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Structured Equity Investment Strategies for Long-Term

Asian InvestorsAugust 2011

An EDHEC-Risk Institute Publication

Sponsored by:

The author would like to thank Frédéric Ducoulombier, Robert Kimmel, and Lionel Martellini at EDHEC-Risk Institute for useful comments and suggestions. Financial support from Societe Generale is acknowledged. This study presents the author’s views and conclusions which are not necessarily those of Societe Generale.

Printed in France/Singapore, August 2011. Copyright© EDHEC 2011The opinions expressed in this study are those of the author and do not necessarily reflect those of EDHEC Business School. The author can be contacted at [email protected].

3An EDHEC-Risk Institute Publication

Structured Equity Investment Strategies for Long-Term Asian Investors — August 2011

Foreword .................................................................................................................. 4

About the Author ................................................................................................. 6

Executive Summary .............................................................................................. 7

1. Introduction .................................................................................................... 13

2. Modelling stochastic volatility through Heston’s model ........................19

3. Empirical characteristics of Asian stock markets ......................................25

4. Comparison of strategies: a simulation study ...........................................29

5. Numerical results .............................................................................................35

6. Summary and conclusions ..............................................................................51

Appendices .............................................................................................................55

References ..............................................................................................................75

About EDHEC-Risk Institute ...............................................................................79

About Societe Generale .......................................................................................83

EDHEC-Risk Institute Publications and Position Papers (2008-2011) .........85

Table of Contents

4 An EDHEC-Risk Institute Publication


This publication presents the results of the latest research on structured forms of investment strategies done at EDHEC-Risk Institute with the support of Societe Generale Corporate & Investment Banking and under the leadership of Stoyan Stoyanov, Head of Research at EDHEC Risk Institute–Asia and Professor of Finance at EDHEC Business School.

The current macroeconomic and regulatory environments are extremely challenging for institutional investors. Prudential and accounting standards encourage investors to invest in low risk assets that are highly correlated with liabilities. At the same time, investors operate in a low interest rate environment where attractive risk premia are offered by asset classes that are poorly correlated with liabilities or command high capital charges due to their volatility. The conundrum for long-term institutional investors is how to extract risk premia while limiting exposure to downside risks.

Structured investments allowing investors to gain access to the upside potential of an asset and at the same time providing protection on the downside could play a significant role in addressing these challenges. Ground-breaking research by EDHEC-Risk Institute1 found that long-term static investors should allocate a sizable fraction of their assets to portfolio insurance strategies, which can be accessed in a buy-and-hold manner via structured investment strategies.

The research presented in this publication extends this earlier work in two important directions.

First, it looks at the control of volatility as an objective and assesses various strategies

to pursue this objective. From a theoretical standpoint, a constant volatility portfolio is a key building block for asset allocation in a dynamic setting with stochastic volatility. Such a portfolio can also be used to efficiently deal with the problem of extreme risk arising from stochastic volatility. From a practical standpoint, there is increased demand for volatility targeting in the wake of recent market disorders as investors recognise that diversification is not a tool for downside risk control2 and that traditional portfolio insurance strategies do not explicitly aim at volatility control and as such may be sub-optimal tools to reduce regulatory capital consumption for market risk.

Second, it focuses on Asian markets and investors, an area where practitioner interest meets a dearth of academic research. Asian equity investing is particularly attractive given the Asian growth story and on the back of the shifting balance of economic power. At the same time, it is particularly challenging from a volatility control point of view: Asian equity markets are more volatile than their counterparts in the developed markets of Europe and North America and volatility risk is hard to hedge due to the limited availability of local volatility derivatives. However, with the region’s equity derivatives markets having developed briskly, designing and managing equity structured products is perfectly feasible for investment banks; structuring volatility targeted equity strategies for Asia is one way to address the conundrum faced by institutional investors. This publication evaluates such strategies within the framework of a stochastic volatility model that has been fitted to Asian data and captures deviations from normality exhibited by equity returns.

Foreword

1 - This pioneering research work was also sponsored by Societe Generale Corporate & Investment Banking; the results are presented in Martellini et al (2005) and Goltz et al (2008).

2 - For more on this, please read Amenc et al (2011).



The research results presented in this publication show that a structured target-volatility strategy significantly improves both the downside and the upside of the return distribution relative to a fixed-mix strategy and also allows investors to benefit more from the upside potential when a capital guarantee overlay is applied. It shows how the explicit management of volatility reduces the cost of the capital protection. It also documents utility gains for risk-averse investors, with and without capital guarantee overlay, and makes the case for significant allocations to structured equity investment strategies with volatility targeting.

These results have important practical implications for long-term investors, first of all for insurance companies implementing new solvency frameworks, which need to access the equity risk premium while controlling for downside risk and minimising capital consumption. While presented in the context of Asian equity markets, these results are of general applicability and could be used in other regions and for asset classes exhibiting comparable dynamics.

We would like to express our gratitude to our partners at Societe Generale Corporate & Investment Banking Group for their support for our research.

Frédéric DucoulombierDirector, EDHEC Risk Institute–Asia

Foreword



About the Author

Stoyan Stoyanov is professor of finance at EDHEC Business School and head of research at EDHEC Risk Institute-Asia. He has ten years of experience in the field of risk and investment management. Prior to joining EDHEC Business School, he worked for over six years as head of quantitative research for FinAnalytica. He has designed and implemented investment and risk management models for financial institutions, co-developed a patented system for portfolio optimisation in the presence of non-normality, and led a team of engineers designing and planning the implementation of advanced models for major financial institutions. His research focuses on probability theory, extreme risk modelling, and optimal portfolio theory. He has published over thirty articles in leading academic and practitioner-oriented scientific journals such as Annals of Operations Research, Journal of Banking and Finance, and the Journal of Portfolio Management, contributed to many professional handbooks and co-authored three books on probability and stochastics, financial risk assessment and portfolio optimisation. He holds a master in science in applied probability and statistic from Sofia University and a Ph.D. in finance from the University of Karlsruhe.


Executive Summary



Executive Summary

Institutional investors face significant challenges currently because of two big factors – the macroeconomic and the regulatory environment. Regulation and accounting standards force them to invest in low risk assets that are highly correlated with liabilities. At the same time, they operate in a low interest rate environment where attractive risk premia are offered by asset classes that exhibit typically low correlation with liabilities. A big challenge for long-term investors and especially for underfunded institutions is, therefore, how to extract risk premia and at the same time have a limited exposure to downside risks.

Over the recent years, it has become clear that structured investments could have a significant role in addressing these challenges. Structured investment strategies are based on derivatives with a pay-off function and features tailored to investor needs. Although a wide variety of strategies exists, not fitting a single definition, usually they contain an asset and a derivative structure designed to provide a particular pay-off at maturity. Some popular structures allow investors to gain access to the upside potential of the distribution of the asset and at the same time provide protection on the downside. The results by Goltz, Martellini and Simsek (2008) imply that buy-and-hold investors should optimally allocate a sizable fraction of their portfolios to portfolio insurance strategies.

Hens and Rieger (2009) indicate that structured products are very popular in Germany and Switzerland and less so in the US. Although demand suffered as a result of the 2008 market crash, demand for structured products is again on the rise

in Asia. The overall level of maturity of the markets in Asia and the heterogeneity of regulation across countries, however, create challenges in designing the derivative structure. The derivatives market that has been growing most rapidly, although non-uniformly across countries, is the equity derivatives market. As a result, structured equity investment products appear less costly to design on a relative basis and, because of the high expected growth of Asian economies, they are also attractive from the standpoint of institutional investors.

The main challenges are in the features of Asian equity markets that distinguish them from the European and the US markets – they are more volatile and, perhaps more importantly, volatility derivatives are difficult to get. The reason for this difficulty is that the equity derivatives market remains relatively immature despite its recent growth and derivatives based on option implied volatility indicators are not widely available, even when such indicators exist.

We explore the empirical characteristics of Asian equity markets3 and document significant departures from normality for all markets whether developed or emerging. This means that returns on Asian equity indices do not conform to a normal distribution with stable mean and volatility. We also find that volatility of Asian markets, excluding Australia and New Zealand, is higher than that of the developed markets in Europe and the US. Because the volatility of Asian equity markets is generally higher and volatility risk is difficult to hedge for Asian investors in the absence of volatility derivatives, it makes sense to consider

3 - We consider the main market indices in Australia, China, Hong Kong India, Indonesia, Japan, New Zealand, Singapore, South Korea, Taiwan, US, and Europe. The calculation is based on the monthly logarithmic returns from March 2004 to May 2011.



Executive Summary

structured equity investment strategies with a target volatility feature.

From a theoretical perspective, target volatility strategies can be rationalised in the framework of dynamic asset allocation models. Extending the framework developed by Merton (1969, 1971) with stochastic volatility leads to a fund separation theorem in which the component responsible for performance generation can be interpreted as a target volatility strategy4. As a side result, dynamic asset allocation models imply that an efficient way to deal with the problem of extreme risk arising from stochastic volatility is to construct a portfolio with a constant volatility5.

In this publication, we develop a formal comparative analysis of different strategies in a framework allowing for stochastic volatility reflecting the reality of Asian equity markets. To this end, we choose to model the economy through a stochastic volatility model known as Heston’s model that can accommodate many of the observed departures from normality of the equity return distribution, most notably the higher probability of extreme events and in particular the higher probability of extreme losses versus extreme profits. These properties are known as excess kurtosis and negative skewness, respectively. The model can also describe the empirically observed volatility clustering and the reversion to a long-run volatility level and it also allows for analytically tractable option pricing.

Assuming the economy is driven by Heston’s model, we compare four different strategies combined in two pairs – a fixed-mix and a target volatility strategy, and a classical option-based portfolio insurance (OBPI)

strategy and a target volatility strategy with an OBPI feature implementing a capital guarantee. In our implementation, the target volatility is achieved through state-dependent rebalancing between the market index and the risk-free asset.

Fixed-mix strategies are traditional investment approaches. Typical examples include the 60-40 fixed mix between stocks and fixed-income, the 50-50 fixed mix, or the 40-60 fixed mix. They can be more or less aggressive depending on the (lower or higher) risk-aversion level of the investor which translates into a higher or lower proportion of stocks. Fixed-mix strategies can be viewed as target volatility strategies in a constant volatility economy and, thus, comparing them to true target volatility strategies measures the consequences of trying to control volatility through a traditional fixed-mix in a stochastic volatility environment. The comparison with the target volatility strategy is done both in terms of utility loss and distributional properties of the pay-off at the investment horizon.

Another traditional strategy that is tested is a long-only investment with a capital guarantee feature implemented through a call option written on the underlying. The option provides a protection on the downside while the performance of the underlying asset provides upside potential. It is compared to a target volatility strategy to which a capital guarantee feature is added. This allows us to see the value added of the improved upside potential of the target volatility underlying within the OBPI structure. The improved upside potential appears as a consequence of the dynamic rebalancing which, apart from controlling

4 - See Appendix 6 for references and further details.

5 - Although this conclusion may not extend to tail risk arising from other factors such as jumps in the price or the volatility process, it is a strong message with practical implications.



Executive Summary

volatility, increases the probability of getting positive returns and also reduces extreme risk by making the downside of the asset return distribution at the investment horizon more Gaussian-like. As far as the capital guarantee is concerned, in this paper we consider a level of 90% but the particular level is a matter of choice; our results are general.

To carry out the comparison in a realistic setting, we fit Heston’s model to Asian market data. The estimation methodology allows us to consider economies that are relatively more integrated with the world economy. Fitting Heston’s model parameters to South Korea’s KOSPI 200 index and Hong Kong’s Hang Seng index reveals a significantly higher long-run level of volatility and also a more negative correlation between instantaneous volatility and the market index returns compared to the US market represented by the Standard and Poor’s 500 Stock Price Index (S&P 500). More importantly, we find that the constant volatility model is strongly rejected for the three markets which, from an academic perspective, is not surprising but provides support for target volatility strategies. Apart from parameter values as estimated from historical data, we also consider a scenario with a higher volatility of volatility representing a bigger deviation from the constant volatility case. A motivation for this is the fact that the volatility of volatility is generally difficult to estimate and it is important to size up the implications in case it is underestimated.

Having fitted Heston’s model, we generate 10,000 sample pairs of sample paths for the market index and the instantaneous volatility for a 10-year investment horizon

using the fitted parameters for the Hang Seng index and, for each pair of paths, we calculate the realised return of the four strategies. Then, we compare the benefits in terms of investors’ expected utility and also in terms of different distributional characteristics of portfolio value at the investment horizon and different path-wise properties. Apart from the value-added, we also look at the cost of implementing the structured product which, in this case, depends on the price of the options. Ultimately, the main objective is to draw a conclusion about the value added of the customised structured equity product in different volatility environments.

Our first important finding is that the return distribution of the fixed-mix at the investment horizon in the stochastic volatility economy under consideration is significantly negatively skewed and has excess kurtosis. These properties imply higher downside risk and unattractive upside potential. In stark contrast, the target volatility strategy has a positive skewness and almost no excess kurtosis which means both the downside and the upside of the return distribution are improved. This conclusion is confirmed by comparing the OBPI strategies where the downside risk is completely removed because of the capital guarantee and the upside potential depends entirely on the features of the underlying asset.

Besides providing new light on the distribution properties of these strategies at the investment horizon, we find a significant appetite for target volatility strategies exhibited by risk-averse investors with and without the option overlay. We consider a two-period setting which is a



realistic assumption because structured investment products are usually held to maturity. Firstly, we observe a higher expected utility for target volatility products which becomes more pronounced for higher levels of risk aversion. In particular, the target volatility strategy is preferred to the fixed mix by risk-averse investors at any level of risk aversion. In a similar vein, the OBPI strategy with a target volatility underlying exhibits a better upside potential and, in the high volatility case, the strategy becomes attractive to a broader set of investors compared to the standard OBPI.

Secondly, when added to the investment universe, the portfolio maximising investors’ expected utility contains a significant allocation to a structured investment strategy. In the base case for example, investors for which the 40-60, 50-50, or 60-40 fixed mix strategies of bonds and stocks are optimal would invest about 90%, 76%, and 62% respectively in a target volatility product with equity as underlying designed to have a constant annualised volatility of 18%6. Further on, if we assume the demand for the structured investment strategy is driven by a set of investors with uniformly distributed risk-aversion levels for which the optimal fixed mix is between 20-80 and 80-20, then the average allocation to the same target volatility strategy is about 57% in both the base case and the high volatility case. The average allocation to the target volatility OBPI product, however, increases from about 36% to about 63% implying that a capital guarantee feature would be much more demanded in environments where the stochastic nature of volatility is more pronounced.

We reach similar conclusions about investor preferences assuming that investors evaluate investment options only in terms of mean and downside risk as measured by conditional value-at-risk (CVaR).

Computing the access to the upside for the OBPI strategies, we find that protecting an underlying with target volatility is generally easier and could be cheaper than protecting an underlying whose volatility is left to its own devices. We demonstrate that the classical Black-Scholes framework can be used to price the corresponding call option. While the implementation of the target volatility itself can be associated with significant turnover, in the context of Heston’s model this can be dealt with by adapting the frequency of rebalancing.

To summarise, our results indicate target volatility products can be attractive to investors for at least four reasons. Firstly, they allow for an explicit management of volatility which is impossible to get with the classical investment approaches or products assuming a stochastic volatility setting. Being able to control volatility is important for any problem involving risk-budgeting. Secondly, they can be combined with options in an insurance strategy providing a pay-off profile with capital guarantee and an improved upside potential. This is especially important for underfunded long-term investors who need an access to equity risk premium while respecting short-term risk budgets. Thirdly, institutional investors required to hold risk based capital can benefit from the stabilising effect of the fixed volatility on the dynamics of the capital charge making it easier to maintain. Finally, from an investment banking perspective, providing

Executive Summary

6 - The value of 18% is below the long-term average of the Hang Seng index but it is close to the long-term average of the volatility of the markets in Europe and the US. At any rate, assuming a particular value for the target volatility parameter is non-restrictive because investing in the target volatility product and the risk-free asset leads to additional control on volatility. The volatility of the overall strategy would depend on the allocation to the target volatility product.



Executive Summary

a capital protection feature is much easier when the underlying strategy has a fixed volatility implying greater affordability of the product for investors.


2. xxxxxxxxxxxxxxxxxx1. Introduction




1. Introduction

The vast majority of institutional investors are currently facing a challenge and a dilemma. On the one hand, the desire to generate performance in a low yield environment leads them to invest significantly in equity markets and other classes that are poorly correlated with liabilities but offer attractive risk premia from a long-term perspective. On the other hand, stricter regulations and accounting standards give them significant incentives to invest mostly in assets that are highly correlated with liabilities and severely punish volatility, with a strong associated opportunity cost. While this challenge applies to a large number of investors throughout the world, it is fair to say that the situation is particularly critical for some Asian investors, in particular insurance companies in Japan and other Asian countries.

In this context, one of the key questions is how to get access to upside potential of equity investments without all the associated downside risk. In recent years, it has become clear that structured forms of investment management play a substantial role in addressing this challenge, since they precisely allow investors to obtain access to the upside potential of performance-seeking assets while offering protection on the downside. As a matter of fact, an increasingly thorough range of structured investment strategies has been developed over the past few years, which allow investor to tailor the risk-return profile of their portfolios in a more efficient way than simple linear exposure to the return on traditional asset classes.

From an academic standpoint, it has in fact early been recognised that structured products7 are natural investment vehicles

for institutional investors. For example, Leland (1980) has shown in a fairly general context that investors whose risk tolerance increases with wealth more rapidly than the average will rationally wish to obtain portfolio insurance8. This is the case in particular for institutional investors whose portfolio value must at all cost exceed a given value e.g. that of their liabilities, but thereafter can accept reasonable risks. In a related paper, Draper and Shimko (1993) have argued that institutional investors are likely to have a particularly strong preference for non-linear payoffs because of the non-linear nature of the liability constraints they face. From a pragmatic standpoint and taking a pension fund example, it is clear that a small change in the probability of extreme contribution rates is typically considered much more important than an equal change in the probability of an extremely high refund. In other words, structured equity investment strategies offering some capital guarantee have the potential to allow investors to profit from the equity risk premium, which is often needed to match the returns on liability-driven benchmarks, without being fully exposed to the downside risk associated with investing in stocks. This is especially important for under-funded institutions, and in the presence of a low interest rate environment that raises concerns of mean-reversion towards higher interest rate levels, and hence lower bond prices, making a dominant exposure to bond markets an uncomfortable decision.

In the same spirit, Goltz, Martellini and Simsek (2008) find that typical static investors should optimally allocate a sizable fraction of their portfolio to portfolio insurance strategies, and the associated

7 - In an attempt to clarify the terminology it should be stated that we use the label “structured products” to refer to any contractual plain vanilla or more exotic non-linear payoff resulting from a large variety of quantitative asset management techniques. This stands in contrast to the use of the term in the US, where “structured products” typically refer to mortgage-backed securities and other forms of asset-backed securities.

8 - Leland (1980) also shows that investors whose return expectations are more optimistic than average will also rationally wish to obtain portfolio insurance.



utility gains are significant, a result that is robust with respect to various parametric assumptions, as well as the presence of realistic levels of market frictions and heterogeneous expectations on volatility. These attempts at analysing the benefits of structured products for institutional investors, however, have been focusing on plain vanilla structures such as asset plus put option (or bond plus call option).

Considering structured equity investment strategies for long-term investors in Asia is relevant at least for two reasons. First, generally, in the past few years the demand for equity structured products in Asia has increased, which is a natural effect given the growth of the region. Second, to be able to construct structured products, it is necessary to rely on a market for derivative products which are building blocks for these products. Alternatively, it should be possible to replicate the building blocks in an efficient way which, in turn, hinges on a liquid market for the underlying securities. Given the relative non-homogeneity of the region with respect to financial markets legislation (see, for example, Dufey (2009) and Hohensee and Lee (2006)) and the fact that equity markets are among the most mature ones in Asia relative to the other financial markets, it makes sense to consider structured equity investment strategies. In fact, it is reported in Hsieh and Nieh (2010) that equity derivatives have seen the most rapid growth of all traded derivative products across Asia even though the growth has been non-uniform with substantial regional variations. Hong Kong, South Korea, and Taiwan have the most developed equity derivative markets while the ones in China, Indonesia, the Philippines and Thailand are least developed or do

not exist at all. In terms of products, index options and index futures are the most widely traded classes of equity derivatives, see Hsieh and Nieh (2010) and Jobst (2008) for additional details.

An important aspect when dealing with equity derivatives is the volatility of the underlying security. Even though volatility is a global concern, Asian stock markets are generally regarded as more volatile than US and European markets and this is something we confirm in Section 3 and Appendix 2. Perhaps more importantly, forward-looking volatility is particularly difficult to measure in Asia. The reason is that equity options markets in the region, generally, are not sufficiently liquid which makes the task of extracting reliable implied volatility estimates from option prices difficult to accomplish perhaps with the exception of the most developed markets. Not surprisingly the supply of volatility products is scarce. Since there is no good way to control volatility through derivatives based on VIX-type indices in a relatively high volatility environment, it makes sense to consider structured equity products customised with specific features such as a pre-specified volatility target.

Apart from the difficulties of measuring volatility, we also have to consider another stylised fact about stock returns – that they exhibit fat tails and excess kurtosis. Intuitively, the less mature the market is, the more frequent extreme profits or losses are and, therefore, the higher the tail thickness. The intuition is not very precise since the relative frequency of an event is also influenced by the value of the scale parameter, or volatility, and it is the interplay between volatility and tail

1. Introduction



thickness that determines the probability of extreme events. Some empirical studies suggest9 that the degree of tail thickness of major market indices in Asia, even though a bit higher, is not dramatically higher than the tail thickness of major market indices in the US implying that the relatively higher frequency of extreme events on Asian markets is due to their relatively higher volatility.

In addition to this empirical fact, there is ample literature in the area of financial econometrics demonstrating that higher excess kurtosis and fat tails of the unconditional return distribution can arise from a stochastic and a dynamic component of volatility10. As a result, taking into account the dynamics and the stochastics of volatility does mitigate, to a degree, extreme tail risk. Certainly, there could be other sources of tail risk such as the dynamics of higher order moments, or an independent, pure jump process superimposed on the Gaussian behaviour, but exploring these alternatives goes beyond the goal of this study. Nevertheless, if the market is more volatile, it makes even more sense to focus on volatility as a first step and explore the value added of introducing a volatility target in a structured equity product.

Finally, from a regulatory viewpoint, target volatility products make good sense for long-term institutional investors. The insurance industry regulation in Europe is moving towards a harmonised framework known as Solvency II under which insurance companies should maintain risk based capital charges. Although the investment horizon is long, international accounting standards force institutions to register the unrealised gains and losses in the income

statements and update capital charges on a regular basis. As a consequence, the portfolio volatility dynamics affect directly the dynamics of solvency margins that have to be maintained.

In contrast to Europe, the corresponding regulation in Asia is country specific. Nevertheless, in some countries such as Hong Kong and Singapore, there are risk-based capital charges and many other countries have expressed agreement that the Solvency II principles make good sense. Because of the generally more volatile markets in Asia, following the Solvency II principles would imply more volatile capital charges. In this context, products with a target volatility feature would exercise a stabilising effect on the solvency margin dynamics which would, in turn, facilitate its maintenance. Since a target volatility strategy can be combined with an option overlay implementing a capital guarantee, the strategies could also reduce the risk-based capital charges and make investment into equity less costly; they would also be attractive to long-term investors, such as pension funds, that wish or need to control their asset to liability ratio.

In this research study, our goal is to develop a formal comparative analysis of different strategies in a framework allowing for stochastic volatility reflecting the reality of Asian markets. Therefore, as a first step, we choose a flexible stochastic model and also a diverse set of strategies. The strategies include simple portfolios of equity and a safe asset, represented by a pure discount bond, with and without a volatility target and also more sophisticated strategies including these simple ones

9 - See, for example, Jondeaua and Rockinger (2003) and Kittiakarasakuna and Tse (2010). Even though both studies are based on extreme value theory and Jondeaua and Rockinger (2003) acknowledge that the number of extremes observed seems insufficient to draw a solid conclusion about the tail thickness across different regions, we adopt the conclusion in Kittiakarasakuna and Tse (2010) that there are no significant differences in the tail thickness between Asia and the US as our working assumption.

10 - See, for example, Clark (1973) for more information on subordinated processes and Francq and Zakoian (2010) for more details on GARCH and GARCH-type processes and applications in finance.

1. Introduction



in combination with options in order to implement additional controls on the downside of the stock return distribution. Our approach is based on simulating an economy with different volatility regimes by assigning realistic values to the parameters of the model. We compare the benefits in terms of investors’ expected utility and also in terms of different distributional characteristics of portfolio value at the investment horizon and different path-wise properties. Apart from the value-added, we also look at the cost of implementing the structured strategy which, in this case, depends on the price of the options. Ultimately, the main objective is to draw a conclusion about the value added of the customised structured equity strategy in different volatility environments.

The remainder of this publication begins with a discussion of the Heston stochastic volatility model and the methods for parameter estimation discussed in the academic literature. Next, we fit the model on the Hang Seng index and the KOSPI 200 index and compare the fitted model parameters to the S&P 500 index. The parameter estimation method relies on an approximation to instantaneous volatility through a forward-looking volatility estimator proxied from the US market. In Section 4, we describe the strategies and in Section 5 we provide the simulated results.

1. Introduction



1. Introduction


2. Modelling stochastic volatility through Heston’s

model



2. Modelling stochastic volatility through Heston’s model

The celebrated option pricing model developed by Black and Scholes (1973) and Merton (1973) assumed that stock prices follow geometric Brownian motion (GBM). The dynamics of GBM implies that the increments of the logarithm of the stock price, or the log-returns, are normally distributed; the mean and variance of the normal distribution depend linearly on the return frequency.

Empirical evidence against the hypothesis of normality had been published as early as the 1960s by Mandelbrot (1963) and Fama (1963, 1965). Evidence against GBM was noted also in Black (1976). Although there is no consensus in the academic literature on the particular functional form of the distribution of stock log-returns, there is agreement on a number of properties often described as stylised facts:

(a) Clustering of volatility – when volatility is high, it is likely to remain high and when it is low, it is likely to remain low.

(b) Auto-regressive behaviour – price changes depend on price changes in the past.

(c) Return distributions are skewed and leptokurtic – there is an asymmetry in the upside and downside potential; return distributions are usually more peaked at the centre than the normal distribution.

(d) Return distributions are fat-tailed – the probability of extreme profits or losses is larger than predicted by the normal distribution. Tail thickness varies from asset to asset.

The first two properties, (a) and (b), concern the time domain, while the last two properties, (c) and (d), concern the

state space. Properties from the time domain can influence the properties concerning the state space. For example, dynamics in volatility can lead to fat-tailed unconditional return distributions, see, for example, Clark (1973).

In fact, the volatility of financial return distributions is a characteristic studied extensively in the academic literature. The celebrated ARCH model by Engle (1982) and the success of the GARCH developed by Bollerslev (1986) indicate that the dynamics of variance of financial time series has a long-term constant component and a dynamic component that is influenced by the past levels of variance and the new information arriving at the current time instant. According to the simplest form of the GARCH model, GARCH(1,1), the variance at time t is a linear function of only the variance at time t-1 and the squared residual at time t. Engle (2003) notes that this simple model for the dynamics of variance is remarkably successful across many asset classes.

A model that integrates these empirical facts into a consistent framework in continuous time was proposed by Heston (1993). It is an extension of the classical GBM with a mean-reverting process for the variance of stock log-returns. According to this model, the equity price is described by the following system of stochastic differential equations (SDEs):

dS S dt v S dW

dv v dt v dW

dW dW dt

t t t t s

t t t v

S v

n

l v

t

i

= +

= - +

=

^ h (1)

where t is the correlation coefficient, l > 0 is the speed of mean reversion,




i> 0 is the long-run level and is also the unconditional mean of vt , and v>0 is the volatility of volatility. The stochastic process for volatility is known as a square root process – the volatility is always non-negative and if 2 > 2l vi , then it is strictly positive (see Mikhailov and Nögel (2003)). The traditional GBM arises from this model with v = 0 and to i= . The resulting dynamics are described by the SDE

dS S dt S dWt t t sn i= +

that is a GBM with variance equal to i .

Insight into the behaviour of variance vtcan be gained by writing the solution to the square root process in discrete time,

The sequence , , ..., , ...v v vt1 2 behaves as an auto-regressive process of order 1 where the residual

e v dWtt

t

t

v11

1

f v= l xx+

- + -

+

^ h#

is a martingale. As a result, the variance at time t + 1 depends on the long-run level, the variance at the previous time instant t, and the new information that has arrived in between the time instants t and t + 1 accumulated in the residual t 1f + . The fact that the error term is a martingale means that the past does not help to predict the future.

The moments of the variance vt conditional on are given by11

(2)

When t increases, the conditional mean and variance converge to

|

| .

lim

lim

E v v

Var v v2

tt

tt

0

0

2

i

liv

=

=

"

"

3

3

^

^

h

h (3)

Apart from providing a mean-reverting process for volatility, the Heston model leads to non-normal equity log-returns. Increasing the parameter v leads to a higher excess kurtosis of the equity log-return distribution which results in tails fatter than the tails of the normal distribution. In addition to that, the correlation coefficient t controls the skewness in the following fashion: positive, zero, or negative skewness results from positive, zero, or negative t (see Heston (1993)). Since empirically the correlation between stock returns and volatility is negative, this model can provide an explanation of the observed negative skewness of stock returns.

Even though it would be difficult to argue that any departure from normality in equity returns can be captured by the classical Heston model, it represents a good starting point. This is also supported by a few extensions of this model available in the academic literature. Heston and Nandi (2000) considered a GARCH-type process, which converges to the continuous time Heston model, arguing that it can be fitted to empirical data more easily. Sepp (2008) considers an extension of the classical model with a pure jump process for both the stock price and volatility. Similar extensions are discussed also by Chacko and Viceira (2003).

The wide application of Heston’s model in the academic literature can be partly

v e e v e v dW1t tt

t

t

t v11

1

i v= - + +l l l x+

- - - + -

+

^ ^h h#

|

| .

E v v v ve e

Var v v v e e e

1

21

tt t

t vt t t

0

0

22

22

i

lv

liv

= = + -

= - + -

l l

l l l

- -

=- - -

^ ^

^ ^ ^

h h

h h h

11 - See Jiang and Knight (2002) for additional details.



explained by the availability of an expression for the price of a European call option written on St that is suitable for numerical work. The system of equations introduced above (Eq. 1) describes the dynamics of St and vt in the physical probability measure P. Under the risk neutral measure Q, the equations change to

dS r d S dt v S dW

dv v dt v dW

dW dW dt

t t t t SQ

t t t vQ

SQ

vQ

l i v

t

= - +

= - +

=

l l

^^hh (4)

where d denotes the dividend yield, 2l l m v= +l and

2l m vli i=+

l in which 2m is an additional parameter determined by the market price of risk of volatility that equals vt2m .

In this paper, we adopt the approach of Carr and Madan (1998)12 according to which the option price can be expressed as

, , , , , ,C s v K T t S c s v K T tt t t t t- = -^ ^h h

where logs St t= and , , ,c s v K T tt t -^ h is a scaled option price given by

, , ,

( )Re

exp

c s v K T t

u iu

w w m w vdu

1 2 1

t t

t t

2

0 1 2

0a a a

- =

+ - + +

+ +3

^

^^e

h

hh o#

a is an arbitrary scaling parameter and

ln

exp

w T t r d r r d iu T t

w iu

w u iu

c c u c u

iu

T t

c

c i

c

1 2

2 1 1 1 1 1

1

12

1

1 2 1 1

2 1 1 2

1

0 2 1 2

1

22

2 1

0 0 1 22

1 0

2

0

1 0

0

2 2 2 2

12

22 2

avl i c c

a

a a ac c

c

c l a tv c

ccc c

l v a l t v v a t

v v a t l t v

v t

= - - + - + - + - -

= +

= - + + + + -

= + +

= - + + +

= +- -

= - + - - + -

=- + - + -

= -

l l

l l

l

^ ^ ^ ^^ ^^

^ ^^ c

^^^c

^ ^ ^ ^ ^^ ^^

^

h h h h h h h

h hh m

hh h m

h h h h hh h h

h

Alternative formulations of the dynamics of variance discussed in the academic literature lead to option pricing frameworks that are very complicated for numerical work.

Even if the Heston model appears to be statistically significant and is not rejected compared to the constant volatility reduction13, it is not clear a priori whether the behaviour of the instantaneous volatility in the real world follows the structure of the Heston model or any other stochastic volatility model. There are several papers in the academic literature that consider extensions of the Heston model and report that, empirically, the extensions are statistically significant. Some extended models include jumps in the price process and/or in the volatility process of Eq. 1 (see Eraker (2004) and Sepp (2008)). Other extensions consider a power function of the instantaneous variance, instead of square root, in front of dWv in Eq. 1 adding one additional parameter for estimation (see Aït-Sahalia and Kimmel (2007)).

It is not the aim of this paper to test Heston’s model against its extensions or other available stochastic volatility models. However, we can think about the possible


12 - Heston (1993) provides an option pricing formula based on a Fourier transform and Carr and Madan (1998) provide an improvement that overcomes numerical difficulties.

13 - The estimation methodology and fitted parameter values for the KOSPI 200 and Hang Seng indices are provided in Appendix 2.



impact of having made an error accepting the model when an extension could be more relevant. First of all, we would expect the basic conclusions to continue to hold in more general models because the fixed-mix strategy, against which the target volatility strategies are benchmarked, is designed to work in a constant volatility world and adding jumps or extending functional forms will not make it more adequate. However, in jump-diffusion models the relationship between daily, weekly, and monthly rebalancing results could change and, also, the spikes in volatility would increase the turnover of the target volatility strategy. The extent to which these effects materialise will depend heavily on the size and intensity of the jumps. Nevertheless, this is one aspect that should be considered in a practical implementation.




2. Modelling Stochastic Volatility through Heston’s Model


3. Empirical characteristics of Asian stock markets




In this section, we calculate a few simple characteristics of the index returns of some of the main market indices in Asia. Our goal is to assess whether the unconditional distribution exhibits negative skewness and excess kurtosis and compare these characteristics to indices from the US and Europe.

In this analysis, we include main market indices in Australia, China, Hong Kong India, Indonesia, Japan, New Zealand, Singapore, South Korea, Taiwan, US, and Europe. The calculation is based on the monthly logarithmic returns from March 2004 to May 201114, which is the longest period for which there is data for all indices. Apart from skewness and kurtosis, we calculate the annualised mean and volatility and also the Jarque-Bera (JB) test, which is a

common test for normality. The JB statistic is defined by

3JB n S

K

6 42

2

= +-^c h m

where n stands for the sample size, S denotes the skewness and K the kurtosis of the distribution. The p-value used in the statistical test is calculated from the asymptotic distribution of the JB statistic which is 22\ ^ h. If the p-value is smaller than 0.05, we reject the hypothesis that sample comes from a normal distribution.

The results are provided in Table 1. We reject normality at 5% significance level for all indices; only the Taiwan SE (TSE) Index and the Chinese (XINHUA 50) Index are anywhere close to the threshold. In fact, the skewness and kurtosis of Asian

14 - We work with monthly data in this section because the methodology for fitting the Heston model described in Appendix 2 is based on monthly data. Further on, finding significant skewness and kurtosis for monthly returns implies that these characteristics will be even more pronounced in daily data.

Mean (ann)Volatility

(ann) Skewness Kurtosis JB-test p-value

AsiaKOSPI 200 12.30% 21.86% -0.75 4.70 18.54 0.00009

Hang Seng 7.34% 22.55% -0.89 6.03 44.81 0.00000

Straits Times 7.39% 20.49% -1.27 9.52 177.60 0.00000

TOPIX -3.52% 19.06% -0.85 5.41 31.53 0.00000

TWSE Index 3.95% 22.38% -0.49 3.87 6.22 0.04452

NIFTY Index 15.55% 28.60% -0.90 5.60 36.30 0.00000

XINHUA 50 6.62% 36.31% -0.62 3.96 9.00 0.01110

JCI 22.31% 25.83% -1.87 11.42 307.63 0.00000

ASX 200 4.65% 14.23% -1.19 4.50 28.67 0.00000

NZX All Index -0.82% 13.23% -0.86 3.82 13.02 0.00149

US and EuropeS&P 500 2.22% 15.71% -1.18 5.960 51.890 0.00000

DAX 30 8.22% 18.20% -0.814 4.876 22.366 0.00001

EURO STOXX 2.22% 15.70% -1.177 5.962 51.888 0.00000

FTSE 100 12.30% 21.90% -0.748 4.696 18.537 0.00009

Table 1. Moment characteristics and Jarque-Bera test for main market indices in AsiaThe calculations are based on the monthly log-returns from March 2004 to May 2011.




markets are quite comparable to those of the US market. Also, with the notable exception of the major indices of Australia and New Zealand, we have indication that the annualised volatility of Asian markets is generally higher than that of the US and European markets.

The rejection of normality can be regarded as a rejection of the geometric Brownian motion (GBM) hypothesis. If GBM were true, then the log-returns at any frequency should be normally distributed. Figure 1 and Figure 2 present another way of gaining

insight about the shape of the log-return distribution.

Both figures show a histogram of the daily log-return distribution of the KOSPI 200 and Hang Seng indices together with the density of a fitted normal distribution. The empirical distributions are more skewed and peaked around the centre and they also have tails heavier than the Gaussian tail.

The last feature is illustrated in Table 2. We calculate the empirical probability of the indices losing more than 3% and 4%

Figure 1. KOSPI 200 daily returnsA histogram of the daily returns of KOSPI 200 from March 2004 to May 2011 with a fitted normal distribution.

Figure 2. Hang Seng daily returnsA histogram of the daily returns of Hang Seng from March 2004 to May 2011 with a fitted normal distribution.

Table 2. Probability of KOSPI 200 and Hang Seng losing more than 3% or 4%Expressed as a frequency calculated from the empirical data and from the fitted normal distribution. Calculations use daily data from March 2004 to May 2011.

Probability of recording a daily loss larger than

KOSPI 200 3% 4%Empirical Once every 2.03 years Once every 4.05 years

Fitted Normal Once every 1219 years Once every 3.9 mln years

Hang SengEmpirical Once every 1.34 years Once every 8 years

Fitted Normal Once every 289 years Once every 331 000 years



15 - Appendix 1 provides an empirical illustration of the clustering of volatility effect in the returns for the KOSPI 200 and Hang Seng indices.


on a daily basis and compare the numbers to the probabilities implied by the fitted normal distribution. The probabilities are provided in terms of frequencies for easier interpretation. Clearly, the fitted normal distribution assigns much smaller probabilities to those extreme events.

These characteristics are not specific to the KOSPI 200 and Hang Seng indices. Rather, they are generic and, as mentioned in Section 2, they represent stylised properties of financial returns in general15. From investors’ viewpoint, the higher volatility on Asian markets reported in Table 1 motivates a demand for target volatility products. Also, given the non-normalities in the data, it makes sense to apply a stochastic volatility model and the Heston model is a natural choice.

Fitting the parameters of Heston’s model is an involved affair. A summary of different approaches available in the academic literature is provided in Appendix 1. For Asian markets, the estimation problem becomes even worse because implied volatility, as measured for example by a VIX-type indicator, cannot be observed in most countries. In Appendix 1, we develop a proxy that can be implemented throughout Asia and fit the model parameters for the KOSPI 200 and Hang Seng indices (for which, incidentally, VIX-type indicators are available, although not for the entire period that we are studying in the case of the KOSPI 200 index). The estimated values are provided in Table 14.


4. Comparison of strategies: a simulation study




A big concern for long-term investors in Asia is volatility of equity markets. Without a forward-looking estimate of volatility, such as the VIX index, and related volatility derivatives, there could be interest in target volatility structured products arising from hedging needs. Our goal in this section is to explore if risk-averse investors would prefer target volatility products and also products with control on the downside in a stochastic volatility environment. This setup seems to be plausible based on the numerical results in Section 3. We proceed with a description of the strategies that provide control on volatility, the plan for the numerical studies, and the results.

4.1. Strategies and investorsThe numerical studies involve four strategies, which we test by simulating the wealth generated by each strategy for a horizon of 10 years. At the end of the 10-year period, we calculate the expected utility of wealth at the investment horizon for different levels of risk-aversion. The strategy with the highest expected utility is most preferred by the corresponding investors.

To describe investor preferences, we use the constant relative risk-aversion (CRRA) utility functions defined as,

,u x u1

11

cc

=--c-^ h

where c > 1 is the risk-aversion parameter. When c approaches 1, the CRRA utility converges to a logarithmic utility, u(x,1) = log(x). We choose values of c ranging from 1 to 6.

In the remaining part of this section, we describe the four strategies.

4.1.1. Target volatility strategies in a deterministic and a stochastic volatility environmentSuppose that St describes the price process of an equity index and the dynamics of St are described by Eq. 1. On the market, there is also a safe asset (cash) yielding a constant risk-free rate r with the dynamics

dB rB dtt t=

Denote by tr the value of a portfolio with an allocation of wt to the risky and (1 – wt ) to the risk-free asset. Then, under the assumption that the strategy is self-financing, using Ito’s lemma we obtain that the dynamics of tr is driven by the following SDE,

d w w r dt w v dW1t t t t t t t S2r n r r= + - +^^ h h

where v w vt t t2=r denotes the instantaneous

volatility of the return of the portfolio tr and the process for vt is given in Eq. 1. In this setting, a constant volatility target for the portfolio, v TVt =

r , can be achieved by setting

wv

TVt

t

= (6)

If volatility is constant, v= 0, then the allocation to the stock becomes

w TVt

i=

In this equation, both the numerator and the denominator are constants and, as a result, the strategy is a fixed mix. In contrast, when v> 0 the allocation to the stock is itself stochastic and depends on the value of the instantaneous volatility.

The first comparison that we consider here involves a fixed-mix and a target volatility




16 - Note that even ift = 0, Eq. 6 would still lead to a constant volatility strategy. In this setting, it would nevertheless be impossible to link weight dynamics to the state of the market.

strategy as defined by Eq. 6 with TV = 18% which is a reasonable choice. In fact, Table 1 indicates that 18% is below the average volatility of Asian markets and the Hang Seng index in particular. This value is, however, close to the average volatility of the markets in Europe and the US and a representative Asian investor would be comfortable with this level. Additional remarks provided in Appendix 3 imply that the particular choice of the TV parameter value is not so significant; the real benefit of target volatility strategies is that they allow for maintaining a constant volatility level in the time domain.

From a practical viewpoint, fixed-mix strategies are fairly popular in the industry. Typical examples include the 60-40 fixed mix between stocks and fixed-income, the 50-50 fixed mix, sometimes referred to as a balanced strategy, and a 40-60 fixed mix, sometimes referred to as a conservative strategy. We are not going to discuss the pros and cons of fixed-mix strategies for long-term investors and the conditions under which they represent an efficient form of investment. In our theoretical construct however, fixed-mix strategies arise as a special case of target volatility strategies in a constant volatility economy.

It is important to note that the negative correlation between dWS and dWv has an important implication about the way Eq. 616 works. During a market crash, the market index decreases and the instantaneous volatility increases. Since the instantaneous volatility is in the denominator in Eq. 6, a higher instantaneous volatility translates into a lower weight of the market index in the strategy. Therefore, in times of increasing volatility, the allocation to the

risk-free asset grows, limiting to a degree the potential for future losses.

That the weight in Eq. 6 is a decreasing function of to makes also economic sense. As mentioned in the introduction, empirical research in finance has confirmed the clustering of volatility property. Thus, the persistence of volatility in times of market crashes rationalises a lower allocation to the risky assets in times of high volatility. For a practical implementation, however, it is important to have a smooth, good-quality forecast of market volatility.

Apart from the fixed volatility property, the advantages of the target volatility strategy are also in the properties of the log-return distribution at a given fixed horizon. The dynamics of the target volatility strategy is driven by the SDE

dr

v

TV r dt TVdWt

t

t

Srr n= + - +^c hm (7)

which is a geometric Brownian motion with a stochastic drift and constant volatility (by construction). The left tail of the distribution of the log-return is Gaussian, which implies no extreme risks are present. In contrast, the fixed-mix strategy is negatively skewed and has excess kurtosis (see Appendix 3 for additional details).

4.1.2. OBPITarget volatility strategies provide control on volatility which is an important parameter. There is, however, no explicit control on drawdowns. For this reason, we also compare option-based portfolio insurance (OBPI) with the market index and also with the target volatility strategy defined through Eq. 6. Generally, OBPI involves a portfolio of



a call option and a bond or, because of the put-call parity, the portfolio can be constructed using a put option and the underlying asset. OBPI is one of the most common type of structures with non-linear pay-offs. From a theoretical viewpoint, OBPI strategies are optimal for an investor facing explicit constraints on terminal wealth (see Basak (1995) or Grossman and Vila (1989)).

We can write the pay-off at maturity of an OBPI product based on a European call option with a lower constraint on terminal wealth equal to K as

,maxPayoff K k S K 0T= + -^ h

where ST denotes the price of the stock market index at expiration and k is a coefficient denoting the access to upside potential. This pay-off can be achieved by constructing a portfolio of the following components at time t = 0:

• A pure discount bond investment of Ke rt-

• An investment in a European call option of kC0 where C0 denotes the price of the call option at time zero.

The participation to the upside potential k is calculated taking into account that at time t = 0, our capital equals I0

kC

I Ke rT

0

0=- -

The price of the call option depends on the assumed dynamics of St . If St is described by the Heston model defined in Eq. 4, then the price of the option is given by

, , ,C C s v K T0 0 0= ^ h (8)

which is defined in Eq. 5. In the numerical calculations, we do not calculate Eq. 5 directly through numerical integration. Rather, we use a more efficient approach based on the fast-Fourier transform method described in Moodley (2005).

Constructing an OBPI product with the target volatility strategy is actually much simpler. An option written on this strategy can be priced using the classical Black-Scholes theory17 and the price of the option at time t = 0 equals

, , , ,C C TV K T rBS0 0r= ^ h (9) which is calculated through the Black-Scholes closed-form expression. In this paper, we choose K = 0.9 and initial value 0r = 1, which implies a maximum drawdown

constraint of 10% at the investment horizon.

4.2. Plan of numerical studiesThe results presented in Appendix 2 indicate that the Heston stochastic volatility model for equity markets in Asia is statistically acceptable. Table 14 contains the corresponding parameter estimates, which are based on approximation on two levels: first, the forward-looking estimate of volatility is proxied from that of the US market and, second, the instantaneous volatility is approximated from the forward-looking estimate through Eq. 12. For the numerical examples, we choose the parameter estimates of the Hang Seng index.

The risk-free asset is assumed to yield a constant annualised rate of 2% which is representative of the average three-month Hong Kong interbank offered rate (HIBOR)


17 - See Appendix 4 for additional details.



from February 2002 to June 2010 which the period used for fitting the Heston model parameters18.

In order to mitigate the inaccuracies because of various approximations in the fitting process, we consider two sets of simulations. The first one uses the parameter estimates for the Hang Seng index provided in Table 14 and represents a “normal” stochastic volatility environment, or the base case. In the second set, we set the value for v to the upper bound of the confidence interval reported in Table 14. This run represents a more uncertain stochastic volatility that we label higher volatility case.

For each set of parameters, we generate 10,000 pairs of sample paths for the index value and the instantaneous volatility for a 10-year period with daily frequency. For each pair, we calculate the dynamic weight according to Eq. 6 and the wealth path of the fixed mix and the target volatility strategies. Having obtained the scenarios at the 10-year horizon, we calculate the expected utility of each strategy and also the distributional characteristics of the realised return.

One general concern with the dynamic weight in Eq. 6 is turnover; if turnover is too high then a practical implementation would be too costly. Furthermore, high-frequency rebalancing could be infeasible. To address these issues, apart from daily rebalancing frequency, we consider also weekly and monthly rebalancing.

Note that reducing the rebalancing frequency has two theoretical impacts. Firstly, because of the lower turnover, the

implementation cost decreases. Secondly, because of the less frequent rebalancing, the volatility of the strategy will deviate from the target. In between two consecutive rebalancing dates, the strategy becomes buy-and-hold.


18 - A risk-free rate of 2% is also representative of the average 3-month Treasury bill rate in the same time period. Treasury bills mature in one year or less and are usually considered riskless because they are short-term and are guaranteed by the US government. The simulated economy in the paper is supposed to be representative of the Hong Kong market and, because of the linked exchange rate of the Hong Kong dollar to the US dollar, US Treasury bills are available with no additional currency risk. Therefore, an average rate of 2% seems to be a reasonable approximation for the risk-free rate. Further on, to keep things simple, we assume a flat yield curve.


5. Numerical results



In this section, we present the numerical results which are divided into two sections: path-wise properties and distributional properties at horizon. The first group of properties concerns the temporal behaviour while the second group concerns the state-space. Since constraints on the temporal behaviour necessarily influence the distribution at a given time instant it is important to determine the impact of the target volatility constraint, which is path-wise, on the distribution at the horizon.

The system of SDEs in Eq. 1 is discretised through Euler’s method. Even though the fitted values satisfy the constraint 2li> 2v and in theory vt is strictly positive, in

practice, because of the error introduced by discretisation, there is a certain probability to obtain negative scenarios for vt . To avoid this, one can resort to a Misltein discretisation scheme (see, for example, Glasserman (2004)). However, we never observed negative values for vt in the 10,000 sample paths with a daily time grid, neither for the base case, nor for the higher volatility case, and we stayed with Euler’s method.

In discrete time, the target volatility strategy is implemented in the following way. Denote the time grid by t1 , t2 , …, tn . To satisfy the self-financing property, the allocation wt given in Eq. 6 is calculated for t tk= from the instantaneous volatility observed at the time instant preceding tk ,

wv

TVt

t

k

k 1

=-

(10)

Therefore, as a consequence of the discretisation, the volatility of the target volatility strategy is not constant but

converges to a constant as the time grid becomes finer. The weekly and monthly rebalancing is also implemented through Eq. 10 but the rebalancing occurs at the beginning of each 5-day (resp. 20-day) period and the strategy is buy-and-hold in between rebalancing periods.

5.1. Path-wise propertiesA few sample paths of the market index value and the corresponding instantaneous volatility (with a matching colour) are provided in Figure 3. The impact of the fitted negative correlation in Table 14 is apparent – an increasing instantaneous volatility leads to a decreasing market index value.

Properties of the distribution of the unconditional daily return volatility are provided in Table 3. For each simulated path of the fixed mix and the target volatility strategies, we calculate the volatility of the daily returns. The table consists of two parts – the upper part corresponds to the base case and the bottom part, to the high volatility case. The first column of the table contains the mean of volatility across the sample paths, in the second column we report the ,q q. % . %2 5 97 56 @ quantile interval having a probability of 95%, and in the third column, we report the standard deviation of volatility.

The mean volatility of the target volatility strategy for daily, weekly, and monthly rebalancing matches the value set for TV. Although not constant, volatility varies in a small range around the mean value. The mean volatility remains almost constant when rebalancing frequency decreases from daily to monthly. The standard deviation,





however, slightly increases; the effect being more pronounced in the high volatility case. This result suggests that the approximation error does not increase too much with the change in the rebalancing frequency from daily to monthly.

As expected, the fixed-mix strategy has a much more unstable volatility for both the base and the high volatility case. Another important path-wise property of the target volatility strategy is the distribution of the average turnover. It is apparent from Eq. 6 that the turnover is directly dependent on

Figure 3. Sample paths of the price and volatility generated from the Heston model with the parameters for the Hang Seng index (base case) provided in Table 14 for an investment horizon of 10 years

Table 3. Characteristics of the unconditional distribution of the daily return volatility (path-wise) for the base case and the higher volatility case

Mean volatility Quantile interval ,q q. % . %2 5 97 56 @

Standard deviation

Base CaseMarket index 26.08% [19.6%, 33.8%] 3.5888%

Fixed-mix 17.80% [13.4%, 22.9%] 2.4647%

Target volatility 18.00% [17.5%, 18.5%] 0.2522%

Target volatility, weekly 18.00% [17.5%, 18.5%] 0.2524%

Target volatility, monthly 18.01% [17.5%, 18.5%] 0.2543%

High volatility caseMarket index 25.07% [17%, 36.3%] 5.05%

Fixed-mix 17.65% [11.7%, 25.1%] 3.4905%

Target volatility 18.00% [17.5%, 18.5%] 0.2561%

Target volatility, weekly 18.00% [17.5%, 18.5%] 0.2570%

Target volatility, monthly 18.03% [17.5%, 18.5%] 0.2639%



the variability of vt . At a given rebalancing date tk2 , the turnover is calculated by

TO w wt t t0

k k k2 2 2= -

where wt0k2 is the initial weight and wtk2 is the target weight calculated using Eq. 6. The initial weight is computed from the target weight at the previous rebalancing date tk1 by

wSwS

t

t

tt

t

t01k

k

k

k

k

2

2

2

1

1

rr

= c m

in which the bracketed expression represents the quantity of the risky asset bought at tk1 . Summing over all rebalancing dates, we obtain the total turnover for a given sample path.

The average turnover calculated from all sample paths and the ,q q. % . %2 5 97 56 @ quantile interval of the turnover distribution are provided in Table 4 for both the base and the high volatility case. Apart from the target volatility strategy, we also include the fixed-mix strategy because maintaining the fixed-mix leads to some turnover as well, which depends on the variability of

the price of the risky asset.

The numbers in Table 4 indicate there is a huge difference in turnover between daily, weekly and monthly rebalancing and also between the base and the high volatility case. The turnover of the fixed-mix remains relatively stable.

Apart from turnover, another interesting characteristic is the average weight of the market index. The weight averaged both across sample paths and rebalancing dates is reported in Table 5.

In the higher volatility case, the average weight of the market index is higher than in the base case and, in both cases, it is higher than the weight of the fixed mix. However, note that

wv

TV TVt

t

#i

=

when vt $ i; that is, when the instantaneous variance is above its long-term average, the strategy is less invested in the risky asset than the fixed mix and vice versa. Therefore, the higher average allocation comes from


Table 4. The mean turnover (annualised) and the ,q q. % . %2 5 97 56 @ quantile interval of the mean turnover distribution

Mean turnover, p.a. Quantile interval ,q q. % . %2 5 97 56 @

Base CaseFixed-mix, daily 71.40% [53.1%, 92.2%]

Target volatility, daily 172.00% [133.7%, 278.0%]

Target volatility, weekly 58.10% [34.7%, 121.1%]

Target volatility, monthly 26.13% [13.3%, 59.9%]

High volatility caseFixed-mix, daily 71.39% [44.9%, 100.5%]

Target volatility, daily 305.70% [158.7%, 780.0%]

Target volatility, weekly 121.20% [42.9%, 362.8%]

Target volatility, monthly 57.04% [17.9%, 177.8%]



a higher allocation to the risky asset in lower volatility periods, which tend to be associated with higher returns.

This behaviour is illustrated on Figure 4 with two sample paths. The top and the middle plots show the dynamics of the index value and the corresponding instantaneous volatility. The bottom plot shows the weight of the index calculated according to Eq. 6. The solid black line shows the long-term average volatility on the middle plot and the weight of the index in the fixed-mix on the bottom plot.

5.2. Distributional properties at the investment horizonFrom an investor viewpoint, the dynamics of the weights in the target volatility strategy is supposed to offset the adverse effects of the stochastic volatility on the distribution of log-return at the investment horizon. We argued in Section 2 that a negative value for t and the volatility of volatility vlead to negatively skewed and leptokurtic log-return distributions.

Histograms of the annualised log-return distribution of the market index, the fixed-mix, and the target volatility strategy

Table 5. The average weight of the market index in the base case and the high volatility case

Base case averagewt

High volatility caseaverage wt

Fixed-mix, daily 68.09% 68.09%

Target volatility, daily 72.48% 78.89%

Target volatility, weekly 72.47% 78.87%

Target volatility, monthly 72.44% 78.81%

Figure 4. Two sample paths together with their instantaneous volatilities and wtcalculated by Eq. 6The solid black line denotes the long-term average volatility (i ) on the middle plot and the weight of the fixed-mix on the bottom plot.




are provided in Figure 5. Visually, the distribution of the market index and the fixed-mix strategy are highly asymmetric.

The corresponding numerical values are provided in Table 6. Note that the kurtosis of the normal distribution is equal to 3 and any distribution with a kurtosis higher than 3 is said to be leptokurtic, or it is said to have excess kurtosis. The fixed-mix strategy and the market index have negative skewness and excess kurtosis. In fact, for the parameter values that we consider, the differences between the skewness and kurtosis of the fixed mix relative to the market index are negligible. In Appendix 3, we demonstrate that if the market index follows Heston’s model, then the wealth process of the fixed-mix strategy also follows Heston’s model but with a different long-run level i and volatility of volatility v . This result implies that the fixed-mix strategy does not improve dramatically the risk profile of the market index as far as departures from normality are concerned.

Both statistics skewness and kurtosis of the fixed mix and the market index deteriorate in the high volatility case. In stark contrast, the log-return distribution of the target volatility strategy has a slightly

positive skewness and a kurtosis close to the normal distribution value of 3. The positive skewness is not a random effect of the Monte Carlo method and appears because of the random term in the drift of Eq. 719. The standard deviation of the target volatility strategy is higher than the value of 18% set for TV. The reason is, again, the random term in the drift of Eq. 7. The positive skewness is, essentially, a by-product of the dynamic reallocation – by stabilising portfolio volatility in the time domain, we neutralise the impact of the stochastic volatility of the market index on the portfolio log-return distribution in the state space.

The impact of the downside protection in the OBPI strategies is clearly visible on Figure 7 and Figure 8. The upside potential of the target volatility strategy is better than that of the market index and, as a result, the OBPI with the target volatility strategy as the underlying has a better right tail. Numerically, it shows up as higher skewness and kurtosis in Table 6.

Another insightful way to compare the strategies at the investment horizon is to look at the joint distribution of their returns. Figure 6 provides such a comparison. On the

Figure 5. Histograms of the annualised log-return distribution of the market index, the fixed mix, and the target volatility strategies at the investment horizon in the base case


19 - See Appendix 2 for additional details.



left plot, we can see the annualised returns of the fixed-mix strategy shown against the returns of the target volatility strategy and on the right plot, the same comparison of the two OBPI strategies. Clearly, the advantage of the target volatility strategies

appears in those states of the world when the market loses a lot or gains a lot. In fact, since the target volatility strategies appear on the vertical axis, the bigger the convexity of the cloud is, the more pronounced the advantage of the target volatility strategy is.

Table 6. Distributional characteristics of the annualised log-return distribution of the five strategies at the investment horizon in the base and the high volatility caseIf the rebalancing frequency is not specified, then it is daily.

Mean Std Dev Skewness Kurtosis

Base CaseMarket index 7.79% 29.04% -0.7058 3.8097

Fixed-mix 6.71% 19.22% -0.7007 3.7993

Target volatility 7.12% 20.61% 0.0712 3.0731

Target volatility, weekly 7.12% 20.61% 0.0719 3.0695

Target volatility, monthly 7.12% 20.62% 0.0738 3.0683

OBPI, standard 6.50% 18.53% 0.3995 2.3087

OBPI, target volatility 6.05% 16.56% 0.6263 3.1112

High volatility caseMarket index 7.80% 29.84% -0.9002 4.1421

Fixed-mix 6.71% 19.51% -0.8922 4.1278

Target volatility 7.71% 22.73% 0.2399 3.0524

Target volatility, weekly 7.71% 22.73% 0.2406 3.0523

Target volatility, monthly 7.70% 22.71% 0.2381 3.0344

OBPI, standard 6.66% 18.37% 0.2581 2.0536

OBPI, target volatility 6.61% 18.65% 0.7421 3.3006

Figure 6. The joint annualised return distribution of the fixed-mix and the target volatility strategies (left) and the standard OBPI and the OBPI, target volatility strategies (right) in the base case




Figure 7. Histograms of the annualised log-return distribution of the two OBPI strategies and their underlying in the base case

Figure 8. Histograms of the annualised log-return distribution of the two OBPI strategies and their underlying in the high volatility case




States of the world in which the market loses or gains a lot are more likely in markets that are generally more volatile. Thus, higher volatility markets would make the advantage of target volatility strategies more pronounced. A comparison of Figure 6 to Figure 9 illustrates this conclusion. The convexity of the cloud of points on both the left and the right plot in Figure 9 is higher than on Figure 6.

The improvement in the downside of OBPI strategies comes at a cost because we have to purchase the derivative instrument20. Other than through the price itself, the cost can also be represented as an implicit opportunity cost. In our example, the

implicit opportunity cost materialises as a lower expected return at the investment horizon: compare the differences between the means of the market index and OBPI, standard and also between target volatility and OBPI, target volatility in Table 6. However, in both the base case and the high volatility case, the implicit opportunity cost of OBPI target volatility is below that of the standard OBPI. The reason is that the option is easier to implement when the volatility of the asset is constant.

This aspect is also illustrated in Table 7 which shows the option prices calculated according to Eq. 8 and Eq. 9, and also the access to the upside potential.

Figure 9. Joint annualised return distribution of the fixed-mix and the target volatility strategies (left) and the standard OBPI and the OBPI, target volatility strategies (right) in the higher volatility case

Table 7. Option prices calculated through Eq. 8 and Eq. 9 and access to the upside potential, k, for OBPI standard and OBPI target volatility in the base case

Option price Access to upside, k

Base CaseOBPI, standard 0.4246 0.6197

OBPI, target volatility 0.3525 0.7465


20 - See Amenc et al (2010, 2011) on the cost of insurance and how it can be compensated.



5.3. Investors’ preferencesAs pointed out in Section 4.1, we consider risk-averse investors whose preferences are described by the family of CRRA utility functions. A given strategy ,T2r is preferred to another strategy ,T1r by investors with a risk-aversion parameter c if

, ,Eu Eu, ,T T1 2#r c r c^ ^h h

where T denotes the investment horizon and in our calculations T = 10 years.

Using the simulated distributions of the strategies at the investment horizon, we calculate the expected utilities for c ranging in [1, 6]. Figure 10 shows the expected utilities of the fixed-mix and the target volatility strategies both in the base and the high volatility case. For all values

of the risk-aversion parameter, the target volatility strategy is preferred to the fixed-mix. The gap widens with the increase in risk-aversion. Further on, the fixed-mix in the high volatility case is less preferred than the same strategy in the base case. In contrast, the target volatility strategy in the high volatility case is more preferred. As a consequence, the gap between the fixed-mix and the target volatility strategy is uniformly wider in the high volatility case.

The difference in the expected utilities of the two OBPI strategies are less pronounced and for this reason, instead of providing the values themselves, we provide percentage difference

,

, ,R

Eu

Eu Eu

,

, ,

OBPI T

OBPITV T OBPI Tc

r cr c r c

=-^ ^

^ ^h hh h (11)

Figure 10. Expected utilities of the fixed-mix and target volatility strategies in the base case and the higher volatility case for different values of the risk aversion parameter.




where OBPI TV stands for OBPI, target volatility and T = 10 years. A positive R c^ hindicates a preference for the OBPI target volatility strategy.

The ratio R c^ h is plotted in Figure 11 in the base and the high volatility case. In the base case, for risk-aversion higher than 2.5 the OBPI with the target volatility underlying is preferred. In the high volatility case, the threshold value for the risk-aversion parameter decreases to about 1.9.

Figure 11 demonstrates that, although the downside is under explicit control in both OBPI strategies, the better upside potential of the target volatility strategy results in higher expected utilities.

From a practical perspective, it is interesting

to examine how the higher preference for structured products would transform to an actual allocation. To explore this question, we consider the following static portfolio allocation problem

,

. . 1

max Eu W

s t w w w

w 0

wT

1 2 3

$

c

+ + =

^ h

where the wealth WT at the investment horizon is given by

W W w wS

S Sw e1 1T

T T rT0 1

0

02

0

03

rr r

= +-

+-

+ -^ h; E

in which r denotes the target volatility strategy, the standard OBPI or OBPI, target volatility. A strictly positive w1 in the optimal solution implies that risk-averse investors demand the corresponding

Figure 11. Percentage difference between the expected utilities of the OBPI strategies for different values of risk-aversion defined in Eq. 11




structured product. The calculation of WT

does not pose any difficulties because in each state of the world we have the joint realisation ( Tr , ST ).

The optimal allocations for the risk aversion level ranging from 1 to 6 in the base case and the high volatility case are provided in Figure 12 and Figure 13, respectively. It is clear that there is significant demand for the structured investment strategy. Further on, the allocation to the structured strategy generally increases in the high volatility case.

We can compare the allocations for a particular choice of c . Assuming that the typical conservative choice of institutional investors can be a portfolio of 60-40, 50-50, or 40-60 mix of stocks and bonds21, we back-out the risk aversion levels leading to these portfolios. As a next step, we calculate the optimal allocation for these levels of risk aversion with the structured strategies added to the investable universe. The plot in the upper left corner in Figure 12 and Figure 13 illustrates how we back out the risk aversion level. The vertical lines on the other plots indicate the corresponding portfolios for the three risk-aversion levels. The particular allocations and risk aversion levels are provided in Table 8, Table 9, and Table 10. The allocation to the structured strategies increases significantly in the high volatility case.

Finally, assume the demand for structured strategies in the economy is driven by investors characterised by the following property – their risk aversion levels are such that the optimal fixed mix in the absence of structured strategies ranges from a 20-80 to a 80-20 allocation of bonds and stocks.

In this way, we exclude from the set the extremely risk-averse and the extremely risk-loving ones. Further on, we assume for simplicity that the risk-aversion levels are uniformly distributed between the two cut-off points c= 1.4 and c= 5.14 resulting in a 20-80 and a 80-20 optimal allocation of bonds and stocks, respectively. Under these assumptions, we can calculate the average allocation to the structured strategies. This information is provided in Table 11. Again, the allocation to the structured strategies is quite significant and increases dramatically in the high volatility case. In fact, the most dramatic change in the average allocation is observed for the structured strategies with a capital guarantee feature indicating that this feature would be desirable in economies characterised by a more significant departure from constant volatility.

Additional analysis is provided in Appendix 5 through a comparison between the mean-variance and the mean-conditional-value-at-risk (mean-CVaR) efficient portfolios. Appendix 5 complements the analysis in this section by demonstrating that if the portfolio construction framework takes into account the non-linear pay-off distribution of the structured product by focusing on the downside of the portfolio return distribution, then the efficient portfolios contain a significant allocation to the structured strategies and the target volatility strategy in particular. This approach is alternative to the expected utility framework in which the non-linearities are taken into account implicitly through the utility function22.

The optimisation problem discussed here is static which makes sense because structured products are usually held until maturity. In

21 - Note that because the risk-free rate is assumed constant, the hedging properties of the risk-free asset are in fact a little overemphasized in the simulated economy. In reality, the risk-free rate is stochastic and the correlation between the log-returns of a government bond and the stock market index is usually modestly positive on average although not very stable through time, see, for example, Baker and Wurgler (2010).

22 - In fact, we can argue that the non-linear pay-off function is taken into account because the expected utility framework implicitly considers the higher-order moments in addition to the mean and variance. This conclusion becomes apparent from the classical Taylor series expansion of the utility function.




Figure 12. Allocation to the three asset classes (bond, market index, and structured strategy) maximising expected utility for different levels of risk aversion in the base caseThe straight lines identify risk-aversion levels leading to 60-40, 50-50, and 40-60 fixed mix of market index and bond.

Figure 13. Allocation to the three asset classes (bond, market index, and structured strategy) maximising expected utility for different levels of risk aversion in the high volatility caseThe straight lines identify risk-aversion levels leading to 60-40, 50-50, and 40-60 fixed mix strategies of market index and bond. 40-60 fixed mix of market index and bond.




Table 8. Optimal allocations for institutional investor with a risk aversion level of c = 1.9596 in the base case and c = 1.8586 in the high volatility case The values of c correspond to an optimal allocation of 60-40 between stocks and bonds in the absence of structured products.

Market index Target volatility OBPI, standard OBPI, target volatility

Base caseProduct 0% 89.78% 73.83% 67.93%

Market index 60% 0% 26.17% 32.07%

Bond 40% 10.22% 0% 0%

High volatility caseProduct 0% 93.84% 74.26% 71.35%

Market index 60% 0% 25.74% 28.65%

Bond 40% 6.16% 0% 0%

Table 9. Optimal allocations for institutional investor with a risk aversion level of c = 2.3131 in the base case and c = 2.2626 in the high volatility case The values of c correspond to an optimal allocation of 50-50 between stocks and bonds in the absence of structured products.



Market index 50% 0% 26.25% 21.55%

Bond 50% 23.69% 22.50% 8.65%


Market index 50% 0% 5.13% 11.05%

Bond 50% 23.04% 0% 0%

Table 10. Optimal allocations for institutional investor with a risk aversion level of c = 2.8182 in the base case and 2.7677 in the high volatility case. The values of c correspond to an optimal allocation of 40-60 between stocks and bonds in the absence of structured products.



Market index 40% 0% 25.50% 21.18%

Bond 60% 37.88% 43.28% 32.64%


Market index 40% 0% 6.76% 1.63%

Bond 60% 38.01% 24.18% 8.66%




this setting, the target volatility strategy is exogenously constructed and is one asset in the investable universe. We can extend the discussion and ask a more general question: is it possible to rationalise target volatility strategies, or, in other words, do they have any optimal properties?

The answer to this question is in the affirmative, details and further references are included in Appendix 6. Extending the dynamic asset allocation model introduced by Merton (1969, 1971) with a stochastic volatility leads to a fund separation theorem in which the component responsible for performance generation can be interpreted as a target volatility portfolio. The value of the target volatility depends on the risk-aversion level. Alternatively, we can isolate a target volatility fund with a unit volatility and argue that each risk-averse investor is willing to hold a fraction of this fund. The allocation to this fund depends, among other things, on the risk-aversion level. In summary, target volatility strategies arise as a component of the portfolio maximising the expected CRRA utility function at the investment horizon and can be rationalised by dynamic portfolio theory.

From the standpoint of extreme risk management, it turns out that an

efficient way to hedge the tail risk arising from stochastic volatility is to construct a portfolio with a fixed volatility target. This conclusion may not extend to tail risk arising from other factors such as jumps in the price or the volatility process, but is a strong message with practical implications.

Table 11. Average allocation to the structured strategy assuming investors are uniformly distributed across different risk-aversion levels ranging from c = 1.4 to c = 5.14 which lead to optimal allocations of 20-80 and 80-20 of bonds and stocks, respectively, in the absence of structured strategies

Market index Target volatility

Target volatility 0% 62.12%

OBPI, standard 40% 0%

OBPI, 60% 37.88%



6. Summary and conclusions




Asian equity markets are different from European and US markets in many ways: they are fragmented, more volatile, and they do not offer uniformly liquid options markets. The last feature has at least two implications for long-term investors. Firstly, it is more difficult to construct a forward-looking estimate of volatility through options implied volatilities such as the VIX index in the US and, secondly, options are not easily available to provide downside protection.

In this environment, target volatility products are attractive at least for four reasons. Firstly, they allow for an explicit control of volatility which is impossible to get with the classical fixed-mix products in a stochastic volatility setting. Secondly, they can be combined with options in an insurance strategy providing principal guarantee with an improved upside potential. This is especially important for underfunded long-term investors. Thirdly, institutional investors required to hold risk based capital charges can benefit from the stabilising effect of the fixed volatility on the dynamics of the capital charge making it easier to maintain. Finally, from an investment banking perspective, providing a capital protection feature is much easier when the underlying strategy has a fixed volatility implying greater affordability of the product.

In particular, we compared four strategies providing different degree of volatility and downside protection: a fixed-mix strategy, a target volatility strategy, a standard OBPI and an OBPI strategy with the target volatility strategy as underlying. The OBPI strategies were constructed with a 90% principal guarantee feature. We

considered these strategies in the context of the Heston stochastic volatility model that we fitted to the Hang Seng index by developing an approximation to the instantaneous volatility through a forward-looking volatility estimator. The fitted parameters revealed a higher long-term volatility and a higher degree of negative correlation between the increments of volatility and index values.

The four strategies were compared in a simulation study in which we considered two regimes: a base case using the fitted values for the Hang Seng index, and a high volatility case in which the volatility of volatility parameter is additionally increased. In both cases, the target volatility strategy exhibits improved distributional characteristics at the 10-year investment horizon compared to the fixed-mix – skewness improves from -0.7 to 0.07 and kurtosis decreases from 3.8 to 3.07. The target volatility strategy is preferred to the fixed mix by risk-averse investors at any level of risk aversion. Further on, in high volatility environment the target volatility strategy becomes relatively more attractive. In a similar vein, the OBPI strategy with a target volatility underlying exhibits a better upside potential and, in the high volatility case, the strategy becomes attractive to a broader set of investors compared to the standard OBPI. The implicit opportunity cost of the classical OBPI appears to be higher than the OBPI with a target volatility underlying.

Additional analysis indicates that optimal portfolios obtained by maximising investor’s expected utility include a significant allocation to the structured strategies with a more sizable allocation




to target volatility strategies in the high-volatility case. In the base case, for example, investors for which the 40-60, 50-50, or 60-40 fixed mix strategies of bonds and stocks are optimal would invest about 90%, 76%, and 62% respectively in a target volatility product with equity as underlying designed to have a constant annualised volatility of 18%. Further on, if we assume the demand for structured products is driven by a set of investors with uniformly distributed risk-aversion levels for which the optimal fixed mix in the absence of a structured product is between 20-80 and 80-20 allocation of fixed-income and stocks, then the average allocation to the target volatility strategy is about 57% in both the base case and the high volatility case. The average allocation to the target volatility OBPI product, however, increases from about 36% to about 63% implying that a capital guarantee feature would be much more demanded in environments where the stochastic nature of volatility is more pronounced.

One drawback of target volatility strategies is the high turnover because of the functional dependence on the value of the instantaneous volatility. The higher turnover can be mitigated by going to weekly or monthly rebalancing that does not impact adversely the distributional characteristics in the Heston world (when these strategies are packaged by an investment bank as structured solutions, the investor has a buy-and-hold product and the drawback mentioned here affects only the bank in its hedging activities).

Our work can be extended in different directions. Empirical research has indicated that Heston’s model, although more realistic

than a constant volatility model and a good first step, ignores the fact that volatility may spike. Therefore, introducing jumps in the instantaneous volatility process or the index value process (or both) is a reasonable extension. Not only would it make the comparisons even more realistic but it could also provide a better rendering of how the turnover of the target volatility strategy changes as a function of the rebalancing frequency and the jump intensity.

From a theoretical viewpoint, target volatility strategies can be rationalised by the framework of dynamic asset allocation models. Extending Merton’s framework with stochastic volatility leads to a fund separation theorem in which the component responsible for performance generation can be interpreted as a target volatility strategy. As a side result, dynamic asset allocation models imply that an efficient way to deal with the problem of extreme risk arising from stochastic volatility is to construct a portfolio with a constant volatility. Thus, another interesting direction for research is to explore further the optimality properties of target volatility strategies in a dynamic asset allocation setting. For example, it is a well-known fact that long-term investors are also subject to short-term constraints. From a practical viewpoint, it would be interesting to see whether introducing constraints in the problem would result in an optimal solution of OBPI-type with an underlying having a target volatility component. We leave these and other interesting questions to further research.


Appendices



Appendices

A1. Clustering of volatility and Asian indicesIn this appendix, we provide additional details on the clustering of volatility property of Asian indices. In order to check if there is clustering of volatility, we use the celebrated GARCH(1,1) model. The model is defined in the following way

K

t t t

t t t

t t t2

12

12

n v f

v f d

v ad bv

= +

=

= + +- -

r

where rt denotes the return observed at discrete time instants, μ denotes the mean of rt , tf are independent and identically distributed random variables, and K > 0, a > 0 and b > 0 are model parameters. This model can take into account the clustering of volatility effect which is apparent from the equation defining the dynamics of tv . The parameter K is related to the

long-run level of volatility, b determines the dependence on the level of volatility at the previous time instant, and a determines the sensitivity to the new information unavailable in t 1v - . Additional information on the properties of the GARCH(1,1) model and other more general models is available in Tsay (2002).

The parameters responsible for the dynamics of volatility are a and b. If both parameters are equal to zero, then volatility is constant. Therefore, one way to check if there is a significant clustering of volatility is to check the significance of these parameters.

Table 12 provides fitted values for the KOSPI 200 and Hang Seng indices using daily log-returns in the period from February 2003 to February 2011. Both parameters are significant for both indices, which means that there is a significant clustering of volatility effect.

The clustering of volatility effect can be visualised through a band defined by a multiple of the fitted volatilities, e.g.

k kt tn v n v+-6 @ where k is a factor. Figure 14 and Figure 15 show the observed historical data together with such a band in which k = 2. Both plots illustrate the dynamic nature of volatility.

Table 12. Estimates of the parameters of GARCH(1,1) fitted on daily log-returns of the KOSPI 200 and Hang Seng indices in the period from February 2003 to February 2011 together with the corresponding 95% confidence intervals.

a b

KOSPI 200Fitted values 0.0773 0.9083

95% Confidence interval [0.0595, 0.0951] [0.8869, 0.9296]

Hang SengFitted values 0.0669 0.9268

95% Confidence interval [0.0519, 0.082] [0.9106, 0.9431]



Appendices

Figure 14. Daily log-returns of the KOSPI 200 index from February 2003 to February 2011 together with the band defined by the interval ,2 2t tn v n v- +6 @ where tv is calculated from a fitted GARCH(1,1) model

Figure 15. Daily log-returns of the Hang Seng index from February 2003 to February 2011 together with the band defined by the interval ,2 2t tn v n v- +6 @ where tv is calculated from a fitted GARCH(1,1) model



Appendices

A2. Fitting Heston’s model to Asian marketsA big practical concern for any model is the statistical problem of parameter estimation. There are two main approaches for Heston’s model – (i) estimate the vector of parameters using only historical data or (ii) extend the information in historical data with the information available in option prices. The general advantage of (i) is that the only data needed is the time series of St . However, the variance of the log-return of St is not observable which implies that the statistical estimation is a filtering problem. The advantage of (ii) is that volatility can be extracted from option prices. Further on, (ii) provides the only way to estimate the market price of risk. A reliable implementation, however, relies on a liquid option market which is a challenging requirement for the markets in Asia. This limitation can be overcome by adopting an approach in which the instantaneous volatility can be proxied through implied volatility. This approach is called the integrated volatility proxy method (see Aït-Sahalia and Kimmel (2007)).

Concerning (i), there are several estimation methods discussed in the academic literature. Moment-based techniques are described by Chacko and Viceira (2003). The approach is based on the characteristic function of log(St )which can be derived in closed form. The parameters of the model are estimated through the generalised methods of moments by minimising the error between the real and the imaginary parts of the theoretical and the empirical characteristic functions. Chacko and Viceira (2003) report parameter estimates for S&P 500 based on daily, weekly, and monthly returns. A similar technique is discussed by

Jiang and Knight (2002). Other methods include approximating the conditional moments of integrated volatility through high-frequency data (see Bollerslev and Zhou (2002)), filtering techniques based on the characteristic function (see Bates (2002)), and a Bayesian approach (see Eraker (2001)).

The approach in (ii) leads to a much more computationally demanding estimation problem. The option price in Eq. 5 is a function of the model parameter, the market price of volatility risk, and the current level of the instantaneous volatility. Since instantaneous variance is not observable, we use the option price to gain inferences about the variable vt . As a result, the estimation method is based on the pair (St , Ct ) where Ct is a European call option on St traded on the market. To avoid problems with liquidity, the option can be selected to be at-the-money with a short time to maturity. Computational complexity arises because the option price has to be calculated for any intermediate approximation of the parameters within the estimation algorithm. Although more demanding, this approach allows full identification of all model parameters.

Different statistical methods can be used in this context. Aït-Sahalia and Kimmel (2007) apply this technique with the method of maximum likelihood while Eraker (2001) uses a Bayesian approach. The reported parameter estimates are based on daily data.

The integrated volatility proxy approach is discussed by Aït-Sahalia and Kimmel (2007). The instantaneous volatility is proxied through implied volatility extracted



from option prices. The goal is to adjust the implied variance of the Black-Scholes model for the mean-reversion effect present in Heston’s model. Suppose that the drift of vt in the risk-neutral world is given by

vtil -l l^ h as in Eq. 4. Denote by ,V T t^ h the average integrated variance,

,V T tT t

v du1u

t

T

=-

^ h #

The expected average integrated variance ,E V T tt ^ h can be approximated by the

implied variance from the Black-Scholes model24. On the other hand, the quantity

,E V T tt ^ h can be derived from the model and is given by

,( )

( )E V T tT t

e v1( )

t

T t

ti il

=-- - +

l- -

ll l

l

^ h

As a consequence, the instantaneous variance vt can be expressed as a function of implied variance and the model parameters in the risk-neutral world,

ve

T t T t

1( )

,t T t

impl t2

.v

il l i

-

- - + -+l- -

l l ll

l

^ ^h h (12)

where ,impl t2v denotes the implied variance

at time t.

The advantage of the integrated volatility proxy method is its computational simplicity. Having obtained a time series of vt , conditional on the parameters determining the drift of vt , we can apply the maximum likelihood estimation algorithm developed by Aït-Sahalia and Kimmel (2007). The disadvantage is that the market price of volatility risk cannot be estimated as it is already incorporated in the drift of vt in the risk neutral world. Therefore, in order to apply this approach in practice, we have to assume a value for 2m .

A2.1 Forward looking estimator of market volatilityParameter estimation of Heston’s model for Asian markets is particularly difficult because option markets in Asia are not as liquid as in the US or Europe. Thus, it is impossible to get a long time series of Ct . Long time series of implied volatilities are difficult to obtain as well.

On the other hand, assuming that no information for vt can be extracted and relying entirely on the time series of St would be too restrictive. Implied volatility indices, such as VIX, provide forward-looking estimates of volatility for the developed markets. Because of globalisation and market integration, we can assume that an estimate of volatility for the developed markets could be informative for the Asian markets which are more integrated in the global economy. If we find a statistically significant relationship between the returns of a market index in Asia and the returns of S&P 500, then we can calculate a forward-looking estimate of volatility of the Asian market index through the VIX index and the estimated regression coefficients.

In fact, significant integration between Asian and the US equity markets has been reported in empirical studies. Burdekin and Siklos (2011) report a significant long-term cointegration between the major Asian equity markets25 and S&P 500 based on data from 1999 to 2010. In addition, Fung et al (2008) report high sensitivity of Asian equity markets to the influence of US markets and also higher cointegration between the more mature markets in the region such as Hong Kong, Japan, Korea, and Singapore.

24 - For additional details and underlying assumptions, see Aït-Sahalia and Kimmel (2007) and the references therein.

25 - The study is based on the following indices – S&P 500 (US), Nikkei 225 (Japan), ASX (Australia), NZX All Index (New Zealand), KOSPI (Korea), Straits Times (Singapore), TWSE (Taiwan), FBMKLCI (Malaysia), JCI (Indonesia), PCOMP (Philippines), SET (Thailand), and Shanghai Composite Index (China).

Appendices



Our approach to fitting the parameters of Heston’s model for Asian markets consists of two steps. First, we use a statistical model to calculate forward-looking estimates of the variance of the market index. Second, we adopt the integrated volatility proxy method and use the estimated variance as a proxy for the instantaneous variance in the maximum likelihood procedure.

In this section, we introduce a forward-looking estimator of market volatility proxied through VIX. We apply the model to the KOSPI 200 and Hang Seng indices because the corresponding economies are in Asia excluding Japan and, at the same time, they are relatively more integrated in the world economy.

Consider the following one-factor model,

r br ea ,t sp t t= + + (13) where rt denotes the return of an Asian market index at time t and r ,sp t denotes the return of S&P 500 at time t and et is the residual. The variance of rt can be expressed as

r b r e,t sp t t2 2 2 2v v v= +t^ ^ ^h h h

We can utilize the squared VIX at time t, VIXt

2 , as an imperfect biased proxy for the volatility of S&P 500. As a consequence, the expression for the variance of rt becomes,

r b eVIXt tt2 2 22v v= +t^ ^h h

where VIXt2 has been transformed to the frequency of rt . In fact the VIX component in the decomposition can be viewed as a global component and et

2v ^ h can be viewed as a local component. The R2

statistic is an indicator of how dominant the global component is.

If we estimate the constant et2v ^ h directly

from the residuals, then our working assumption would be that the dynamics of rt

2v ^ h is a linear function of the dynamics of VIXt2 : the estimated b2t is a scale factor and the constant et

2v ^ h is a shift factor. In order to go beyond this assumption, we assume that et

2v ^ h has its own dynamics that can be extracted cross-sectionally from the constituents of the market index. To this end, we adopt the cross-sectional volatility index introduced by Goltz et al (2011) of the corresponding market index as the driver of the specific volatility of rt . Therefore, our model is

r b VIX ,t t s t2 2 2 2v v= +t^ h (14)

where ,s t

2v is a time dependent specific component calculated through the cross-sectional volatility index. The general idea behind the cross-sectional calculation of the specific volatility is based on the cross-sectional one-factor model representation of the market index as a stock portfolio. Assuming a one-factor model holds for all index constituents, then at any time t, the cross-sectional volatility index is a biased estimator of the average of the cross-sectional specific volatilities which is itself a biased estimator of the specific volatility of the cap-weighted portfolio representing the market index.

The success of this approach greatly depends on the degree of integration of the corresponding Asian market to the global market. We apply the approach to the main market indices of Hong Kong and South Korea.

Appendices



Table 13. Intercept, the slope, and the R2 statistic of the regression model for the Hang Seng and KOSPI 200 indices The 95% confidence intervals are provided below the corresponding estimate.

Intercept Slope R2

KOSPI 200 0.0084 1.039 51%

95% CI (-0.001, 0.0178) (0.8282, 1.2337)

Hang Seng 0.0072 0.9973 51.7%

95% CI (-0.0018, 0.0162) (0.8043, 1.1903)

Figure 16. Scaled index values of KOSPI 200, Hang Seng and S&P 500 from February 2002 to June 2010

!

!"#

$

$"#

%

%"#

&

&"#

!"#$%&'()*$+&,+-()$#.

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Figure 17. Scatter-plot diagrams of the monthly returns of the KOSPI 200 and S&P 500 indices (left) and the Hang Seng and S&P 500 indices (right) from February 2002 to June 2010There is a clear positive relationship between the corresponding indices.

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!"#$%&'#

$

&()*++

!"#$%

!"#$

!"#&%

!"#&

!"#"%

"

"#"%

"#&

"#&%

"#$

!"#$ !"#&% !"#& !"#"% " "#"% "#& "#&%

!"#$%&'((

#)$*((

Appendices



The regression results are reported in Table 13 and are based on monthly returns from February 2002 to June 2010. The scaled values of the three indices are shown in Figure 16 and scatter-plots of the returns are shown in Figure 17.

The results in Table 13 indicate a satisfactory explanatory power of the regression model. Before calculating the variance of rt from Eq. 14, we have to compute the specific variance. We calculate it in two steps: first, we calculate the cross-sectional volatility index for the Hang Seng and KOSPI 200 indices for the period from February 2002 to June 2010 using monthly returns. Since the cross-sectional volatility index itself is a biased estimator of the specific volatility of the market index, we have to apply a correction. However, the only information that can be extracted from the factor model is the average specific volatility in the corresponding time period by computing the volatility of the residuals. We use this information to match the scale of the average cross-sectional volatility index in this period. Therefore, we set the specific

variance in Eq. 14 to

T

e

1,

,

,s t t

CR t

t

T

tCR

2

1

2

vv

vv=

=

^ h> H/

where etv^ h is the standard deviation of the residual in the one-factor model and ,CR tv denotes the cross-sectional volatility index calculated at time t. Thus, the dynamics of rtv^ h is essentially determined by the dynamics of the cross-sectional volatility index.

Plots of the annualised forward-looking volatilities rtv^ h of the monthly returns of the KOSPI 200 and Hang Seng indices as calculated from Eq. 14 are provided in Figure 18. The volatility of the monthly return of S&P 500 is the VIX index.

The volatilities of the Hang Seng and KOSPI 200 indices are generally bigger than VIX which is expected. The similarity in the dynamics of the three time series is due to the common global component but there is also a noticeable specific dynamic for each index.

Figure 18. Annualised forward-looking estimates of volatility of the monthly returns of the KOSPI 200 and Hang Seng indices calculated through Eq. 14The graph for S&P 500 is the VIX index

!"

#!"

$!"

%!"

&!"

'!"

(!"

)!"

!"#$%&'()*+,),+-

*+,-./$!!

0123/,423

,5-'!!

Appendices



A2.2 Estimating the parameters of Heston’s modelIn our approach to fitting the parameters of Heston’s model, we assume that the forward-looking variances estimated through Eq. 14 can be used as a proxy for the instantaneous variance in Heston’s model via Eq. 12. In this way, we can use the method of maximum likelihood; however, we need an assumption for the market price of volatility risk because it appears in the model parameters in the risk neutral world and cannot be estimated without incorporating data from traded instruments. Therefore, to calculate vtthrough Eq. 12, we assume 2m = 0.0455 which is the value fitted in Eraker (2004).

We employ the following algorithm based on the method of maximum likelihood.

Step 0. Initialised the parameter estimates, , , , , .0 0 0 0 0n l i v v^ h This is the current

best approximation. Set m = 1 which is an integer counting the iterations.

Step 1. Calculate the vt from Eq. 12 using the current best approximation of the

parameter values and 2m = 0.0455.

Step 2. Use the pair of time series (St , vt ), where St is the value of the corresponding market index, in the maximum likelihood method to obtain the next best approximation, , , , .m m m m mn l i v t^ h

Step 3. Check if the Euclidean distance between , , , ,m m m m m1 1 1 1 1n l i v t- - - - -^ h , , , ,m m m m mn l i v t^ h and smaller than a

certain tolerance level (in this case 0.001). If yes, then the current best approximation is accepted as the maximum likelihood estimate. If not, then increment m and go back to Step 1.

Parameter estimates of Heston’s model for the KOSPI 200, Hang Seng, and S&P 500 indices fitted according to the algorithm outlined above and the corresponding 95% confidence intervals are provided in Table 14. We use monthly data for the index values from February 2002 to June 2010 and the calculated implied volatility proxies.

Table 14. Parameter estimates of Heston’s model for KOSPI 200, Hang Seng, and S&P 500 indices using monthly data from February 2002 to June 2010

l i v t

KOSPI 200 0.3064 0.0641 0.097 -0.92

95% CI (0.2307, 0.3821) (0.0481, 0.0801) (0.0687, 0.1253) (-1, -0.77)

Hang Seng 0.2157 0.0696 0.0778 -0.86

95% CI (0.1480, 0.2834) (0.0476, 0.0915) (0.0423, 0.1131) (-1, -0.59)

S&P 500 0.2147 0.0314 0.0955 -0.65

95% CI (0.0677, 0.3617) (0.0027, 0.0601) (0.0778, 0.1131) (-0.76, -0.54)

Appendices



To simulate sample paths from Heston’s model, we need an estimate of the parameter μ. The SDE of the index log-returns is given by

2

lnd Svdt v dWt

tt sn= - +` j

and the mean of the log-return distribution at a given frequency tD depends only on the drift. To estimate μ, we use the estimator

tx

2n i

D= +t r t

where is the average log-return at frequency and tD is the unconditional mean of vtgiven in Table 14. The estimated values of μ for the three indices are provided in Table 15.

Table 15. Estimates for the drift parameter for the three indices

μKOSPI 200 0.1240

Hang Seng 0.1131

S&P 500 0.0072

The estimates of the volatility of volatility v provided in Table 14 are significant for the three indices and we cannot reject the hypothesis that the instantaneous volatility of the log-returns of the market indices is constant. This is an argument against the

assumption of GBM in the classical models and supports the hypothesis that volatility is itself stochastic.

There are several empirical studies in the academic literature that report parameter estimates of Heston’s model for the S&P 500 index. It is, however, difficult to compare the values across papers because different authors chose different samples and also different frequencies. Another reason, as reported by Chacko and Viceira (2003), is the interplay between the model parameters. For example, conditioning the variance of vt on v0 = i in Eq. 2 leads to the following simpler expression

|Var v v e210t

t2

2ilv= = - l-^ ^h h

that shows the conditional variance of vtdepends on the ratio /22v l . As a result, a higher value of v can be compensated by a higher value of l in order to match the conditional variance in the data. In the same manner, the long-run variance given in Eq. 3, which is also the unconditional variance of vt , depends not only on v and l but also on i .

To have some benchmark, we compare the unconditional standard deviation of vt calculated from the fitted parameters for S&P 500 in Table 14 and also from the fitted

Table 16. Unconditional standard deviation of vt calculated from estimates reported in different studies

Chacko and Viceira (2003), Table 1

Aït-Sahalia and Kimmel (2007), Table 6

This paper, Table 14

Monthly returns from

1926 to 1997

Daily returns from

1990 to 2004

Monthly returns from

2002 to 2010

StDev v2

t vli=^ h 0.0448 0.0339 0.0258

Appendices



parameters in Table 1 in Chacko and Viceira (2003) and also in Table 6 in Aït-Sahalia and Kimmel (2007). Chacko and Viceira (2003) use monthly returns from 1926 to 1997 and the generalised method of moments while Aït-Sahalia and Kimmel (2007) use daily returns from 1990 to 2004 and the method of integrated proxy combined with maximum likelihood estimation.

The comparison is provided in Table 16. Although the time periods, the estimation methods, and the estimated values are different, the calculated unconditional standard deviation does not vary dramatically. In fact, the number calculated from the values in Table 14 in this paper is very close to the number in the second column which is expected because the estimation methodology is essentially the same. The number in the first column is almost twice as big but this is not surprising since the period 1926-1997 includes market crashes bigger than the ones from the last decade.

If we compare the fitted parameters across markets, we find three interesting

observations. Firstly, the estimated correlation coefficients are all negative which implies negative skewness of the unconditional distribution. The ones corresponding to the Hang Seng and KOSPI 200 indices are significantly larger in absolute value, even though the confidence bounds are relatively larger if compared to the ones corresponding to the S&P 500 index. For example, we cannot accept that the correlation coefficient of the Hang Seng index is significantly bigger than that of S&P 500 index. We can, however, accept this hypothesis for the KOSPI 200 index.

Secondly, the estimated values of the volatility of volatility parameter are close to each other and we can safely assume that they are not materially different across the three markets.

Finally, there is a significant difference in the unconditional mean of vt of the three markets – the i of the two Asian markets is twice the i of the US market. Since irepresents the long-term level of volatility, we can conclude that the long-term level of volatility in Asia is significantly higher than

Table 17. Parameter estimates of Heston’s model for the KOSPI 200, Hang Seng, and S&P 500 indices using monthly data from February 2002 to June 2010 assuming a ten times larger value for the market price of volatility risk, 2m = 0.455

l i v t

KOSPI 200 0.3275 0.0599 0.0968 -0.92

95% CI (0.2477, 0.4073) (0.0452, 0.0747) (0.0688, 0.1248) (-1, -0.77)

Hang Seng 0.2315 0.0649 0.0775 -0.87

95% CI (0.1586, 0.3044) (0.0443, 0.0854) (0.0419, 0.1132) (-1, -0.59)

S&P 500 0.1869 0.0362 0.0954 -0.65

95% CI (0.0571 0.3167) (0.0038, 0.0685) (0.0778, 0.1130) (-0.76, -0.54)

Appendices



the long-term volatility for the US market. In fact, the estimate of i for S&P 500 is outside of the confidence bounds of the estimates of i for the Asian markets.

The last question that needs attention is the marginal impact of the value of 2m on the parameter estimates. Table 14 is produced assuming 2m = 0.0455 fitted in Eraker (2004) and it makes sense to re-estimate the parameters of the model assuming a different value. It turns out the parameter estimates are not very sensitive to the value of 2m .

Fitted values for a market price of volatility risk 2m = 0.455 are provided in Table 17. Comparing the values in Table 14 and Table 17, we conclude that there are small changes in the estimated values for l and .i The estimates of v and t remain virtually the same. Finally, the main conclusions drawn on the basis of Table 14 continue to hold.

Apart from the total volatility proxy developed in this appendix, there are two other methods for obtaining an approximation for the volatility of the index log-returns: (i) use the implied volatility time series provided it is available and (ii)

directly use the cross-sectional volatility as a proxy. The latter approach is motivated by Goltz et al (2011), who report high correlations between the implied volatility and cross-sectional volatility time series.

Table 18 provides fitted values for the parameters of Heston’s model for the Hang Seng Index based on daily data. The values in the first row are calculated using the corresponding implied volatility index (VHSI) and the values in the second row are calculated with the cross-sectional volatility computed according to the methodology described in Goltz et al (2011). Comparing these estimates to the numbers in Table 17, we find that the fitted value for the volatility of volatility parameter is higher when using daily data. To be conservative, we use the values in Table 17 as they imply a smaller deviation from the constant volatility assumption.

Determining which of the above approaches would be best suited in practice goes beyond the scope of this paper but one general method would be to back-test the strategy of interest under each of these approaches and perform a cost and benefit (control of volatility achieved) analysis.

Appendices

Table 18. Parameter estimates of Heston’s model for the Hang Seng Index using daily data for the HSI Volatility index (VHSI) and the cross-sectional volatility The time period is from February 2002 to June 2010

l i v t

VHSI 0.3765 0.0426 0.1714 -0.8235

Cross Sectional Volatility 1.0753 0.0621 0.2222 -0.3783



A3. Properties of the fixed-mix and target volatility strategies in Heston’s modelIn this appendix, we demonstrate that the left tail of the log-return distribution of an asset evolving according to Eq. 7 is Gaussian. Applying Ito’s lemma leads to the following equation for the process

logYt tr= ^ h

dY rv

TV r TV dt TVdW2

t

t

S

2

n= + - - +^c h m

Integrating this equation leads to

where the random variable dWt

T

x# follows a normal distribution with a zero mean and variance equal to T and the random variable

vd1

t

T

xx

# is non-negative almost surely. Non-negativity holds because vt follows a square-root process and is non-negative almost surely for any t. The main result is stated in the next proposition.

Proposition 1. Suppose that vt is a square-root process defined in Eq. 1 such that 2li > 2v . If μ ≥ r, then the random variable derived in Eq. 15 has a left tail which is asymptotically not heavier than a Gaussian tail

, ]limP Z x

P Y Y x0 1

x

T t

#!

#-" 3- ^

^ ^hh

where Z has a normal distribution.

Proof. The assumptions in the proposition guarantee that vt is strictly positive for any t and that the coefficient TV(μ – r) is non-negative. The proof follows directly from Eq. 15 that can be rewritten as Y Y aX ZT t- = +

where a ≥ 0 and Z has a normal distribution. The probability in the numerator can be bounded in the following way

( | ) ( )

( | ) ( ) ( )

P Y Y x P Z x aX

P Z x ay X y dP X y

P Z x X y dP X y P Z x

T t

0

0

# #

# #

# # # #

- = -

= - =

= =

3

3

^ ^h h

#

#

because the integrand can be bounded, ( | ) ( | ) .P Z x ay X y P Z x X y# # #- = =

As a result

( )P Z x

P Y Y x0 1<

T t

#

##

-^ h

for any x . Q.E.D.

Because of the non-negativity of the first stochastic term in Eq. 15, the distribution of Y YT t- is expected to have a positive skewness. Further on, the annualised volatility of Y YT t- is bigger than TV /1 2 because of the same term.

As far as the fixed-mix is concerned, the wealth of the strategy is described by the Heston model, however, with different parameters. The SDE describing the dynamics of the wealth process is given by

( )d w w r dt w v dW1 t t t S2r n r r= + - +^ h

where w is a positive constant and vt follows the square-root process. The SDE for the scaled vt is derived using Ito’s lemma

d w v w v dt w v dWt t t v2 2 2l i v= - +^ ^h h

which can be rewritten as

( )d w v w w v dt w w v dWt t t v2 2 2 2l i v= - +^ h

Y Y r TV T t TV rvd TV dW

21

T t

t

T

t

T

n x- = - - + - +x

xc ^ ^m h h # # (15)

Appendices



The dynamics of the wealth process can be restated as a system of two SDEs d dt v dW

dv v dt v dW

dW dW dt

t w t tw

t S

tw

w tw

w tw

v

S v

r n r r

l i v

t

= +

= - +

=

^ h

where wn = ww r1n -+ ^^ h h, wi =w2i and wwv v= . As a result, the distribution of the log-return of the fixed-mix strategy is skewed to the left (if the correlation between the two Wiener processes is negative) and has excess kurtosis and its left tail is thicker than the normal tail. Consequently, it is thicker than the tail of the log-return of the target volatility strategy.

Finally, a fixed mix between a target volatility strategy and the risk-free asset results in a portfolio which is itself a target volatility strategy, however, with a smaller target volatility parameter, proportional to the allocation to the initial strategy. The proof is straightforward.

A4. Pricing options on a target volatility strategy in Heston’s modelSuppose that the price process of a risky asset is described by the following systems of SDEs

dr

v

TV r

dv v

dW dW dt

dt TVdW

dt v dW

t

t

t

t t

S v

S

t v

rr n

l i

t

= + -

= -

=

+

+

^c

^

hm

h

in which TV is a constant. In fact, this is the original Heston-model setup in which the risky asset is a constant volatility strategy.

The model can be equivalently restated in the following way

dr

v

TV r dt

TV dW dW

dv v dt v dW

1

t

t

t

t t

t t t t

12

2

1

rr n

t t

l i v

= + -

+ + -

= - +

^c

^^

hm

hh

where dW t1 and dW t2 are independent.

It is possible to demonstrate that the conditions which guarantee existence of an equivalent martingale measure (EMM) in the Heston model are also sufficient conditions for the model in Eq. 16. Under the EMM, the equation for tr becomes an ordinary GBM implying that the classical Black-Scholes formula can be used to price an option with tr as underlying. The main result is included in the following proposition.

Proposition 2. Suppose that the real-world dynamics in the economy are described by the system in Eq. 16. If / <2 3#l v m- and 2 2$li v , then the Q-measure defined by

( )

21

expdP

dQL T

u dW u dW

u du u du

1

0

1 2 2

0

12

0

22

0

T

u u

T

T T

c c

c c

= =

+

- +

- ^ ^e

^ ^e

h h o

h h o

Z

[

\

]]]

]]

_

`

a

bbb

bb

# #

# #

where

t

t

v

v

rv

1

1

t

t

t

1

2

2

22

c

c

m

t

nm t

=

=-

--

^

^ c

h

h m

is an equivalent local martingale measure. Under the measure Q, the discounted trprocess is a martingale and is invariant of the choice of 2m .

Proof: Wong and Heyde (2006) demonstrate that under the conditions stated in the theorem

Appendices

(16)



E(L(T)) = 1 and L(T) is a martingale. As a consequence, Girsanov’s theorem can be applied and the measure Q is an equivalent local martingale measure. It remains to verify if the discounted tr process is a martingale under Q. To this end, define

dW t

dW t dt dW

dt dWtQ

tQ

t

t1 1

2 22

1c

c

=

= +

+^^hh

A direct calculation shows that

drdt TV dW dW1

t

ttQ

tQ

12

2rr t t= + + -^ h

The discounted price process equals

.

expZ T rT

TV W TV W TV T12

T

TQ

TQ

12

2

2

r

t t

= -

+ - -exp=

^ ^

c

h h

m

The mathematical expectation of Z(T) is easy

to calculate directly bearing in mind that the two random variables in the exponent are independent

.

1

exp exp

E Z T

E TV W E TV W TV T12

TQ

TQ

12

2

2

t t= - -

=

^

^ c

h

h m

6 @

As a result, the discounted price process Z(T) is a martingale under the measure Q.

Q.E.D.

A5. Asset allocation in mean-variance and mean-CVaR frameworksApart from the preferences of risk-averse investors as described by CRRA utility functions, we consider also the allocations among the bond, the index, and the structured strategy along the efficient frontiers generated by two frameworks: the

Figure 19. Efficient portfolios along the mean-variance efficient frontier. The allocations are the same for the base case and the high volatility case. Smaller portfolio labels correspond to higher risk aversion.

Appendices



mean-variance analysis and the mean-CVaR analysis where CVaR stands for conditional value-at-risk. The mean-variance framework is a crude second-order approximation to the utility function approach which ignores the impact of higher-order moments. The mean-CVaR framework is an alternative in which the risk-measure is focused on the downside and takes into account the non-linearities in the pay-off distribution of the structured product. CVaR also satisfies a set of sensible properties which qualify it as a coherent risk measure (see Artzner et al (1998)).

CVaR at a tail probability f is defined as the average loss provided that the loss is larger than a given quantile level,

CVaR X q X dp1p

0f

=-f

f

^ ^h h#

where q Xp ^ h stands for the p-quantile of the return described by the random variable X. Being an average of extreme losses, CVaR focuses entirely on the left tail of the distribution. In our calculations, we choose f = 5% which is a standard choice26.

The efficient portfolios of both frameworks are obtained by solving an optimisation problem of the following generic type

. .

max w Er Risk w r

s t w w w

w

1

0

1 2 3

#

$

h-

+ + =

l l^ h6 @

where Risk in the objective function is either portfolio variance or portfolio CVaR and h represents the degree of risk aversion. The set of all efficient portfolios is obtained by varying h . If h is very large, then the optimal solution converges to the global minimum risk portfolio and, if h = 0, then the solution is the maximum expected return portfolio.

The mean-variance efficient portfolios are provided on Figure 19. Portfolios with a smaller label correspond to a higher level of risk aversion. The efficient portfolios are one and the same in the base case and the high-volatility case. Apparently, irrespective of the structured strategy, the optimal portfolios are allocated to the bond and

Table 19. Allocations in the model portfolios and corresponding risk and return characteristics


Base caseProduct 0.00% 57.70% 70.43% 82.81%

Market index 40.00% 0.00% 0.00% 0.00%

Bond 60.00% 42.30% 29.57% 17.19%

Mean 9.70% 9.70% 9.70% 9.70%

CVaR 16.89% 16.27% 0.50% 4.48%

High volatility caseProduct 0.00% 46.64% 69.34% 65.98%

Market index 40.00% 0.00% 0.00% 0.00%

Bond 60.00% 53.36% 30.66% 34.02%

Mean 9.77% 9.77% 9.77% 9.77%

CVaR 17.97% 8.88% 0.15% -0.93%

Appendices

26 - The choice f= 5% corresponds to a threshold quantile equal to the 95% value-at-risk of the return distribution. Thus CVaR5% measures the average loss beyond that value-at-risk level.



Figure 20. Efficient portfolios along the mean-CVaR efficient frontier in the base caseSmaller portfolio labels correspond to higher risk aversion.

Figure 21. Efficient portfolios along the mean-CVaR efficient frontier in the high volatility caseSmaller portfolio labels correspond to higher risk aversion.

Appendices



the stock only. The reason is that the better properties of the non-linear pay-off of the structured strategy are not recognised by the mean-variance framework and the positive skewness is in fact penalised.

The mean-CVaR efficient portfolios in the base case and the high volatility case are provided on Figure 20 and Figure 21, respectively. Firstly, the difference from the mean-variance optimal allocations is obvious. Separating the upside from the downside in the risk measure leads to a significant preference for the corresponding structured strategy. Secondly, at a given level of risk aversion, the market index has a smaller allocation in the high-volatility case.

Choosing the 40-60 allocation in the market index and the bond as a model portfolio, we can compare the corresponding mean-CVaR optimal portfolios yielding the return of the model portfolio. The corresponding allocations are provided in Table 19. The benefit of including a structured strategy in the portfolio is quite significant in the high-volatility case.

The same analysis can be performed for other model portfolios. The main conclusion, however, is similar to the one drawn from the utility function based framework – the better distributional characteristics translate to significant allocations in the target volatility strategy. Intuitively, the capital guarantee feature leads to more significant benefits in economies characterised by a more significant departure from a constant volatility economy.

A6. Optimal properties of target volatility strategiesA general and consistent framework for asset allocation of long-term investors is provided by the dynamic portfolio theory posited by Merton (1969, 1971). The theory presents the most natural form of asset management generalising substantially the static portfolio selection model developed by Markowitz (1952). Merton (1971) demonstrated that in addition to the standard speculative motive, non-myopic long-term investors include intertemporal hedging demands in the presence of a stochastic opportunity set. The model has been extended in several directions: with stochastic interest rates only (Lioui and Poncet 2001; Munk and Sørensen 2004), with a stochastic, mean-reverting equity risk premium and non-stochastic interest rates (Kim and Omberg 1996; Wachter 2002), with both variables stochastic (Brennan et al. 1997; Munk et al. 2004).

Other papers have considered the impact of stochastic volatility, see for example Deguest et al (2011) and the references therein. Deguest et al (2011) prove that in a setting allowing for stochastic interest rates driven by the Vasicek (1977) model, an equity index driven by a GBM with instantaneous variance driven by Heston’s model, a four fund separation theorem holds for the optimal solution of risk-averse investors whose preferences are described by a CRRA utility function. The building block of the optimal portfolio responsible for speculative demand, also known as the performance seeking component, has the same structure as the speculative demand in Merton’s model that assumes constant investment opportunities

Appendices

w wtSD

tm

tm

tm

cvm

=



where wtm denotes the portfolio maximising the Sharpe ratio over the next trading interval, t

mm is the Sharpe ratio of wtm , tmv

denotes the volatility of wtm , and c is the risk-aversion parameter of the CRRA utility function. The difference from an economy with constant opportunities is that the maximum Sharpe ratio portfolio depends on the current Sharpe ratio and the volatility of the equity index and the allocation to this building block in the optimal portfolio,

tm

tm

cvm , is also a function of these inputs.

The speculative demand component wtSD

can also be interpreted as a target volatility strategy. To do that, we re-arrange the terms

/w w

1

target volatility strategy

tSD

tm

tm t

mmvc

= c m; E

where /w

1

tm t

m

vcc m; E is in fact a target

volatility strategy (compare to Eq. 1027) with a target volatility equal to 1⁄c28 .

As a consequence, the value of the target volatility parameter is inversely proportional to the risk-aversion parameter implying that different investors would aim at different levels of volatility but their overall strategy is essentially the same – it is to maintain a constant volatility across time of the maximum Sharpe ratio portfolio.

Alternatively, the solution can also be represented as

w w1 1

arg

tSD

tm

tm t

m

t et volatility strategy

mc v

= c m; E

where the target volatility strategy has a unit volatility and the risk-aversion level controls its relative weight in the optimal portfolio. Since one of the four funds in

the optimal solution is cash, by allocating 1c to the target volatility strategy and (1- 1c ) to cash, we obtain a portfolio with volatility equal to 1⁄c . The difference in this interpretation is that the target volatility strategy defined by w1

tm t

m

vc m is one and the

same for all investors because it is invariant of the risk-aversion level. In other words, any risk-averse investor would be willing to hold this strategy in their portfolio.

The parameter tmm is the Sharpe ratio of

the maximum Sharpe ratio portfolio but, because at each time t the target volatility strategy is a portfolio of wtm and cash, it follows that t

mm is also equal to the Sharpe ratio of the target volatility portfolio. Formally, this is evident from Eq. 7; the Sharpe ratio of the target volatility portfolio at each time t is equal to the Sharpe ratio of the underlying equity index.

From the standpoint of extreme risk management, it is amazing that without an explicit objective to hedge the tail risk arising from the stochastic volatility, the optimal solution implies that all investors try to do so by maintaining a constant volatility of their performance seeking portfolio. As demonstrated in Appendix 3, this goal results in a portfolio the log-return distribution of which has a light left tail implying the probabilities of extreme events are reduced.

27 - In the denominator in Eq. 10, there is the square root of instantaneous variance while here we use directly the instantaneous volatility. Note that the portfolio wt

m consists of

several assets while Eq. 10 refers to only one risky asset. This is, however, not restrictive because re-normalising the weights by

tmv , we obtain a portfolio

with volatility equal to 1 for any time t which represents a target volatility strategy.

28 - There is an investment in cash which is missing in the equation but is generally present in the optimal solution and represents one of the four funds in the fund separation result by Deguest et al (2011).

Appendices



Appendices


References



References

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• Amenc, N., F. Goltz, and S. Stoyanov. 2011. A post crisis perspective on diversification for risk management. EDHEC Risk-Institute publication (May).

• Amenc, N., L. Martellini, F. Goltz, and V. Milhau. 2010. New frontiers in benchmarking and liability-driven investing. EDHEC Risk-Institute publication (September).

• Artzner, P., F. Delbaen, J.-M. Eber and D. Heath. 1998, Coherent measures of risk, Mathematical Finance 6: 203–228.

• Baker, M., and J. Wurgler. 2010. Co-movement and predictability relationships between bonds and the cross-section of stocks, NYU Working Paper No. FIN-10-003, available online at: http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1688191

• Basak, S. 1995. A general equilibrium model of portfolio insurance. Review of Financial Studies 8: 1059-90.

• Bates, D.S. 2002. Maximum likelihood estimation of latent affine processes. Working Paper, University of Iowa.

• Bollerslev, T. 1986. Generalized Auto-regressive Conditional Heteroskedasticity. Journal of Econometrics 31: 307-327.

• Bollerslev, T., and H. Zhou. 2002. Estimating stochastic volatility diffusions using conditional moments of integrated volatility. Journal of Econometrics 109: 33–65.

• Brennan, M., E. Schwartz, and R. Lagnado. 1997. Strategic asset allocation. Journal of Economic Dynamics and Control 21 (8-9):1377-1403.

• Burdekin, R., and P. Siklos. 2011. Enter the dragon: Interactions between Chinese, US, and Asia-Pacific equity markets, 1995-2010, HKIMR Working Paper 23/2011.

• Carr, P., and D. Madan. 1999. Option valuation using the fast Fourier transform. Journal of Computational Finance 2(4): 61-73.

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• Clark, P. 1973. A subordinated stochastic process model with finite variance for speculative prices. Econometrica 41(1): 135-155.

• Deguest, R., L. Martellini, and V. Milhau. 2011. Life-Cycle Investing for Individual Investors - When Wall Street meets Main Street. EDHEC-Risk Institute, working paper.

• Draper, D., and D. Shimko. 1993. On the Existence of ‘Redundant Securities’, Working Paper, University of Southern California.



References

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• Engle, R. 1982. Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation. Econometrica 50(4): 987-1007.

• Engle, R. 2003. Risk and Volatility: Econometric models and financial practice. Nobel Lecture.

• Eraker, B. 2001. MCMC analysis of diffusion models with application to finance. Journal of Business and Economic Statistics 19: 177–191.

• Eraker, B. 2004. Do stock prices and volatility jump? Reconciling evidence from spot and option prices. The Journal of Finance 59(3): 1367-1402.

• Fama, E. 1963. Mandelbrot and the stable Paretian hypothesis. Journal of Business 36: 420–429.

• Fama, E. 1965. The behavior of stock market prices. Journal of Business 38: 34–105.

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• Fung, L., C-S. Tam and I. Yu. 2008. Assessing the Integration of Asia's Equity and Bond Markets. Bank of International Settlement Papers (42).

• Glasserman, P. 2004. Monte Carlo methods in financial engineering, Springer-Verlag, New York.

• Goltz, F., L. Martellini, and K. Simsek. 2008. Static Allocation Decisions in the Presence of Portfolio Insurance. Journal of Investment Management, Vol. 6, Issue 2, p37-56.

• Goltz, F., R. Guobuzaite, L. Martellini, and S. Stoyanov. 2011. Introducing a New Form of Volatility Index: the Cross-Sectional Volatility Index, forthcoming in Bankers, Markets & Investors.

• Grossman, S. J., and J.-L. Vila. 1989. Portfolio insurance in complete markets: A note. Journal of Business 62:473-76.

• Hens, T., and M. Rieger. 2009. The dark side of the moon: structured products from the customer’s perspective. EFA 2009 Bergen Meetings Paper. Available at SSRN: http://ssrn.com/abstract=1342360

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• Jobst, Andreas A. 2008. The development of equity derivative markets in emerging Asia. International Journal of Emerging Markets 3(2): 163:180.



References

• Jondeaua, E., and M. Rockinger. 2003. Testing for differences in the tails of stock-market returns. Journal of Empirical Finance 10: 559-581.

• Kim, T., and E. Omberg. 1996. Dynamic nonmyopic portfolio behavior. Review of Financial Studies 9 (1): 141-61.

• Kittiakarasakuna, J., and Y. Tse. 2010. Modeling the fat tails in Asian stock markets, forthcoming in International Review of Economics and Finance.

• Leland, H. 1980. Who should buy portfolio insurance? Journal of Finance 35: 581-94.

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About EDHEC-Risk Institute




The Choice of Asset Allocation and Risk ManagementEDHEC-Risk structures all of its research work around asset allocation and risk management. This issue corresponds to a genuine expectation from the market.

On the one hand, the prevailing stock market situation in recent years has shown the limitations of diversification alone as a risk management technique and the usefulness of approaches based on dynamic portfolio allocation.

On the other, the appearance of new asset classes (hedge funds, private equity, real assets), with risk profiles that are very different from those of the traditional investment universe, constitutes a new opportunity and challenge for the implementation of allocation in an asset management or asset-liability management context.

This strategic choice is applied to all of the Institute's research programmes, whether they involve proposing new methods of strategic allocation, which integrate the alternative class; taking extreme risks into account in portfolio construction; studying the usefulness of derivatives in implementing asset-liability management approaches; or orienting the concept of dynamic “core-satellite” investment management in the framework of absolute return or target-date funds.

An Applied Research ApproachIn an attempt to ensure that the research it carries out is truly applicable, EDHEC has implemented a dual validation system for the work of EDHEC-Risk. All research work must be part of a research

programme, the relevance and goals of which have been validated from both an academic and a business viewpoint by the Institute's advisory board. This board is made up of internationally recognised researchers, the Institute's business partners, and representatives of major international institutional investors. Management of the research programmes respects a rigorous validation process, which guarantees the scientific quality and the operational usefulness of the programmes.

Six research programmes have been conducted by the centre to date: • Asset allocation and alternative diversification• Style and performance analysis • Indices and benchmarking• Operational risks and performance• Asset allocation and derivative instruments• ALM and asset management

These programmes receive the support of a large number of financial companies. The results of the research programmes are disseminated through the EDHEC-Risk locations in London, Nice, and Singapore.

In addition, EDHEC-Risk has developed a close partnership with a small number of sponsors within the framework of research chairs or major research projects:• Regulation and Institutional Investment,in partnership with AXA Investment Managers• Asset-Liability Management and Institutional Investment Management, in partnership with BNP Paribas Investment Partners• Risk and Regulation in the European Fund Management Industry, in partnership with CACEIS

Founded in 1906, EDHEC is one of the foremost French

business schools. Accredited by the three main international

academic organisations, EQUIS, AACSB, and Association

of MBAs, EDHEC has for a number of years been pursuing

a strategy for international excellence that led it to set up

EDHEC-Risk in 2001. With sixty-six professors,

research engineers, and research associates, EDHEC-Risk has

the largest asset management research team in Europe.




• Structured Products and Derivative Instruments, sponsored by the French Banking Federation (FBF)• Dynamic Allocation Models and New Forms of Target-Date Funds,in partnership with UFG-LFP• Advanced Modelling for Alternative Investments, in partnership with Newedge Prime Brokerage• Asset-Liability Management Techniques for Sovereign Wealth Fund Management, in partnership with Deutsche Bank• Core-Satellite and ETF Investment, in partnership with Amundi ETF• The Case for Inflation-Linked Corporate Bonds: Issuers’ and Investors’ Perspectives, in partnership with Rothschild & Cie• Advanced Investment Solutions for Liability Hedging for Inflation Risk, in partnership with Ontario Teachers’Pension Plan• Exploring the Commodity Futures Risk Premium: Implications for Asset Allocation and Regulation, in partnership with CME Group• Structured Equity Investment Strategies for Long-Term Asian Investors, in partnership with Société Générale Corporate & Investment Banking• The Benefits of Volatility Derivatives in Equity Portfolio Management, in partnership with Eurex • Solvency II Benchmarks,in partnership with Russell Investments

The philosophy of the Institute is to validate its work by publication in international journals, as well as to make it available to the sector through its position papers, published studies, and conferences.

Each year, EDHEC-Risk organises a major international conference for institutional investors and investment management professionals with a view to presenting the results of its research: EDHEC-Risk Institutional Days.

EDHEC also provides professionals with access to its website, www.edhec-risk.com, which is entirely devoted to international asset management research. The website, which has more than 42,000 regular visitors, is aimed at professionals who wish to benefit from EDHEC’s analysis and expertise in the area of applied portfolio management research. Its monthly newsletter is distributed to close to one million readers.

Research for BusinessThe Institute’s activities have also given rise to executive education and research service offshoots. EDHEC-Risk's executive education programmes help investment professionals to upgrade their skills with advanced risk and asset management training across traditional and alternative classes.

The EDHEC-Risk Institute PhD in Financewww.edhec-risk.com/AIeducation/PhD_Finance

The EDHEC-Risk Institute PhD in Finance is designed for professionals who aspire

EDHEC-Risk Institute: Key Figures, 2009-2010

Nbr of permanent staff 66

Nbr of research associates 18

Nbr of affiliate professors 6

Overall budget €9,600,000

External financing €6,345,000

Nbr of conference delegates 2,300

Nbr of participants at EDHEC-Risk Indices & Benchmarks seminars 582

Nbr of participants at EDHEC-Risk Institute Risk Management seminars 512

Nbr of participants at EDHEC-Risk Institute Executive Education seminars 247




to higher intellectual levels and aim to redefine the investment banking and asset management industries. It is offered in two tracks: a residential track for high-potential graduate students, who hold part-time positions at EDHEC, and an executive track for practitioners who keep their full-time jobs. Drawing its faculty from the world’s best universities and enjoying the support of the research centre with the greatest impact on the financial industry, the EDHEC-Risk Institute PhD in Finance creates an extraordinary platform for professional development and industry innovation.

FTSE EDHEC-Risk Efficient Indiceswww.edhec-risk.com/indexes/efficient

FTSE Group, the award winning global index provider, and EDHEC-Risk Institute launched the first set of FTSE EDHEC-Risk Efficient Indices at the beginning of 2010. Offered for a full global range, including All World, All World ex US, All World ex UK, Developed, Emerging, USA, UK, Eurobloc, Developed Europe, Developed Europe ex UK, Japan, Developed Asia Pacific ex Japan, Asia Pacific, Asia Pacific ex Japan, and Japan, the index series aims to capture equity market returns with an improved risk/reward efficiency compared to cap-weighted indices. The weighting of the portfolio of constituents achieves the highest possible return-to-risk efficiency by maximising the Sharpe ratio (the reward of an investment per unit of risk). These indices provide investors with an enhanced risk-adjusted strategy in comparison to cap-weighted indices, which have been the subject of numerous critiques, both theoretical and practical, over the last few years. The index series is based on all constituent securities in the FTSE All-World Index Series. Constituents

are weighted in accordance with EDHEC-Risk’s portfolio optimisation, reflecting their ability to maximise the reward-to-risk ratio for a broad market index. The index series is rebalanced quarterly at the same time as the review of the underlying FTSE All-World Index Series. The performances of the EDHEC-Risk Efficient Indices are published monthly on www.edhec-risk.com.

EDHEC-Risk Alternative Indexeswww.edhec-risk.com/indexes/pure_style

The different hedge fund indexes available on the market are computed from different data, according to diverse fund selection criteria and index construction methods; they unsurprisingly tell very different stories. Challenged by this heterogeneity, investors cannot rely on competing hedge fund indexes to obtain a “true and fair” view of performance and are at a loss when selecting benchmarks. To address this issue, EDHEC Risk was the first to launch composite hedge fund strategy indexes as early as 2003. The thirteen EDHEC-Risk Alternative Indexes are published monthly on www.edhec-risk.com and are freely available to managers and investors.


About Societe Generale



Societe Generale Corporate & Investment BankingAt the core of Societe Generale’s universal banking business model, the Corporate & Investment Bank is a well-diversified and leading player with nearly 12,000 professionals present in 33 countries across Europe, the Americas and Asia-Pacific.

Standing by its clients across sectors, the Corporate & Investment Bank tailors solutions for them by capitalising on its worldwide expertise in investment banking, global finance, and global markets.

• For Corporates, Financial Institutions and public sector: providing a global advisory approach (M&A, debt, equity transaction, capital structure, and asset & liability management), as well as quality capital raising solutions across the debt and equity spectrum, optimised financing and expert risk management responses notably in the realm of foreign exchange and rates derivatives.

• For investors: offering reliable and sound investment opportunities and risk management solutions through its integrated global markets platform delivering seamless access to markets (equity, rates, credit, currencies, commodities and derivatives), as well as advice and solid financial engineering, quality of execution and forward looking research across asset classes.

Societe Generale Corporate & Investment Banking in Asia-PacificBacked by worldwide expertise and an extensive network in Asia Pacific, Societe Generale Corporate & Investment Banking has built prime corporate & investment banking operations in the region to become a leading regional player in investment banking, global finance, and global markets. Societe Generale Corporate & Investment Banking combines in

Asia Pacific both global and local strengths to provide corporate clients, financial institutions and private investors with value-added integrated financial solutions.

www.sgcib.com

About Societe Generale


EDHEC-Risk Institute Publications and Position

Papers (2008-2011)



EDHEC-Risk Institute Publications (2008-2011)

2011• Miffre, Joelle. Long-Short Commodity Investing: Implications for Portfolio Risk and Market Regulation (August).

• Charbit, E., Giraud J. R., Goltz. F. and L.Tang. Capturing the Market, Value, or Momentum Premium with Downside Risk Control: Dynamic Allocation Strategies with Exchange-Traded Funds (July).

• Campani, C.H. and F. Goltz. A Review of corporate bond indices: Construction principles, return heterogeneity, and fluctuations in risk exposures (June).

• Martellini, L., and V. Milhau. Capital structure choices, pension fund allocation decisions, and the rational pricing of liability streams (June).

• Amenc, N., F. Goltz, and S. Stoyanov. A post-crisis perspective on diversification for risk management (May).

• Amenc, N., F. Goltz, Martellini, L., and L. Tang. Improved beta? A comparison of index-weighting schemes (April).

• Amenc, N., F. Goltz, Martellini, L., and D. Sahoo. Is there a risk/return tradeoff across stocks? An answer from a long-horizon perspective (April).

• Sender, S. The elephant in the room: Accounting and sponsor risks in corporate pension plans (March).

• Martellini, L., and V. Milhau. Optimal design of corporate market debt programmes in the presence of interest-rate and inflation risks (February).

2010• Amenc, N., and S. Sender. The European fund management industry needs a better grasp of non-financial risks (December).

• Amenc, N., S, Focardi, F. Goltz, D. Schröder, and L. Tang. EDHEC-Risk European private wealth management survey (November).

• Amenc, N., F. Goltz, and L. Tang. Adoption of green investing by institutional investors: A European survey (November).

• Martellini, L., and V. Milhau. An integrated approach to asset-liability management: Capital structure choices, pension fund allocation decisions and the rational pricing of liability streams (November).

• Hitaj, A., L. Martellini, and G. Zambruno. Optimal Hedge Fund Allocation with Improved Estimates for Coskewness and Cokurtosis Parameters (October).

• Amenc, N., F. Goltz, Martellini, L., and V. Milhau. New frontiers in benchmarking and liability-driven investing (September).

• Martellini, L., and V. Milhau. From deterministic to stochastic life-cycle investing: Implications for the design of improved forms of target date funds (September).

• Martellini, L., and V. Milhau. Capital structure choices, pension fund allocation decisions and the rational pricing of liability streams (July).




• Sender, S. EDHEC survey of the asset and liability management practices of European pension funds (June).

• Goltz, F., A. Grigoriu, and L. Tang. The EDHEC European ETF survey 2010 (May).• Martellini, L., and V. Milhau. Asset-liability management decisions for sovereign wealth funds (May).

• Amenc, N., and S. Sender. Are hedge-fund UCITS the cure-all? (March).

• Amenc, N., F. Goltz, and A. Grigoriu. Risk control through dynamic core-satellite portfolios of ETFs: Applications to absolute return funds and tactical asset allocation (January).

• Amenc, N., F. Goltz, and P. Retkowsky. Efficient indexation: An alternative to cap-weighted indices (January).

• Goltz, F., and V. Le Sourd. Does finance theory make the case for capitalisation-weighted indexing? (January).

2009• Sender, S. Reactions to an EDHEC study on the impact of regulatory constraints on the ALM of pension funds (October).

• Amenc, N., L. Martellini, V. Milhau, and V. Ziemann. Asset-liability management in private wealth management (September).

• Amenc, N., F. Goltz, A. Grigoriu, and D. Schroeder. The EDHEC European ETF survey (May).

• Sender, S. The European pension fund industry again beset by deficits (May).

• Martellini, L., and V. Milhau. Measuring the benefits of dynamic asset allocation strategies in the presence of liability constraints (March).

• Le Sourd, V. Hedge fund performance in 2008 (February).

• La gestion indicielle dans l'immobilier et l'indice EDHEC IEIF Immobilier d'Entreprise France (February).

• Real estate indexing and the EDHEC IEIF Commercial Property (France) Index (February).

• Amenc, N., L. Martellini, and S. Sender. Impact of regulations on the ALM of European pension funds (January).

• Goltz, F. A long road ahead for portfolio construction: Practitioners' views of an EDHEC survey. (January).

2008• Amenc, N., L. Martellini, and V. Ziemann. Alternative investments for institutional investors: Risk budgeting techniques in asset management and asset-liability management (December).

• Goltz, F., and D. Schroeder. Hedge fund reporting survey (November).




• D’Hondt, C., and J.-R. Giraud. Transaction cost analysis A-Z: A step towards best execution in the post-MiFID landscape (November).

• Amenc, N., and D. Schroeder. The pros and cons of passive hedge fund replication (October). • Amenc, N., F. Goltz, and D. Schroeder. Reactions to an EDHEC study on asset-liability management decisions in wealth management (September).

• Amenc, N., F. Goltz, A. Grigoriu, V. Le Sourd, and L. Martellini. The EDHEC European ETF survey 2008 (June).

• Amenc, N., F. Goltz, and V. Le Sourd. Fundamental differences? Comparing alternative index weighting mechanisms (April).

• Le Sourd, V. Hedge fund performance in 2007 (February).

• Amenc, N., F. Goltz, V. Le Sourd, and L. Martellini. The EDHEC European investment practices survey 2008 (January).



EDHEC-Risk Institute Position Papers (2008-2011)

2011• Uppal, R. A Short Note on the Tobin Tax: The Costs and Benefits of a Tax on Financial Transactions (July).

• Till, H. A Review of the G20 Meeting on Agriculture: Addressing Price Volatility in the Food Markets (July).

2010• Amenc, N., and V. Le Sourd. The performance of socially responsible investment and sustainable development in France: An update after the financial crisis (September).

• Amenc, N., A. Chéron, S. Gregoir, and L. Martellini. Il faut préserver le Fonds de Réserve pour les Retraites (July).

• Amenc, N., P. Schoefler, and P. Lasserre. Organisation optimale de la liquidité des fonds d’investissement (March).

• Lioui, A. Spillover effects of counter-cyclical market regulation: Evidence from the 2008 ban on short sales (March).

2009• Till, H. Has there been excessive speculation in the US oil futures markets? (November).

• Amenc, N., and S. Sender. A welcome European Commission consultation on the UCITS depositary function, a hastily considered proposal (September).

• Sender, S. IAS 19: Penalising changes ahead (September).

• Amenc, N. Quelques réflexions sur la régulation de la gestion d'actifs (June).

• Giraud, J.-R. MiFID: One year on (May).

• Lioui, A. The undesirable effects of banning short sales (April).

• Gregoriou, G., and F.-S. Lhabitant. Madoff: A riot of red flags (January).

2008 • Amenc, N., and S. Sender. Assessing the European banking sector bailout plans (December).

• Amenc, N., and S. Sender. Les mesures de recapitalisation et de soutien à la liquidité du secteur bancaire européen (December).

• Amenc, N., F. Ducoulombier, and P. Foulquier. Reactions to an EDHEC study on the fair value controversy (December). With the EDHEC Financial Analysis and Accounting Research Centre.

• Amenc, N., F. Ducoulombier, and P. Foulquier. Réactions après l’étude. Juste valeur ou non : un débat mal posé (December). With the EDHEC Financial Analysis and Accounting Research Centre.

• Amenc, N., and V. Le Sourd. Les performances de l’investissement socialement responsable en France (December).



EDHEC-Risk Institute Position Papers (2008-2011)

• Amenc, N., and V. Le Sourd. Socially responsible investment performance in France (December).

• Amenc, N., B. Maffei, and H. Till. Les causes structurelles du troisième choc pétrolier (November).

• Amenc, N., B. Maffei, and H. Till. Oil prices: The true role of speculation (November).

• Sender, S. Banking: Why does regulation alone not suffice? Why must governments intervene? (November).

• Till, H. The oil markets: Let the data speak for itself (October).

• Amenc, N., F. Goltz, and V. Le Sourd. A comparison of fundamentally weighted indices: Overview and performance analysis (March).

• Sender, S. QIS4: Significant improvements, but the main risk for life insurance is not taken into account in the standard formula (February). With the EDHEC Financial Analysis and Accounting Research Centre.

For more information, please contact: Carolyn Essid on +33 493 187 824 or by e-mail to: [email protected]

EDHEC-Risk Institute393 promenade des AnglaisBP 311606202 Nice Cedex 3 — France

EDHEC Risk Institute—Europe10 Fleet Place - LudgateLondon EC4M 7RB - United Kingdom

EDHEC Risk Institute—Asia1 George Street - #07-02Singapore 049145

www.edhec-risk.com

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