁎− 0.3990 0.1076 0.3062 0.2026 0.3682 0.3362 0.3329

0.18% 0.30% 0.12% 0.31% 0.08% 0.15% 0.25% 0.24% 0.26% 0.26% 0.23% 0.23% 0.24%

2.4493 3.6500 3.5779 2.5919 4.4165 3.1048 4.2095 2.5497 3.0528 2.2078 3.3802 2.6954 3.2413

2.7268 12.9409 1.4304 0.6721 21.9827 2.2749 8.3987 0.9961 1.5110 3.1673 2.7467 2.1795 2.0062

B) Descriptive statistics for the returns of the MSCI stock and CTA indexes for MSCI Canada index 1.52% 20.88% MSCI Japan Index 4.20% 32.64% MSCI Switzerland Index 0.40% 10.57% MSCI US Index 1.59% 15.53% MSCI UK Index − 0.81% 7.92% MSCI France Index 0.09% 13.94% MSCI Germany Index − 0.54% 19.77% Currency CTA 2.48% 17.00% Diversified CTA 2.97% 22.22% Equity CTA 2.63% 14.95% Finance CTA 2.81% 23.17% Discretionary CTA 3.03% 16.62% Systematic CTA 2.64% 22.45%

the in-sample period — 1999 to 2002 − 10.70% 8.18% 0.4465 − 17.67% 13.79% 0.3063 − 17.50% 6.07% − 0.4815 − 16.56% 7.26% − 0.3158 − 12.77% 4.54% − 0.5628 − 15.20% 6.90% − 0.1760 −24.23% 8.60% − 0.3213 − 8.39% 5.61% 0.1726 −12.11% 6.06% ⁎⁎0.5472 − 13.15% 6.21% − 0.3428 − 11.77% 6.06% ⁎⁎⁎0.6374 − 9.88% 5.59% 0.0739 −11.16% 5.86% ⁎⁎⁎0.6735

0.20% 0.43% 0.18% 0.20% 0.15% 0.24% 0.43% 0.07% 0.06% 0.11% 0.07% 0.06% 0.06%

2.4984 2.1420 2.9377 2.9063 2.9522 2.2989 3.1226 2.9459 4.4674 2.8284 5.1438 2.9347 4.9489

2.0977 2.2227 1.8624 0.8152 2.5389 1.2309 0.8558 0.2442 6.7021 0.9987 12.4428 0.0522 11.2251

C) Descriptive statistics for the returns of the MSCI stock and CTA indexes for MSCI Canada index 4.67% 25.74% MSCI Japan Index −0.83% 9.85% MSCI Switzerland Index 2.04% 19.37% MSCI US Index − 2.81% 14.81% MSCI UK Index 1.15% 8.47% MSCI France Index 1.42% 11.55% MSCI Germany Index 1.78% 18.41% Currency CTA −3.40% 11.82% Diversified CTA −3.38% 13.94% Equity CTA −3.13% 16.04% Finance CTA − 2.90% 14.74% Discretionary CTA − 2.94% 19.45% Systematic CTA − 3.27% 12.57%

the out-sample period — 2003 to 2006 − 11.08% 9.48% 0.1101 − 11.59% 5.10% 0.0125 − 10.25% 5.59% 0.5141 − 15.84% 7.05% 0.3334 − 5.25% 2.35% − 0.0608 −9.52% 3.98% − 0.2266 − 11.49% 5.23% ⁎⁎0.4785 − 14.52% 6.45% 0.4664 − 14.84% 7.11% 0.7063 − 12.68% 7.17% ⁎⁎0.9567 − 14.87% 7.01% 0.6833 −13.21% 7.60% ⁎⁎⁎1.1173 − 13.70% 6.78% 0.6349

0.17% 0.16% 0.07% 0.42% 0.01% 0.05% 0.07% 0.42% 0.45% 0.41% 0.40% 0.41% 0.42%

2.4087 2.5091 3.7984 2.9540 4.1773 3.6870 4.8079 2.7009 3.0629 3.3123 3.0934 4.0014 2.8715

0.7963 0.4833 3.3895 0.8937 2.8018 1.3548 8.3688 1.9194 3.9987 7.5167 3.7529 11.9920 3.2582

A) Descriptive statistics for the returns of the MSCI Canada index MSCI Japan Index MSCI Switzerland Index MSCI US Index MSCI UK Index MSCI France Index MSCI Germany Index Currency CTA Diversified CTA Equity CTA Finance CTA Discretionary CTA Systematic CTA

Mean

Max

Min

Std. dev.

MSCI stock and CTA indexes for the full period — 1999 to 2006 3.09% 25.74% − 11.08% 8.95% 1.68% 32.64% − 17.67% 10.64% 1.22% 19.37% − 17.50% 5.86% −0.61% 15.53% − 16.56% 7.45% 0.17% 8.47% − 12.77% 3.73% 0.76% 13.94% − 15.20% 5.64% 0.62% 19.77% − 24.23% 7.17% −0.46% 17.00% − 14.52% 6.70% − 0.21% 22.22% − 14.84% 7.31% −0.25% 16.04% −13.15% 7.28% − 0.04% 23.17% − 14.87% 7.12% 0.04% 19.45% − 13.21% 7.28% − 0.32% 22.45% −13.70% 6.97%

Note: 1) The Jarque–Bera (JB) statistics tests for skewness by taking into account kurtosis. It is estimated as JB = N[s2 / 6 + (k − 3)2 / 24], where S denotes the value of skewness and k denotes the value of kurtosis, N denotes the number of data used for the test. The JB test follows a chi square distribution with 2 degree of freedom. 2) (⁎⁎⁎) indicates 1% level significance (critical value for chi square is 9.21) and (⁎⁎) indicates 5% level significance (critical value for chi square is 5.991). 3) The return of the index values are all in UK£, converted from foreign currency using the one-month currency forward contract. 4) LPM measures the below-target variance. It is based on the Formula LPMðn; tÞ =

1 k

k

∑ ½Maxð0; ðt−RT Þn where K is the number of observations, t is the target return, n is the degree of the lower partial moment, RT is the return for the asset during

i=1

time period T, and Max is a maximisation function which chooses the larger of two numbers, 0 or (t − RT).

4. Discussion of results Table 3, which is divided into sections 3A and 3B, shows the main findings. Table 3A shows the asset allocations for the minimum variance and the n-degree LPM models (of n=1 to n=4, denoted as LPM1, LPM2, LPM3 and LPM4), and the underlying individual asset skewness for the out-sample periods. Table 3B compares the out of sample performance performances, comparing the average monthly returns, 4 years holding period returns (thereafter, as HPR) and the portfolio skewness for the minimum variance and the n-degrees LPM models. Table 4 shows the covariance and the CLPM of the main assets 20 allocated in the minimum variance and the various LPM models out of sample. Table 3A shows that the minimum variance model has the highest number of assets allocated. This has the best diversification effects of all the allocation models. The minimum variance model produces the 20 Due to lack of spaces, we only discuss and report CLPM or covariance among the 4 or 5 assets, that are weighted the highest of all assets allocated and in total represent more than 50% in the portfolio. See Table 3 for full lists of assets allocated in the respective portfolios.

smallest standard deviation when compared to the other LPM models. However, it has also the lowest monthly out of sample return of 0.75%. Table 4 shows some values of covariance to be negative, such as those of UK/France stocks and the UK/Germany stocks. These resulted in lower standard deviation, but also lower returns for the portfolio. The allocation of assets in the minimum variance model is based on the assets' variance and covariance of the entire return distribution. The asset allocation outcome could become sub-optimal especially when three of the allocated assets are of significant positive skewness in sample, as was shown to be the case in Table 2. The LPM model's specification, unlike the minimum variance models, accommodates return distributions of varying characteristics. The returns series of the underlying assets are transformed21 to capture the below-target variations of the assets, despite their distributional characteristics, for the purpose of minimising the downside risk of the

21

See Footnote 6.

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Table 3 Portfolio allocations and the out-sample portfolio results of the minimum variance and minimum LPM models. Portfolio assets A) Portfolio assets allocations MSCI Canada index MSCI Japan Index MSCI Switzerland Index MSCI US Index MSCI UK Index MSCI France Index MSCI Germany Index Currency CTA Diversified CTA Equity CTA Finance CTA Discretionary CTA Systematic CTA Total allocation No. of assets allocated

Assets' out-sample skewness

MV

LPM (n = 1)

LPM (n = 2)

LPM (n = 3)

LPM (n = 4)

0.1101 0.0125 0.5141 0.3334 −0.0608 − 0.2266 **0.4785 0.4664 0.7063 **0.9567 0.6833 ***1.1173 0.6349

9% 0% 25% 6% 13% 16% 12% 3% 3% 4% 4% 1% 4% 100% 12

40% 2% 8% 0% 4% 20% 1% 0% 13% 2% 4% 1% 3% 100% 11

8% 0% 19% 5% 32% 18% 16% 1% 0% 0% 1% 0% 0% 100% 8

7% 5% 19% 0% 31% 17% 15% 0% 0% 0% 5% 0% 0% 100% 7

0% 7% 2% 0% 6% 0% 80% 0% 5% 0% 0% 0% 0% 100% 5

MV

LPM 1

LPM 2

LPM 3

LPM 4

0.75% 38.35% 4.20% − 0.19 0.07% 5.44

⁎1.59% 97.76% 5.03% 0.05 0.17% 7.31

⁎⁎⁎1.44% 94.72% 4.84% 0.12 0.05% 23.78

⁎⁎⁎1.37% 88.86% 4.70% 0.19 0.04% 24.69

⁎⁎1.31% 78.59% 7.50% ⁎0.40 0.10% 9.64

B) Out-sample results Average monthly returns 4 years (2003 to 2006) holding periods returns Standard deviation Skewness Semi-variance (SV) Reward-to-semi-variance (R/SV ratio)

Note: 1) The critical values for testing the differences of the portfolio returns among the various allocation methods (i.e., LPM of n = 1, n = 2, n = 3, n = 4 and the MV) are 2.43 (5% significant level), 3.45 (1% significant level). As the F critical value (p value) is about 1.428 (0.000), the null hypothesis of no difference among the monthly returns series generated by the various allocation methods can therefore be rejected. These portfolios returns are therefore significantly different from one another. Our results are similar to that in Nawrocki (1992), which show that portfolio out-sample returns do also decrease, from 2.514 to 2.4849, as the LPM is adjusted from n = 1 to n = 4. 2) (***) indicates 1% level significance (critical value for chi square is 9.21), (⁎⁎) indicates 5% level significance (critical value for chi square is 5.991) and (⁎) indicates 10% level significance (critical value for chi square). The estimation of skewness is based on the Jarque–Bera statistics test. Ri − Rf , with Ri being the average monthly portfolio returns, for i = 1 to 5, for minimum variance, LPM1 to LPM4 3) We defined Reward-to-semi-variance ratio, R/SV, as: Semi  variance portfolio, and Rf as the risk free rate, which we used the UK 3 month treasury bills.

portfolio. The presence of skewness underlying the return distributions does not affect the allocation process as much as in the minimum variance portfolio model. The n-degrees LPM models in Table 3 show different degrees of downside risk tolerance, with n = 1 as the “most tolerant” or more precisely, indifference towards below-target variations and n = 4 as the “least tolerant” in this regard. Table 3 shows the number of assets allocated falls as the value of n increases. It also reveals skewness increases as a result, implying that, while skewness is not diversified away or reduced by the relatively smaller number of asset allocation, preference for portfolio skewness, however, appears to increase with the value of n or when downside risk tolerance level is reduced. Furthermore, it is also observed in Table 3 that the average monthly returns and the HPR are all reduced following changes to the value of n in the LPM models. However, the reduction in diversification following from the reduced assets allocated, as observed in Table 3, does not increase standard deviation as much as expected. Apart from increasing to 5.03% when switching from the minimum variance to LPM1 model, all the other n-degree LPM models do not have standard deviations consistent with the level of diversification. This leads to a rather high reward-to-semi-variance ratios as shown in Table 3. The reward-to-semi-variance ratio increases from 7.31 (LPM1) to 24.69 (LPM3), and then falls to 9.64 (LPM4), a pattern that appears to be remarkably similar to that reported by Nawrocki (1992). These portfolio returns are also affected by the CLPM underlying the portfolios optimized by the n-degrees LPM models. Equivalent to

the use of covariance in the minimum variance model, in the LPM models, CLPM of the various asset pairs is captured instead. This measures the extent of the interactions of the LPM underlying the asset pairs. Table 4 shows some negative or relatively lower CLPM. For the LPM1 portfolio, Table 4 shows the values of the out samples CLPM to be 0.00704% (Canada/France stock), 0.0782% (Canada/Diversified CTA) and 0.0073% (France stock/Diversified CTA). These relatively low CLPM are indicative of good diversifiers in terms of the belowtarget variations of the portfolio. The out of sample portfolio produced a monthly return of 1.59%. It is notable that as a result of switching from the minimum variance model to the LPM1 model, an increase in skewness occurs (from − 0.19 to 0.05). Similar patterns of CLPM are present in the LPM2 portfolio. An example is the values of CLPM associated with the Swiss stock, which as shown in Table 4, are mostly negative except for the Swiss/ Germany asset pairs. These minimised the below-target variation of the LPM2 portfolios. However, as n increases and lesser risk tolerance is incorporated in the LPM model, fewer assets are allocated, which adversely affects diversification — especially in relation to the LPM4 portfolio. The numbers of assets allocated to the LPM4 portfolio are further reduced. Analysing the CLPM values in Table 4 indicates that the German stock is lowly correlated with most assets (between −0.006% and 0.018%) in the LPM4 portfolio. However, the CLPM of the Diversified CTA does not indicate weak (but good) correlations, with the Japanese stock (0.049%) and the Swiss Stock (0.166%) on the below-target variation in LPM4 portfolio out sample. These CLPM

K.-H. Tee / International Review of Financial Analysis 18 (2009) 303–310 Table 4 Analysis of covariance and CLPM underlying the assets in the models. Minimum variance Assets allocated (weighting) Switzerland stock (25%), UK stock (13%), France stock (16%), Germany stock (12%) Assets' pairs

Out-sample covariance

Swiss/UK stock Swiss/France stock Swiss/Germany stock UK/France stock UK/Germany stock France/Germany stock

0.0165 0.0978 0.1555 − 0.0175 − 0.0115 0.1948

LPM1 Assets allocated (weighting) Canada stock (40%); France stock (20%); Diversified CTA (13%); Switzerland stock (8%) Assets' pairs

Out-sample CLPM

Canada/France stock Canada/diversified CTA France stock/diversified CTA Swiss/Canada stock Swiss/diversified CTA Swiss/France stock

0.0070 0.0782 0.0073 0.0092 0.0073 0.0048

LPM2 Assets allocated (weighting) Switzerland stock (19%), UK stock (32%); France stock (18%), Germany (16%) Assets' pairs

Out-sample CLPM

Swiss/UK stock Swiss/France stock Swiss/Germany stock UK/France stock UK/Germany stock France/Germany stock

−0.0016 −0.0017 0.0024 −0.0015 −0.0018 0.0576

LPM3 Assets allocated (weighting) Switzerland stock (19%), UK stock (31%), France stock (17%), Germany stock (15%) Assets' pairs

Out-sample CLPM

Swiss/UK stock Swiss/France stock Swiss/Germany stock UK/France stock UK/Germany stock France/Germany stock

−0.0029 −0.0055 −0.0025 −0.0025 −0.0027 0.1203

LPM4 Assets allocated (weighting) Japan stock (7%), Switzerland stock (2%), UK stock (6%), Germany stock (80%), diversified CTA (5%) Assets' pairs

Out-sample CLPM

Japan/Swiss stock Japan/UK stock Japan/Germany stock Japan/diversified CTA Swiss/UK stock Swiss/Germany stock Swiss/diversified UK/Germany stock UK/diversified CTA Germany/diversified CTA

0.0066 −0.0067 0.0077 0.0494 − 0.0035 − 0.0062 0.1659 − 0.0034 −0.0139 0.0183

309

values are 5 to 15 times higher than the CLPM values the German stock is associated with, which seriously reduces the positive downside risk diversification effects and causes a huge increase in the LPM4 portfolio's standard deviation to 7.5%, though their weightings may still be smaller compared to the German Stock. The German stock constitutes 80% of the LPM4 portfolio and is significantly skewed out-sample, which subsequently contributes to increasing LPM4 portfolio's skewness (significant) to 0.4. 5. Concluding remarks The concept and applications of asymmetric risk estimation have gained in popularity following the use of VaR methodologies in risk management. The discussion and comparison of VaR in relation to lower partial moment (LPM) based on recent research, such as Danielsson et al (2006) and Hyung and de Vries (2005), indicated the much greater applicability of LPM to portfolio allocation problems when investors exhibit a wide range of risk-averse behaviours in relation to below-target returns. Indeed, using the n-degree LPM models, it was shown analytically that Value-at-risk is simply a special case of LPM when n = 0. It turns out that VaR involves much greater restrictions compared to the n-degree LPM models in explaining risk tolerance levels. The empirical analysis of the paper focused on the ndegree LPM models to analyse the effect of downside risk reduction on UK portfolio diversification and returns and the effect of setting a target threshold return and allowing for the adjustment of risk tolerance level in the n-degree LPM models. The minimum variance (MV) model was used alongside the ndegree LPM models in the portfolio asset allocation process, involving both the Stock Market and the Managed Futures Indexes, due to its general application in portfolio theory. The n-degree LPM models' allocation is based on assets' returns, that are transformed by the algorithm in the models, to capture downside risks. Therefore, unlike the minimum variance portfolio, skewness did not cause any significant problems for the LPM models. However, it was observed that the effect of varying the tolerance of downside risk in the ndegree LPM models was a reduction in the portfolio returns. This increased the skewness preference of investors which also resulted in a large reduction in the number of assets allocated. The LPM model increased the returns to investors due largely to the relatively lower diversification underlying the allocation process as compared to the minimum variance model. This allocation process places the portfolios towards the higher end of the risk-return area of the efficient frontier. Moreover, as lower downside risk tolerance was incorporated in the LPM model following an increase in the n degree, the portfolio returns were further reduced, implying a significant premium associated with the reluctance to tolerate additional downside risk. Portfolio skewness was increased as a result, indicating the existence of a trade-off between portfolio returns and skewness, a result that was also found by Simkowitz and Beedles (1978) and more recently, by Huang and Yau (2006). To conclude, this paper used the n-degree LPM models to analyse the portfolio diversification outcomes for investors with some tolerance of downside risk. This paper used data on Managed Futures Funds that are already diversified. Even so, the n-degree LPM models were still of significant importance, especially as it is increasingly

Notes to Table 4: Note: 1) We only discuss CLPM or covariance among the 4 or 5 assets, that are weighted the highest of all assets allocated and in total represent more than 50% of the portfolio. See Table 3 for the full lists of assets allocated in the respective portfolios. 2) CLPM is calculated as CLPM = (SDni)(SDnj)(rij) where r is the correlation coefficient between for security i andj for the period securities i and j, SDni and SDnj are the semi deviations  n. The semi deviation is calculated as SDni =

1 m

m

∑ ½Maxð0; h−Rit Þn

1=n

t =1

the target return, m is the number of observation and n is the LPM degree.

where h is

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popular for investors to include hedge funds and fund-of-funds type investments within equity portfolios. The findings have implications for the use of Managed Futures Funds within UK equity portfolios, and how the potential benefits of fund diversification could deteriorate following reductions in downside risk tolerance levels. This reinforces the importance of risk (tolerance) perception (see for example, Veld & Veld-Merkoulova (2008)), particularly downside risk, when making decisions to include Managed Futures Funds in UK equity portfolios, which our findings reveal, could also adversely affect portfolio returns. Acknowledgements The author acknowledges helpful comments and suggestions from the reviewer, from Professor Robert Watson, who help proof read the final version of the article, and from Professor Roger Buckland, who help with the earlier working paper version of the article. Any remaining errors are the author’s sole responsibility. Appendix A. Derivation of Value-at-risk from lower partial moment of zero (developed partially from Guthoff et al. (1997)) Defining n-degree LPM as: t

n

LPMn ðtÞ = ∫−∞ ðt  xÞ df ðxÞ where t is the target rate of return, f(x) is the probability of getting a return less than t, x is the security returns and “n” is the power or exponential variable that determines the weights investors place on deviations. If n = 0, n-degree LPM derived as follows: t

0

t

LPM0 ðtÞ = ∫−∞ ðt  xÞ df ðxÞ = ∫−∞ 1⋅f ðxÞdðxÞ = FðtÞ:

ð1Þ

Therefore, when n = 0, i.e., when no weight is placed on the derivation from the target return, t, LPM is a cumulative distribution function of normality (F(t)). Setting the target as zero and then minus the “value” at risk, (i.e., the pre-determined worst expected loss of the security), we get the following From ð1Þ LPM0 ðtÞ = FðtÞ; therefore LPM0 ð−VaRðpÞÞ = Fð−VaRðpÞÞ Equivalent to : VaRðpÞ = −F

−1

ðLPM0 ð−VaRðpÞÞÞ:

Proof. Value-at-risk is transformed into lower partial moment of zero, LPM0 (−VaR(p)), of target t = −VaR(p), giving the probability that the actual loss to be greater than −Var(p).

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